All the generating functions (and statistical information) Enumerating compositions by number of occurrences, as containments of all possible pairs of offending compositions of, 7, with , 2, parts and of , 8, also with , 2, parts By Shalosh B. Ekhad Theorem Number, 1, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 6], nor the composition, [3, 5] Then infinity ----- 5 \ n x - x + 1 ) a(n) x = ----------------------------------------- / 7 6 4 3 2 ----- (-1 + x) (x + x + x + x + x + x - 1) n = 0 and in Maple format (x^5-x+1)/(-1+x)/(x^7+x^6+x^4+x^3+x^2+x-1) The asymptotic expression for a(n) is, n 0.57122365827984603579 1.9576152606398896277 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 6] and d occurrences (as containment) of the composition, [3, 5], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| ) | ) | ) A(n, c, d) x X1 X2 || = - ( / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 6 6 5 5 / 8 7 8 X1 X2 x - X2 x + X2 x - x + x - 1) / (X1 X2 x + X1 X2 x - X1 x / 8 6 7 8 6 5 6 5 - X2 x - X1 X2 x - X1 x + x + 2 X2 x - X2 x - x + x - 2 x + 1) and in Maple format -(X1*X2*x^6-X2*x^6+X2*x^5-x^5+x-1)/(X1*X2*x^8+X1*X2*x^7-X1*x^8-X2*x^8-X1*X2*x^6 -X1*x^7+x^8+2*X2*x^6-X2*x^5-x^6+x^5-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 6], are n 53 n - 3/32 + ----, and , - 9/128 + ----, respectively, while 64 4096 Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 5], are n 911 127 n - 7/128 + ---, and , - ----- + -----, respectively, while the asymptotic cor\ 128 16384 16384 relation between these two random variables is 1/2 1/2 26 53 127 --------------- 6731 and in floating point 0.3169082555 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 26 6731 ate normal pair with correlation, ---------- 6731 1/2 26 6731 8083 i.e. , [[----------, 0], [0, ----]] 6731 6731 ------------------------------------------------- Theorem Number, 2, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 6], nor the composition, [4, 4] Then infinity ----- 4 \ n x - x + 1 ) a(n) x = -------------------------------------------------- / 8 7 6 5 3 2 ----- (-1 + x) (x + 2 x + 2 x + x + x + x + x - 1) n = 0 and in Maple format (x^4-x+1)/(-1+x)/(x^8+2*x^7+2*x^6+x^5+x^3+x^2+x-1) The asymptotic expression for a(n) is, n 0.57292931369834866970 1.9558149899357794253 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 6] and d occurrences (as containment) of the composition, [4, 4], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| ) | ) | ) A(n, c, d) x X1 X2 || = - ( / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 6 6 4 4 / 9 8 9 X1 X2 x - X2 x + X2 x - x + x - 1) / (X1 X2 x + X1 X2 x - X1 x / 9 7 8 8 9 6 7 8 6 - X2 x + X1 X2 x - X1 x - X2 x + x - X1 X2 x - X1 x + x + 2 X2 x 5 6 4 5 4 + X2 x - x - X2 x - x + x - 2 x + 1) and in Maple format -(X1*X2*x^6-X2*x^6+X2*x^4-x^4+x-1)/(X1*X2*x^9+X1*X2*x^8-X1*x^9-X2*x^9+X1*X2*x^7 -X1*x^8-X2*x^8+x^9-X1*X2*x^6-X1*x^7+x^8+2*X2*x^6+X2*x^5-x^6-X2*x^4-x^5+x^4-2*x+ 1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 6], are n 53 n - 3/32 + ----, and , - 9/128 + ----, respectively, while 64 4096 Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 4], are n 1087 143 n - 7/128 + ---, and , - ----- + -----, respectively, while the asymptotic cor\ 128 16384 16384 relation between these two random variables is 1/2 1/2 18 53 143 --------------- 7579 and in floating point 0.2067600133 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 18 7579 ate normal pair with correlation, ---------- 7579 1/2 18 7579 8227 i.e. , [[----------, 0], [0, ----]] 7579 7579 ------------------------------------------------- Theorem Number, 3, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 6], nor the composition, [5, 3] Then infinity ----- 10 9 5 \ n x + x - x + x - 1 ) a(n) x = - ----------------------------------------- / 7 6 4 3 2 ----- (-1 + x) (x + x + x + x + x + x - 1) n = 0 and in Maple format -(x^10+x^9-x^5+x-1)/(-1+x)/(x^7+x^6+x^4+x^3+x^2+x-1) The asymptotic expression for a(n) is, n 0.56732263340074550724 1.9576152606398896277 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 6] and d occurrences (as containment) of the composition, [5, 3], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 10 9 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x + X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 10 10 9 9 10 9 6 6 5 - X1 x - X2 x - X1 x - X2 x + x + x + X1 X2 x - X2 x + X2 x 5 / 8 7 8 8 6 7 - x + x - 1) / (X1 X2 x + X1 X2 x - X1 x - X2 x - X1 X2 x - X1 x / 8 6 5 6 5 + x + 2 X2 x - X2 x - x + x - 2 x + 1) and in Maple format -(X1*X2*x^10+X1*X2*x^9-X1*x^10-X2*x^10-X1*x^9-X2*x^9+x^10+x^9+X1*X2*x^6-X2*x^6+ X2*x^5-x^5+x-1)/(X1*X2*x^8+X1*X2*x^7-X1*x^8-X2*x^8-X1*X2*x^6-X1*x^7+x^8+2*X2*x^ 6-X2*x^5-x^6+x^5-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 6], are n 53 n - 3/32 + ----, and , - 9/128 + ----, respectively, while 64 4096 Furthermore the average and variance of the random variable: Number of occur\ rences of, [5, 3], are n 911 127 n - 7/128 + ---, and , - ----- + -----, respectively, while the asymptotic cor\ 128 16384 16384 relation between these two random variables is 1/2 1/2 26 53 127 --------------- 6731 and in floating point 0.3169082555 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 26 6731 ate normal pair with correlation, ---------- 6731 1/2 26 6731 8083 i.e. , [[----------, 0], [0, ----]] 6731 6731 ------------------------------------------------- Theorem Number, 4, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 6], nor the composition, [6, 2] Then infinity ----- 11 10 9 8 6 \ n x + x + x + x - x + x - 1 ) a(n) x = - ------------------------------------ / 5 4 3 2 ----- (-1 + x) (x + x + x + x + x - 1) n = 0 and in Maple format -(x^11+x^10+x^9+x^8-x^6+x-1)/(-1+x)/(x^5+x^4+x^3+x^2+x-1) The asymptotic expression for a(n) is, n 0.54757187353650689312 1.9659482366454853372 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 6] and d occurrences (as containment) of the composition, [6, 2], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 11 10 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x + X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 11 11 9 10 10 11 8 9 - X1 x - X2 x + X1 X2 x - X1 x - X2 x + x + X1 X2 x - X1 x 9 10 8 8 9 6 8 6 / - X2 x + x - X1 x - X2 x + x + X1 X2 x + x - x + x - 1) / ( / 7 6 7 6 X1 X2 x - X1 X2 x - X1 x + x - 2 x + 1) and in Maple format -(X1*X2*x^11+X1*X2*x^10-X1*x^11-X2*x^11+X1*X2*x^9-X1*x^10-X2*x^10+x^11+X1*X2*x^ 8-X1*x^9-X2*x^9+x^10-X1*x^8-X2*x^8+x^9+X1*X2*x^6+x^8-x^6+x-1)/(X1*X2*x^7-X1*X2* x^6-X1*x^7+x^6-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 6], are n 53 n - 3/32 + ----, and , - 9/128 + ----, respectively, while 64 4096 Furthermore the average and variance of the random variable: Number of occur\ rences of, [6, 2], are n 815 119 n - 7/128 + ---, and , - ----- + -----, respectively, while the asymptotic cor\ 128 16384 16384 relation between these two random variables is 1/2 1/2 54 53 119 --------------- 6307 and in floating point 0.6799584013 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 54 6307 ate normal pair with correlation, ---------- 6307 1/2 54 6307 12139 i.e. , [[----------, 0], [0, -----]] 6307 6307 ------------------------------------------------- Theorem Number, 5, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 6], nor the composition, [7, 1] Then infinity ----- 12 11 10 9 8 6 \ n x + x + x + x + x - x + x - 1 ) a(n) x = - -------------------------------------- / 5 4 3 2 ----- (-1 + x) (x + x + x + x + x - 1) n = 0 and in Maple format -(x^12+x^11+x^10+x^9+x^8-x^6+x-1)/(-1+x)/(x^5+x^4+x^3+x^2+x-1) The asymptotic expression for a(n) is, n 0.54724341851139256920 1.9659482366454853372 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 6] and d occurrences (as containment) of the composition, [7, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 12 11 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x + X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 12 12 10 11 11 12 9 10 - X1 x - X2 x + X1 X2 x - X1 x - X2 x + x + X1 X2 x - X1 x 10 11 8 9 9 10 7 8 8 - X2 x + x + X1 X2 x - X1 x - X2 x + x + X1 X2 x - X1 x - X2 x 9 7 8 6 6 / + x - X1 x + x + X1 x - x + x - 1) / ( / 7 7 6 6 X1 X2 x - X1 x + X1 x - x + 2 x - 1) and in Maple format (X1*X2*x^12+X1*X2*x^11-X1*x^12-X2*x^12+X1*X2*x^10-X1*x^11-X2*x^11+x^12+X1*X2*x^ 9-X1*x^10-X2*x^10+x^11+X1*X2*x^8-X1*x^9-X2*x^9+x^10+X1*X2*x^7-X1*x^8-X2*x^8+x^9 -X1*x^7+x^8+X1*x^6-x^6+x-1)/(X1*X2*x^7-X1*x^7+X1*x^6-x^6+2*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 6], are n 53 n - 3/32 + ----, and , - 9/128 + ----, respectively, while 64 4096 Furthermore the average and variance of the random variable: Number of occur\ rences of, [7, 1], are n 763 115 n - 7/128 + ---, and , - ----- + -----, respectively, while the asymptotic cor\ 128 16384 16384 relation between these two random variables is 1/2 1/2 52 53 115 --------------- 6095 and in floating point 0.6660648087 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 52 6095 ate normal pair with correlation, ---------- 6095 1/2 52 6095 11503 i.e. , [[----------, 0], [0, -----]] 6095 6095 ------------------------------------------------- Theorem Number, 6, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 5], nor the composition, [1, 7] Then infinity ----- 5 \ n x - x + 1 ) a(n) x = ----------------------------------------- / 7 6 4 3 2 ----- (-1 + x) (x + x + x + x + x + x - 1) n = 0 and in Maple format (x^5-x+1)/(-1+x)/(x^7+x^6+x^4+x^3+x^2+x-1) The asymptotic expression for a(n) is, n 0.57122365827984603579 1.9576152606398896277 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 5] and d occurrences (as containment) of the composition, [1, 7], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| ) | ) | ) A(n, c, d) x X1 X2 || = - ( / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 7 7 5 5 / 8 7 8 X1 X2 x - X1 x + X1 x - x + x - 1) / (X1 X2 x - X1 X2 x - X1 x / 8 7 8 6 5 6 5 - X2 x + X1 x + x + X1 x - X1 x - x + x - 2 x + 1) and in Maple format -(X1*X2*x^7-X1*x^7+X1*x^5-x^5+x-1)/(X1*X2*x^8-X1*X2*x^7-X1*x^8-X2*x^8+X1*x^7+x^ 8+X1*x^6-X1*x^5-x^6+x^5-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 5], are n 83 57 n - 3/32 + ----, and , - ---- + ----, respectively, while 64 1024 4096 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 7], are n 763 115 n - 7/128 + ---, and , - ----- + -----, respectively, while the asymptotic cor\ 128 16384 16384 relation between these two random variables is 1/2 1/2 22 57 115 --------------- 6555 and in floating point 0.2717292141 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 22 6555 ate normal pair with correlation, ---------- 6555 1/2 22 6555 7523 i.e. , [[----------, 0], [0, ----]] 6555 6555 ------------------------------------------------- Theorem Number, 7, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 5], nor the composition, [4, 4] Then infinity ----- 4 \ n x - x + 1 ) a(n) x = --------------------------- / 8 6 5 4 ----- x - x - x + x - 2 x + 1 n = 0 and in Maple format (x^4-x+1)/(x^8-x^6-x^5+x^4-2*x+1) The asymptotic expression for a(n) is, n 0.56267477460473740258 1.9605271155383154604 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 5] and d occurrences (as containment) of the composition, [4, 4], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| ) | ) | ) A(n, c, d) x X1 X2 || = - ( / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 5 5 4 4 / 8 7 8 X1 X2 x - X2 x + X2 x - x + x - 1) / (X1 X2 x + X1 X2 x - X1 x / 8 6 7 8 5 5 6 4 5 - X2 x + X1 X2 x - X1 x + x - X1 X2 x + 2 X2 x - x - X2 x - x 4 + x - 2 x + 1) and in Maple format -(X1*X2*x^5-X2*x^5+X2*x^4-x^4+x-1)/(X1*X2*x^8+X1*X2*x^7-X1*x^8-X2*x^8+X1*X2*x^6 -X1*x^7+x^8-X1*X2*x^5+2*X2*x^5-x^6-X2*x^4-x^5+x^4-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 5], are n 83 57 n - 3/32 + ----, and , - ---- + ----, respectively, while 64 1024 4096 Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 4], are n 1087 143 n - 7/128 + ---, and , - ----- + -----, respectively, while the asymptotic cor\ 128 16384 16384 relation between these two random variables is 1/2 1/2 12 57 143 --------------- 2717 and in floating point 0.3987466561 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 1/2 12 3 2717 ate normal pair with correlation, --------------- 2717 1/2 1/2 12 3 2717 3581 i.e. , [[---------------, 0], [0, ----]] 2717 2717 ------------------------------------------------- Theorem Number, 8, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 5], nor the composition, [5, 3] Then infinity ----- 9 8 5 \ n x + x - x + x - 1 ) a(n) x = - ---------------------------- / 10 9 6 5 ----- x + x - x + x - 2 x + 1 n = 0 and in Maple format -(x^9+x^8-x^5+x-1)/(x^10+x^9-x^6+x^5-2*x+1) The asymptotic expression for a(n) is, n 0.56216678634802894846 1.9598665787705074180 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 5] and d occurrences (as containment) of the composition, [5, 3], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 9 8 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x + X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 9 9 8 8 9 8 5 5 / - X1 x - X2 x - X1 x - X2 x + x + x + X1 X2 x - x + x - 1) / ( / 10 9 10 10 9 9 10 7 X1 X2 x + X1 X2 x - X1 x - X2 x - X1 x - X2 x + x + X1 X2 x 9 6 7 5 6 5 + x + X1 X2 x - X1 x - X1 X2 x - x + x - 2 x + 1) and in Maple format -(X1*X2*x^9+X1*X2*x^8-X1*x^9-X2*x^9-X1*x^8-X2*x^8+x^9+x^8+X1*X2*x^5-x^5+x-1)/( X1*X2*x^10+X1*X2*x^9-X1*x^10-X2*x^10-X1*x^9-X2*x^9+x^10+X1*X2*x^7+x^9+X1*X2*x^6 -X1*x^7-X1*X2*x^5-x^6+x^5-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 5], are n 83 57 n - 3/32 + ----, and , - ---- + ----, respectively, while 64 1024 4096 Furthermore the average and variance of the random variable: Number of occur\ rences of, [5, 3], are n 911 127 n - 7/128 + ---, and , - ----- + -----, respectively, while the asymptotic cor\ 128 16384 16384 relation between these two random variables is 1/2 1/2 12 57 127 --------------- 2413 and in floating point 0.4231196673 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 1/2 12 3 2413 ate normal pair with correlation, --------------- 2413 1/2 1/2 12 3 2413 3277 i.e. , [[---------------, 0], [0, ----]] 2413 2413 ------------------------------------------------- Theorem Number, 9, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 5], nor the composition, [6, 2] Then infinity ----- 9 7 6 5 4 3 2 \ n x + x - x + x - 2 x + 2 x - 2 x + 2 x - 1 ) a(n) x = - -------------------------------------------------------- / 10 8 7 6 5 4 3 2 ----- x + x - x + x - 2 x + 3 x - 3 x + 3 x - 3 x + 1 n = 0 and in Maple format -(x^9+x^7-x^6+x^5-2*x^4+2*x^3-2*x^2+2*x-1)/(x^10+x^8-x^7+x^6-2*x^5+3*x^4-3*x^3+ 3*x^2-3*x+1) The asymptotic expression for a(n) is, n 0.56452255853619848474 1.9585581874012132494 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 5] and d occurrences (as containment) of the composition, [6, 2], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 10 9 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x + X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 10 10 8 9 9 10 8 8 9 - X1 x - X2 x + X1 X2 x - X1 x - X2 x + x - X1 x - X2 x + x 6 8 6 5 5 / 11 10 + X1 X2 x + x - X1 x + X1 x - x + x - 1) / (X1 X2 x + X1 X2 x / 11 11 9 10 10 11 9 9 10 - X1 x - X2 x + X1 X2 x - X1 x - X2 x + x - X1 x - X2 x + x 7 9 6 7 6 5 6 5 + X1 X2 x + x - X1 X2 x - X1 x + 2 X1 x - X1 x - x + x - 2 x + 1) and in Maple format -(X1*X2*x^10+X1*X2*x^9-X1*x^10-X2*x^10+X1*X2*x^8-X1*x^9-X2*x^9+x^10-X1*x^8-X2*x ^8+x^9+X1*X2*x^6+x^8-X1*x^6+X1*x^5-x^5+x-1)/(X1*X2*x^11+X1*X2*x^10-X1*x^11-X2*x ^11+X1*X2*x^9-X1*x^10-X2*x^10+x^11-X1*x^9-X2*x^9+x^10+X1*X2*x^7+x^9-X1*X2*x^6- X1*x^7+2*X1*x^6-X1*x^5-x^6+x^5-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 5], are n 83 57 n - 3/32 + ----, and , - ---- + ----, respectively, while 64 1024 4096 Furthermore the average and variance of the random variable: Number of occur\ rences of, [6, 2], are n 815 119 n - 7/128 + ---, and , - ----- + -----, respectively, while the asymptotic cor\ 128 16384 16384 relation between these two random variables is 1/2 1/2 4 57 119 -------------- 969 and in floating point 0.3399751097 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 1/2 4 7 969 ate normal pair with correlation, ------------- 969 1/2 1/2 4 7 969 1193 i.e. , [[-------------, 0], [0, ----]] 969 969 ------------------------------------------------- Theorem Number, 10, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 5], nor the composition, [7, 1] Then infinity ----- 11 10 9 5 \ n x + x + x - x + x - 1 ) a(n) x = - ----------------------------------------- / 7 6 4 3 2 ----- (-1 + x) (x + x + x + x + x + x - 1) n = 0 and in Maple format -(x^11+x^10+x^9-x^5+x-1)/(-1+x)/(x^7+x^6+x^4+x^3+x^2+x-1) The asymptotic expression for a(n) is, n 0.56664886645723598634 1.9576152606398896277 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 5] and d occurrences (as containment) of the composition, [7, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 11 10 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x + X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 11 11 9 10 10 11 9 9 10 - X1 x - X2 x + X1 X2 x - X1 x - X2 x + x - X1 x - X2 x + x 7 9 7 5 5 / 8 7 + X1 X2 x + x - X1 x + X1 x - x + x - 1) / (X1 X2 x - X1 X2 x / 8 8 7 8 6 5 6 5 - X1 x - X2 x + X1 x + x + X1 x - X1 x - x + x - 2 x + 1) and in Maple format -(X1*X2*x^11+X1*X2*x^10-X1*x^11-X2*x^11+X1*X2*x^9-X1*x^10-X2*x^10+x^11-X1*x^9- X2*x^9+x^10+X1*X2*x^7+x^9-X1*x^7+X1*x^5-x^5+x-1)/(X1*X2*x^8-X1*X2*x^7-X1*x^8-X2 *x^8+X1*x^7+x^8+X1*x^6-X1*x^5-x^6+x^5-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 5], are n 83 57 n - 3/32 + ----, and , - ---- + ----, respectively, while 64 1024 4096 Furthermore the average and variance of the random variable: Number of occur\ rences of, [7, 1], are n 763 115 n - 7/128 + ---, and , - ----- + -----, respectively, while the asymptotic cor\ 128 16384 16384 relation between these two random variables is 1/2 1/2 22 57 115 --------------- 6555 and in floating point 0.2717292141 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 22 6555 ate normal pair with correlation, ---------- 6555 1/2 22 6555 7523 i.e. , [[----------, 0], [0, ----]] 6555 6555 ------------------------------------------------- Theorem Number, 11, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [3, 4], nor the composition, [1, 7] Then infinity ----- 4 \ n x - x + 1 ) a(n) x = -------------------------------------------------- / 8 7 6 5 3 2 ----- (-1 + x) (x + 2 x + 2 x + x + x + x + x - 1) n = 0 and in Maple format (x^4-x+1)/(-1+x)/(x^8+2*x^7+2*x^6+x^5+x^3+x^2+x-1) The asymptotic expression for a(n) is, n 0.57292931369834866970 1.9558149899357794253 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [3, 4] and d occurrences (as containment) of the composition, [1, 7], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| ) | ) | ) A(n, c, d) x X1 X2 || = - ( / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 7 7 4 4 / 9 8 9 X1 X2 x - X1 x + X1 x - x + x - 1) / (X1 X2 x + X1 X2 x - X1 x / 9 7 8 8 9 7 8 6 5 6 - X2 x - X1 X2 x - X1 x - X2 x + x + X1 x + x + X1 x + X1 x - x 4 5 4 - X1 x - x + x - 2 x + 1) and in Maple format -(X1*X2*x^7-X1*x^7+X1*x^4-x^4+x-1)/(X1*X2*x^9+X1*X2*x^8-X1*x^9-X2*x^9-X1*X2*x^7 -X1*x^8-X2*x^8+x^9+X1*x^7+x^8+X1*x^6+X1*x^5-x^6-X1*x^4-x^5+x^4-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 4], are n 103 65 n - 3/32 + ----, and , - ---- + ----, respectively, while 64 1024 4096 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 7], are n 763 115 n - 7/128 + ---, and , - ----- + -----, respectively, while the asymptotic cor\ 128 16384 16384 relation between these two random variables is 1/2 1/2 2 13 23 ------------- 299 and in floating point 0.1156629864 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 2 299 ate normal pair with correlation, -------- 299 1/2 2 299 307 i.e. , [[--------, 0], [0, ---]] 299 299 ------------------------------------------------- Theorem Number, 12, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [3, 4], nor the composition, [2, 6] Then infinity ----- 4 \ n x - x + 1 ) a(n) x = --------------------------- / 8 6 5 4 ----- x - x - x + x - 2 x + 1 n = 0 and in Maple format (x^4-x+1)/(x^8-x^6-x^5+x^4-2*x+1) The asymptotic expression for a(n) is, n 0.56267477460473740258 1.9605271155383154604 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [3, 4] and d occurrences (as containment) of the composition, [2, 6], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| ) | ) | ) A(n, c, d) x X1 X2 || = - ( / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 6 6 4 4 / 8 7 8 X1 X2 x - X1 x + X1 x - x + x - 1) / (X1 X2 x + X1 X2 x - X1 x / 8 6 7 8 6 5 6 4 5 4 - X2 x - X1 X2 x - X1 x + x + 2 X1 x + X1 x - x - X1 x - x + x - 2 x + 1) and in Maple format -(X1*X2*x^6-X1*x^6+X1*x^4-x^4+x-1)/(X1*X2*x^8+X1*X2*x^7-X1*x^8-X2*x^8-X1*X2*x^6 -X1*x^7+x^8+2*X1*x^6+X1*x^5-x^6-X1*x^4-x^5+x^4-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 4], are n 103 65 n - 3/32 + ----, and , - ---- + ----, respectively, while 64 1024 4096 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 6], are n 815 119 n - 7/128 + ---, and , - ----- + -----, respectively, while the asymptotic cor\ 128 16384 16384 relation between these two random variables is 1/2 1/2 4 65 119 -------------- 1105 and in floating point 0.3183668732 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 1/2 4 7 1105 ate normal pair with correlation, -------------- 1105 1/2 1/2 4 7 1105 1329 i.e. , [[--------------, 0], [0, ----]] 1105 1105 ------------------------------------------------- Theorem Number, 13, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [3, 4], nor the composition, [5, 3] Then infinity ----- 7 6 5 4 \ n x + x + x + x + 1 ) a(n) x = - ---------------------------------------------- / 9 8 7 6 5 3 2 ----- x + 2 x + 2 x + 2 x + x + x + x + x - 1 n = 0 and in Maple format -(x^7+x^6+x^5+x^4+1)/(x^9+2*x^8+2*x^7+2*x^6+x^5+x^3+x^2+x-1) The asymptotic expression for a(n) is, n 0.55693630447221345149 1.9626042453349608416 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [3, 4] and d occurrences (as containment) of the composition, [5, 3], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| ) | ) | ) A(n, c, d) x X1 X2 || = - ( / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 8 8 8 8 5 5 4 4 / X1 X2 x - X1 x - X2 x + x + X1 X2 x - X1 x + X1 x - x + x - 1) / / 10 9 10 10 9 9 10 7 (X1 X2 x + X1 X2 x - X1 x - X2 x - X1 x - X2 x + x + X1 X2 x 9 6 7 5 5 6 4 5 4 + x + X1 X2 x - X1 x - X1 X2 x + 2 X1 x - x - X1 x - x + x - 2 x + 1) and in Maple format -(X1*X2*x^8-X1*x^8-X2*x^8+x^8+X1*X2*x^5-X1*x^5+X1*x^4-x^4+x-1)/(X1*X2*x^10+X1* X2*x^9-X1*x^10-X2*x^10-X1*x^9-X2*x^9+x^10+X1*X2*x^7+x^9+X1*X2*x^6-X1*x^7-X1*X2* x^5+2*X1*x^5-x^6-X1*x^4-x^5+x^4-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 4], are n 103 65 n - 3/32 + ----, and , - ---- + ----, respectively, while 64 1024 4096 Furthermore the average and variance of the random variable: Number of occur\ rences of, [5, 3], are n 911 127 n - 7/128 + ---, and , - ----- + -----, respectively, while the asymptotic cor\ 128 16384 16384 relation between these two random variables is 1/2 1/2 8 65 127 -------------- 1651 and in floating point 0.4402521166 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 1/2 8 5 1651 ate normal pair with correlation, -------------- 1651 1/2 1/2 8 5 1651 2291 i.e. , [[--------------, 0], [0, ----]] 1651 1651 ------------------------------------------------- Theorem Number, 14, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [3, 4], nor the composition, [6, 2] Then infinity ----- 6 3 \ n x + x - x + 1 ) a(n) x = - --------------------------- / 7 5 4 3 ----- x + x + x - x + 2 x - 1 n = 0 and in Maple format -(x^6+x^3-x+1)/(x^7+x^5+x^4-x^3+2*x-1) The asymptotic expression for a(n) is, n 0.56609559480083299502 1.9581432687881222265 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [3, 4] and d occurrences (as containment) of the composition, [6, 2], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| ) | ) | ) A(n, c, d) x X1 X2 || = - ( / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 9 9 9 9 6 6 4 4 / X1 X2 x - X1 x - X2 x + x + X1 X2 x - X1 x + X1 x - x + x - 1) / / 10 10 10 8 10 7 8 8 (X1 X2 x - X1 x - X2 x + X1 X2 x + x + X1 X2 x - X1 x - X2 x 6 7 8 6 5 6 4 5 4 - X1 X2 x - X1 x + x + 2 X1 x + X1 x - x - X1 x - x + x - 2 x + 1 ) and in Maple format -(X1*X2*x^9-X1*x^9-X2*x^9+x^9+X1*X2*x^6-X1*x^6+X1*x^4-x^4+x-1)/(X1*X2*x^10-X1*x ^10-X2*x^10+X1*X2*x^8+x^10+X1*X2*x^7-X1*x^8-X2*x^8-X1*X2*x^6-X1*x^7+x^8+2*X1*x^ 6+X1*x^5-x^6-X1*x^4-x^5+x^4-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 4], are n 103 65 n - 3/32 + ----, and , - ---- + ----, respectively, while 64 1024 4096 Furthermore the average and variance of the random variable: Number of occur\ rences of, [6, 2], are n 815 119 n - 7/128 + ---, and , - ----- + -----, respectively, while the asymptotic cor\ 128 16384 16384 relation between these two random variables is 1/2 1/2 4 65 119 -------------- 1547 and in floating point 0.2274049094 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 1/2 4 5 1547 ate normal pair with correlation, -------------- 1547 1/2 1/2 4 5 1547 1707 i.e. , [[--------------, 0], [0, ----]] 1547 1547 ------------------------------------------------- Theorem Number, 15, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [3, 4], nor the composition, [7, 1] Then infinity ----- 9 8 7 6 5 4 \ n x + x + x + x + x + x + 1 ) a(n) x = - --------------------------------------- / 8 7 6 5 3 2 ----- x + 2 x + 2 x + x + x + x + x - 1 n = 0 and in Maple format -(x^9+x^8+x^7+x^6+x^5+x^4+1)/(x^8+2*x^7+2*x^6+x^5+x^3+x^2+x-1) The asymptotic expression for a(n) is, n 0.57167348045170575921 1.9558149899357794253 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [3, 4] and d occurrences (as containment) of the composition, [7, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| ) | ) | ) A(n, c, d) x X1 X2 || = - ( / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 10 10 10 10 7 7 4 4 X1 X2 x - X1 x - X2 x + x + X1 X2 x - X1 x + X1 x - x + x - 1) / 9 8 9 9 7 8 8 9 / (X1 X2 x + X1 X2 x - X1 x - X2 x - X1 X2 x - X1 x - X2 x + x / 7 8 6 5 6 4 5 4 + X1 x + x + X1 x + X1 x - x - X1 x - x + x - 2 x + 1) and in Maple format -(X1*X2*x^10-X1*x^10-X2*x^10+x^10+X1*X2*x^7-X1*x^7+X1*x^4-x^4+x-1)/(X1*X2*x^9+ X1*X2*x^8-X1*x^9-X2*x^9-X1*X2*x^7-X1*x^8-X2*x^8+x^9+X1*x^7+x^8+X1*x^6+X1*x^5-x^ 6-X1*x^4-x^5+x^4-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 4], are n 103 65 n - 3/32 + ----, and , - ---- + ----, respectively, while 64 1024 4096 Furthermore the average and variance of the random variable: Number of occur\ rences of, [7, 1], are n 763 115 n - 7/128 + ---, and , - ----- + -----, respectively, while the asymptotic cor\ 128 16384 16384 relation between these two random variables is 1/2 1/2 2 13 23 ------------- 299 and in floating point 0.1156629864 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 2 299 ate normal pair with correlation, -------- 299 1/2 2 299 307 i.e. , [[--------, 0], [0, ---]] 299 299 ------------------------------------------------- Theorem Number, 16, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [4, 3], nor the composition, [1, 7] Then infinity ----- 9 8 7 6 5 4 \ n x + x + x + x + x + x + 1 ) a(n) x = - --------------------------------------- / 8 7 6 5 3 2 ----- x + 2 x + 2 x + x + x + x + x - 1 n = 0 and in Maple format -(x^9+x^8+x^7+x^6+x^5+x^4+1)/(x^8+2*x^7+2*x^6+x^5+x^3+x^2+x-1) The asymptotic expression for a(n) is, n 0.57167348045170575921 1.9558149899357794253 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [4, 3] and d occurrences (as containment) of the composition, [1, 7], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| ) | ) | ) A(n, c, d) x X1 X2 || = - ( / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 10 10 10 10 7 7 4 4 X1 X2 x - X1 x - X2 x + x + X1 X2 x - X1 x + X1 x - x + x - 1) / 9 8 9 9 7 8 8 9 / (X1 X2 x + X1 X2 x - X1 x - X2 x - X1 X2 x - X1 x - X2 x + x / 7 8 6 5 6 4 5 4 + X1 x + x + X1 x + X1 x - x - X1 x - x + x - 2 x + 1) and in Maple format -(X1*X2*x^10-X1*x^10-X2*x^10+x^10+X1*X2*x^7-X1*x^7+X1*x^4-x^4+x-1)/(X1*X2*x^9+ X1*X2*x^8-X1*x^9-X2*x^9-X1*X2*x^7-X1*x^8-X2*x^8+x^9+X1*x^7+x^8+X1*x^6+X1*x^5-x^ 6-X1*x^4-x^5+x^4-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 3], are n 103 65 n - 3/32 + ----, and , - ---- + ----, respectively, while 64 1024 4096 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 7], are n 763 115 n - 7/128 + ---, and , - ----- + -----, respectively, while the asymptotic cor\ 128 16384 16384 relation between these two random variables is 1/2 1/2 2 13 23 ------------- 299 and in floating point 0.1156629864 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 2 299 ate normal pair with correlation, -------- 299 1/2 2 299 307 i.e. , [[--------, 0], [0, ---]] 299 299 ------------------------------------------------- Theorem Number, 17, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [4, 3], nor the composition, [2, 6] Then infinity ----- 6 3 \ n x + x - x + 1 ) a(n) x = - --------------------------- / 7 5 4 3 ----- x + x + x - x + 2 x - 1 n = 0 and in Maple format -(x^6+x^3-x+1)/(x^7+x^5+x^4-x^3+2*x-1) The asymptotic expression for a(n) is, n 0.56609559480083299502 1.9581432687881222265 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [4, 3] and d occurrences (as containment) of the composition, [2, 6], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| ) | ) | ) A(n, c, d) x X1 X2 || = - ( / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 9 9 9 9 6 6 4 4 / X1 X2 x - X1 x - X2 x + x + X1 X2 x - X1 x + X1 x - x + x - 1) / / 10 10 10 8 10 7 8 8 (X1 X2 x - X1 x - X2 x + X1 X2 x + x + X1 X2 x - X1 x - X2 x 6 7 8 6 5 6 4 5 4 - X1 X2 x - X1 x + x + 2 X1 x + X1 x - x - X1 x - x + x - 2 x + 1 ) and in Maple format -(X1*X2*x^9-X1*x^9-X2*x^9+x^9+X1*X2*x^6-X1*x^6+X1*x^4-x^4+x-1)/(X1*X2*x^10-X1*x ^10-X2*x^10+X1*X2*x^8+x^10+X1*X2*x^7-X1*x^8-X2*x^8-X1*X2*x^6-X1*x^7+x^8+2*X1*x^ 6+X1*x^5-x^6-X1*x^4-x^5+x^4-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 3], are n 103 65 n - 3/32 + ----, and , - ---- + ----, respectively, while 64 1024 4096 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 6], are n 815 119 n - 7/128 + ---, and , - ----- + -----, respectively, while the asymptotic cor\ 128 16384 16384 relation between these two random variables is 1/2 1/2 4 65 119 -------------- 1547 and in floating point 0.2274049094 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 1/2 4 5 1547 ate normal pair with correlation, -------------- 1547 1/2 1/2 4 5 1547 1707 i.e. , [[--------------, 0], [0, ----]] 1547 1547 ------------------------------------------------- Theorem Number, 18, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [4, 3], nor the composition, [3, 5] Then infinity ----- 7 6 5 4 \ n x + x + x + x + 1 ) a(n) x = - ---------------------------------------------- / 9 8 7 6 5 3 2 ----- x + 2 x + 2 x + 2 x + x + x + x + x - 1 n = 0 and in Maple format -(x^7+x^6+x^5+x^4+1)/(x^9+2*x^8+2*x^7+2*x^6+x^5+x^3+x^2+x-1) The asymptotic expression for a(n) is, n 0.55693630447221345149 1.9626042453349608416 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [4, 3] and d occurrences (as containment) of the composition, [3, 5], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| ) | ) | ) A(n, c, d) x X1 X2 || = - ( / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 8 8 8 8 5 5 4 4 / X1 X2 x - X1 x - X2 x + x + X1 X2 x - X1 x + X1 x - x + x - 1) / / 10 9 10 10 9 9 10 7 (X1 X2 x + X1 X2 x - X1 x - X2 x - X1 x - X2 x + x + X1 X2 x 9 6 7 5 5 6 4 5 4 + x + X1 X2 x - X1 x - X1 X2 x + 2 X1 x - x - X1 x - x + x - 2 x + 1) and in Maple format -(X1*X2*x^8-X1*x^8-X2*x^8+x^8+X1*X2*x^5-X1*x^5+X1*x^4-x^4+x-1)/(X1*X2*x^10+X1* X2*x^9-X1*x^10-X2*x^10-X1*x^9-X2*x^9+x^10+X1*X2*x^7+x^9+X1*X2*x^6-X1*x^7-X1*X2* x^5+2*X1*x^5-x^6-X1*x^4-x^5+x^4-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 3], are n 103 65 n - 3/32 + ----, and , - ---- + ----, respectively, while 64 1024 4096 Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 5], are n 911 127 n - 7/128 + ---, and , - ----- + -----, respectively, while the asymptotic cor\ 128 16384 16384 relation between these two random variables is 1/2 1/2 8 65 127 -------------- 1651 and in floating point 0.4402521166 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 1/2 8 5 1651 ate normal pair with correlation, -------------- 1651 1/2 1/2 8 5 1651 2291 i.e. , [[--------------, 0], [0, ----]] 1651 1651 ------------------------------------------------- Theorem Number, 19, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [4, 3], nor the composition, [6, 2] Then infinity ----- 4 \ n x - x + 1 ) a(n) x = --------------------------- / 8 6 5 4 ----- x - x - x + x - 2 x + 1 n = 0 and in Maple format (x^4-x+1)/(x^8-x^6-x^5+x^4-2*x+1) The asymptotic expression for a(n) is, n 0.56267477460473740258 1.9605271155383154604 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [4, 3] and d occurrences (as containment) of the composition, [6, 2], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| ) | ) | ) A(n, c, d) x X1 X2 || = - ( / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 6 6 4 4 / 8 7 8 X1 X2 x - X1 x + X1 x - x + x - 1) / (X1 X2 x + X1 X2 x - X1 x / 8 6 7 8 6 5 6 4 5 4 - X2 x - X1 X2 x - X1 x + x + 2 X1 x + X1 x - x - X1 x - x + x - 2 x + 1) and in Maple format -(X1*X2*x^6-X1*x^6+X1*x^4-x^4+x-1)/(X1*X2*x^8+X1*X2*x^7-X1*x^8-X2*x^8-X1*X2*x^6 -X1*x^7+x^8+2*X1*x^6+X1*x^5-x^6-X1*x^4-x^5+x^4-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 3], are n 103 65 n - 3/32 + ----, and , - ---- + ----, respectively, while 64 1024 4096 Furthermore the average and variance of the random variable: Number of occur\ rences of, [6, 2], are n 815 119 n - 7/128 + ---, and , - ----- + -----, respectively, while the asymptotic cor\ 128 16384 16384 relation between these two random variables is 1/2 1/2 4 65 119 -------------- 1105 and in floating point 0.3183668732 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 1/2 4 7 1105 ate normal pair with correlation, -------------- 1105 1/2 1/2 4 7 1105 1329 i.e. , [[--------------, 0], [0, ----]] 1105 1105 ------------------------------------------------- Theorem Number, 20, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [4, 3], nor the composition, [7, 1] Then infinity ----- 4 \ n x - x + 1 ) a(n) x = -------------------------------------------------- / 8 7 6 5 3 2 ----- (-1 + x) (x + 2 x + 2 x + x + x + x + x - 1) n = 0 and in Maple format (x^4-x+1)/(-1+x)/(x^8+2*x^7+2*x^6+x^5+x^3+x^2+x-1) The asymptotic expression for a(n) is, n 0.57292931369834866970 1.9558149899357794253 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [4, 3] and d occurrences (as containment) of the composition, [7, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| ) | ) | ) A(n, c, d) x X1 X2 || = - ( / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 7 7 4 4 / 9 8 9 X1 X2 x - X1 x + X1 x - x + x - 1) / (X1 X2 x + X1 X2 x - X1 x / 9 7 8 8 9 7 8 6 5 6 - X2 x - X1 X2 x - X1 x - X2 x + x + X1 x + x + X1 x + X1 x - x 4 5 4 - X1 x - x + x - 2 x + 1) and in Maple format -(X1*X2*x^7-X1*x^7+X1*x^4-x^4+x-1)/(X1*X2*x^9+X1*X2*x^8-X1*x^9-X2*x^9-X1*X2*x^7 -X1*x^8-X2*x^8+x^9+X1*x^7+x^8+X1*x^6+X1*x^5-x^6-X1*x^4-x^5+x^4-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 3], are n 103 65 n - 3/32 + ----, and , - ---- + ----, respectively, while 64 1024 4096 Furthermore the average and variance of the random variable: Number of occur\ rences of, [7, 1], are n 763 115 n - 7/128 + ---, and , - ----- + -----, respectively, while the asymptotic cor\ 128 16384 16384 relation between these two random variables is 1/2 1/2 2 13 23 ------------- 299 and in floating point 0.1156629864 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 2 299 ate normal pair with correlation, -------- 299 1/2 2 299 307 i.e. , [[--------, 0], [0, ---]] 299 299 ------------------------------------------------- Theorem Number, 21, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [5, 2], nor the composition, [1, 7] Then infinity ----- 11 10 9 5 \ n x + x + x - x + x - 1 ) a(n) x = - ----------------------------------------- / 7 6 4 3 2 ----- (-1 + x) (x + x + x + x + x + x - 1) n = 0 and in Maple format -(x^11+x^10+x^9-x^5+x-1)/(-1+x)/(x^7+x^6+x^4+x^3+x^2+x-1) The asymptotic expression for a(n) is, n 0.56664886645723598634 1.9576152606398896277 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [5, 2] and d occurrences (as containment) of the composition, [1, 7], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 11 10 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x + X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 11 11 9 10 10 11 9 9 10 - X1 x - X2 x + X1 X2 x - X1 x - X2 x + x - X1 x - X2 x + x 7 9 7 5 5 / 8 7 + X1 X2 x + x - X1 x + X1 x - x + x - 1) / (X1 X2 x - X1 X2 x / 8 8 7 8 6 5 6 5 - X1 x - X2 x + X1 x + x + X1 x - X1 x - x + x - 2 x + 1) and in Maple format -(X1*X2*x^11+X1*X2*x^10-X1*x^11-X2*x^11+X1*X2*x^9-X1*x^10-X2*x^10+x^11-X1*x^9- X2*x^9+x^10+X1*X2*x^7+x^9-X1*x^7+X1*x^5-x^5+x-1)/(X1*X2*x^8-X1*X2*x^7-X1*x^8-X2 *x^8+X1*x^7+x^8+X1*x^6-X1*x^5-x^6+x^5-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [5, 2], are n 83 57 n - 3/32 + ----, and , - ---- + ----, respectively, while 64 1024 4096 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 7], are n 763 115 n - 7/128 + ---, and , - ----- + -----, respectively, while the asymptotic cor\ 128 16384 16384 relation between these two random variables is 1/2 1/2 22 57 115 --------------- 6555 and in floating point 0.2717292141 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 22 6555 ate normal pair with correlation, ---------- 6555 1/2 22 6555 7523 i.e. , [[----------, 0], [0, ----]] 6555 6555 ------------------------------------------------- Theorem Number, 22, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [5, 2], nor the composition, [2, 6] Then infinity ----- 9 7 6 5 4 3 2 \ n x + x - x + x - 2 x + 2 x - 2 x + 2 x - 1 ) a(n) x = - -------------------------------------------------------- / 10 8 7 6 5 4 3 2 ----- x + x - x + x - 2 x + 3 x - 3 x + 3 x - 3 x + 1 n = 0 and in Maple format -(x^9+x^7-x^6+x^5-2*x^4+2*x^3-2*x^2+2*x-1)/(x^10+x^8-x^7+x^6-2*x^5+3*x^4-3*x^3+ 3*x^2-3*x+1) The asymptotic expression for a(n) is, n 0.56452255853619848474 1.9585581874012132494 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [5, 2] and d occurrences (as containment) of the composition, [2, 6], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 10 9 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x + X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 10 10 8 9 9 10 8 8 9 - X1 x - X2 x + X1 X2 x - X1 x - X2 x + x - X1 x - X2 x + x 6 8 6 5 5 / 11 10 + X1 X2 x + x - X1 x + X1 x - x + x - 1) / (X1 X2 x + X1 X2 x / 11 11 9 10 10 11 9 9 10 - X1 x - X2 x + X1 X2 x - X1 x - X2 x + x - X1 x - X2 x + x 7 9 6 7 6 5 6 5 + X1 X2 x + x - X1 X2 x - X1 x + 2 X1 x - X1 x - x + x - 2 x + 1) and in Maple format -(X1*X2*x^10+X1*X2*x^9-X1*x^10-X2*x^10+X1*X2*x^8-X1*x^9-X2*x^9+x^10-X1*x^8-X2*x ^8+x^9+X1*X2*x^6+x^8-X1*x^6+X1*x^5-x^5+x-1)/(X1*X2*x^11+X1*X2*x^10-X1*x^11-X2*x ^11+X1*X2*x^9-X1*x^10-X2*x^10+x^11-X1*x^9-X2*x^9+x^10+X1*X2*x^7+x^9-X1*X2*x^6- X1*x^7+2*X1*x^6-X1*x^5-x^6+x^5-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [5, 2], are n 83 57 n - 3/32 + ----, and , - ---- + ----, respectively, while 64 1024 4096 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 6], are n 815 119 n - 7/128 + ---, and , - ----- + -----, respectively, while the asymptotic cor\ 128 16384 16384 relation between these two random variables is 1/2 1/2 4 57 119 -------------- 969 and in floating point 0.3399751097 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 1/2 4 7 969 ate normal pair with correlation, ------------- 969 1/2 1/2 4 7 969 1193 i.e. , [[-------------, 0], [0, ----]] 969 969 ------------------------------------------------- Theorem Number, 23, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [5, 2], nor the composition, [3, 5] Then infinity ----- 9 8 5 \ n x + x - x + x - 1 ) a(n) x = - ---------------------------- / 10 9 6 5 ----- x + x - x + x - 2 x + 1 n = 0 and in Maple format -(x^9+x^8-x^5+x-1)/(x^10+x^9-x^6+x^5-2*x+1) The asymptotic expression for a(n) is, n 0.56216678634802894846 1.9598665787705074180 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [5, 2] and d occurrences (as containment) of the composition, [3, 5], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 9 8 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x + X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 9 9 8 8 9 8 5 5 / - X1 x - X2 x - X1 x - X2 x + x + x + X1 X2 x - x + x - 1) / ( / 10 9 10 10 9 9 10 7 X1 X2 x + X1 X2 x - X1 x - X2 x - X1 x - X2 x + x + X1 X2 x 9 6 7 5 6 5 + x + X1 X2 x - X1 x - X1 X2 x - x + x - 2 x + 1) and in Maple format -(X1*X2*x^9+X1*X2*x^8-X1*x^9-X2*x^9-X1*x^8-X2*x^8+x^9+x^8+X1*X2*x^5-x^5+x-1)/( X1*X2*x^10+X1*X2*x^9-X1*x^10-X2*x^10-X1*x^9-X2*x^9+x^10+X1*X2*x^7+x^9+X1*X2*x^6 -X1*x^7-X1*X2*x^5-x^6+x^5-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [5, 2], are n 83 57 n - 3/32 + ----, and , - ---- + ----, respectively, while 64 1024 4096 Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 5], are n 911 127 n - 7/128 + ---, and , - ----- + -----, respectively, while the asymptotic cor\ 128 16384 16384 relation between these two random variables is 1/2 1/2 12 57 127 --------------- 2413 and in floating point 0.4231196673 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 1/2 12 3 2413 ate normal pair with correlation, --------------- 2413 1/2 1/2 12 3 2413 3277 i.e. , [[---------------, 0], [0, ----]] 2413 2413 ------------------------------------------------- Theorem Number, 24, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [5, 2], nor the composition, [4, 4] Then infinity ----- 4 \ n x - x + 1 ) a(n) x = --------------------------- / 8 6 5 4 ----- x - x - x + x - 2 x + 1 n = 0 and in Maple format (x^4-x+1)/(x^8-x^6-x^5+x^4-2*x+1) The asymptotic expression for a(n) is, n 0.56267477460473740258 1.9605271155383154604 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [5, 2] and d occurrences (as containment) of the composition, [4, 4], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| ) | ) | ) A(n, c, d) x X1 X2 || = - ( / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 5 5 4 4 / 8 7 8 X1 X2 x - X2 x + X2 x - x + x - 1) / (X1 X2 x + X1 X2 x - X1 x / 8 6 7 8 5 5 6 4 5 - X2 x + X1 X2 x - X1 x + x - X1 X2 x + 2 X2 x - x - X2 x - x 4 + x - 2 x + 1) and in Maple format -(X1*X2*x^5-X2*x^5+X2*x^4-x^4+x-1)/(X1*X2*x^8+X1*X2*x^7-X1*x^8-X2*x^8+X1*X2*x^6 -X1*x^7+x^8-X1*X2*x^5+2*X2*x^5-x^6-X2*x^4-x^5+x^4-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [5, 2], are n 83 57 n - 3/32 + ----, and , - ---- + ----, respectively, while 64 1024 4096 Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 4], are n 1087 143 n - 7/128 + ---, and , - ----- + -----, respectively, while the asymptotic cor\ 128 16384 16384 relation between these two random variables is 1/2 1/2 12 57 143 --------------- 2717 and in floating point 0.3987466561 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 1/2 12 3 2717 ate normal pair with correlation, --------------- 2717 1/2 1/2 12 3 2717 3581 i.e. , [[---------------, 0], [0, ----]] 2717 2717 ------------------------------------------------- Theorem Number, 25, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [5, 2], nor the composition, [7, 1] Then infinity ----- 5 \ n x - x + 1 ) a(n) x = ----------------------------------------- / 7 6 4 3 2 ----- (-1 + x) (x + x + x + x + x + x - 1) n = 0 and in Maple format (x^5-x+1)/(-1+x)/(x^7+x^6+x^4+x^3+x^2+x-1) The asymptotic expression for a(n) is, n 0.57122365827984603579 1.9576152606398896277 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [5, 2] and d occurrences (as containment) of the composition, [7, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| ) | ) | ) A(n, c, d) x X1 X2 || = - ( / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 7 7 5 5 / 8 7 8 X1 X2 x - X1 x + X1 x - x + x - 1) / (X1 X2 x - X1 X2 x - X1 x / 8 7 8 6 5 6 5 - X2 x + X1 x + x + X1 x - X1 x - x + x - 2 x + 1) and in Maple format -(X1*X2*x^7-X1*x^7+X1*x^5-x^5+x-1)/(X1*X2*x^8-X1*X2*x^7-X1*x^8-X2*x^8+X1*x^7+x^ 8+X1*x^6-X1*x^5-x^6+x^5-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [5, 2], are n 83 57 n - 3/32 + ----, and , - ---- + ----, respectively, while 64 1024 4096 Furthermore the average and variance of the random variable: Number of occur\ rences of, [7, 1], are n 763 115 n - 7/128 + ---, and , - ----- + -----, respectively, while the asymptotic cor\ 128 16384 16384 relation between these two random variables is 1/2 1/2 22 57 115 --------------- 6555 and in floating point 0.2717292141 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 22 6555 ate normal pair with correlation, ---------- 6555 1/2 22 6555 7523 i.e. , [[----------, 0], [0, ----]] 6555 6555 ------------------------------------------------- Theorem Number, 26, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [6, 1], nor the composition, [1, 7] Then infinity ----- 12 11 10 9 8 6 \ n x + x + x + x + x - x + x - 1 ) a(n) x = - -------------------------------------- / 5 4 3 2 ----- (-1 + x) (x + x + x + x + x - 1) n = 0 and in Maple format -(x^12+x^11+x^10+x^9+x^8-x^6+x-1)/(-1+x)/(x^5+x^4+x^3+x^2+x-1) The asymptotic expression for a(n) is, n 0.54724341851139256920 1.9659482366454853372 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [6, 1] and d occurrences (as containment) of the composition, [1, 7], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 12 11 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x + X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 12 12 10 11 11 12 9 10 - X1 x - X2 x + X1 X2 x - X1 x - X2 x + x + X1 X2 x - X1 x 10 11 8 9 9 10 7 8 8 - X2 x + x + X1 X2 x - X1 x - X2 x + x + X1 X2 x - X1 x - X2 x 9 7 8 6 6 / + x - X1 x + x + X1 x - x + x - 1) / ( / 7 7 6 6 X1 X2 x - X1 x + X1 x - x + 2 x - 1) and in Maple format (X1*X2*x^12+X1*X2*x^11-X1*x^12-X2*x^12+X1*X2*x^10-X1*x^11-X2*x^11+x^12+X1*X2*x^ 9-X1*x^10-X2*x^10+x^11+X1*X2*x^8-X1*x^9-X2*x^9+x^10+X1*X2*x^7-X1*x^8-X2*x^8+x^9 -X1*x^7+x^8+X1*x^6-x^6+x-1)/(X1*X2*x^7-X1*x^7+X1*x^6-x^6+2*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [6, 1], are n 53 n - 3/32 + ----, and , - 9/128 + ----, respectively, while 64 4096 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 7], are n 763 115 n - 7/128 + ---, and , - ----- + -----, respectively, while the asymptotic cor\ 128 16384 16384 relation between these two random variables is 1/2 1/2 52 53 115 --------------- 6095 and in floating point 0.6660648087 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 52 6095 ate normal pair with correlation, ---------- 6095 1/2 52 6095 11503 i.e. , [[----------, 0], [0, -----]] 6095 6095 ------------------------------------------------- Theorem Number, 27, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [6, 1], nor the composition, [2, 6] Then infinity ----- 11 10 9 8 6 \ n x + x + x + x - x + x - 1 ) a(n) x = - ------------------------------------ / 5 4 3 2 ----- (-1 + x) (x + x + x + x + x - 1) n = 0 and in Maple format -(x^11+x^10+x^9+x^8-x^6+x-1)/(-1+x)/(x^5+x^4+x^3+x^2+x-1) The asymptotic expression for a(n) is, n 0.54757187353650689312 1.9659482366454853372 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [6, 1] and d occurrences (as containment) of the composition, [2, 6], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 11 10 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x + X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 11 11 9 10 10 11 8 9 - X1 x - X2 x + X1 X2 x - X1 x - X2 x + x + X1 X2 x - X1 x 9 10 8 8 9 6 8 6 / - X2 x + x - X1 x - X2 x + x + X1 X2 x + x - x + x - 1) / ( / 7 6 7 6 X1 X2 x - X1 X2 x - X1 x + x - 2 x + 1) and in Maple format -(X1*X2*x^11+X1*X2*x^10-X1*x^11-X2*x^11+X1*X2*x^9-X1*x^10-X2*x^10+x^11+X1*X2*x^ 8-X1*x^9-X2*x^9+x^10-X1*x^8-X2*x^8+x^9+X1*X2*x^6+x^8-x^6+x-1)/(X1*X2*x^7-X1*X2* x^6-X1*x^7+x^6-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [6, 1], are n 53 n - 3/32 + ----, and , - 9/128 + ----, respectively, while 64 4096 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 6], are n 815 119 n - 7/128 + ---, and , - ----- + -----, respectively, while the asymptotic cor\ 128 16384 16384 relation between these two random variables is 1/2 1/2 54 53 119 --------------- 6307 and in floating point 0.6799584013 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 54 6307 ate normal pair with correlation, ---------- 6307 1/2 54 6307 12139 i.e. , [[----------, 0], [0, -----]] 6307 6307 ------------------------------------------------- Theorem Number, 28, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [6, 1], nor the composition, [3, 5] Then infinity ----- 10 9 5 \ n x + x - x + x - 1 ) a(n) x = - ----------------------------------------- / 7 6 4 3 2 ----- (-1 + x) (x + x + x + x + x + x - 1) n = 0 and in Maple format -(x^10+x^9-x^5+x-1)/(-1+x)/(x^7+x^6+x^4+x^3+x^2+x-1) The asymptotic expression for a(n) is, n 0.56732263340074550724 1.9576152606398896277 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [6, 1] and d occurrences (as containment) of the composition, [3, 5], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 10 9 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x + X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 10 10 9 9 10 9 6 6 5 - X1 x - X2 x - X1 x - X2 x + x + x + X1 X2 x - X2 x + X2 x 5 / 8 7 8 8 6 7 - x + x - 1) / (X1 X2 x + X1 X2 x - X1 x - X2 x - X1 X2 x - X1 x / 8 6 5 6 5 + x + 2 X2 x - X2 x - x + x - 2 x + 1) and in Maple format -(X1*X2*x^10+X1*X2*x^9-X1*x^10-X2*x^10-X1*x^9-X2*x^9+x^10+x^9+X1*X2*x^6-X2*x^6+ X2*x^5-x^5+x-1)/(X1*X2*x^8+X1*X2*x^7-X1*x^8-X2*x^8-X1*X2*x^6-X1*x^7+x^8+2*X2*x^ 6-X2*x^5-x^6+x^5-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [6, 1], are n 53 n - 3/32 + ----, and , - 9/128 + ----, respectively, while 64 4096 Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 5], are n 911 127 n - 7/128 + ---, and , - ----- + -----, respectively, while the asymptotic cor\ 128 16384 16384 relation between these two random variables is 1/2 1/2 26 53 127 --------------- 6731 and in floating point 0.3169082555 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 26 6731 ate normal pair with correlation, ---------- 6731 1/2 26 6731 8083 i.e. , [[----------, 0], [0, ----]] 6731 6731 ------------------------------------------------- Theorem Number, 29, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [6, 1], nor the composition, [4, 4] Then infinity ----- 4 \ n x - x + 1 ) a(n) x = -------------------------------------------------- / 8 7 6 5 3 2 ----- (-1 + x) (x + 2 x + 2 x + x + x + x + x - 1) n = 0 and in Maple format (x^4-x+1)/(-1+x)/(x^8+2*x^7+2*x^6+x^5+x^3+x^2+x-1) The asymptotic expression for a(n) is, n 0.57292931369834866970 1.9558149899357794253 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [6, 1] and d occurrences (as containment) of the composition, [4, 4], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| ) | ) | ) A(n, c, d) x X1 X2 || = - ( / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 6 6 4 4 / 9 8 9 X1 X2 x - X2 x + X2 x - x + x - 1) / (X1 X2 x + X1 X2 x - X1 x / 9 7 8 8 9 6 7 8 6 - X2 x + X1 X2 x - X1 x - X2 x + x - X1 X2 x - X1 x + x + 2 X2 x 5 6 4 5 4 + X2 x - x - X2 x - x + x - 2 x + 1) and in Maple format -(X1*X2*x^6-X2*x^6+X2*x^4-x^4+x-1)/(X1*X2*x^9+X1*X2*x^8-X1*x^9-X2*x^9+X1*X2*x^7 -X1*x^8-X2*x^8+x^9-X1*X2*x^6-X1*x^7+x^8+2*X2*x^6+X2*x^5-x^6-X2*x^4-x^5+x^4-2*x+ 1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [6, 1], are n 53 n - 3/32 + ----, and , - 9/128 + ----, respectively, while 64 4096 Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 4], are n 1087 143 n - 7/128 + ---, and , - ----- + -----, respectively, while the asymptotic cor\ 128 16384 16384 relation between these two random variables is 1/2 1/2 18 53 143 --------------- 7579 and in floating point 0.2067600133 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 18 7579 ate normal pair with correlation, ---------- 7579 1/2 18 7579 8227 i.e. , [[----------, 0], [0, ----]] 7579 7579 ------------------------------------------------- Theorem Number, 30, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [6, 1], nor the composition, [5, 3] Then infinity ----- 5 \ n x - x + 1 ) a(n) x = ----------------------------------------- / 7 6 4 3 2 ----- (-1 + x) (x + x + x + x + x + x - 1) n = 0 and in Maple format (x^5-x+1)/(-1+x)/(x^7+x^6+x^4+x^3+x^2+x-1) The asymptotic expression for a(n) is, n 0.57122365827984603579 1.9576152606398896277 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [6, 1] and d occurrences (as containment) of the composition, [5, 3], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| ) | ) | ) A(n, c, d) x X1 X2 || = - ( / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 6 6 5 5 / 8 7 8 X1 X2 x - X2 x + X2 x - x + x - 1) / (X1 X2 x + X1 X2 x - X1 x / 8 6 7 8 6 5 6 5 - X2 x - X1 X2 x - X1 x + x + 2 X2 x - X2 x - x + x - 2 x + 1) and in Maple format -(X1*X2*x^6-X2*x^6+X2*x^5-x^5+x-1)/(X1*X2*x^8+X1*X2*x^7-X1*x^8-X2*x^8-X1*X2*x^6 -X1*x^7+x^8+2*X2*x^6-X2*x^5-x^6+x^5-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [6, 1], are n 53 n - 3/32 + ----, and , - 9/128 + ----, respectively, while 64 4096 Furthermore the average and variance of the random variable: Number of occur\ rences of, [5, 3], are n 911 127 n - 7/128 + ---, and , - ----- + -----, respectively, while the asymptotic cor\ 128 16384 16384 relation between these two random variables is 1/2 1/2 26 53 127 --------------- 6731 and in floating point 0.3169082555 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 26 6731 ate normal pair with correlation, ---------- 6731 1/2 26 6731 8083 i.e. , [[----------, 0], [0, ----]] 6731 6731 ------------------------------------------------- ------------------------ This ends this article, that took, 3.294, to generate. ---------------------------------- All the generating functions (and statistical information) Enumerating compo\ sitions by number of occurrences, as containments of all possible pairs of offending compositions of, 7, with , 3, parts and of , 8, also with , 3, parts By Shalosh B. Ekhad Theorem Number, 1, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 1, 5], nor the composition, [1, 3, 4] Then infinity ----- 8 7 6 5 4 2 \ n x - x - x + x - x - x + 2 x - 1 ) a(n) x = ------------------------------------- / 6 5 3 2 2 ----- (x + x + x + x + x - 1) (-1 + x) n = 0 and in Maple format (x^8-x^7-x^6+x^5-x^4-x^2+2*x-1)/(x^6+x^5+x^3+x^2+x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.69790391550484586467 1.9111834366854762350 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 1, 5] and d occurrences (as containment) of the composition, [1, 3, 4], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 8 8 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x - X1 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 8 6 7 8 5 6 6 7 5 - X2 x - X1 X2 x + X2 x + x + X1 X2 x + X1 x + X2 x - x - 2 X2 x 6 4 5 4 2 / 7 6 - x + X2 x + x - x - x + 2 x - 1) / ((-1 + x) (X1 X2 x + X1 X2 x / 7 7 5 6 7 5 4 5 4 - X1 x - X2 x - X1 X2 x - X1 x + x + 2 X2 x - X2 x - x + x - 2 x + 1)) and in Maple format (X1*X2*x^8-X1*x^8-X2*x^8-X1*X2*x^6+X2*x^7+x^8+X1*X2*x^5+X1*x^6+X2*x^6-x^7-2*X2* x^5-x^6+X2*x^4+x^5-x^4-x^2+2*x-1)/(-1+x)/(X1*X2*x^7+X1*X2*x^6-X1*x^7-X2*x^7-X1* X2*x^5-X1*x^6+x^7+2*X2*x^5-X2*x^4-x^5+x^4-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 1, 5], are n 139 23 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 3, 4], are n 135 65 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 12 23 65 -------------- 1495 and in floating point 0.3103563600 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 12 1495 ate normal pair with correlation, ---------- 1495 1/2 12 1495 1783 i.e. , [[----------, 0], [0, ----]] 1495 1495 ------------------------------------------------- Theorem Number, 2, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 1, 5], nor the composition, [1, 4, 3] Then infinity ----- 8 7 6 4 \ n x + 2 x + x + x - x + 1 ) a(n) x = ------------------------------------ / 6 5 3 2 ----- (-1 + x) (x + x + x + x + x - 1) n = 0 and in Maple format (x^8+2*x^7+x^6+x^4-x+1)/(-1+x)/(x^6+x^5+x^3+x^2+x-1) The asymptotic expression for a(n) is, n 0.69079734461128122568 1.9111834366854762350 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 1, 5] and d occurrences (as containment) of the composition, [1, 4, 3], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 9 8 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x + X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 9 9 8 8 9 6 7 8 5 - X1 x - X2 x - X1 x - X2 x + x - X1 X2 x + X2 x + x + X1 X2 x 6 6 7 5 6 4 5 4 2 / + X1 x + X2 x - x - 2 X2 x - x + X2 x + x - x - x + 2 x - 1) / / 7 6 7 7 5 6 7 ((-1 + x) (X1 X2 x + X1 X2 x - X1 x - X2 x - X1 X2 x - X1 x + x 5 4 5 4 + 2 X2 x - X2 x - x + x - 2 x + 1)) and in Maple format (X1*X2*x^9+X1*X2*x^8-X1*x^9-X2*x^9-X1*x^8-X2*x^8+x^9-X1*X2*x^6+X2*x^7+x^8+X1*X2 *x^5+X1*x^6+X2*x^6-x^7-2*X2*x^5-x^6+X2*x^4+x^5-x^4-x^2+2*x-1)/(-1+x)/(X1*X2*x^7 +X1*X2*x^6-X1*x^7-X2*x^7-X1*X2*x^5-X1*x^6+x^7+2*X2*x^5-X2*x^4-x^5+x^4-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 1, 5], are n 139 23 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 4, 3], are n 135 65 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 12 23 65 -------------- 1495 and in floating point 0.3103563600 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 12 1495 ate normal pair with correlation, ---------- 1495 1/2 12 1495 1783 i.e. , [[----------, 0], [0, ----]] 1495 1495 ------------------------------------------------- Theorem Number, 3, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 1, 5], nor the composition, [1, 5, 2] Then infinity ----- 10 9 8 5 2 \ n x + x + x - x - x + 2 x - 1 ) a(n) x = --------------------------------- / 4 3 2 2 ----- (x + x + x + x - 1) (-1 + x) n = 0 and in Maple format (x^10+x^9+x^8-x^5-x^2+2*x-1)/(x^4+x^3+x^2+x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.63530875266095142903 1.9275619754829253043 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 1, 5] and d occurrences (as containment) of the composition, [1, 5, 2], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 10 9 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x + X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 10 10 8 9 9 10 7 8 - X1 x - X2 x + X1 X2 x - X1 x - X2 x + x + X1 X2 x - X1 x 8 9 6 7 8 5 6 5 2 - X2 x + x - X1 X2 x - X1 x + x + X1 X2 x + X1 x - x - x + 2 x / 6 5 6 5 - 1) / ((-1 + x) (X1 X2 x - X1 X2 x - X1 x + x - 2 x + 1)) / and in Maple format (X1*X2*x^10+X1*X2*x^9-X1*x^10-X2*x^10+X1*X2*x^8-X1*x^9-X2*x^9+x^10+X1*X2*x^7-X1 *x^8-X2*x^8+x^9-X1*X2*x^6-X1*x^7+x^8+X1*X2*x^5+X1*x^6-x^5-x^2+2*x-1)/(-1+x)/(X1 *X2*x^6-X1*X2*x^5-X1*x^6+x^5-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 1, 5], are n 139 23 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 5, 2], are n 111 57 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 8 23 57 ------------- 437 and in floating point 0.6628418121 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 1/2 8 3 437 ate normal pair with correlation, ------------- 437 1/2 1/2 8 3 437 821 i.e. , [[-------------, 0], [0, ---]] 437 437 ------------------------------------------------- Theorem Number, 4, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 1, 5], nor the composition, [1, 6, 1] Then infinity ----- 11 10 9 8 5 2 \ n x + x + x + x - x - x + 2 x - 1 ) a(n) x = --------------------------------------- / 4 3 2 2 ----- (x + x + x + x - 1) (-1 + x) n = 0 and in Maple format (x^11+x^10+x^9+x^8-x^5-x^2+2*x-1)/(x^4+x^3+x^2+x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.63351683346155297547 1.9275619754829253043 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 1, 5] and d occurrences (as containment) of the composition, [1, 6, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 11 10 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x + X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 11 11 9 10 10 11 8 9 - X1 x - X2 x + X1 X2 x - X1 x - X2 x + x + X1 X2 x - X1 x 9 10 8 8 9 6 8 6 5 5 - X2 x + x - X1 x - X2 x + x + X1 X2 x + x - X1 x + X1 x - x 2 / 6 6 5 5 - x + 2 x - 1) / ((-1 + x) (X1 X2 x - X1 x + X1 x - x + 2 x - 1)) / and in Maple format -(X1*X2*x^11+X1*X2*x^10-X1*x^11-X2*x^11+X1*X2*x^9-X1*x^10-X2*x^10+x^11+X1*X2*x^ 8-X1*x^9-X2*x^9+x^10-X1*x^8-X2*x^8+x^9+X1*X2*x^6+x^8-X1*x^6+X1*x^5-x^5-x^2+2*x-\ 1)/(-1+x)/(X1*X2*x^6-X1*x^6+X1*x^5-x^5+2*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 1, 5], are n 139 23 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 6, 1], are n 49 53 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 22 23 53 -------------- 1219 and in floating point 0.6301164650 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 22 1219 ate normal pair with correlation, ---------- 1219 1/2 22 1219 2187 i.e. , [[----------, 0], [0, ----]] 1219 1219 ------------------------------------------------- Theorem Number, 5, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 1, 5], nor the composition, [2, 2, 4] Then infinity ----- 11 10 7 6 5 4 2 \ n x - x + x - 2 x + x - x - x + 2 x - 1 ) a(n) x = ----------------------------------------------- / 9 8 7 5 3 2 2 ----- (x + x + x + x + x + x + x - 1) (-1 + x) n = 0 and in Maple format (x^11-x^10+x^7-2*x^6+x^5-x^4-x^2+2*x-1)/(x^9+x^8+x^7+x^5+x^3+x^2+x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.70132476238170317564 1.9099139227361088013 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 1, 5] and d occurrences (as containment) of the composition, [2, 2, 4], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 11 2 11 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x - X1 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 11 10 11 11 10 10 11 - 2 X1 X2 x + X1 X2 x + 2 X1 x + X2 x - X1 x - X2 x - x 10 7 6 7 7 5 6 6 + x - X1 X2 x + X1 X2 x + X1 x + X2 x - X1 X2 x - X1 x - 2 X2 x 7 5 6 4 5 4 2 / 2 11 - x + 2 X2 x + 2 x - X2 x - x + x + x - 2 x + 1) / (X1 X2 x / 2 11 11 10 11 11 10 10 - X1 x - 2 X1 X2 x + X1 X2 x + 2 X1 x + X2 x - X1 x - X2 x 11 8 10 7 8 8 6 7 - x + X1 X2 x + x - X1 X2 x - X1 x - X2 x + 2 X1 X2 x + X1 x 7 8 5 6 6 7 5 6 + 2 X2 x + x - X1 X2 x - X1 x - 3 X2 x - 2 x + 3 X2 x + 2 x 4 5 4 2 - X2 x - 2 x + x + 2 x - 3 x + 1) and in Maple format (X1^2*X2*x^11-X1^2*x^11-2*X1*X2*x^11+X1*X2*x^10+2*X1*x^11+X2*x^11-X1*x^10-X2*x^ 10-x^11+x^10-X1*X2*x^7+X1*X2*x^6+X1*x^7+X2*x^7-X1*X2*x^5-X1*x^6-2*X2*x^6-x^7+2* X2*x^5+2*x^6-X2*x^4-x^5+x^4+x^2-2*x+1)/(X1^2*X2*x^11-X1^2*x^11-2*X1*X2*x^11+X1* X2*x^10+2*X1*x^11+X2*x^11-X1*x^10-X2*x^10-x^11+X1*X2*x^8+x^10-X1*X2*x^7-X1*x^8- X2*x^8+2*X1*X2*x^6+X1*x^7+2*X2*x^7+x^8-X1*X2*x^5-X1*x^6-3*X2*x^6-2*x^7+3*X2*x^5 +2*x^6-X2*x^4-2*x^5+x^4+2*x^2-3*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 1, 5], are n 139 23 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 2, 4], are n 75 69 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 10 3 ------- 69 and in floating point 0.2510218562 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 10 3 ate normal pair with correlation, ------- 69 1/2 10 3 1787 i.e. , [[-------, 0], [0, ----]] 69 1587 ------------------------------------------------- Theorem Number, 6, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 1, 5], nor the composition, [2, 3, 3] Then infinity ----- 2 9 8 7 6 4 3 2 \ n (x - x + 1) (x + x - x - x - x - x + x + x - 1) ) a(n) x = -------------------------------------------------------- / 9 8 7 6 5 4 2 2 ----- (x + 2 x + x + x + 2 x + x + x + x - 1) (-1 + x) n = 0 and in Maple format (x^2-x+1)*(x^9+x^8-x^7-x^6-x^4-x^3+x^2+x-1)/(x^9+2*x^8+x^7+x^6+2*x^5+x^4+x^2+x-\ 1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.70325653317007746417 1.9073680513412163549 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 1, 5] and d occurrences (as containment) of the composition, [2, 3, 3], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 11 2 11 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x - X1 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 11 11 11 9 11 8 9 - 2 X1 X2 x + 2 X1 x + X2 x + X1 X2 x - x - X1 X2 x - X1 x 9 8 8 9 6 8 5 6 6 - X2 x + X1 x + X2 x + x + X1 X2 x - x - X1 X2 x - X1 x - 2 X2 x 5 6 4 3 4 3 2 / 2 11 + X2 x + 2 x + X2 x - X2 x - x + x + x - 2 x + 1) / (X1 X2 x / 2 11 11 11 11 9 11 8 - X1 x - 2 X1 X2 x + 2 X1 x + X2 x + 2 X1 X2 x - x - X1 X2 x 9 9 8 8 9 6 7 8 - 2 X1 x - 2 X2 x + X1 x + X2 x + 2 x + 2 X1 X2 x + X2 x - x 5 6 6 7 5 6 4 3 4 - X1 X2 x - X1 x - 3 X2 x - x + X2 x + 2 x + 2 X2 x - X2 x - 2 x 3 2 + x + 2 x - 3 x + 1) and in Maple format (X1^2*X2*x^11-X1^2*x^11-2*X1*X2*x^11+2*X1*x^11+X2*x^11+X1*X2*x^9-x^11-X1*X2*x^8 -X1*x^9-X2*x^9+X1*x^8+X2*x^8+x^9+X1*X2*x^6-x^8-X1*X2*x^5-X1*x^6-2*X2*x^6+X2*x^5 +2*x^6+X2*x^4-X2*x^3-x^4+x^3+x^2-2*x+1)/(X1^2*X2*x^11-X1^2*x^11-2*X1*X2*x^11+2* X1*x^11+X2*x^11+2*X1*X2*x^9-x^11-X1*X2*x^8-2*X1*x^9-2*X2*x^9+X1*x^8+X2*x^8+2*x^ 9+2*X1*X2*x^6+X2*x^7-x^8-X1*X2*x^5-X1*x^6-3*X2*x^6-x^7+X2*x^5+2*x^6+2*X2*x^4-X2 *x^3-2*x^4+x^3+2*x^2-3*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 1, 5], are n 139 23 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 3, 3], are n 97 85 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 6 23 85 ------------- 1955 and in floating point 0.1356993819 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 6 1955 ate normal pair with correlation, --------- 1955 1/2 6 1955 2027 i.e. , [[---------, 0], [0, ----]] 1955 1955 ------------------------------------------------- Theorem Number, 7, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 1, 5], nor the composition, [2, 4, 2] Then infinity ----- 11 9 7 6 5 4 2 \ n x + x + x - 2 x + x - x - x + 2 x - 1 ) a(n) x = ----------------------------------------------- / 9 8 7 5 3 2 2 ----- (x + x + x + x + x + x + x - 1) (-1 + x) n = 0 and in Maple format (x^11+x^9+x^7-2*x^6+x^5-x^4-x^2+2*x-1)/(x^9+x^8+x^7+x^5+x^3+x^2+x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.69057460186873174687 1.9099139227361088013 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 1, 5] and d occurrences (as containment) of the composition, [2, 4, 2], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 11 2 11 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x - X1 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 11 11 11 9 11 9 9 - 2 X1 X2 x + 2 X1 x + X2 x - X1 X2 x - x + X1 x + X2 x 7 9 6 7 7 5 6 6 - X1 X2 x - x + X1 X2 x + X1 x + X2 x - X1 X2 x - X1 x - 2 X2 x 7 5 6 4 5 4 2 / 2 11 - x + 2 X2 x + 2 x - X2 x - x + x + x - 2 x + 1) / (X1 X2 x / 2 11 11 10 11 11 10 10 - X1 x - 2 X1 X2 x + X1 X2 x + 2 X1 x + X2 x - X1 x - X2 x 11 8 10 7 8 8 6 7 - x + X1 X2 x + x - X1 X2 x - X1 x - X2 x + 2 X1 X2 x + X1 x 7 8 5 6 6 7 5 6 + 2 X2 x + x - X1 X2 x - X1 x - 3 X2 x - 2 x + 3 X2 x + 2 x 4 5 4 2 - X2 x - 2 x + x + 2 x - 3 x + 1) and in Maple format (X1^2*X2*x^11-X1^2*x^11-2*X1*X2*x^11+2*X1*x^11+X2*x^11-X1*X2*x^9-x^11+X1*x^9+X2 *x^9-X1*X2*x^7-x^9+X1*X2*x^6+X1*x^7+X2*x^7-X1*X2*x^5-X1*x^6-2*X2*x^6-x^7+2*X2*x ^5+2*x^6-X2*x^4-x^5+x^4+x^2-2*x+1)/(X1^2*X2*x^11-X1^2*x^11-2*X1*X2*x^11+X1*X2*x ^10+2*X1*x^11+X2*x^11-X1*x^10-X2*x^10-x^11+X1*X2*x^8+x^10-X1*X2*x^7-X1*x^8-X2*x ^8+2*X1*X2*x^6+X1*x^7+2*X2*x^7+x^8-X1*X2*x^5-X1*x^6-3*X2*x^6-2*x^7+3*X2*x^5+2*x ^6-X2*x^4-2*x^5+x^4+2*x^2-3*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 1, 5], are n 139 23 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 4, 2], are n 75 69 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 10 3 ------- 69 and in floating point 0.2510218562 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 10 3 ate normal pair with correlation, ------- 69 1/2 10 3 1787 i.e. , [[-------, 0], [0, ----]] 69 1587 ------------------------------------------------- Theorem Number, 8, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 1, 5], nor the composition, [2, 5, 1] Then infinity ----- 11 10 9 8 5 2 \ n x + x + x + x - x - x + 2 x - 1 ) a(n) x = --------------------------------------- / 4 3 2 2 ----- (x + x + x + x - 1) (-1 + x) n = 0 and in Maple format (x^11+x^10+x^9+x^8-x^5-x^2+2*x-1)/(x^4+x^3+x^2+x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.63351683346155297547 1.9275619754829253043 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 1, 5] and d occurrences (as containment) of the composition, [2, 5, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 12 2 12 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x - X1 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 12 12 12 12 7 8 6 - 2 X1 X2 x + 2 X1 x + X2 x - x + 2 X1 X2 x - X2 x - 2 X1 X2 x 7 8 5 6 6 5 3 2 / - 2 X1 x + x + X1 X2 x + X1 x + x - x + x - 3 x + 3 x - 1) / ( / 2 6 5 6 5 (-1 + x) (X1 X2 x - X1 X2 x - X1 x + x - 2 x + 1)) and in Maple format -(X1^2*X2*x^12-X1^2*x^12-2*X1*X2*x^12+2*X1*x^12+X2*x^12-x^12+2*X1*X2*x^7-X2*x^8 -2*X1*X2*x^6-2*X1*x^7+x^8+X1*X2*x^5+X1*x^6+x^6-x^5+x^3-3*x^2+3*x-1)/(-1+x)^2/( X1*X2*x^6-X1*X2*x^5-X1*x^6+x^5-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 1, 5], are n 139 23 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 5, 1], are n 111 57 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 8 23 57 ------------- 437 and in floating point 0.6628418121 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 1/2 8 3 437 ate normal pair with correlation, ------------- 437 1/2 1/2 8 3 437 821 i.e. , [[-------------, 0], [0, ---]] 437 437 ------------------------------------------------- Theorem Number, 9, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 1, 5], nor the composition, [3, 1, 4] Then infinity ----- 11 9 4 2 \ n x - x - x - x + 2 x - 1 ) a(n) x = --------------------------------------------------- / 6 5 3 2 6 5 2 ----- (x + x + x + x + x - 1) (x + x + 1) (-1 + x) n = 0 and in Maple format (x^11-x^9-x^4-x^2+2*x-1)/(x^6+x^5+x^3+x^2+x-1)/(x^6+x^5+1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.69440972834504952286 1.9111834366854762350 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 1, 5] and d occurrences (as containment) of the composition, [3, 1, 4], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 12 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 2 11 2 12 2 12 2 11 12 + X1 X2 x - X1 X2 x - 2 X1 X2 x - X1 X2 x + 2 X1 X2 x 2 12 2 10 11 2 11 12 10 + X2 x + 2 X1 X2 x - X1 X2 x - X2 x - X2 x - 3 X1 X2 x 11 2 10 11 10 2 9 10 11 9 + X1 x - X2 x + 2 X2 x + X1 x + X2 x + X2 x - x - 2 X2 x 9 5 5 4 4 2 / 7 + x - X1 X2 x + X2 x - X2 x + x + x - 2 x + 1) / ((X1 X2 x / 6 7 7 5 6 7 5 4 5 + X1 X2 x - X1 x - X2 x - X1 X2 x - X1 x + x + 2 X2 x - X2 x - x 4 + x - 2 x + 1) 7 6 7 7 6 7 5 5 (X1 X2 x + X1 X2 x - X1 x - X2 x - X1 x + x + X2 x - x + x - 1)) and in Maple format -(X1^2*X2^2*x^12+X1^2*X2^2*x^11-X1^2*X2*x^12-2*X1*X2^2*x^12-X1^2*X2*x^11+2*X1* X2*x^12+X2^2*x^12+2*X1*X2^2*x^10-X1*X2*x^11-X2^2*x^11-X2*x^12-3*X1*X2*x^10+X1*x ^11-X2^2*x^10+2*X2*x^11+X1*x^10+X2^2*x^9+X2*x^10-x^11-2*X2*x^9+x^9-X1*X2*x^5+X2 *x^5-X2*x^4+x^4+x^2-2*x+1)/(X1*X2*x^7+X1*X2*x^6-X1*x^7-X2*x^7-X1*X2*x^5-X1*x^6+ x^7+2*X2*x^5-X2*x^4-x^5+x^4-2*x+1)/(X1*X2*x^7+X1*X2*x^6-X1*x^7-X2*x^7-X1*x^6+x^ 7+X2*x^5-x^5+x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 1, 5], are n 139 23 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 1, 4], are n 141 65 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 12 23 65 -------------- 1495 and in floating point 0.3103563600 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 12 1495 ate normal pair with correlation, ---------- 1495 1/2 12 1495 1783 i.e. , [[----------, 0], [0, ----]] 1495 1495 ------------------------------------------------- Theorem Number, 10, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 1, 5], nor the composition, [3, 2, 3] Then infinity ----- \ n 11 10 9 8 7 6 5 4 3 2 ) a(n) x = (x + x - x - x + x - x - x + x - x - x + 2 x - 1 / ----- n = 0 / ) / ( / 12 11 10 9 8 7 6 5 4 2 (x + 3 x + 3 x + 2 x + 3 x + 3 x + x + x + x + x + x - 1) 2 (-1 + x) ) and in Maple format (x^11+x^10-x^9-x^8+x^7-x^6-x^5+x^4-x^3-x^2+2*x-1)/(x^12+3*x^11+3*x^10+2*x^9+3*x ^8+3*x^7+x^6+x^5+x^4+x^2+x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.70407803205746399354 1.9060446822436648581 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 1, 5] and d occurrences (as containment) of the composition, [3, 2, 3], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 12 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 12 2 12 2 11 12 2 12 2 11 - X1 X2 x - 2 X1 X2 x + X1 X2 x + 2 X1 X2 x + X2 x - X1 x 2 10 11 12 10 11 2 10 + 2 X1 X2 x - 2 X1 X2 x - X2 x - 3 X1 X2 x + 2 X1 x - 2 X2 x 11 9 10 10 11 9 2 8 9 + X2 x + X1 X2 x + X1 x + 3 X2 x - x - X1 x + X2 x - X2 x 10 7 8 9 6 7 7 8 5 - x - X1 X2 x - 2 X2 x + x + X1 X2 x + X1 x + X2 x + x - X1 X2 x 6 6 7 6 4 5 3 4 3 2 - X1 x - X2 x - x + x + X2 x + x - X2 x - x + x + x - 2 x + 1) / 2 2 14 2 2 13 2 14 2 14 2 2 12 / (X1 X2 x + X1 X2 x - 2 X1 X2 x - 2 X1 X2 x - X1 X2 x / 2 13 2 14 2 13 14 2 14 - 2 X1 X2 x + X1 x - 2 X1 X2 x + 4 X1 X2 x + X2 x 2 12 2 13 2 12 13 14 2 13 + X1 X2 x + X1 x + 4 X1 X2 x + 4 X1 X2 x - 2 X1 x + X2 x 14 2 11 2 11 12 13 - 2 X2 x - X1 X2 x + 2 X1 X2 x - 6 X1 X2 x - 2 X1 x 2 12 13 14 2 11 2 10 11 - 3 X2 x - 2 X2 x + x + X1 x - 2 X1 X2 x - 2 X1 X2 x 12 2 11 12 13 10 2 10 + 2 X1 x - 2 X2 x + 5 X2 x + x + 3 X1 X2 x + 3 X2 x 11 12 9 10 2 9 10 11 + 3 X2 x - 2 x - 2 X1 X2 x - X1 x + X2 x - 5 X2 x - x 8 9 2 8 10 7 8 8 9 - X1 X2 x + 2 X1 x - X2 x + 2 x + X1 X2 x + X1 x + 3 X2 x - x 6 7 7 8 5 6 6 7 - 2 X1 X2 x - X1 x - 2 X2 x - 2 x + X1 X2 x + X1 x + X2 x + 2 x 4 5 3 4 3 2 - 2 X2 x - x + X2 x + 2 x - x - 2 x + 3 x - 1) and in Maple format -(X1^2*X2^2*x^12-X1^2*X2*x^12-2*X1*X2^2*x^12+X1^2*X2*x^11+2*X1*X2*x^12+X2^2*x^ 12-X1^2*x^11+2*X1*X2^2*x^10-2*X1*X2*x^11-X2*x^12-3*X1*X2*x^10+2*X1*x^11-2*X2^2* x^10+X2*x^11+X1*X2*x^9+X1*x^10+3*X2*x^10-x^11-X1*x^9+X2^2*x^8-X2*x^9-x^10-X1*X2 *x^7-2*X2*x^8+x^9+X1*X2*x^6+X1*x^7+X2*x^7+x^8-X1*X2*x^5-X1*x^6-X2*x^6-x^7+x^6+ X2*x^4+x^5-X2*x^3-x^4+x^3+x^2-2*x+1)/(X1^2*X2^2*x^14+X1^2*X2^2*x^13-2*X1^2*X2*x ^14-2*X1*X2^2*x^14-X1^2*X2^2*x^12-2*X1^2*X2*x^13+X1^2*x^14-2*X1*X2^2*x^13+4*X1* X2*x^14+X2^2*x^14+X1^2*X2*x^12+X1^2*x^13+4*X1*X2^2*x^12+4*X1*X2*x^13-2*X1*x^14+ X2^2*x^13-2*X2*x^14-X1^2*X2*x^11+2*X1*X2^2*x^11-6*X1*X2*x^12-2*X1*x^13-3*X2^2*x ^12-2*X2*x^13+x^14+X1^2*x^11-2*X1*X2^2*x^10-2*X1*X2*x^11+2*X1*x^12-2*X2^2*x^11+ 5*X2*x^12+x^13+3*X1*X2*x^10+3*X2^2*x^10+3*X2*x^11-2*x^12-2*X1*X2*x^9-X1*x^10+X2 ^2*x^9-5*X2*x^10-x^11-X1*X2*x^8+2*X1*x^9-X2^2*x^8+2*x^10+X1*X2*x^7+X1*x^8+3*X2* x^8-x^9-2*X1*X2*x^6-X1*x^7-2*X2*x^7-2*x^8+X1*X2*x^5+X1*x^6+X2*x^6+2*x^7-2*X2*x^ 4-x^5+X2*x^3+2*x^4-x^3-2*x^2+3*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 1, 5], are n 139 23 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 2, 3], are n 89 77 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 6 23 77 ------------- 1771 and in floating point 0.1425745369 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 6 1771 ate normal pair with correlation, --------- 1771 1/2 6 1771 1843 i.e. , [[---------, 0], [0, ----]] 1771 1771 ------------------------------------------------- Theorem Number, 11, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 1, 5], nor the composition, [3, 3, 2] Then infinity ----- 12 11 10 9 8 6 4 3 2 \ n x + 2 x + x - x + x - 2 x + x - x - x + 2 x - 1 ) a(n) x = ----------------------------------------------------------- / 9 8 7 6 5 4 2 2 ----- (x + 2 x + x + x + 2 x + x + x + x - 1) (-1 + x) n = 0 and in Maple format (x^12+2*x^11+x^10-x^9+x^8-2*x^6+x^4-x^3-x^2+2*x-1)/(x^9+2*x^8+x^7+x^6+2*x^5+x^4 +x^2+x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.69729201967249542857 1.9073680513412163549 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 1, 5] and d occurrences (as containment) of the composition, [3, 3, 2], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 13 2 12 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x + X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 13 13 2 11 2 12 12 13 - X1 x - 2 X1 X2 x - X1 X2 x - X1 x - 2 X1 X2 x + 2 X1 x 13 2 11 11 12 12 13 10 + X2 x + X1 x + 2 X1 X2 x + 2 X1 x + X2 x - x + 2 X1 X2 x 11 11 12 9 10 10 11 - 2 X1 x - X2 x - x - 2 X1 X2 x - 2 X1 x - 2 X2 x + x 8 9 9 10 7 8 8 9 + X1 X2 x + 2 X1 x + 2 X2 x + 2 x + X1 X2 x - X1 x - X2 x - 2 x 6 7 7 8 5 6 6 7 - 2 X1 X2 x - X1 x - 2 X2 x + x + X1 X2 x + X1 x + 3 X2 x + 2 x 6 4 5 3 4 2 / - 2 x - 2 X2 x - x + X2 x + 2 x - 3 x + 3 x - 1) / ((-1 + x) ( / 2 11 2 11 11 11 11 9 11 X1 X2 x - X1 x - 2 X1 X2 x + 2 X1 x + X2 x + 2 X1 X2 x - x 8 9 9 8 8 9 6 7 - X1 X2 x - 2 X1 x - 2 X2 x + X1 x + X2 x + 2 x + 2 X1 X2 x + X2 x 8 5 6 6 7 5 6 4 3 - x - X1 X2 x - X1 x - 3 X2 x - x + X2 x + 2 x + 2 X2 x - X2 x 4 3 2 - 2 x + x + 2 x - 3 x + 1)) and in Maple format (X1^2*X2*x^13+X1^2*X2*x^12-X1^2*x^13-2*X1*X2*x^13-X1^2*X2*x^11-X1^2*x^12-2*X1* X2*x^12+2*X1*x^13+X2*x^13+X1^2*x^11+2*X1*X2*x^11+2*X1*x^12+X2*x^12-x^13+2*X1*X2 *x^10-2*X1*x^11-X2*x^11-x^12-2*X1*X2*x^9-2*X1*x^10-2*X2*x^10+x^11+X1*X2*x^8+2* X1*x^9+2*X2*x^9+2*x^10+X1*X2*x^7-X1*x^8-X2*x^8-2*x^9-2*X1*X2*x^6-X1*x^7-2*X2*x^ 7+x^8+X1*X2*x^5+X1*x^6+3*X2*x^6+2*x^7-2*x^6-2*X2*x^4-x^5+X2*x^3+2*x^4-3*x^2+3*x -1)/(-1+x)/(X1^2*X2*x^11-X1^2*x^11-2*X1*X2*x^11+2*X1*x^11+X2*x^11+2*X1*X2*x^9-x ^11-X1*X2*x^8-2*X1*x^9-2*X2*x^9+X1*x^8+X2*x^8+2*x^9+2*X1*X2*x^6+X2*x^7-x^8-X1* X2*x^5-X1*x^6-3*X2*x^6-x^7+X2*x^5+2*x^6+2*X2*x^4-X2*x^3-2*x^4+x^3+2*x^2-3*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 1, 5], are n 139 23 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 3, 2], are n 97 85 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 6 23 85 ------------- 1955 and in floating point 0.1356993819 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 6 1955 ate normal pair with correlation, --------- 1955 1/2 6 1955 2027 i.e. , [[---------, 0], [0, ----]] 1955 1955 ------------------------------------------------- Theorem Number, 12, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 1, 5], nor the composition, [3, 4, 1] Then infinity ----- \ n ) a(n) x = / ----- n = 0 12 11 10 9 8 7 6 5 4 2 x + 2 x + x + x + x - x - x + x - x - x + 2 x - 1 -------------------------------------------------------------- 6 5 3 2 2 (x + x + x + x + x - 1) (-1 + x) and in Maple format (x^12+2*x^11+x^10+x^9+x^8-x^7-x^6+x^5-x^4-x^2+2*x-1)/(x^6+x^5+x^3+x^2+x-1)/(-1+ x)^2 The asymptotic expression for a(n) is, n 0.68216970258446089521 1.9111834366854762350 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 1, 5] and d occurrences (as containment) of the composition, [3, 4, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 13 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 12 2 13 13 2 12 12 13 + X1 X2 x - X1 x - 2 X1 X2 x - X1 x - 2 X1 X2 x + 2 X1 x 13 11 12 12 13 11 11 12 + X2 x + X1 X2 x + 2 X1 x + X2 x - x - X1 x - X2 x - x 11 8 7 8 8 6 7 8 + x + X1 X2 x + X1 X2 x - X1 x - 2 X2 x - 2 X1 X2 x - X1 x + 2 x 5 6 6 5 6 4 5 4 3 + X1 X2 x + X1 x + 3 X2 x - 3 X2 x - 2 x + X2 x + 2 x - x + x 2 / 2 7 6 7 7 - 3 x + 3 x - 1) / ((-1 + x) (X1 X2 x + X1 X2 x - X1 x - X2 x / 5 6 7 5 4 5 4 - X1 X2 x - X1 x + x + 2 X2 x - X2 x - x + x - 2 x + 1)) and in Maple format -(X1^2*X2*x^13+X1^2*X2*x^12-X1^2*x^13-2*X1*X2*x^13-X1^2*x^12-2*X1*X2*x^12+2*X1* x^13+X2*x^13+X1*X2*x^11+2*X1*x^12+X2*x^12-x^13-X1*x^11-X2*x^11-x^12+x^11+X1*X2* x^8+X1*X2*x^7-X1*x^8-2*X2*x^8-2*X1*X2*x^6-X1*x^7+2*x^8+X1*X2*x^5+X1*x^6+3*X2*x^ 6-3*X2*x^5-2*x^6+X2*x^4+2*x^5-x^4+x^3-3*x^2+3*x-1)/(-1+x)^2/(X1*X2*x^7+X1*X2*x^ 6-X1*x^7-X2*x^7-X1*X2*x^5-X1*x^6+x^7+2*X2*x^5-X2*x^4-x^5+x^4-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 1, 5], are n 139 23 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 4, 1], are n 135 65 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 12 23 65 -------------- 1495 and in floating point 0.3103563600 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 12 1495 ate normal pair with correlation, ---------- 1495 1/2 12 1495 1783 i.e. , [[----------, 0], [0, ----]] 1495 1495 ------------------------------------------------- Theorem Number, 13, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 1, 5], nor the composition, [4, 1, 3] Then infinity ----- \ n ) a(n) x = / ----- n = 0 16 15 14 13 12 11 10 4 2 x + x - x - 2 x - x - 2 x - x + x + x - 2 x + 1 - --------------------------------------------------------------- 6 5 3 2 6 5 2 (x + x + x + x + x - 1) (x + x + 1) (-1 + x) and in Maple format -(x^16+x^15-x^14-2*x^13-x^12-2*x^11-x^10+x^4+x^2-2*x+1)/(x^6+x^5+x^3+x^2+x-1)/( x^6+x^5+1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.68033974934962079394 1.9111834366854762350 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 1, 5] and d occurrences (as containment) of the composition, [4, 1, 3], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 17 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 17 2 17 2 2 15 2 17 17 - 2 X1 X2 x - 2 X1 X2 x - X1 X2 x + X1 x + 4 X1 X2 x 2 17 2 2 14 2 15 2 15 17 + X2 x - X1 X2 x + 2 X1 X2 x + 3 X1 X2 x - 2 X1 x 17 2 2 13 2 14 2 15 2 14 - 2 X2 x + X1 X2 x + 3 X1 X2 x - X1 x + X1 X2 x 15 2 15 17 2 13 2 14 2 13 - 6 X1 X2 x - 2 X2 x + x - X1 X2 x - 2 X1 x - 3 X1 X2 x 14 15 15 2 2 11 2 12 - 4 X1 X2 x + 3 X1 x + 4 X2 x - X1 X2 x + 2 X1 X2 x 13 14 2 13 14 15 2 11 + 4 X1 X2 x + 3 X1 x + 2 X2 x + X2 x - 2 x + X1 X2 x 2 11 12 13 2 12 13 14 + 2 X1 X2 x - 3 X1 X2 x - X1 x - 2 X2 x - 3 X2 x - x 2 10 11 12 12 13 10 - 2 X1 X2 x - 2 X1 X2 x + X1 x + 3 X2 x + x + 3 X1 X2 x 2 10 11 12 9 10 2 9 10 11 + 2 X2 x - X2 x - x + X1 X2 x - X1 x - X2 x - 3 X2 x + x 9 9 10 6 5 6 5 4 5 - X1 x + X2 x + x - X1 X2 x + X1 X2 x + X2 x - 2 X2 x + X2 x + x 4 3 2 / 7 6 7 - x + x - 3 x + 3 x - 1) / ((-1 + x) (X1 X2 x + X1 X2 x - X1 x / 7 5 6 7 5 4 5 4 - X2 x - X1 X2 x - X1 x + x + 2 X2 x - X2 x - x + x - 2 x + 1) 7 6 7 7 6 7 5 5 (X1 X2 x + X1 X2 x - X1 x - X2 x - X1 x + x + X2 x - x + x - 1)) and in Maple format -(X1^2*X2^2*x^17-2*X1^2*X2*x^17-2*X1*X2^2*x^17-X1^2*X2^2*x^15+X1^2*x^17+4*X1*X2 *x^17+X2^2*x^17-X1^2*X2^2*x^14+2*X1^2*X2*x^15+3*X1*X2^2*x^15-2*X1*x^17-2*X2*x^ 17+X1^2*X2^2*x^13+3*X1^2*X2*x^14-X1^2*x^15+X1*X2^2*x^14-6*X1*X2*x^15-2*X2^2*x^ 15+x^17-X1^2*X2*x^13-2*X1^2*x^14-3*X1*X2^2*x^13-4*X1*X2*x^14+3*X1*x^15+4*X2*x^ 15-X1^2*X2^2*x^11+2*X1*X2^2*x^12+4*X1*X2*x^13+3*X1*x^14+2*X2^2*x^13+X2*x^14-2*x ^15+X1^2*X2*x^11+2*X1*X2^2*x^11-3*X1*X2*x^12-X1*x^13-2*X2^2*x^12-3*X2*x^13-x^14 -2*X1*X2^2*x^10-2*X1*X2*x^11+X1*x^12+3*X2*x^12+x^13+3*X1*X2*x^10+2*X2^2*x^10-X2 *x^11-x^12+X1*X2*x^9-X1*x^10-X2^2*x^9-3*X2*x^10+x^11-X1*x^9+X2*x^9+x^10-X1*X2*x ^6+X1*X2*x^5+X2*x^6-2*X2*x^5+X2*x^4+x^5-x^4+x^3-3*x^2+3*x-1)/(-1+x)/(X1*X2*x^7+ X1*X2*x^6-X1*x^7-X2*x^7-X1*X2*x^5-X1*x^6+x^7+2*X2*x^5-X2*x^4-x^5+x^4-2*x+1)/(X1 *X2*x^7+X1*X2*x^6-X1*x^7-X2*x^7-X1*x^6+x^7+X2*x^5-x^5+x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 1, 5], are n 139 23 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 1, 3], are n 141 65 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 12 23 65 -------------- 1495 and in floating point 0.3103563600 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 12 1495 ate normal pair with correlation, ---------- 1495 1/2 12 1495 1783 i.e. , [[----------, 0], [0, ----]] 1495 1495 ------------------------------------------------- Theorem Number, 14, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 1, 5], nor the composition, [4, 2, 2] Then infinity ----- \ n 16 15 14 12 11 10 9 7 ) a(n) x = - (x + 2 x + 2 x - 2 x - 3 x - x - x - x / ----- n = 0 6 5 4 2 / + 2 x - x + x + x - 2 x + 1) / ( / 9 8 7 5 3 2 2 (x + x + x + x + x + x + x - 1) (-1 + x) ) and in Maple format -(x^16+2*x^15+2*x^14-2*x^12-3*x^11-x^10-x^9-x^7+2*x^6-x^5+x^4+x^2-2*x+1)/(x^9+x ^8+x^7+x^5+x^3+x^2+x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.68190830405220103442 1.9099139227361088013 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 1, 5] and d occurrences (as containment) of the composition, [4, 2, 2], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 17 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 2 16 2 17 2 17 2 16 2 17 + X1 X2 x - 2 X1 X2 x - 2 X1 X2 x - 2 X1 X2 x + X1 x 2 16 17 2 17 2 16 16 17 - 2 X1 X2 x + 4 X1 X2 x + X2 x + X1 x + 4 X1 X2 x - 2 X1 x 2 16 17 2 14 2 14 16 16 17 + X2 x - 2 X2 x + X1 X2 x + X1 X2 x - 2 X1 x - 2 X2 x + x 2 13 2 14 2 13 14 2 14 16 + X1 X2 x - X1 x + X1 X2 x - 4 X1 X2 x - X2 x + x 2 12 2 13 13 14 2 13 14 + X1 X2 x - X1 x - 4 X1 X2 x + 3 X1 x - X2 x + 3 X2 x 2 11 2 12 12 13 13 14 - X1 X2 x - X1 x - 2 X1 X2 x + 3 X1 x + 3 X2 x - 2 x 2 11 11 12 12 13 11 11 + X1 x + 3 X1 X2 x + 2 X1 x + X2 x - 2 x - 3 X1 x - 2 X2 x 12 9 11 8 9 9 7 8 - x + X1 X2 x + 2 x - X1 X2 x - X1 x - X2 x + 2 X1 X2 x + X1 x 8 9 6 7 7 8 5 6 + X2 x + x - 2 X1 X2 x - 2 X1 x - 3 X2 x - x + X1 X2 x + X1 x 6 7 5 6 4 5 4 3 2 + 4 X2 x + 3 x - 3 X2 x - 3 x + X2 x + 2 x - x + x - 3 x + 3 x / 2 11 2 11 11 10 - 1) / ((-1 + x) (X1 X2 x - X1 x - 2 X1 X2 x + X1 X2 x / 11 11 10 10 11 8 10 7 + 2 X1 x + X2 x - X1 x - X2 x - x + X1 X2 x + x - X1 X2 x 8 8 6 7 7 8 5 6 - X1 x - X2 x + 2 X1 X2 x + X1 x + 2 X2 x + x - X1 X2 x - X1 x 6 7 5 6 4 5 4 2 - 3 X2 x - 2 x + 3 X2 x + 2 x - X2 x - 2 x + x + 2 x - 3 x + 1)) and in Maple format (X1^2*X2^2*x^17+X1^2*X2^2*x^16-2*X1^2*X2*x^17-2*X1*X2^2*x^17-2*X1^2*X2*x^16+X1^ 2*x^17-2*X1*X2^2*x^16+4*X1*X2*x^17+X2^2*x^17+X1^2*x^16+4*X1*X2*x^16-2*X1*x^17+ X2^2*x^16-2*X2*x^17+X1^2*X2*x^14+X1*X2^2*x^14-2*X1*x^16-2*X2*x^16+x^17+X1^2*X2* x^13-X1^2*x^14+X1*X2^2*x^13-4*X1*X2*x^14-X2^2*x^14+x^16+X1^2*X2*x^12-X1^2*x^13-\ 4*X1*X2*x^13+3*X1*x^14-X2^2*x^13+3*X2*x^14-X1^2*X2*x^11-X1^2*x^12-2*X1*X2*x^12+ 3*X1*x^13+3*X2*x^13-2*x^14+X1^2*x^11+3*X1*X2*x^11+2*X1*x^12+X2*x^12-2*x^13-3*X1 *x^11-2*X2*x^11-x^12+X1*X2*x^9+2*x^11-X1*X2*x^8-X1*x^9-X2*x^9+2*X1*X2*x^7+X1*x^ 8+X2*x^8+x^9-2*X1*X2*x^6-2*X1*x^7-3*X2*x^7-x^8+X1*X2*x^5+X1*x^6+4*X2*x^6+3*x^7-\ 3*X2*x^5-3*x^6+X2*x^4+2*x^5-x^4+x^3-3*x^2+3*x-1)/(-1+x)/(X1^2*X2*x^11-X1^2*x^11 -2*X1*X2*x^11+X1*X2*x^10+2*X1*x^11+X2*x^11-X1*x^10-X2*x^10-x^11+X1*X2*x^8+x^10- X1*X2*x^7-X1*x^8-X2*x^8+2*X1*X2*x^6+X1*x^7+2*X2*x^7+x^8-X1*X2*x^5-X1*x^6-3*X2*x ^6-2*x^7+3*X2*x^5+2*x^6-X2*x^4-2*x^5+x^4+2*x^2-3*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 1, 5], are n 139 23 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 2, 2], are n 75 69 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 10 3 ------- 69 and in floating point 0.2510218562 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 10 3 ate normal pair with correlation, ------- 69 1/2 10 3 1787 i.e. , [[-------, 0], [0, ----]] 69 1587 ------------------------------------------------- Theorem Number, 15, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 1, 5], nor the composition, [4, 3, 1] Then infinity ----- \ n ) a(n) x = / ----- n = 0 3 2 13 11 9 7 6 4 3 2 (x + x - 1) (x + x - 2 x - 2 x - x - x + x - 2 x + 2 x - 1) - ----------------------------------------------------------------------- 6 5 3 2 2 (x + x + x + x + x - 1) (-1 + x) and in Maple format -(x^3+x^2-1)*(x^13+x^11-2*x^9-2*x^7-x^6-x^4+x^3-2*x^2+2*x-1)/(x^6+x^5+x^3+x^2+x -1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.67598850878507360158 1.9111834366854762350 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 1, 5] and d occurrences (as containment) of the composition, [4, 3, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 17 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 17 2 17 2 17 17 2 17 17 - 2 X1 X2 x - 2 X1 X2 x + X1 x + 4 X1 X2 x + X2 x - 2 X1 x 17 2 14 17 2 13 2 14 2 13 - 2 X2 x + X1 X2 x + x + X1 X2 x - X1 x + X1 X2 x 14 2 12 2 13 13 14 2 13 - 2 X1 X2 x + X1 X2 x - X1 x - 4 X1 X2 x + 2 X1 x - X2 x 14 2 12 12 13 13 14 11 + X2 x - X1 x - 2 X1 X2 x + 3 X1 x + 3 X2 x - x + X1 X2 x 12 12 13 10 11 11 12 10 + 2 X1 x + X2 x - 2 x + X1 X2 x - X1 x - X2 x - x - X1 x 10 11 8 10 7 8 8 6 - X2 x + x + X1 X2 x + x + X1 X2 x - X1 x - 2 X2 x - 2 X1 X2 x 7 8 5 6 6 5 6 4 - X1 x + 2 x + X1 X2 x + X1 x + 3 X2 x - 3 X2 x - 2 x + X2 x 5 4 3 2 / 2 7 6 + 2 x - x + x - 3 x + 3 x - 1) / ((-1 + x) (X1 X2 x + X1 X2 x / 7 7 5 6 7 5 4 5 4 - X1 x - X2 x - X1 X2 x - X1 x + x + 2 X2 x - X2 x - x + x - 2 x + 1)) and in Maple format -(X1^2*X2^2*x^17-2*X1^2*X2*x^17-2*X1*X2^2*x^17+X1^2*x^17+4*X1*X2*x^17+X2^2*x^17 -2*X1*x^17-2*X2*x^17+X1^2*X2*x^14+x^17+X1^2*X2*x^13-X1^2*x^14+X1*X2^2*x^13-2*X1 *X2*x^14+X1^2*X2*x^12-X1^2*x^13-4*X1*X2*x^13+2*X1*x^14-X2^2*x^13+X2*x^14-X1^2*x ^12-2*X1*X2*x^12+3*X1*x^13+3*X2*x^13-x^14+X1*X2*x^11+2*X1*x^12+X2*x^12-2*x^13+ X1*X2*x^10-X1*x^11-X2*x^11-x^12-X1*x^10-X2*x^10+x^11+X1*X2*x^8+x^10+X1*X2*x^7- X1*x^8-2*X2*x^8-2*X1*X2*x^6-X1*x^7+2*x^8+X1*X2*x^5+X1*x^6+3*X2*x^6-3*X2*x^5-2*x ^6+X2*x^4+2*x^5-x^4+x^3-3*x^2+3*x-1)/(-1+x)^2/(X1*X2*x^7+X1*X2*x^6-X1*x^7-X2*x^ 7-X1*X2*x^5-X1*x^6+x^7+2*X2*x^5-X2*x^4-x^5+x^4-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 1, 5], are n 139 23 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 3, 1], are n 135 65 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 12 23 65 -------------- 1495 and in floating point 0.3103563600 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 12 1495 ate normal pair with correlation, ---------- 1495 1/2 12 1495 1783 i.e. , [[----------, 0], [0, ----]] 1495 1495 ------------------------------------------------- Theorem Number, 16, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 1, 5], nor the composition, [5, 1, 2] Then infinity ----- \ n 3 15 14 13 12 10 9 ) a(n) x = - (x + x - 1) (x + 2 x + 2 x + x - x - 2 x / ----- n = 0 8 7 6 5 3 / 4 3 2 - 2 x - 2 x - 2 x - x - x + x - 1) / ((x + x + x + x - 1) / 2 (-1 + x) ) and in Maple format -(x^3+x-1)*(x^15+2*x^14+2*x^13+x^12-x^10-2*x^9-2*x^8-2*x^7-2*x^6-x^5-x^3+x-1)/( x^4+x^3+x^2+x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.61316322549247699946 1.9275619754829253043 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 1, 5] and d occurrences (as containment) of the composition, [5, 1, 2], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 19 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 2 18 2 19 2 19 2 2 17 2 18 + X1 X2 x - 2 X1 X2 x - 2 X1 X2 x + X1 X2 x - 2 X1 X2 x 2 19 2 18 19 2 19 2 17 2 18 + X1 x - 2 X1 X2 x + 4 X1 X2 x + X2 x - 2 X1 X2 x + X1 x 2 17 18 19 2 18 19 2 2 15 - 2 X1 X2 x + 4 X1 X2 x - 2 X1 x + X2 x - 2 X2 x - X1 X2 x 2 17 2 16 17 18 2 17 18 19 + X1 x + X1 X2 x + 4 X1 X2 x - 2 X1 x + X2 x - 2 X2 x + x 2 2 14 2 15 2 15 16 17 - X1 X2 x + 3 X1 X2 x + 2 X1 X2 x - 2 X1 X2 x - 2 X1 x 2 16 17 18 2 2 13 2 14 2 15 - X2 x - 2 X2 x + x - X1 X2 x + 3 X1 X2 x - 2 X1 x 2 14 15 16 2 15 16 17 + 2 X1 X2 x - 6 X1 X2 x + X1 x - X2 x + 2 X2 x + x 2 2 12 2 13 2 14 2 13 14 + X1 X2 x + 3 X1 X2 x - 2 X1 x + X1 X2 x - 6 X1 X2 x 15 2 14 15 16 2 2 11 2 12 + 4 X1 x - X2 x + 3 X2 x - x - X1 X2 x - X1 X2 x 2 13 13 14 14 15 2 11 - 2 X1 x - 4 X1 X2 x + 4 X1 x + 3 X2 x - 2 x + X1 X2 x 12 13 13 14 11 12 13 - X1 X2 x + 3 X1 x + X2 x - 2 x + X1 X2 x + X1 x - x 10 11 9 10 10 8 9 + X1 X2 x - X1 x + X1 X2 x - X1 x - X2 x + X1 X2 x - X1 x 9 10 8 8 9 6 8 5 6 5 - X2 x + x - X1 x - X2 x + x - X1 X2 x + x + X1 X2 x + x - x 3 2 / 6 6 + x - 3 x + 3 x - 1) / ((-1 + x) (X1 X2 x - X1 x + x - 1) / 6 5 6 5 (X1 X2 x - X1 X2 x - X1 x + x - 2 x + 1)) and in Maple format -(X1^2*X2^2*x^19+X1^2*X2^2*x^18-2*X1^2*X2*x^19-2*X1*X2^2*x^19+X1^2*X2^2*x^17-2* X1^2*X2*x^18+X1^2*x^19-2*X1*X2^2*x^18+4*X1*X2*x^19+X2^2*x^19-2*X1^2*X2*x^17+X1^ 2*x^18-2*X1*X2^2*x^17+4*X1*X2*x^18-2*X1*x^19+X2^2*x^18-2*X2*x^19-X1^2*X2^2*x^15 +X1^2*x^17+X1*X2^2*x^16+4*X1*X2*x^17-2*X1*x^18+X2^2*x^17-2*X2*x^18+x^19-X1^2*X2 ^2*x^14+3*X1^2*X2*x^15+2*X1*X2^2*x^15-2*X1*X2*x^16-2*X1*x^17-X2^2*x^16-2*X2*x^ 17+x^18-X1^2*X2^2*x^13+3*X1^2*X2*x^14-2*X1^2*x^15+2*X1*X2^2*x^14-6*X1*X2*x^15+ X1*x^16-X2^2*x^15+2*X2*x^16+x^17+X1^2*X2^2*x^12+3*X1^2*X2*x^13-2*X1^2*x^14+X1* X2^2*x^13-6*X1*X2*x^14+4*X1*x^15-X2^2*x^14+3*X2*x^15-x^16-X1^2*X2^2*x^11-X1^2* X2*x^12-2*X1^2*x^13-4*X1*X2*x^13+4*X1*x^14+3*X2*x^14-2*x^15+X1^2*X2*x^11-X1*X2* x^12+3*X1*x^13+X2*x^13-2*x^14+X1*X2*x^11+X1*x^12-x^13+X1*X2*x^10-X1*x^11+X1*X2* x^9-X1*x^10-X2*x^10+X1*X2*x^8-X1*x^9-X2*x^9+x^10-X1*x^8-X2*x^8+x^9-X1*X2*x^6+x^ 8+X1*X2*x^5+x^6-x^5+x^3-3*x^2+3*x-1)/(-1+x)/(X1*X2*x^6-X1*x^6+x-1)/(X1*X2*x^6- X1*X2*x^5-X1*x^6+x^5-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 1, 5], are n 139 23 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [5, 1, 2], are n 113 57 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 8 23 57 ------------- 437 and in floating point 0.6628418121 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 1/2 8 3 437 ate normal pair with correlation, ------------- 437 1/2 1/2 8 3 437 821 i.e. , [[-------------, 0], [0, ---]] 437 437 ------------------------------------------------- Theorem Number, 17, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 1, 5], nor the composition, [5, 2, 1] Then infinity ----- \ n 3 15 14 13 12 11 10 ) a(n) x = - (x + x - 1) (x + 2 x + 2 x + 2 x + x - x / ----- n = 0 9 8 7 6 5 3 / 4 3 2 - 2 x - 2 x - 2 x - 2 x - x - x + x - 1) / ((x + x + x + x - 1) / 2 (-1 + x) ) and in Maple format -(x^3+x-1)*(x^15+2*x^14+2*x^13+2*x^12+x^11-x^10-2*x^9-2*x^8-2*x^7-2*x^6-x^5-x^3 +x-1)/(x^4+x^3+x^2+x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.61223359561691903626 1.9275619754829253043 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 1, 5] and d occurrences (as containment) of the composition, [5, 2, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 19 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 2 18 2 19 2 19 2 2 17 2 18 + X1 X2 x - 2 X1 X2 x - 2 X1 X2 x + X1 X2 x - 2 X1 X2 x 2 19 2 18 19 2 19 2 17 2 18 + X1 x - 2 X1 X2 x + 4 X1 X2 x + X2 x - 2 X1 X2 x + X1 x 2 17 18 19 2 18 19 2 17 - 2 X1 X2 x + 4 X1 X2 x - 2 X1 x + X2 x - 2 X2 x + X1 x 17 18 2 17 18 19 2 15 + 4 X1 X2 x - 2 X1 x + X2 x - 2 X2 x + x + X1 X2 x 2 15 17 17 18 2 14 2 15 + X1 X2 x - 2 X1 x - 2 X2 x + x + X1 X2 x - X1 x 2 14 15 2 15 17 2 13 2 14 + X1 X2 x - 4 X1 X2 x - X2 x + x + X1 X2 x - X1 x 2 13 14 15 2 14 15 2 12 + X1 X2 x - 4 X1 X2 x + 3 X1 x - X2 x + 3 X2 x + X1 X2 x 2 13 13 14 2 13 14 15 2 12 - X1 x - 4 X1 X2 x + 3 X1 x - X2 x + 3 X2 x - 2 x - X1 x 12 13 13 14 11 12 - 2 X1 X2 x + 3 X1 x + 3 X2 x - 2 x + X1 X2 x + 2 X1 x 12 13 10 11 11 12 9 10 + X2 x - 2 x + X1 X2 x - X1 x - X2 x - x + X1 X2 x - X1 x 10 11 9 9 10 7 8 9 - X2 x + x - X1 x - X2 x + x + 2 X1 X2 x - X2 x + x 6 7 8 5 6 6 5 3 2 - 2 X1 X2 x - 2 X1 x + x + X1 X2 x + X1 x + x - x + x - 3 x + 3 x / 2 6 5 6 5 - 1) / ((-1 + x) (X1 X2 x - X1 X2 x - X1 x + x - 2 x + 1)) / and in Maple format -(X1^2*X2^2*x^19+X1^2*X2^2*x^18-2*X1^2*X2*x^19-2*X1*X2^2*x^19+X1^2*X2^2*x^17-2* X1^2*X2*x^18+X1^2*x^19-2*X1*X2^2*x^18+4*X1*X2*x^19+X2^2*x^19-2*X1^2*X2*x^17+X1^ 2*x^18-2*X1*X2^2*x^17+4*X1*X2*x^18-2*X1*x^19+X2^2*x^18-2*X2*x^19+X1^2*x^17+4*X1 *X2*x^17-2*X1*x^18+X2^2*x^17-2*X2*x^18+x^19+X1^2*X2*x^15+X1*X2^2*x^15-2*X1*x^17 -2*X2*x^17+x^18+X1^2*X2*x^14-X1^2*x^15+X1*X2^2*x^14-4*X1*X2*x^15-X2^2*x^15+x^17 +X1^2*X2*x^13-X1^2*x^14+X1*X2^2*x^13-4*X1*X2*x^14+3*X1*x^15-X2^2*x^14+3*X2*x^15 +X1^2*X2*x^12-X1^2*x^13-4*X1*X2*x^13+3*X1*x^14-X2^2*x^13+3*X2*x^14-2*x^15-X1^2* x^12-2*X1*X2*x^12+3*X1*x^13+3*X2*x^13-2*x^14+X1*X2*x^11+2*X1*x^12+X2*x^12-2*x^ 13+X1*X2*x^10-X1*x^11-X2*x^11-x^12+X1*X2*x^9-X1*x^10-X2*x^10+x^11-X1*x^9-X2*x^9 +x^10+2*X1*X2*x^7-X2*x^8+x^9-2*X1*X2*x^6-2*X1*x^7+x^8+X1*X2*x^5+X1*x^6+x^6-x^5+ x^3-3*x^2+3*x-1)/(-1+x)^2/(X1*X2*x^6-X1*X2*x^5-X1*x^6+x^5-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 1, 5], are n 139 23 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [5, 2, 1], are n 111 57 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 8 23 57 ------------- 437 and in floating point 0.6628418121 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 1/2 8 3 437 ate normal pair with correlation, ------------- 437 1/2 1/2 8 3 437 821 i.e. , [[-------------, 0], [0, ---]] 437 437 ------------------------------------------------- Theorem Number, 18, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 1, 5], nor the composition, [6, 1, 1] Then infinity ----- \ n 20 19 18 17 16 15 14 13 ) a(n) x = - (x + 2 x + 3 x + 4 x + 3 x + x - x - 3 x / ----- n = 0 12 11 10 9 8 5 2 / - 4 x - 4 x - 3 x - 2 x - x + x + x - 2 x + 1) / ( / 4 3 2 2 (x + x + x + x - 1) (-1 + x) ) and in Maple format -(x^20+2*x^19+3*x^18+4*x^17+3*x^16+x^15-x^14-3*x^13-4*x^12-4*x^11-3*x^10-2*x^9- x^8+x^5+x^2-2*x+1)/(x^4+x^3+x^2+x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.60970919018057873430 1.9275619754829253043 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 1, 5] and d occurrences (as containment) of the composition, [6, 1, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 21 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 2 20 2 21 2 21 2 2 19 2 20 + X1 X2 x - 2 X1 X2 x - 2 X1 X2 x + X1 X2 x - 2 X1 X2 x 2 21 2 20 21 2 21 2 2 18 + X1 x - 2 X1 X2 x + 4 X1 X2 x + X2 x + X1 X2 x 2 19 2 20 2 19 20 21 2 20 - 2 X1 X2 x + X1 x - 2 X1 X2 x + 4 X1 X2 x - 2 X1 x + X2 x 21 2 18 2 19 2 18 19 - 2 X2 x - 2 X1 X2 x + X1 x - 2 X1 X2 x + 4 X1 X2 x 20 2 19 20 21 2 18 2 17 18 - 2 X1 x + X2 x - 2 X2 x + x + X1 x + X1 X2 x + 4 X1 X2 x 19 2 18 19 20 2 16 2 16 - 2 X1 x + X2 x - 2 X2 x + x + X1 X2 x + X1 X2 x 17 18 2 17 18 19 2 15 2 16 - 2 X1 X2 x - 2 X1 x - X2 x - 2 X2 x + x + X1 X2 x - X1 x 2 15 16 17 2 16 17 18 + X1 X2 x - 4 X1 X2 x + X1 x - X2 x + 2 X2 x + x 2 14 2 15 2 14 15 16 2 15 + X1 X2 x - X1 x + X1 X2 x - 4 X1 X2 x + 3 X1 x - X2 x 16 17 2 13 2 14 14 15 2 14 + 3 X2 x - x + X1 X2 x - X1 x - 4 X1 X2 x + 3 X1 x - X2 x 15 16 2 13 13 14 14 15 + 3 X2 x - 2 x - X1 x - 2 X1 X2 x + 3 X1 x + 3 X2 x - 2 x 13 13 14 11 13 10 11 + 2 X1 x + X2 x - 2 x + X1 X2 x - x + X1 X2 x - X1 x 11 9 10 10 11 8 9 9 - X2 x + X1 X2 x - X1 x - X2 x + x + X1 X2 x - X1 x - X2 x 10 7 8 8 9 6 7 8 6 + x - X1 X2 x - X1 x - X2 x + x + X1 X2 x + X1 x + x - 2 X1 x 5 6 5 3 2 / 2 + X1 x + x - x + x - 3 x + 3 x - 1) / ((-1 + x) / 6 6 5 5 (X1 X2 x - X1 x + X1 x - x + 2 x - 1)) and in Maple format (X1^2*X2^2*x^21+X1^2*X2^2*x^20-2*X1^2*X2*x^21-2*X1*X2^2*x^21+X1^2*X2^2*x^19-2* X1^2*X2*x^20+X1^2*x^21-2*X1*X2^2*x^20+4*X1*X2*x^21+X2^2*x^21+X1^2*X2^2*x^18-2* X1^2*X2*x^19+X1^2*x^20-2*X1*X2^2*x^19+4*X1*X2*x^20-2*X1*x^21+X2^2*x^20-2*X2*x^ 21-2*X1^2*X2*x^18+X1^2*x^19-2*X1*X2^2*x^18+4*X1*X2*x^19-2*X1*x^20+X2^2*x^19-2* X2*x^20+x^21+X1^2*x^18+X1*X2^2*x^17+4*X1*X2*x^18-2*X1*x^19+X2^2*x^18-2*X2*x^19+ x^20+X1^2*X2*x^16+X1*X2^2*x^16-2*X1*X2*x^17-2*X1*x^18-X2^2*x^17-2*X2*x^18+x^19+ X1^2*X2*x^15-X1^2*x^16+X1*X2^2*x^15-4*X1*X2*x^16+X1*x^17-X2^2*x^16+2*X2*x^17+x^ 18+X1^2*X2*x^14-X1^2*x^15+X1*X2^2*x^14-4*X1*X2*x^15+3*X1*x^16-X2^2*x^15+3*X2*x^ 16-x^17+X1^2*X2*x^13-X1^2*x^14-4*X1*X2*x^14+3*X1*x^15-X2^2*x^14+3*X2*x^15-2*x^ 16-X1^2*x^13-2*X1*X2*x^13+3*X1*x^14+3*X2*x^14-2*x^15+2*X1*x^13+X2*x^13-2*x^14+ X1*X2*x^11-x^13+X1*X2*x^10-X1*x^11-X2*x^11+X1*X2*x^9-X1*x^10-X2*x^10+x^11+X1*X2 *x^8-X1*x^9-X2*x^9+x^10-X1*X2*x^7-X1*x^8-X2*x^8+x^9+X1*X2*x^6+X1*x^7+x^8-2*X1*x ^6+X1*x^5+x^6-x^5+x^3-3*x^2+3*x-1)/(-1+x)^2/(X1*X2*x^6-X1*x^6+X1*x^5-x^5+2*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 1, 5], are n 139 23 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [6, 1, 1], are n 49 53 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 22 23 53 -------------- 1219 and in floating point 0.6301164650 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 22 1219 ate normal pair with correlation, ---------- 1219 1/2 22 1219 2187 i.e. , [[----------, 0], [0, ----]] 1219 1219 ------------------------------------------------- Theorem Number, 19, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 2, 4], nor the composition, [1, 1, 6] Then infinity ----- 8 7 6 5 4 2 \ n x - x - x + x - x - x + 2 x - 1 ) a(n) x = ------------------------------------- / 6 5 3 2 2 ----- (x + x + x + x + x - 1) (-1 + x) n = 0 and in Maple format (x^8-x^7-x^6+x^5-x^4-x^2+2*x-1)/(x^6+x^5+x^3+x^2+x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.69790391550484586467 1.9111834366854762350 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 2, 4] and d occurrences (as containment) of the composition, [1, 1, 6], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 8 7 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x - X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 8 8 6 7 7 8 7 5 6 4 - X1 x - X2 x + X1 X2 x + X1 x + X2 x + x - x - X1 x - x + X1 x 5 4 2 / 7 6 7 + x - x - x + 2 x - 1) / ((-1 + x) (X1 X2 x - X1 X2 x - X1 x / 7 6 7 5 4 5 4 - X2 x + X1 x + x + X1 x - X1 x - x + x - 2 x + 1)) and in Maple format (X1*X2*x^8-X1*X2*x^7-X1*x^8-X2*x^8+X1*X2*x^6+X1*x^7+X2*x^7+x^8-x^7-X1*x^5-x^6+ X1*x^4+x^5-x^4-x^2+2*x-1)/(-1+x)/(X1*X2*x^7-X1*X2*x^6-X1*x^7-X2*x^7+X1*x^6+x^7+ X1*x^5-X1*x^4-x^5+x^4-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 2, 4], are n 183 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 1, 6], are n 49 53 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 8 3 53 ------------ 477 and in floating point 0.2114804230 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 8 159 ate normal pair with correlation, -------- 477 1/2 8 159 1559 i.e. , [[--------, 0], [0, ----]] 477 1431 ------------------------------------------------- Theorem Number, 20, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 2, 4], nor the composition, [1, 4, 3] Then infinity ----- 8 7 6 4 \ n x + x + x + x - x + 1 ) a(n) x = ----------------------------------------- / 7 6 5 3 2 ----- (-1 + x) (x + x + x + x + x + x - 1) n = 0 and in Maple format (x^8+x^7+x^6+x^4-x+1)/(-1+x)/(x^7+x^6+x^5+x^3+x^2+x-1) The asymptotic expression for a(n) is, n 0.65748289688935283811 1.9221280043552247737 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 2, 4] and d occurrences (as containment) of the composition, [1, 4, 3], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 9 9 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x - X1 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 9 7 9 7 5 6 4 6 5 - X2 x + X1 X2 x + x - X1 x - X1 X2 x + X1 x + X1 X2 x - x + x 4 2 / 8 8 8 6 - x - x + 2 x - 1) / ((-1 + x) (X1 X2 x - X1 x - X2 x + X1 X2 x / 8 5 6 4 5 4 + x + X1 X2 x - X1 x - X1 X2 x - x + x - 2 x + 1)) and in Maple format (X1*X2*x^9-X1*x^9-X2*x^9+X1*X2*x^7+x^9-X1*x^7-X1*X2*x^5+X1*x^6+X1*X2*x^4-x^6+x^ 5-x^4-x^2+2*x-1)/(-1+x)/(X1*X2*x^8-X1*x^8-X2*x^8+X1*X2*x^6+x^8+X1*X2*x^5-X1*x^6 -X1*X2*x^4-x^5+x^4-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 2, 4], are n 183 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 4, 3], are n 135 65 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 22 3 65 ------------- 585 and in floating point 0.5251509079 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 22 195 ate normal pair with correlation, --------- 585 1/2 22 195 2723 i.e. , [[---------, 0], [0, ----]] 585 1755 ------------------------------------------------- Theorem Number, 21, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 2, 4], nor the composition, [1, 5, 2] Then infinity ----- 10 9 6 5 4 2 \ n x + x - x + x - x - x + 2 x - 1 ) a(n) x = -------------------------------------- / 9 8 5 4 ----- (-1 + x) (x + x - x + x - 2 x + 1) n = 0 and in Maple format (x^10+x^9-x^6+x^5-x^4-x^2+2*x-1)/(-1+x)/(x^9+x^8-x^5+x^4-2*x+1) The asymptotic expression for a(n) is, n 0.67229840142428698322 1.9159714162121856417 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 2, 4] and d occurrences (as containment) of the composition, [1, 5, 2], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 10 9 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x + X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 10 10 9 9 10 7 9 6 7 - X1 x - X2 x - X1 x - X2 x + x + X1 X2 x + x - X1 X2 x - X1 x 5 6 5 6 4 5 4 2 / + X1 X2 x + 2 X1 x - 2 X1 x - x + X1 x + x - x - x + 2 x - 1) / / 9 8 9 9 8 8 9 ((-1 + x) (X1 X2 x + X1 X2 x - X1 x - X2 x - X1 x - X2 x + x 6 8 5 6 5 4 5 4 + X1 X2 x + x - X1 X2 x - X1 x + 2 X1 x - X1 x - x + x - 2 x + 1)) and in Maple format (X1*X2*x^10+X1*X2*x^9-X1*x^10-X2*x^10-X1*x^9-X2*x^9+x^10+X1*X2*x^7+x^9-X1*X2*x^ 6-X1*x^7+X1*X2*x^5+2*X1*x^6-2*X1*x^5-x^6+X1*x^4+x^5-x^4-x^2+2*x-1)/(-1+x)/(X1* X2*x^9+X1*X2*x^8-X1*x^9-X2*x^9-X1*x^8-X2*x^8+x^9+X1*X2*x^6+x^8-X1*X2*x^5-X1*x^6 +2*X1*x^5-X1*x^4-x^5+x^4-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 2, 4], are n 183 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 5, 2], are n 111 57 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 14 19 -------- 171 and in floating point 0.3568689194 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 14 19 ate normal pair with correlation, -------- 171 1/2 14 19 1931 i.e. , [[--------, 0], [0, ----]] 171 1539 ------------------------------------------------- Theorem Number, 22, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 2, 4], nor the composition, [1, 6, 1] Then infinity ----- 4 2 6 5 \ n (x + x - 2 x + 1) (x + x - 1) ) a(n) x = ------------------------------------- / 6 5 3 2 2 ----- (x + x + x + x + x - 1) (-1 + x) n = 0 and in Maple format (x^4+x^2-2*x+1)*(x^6+x^5-1)/(x^6+x^5+x^3+x^2+x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.68707893079649356212 1.9111834366854762350 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 2, 4] and d occurrences (as containment) of the composition, [1, 6, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 10 9 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x + X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 10 10 8 9 9 10 7 8 - X1 x - X2 x + X1 X2 x - X1 x - X2 x + x - X1 X2 x - X1 x 8 9 6 7 7 8 7 5 6 4 - X2 x + x + X1 X2 x + X1 x + X2 x + x - x - X1 x - x + X1 x 5 4 2 / 7 6 7 + x - x - x + 2 x - 1) / ((-1 + x) (X1 X2 x - X1 X2 x - X1 x / 7 6 7 5 4 5 4 - X2 x + X1 x + x + X1 x - X1 x - x + x - 2 x + 1)) and in Maple format (X1*X2*x^10+X1*X2*x^9-X1*x^10-X2*x^10+X1*X2*x^8-X1*x^9-X2*x^9+x^10-X1*X2*x^7-X1 *x^8-X2*x^8+x^9+X1*X2*x^6+X1*x^7+X2*x^7+x^8-x^7-X1*x^5-x^6+X1*x^4+x^5-x^4-x^2+2 *x-1)/(-1+x)/(X1*X2*x^7-X1*X2*x^6-X1*x^7-X2*x^7+X1*x^6+x^7+X1*x^5-X1*x^4-x^5+x^ 4-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 2, 4], are n 183 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 6, 1], are n 49 53 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 8 3 53 ------------ 477 and in floating point 0.2114804230 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 8 159 ate normal pair with correlation, -------- 477 1/2 8 159 1559 i.e. , [[--------, 0], [0, ----]] 477 1431 ------------------------------------------------- Theorem Number, 23, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 2, 4], nor the composition, [2, 1, 5] Then infinity ----- 7 6 5 4 2 \ n x - 2 x + x - x - x + 2 x - 1 ) a(n) x = - -------------------------------------- / 7 6 5 4 ----- (-1 + x) (x - x + x - x + 2 x - 1) n = 0 and in Maple format -(x^7-2*x^6+x^5-x^4-x^2+2*x-1)/(-1+x)/(x^7-x^6+x^5-x^4+2*x-1) The asymptotic expression for a(n) is, n 0.68788768175297946158 1.9132221246804735080 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 2, 4] and d occurrences (as containment) of the composition, [2, 1, 5], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 11 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 11 11 7 6 7 7 5 - 2 X1 X2 x + X1 x - X1 X2 x + X1 X2 x + X1 x + X2 x - X1 X2 x 6 6 7 5 6 4 5 4 2 - 2 X1 x - X2 x - x + 2 X1 x + 2 x - X1 x - x + x + x - 2 x + 1) / 2 12 2 11 12 11 12 11 / (X1 X2 x - X1 X2 x - 2 X1 X2 x + 2 X1 X2 x + X1 x - X1 x / 8 7 8 8 6 7 7 8 - X1 X2 x + X1 X2 x + X1 x + X2 x - 2 X1 X2 x - X1 x - 2 X2 x - x 5 6 6 7 5 6 4 5 4 + X1 X2 x + 3 X1 x + X2 x + 2 x - 3 X1 x - 2 x + X1 x + 2 x - x 2 - 2 x + 3 x - 1) and in Maple format -(X1*X2^2*x^11-2*X1*X2*x^11+X1*x^11-X1*X2*x^7+X1*X2*x^6+X1*x^7+X2*x^7-X1*X2*x^5 -2*X1*x^6-X2*x^6-x^7+2*X1*x^5+2*x^6-X1*x^4-x^5+x^4+x^2-2*x+1)/(X1*X2^2*x^12-X1* X2^2*x^11-2*X1*X2*x^12+2*X1*X2*x^11+X1*x^12-X1*x^11-X1*X2*x^8+X1*X2*x^7+X1*x^8+ X2*x^8-2*X1*X2*x^6-X1*x^7-2*X2*x^7-x^8+X1*X2*x^5+3*X1*x^6+X2*x^6+2*x^7-3*X1*x^5 -2*x^6+X1*x^4+2*x^5-x^4-2*x^2+3*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 2, 4], are n 183 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 1, 5], are n 113 57 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 4 19 ------- 57 and in floating point 0.3058876452 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 4 19 ate normal pair with correlation, ------- 57 1/2 4 19 203 i.e. , [[-------, 0], [0, ---]] 57 171 ------------------------------------------------- Theorem Number, 24, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 2, 4], nor the composition, [2, 3, 3] Then infinity ----- 7 6 4 3 2 \ n x - 2 x + x - x - x + 2 x - 1 ) a(n) x = - ------------------------------------------- / 7 6 5 4 3 ----- (-1 + x) (x - x + x + x - x + 2 x - 1) n = 0 and in Maple format -(x^7-2*x^6+x^4-x^3-x^2+2*x-1)/(-1+x)/(x^7-x^6+x^5+x^4-x^3+2*x-1) The asymptotic expression for a(n) is, n 0.64732625001158940098 1.9249743097683141877 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 2, 4] and d occurrences (as containment) of the composition, [2, 3, 3], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 7 7 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x - X1 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 7 5 6 6 7 4 5 6 - X2 x - X1 X2 x + X1 x + X2 x + x + X1 X2 x + X2 x - 2 x 4 3 4 3 2 / 8 7 - 2 X2 x + X2 x + x - x - x + 2 x - 1) / (X1 X2 x - X1 X2 x / 8 8 7 7 8 5 6 6 7 - X1 x - X2 x + X1 x + 2 X2 x + x + 2 X1 X2 x - X1 x - X2 x - 2 x 4 5 6 4 3 4 3 2 - X1 X2 x - 2 X2 x + 2 x + 3 X2 x - X2 x - 2 x + x + 2 x - 3 x + 1 ) and in Maple format -(X1*X2*x^7-X1*x^7-X2*x^7-X1*X2*x^5+X1*x^6+X2*x^6+x^7+X1*X2*x^4+X2*x^5-2*x^6-2* X2*x^4+X2*x^3+x^4-x^3-x^2+2*x-1)/(X1*X2*x^8-X1*X2*x^7-X1*x^8-X2*x^8+X1*x^7+2*X2 *x^7+x^8+2*X1*X2*x^5-X1*x^6-X2*x^6-2*x^7-X1*X2*x^4-2*X2*x^5+2*x^6+3*X2*x^4-X2*x ^3-2*x^4+x^3+2*x^2-3*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 2, 4], are n 183 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 3, 3], are n 97 85 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 8 3 85 ------------ 255 and in floating point 0.5009794331 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 8 255 ate normal pair with correlation, -------- 255 1/2 8 255 383 i.e. , [[--------, 0], [0, ---]] 255 255 ------------------------------------------------- Theorem Number, 25, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 2, 4], nor the composition, [2, 4, 2] Then infinity ----- 9 8 6 5 4 2 \ n x + x - x + x - x - x + 2 x - 1 ) a(n) x = - ------------------------------------- / 5 4 ----- (-1 + x) (x - x + 2 x - 1) n = 0 and in Maple format -(x^9+x^8-x^6+x^5-x^4-x^2+2*x-1)/(-1+x)/(x^5-x^4+2*x-1) The asymptotic expression for a(n) is, n 0.61915988369516142914 1.9331849818995204468 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 2, 4] and d occurrences (as containment) of the composition, [2, 4, 2], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 9 8 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x + X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 9 9 8 8 9 6 8 5 4 - X1 x - X2 x - X1 x - X2 x + x + X1 X2 x + x - X1 X2 x + X1 X2 x 6 5 4 2 / 7 6 7 - x + x - x - x + 2 x - 1) / (X1 X2 x - X1 X2 x - X1 x / 5 4 6 5 4 2 + 2 X1 X2 x - X1 X2 x + x - 2 x + x + 2 x - 3 x + 1) and in Maple format -(X1*X2*x^9+X1*X2*x^8-X1*x^9-X2*x^9-X1*x^8-X2*x^8+x^9+X1*X2*x^6+x^8-X1*X2*x^5+ X1*X2*x^4-x^6+x^5-x^4-x^2+2*x-1)/(X1*X2*x^7-X1*X2*x^6-X1*x^7+2*X1*X2*x^5-X1*X2* x^4+x^6-2*x^5+x^4+2*x^2-3*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 2, 4], are n 183 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 4, 2], are n 75 69 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 32 23 -------- 207 and in floating point 0.7413845834 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 32 23 ate normal pair with correlation, -------- 207 1/2 32 23 3911 i.e. , [[--------, 0], [0, ----]] 207 1863 ------------------------------------------------- Theorem Number, 26, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 2, 4], nor the composition, [2, 5, 1] Then infinity ----- 12 11 8 7 6 5 4 3 2 \ n x - x + x + x - 2 x + 2 x - x + x - 3 x + 3 x - 1 ) a(n) x = ------------------------------------------------------------ / 5 4 2 ----- (x - x + 2 x - 1) (-1 + x) n = 0 and in Maple format (x^12-x^11+x^8+x^7-2*x^6+2*x^5-x^4+x^3-3*x^2+3*x-1)/(x^5-x^4+2*x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.61442764446410597890 1.9331849818995204468 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 2, 4] and d occurrences (as containment) of the composition, [2, 5, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 12 11 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x - X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 12 12 11 11 12 11 7 8 - X1 x - X2 x + X1 x + X2 x + x - x + 2 X1 X2 x - X2 x 6 7 8 5 6 7 5 6 - 2 X1 X2 x - 3 X1 x + x + X1 X2 x + 4 X1 x + x - 3 X1 x - 2 x 4 5 4 3 2 / 2 + X1 x + 2 x - x + x - 3 x + 3 x - 1) / ((-1 + x) / 6 5 6 5 4 5 4 (X1 X2 x - X1 X2 x - X1 x + 2 X1 x - X1 x - x + x - 2 x + 1)) and in Maple format -(X1*X2*x^12-X1*X2*x^11-X1*x^12-X2*x^12+X1*x^11+X2*x^11+x^12-x^11+2*X1*X2*x^7- X2*x^8-2*X1*X2*x^6-3*X1*x^7+x^8+X1*X2*x^5+4*X1*x^6+x^7-3*X1*x^5-2*x^6+X1*x^4+2* x^5-x^4+x^3-3*x^2+3*x-1)/(-1+x)^2/(X1*X2*x^6-X1*X2*x^5-X1*x^6+2*X1*x^5-X1*x^4-x ^5+x^4-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 2, 4], are n 183 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 5, 1], are n 111 57 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 26 19 -------- 171 and in floating point 0.6627565645 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 26 19 ate normal pair with correlation, -------- 171 1/2 26 19 2891 i.e. , [[--------, 0], [0, ----]] 171 1539 ------------------------------------------------- Theorem Number, 27, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 2, 4], nor the composition, [3, 1, 4] Then infinity ----- 4 2 \ n x + x - 2 x + 1 ) a(n) x = --------------------------------- / 7 6 4 ----- (-1 + x) (x + x - x + 2 x - 1) n = 0 and in Maple format (x^4+x^2-2*x+1)/(-1+x)/(x^7+x^6-x^4+2*x-1) The asymptotic expression for a(n) is, n 0.67238389941451345857 1.9168163613635124211 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 2, 4] and d occurrences (as containment) of the composition, [3, 1, 4], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 10 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 10 2 9 10 9 10 9 - X1 X2 x + X1 X2 x - X1 X2 x - 2 X1 X2 x + X1 x + X1 x 6 5 6 4 5 4 2 / - X1 X2 x + X1 X2 x + X2 x - X1 X2 x - X2 x + x + x - 2 x + 1) / / 2 2 12 2 2 11 2 12 2 2 10 2 11 (X1 X2 x + X1 X2 x - 2 X1 X2 x - X1 X2 x - 2 X1 X2 x 2 12 2 11 2 10 2 11 2 10 11 + X1 x + X1 X2 x + X1 X2 x + X1 x + X1 X2 x - 2 X1 X2 x 2 9 10 11 9 8 9 - X1 X2 x - X1 X2 x + X1 x + 2 X1 X2 x - X1 X2 x - X1 x 7 8 8 6 7 8 5 6 - X1 X2 x + X1 x + X2 x + X1 X2 x + X1 x - x - 2 X1 X2 x - 2 X2 x 4 5 6 5 4 2 + X1 X2 x + X2 x + x + x - x - 2 x + 3 x - 1) and in Maple format -(X1^2*X2^2*x^10-X1^2*X2*x^10+X1*X2^2*x^9-X1*X2*x^10-2*X1*X2*x^9+X1*x^10+X1*x^9 -X1*X2*x^6+X1*X2*x^5+X2*x^6-X1*X2*x^4-X2*x^5+x^4+x^2-2*x+1)/(X1^2*X2^2*x^12+X1^ 2*X2^2*x^11-2*X1^2*X2*x^12-X1^2*X2^2*x^10-2*X1^2*X2*x^11+X1^2*x^12+X1*X2^2*x^11 +X1^2*X2*x^10+X1^2*x^11+X1*X2^2*x^10-2*X1*X2*x^11-X1*X2^2*x^9-X1*X2*x^10+X1*x^ 11+2*X1*X2*x^9-X1*X2*x^8-X1*x^9-X1*X2*x^7+X1*x^8+X2*x^8+X1*X2*x^6+X1*x^7-x^8-2* X1*X2*x^5-2*X2*x^6+X1*X2*x^4+X2*x^5+x^6+x^5-x^4-2*x^2+3*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 2, 4], are n 183 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 1, 4], are n 141 65 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 16 3 65 ------------- 585 and in floating point 0.3819279330 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 16 195 ate normal pair with correlation, --------- 585 1/2 16 195 2267 i.e. , [[---------, 0], [0, ----]] 585 1755 ------------------------------------------------- Theorem Number, 28, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 2, 4], nor the composition, [3, 2, 3] Then infinity ----- 2 7 5 4 3 2 \ n (x - x + 1) (x - x - x - x + x + x - 1) ) a(n) x = ------------------------------------------------- / 8 4 3 2 3 ----- (-1 + x) (x - x - x - x - x + 1) (x - x + 1) n = 0 and in Maple format (x^2-x+1)*(x^7-x^5-x^4-x^3+x^2+x-1)/(-1+x)/(x^8-x^4-x^3-x^2-x+1)/(x^3-x+1) The asymptotic expression for a(n) is, n 0.65527718001580393906 1.9217220658969757404 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 2, 4] and d occurrences (as containment) of the composition, [3, 2, 3], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 10 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 10 2 10 2 9 10 2 10 - X1 X2 x - 2 X1 X2 x + 2 X1 X2 x + 2 X1 X2 x + X2 x 9 2 9 10 9 2 8 9 8 9 - 3 X1 X2 x - 2 X2 x - X2 x + X1 x + X2 x + 3 X2 x - 2 X2 x - x 6 8 5 6 4 5 4 5 - X1 X2 x + x + X1 X2 x + X2 x - X1 X2 x - 2 X2 x + 2 X2 x + x 3 4 3 2 / 2 2 12 2 2 11 - X2 x - x + x + x - 2 x + 1) / (X1 X2 x + X1 X2 x / 2 12 2 12 2 2 10 2 11 2 12 - 2 X1 X2 x - 2 X1 X2 x - X1 X2 x - 2 X1 X2 x + X1 x 12 2 12 2 10 2 11 2 10 12 + 4 X1 X2 x + X2 x + X1 X2 x + X1 x + 4 X1 X2 x - 2 X1 x 2 11 12 2 9 10 2 10 11 - X2 x - 2 X2 x - 2 X1 X2 x - 6 X1 X2 x - 2 X2 x + 2 X2 x 12 9 10 2 9 10 11 8 + x + 3 X1 X2 x + 2 X1 x + 3 X2 x + 3 X2 x - x - X1 X2 x 9 2 8 9 10 7 8 8 9 - X1 x - X2 x - 5 X2 x - x - X1 X2 x + X1 x + 3 X2 x + 2 x 6 7 8 5 6 4 5 6 + X1 X2 x + X1 x - 2 x - 2 X1 X2 x - 2 X2 x + X1 X2 x + 3 X2 x + x 4 5 3 4 3 2 - 3 X2 x - x + X2 x + 2 x - x - 2 x + 3 x - 1) and in Maple format -(X1^2*X2^2*x^10-X1^2*X2*x^10-2*X1*X2^2*x^10+2*X1*X2^2*x^9+2*X1*X2*x^10+X2^2*x^ 10-3*X1*X2*x^9-2*X2^2*x^9-X2*x^10+X1*x^9+X2^2*x^8+3*X2*x^9-2*X2*x^8-x^9-X1*X2*x ^6+x^8+X1*X2*x^5+X2*x^6-X1*X2*x^4-2*X2*x^5+2*X2*x^4+x^5-X2*x^3-x^4+x^3+x^2-2*x+ 1)/(X1^2*X2^2*x^12+X1^2*X2^2*x^11-2*X1^2*X2*x^12-2*X1*X2^2*x^12-X1^2*X2^2*x^10-\ 2*X1^2*X2*x^11+X1^2*x^12+4*X1*X2*x^12+X2^2*x^12+X1^2*X2*x^10+X1^2*x^11+4*X1*X2^ 2*x^10-2*X1*x^12-X2^2*x^11-2*X2*x^12-2*X1*X2^2*x^9-6*X1*X2*x^10-2*X2^2*x^10+2* X2*x^11+x^12+3*X1*X2*x^9+2*X1*x^10+3*X2^2*x^9+3*X2*x^10-x^11-X1*X2*x^8-X1*x^9- X2^2*x^8-5*X2*x^9-x^10-X1*X2*x^7+X1*x^8+3*X2*x^8+2*x^9+X1*X2*x^6+X1*x^7-2*x^8-2 *X1*X2*x^5-2*X2*x^6+X1*X2*x^4+3*X2*x^5+x^6-3*X2*x^4-x^5+X2*x^3+2*x^4-x^3-2*x^2+ 3*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 2, 4], are n 183 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 2, 3], are n 89 77 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 20 3 77 ------------- 693 and in floating point 0.4386344635 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 20 231 ate normal pair with correlation, --------- 693 1/2 20 231 2879 i.e. , [[---------, 0], [0, ----]] 693 2079 ------------------------------------------------- Theorem Number, 29, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 2, 4], nor the composition, [3, 3, 2] Then infinity ----- 11 9 7 6 4 3 2 \ n x + x + x - 2 x + x - x - x + 2 x - 1 ) a(n) x = ------------------------------------------------- / 10 7 6 5 4 3 ----- (-1 + x) (x - x + x - x - x + x - 2 x + 1) n = 0 and in Maple format (x^11+x^9+x^7-2*x^6+x^4-x^3-x^2+2*x-1)/(-1+x)/(x^10-x^7+x^6-x^5-x^4+x^3-2*x+1) The asymptotic expression for a(n) is, n 0.64822251981642343554 1.9223246520768555496 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 2, 4] and d occurrences (as containment) of the composition, [3, 3, 2], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 12 2 12 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x - X1 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 12 11 12 12 10 11 - 2 X1 X2 x + X1 X2 x + 2 X1 x + X2 x - X1 X2 x - X1 x 11 12 9 10 10 11 8 9 - X2 x - x + X1 X2 x + X1 x + X2 x + x - X1 X2 x - X1 x 9 10 7 8 8 9 6 7 - X2 x - x + X1 X2 x + X1 x + X2 x + x + X1 X2 x - 2 X1 x 7 8 5 6 7 4 5 6 - 2 X2 x - x - 2 X1 X2 x + X1 x + 3 x + X1 X2 x + 3 X2 x - 2 x 4 5 3 4 2 / 2 11 - 3 X2 x - x + X2 x + 2 x - 3 x + 3 x - 1) / ((-1 + x) (X1 X2 x / 2 11 11 10 11 11 10 10 - X1 x - 2 X1 X2 x + X1 X2 x + 2 X1 x + X2 x - X1 x - X2 x 11 8 10 7 8 8 7 7 8 - x + X1 X2 x + x - X1 X2 x - X1 x - X2 x + X1 x + 2 X2 x + x 5 6 6 7 4 5 6 4 + 2 X1 X2 x - X1 x - X2 x - 2 x - X1 X2 x - 2 X2 x + 2 x + 3 X2 x 3 4 3 2 - X2 x - 2 x + x + 2 x - 3 x + 1)) and in Maple format (X1^2*X2*x^12-X1^2*x^12-2*X1*X2*x^12+X1*X2*x^11+2*X1*x^12+X2*x^12-X1*X2*x^10-X1 *x^11-X2*x^11-x^12+X1*X2*x^9+X1*x^10+X2*x^10+x^11-X1*X2*x^8-X1*x^9-X2*x^9-x^10+ X1*X2*x^7+X1*x^8+X2*x^8+x^9+X1*X2*x^6-2*X1*x^7-2*X2*x^7-x^8-2*X1*X2*x^5+X1*x^6+ 3*x^7+X1*X2*x^4+3*X2*x^5-2*x^6-3*X2*x^4-x^5+X2*x^3+2*x^4-3*x^2+3*x-1)/(-1+x)/( X1^2*X2*x^11-X1^2*x^11-2*X1*X2*x^11+X1*X2*x^10+2*X1*x^11+X2*x^11-X1*x^10-X2*x^ 10-x^11+X1*X2*x^8+x^10-X1*X2*x^7-X1*x^8-X2*x^8+X1*x^7+2*X2*x^7+x^8+2*X1*X2*x^5- X1*x^6-X2*x^6-2*x^7-X1*X2*x^4-2*X2*x^5+2*x^6+3*X2*x^4-X2*x^3-2*x^4+x^3+2*x^2-3* x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 2, 4], are n 183 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 3, 2], are n 97 85 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 4 3 85 ------------ 153 and in floating point 0.4174828609 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 1/2 4 5 51 ate normal pair with correlation, ------------ 153 1/2 1/2 4 5 51 619 i.e. , [[------------, 0], [0, ---]] 153 459 ------------------------------------------------- Theorem Number, 30, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 2, 4], nor the composition, [3, 4, 1] Then infinity ----- 12 11 8 7 6 5 4 3 2 \ n x - x + x + x - 2 x + 2 x - x + x - 3 x + 3 x - 1 ) a(n) x = ------------------------------------------------------------ / 5 4 2 ----- (x - x + 2 x - 1) (-1 + x) n = 0 and in Maple format (x^12-x^11+x^8+x^7-2*x^6+2*x^5-x^4+x^3-3*x^2+3*x-1)/(x^5-x^4+2*x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.61442764446410597890 1.9331849818995204468 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 2, 4] and d occurrences (as containment) of the composition, [3, 4, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 11 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 12 2 11 11 12 12 11 11 + X1 X2 x - X1 x - 2 X1 X2 x - X1 x - X2 x + 2 X1 x + X2 x 12 11 7 8 6 7 8 5 + x - x + X1 X2 x - X2 x + X1 X2 x - 2 X1 x + x - 2 X1 X2 x 6 7 4 6 5 4 3 2 / + X1 x + x + X1 X2 x - 2 x + 2 x - x + x - 3 x + 3 x - 1) / ( / 2 6 5 6 4 5 4 (-1 + x) (X1 X2 x + X1 X2 x - X1 x - X1 X2 x - x + x - 2 x + 1)) and in Maple format -(X1^2*X2*x^11+X1*X2*x^12-X1^2*x^11-2*X1*X2*x^11-X1*x^12-X2*x^12+2*X1*x^11+X2*x ^11+x^12-x^11+X1*X2*x^7-X2*x^8+X1*X2*x^6-2*X1*x^7+x^8-2*X1*X2*x^5+X1*x^6+x^7+X1 *X2*x^4-2*x^6+2*x^5-x^4+x^3-3*x^2+3*x-1)/(-1+x)^2/(X1*X2*x^6+X1*X2*x^5-X1*x^6- X1*X2*x^4-x^5+x^4-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 2, 4], are n 183 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 4, 1], are n 135 65 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 2 3 65 ------------ 39 and in floating point 0.7161148743 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 1/2 2 5 39 ate normal pair with correlation, ------------ 39 1/2 1/2 2 5 39 79 i.e. , [[------------, 0], [0, --]] 39 39 ------------------------------------------------- Theorem Number, 31, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 2, 4], nor the composition, [4, 1, 3] Then infinity ----- \ n ) a(n) x = / ----- n = 0 18 17 16 14 12 11 10 8 4 2 x + x + x - x + x + x + x + x - x - x + 2 x - 1 ---------------------------------------------------------------- 10 9 8 7 6 3 2 2 (x + 2 x + 3 x + 2 x + x + x + x + x - 1) (-1 + x) and in Maple format (x^18+x^17+x^16-x^14+x^12+x^11+x^10+x^8-x^4-x^2+2*x-1)/(x^10+2*x^9+3*x^8+2*x^7+ x^6+x^3+x^2+x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.65835090509051371772 1.9175371779461324874 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 2, 4] and d occurrences (as containment) of the composition, [4, 1, 3], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 3 19 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 2 19 3 19 2 19 2 19 3 19 - 3 X1 X2 x - 2 X1 X2 x + 3 X1 X2 x + 6 X1 X2 x + X2 x 2 19 19 2 19 2 2 16 19 19 - X1 x - 6 X1 X2 x - 3 X2 x + X1 X2 x + 2 X1 x + 3 X2 x 2 16 2 16 19 2 16 2 15 16 - 2 X1 X2 x - 2 X1 X2 x - x + X1 x - X1 X2 x + 4 X1 X2 x 2 16 2 14 15 16 2 15 16 + X2 x + X1 X2 x + 2 X1 X2 x - 2 X1 x + X2 x - 2 X2 x 2 2 12 2 13 14 15 2 14 15 - X1 X2 x + X1 X2 x - 2 X1 X2 x - X1 x - X2 x - 2 X2 x 16 2 2 11 2 12 2 13 2 12 13 + x + X1 X2 x + 2 X1 X2 x - X1 x + X1 X2 x - 2 X1 X2 x 14 14 15 2 2 10 2 11 2 12 + X1 x + 2 X2 x + x - X1 X2 x - X1 X2 x - X1 x 12 13 13 14 2 10 2 10 - 2 X1 X2 x + 2 X1 x + X2 x - x + X1 X2 x + X1 X2 x 11 12 13 2 9 11 9 10 - X1 X2 x + X1 x - x - X1 X2 x + X1 x + X1 X2 x - X1 x 10 8 9 10 7 8 8 9 - X2 x + X1 X2 x + X2 x + x - X1 X2 x - X1 x - X2 x - x 6 7 8 5 6 4 5 5 + 2 X1 X2 x + X2 x + x - 2 X1 X2 x - 2 X2 x + X1 X2 x + X2 x + x 4 3 2 / 2 2 14 2 2 13 - x + x - 3 x + 3 x - 1) / ((-1 + x) (X1 X2 x + X1 X2 x / 2 14 2 14 2 13 2 14 2 13 - 2 X1 X2 x - X1 X2 x - 2 X1 X2 x + X1 x - X1 X2 x 14 2 2 11 2 12 2 13 2 12 + 2 X1 X2 x + X1 X2 x - X1 X2 x + X1 x + X1 X2 x 13 14 2 2 10 2 11 2 12 2 11 + 2 X1 X2 x - X1 x - X1 X2 x - 2 X1 X2 x + X1 x + X1 X2 x 13 2 10 2 11 2 10 11 12 - X1 x + X1 X2 x + X1 x + X1 X2 x - 2 X1 X2 x - X1 x 12 2 9 10 11 12 9 9 - X2 x - X1 X2 x - X1 X2 x + X1 x + x + X1 x + 2 X2 x 7 9 6 7 5 6 4 - X1 X2 x - 2 x + X1 X2 x + X1 x - 2 X1 X2 x - 2 X2 x + X1 X2 x 5 6 5 4 2 + X2 x + x + x - x - 2 x + 3 x - 1)) and in Maple format -(X1^2*X2^3*x^19-3*X1^2*X2^2*x^19-2*X1*X2^3*x^19+3*X1^2*X2*x^19+6*X1*X2^2*x^19+ X2^3*x^19-X1^2*x^19-6*X1*X2*x^19-3*X2^2*x^19+X1^2*X2^2*x^16+2*X1*x^19+3*X2*x^19 -2*X1^2*X2*x^16-2*X1*X2^2*x^16-x^19+X1^2*x^16-X1*X2^2*x^15+4*X1*X2*x^16+X2^2*x^ 16+X1*X2^2*x^14+2*X1*X2*x^15-2*X1*x^16+X2^2*x^15-2*X2*x^16-X1^2*X2^2*x^12+X1^2* X2*x^13-2*X1*X2*x^14-X1*x^15-X2^2*x^14-2*X2*x^15+x^16+X1^2*X2^2*x^11+2*X1^2*X2* x^12-X1^2*x^13+X1*X2^2*x^12-2*X1*X2*x^13+X1*x^14+2*X2*x^14+x^15-X1^2*X2^2*x^10- X1^2*X2*x^11-X1^2*x^12-2*X1*X2*x^12+2*X1*x^13+X2*x^13-x^14+X1^2*X2*x^10+X1*X2^2 *x^10-X1*X2*x^11+X1*x^12-x^13-X1*X2^2*x^9+X1*x^11+X1*X2*x^9-X1*x^10-X2*x^10+X1* X2*x^8+X2*x^9+x^10-X1*X2*x^7-X1*x^8-X2*x^8-x^9+2*X1*X2*x^6+X2*x^7+x^8-2*X1*X2*x ^5-2*X2*x^6+X1*X2*x^4+X2*x^5+x^5-x^4+x^3-3*x^2+3*x-1)/(-1+x)/(X1^2*X2^2*x^14+X1 ^2*X2^2*x^13-2*X1^2*X2*x^14-X1*X2^2*x^14-2*X1^2*X2*x^13+X1^2*x^14-X1*X2^2*x^13+ 2*X1*X2*x^14+X1^2*X2^2*x^11-X1^2*X2*x^12+X1^2*x^13+X1*X2^2*x^12+2*X1*X2*x^13-X1 *x^14-X1^2*X2^2*x^10-2*X1^2*X2*x^11+X1^2*x^12+X1*X2^2*x^11-X1*x^13+X1^2*X2*x^10 +X1^2*x^11+X1*X2^2*x^10-2*X1*X2*x^11-X1*x^12-X2*x^12-X1*X2^2*x^9-X1*X2*x^10+X1* x^11+x^12+X1*x^9+2*X2*x^9-X1*X2*x^7-2*x^9+X1*X2*x^6+X1*x^7-2*X1*X2*x^5-2*X2*x^6 +X1*X2*x^4+X2*x^5+x^6+x^5-x^4-2*x^2+3*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 2, 4], are n 183 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 1, 3], are n 141 65 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 16 3 65 ------------- 585 and in floating point 0.3819279330 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 16 195 ate normal pair with correlation, --------- 585 1/2 16 195 2267 i.e. , [[---------, 0], [0, ----]] 585 1755 ------------------------------------------------- Theorem Number, 32, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 2, 4], nor the composition, [4, 2, 2] Then infinity ----- \ n ) a(n) x = / ----- n = 0 14 13 11 10 9 8 6 5 4 2 x + 2 x - 2 x - x - x - x + x - x + x + x - 2 x + 1 - ----------------------------------------------------------------- 11 10 9 5 4 (-1 + x) (x + 2 x + x - x + x - 2 x + 1) and in Maple format -(x^14+2*x^13-2*x^11-x^10-x^9-x^8+x^6-x^5+x^4+x^2-2*x+1)/(-1+x)/(x^11+2*x^10+x^ 9-x^5+x^4-2*x+1) The asymptotic expression for a(n) is, n 0.65315694203017104538 1.9197449317290998989 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 2, 4] and d occurrences (as containment) of the composition, [4, 2, 2], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 15 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 2 14 2 15 2 15 2 14 2 15 + X1 X2 x - 2 X1 X2 x - 2 X1 X2 x - 2 X1 X2 x + X1 x 2 14 15 2 15 2 13 2 14 2 13 - 2 X1 X2 x + 4 X1 X2 x + X2 x + X1 X2 x + X1 x + X1 X2 x 14 15 2 14 15 2 12 2 13 + 4 X1 X2 x - 2 X1 x + X2 x - 2 X2 x + X1 X2 x - X1 x 2 12 13 14 2 13 14 15 2 12 + X1 X2 x - 4 X1 X2 x - 2 X1 x - X2 x - 2 X2 x + x - X1 x 12 13 2 12 13 14 11 12 - 4 X1 X2 x + 3 X1 x - X2 x + 3 X2 x + x + X1 X2 x + 3 X1 x 12 13 11 11 12 11 8 7 + 3 X2 x - 2 x - X1 x - X2 x - 2 x + x + X1 X2 x - X1 X2 x 8 8 6 8 5 7 4 6 - X1 x - X2 x + 2 X1 X2 x + x - 2 X1 X2 x + x + X1 X2 x - 2 x 5 4 3 2 / 2 12 2 11 + 2 x - x + x - 3 x + 3 x - 1) / ((-1 + x) (X1 X2 x + X1 X2 x / 2 12 12 2 11 11 12 12 - X1 x - 2 X1 X2 x - X1 x - 2 X1 X2 x + 2 X1 x + X2 x 10 11 11 12 9 10 10 11 + X1 X2 x + 2 X1 x + X2 x - x + X1 X2 x - X1 x - X2 x - x 9 9 10 7 9 6 7 5 - X1 x - X2 x + x + X1 X2 x + x - X1 X2 x - X1 x + 2 X1 X2 x 4 6 5 4 2 - X1 X2 x + x - 2 x + x + 2 x - 3 x + 1)) and in Maple format (X1^2*X2^2*x^15+X1^2*X2^2*x^14-2*X1^2*X2*x^15-2*X1*X2^2*x^15-2*X1^2*X2*x^14+X1^ 2*x^15-2*X1*X2^2*x^14+4*X1*X2*x^15+X2^2*x^15+X1^2*X2*x^13+X1^2*x^14+X1*X2^2*x^ 13+4*X1*X2*x^14-2*X1*x^15+X2^2*x^14-2*X2*x^15+X1^2*X2*x^12-X1^2*x^13+X1*X2^2*x^ 12-4*X1*X2*x^13-2*X1*x^14-X2^2*x^13-2*X2*x^14+x^15-X1^2*x^12-4*X1*X2*x^12+3*X1* x^13-X2^2*x^12+3*X2*x^13+x^14+X1*X2*x^11+3*X1*x^12+3*X2*x^12-2*x^13-X1*x^11-X2* x^11-2*x^12+x^11+X1*X2*x^8-X1*X2*x^7-X1*x^8-X2*x^8+2*X1*X2*x^6+x^8-2*X1*X2*x^5+ x^7+X1*X2*x^4-2*x^6+2*x^5-x^4+x^3-3*x^2+3*x-1)/(-1+x)/(X1^2*X2*x^12+X1^2*X2*x^ 11-X1^2*x^12-2*X1*X2*x^12-X1^2*x^11-2*X1*X2*x^11+2*X1*x^12+X2*x^12+X1*X2*x^10+2 *X1*x^11+X2*x^11-x^12+X1*X2*x^9-X1*x^10-X2*x^10-x^11-X1*x^9-X2*x^9+x^10+X1*X2*x ^7+x^9-X1*X2*x^6-X1*x^7+2*X1*X2*x^5-X1*X2*x^4+x^6-2*x^5+x^4+2*x^2-3*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 2, 4], are n 183 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 2, 2], are n 75 69 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 20 23 -------- 207 and in floating point 0.4633653646 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 20 23 ate normal pair with correlation, -------- 207 1/2 20 23 2663 i.e. , [[--------, 0], [0, ----]] 207 1863 ------------------------------------------------- Theorem Number, 33, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 2, 4], nor the composition, [4, 3, 1] Then infinity ----- 15 11 10 9 6 5 4 2 \ n x + x + 2 x + 2 x - x + x - x - x + 2 x - 1 ) a(n) x = ------------------------------------------------------ / 7 6 5 3 2 2 ----- (x + x + x + x + x + x - 1) (-1 + x) n = 0 and in Maple format (x^15+x^11+2*x^10+2*x^9-x^6+x^5-x^4-x^2+2*x-1)/(x^7+x^6+x^5+x^3+x^2+x-1)/(-1+x) ^2 The asymptotic expression for a(n) is, n 0.64232565556525224311 1.9221280043552247737 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 2, 4] and d occurrences (as containment) of the composition, [4, 3, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 16 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 2 15 2 16 2 16 2 15 2 16 - X1 X2 x - 2 X1 X2 x - 2 X1 X2 x + 2 X1 X2 x + X1 x 2 15 16 2 16 2 15 15 16 + 2 X1 X2 x + 4 X1 X2 x + X2 x - X1 x - 4 X1 X2 x - 2 X1 x 2 15 16 15 15 16 2 12 2 12 - X2 x - 2 X2 x + 2 X1 x + 2 X2 x + x - X1 X2 x - X1 X2 x 15 2 11 2 12 12 2 12 2 11 - x - X1 X2 x + X1 x + 3 X1 X2 x + X2 x + X1 x 11 12 12 11 11 12 9 + 2 X1 X2 x - 2 X1 x - 2 X2 x - 2 X1 x - X2 x + x - 2 X1 X2 x 11 8 9 9 7 8 9 6 + x + X1 X2 x + 2 X1 x + 2 X2 x - X1 X2 x - X1 x - 2 x - X1 X2 x 7 5 6 7 4 6 5 4 3 + 2 X1 x + 2 X1 X2 x - X1 x - x - X1 X2 x + 2 x - 2 x + x - x 2 / 2 8 8 8 6 8 + 3 x - 3 x + 1) / ((-1 + x) (X1 X2 x - X1 x - X2 x + X1 X2 x + x / 5 6 4 5 4 + X1 X2 x - X1 x - X1 X2 x - x + x - 2 x + 1)) and in Maple format (X1^2*X2^2*x^16-X1^2*X2^2*x^15-2*X1^2*X2*x^16-2*X1*X2^2*x^16+2*X1^2*X2*x^15+X1^ 2*x^16+2*X1*X2^2*x^15+4*X1*X2*x^16+X2^2*x^16-X1^2*x^15-4*X1*X2*x^15-2*X1*x^16- X2^2*x^15-2*X2*x^16+2*X1*x^15+2*X2*x^15+x^16-X1^2*X2*x^12-X1*X2^2*x^12-x^15-X1^ 2*X2*x^11+X1^2*x^12+3*X1*X2*x^12+X2^2*x^12+X1^2*x^11+2*X1*X2*x^11-2*X1*x^12-2* X2*x^12-2*X1*x^11-X2*x^11+x^12-2*X1*X2*x^9+x^11+X1*X2*x^8+2*X1*x^9+2*X2*x^9-X1* X2*x^7-X1*x^8-2*x^9-X1*X2*x^6+2*X1*x^7+2*X1*X2*x^5-X1*x^6-x^7-X1*X2*x^4+2*x^6-2 *x^5+x^4-x^3+3*x^2-3*x+1)/(-1+x)^2/(X1*X2*x^8-X1*x^8-X2*x^8+X1*X2*x^6+x^8+X1*X2 *x^5-X1*x^6-X1*X2*x^4-x^5+x^4-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 2, 4], are n 183 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 3, 1], are n 135 65 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 22 3 65 ------------- 585 and in floating point 0.5251509079 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 22 195 ate normal pair with correlation, --------- 585 1/2 22 195 2723 i.e. , [[---------, 0], [0, ----]] 585 1755 ------------------------------------------------- Theorem Number, 34, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 2, 4], nor the composition, [5, 1, 2] Then infinity ----- \ n 22 21 20 18 17 16 15 14 13 ) a(n) x = (x + 2 x + x - x - 2 x - x + x + 2 x + x / ----- n = 0 11 7 6 5 4 3 2 / - x - 3 x + 3 x - 2 x + x - x + 3 x - 3 x + 1) / ( / 12 11 10 8 7 6 5 4 2 (x + 2 x + 2 x - x - x + x - x + x - 2 x + 1) (-1 + x) ) and in Maple format (x^22+2*x^21+x^20-x^18-2*x^17-x^16+x^15+2*x^14+x^13-x^11-3*x^7+3*x^6-2*x^5+x^4- x^3+3*x^2-3*x+1)/(x^12+2*x^11+2*x^10-x^8-x^7+x^6-x^5+x^4-2*x+1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.66523001633931812792 1.9143463083540318048 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 2, 4] and d occurrences (as containment) of the composition, [5, 1, 2], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 3 22 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 3 21 2 2 22 3 22 2 3 20 + 2 X1 X2 x - 3 X1 X2 x - 2 X1 X2 x + X1 X2 x 2 2 21 2 22 3 21 2 22 3 22 - 6 X1 X2 x + 3 X1 X2 x - 4 X1 X2 x + 6 X1 X2 x + X2 x 2 2 20 2 21 2 22 3 20 2 21 - 3 X1 X2 x + 6 X1 X2 x - X1 x - 2 X1 X2 x + 12 X1 X2 x 22 3 21 2 22 2 20 2 21 - 6 X1 X2 x + 2 X2 x - 3 X2 x + 3 X1 X2 x - 2 X1 x 2 20 21 22 3 20 2 21 22 + 6 X1 X2 x - 12 X1 X2 x + 2 X1 x + X2 x - 6 X2 x + 3 X2 x 2 2 18 2 20 20 21 2 20 21 + X1 X2 x - X1 x - 6 X1 X2 x + 4 X1 x - 3 X2 x + 6 X2 x 22 2 2 17 2 18 2 18 20 20 - x + 2 X1 X2 x - 2 X1 X2 x - 2 X1 X2 x + 2 X1 x + 3 X2 x 21 2 17 2 18 2 17 18 2 18 - 2 x - 4 X1 X2 x + X1 x - 4 X1 X2 x + 4 X1 X2 x + X2 x 20 2 17 2 16 17 18 2 17 - x + 2 X1 x - X1 X2 x + 8 X1 X2 x - 2 X1 x + 2 X2 x 18 2 15 16 17 2 16 17 - 2 X2 x + X1 X2 x + 2 X1 X2 x - 4 X1 x + X2 x - 4 X2 x 18 2 2 13 2 14 2 14 15 16 + x - X1 X2 x + X1 X2 x + X1 X2 x - 2 X1 X2 x - X1 x 2 15 16 17 2 13 2 14 2 13 - X2 x - 2 X2 x + 2 x + 3 X1 X2 x - X1 x + X1 X2 x 14 15 2 14 15 16 2 13 2 12 - 4 X1 X2 x + X1 x - X2 x + 2 X2 x + x - 2 X1 x + X1 X2 x 13 14 14 15 2 11 12 - 4 X1 X2 x + 3 X1 x + 3 X2 x - x - X1 X2 x - 2 X1 X2 x 13 13 14 11 12 13 11 + 3 X1 x + X2 x - 2 x + 3 X1 X2 x + X1 x - x - 2 X1 x 11 11 7 6 7 7 5 - X2 x + x + 2 X1 X2 x - 2 X1 X2 x - 3 X1 x - 2 X2 x + X1 X2 x 6 6 7 5 6 4 5 4 3 2 + 4 X1 x + X2 x + 3 x - 3 X1 x - 3 x + X1 x + 2 x - x + x - 3 x / 2 2 15 2 15 2 15 + 3 x - 1) / ((-1 + x) (X1 X2 x - 2 X1 X2 x - X1 X2 x / 2 2 13 2 15 15 2 13 2 13 15 - X1 X2 x + X1 x + 2 X1 X2 x + X1 X2 x + X1 X2 x - X1 x 2 12 2 12 2 12 2 11 13 13 - X1 X2 x + X1 X2 x + X1 x - X1 X2 x - X1 x - X2 x 11 12 12 13 10 11 12 + 2 X1 X2 x - X1 x - X2 x + x - 2 X1 X2 x - X1 x + x 9 10 10 9 9 10 7 9 - X1 X2 x + 2 X1 x + 2 X2 x + X1 x + X2 x - 2 x + X1 X2 x - x 6 7 7 5 6 6 7 - 2 X1 X2 x - X1 x - 2 X2 x + X1 X2 x + 3 X1 x + X2 x + 2 x 5 6 4 5 4 2 - 3 X1 x - 2 x + X1 x + 2 x - x - 2 x + 3 x - 1)) and in Maple format -(X1^2*X2^3*x^22+2*X1^2*X2^3*x^21-3*X1^2*X2^2*x^22-2*X1*X2^3*x^22+X1^2*X2^3*x^ 20-6*X1^2*X2^2*x^21+3*X1^2*X2*x^22-4*X1*X2^3*x^21+6*X1*X2^2*x^22+X2^3*x^22-3*X1 ^2*X2^2*x^20+6*X1^2*X2*x^21-X1^2*x^22-2*X1*X2^3*x^20+12*X1*X2^2*x^21-6*X1*X2*x^ 22+2*X2^3*x^21-3*X2^2*x^22+3*X1^2*X2*x^20-2*X1^2*x^21+6*X1*X2^2*x^20-12*X1*X2*x ^21+2*X1*x^22+X2^3*x^20-6*X2^2*x^21+3*X2*x^22+X1^2*X2^2*x^18-X1^2*x^20-6*X1*X2* x^20+4*X1*x^21-3*X2^2*x^20+6*X2*x^21-x^22+2*X1^2*X2^2*x^17-2*X1^2*X2*x^18-2*X1* X2^2*x^18+2*X1*x^20+3*X2*x^20-2*x^21-4*X1^2*X2*x^17+X1^2*x^18-4*X1*X2^2*x^17+4* X1*X2*x^18+X2^2*x^18-x^20+2*X1^2*x^17-X1*X2^2*x^16+8*X1*X2*x^17-2*X1*x^18+2*X2^ 2*x^17-2*X2*x^18+X1*X2^2*x^15+2*X1*X2*x^16-4*X1*x^17+X2^2*x^16-4*X2*x^17+x^18- X1^2*X2^2*x^13+X1^2*X2*x^14+X1*X2^2*x^14-2*X1*X2*x^15-X1*x^16-X2^2*x^15-2*X2*x^ 16+2*x^17+3*X1^2*X2*x^13-X1^2*x^14+X1*X2^2*x^13-4*X1*X2*x^14+X1*x^15-X2^2*x^14+ 2*X2*x^15+x^16-2*X1^2*x^13+X1*X2^2*x^12-4*X1*X2*x^13+3*X1*x^14+3*X2*x^14-x^15- X1*X2^2*x^11-2*X1*X2*x^12+3*X1*x^13+X2*x^13-2*x^14+3*X1*X2*x^11+X1*x^12-x^13-2* X1*x^11-X2*x^11+x^11+2*X1*X2*x^7-2*X1*X2*x^6-3*X1*x^7-2*X2*x^7+X1*X2*x^5+4*X1*x ^6+X2*x^6+3*x^7-3*X1*x^5-3*x^6+X1*x^4+2*x^5-x^4+x^3-3*x^2+3*x-1)/(-1+x)/(X1^2* X2^2*x^15-2*X1^2*X2*x^15-X1*X2^2*x^15-X1^2*X2^2*x^13+X1^2*x^15+2*X1*X2*x^15+X1^ 2*X2*x^13+X1*X2^2*x^13-X1*x^15-X1^2*X2*x^12+X1*X2^2*x^12+X1^2*x^12-X1*X2^2*x^11 -X1*x^13-X2*x^13+2*X1*X2*x^11-X1*x^12-X2*x^12+x^13-2*X1*X2*x^10-X1*x^11+x^12-X1 *X2*x^9+2*X1*x^10+2*X2*x^10+X1*x^9+X2*x^9-2*x^10+X1*X2*x^7-x^9-2*X1*X2*x^6-X1*x ^7-2*X2*x^7+X1*X2*x^5+3*X1*x^6+X2*x^6+2*x^7-3*X1*x^5-2*x^6+X1*x^4+2*x^5-x^4-2*x ^2+3*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 2, 4], are n 183 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [5, 1, 2], are n 113 57 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 4 19 ------- 57 and in floating point 0.3058876452 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 4 19 ate normal pair with correlation, ------- 57 1/2 4 19 203 i.e. , [[-------, 0], [0, ---]] 57 171 ------------------------------------------------- Theorem Number, 35, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 2, 4], nor the composition, [5, 2, 1] Then infinity ----- \ n 18 16 14 13 11 10 9 7 6 ) a(n) x = (x - x + x + 2 x + x - 2 x - x - x + 2 x / ----- n = 0 5 4 3 2 / 9 8 5 4 - 2 x + x - x + 3 x - 3 x + 1) / ((x + x - x + x - 2 x + 1) / 2 (-1 + x) ) and in Maple format (x^18-x^16+x^14+2*x^13+x^11-2*x^10-x^9-x^7+2*x^6-2*x^5+x^4-x^3+3*x^2-3*x+1)/(x^ 9+x^8-x^5+x^4-2*x+1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.65981530479575843941 1.9159714162121856417 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 2, 4] and d occurrences (as containment) of the composition, [5, 2, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 18 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 18 2 18 2 2 16 2 18 18 - 2 X1 X2 x - 2 X1 X2 x - X1 X2 x + X1 x + 4 X1 X2 x 2 18 2 16 2 16 18 18 2 16 + X2 x + 2 X1 X2 x + 2 X1 X2 x - 2 X1 x - 2 X2 x - X1 x 16 2 16 18 2 14 16 16 - 4 X1 X2 x - X2 x + x - X1 X2 x + 2 X1 x + 2 X2 x 2 13 2 13 14 2 14 16 2 12 - X1 X2 x - X1 X2 x + 2 X1 X2 x + X2 x - x - X1 X2 x 2 13 13 14 2 13 14 2 12 + X1 x + 4 X1 X2 x - X1 x + X2 x - 2 X2 x + X1 x 12 13 13 14 11 12 13 + X1 X2 x - 3 X1 x - 3 X2 x + x + X1 X2 x - X1 x + 2 x 10 11 11 9 10 10 11 - 2 X1 X2 x - X1 x - X2 x - X1 X2 x + 2 X1 x + 2 X2 x + x 8 9 9 10 7 8 9 6 + X1 X2 x + X1 x + X2 x - 2 x - 2 X1 X2 x - X1 x - x + 2 X1 X2 x 7 5 6 7 5 6 4 5 4 + 3 X1 x - X1 X2 x - 4 X1 x - x + 3 X1 x + 2 x - X1 x - 2 x + x 3 2 / 2 9 8 9 9 - x + 3 x - 3 x + 1) / ((-1 + x) (X1 X2 x + X1 X2 x - X1 x - X2 x / 8 8 9 6 8 5 6 5 4 - X1 x - X2 x + x + X1 X2 x + x - X1 X2 x - X1 x + 2 X1 x - X1 x 5 4 - x + x - 2 x + 1)) and in Maple format (X1^2*X2^2*x^18-2*X1^2*X2*x^18-2*X1*X2^2*x^18-X1^2*X2^2*x^16+X1^2*x^18+4*X1*X2* x^18+X2^2*x^18+2*X1^2*X2*x^16+2*X1*X2^2*x^16-2*X1*x^18-2*X2*x^18-X1^2*x^16-4*X1 *X2*x^16-X2^2*x^16+x^18-X1*X2^2*x^14+2*X1*x^16+2*X2*x^16-X1^2*X2*x^13-X1*X2^2*x ^13+2*X1*X2*x^14+X2^2*x^14-x^16-X1^2*X2*x^12+X1^2*x^13+4*X1*X2*x^13-X1*x^14+X2^ 2*x^13-2*X2*x^14+X1^2*x^12+X1*X2*x^12-3*X1*x^13-3*X2*x^13+x^14+X1*X2*x^11-X1*x^ 12+2*x^13-2*X1*X2*x^10-X1*x^11-X2*x^11-X1*X2*x^9+2*X1*x^10+2*X2*x^10+x^11+X1*X2 *x^8+X1*x^9+X2*x^9-2*x^10-2*X1*X2*x^7-X1*x^8-x^9+2*X1*X2*x^6+3*X1*x^7-X1*X2*x^5 -4*X1*x^6-x^7+3*X1*x^5+2*x^6-X1*x^4-2*x^5+x^4-x^3+3*x^2-3*x+1)/(-1+x)^2/(X1*X2* x^9+X1*X2*x^8-X1*x^9-X2*x^9-X1*x^8-X2*x^8+x^9+X1*X2*x^6+x^8-X1*X2*x^5-X1*x^6+2* X1*x^5-X1*x^4-x^5+x^4-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 2, 4], are n 183 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [5, 2, 1], are n 111 57 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 14 19 -------- 171 and in floating point 0.3568689194 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 14 19 ate normal pair with correlation, -------- 171 1/2 14 19 1931 i.e. , [[--------, 0], [0, ----]] 171 1539 ------------------------------------------------- Theorem Number, 36, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 2, 4], nor the composition, [6, 1, 1] Then infinity ----- \ n 18 17 16 15 13 12 11 10 ) a(n) x = - (x + 2 x + 2 x + x - 2 x - 3 x - 3 x - 2 x / ----- n = 0 9 8 7 6 5 4 2 / - x - x + x + x - x + x + x - 2 x + 1) / ( / 6 5 3 2 2 (x + x + x + x + x - 1) (-1 + x) ) and in Maple format -(x^18+2*x^17+2*x^16+x^15-2*x^13-3*x^12-3*x^11-2*x^10-x^9-x^8+x^7+x^6-x^5+x^4+x ^2-2*x+1)/(x^6+x^5+x^3+x^2+x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.67380351075430772877 1.9111834366854762350 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 2, 4] and d occurrences (as containment) of the composition, [6, 1, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 19 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 2 18 2 19 2 19 2 18 2 19 + X1 X2 x - 2 X1 X2 x - 2 X1 X2 x - 2 X1 X2 x + X1 x 2 18 19 2 19 2 18 18 19 - 2 X1 X2 x + 4 X1 X2 x + X2 x + X1 x + 4 X1 X2 x - 2 X1 x 2 18 19 2 16 18 18 19 2 15 + X2 x - 2 X2 x + X1 X2 x - 2 X1 x - 2 X2 x + x + X1 X2 x 16 2 16 18 2 14 2 14 15 - 2 X1 X2 x - X2 x + x + X1 X2 x + X1 X2 x - 2 X1 X2 x 16 2 15 16 2 13 2 14 14 + X1 x - X2 x + 2 X2 x + X1 X2 x - X1 x - 4 X1 X2 x 15 2 14 15 16 2 13 13 14 + X1 x - X2 x + 2 X2 x - x - X1 x - 2 X1 X2 x + 3 X1 x 14 15 13 13 14 11 13 10 + 3 X2 x - x + 2 X1 x + X2 x - 2 x + X1 X2 x - x + X1 X2 x 11 11 10 10 11 8 10 7 - X1 x - X2 x - X1 x - X2 x + x + 2 X1 X2 x + x - 2 X1 X2 x 8 8 6 7 7 8 6 5 - 2 X1 x - 2 X2 x + X1 X2 x + X1 x + X2 x + 2 x + X1 x - 2 X1 x 6 4 5 4 3 2 / 2 7 - 2 x + X1 x + 2 x - x + x - 3 x + 3 x - 1) / ((-1 + x) (X1 X2 x / 6 7 7 6 7 5 4 5 4 - X1 X2 x - X1 x - X2 x + X1 x + x + X1 x - X1 x - x + x - 2 x + 1)) and in Maple format -(X1^2*X2^2*x^19+X1^2*X2^2*x^18-2*X1^2*X2*x^19-2*X1*X2^2*x^19-2*X1^2*X2*x^18+X1 ^2*x^19-2*X1*X2^2*x^18+4*X1*X2*x^19+X2^2*x^19+X1^2*x^18+4*X1*X2*x^18-2*X1*x^19+ X2^2*x^18-2*X2*x^19+X1*X2^2*x^16-2*X1*x^18-2*X2*x^18+x^19+X1*X2^2*x^15-2*X1*X2* x^16-X2^2*x^16+x^18+X1^2*X2*x^14+X1*X2^2*x^14-2*X1*X2*x^15+X1*x^16-X2^2*x^15+2* X2*x^16+X1^2*X2*x^13-X1^2*x^14-4*X1*X2*x^14+X1*x^15-X2^2*x^14+2*X2*x^15-x^16-X1 ^2*x^13-2*X1*X2*x^13+3*X1*x^14+3*X2*x^14-x^15+2*X1*x^13+X2*x^13-2*x^14+X1*X2*x^ 11-x^13+X1*X2*x^10-X1*x^11-X2*x^11-X1*x^10-X2*x^10+x^11+2*X1*X2*x^8+x^10-2*X1* X2*x^7-2*X1*x^8-2*X2*x^8+X1*X2*x^6+X1*x^7+X2*x^7+2*x^8+X1*x^6-2*X1*x^5-2*x^6+X1 *x^4+2*x^5-x^4+x^3-3*x^2+3*x-1)/(-1+x)^2/(X1*X2*x^7-X1*X2*x^6-X1*x^7-X2*x^7+X1* x^6+x^7+X1*x^5-X1*x^4-x^5+x^4-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 2, 4], are n 183 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [6, 1, 1], are n 49 53 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 8 3 53 ------------ 477 and in floating point 0.2114804230 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 8 159 ate normal pair with correlation, -------- 477 1/2 8 159 1559 i.e. , [[--------, 0], [0, ----]] 477 1431 ------------------------------------------------- Theorem Number, 37, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 3, 3], nor the composition, [1, 1, 6] Then infinity ----- 9 7 6 4 3 2 \ n x - x - x + x - x - x + 2 x - 1 ) a(n) x = ---------------------------------------------- / 7 6 5 4 2 2 ----- (x + 2 x + 2 x + x + x + x - 1) (-1 + x) n = 0 and in Maple format (x^9-x^7-x^6+x^4-x^3-x^2+2*x-1)/(x^7+2*x^6+2*x^5+x^4+x^2+x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.68493773070922259842 1.9131864632213745587 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 3, 3] and d occurrences (as containment) of the composition, [1, 1, 6], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 9 9 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x - X1 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 9 7 9 6 7 7 7 6 4 - X2 x - X1 X2 x + x + X1 X2 x + X1 x + X2 x - x - x - X1 x 3 4 3 2 / 8 7 + X1 x + x - x - x + 2 x - 1) / ((-1 + x) (X1 X2 x + X1 X2 x / 8 8 6 7 7 8 6 7 5 - X1 x - X2 x - X1 X2 x - X1 x - X2 x + x + X1 x + x + X1 x 4 5 3 4 3 + X1 x - x - X1 x - x + x - 2 x + 1)) and in Maple format (X1*X2*x^9-X1*x^9-X2*x^9-X1*X2*x^7+x^9+X1*X2*x^6+X1*x^7+X2*x^7-x^7-x^6-X1*x^4+ X1*x^3+x^4-x^3-x^2+2*x-1)/(-1+x)/(X1*X2*x^8+X1*X2*x^7-X1*x^8-X2*x^8-X1*X2*x^6- X1*x^7-X2*x^7+x^8+X1*x^6+x^7+X1*x^5+X1*x^4-x^5-X1*x^3-x^4+x^3-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 3, 3], are n 263 35 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 1, 6], are n 49 53 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 4 35 53 ------------- 1855 and in floating point 0.09287269204 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 4 1855 ate normal pair with correlation, --------- 1855 1/2 4 1855 1887 i.e. , [[---------, 0], [0, ----]] 1855 1855 ------------------------------------------------- Theorem Number, 38, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 3, 3], nor the composition, [1, 2, 5] Then infinity ----- 8 7 6 4 3 2 \ n x - x - x + x - x - x + 2 x - 1 ) a(n) x = -------------------------------------- / 7 5 4 3 ----- (-1 + x) (x - x - x + x - 2 x + 1) n = 0 and in Maple format (x^8-x^7-x^6+x^4-x^3-x^2+2*x-1)/(-1+x)/(x^7-x^5-x^4+x^3-2*x+1) The asymptotic expression for a(n) is, n 0.65342844522305416418 1.9235403491125894929 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 3, 3] and d occurrences (as containment) of the composition, [1, 2, 5], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 8 8 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x - X1 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 8 6 7 8 5 6 7 5 6 - X2 x - X1 X2 x + X2 x + x + X1 X2 x + 2 X1 x - x - X1 x - x 4 3 4 3 2 / 7 - X1 x + X1 x + x - x - x + 2 x - 1) / ((-1 + x) (X1 X2 x / 6 7 7 5 6 7 5 4 5 + X1 X2 x - X1 x - X2 x - X1 X2 x - X1 x + x + 2 X1 x + X1 x - x 3 4 3 - X1 x - x + x - 2 x + 1)) and in Maple format (X1*X2*x^8-X1*x^8-X2*x^8-X1*X2*x^6+X2*x^7+x^8+X1*X2*x^5+2*X1*x^6-x^7-X1*x^5-x^6 -X1*x^4+X1*x^3+x^4-x^3-x^2+2*x-1)/(-1+x)/(X1*X2*x^7+X1*X2*x^6-X1*x^7-X2*x^7-X1* X2*x^5-X1*x^6+x^7+2*X1*x^5+X1*x^4-x^5-X1*x^3-x^4+x^3-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 3, 3], are n 263 35 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 2, 5], are n 111 57 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 2 35 57 ------------- 285 and in floating point 0.3134415640 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 1/2 2 7 285 ate normal pair with correlation, ------------- 285 1/2 1/2 2 7 285 341 i.e. , [[-------------, 0], [0, ---]] 285 285 ------------------------------------------------- Theorem Number, 39, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 3, 3], nor the composition, [1, 5, 2] Then infinity ----- 8 7 6 4 3 2 \ n x - x - x + x - x - x + 2 x - 1 ) a(n) x = -------------------------------------- / 7 5 4 3 ----- (-1 + x) (x - x - x + x - 2 x + 1) n = 0 and in Maple format (x^8-x^7-x^6+x^4-x^3-x^2+2*x-1)/(-1+x)/(x^7-x^5-x^4+x^3-2*x+1) The asymptotic expression for a(n) is, n 0.65342844522305416418 1.9235403491125894929 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 3, 3] and d occurrences (as containment) of the composition, [1, 5, 2], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 8 8 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x - X1 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 8 6 7 8 5 6 7 5 6 - X2 x - X1 X2 x + X2 x + x + X1 X2 x + 2 X1 x - x - X1 x - x 4 3 4 3 2 / 7 - X1 x + X1 x + x - x - x + 2 x - 1) / ((-1 + x) (X1 X2 x / 6 7 7 5 6 7 5 4 5 + X1 X2 x - X1 x - X2 x - X1 X2 x - X1 x + x + 2 X1 x + X1 x - x 3 4 3 - X1 x - x + x - 2 x + 1)) and in Maple format (X1*X2*x^8-X1*x^8-X2*x^8-X1*X2*x^6+X2*x^7+x^8+X1*X2*x^5+2*X1*x^6-x^7-X1*x^5-x^6 -X1*x^4+X1*x^3+x^4-x^3-x^2+2*x-1)/(-1+x)/(X1*X2*x^7+X1*X2*x^6-X1*x^7-X2*x^7-X1* X2*x^5-X1*x^6+x^7+2*X1*x^5+X1*x^4-x^5-X1*x^3-x^4+x^3-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 3, 3], are n 263 35 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 5, 2], are n 111 57 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 2 35 57 ------------- 285 and in floating point 0.3134415640 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 1/2 2 7 285 ate normal pair with correlation, ------------- 285 1/2 1/2 2 7 285 341 i.e. , [[-------------, 0], [0, ---]] 285 285 ------------------------------------------------- Theorem Number, 40, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 3, 3], nor the composition, [1, 6, 1] Then infinity ----- 9 7 6 4 3 2 \ n x - x - x + x - x - x + 2 x - 1 ) a(n) x = ---------------------------------------------- / 7 6 5 4 2 2 ----- (x + 2 x + 2 x + x + x + x - 1) (-1 + x) n = 0 and in Maple format (x^9-x^7-x^6+x^4-x^3-x^2+2*x-1)/(x^7+2*x^6+2*x^5+x^4+x^2+x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.68493773070922259842 1.9131864632213745587 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 3, 3] and d occurrences (as containment) of the composition, [1, 6, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 9 9 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x - X1 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 9 7 9 6 7 7 7 6 4 - X2 x - X1 X2 x + x + X1 X2 x + X1 x + X2 x - x - x - X1 x 3 4 3 2 / 8 7 + X1 x + x - x - x + 2 x - 1) / ((-1 + x) (X1 X2 x + X1 X2 x / 8 8 6 7 7 8 6 7 5 - X1 x - X2 x - X1 X2 x - X1 x - X2 x + x + X1 x + x + X1 x 4 5 3 4 3 + X1 x - x - X1 x - x + x - 2 x + 1)) and in Maple format (X1*X2*x^9-X1*x^9-X2*x^9-X1*X2*x^7+x^9+X1*X2*x^6+X1*x^7+X2*x^7-x^7-x^6-X1*x^4+ X1*x^3+x^4-x^3-x^2+2*x-1)/(-1+x)/(X1*X2*x^8+X1*X2*x^7-X1*x^8-X2*x^8-X1*X2*x^6- X1*x^7-X2*x^7+x^8+X1*x^6+x^7+X1*x^5+X1*x^4-x^5-X1*x^3-x^4+x^3-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 3, 3], are n 263 35 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 6, 1], are n 49 53 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 4 35 53 ------------- 1855 and in floating point 0.09287269204 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 4 1855 ate normal pair with correlation, --------- 1855 1/2 4 1855 1887 i.e. , [[---------, 0], [0, ----]] 1855 1855 ------------------------------------------------- Theorem Number, 41, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 3, 3], nor the composition, [2, 1, 5] Then infinity ----- 8 6 4 3 2 \ n x - 2 x + x - x - x + 2 x - 1 ) a(n) x = - ------------------------------------------- / 8 6 5 4 3 ----- (-1 + x) (x - x + x + x - x + 2 x - 1) n = 0 and in Maple format -(x^8-2*x^6+x^4-x^3-x^2+2*x-1)/(-1+x)/(x^8-x^6+x^5+x^4-x^3+2*x-1) The asymptotic expression for a(n) is, n 0.67267896620284337780 1.9158008597433206753 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 3, 3] and d occurrences (as containment) of the composition, [2, 1, 5], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 11 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 11 11 8 8 8 6 8 - 2 X1 X2 x + X1 x - X1 X2 x + X1 x + X2 x + X1 X2 x - x 5 6 6 5 6 4 3 4 3 2 - X1 X2 x - 2 X1 x - X2 x + X1 x + 2 x + X1 x - X1 x - x + x + x / 2 12 2 11 12 11 - 2 x + 1) / (X1 X2 x - X1 X2 x - 2 X1 X2 x + 2 X1 X2 x / 12 11 9 8 9 9 8 8 + X1 x - X1 x - X1 X2 x + X1 X2 x + X1 x + X2 x - X1 x - X2 x 9 6 7 8 5 6 6 7 5 - x - 2 X1 X2 x - X2 x + x + X1 X2 x + 3 X1 x + X2 x + x - X1 x 6 4 3 4 3 2 - 2 x - 2 X1 x + X1 x + 2 x - x - 2 x + 3 x - 1) and in Maple format -(X1*X2^2*x^11-2*X1*X2*x^11+X1*x^11-X1*X2*x^8+X1*x^8+X2*x^8+X1*X2*x^6-x^8-X1*X2 *x^5-2*X1*x^6-X2*x^6+X1*x^5+2*x^6+X1*x^4-X1*x^3-x^4+x^3+x^2-2*x+1)/(X1*X2^2*x^ 12-X1*X2^2*x^11-2*X1*X2*x^12+2*X1*X2*x^11+X1*x^12-X1*x^11-X1*X2*x^9+X1*X2*x^8+ X1*x^9+X2*x^9-X1*x^8-X2*x^8-x^9-2*X1*X2*x^6-X2*x^7+x^8+X1*X2*x^5+3*X1*x^6+X2*x^ 6+x^7-X1*x^5-2*x^6-2*X1*x^4+X1*x^3+2*x^4-x^3-2*x^2+3*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 3, 3], are n 263 35 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 1, 5], are n 113 57 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 8 35 57 ------------- 1995 and in floating point 0.1791094652 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 8 1995 ate normal pair with correlation, --------- 1995 1/2 8 1995 2123 i.e. , [[---------, 0], [0, ----]] 1995 1995 ------------------------------------------------- Theorem Number, 42, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 3, 3], nor the composition, [2, 2, 4] Then infinity ----- 7 6 4 3 2 \ n x - 2 x + x - x - x + 2 x - 1 ) a(n) x = - ------------------------------------------- / 7 6 5 4 3 ----- (-1 + x) (x - x + x + x - x + 2 x - 1) n = 0 and in Maple format -(x^7-2*x^6+x^4-x^3-x^2+2*x-1)/(-1+x)/(x^7-x^6+x^5+x^4-x^3+2*x-1) The asymptotic expression for a(n) is, n 0.64732625001158940098 1.9249743097683141877 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 3, 3] and d occurrences (as containment) of the composition, [2, 2, 4], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 7 7 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x - X1 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 7 5 6 6 7 4 5 6 - X2 x - X1 X2 x + X1 x + X2 x + x + X1 X2 x + X1 x - 2 x 4 3 4 3 2 / 8 7 - 2 X1 x + X1 x + x - x - x + 2 x - 1) / (X1 X2 x - X1 X2 x / 8 8 7 7 8 5 6 6 7 - X1 x - X2 x + X1 x + 2 X2 x + x + 2 X1 X2 x - X1 x - X2 x - 2 x 4 5 6 4 3 4 3 2 - X1 X2 x - 2 X1 x + 2 x + 3 X1 x - X1 x - 2 x + x + 2 x - 3 x + 1 ) and in Maple format -(X1*X2*x^7-X1*x^7-X2*x^7-X1*X2*x^5+X1*x^6+X2*x^6+x^7+X1*X2*x^4+X1*x^5-2*x^6-2* X1*x^4+X1*x^3+x^4-x^3-x^2+2*x-1)/(X1*X2*x^8-X1*X2*x^7-X1*x^8-X2*x^8+X1*x^7+2*X2 *x^7+x^8+2*X1*X2*x^5-X1*x^6-X2*x^6-2*x^7-X1*X2*x^4-2*X1*x^5+2*x^6+3*X1*x^4-X1*x ^3-2*x^4+x^3+2*x^2-3*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 3, 3], are n 263 35 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 2, 4], are n 75 69 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 4 35 69 ------------- 483 and in floating point 0.4069784638 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 1/2 4 5 483 ate normal pair with correlation, ------------- 483 1/2 1/2 4 5 483 643 i.e. , [[-------------, 0], [0, ---]] 483 483 ------------------------------------------------- Theorem Number, 43, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 3, 3], nor the composition, [2, 4, 2] Then infinity ----- 7 6 4 3 2 \ n x - 2 x + x - x - x + 2 x - 1 ) a(n) x = --------------------------------------------------- / 10 8 7 6 4 3 2 ----- x - x + 2 x - 2 x + 2 x - x - 2 x + 3 x - 1 n = 0 and in Maple format (x^7-2*x^6+x^4-x^3-x^2+2*x-1)/(x^10-x^8+2*x^7-2*x^6+2*x^4-x^3-2*x^2+3*x-1) The asymptotic expression for a(n) is, n 0.63012861965149722353 1.9302781753610965686 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 3, 3] and d occurrences (as containment) of the composition, [2, 4, 2], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 7 7 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x - X1 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 7 5 6 6 7 4 5 6 - X2 x - X1 X2 x + X1 x + X2 x + x + X1 X2 x + X1 x - 2 x 4 3 4 3 2 / 10 10 - 2 X1 x + X1 x + x - x - x + 2 x - 1) / (X1 X2 x - X1 x / 10 8 10 7 8 8 7 7 - X2 x - X1 X2 x + x + X1 X2 x + X1 x + X2 x - X1 x - 2 X2 x 8 5 6 6 7 4 5 6 - x - 2 X1 X2 x + X1 x + X2 x + 2 x + X1 X2 x + 2 X1 x - 2 x 4 3 4 3 2 - 3 X1 x + X1 x + 2 x - x - 2 x + 3 x - 1) and in Maple format (X1*X2*x^7-X1*x^7-X2*x^7-X1*X2*x^5+X1*x^6+X2*x^6+x^7+X1*X2*x^4+X1*x^5-2*x^6-2* X1*x^4+X1*x^3+x^4-x^3-x^2+2*x-1)/(X1*X2*x^10-X1*x^10-X2*x^10-X1*X2*x^8+x^10+X1* X2*x^7+X1*x^8+X2*x^8-X1*x^7-2*X2*x^7-x^8-2*X1*X2*x^5+X1*x^6+X2*x^6+2*x^7+X1*X2* x^4+2*X1*x^5-2*x^6-3*X1*x^4+X1*x^3+2*x^4-x^3-2*x^2+3*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 3, 3], are n 263 35 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 4, 2], are n 75 69 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 8 35 69 ------------- 805 and in floating point 0.4883741565 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 1/2 8 3 805 ate normal pair with correlation, ------------- 805 1/2 1/2 8 3 805 1189 i.e. , [[-------------, 0], [0, ----]] 805 805 ------------------------------------------------- Theorem Number, 44, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 3, 3], nor the composition, [2, 5, 1] Then infinity ----- 12 10 8 6 5 4 2 \ n x - x + 2 x - x - x + 2 x - 3 x + 3 x - 1 ) a(n) x = - -------------------------------------------------- / 7 5 4 3 2 ----- (x - x - x + x - 2 x + 1) (-1 + x) n = 0 and in Maple format -(x^12-x^10+2*x^8-x^6-x^5+2*x^4-3*x^2+3*x-1)/(x^7-x^5-x^4+x^3-2*x+1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.64616683293975167207 1.9235403491125894929 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 3, 3] and d occurrences (as containment) of the composition, [2, 5, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 12 12 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x - X1 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 12 10 12 10 10 8 10 7 - X2 x - X1 X2 x + x + X1 x + X2 x + X1 X2 x - x + X1 X2 x 8 8 6 7 7 8 5 - X1 x - 2 X2 x - 2 X1 X2 x - 2 X1 x + X2 x + 2 x + X1 X2 x 6 6 4 5 3 4 2 / + 3 X1 x - x - 2 X1 x - x + X1 x + 2 x - 3 x + 3 x - 1) / ( / 2 7 6 7 7 5 6 7 (-1 + x) (X1 X2 x + X1 X2 x - X1 x - X2 x - X1 X2 x - X1 x + x 5 4 5 3 4 3 + 2 X1 x + X1 x - x - X1 x - x + x - 2 x + 1)) and in Maple format -(X1*X2*x^12-X1*x^12-X2*x^12-X1*X2*x^10+x^12+X1*x^10+X2*x^10+X1*X2*x^8-x^10+X1* X2*x^7-X1*x^8-2*X2*x^8-2*X1*X2*x^6-2*X1*x^7+X2*x^7+2*x^8+X1*X2*x^5+3*X1*x^6-x^6 -2*X1*x^4-x^5+X1*x^3+2*x^4-3*x^2+3*x-1)/(-1+x)^2/(X1*X2*x^7+X1*X2*x^6-X1*x^7-X2 *x^7-X1*X2*x^5-X1*x^6+x^7+2*X1*x^5+X1*x^4-x^5-X1*x^3-x^4+x^3-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 3, 3], are n 263 35 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 5, 1], are n 111 57 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 2 35 57 ------------- 285 and in floating point 0.3134415640 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 1/2 2 7 285 ate normal pair with correlation, ------------- 285 1/2 1/2 2 7 285 341 i.e. , [[-------------, 0], [0, ---]] 285 285 ------------------------------------------------- Theorem Number, 45, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 3, 3], nor the composition, [3, 1, 4] Then infinity ----- 7 6 5 4 3 2 \ n x - x - x + x - x - x + 2 x - 1 ) a(n) x = - ---------------------------------------- / 8 7 4 3 ----- (-1 + x) (x + 2 x + x - x + 2 x - 1) n = 0 and in Maple format -(x^7-x^6-x^5+x^4-x^3-x^2+2*x-1)/(-1+x)/(x^8+2*x^7+x^4-x^3+2*x-1) The asymptotic expression for a(n) is, n 0.65444189970384155768 1.9203064027137528069 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 3, 3] and d occurrences (as containment) of the composition, [3, 1, 4], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 10 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 9 10 9 10 9 7 7 + X1 X2 x - 2 X1 X2 x - 2 X1 X2 x + X1 x + X1 x - X1 X2 x + X1 x 7 5 6 7 4 5 5 6 4 + X2 x + X1 X2 x - X1 x - x - X1 X2 x - X1 x - X2 x + x + 2 X1 x 5 3 4 3 2 / 2 12 2 11 + x - X1 x - x + x + x - 2 x + 1) / (X1 X2 x + 2 X1 X2 x / 12 11 12 2 9 11 9 - 2 X1 X2 x - 4 X1 X2 x + X1 x - X1 X2 x + 2 X1 x + X1 X2 x 8 9 7 8 8 9 7 7 8 - X1 X2 x + X2 x + X1 X2 x + X1 x + X2 x - x - X1 x - 2 X2 x - x 5 6 6 7 4 5 5 4 - 2 X1 X2 x + X1 x - X2 x + 2 x + X1 X2 x + 2 X1 x + X2 x - 3 X1 x 5 3 4 3 2 - x + X1 x + 2 x - x - 2 x + 3 x - 1) and in Maple format -(X1*X2^2*x^10+X1*X2^2*x^9-2*X1*X2*x^10-2*X1*X2*x^9+X1*x^10+X1*x^9-X1*X2*x^7+X1 *x^7+X2*x^7+X1*X2*x^5-X1*x^6-x^7-X1*X2*x^4-X1*x^5-X2*x^5+x^6+2*X1*x^4+x^5-X1*x^ 3-x^4+x^3+x^2-2*x+1)/(X1*X2^2*x^12+2*X1*X2^2*x^11-2*X1*X2*x^12-4*X1*X2*x^11+X1* x^12-X1*X2^2*x^9+2*X1*x^11+X1*X2*x^9-X1*X2*x^8+X2*x^9+X1*X2*x^7+X1*x^8+X2*x^8-x ^9-X1*x^7-2*X2*x^7-x^8-2*X1*X2*x^5+X1*x^6-X2*x^6+2*x^7+X1*X2*x^4+2*X1*x^5+X2*x^ 5-3*X1*x^4-x^5+X1*x^3+2*x^4-x^3-2*x^2+3*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 3, 3], are n 263 35 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 1, 4], are n 141 65 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 16 7 13 ------------- 455 and in floating point 0.3354511477 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 16 91 ate normal pair with correlation, -------- 455 1/2 16 91 2787 i.e. , [[--------, 0], [0, ----]] 455 2275 ------------------------------------------------- Theorem Number, 46, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 3, 3], nor the composition, [3, 2, 3] Then infinity ----- 5 4 3 2 \ n x - x + x + x - 2 x + 1 ) a(n) x = -------------------------------------------- / 3 4 3 2 ----- (-1 + x) (x - x + 1) (x + x + x + x - 1) n = 0 and in Maple format (x^5-x^4+x^3+x^2-2*x+1)/(-1+x)/(x^3-x+1)/(x^4+x^3+x^2+x-1) The asymptotic expression for a(n) is, n 0.63758295460268560269 1.9275619754829253043 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 3, 3] and d occurrences (as containment) of the composition, [3, 2, 3], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 8 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 8 8 6 6 4 5 3 5 - 2 X1 X2 x + X1 x - X1 X2 x + X2 x + X1 X2 x - X2 x - X1 X2 x + x 4 3 2 / 2 10 2 9 10 - x + x + x - 2 x + 1) / (X1 X2 x + X1 X2 x - 2 X1 X2 x / 2 8 9 10 8 9 7 8 - X1 X2 x - 2 X1 X2 x + X1 x + X1 X2 x + X1 x - X1 X2 x + X2 x 6 7 8 6 4 5 6 3 + X1 X2 x + X1 x - x - 2 X2 x - 2 X1 X2 x + X2 x + x + X1 X2 x 5 4 3 2 - x + 2 x - x - 2 x + 3 x - 1) and in Maple format -(X1*X2^2*x^8-2*X1*X2*x^8+X1*x^8-X1*X2*x^6+X2*x^6+X1*X2*x^4-X2*x^5-X1*X2*x^3+x^ 5-x^4+x^3+x^2-2*x+1)/(X1*X2^2*x^10+X1*X2^2*x^9-2*X1*X2*x^10-X1*X2^2*x^8-2*X1*X2 *x^9+X1*x^10+X1*X2*x^8+X1*x^9-X1*X2*x^7+X2*x^8+X1*X2*x^6+X1*x^7-x^8-2*X2*x^6-2* X1*X2*x^4+X2*x^5+x^6+X1*X2*x^3-x^5+2*x^4-x^3-2*x^2+3*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 3, 3], are n 263 35 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 2, 3], are n 89 77 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 32 5 11 ------------- 385 and in floating point 0.6164113027 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 32 55 ate normal pair with correlation, -------- 385 1/2 32 55 4743 i.e. , [[--------, 0], [0, ----]] 385 2695 ------------------------------------------------- Theorem Number, 47, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 3, 3], nor the composition, [3, 3, 2] Then infinity ----- 11 9 8 7 6 5 4 2 \ n x - x + x + x - x - x + 2 x - 3 x + 3 x - 1 ) a(n) x = ---------------------------------------------------- / 5 4 3 2 ----- (x + x - x + 2 x - 1) (-1 + x) n = 0 and in Maple format (x^11-x^9+x^8+x^7-x^6-x^5+2*x^4-3*x^2+3*x-1)/(x^5+x^4-x^3+2*x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.59752426117277830607 1.9417130342786384772 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 3, 3] and d occurrences (as containment) of the composition, [3, 3, 2], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 11 11 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x - X1 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 11 9 11 8 9 9 7 8 - X2 x - X1 X2 x + x + X1 X2 x + X1 x + X2 x - X1 X2 x - X1 x 8 9 6 8 5 7 4 6 3 - X2 x - x + X1 X2 x + x + X1 X2 x + x - 2 X1 X2 x - x + X1 X2 x 5 4 2 / 7 6 7 - x + 2 x - 3 x + 3 x - 1) / ((-1 + x) (X1 X2 x - X1 X2 x - X1 x / 4 6 3 4 3 2 + 2 X1 X2 x + x - X1 X2 x - 2 x + x + 2 x - 3 x + 1)) and in Maple format (X1*X2*x^11-X1*x^11-X2*x^11-X1*X2*x^9+x^11+X1*X2*x^8+X1*x^9+X2*x^9-X1*X2*x^7-X1 *x^8-X2*x^8-x^9+X1*X2*x^6+x^8+X1*X2*x^5+x^7-2*X1*X2*x^4-x^6+X1*X2*x^3-x^5+2*x^4 -3*x^2+3*x-1)/(-1+x)/(X1*X2*x^7-X1*X2*x^6-X1*x^7+2*X1*X2*x^4+x^6-X1*X2*x^3-2*x^ 4+x^3+2*x^2-3*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 3, 3], are n 263 35 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 3, 2], are n 97 85 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 44 7 17 ------------- 595 and in floating point 0.8066946778 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 44 119 ate normal pair with correlation, --------- 595 1/2 44 119 6847 i.e. , [[---------, 0], [0, ----]] 595 2975 ------------------------------------------------- Theorem Number, 48, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 3, 3], nor the composition, [3, 4, 1] Then infinity ----- 12 11 10 8 7 6 5 4 2 \ n x + x - x + x + x - x - x + 2 x - 3 x + 3 x - 1 ) a(n) x = ----------------------------------------------------------- / 5 4 3 2 ----- (x + x - x + 2 x - 1) (-1 + x) n = 0 and in Maple format (x^12+x^11-x^10+x^8+x^7-x^6-x^5+2*x^4-3*x^2+3*x-1)/(x^5+x^4-x^3+2*x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.59130364901708252896 1.9417130342786384772 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 3, 3] and d occurrences (as containment) of the composition, [3, 4, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 12 11 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x + X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 12 12 10 11 11 12 10 10 - X1 x - X2 x - X1 X2 x - X1 x - X2 x + x + X1 x + X2 x 11 10 7 8 6 7 8 5 7 + x - x + X1 X2 x - X2 x + X1 X2 x - 2 X1 x + x - 2 X1 X2 x + x 4 5 6 4 5 3 4 2 + X1 X2 x + 3 X1 x - x - 3 X1 x - x + X1 x + 2 x - 3 x + 3 x - 1) / 2 6 5 6 4 4 5 / ((-1 + x) (X1 X2 x + X1 X2 x - X1 x - X1 X2 x + 2 X1 x - x / 3 4 3 - X1 x - x + x - 2 x + 1)) and in Maple format -(X1*X2*x^12+X1*X2*x^11-X1*x^12-X2*x^12-X1*X2*x^10-X1*x^11-X2*x^11+x^12+X1*x^10 +X2*x^10+x^11-x^10+X1*X2*x^7-X2*x^8+X1*X2*x^6-2*X1*x^7+x^8-2*X1*X2*x^5+x^7+X1* X2*x^4+3*X1*x^5-x^6-3*X1*x^4-x^5+X1*x^3+2*x^4-3*x^2+3*x-1)/(-1+x)^2/(X1*X2*x^6+ X1*X2*x^5-X1*x^6-X1*X2*x^4+2*X1*x^4-x^5-X1*x^3-x^4+x^3-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 3, 3], are n 263 35 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 4, 1], are n 135 65 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 34 7 13 ------------- 455 and in floating point 0.7128336889 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 34 91 ate normal pair with correlation, -------- 455 1/2 34 91 4587 i.e. , [[--------, 0], [0, ----]] 455 2275 ------------------------------------------------- Theorem Number, 49, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 3, 3], nor the composition, [4, 1, 3] Then infinity ----- 7 6 5 4 3 2 \ n x - x - x + x - x - x + 2 x - 1 ) a(n) x = - ---------------------------------------- / 8 7 4 3 ----- (-1 + x) (x + 2 x + x - x + 2 x - 1) n = 0 and in Maple format -(x^7-x^6-x^5+x^4-x^3-x^2+2*x-1)/(-1+x)/(x^8+2*x^7+x^4-x^3+2*x-1) The asymptotic expression for a(n) is, n 0.65444189970384155768 1.9203064027137528069 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 3, 3] and d occurrences (as containment) of the composition, [4, 1, 3], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 10 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 9 10 9 10 9 7 7 + X1 X2 x - 2 X1 X2 x - 2 X1 X2 x + X1 x + X1 x - X1 X2 x + X1 x 7 5 6 7 4 5 5 6 4 + X2 x + X1 X2 x - X1 x - x - X1 X2 x - X1 x - X2 x + x + 2 X1 x 5 3 4 3 2 / 2 12 2 11 + x - X1 x - x + x + x - 2 x + 1) / (X1 X2 x + 2 X1 X2 x / 12 11 12 2 9 11 9 - 2 X1 X2 x - 4 X1 X2 x + X1 x - X1 X2 x + 2 X1 x + X1 X2 x 8 9 7 8 8 9 7 7 8 - X1 X2 x + X2 x + X1 X2 x + X1 x + X2 x - x - X1 x - 2 X2 x - x 5 6 6 7 4 5 5 4 - 2 X1 X2 x + X1 x - X2 x + 2 x + X1 X2 x + 2 X1 x + X2 x - 3 X1 x 5 3 4 3 2 - x + X1 x + 2 x - x - 2 x + 3 x - 1) and in Maple format -(X1*X2^2*x^10+X1*X2^2*x^9-2*X1*X2*x^10-2*X1*X2*x^9+X1*x^10+X1*x^9-X1*X2*x^7+X1 *x^7+X2*x^7+X1*X2*x^5-X1*x^6-x^7-X1*X2*x^4-X1*x^5-X2*x^5+x^6+2*X1*x^4+x^5-X1*x^ 3-x^4+x^3+x^2-2*x+1)/(X1*X2^2*x^12+2*X1*X2^2*x^11-2*X1*X2*x^12-4*X1*X2*x^11+X1* x^12-X1*X2^2*x^9+2*X1*x^11+X1*X2*x^9-X1*X2*x^8+X2*x^9+X1*X2*x^7+X1*x^8+X2*x^8-x ^9-X1*x^7-2*X2*x^7-x^8-2*X1*X2*x^5+X1*x^6-X2*x^6+2*x^7+X1*X2*x^4+2*X1*x^5+X2*x^ 5-3*X1*x^4-x^5+X1*x^3+2*x^4-x^3-2*x^2+3*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 3, 3], are n 263 35 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 1, 3], are n 141 65 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 16 7 13 ------------- 455 and in floating point 0.3354511477 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 16 91 ate normal pair with correlation, -------- 455 1/2 16 91 2787 i.e. , [[--------, 0], [0, ----]] 455 2275 ------------------------------------------------- Theorem Number, 50, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 3, 3], nor the composition, [4, 2, 2] Then infinity ----- 11 9 7 6 4 3 2 \ n x + x + x - 2 x + x - x - x + 2 x - 1 ) a(n) x = - --------------------------------------------- / 7 6 5 4 3 ----- (-1 + x) (x - x + x + x - x + 2 x - 1) n = 0 and in Maple format -(x^11+x^9+x^7-2*x^6+x^4-x^3-x^2+2*x-1)/(-1+x)/(x^7-x^6+x^5+x^4-x^3+2*x-1) The asymptotic expression for a(n) is, n 0.64040732435378480241 1.9249743097683141877 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 3, 3] and d occurrences (as containment) of the composition, [4, 2, 2], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 12 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 12 2 12 11 12 12 10 - 2 X1 X2 x - X2 x + X1 X2 x + X1 x + 2 X2 x - X1 X2 x 11 11 12 9 10 10 11 8 - X1 x - X2 x - x + X1 X2 x + X1 x + X2 x + x - X1 X2 x 9 9 10 7 8 8 9 6 7 - X1 x - X2 x - x + X1 X2 x + X1 x + X2 x + x + X1 X2 x - 2 X1 x 7 8 5 6 7 4 5 6 - 2 X2 x - x - 2 X1 X2 x + X2 x + 3 x + X1 X2 x + 3 X1 x - 2 x 4 5 3 4 2 / 8 - 3 X1 x - x + X1 x + 2 x - 3 x + 3 x - 1) / ((-1 + x) (X1 X2 x / 7 8 8 7 7 8 5 6 - X1 X2 x - X1 x - X2 x + X1 x + 2 X2 x + x + 2 X1 X2 x - X1 x 6 7 4 5 6 4 3 4 3 - X2 x - 2 x - X1 X2 x - 2 X1 x + 2 x + 3 X1 x - X1 x - 2 x + x 2 + 2 x - 3 x + 1)) and in Maple format (X1*X2^2*x^12-2*X1*X2*x^12-X2^2*x^12+X1*X2*x^11+X1*x^12+2*X2*x^12-X1*X2*x^10-X1 *x^11-X2*x^11-x^12+X1*X2*x^9+X1*x^10+X2*x^10+x^11-X1*X2*x^8-X1*x^9-X2*x^9-x^10+ X1*X2*x^7+X1*x^8+X2*x^8+x^9+X1*X2*x^6-2*X1*x^7-2*X2*x^7-x^8-2*X1*X2*x^5+X2*x^6+ 3*x^7+X1*X2*x^4+3*X1*x^5-2*x^6-3*X1*x^4-x^5+X1*x^3+2*x^4-3*x^2+3*x-1)/(-1+x)/( X1*X2*x^8-X1*X2*x^7-X1*x^8-X2*x^8+X1*x^7+2*X2*x^7+x^8+2*X1*X2*x^5-X1*x^6-X2*x^6 -2*x^7-X1*X2*x^4-2*X1*x^5+2*x^6+3*X1*x^4-X1*x^3-2*x^4+x^3+2*x^2-3*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 3, 3], are n 263 35 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 2, 2], are n 75 69 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 4 35 69 ------------- 483 and in floating point 0.4069784638 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 1/2 4 5 483 ate normal pair with correlation, ------------- 483 1/2 1/2 4 5 483 643 i.e. , [[-------------, 0], [0, ---]] 483 483 ------------------------------------------------- Theorem Number, 51, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 3, 3], nor the composition, [4, 3, 1] Then infinity ----- 12 11 10 8 7 6 5 4 2 \ n x + x - x + x + x - x - x + 2 x - 3 x + 3 x - 1 ) a(n) x = ----------------------------------------------------------- / 5 4 3 2 ----- (x + x - x + 2 x - 1) (-1 + x) n = 0 and in Maple format (x^12+x^11-x^10+x^8+x^7-x^6-x^5+2*x^4-3*x^2+3*x-1)/(x^5+x^4-x^3+2*x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.59130364901708252896 1.9417130342786384772 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 3, 3] and d occurrences (as containment) of the composition, [4, 3, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 12 11 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x + X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 12 12 10 11 11 12 10 10 - X1 x - X2 x - X1 X2 x - X1 x - X2 x + x + X1 x + X2 x 11 10 7 8 6 7 8 5 7 + x - x + X1 X2 x - X2 x + X1 X2 x - 2 X1 x + x - 2 X1 X2 x + x 4 5 6 4 5 3 4 2 + X1 X2 x + 3 X1 x - x - 3 X1 x - x + X1 x + 2 x - 3 x + 3 x - 1) / 2 6 5 6 4 4 5 / ((-1 + x) (X1 X2 x + X1 X2 x - X1 x - X1 X2 x + 2 X1 x - x / 3 4 3 - X1 x - x + x - 2 x + 1)) and in Maple format -(X1*X2*x^12+X1*X2*x^11-X1*x^12-X2*x^12-X1*X2*x^10-X1*x^11-X2*x^11+x^12+X1*x^10 +X2*x^10+x^11-x^10+X1*X2*x^7-X2*x^8+X1*X2*x^6-2*X1*x^7+x^8-2*X1*X2*x^5+x^7+X1* X2*x^4+3*X1*x^5-x^6-3*X1*x^4-x^5+X1*x^3+2*x^4-3*x^2+3*x-1)/(-1+x)^2/(X1*X2*x^6+ X1*X2*x^5-X1*x^6-X1*X2*x^4+2*X1*x^4-x^5-X1*x^3-x^4+x^3-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 3, 3], are n 263 35 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 3, 1], are n 135 65 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 34 7 13 ------------- 455 and in floating point 0.7128336889 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 34 91 ate normal pair with correlation, -------- 455 1/2 34 91 4587 i.e. , [[--------, 0], [0, ----]] 455 2275 ------------------------------------------------- Theorem Number, 52, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 3, 3], nor the composition, [5, 1, 2] Then infinity ----- \ n ) a(n) x = / ----- n = 0 16 14 10 9 8 7 6 5 4 2 x - x + x - x + x + 2 x - 2 x - x + 2 x - 3 x + 3 x - 1 -------------------------------------------------------------------- 8 6 5 4 3 2 (x - x + x + x - x + 2 x - 1) (-1 + x) and in Maple format (x^16-x^14+x^10-x^9+x^8+2*x^7-2*x^6-x^5+2*x^4-3*x^2+3*x-1)/(x^8-x^6+x^5+x^4-x^3 +2*x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.66664854364473939498 1.9158008597433206753 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 3, 3] and d occurrences (as containment) of the composition, [5, 1, 2], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 16 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 16 2 16 2 14 16 16 14 - 2 X1 X2 x - X2 x - X1 X2 x + X1 x + 2 X2 x + 2 X1 X2 x 2 14 16 2 12 14 14 2 11 + X2 x - x - X1 X2 x - X1 x - 2 X2 x + X1 X2 x 12 14 11 12 10 11 9 + 2 X1 X2 x + x - 2 X1 X2 x - X1 x - X1 X2 x + X1 x + X1 X2 x 10 10 8 9 9 10 7 8 + X1 x + X2 x - X1 X2 x - X1 x - X2 x - x - X1 X2 x + X1 x 8 9 6 7 7 8 5 6 + X2 x + x + 2 X1 X2 x + 2 X1 x + X2 x - x - X1 X2 x - 3 X1 x 6 7 6 4 5 3 4 2 / - X2 x - 2 x + 2 x + 2 X1 x + x - X1 x - 2 x + 3 x - 3 x + 1) / / 2 12 2 11 12 11 12 ((-1 + x) (X1 X2 x - X1 X2 x - 2 X1 X2 x + 2 X1 X2 x + X1 x 11 9 8 9 9 8 8 9 - X1 x - X1 X2 x + X1 X2 x + X1 x + X2 x - X1 x - X2 x - x 6 7 8 5 6 6 7 5 6 - 2 X1 X2 x - X2 x + x + X1 X2 x + 3 X1 x + X2 x + x - X1 x - 2 x 4 3 4 3 2 - 2 X1 x + X1 x + 2 x - x - 2 x + 3 x - 1)) and in Maple format (X1*X2^2*x^16-2*X1*X2*x^16-X2^2*x^16-X1*X2^2*x^14+X1*x^16+2*X2*x^16+2*X1*X2*x^ 14+X2^2*x^14-x^16-X1*X2^2*x^12-X1*x^14-2*X2*x^14+X1*X2^2*x^11+2*X1*X2*x^12+x^14 -2*X1*X2*x^11-X1*x^12-X1*X2*x^10+X1*x^11+X1*X2*x^9+X1*x^10+X2*x^10-X1*X2*x^8-X1 *x^9-X2*x^9-x^10-X1*X2*x^7+X1*x^8+X2*x^8+x^9+2*X1*X2*x^6+2*X1*x^7+X2*x^7-x^8-X1 *X2*x^5-3*X1*x^6-X2*x^6-2*x^7+2*x^6+2*X1*x^4+x^5-X1*x^3-2*x^4+3*x^2-3*x+1)/(-1+ x)/(X1*X2^2*x^12-X1*X2^2*x^11-2*X1*X2*x^12+2*X1*X2*x^11+X1*x^12-X1*x^11-X1*X2*x ^9+X1*X2*x^8+X1*x^9+X2*x^9-X1*x^8-X2*x^8-x^9-2*X1*X2*x^6-X2*x^7+x^8+X1*X2*x^5+3 *X1*x^6+X2*x^6+x^7-X1*x^5-2*x^6-2*X1*x^4+X1*x^3+2*x^4-x^3-2*x^2+3*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 3, 3], are n 263 35 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [5, 1, 2], are n 113 57 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 8 35 57 ------------- 1995 and in floating point 0.1791094652 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 8 1995 ate normal pair with correlation, --------- 1995 1/2 8 1995 2123 i.e. , [[---------, 0], [0, ----]] 1995 1995 ------------------------------------------------- Theorem Number, 53, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 3, 3], nor the composition, [5, 2, 1] Then infinity ----- 13 12 8 6 5 4 2 \ n x - x - 2 x + x + x - 2 x + 3 x - 3 x + 1 ) a(n) x = -------------------------------------------------- / 7 5 4 3 2 ----- (x - x - x + x - 2 x + 1) (-1 + x) n = 0 and in Maple format (x^13-x^12-2*x^8+x^6+x^5-2*x^4+3*x^2-3*x+1)/(x^7-x^5-x^4+x^3-2*x+1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.64093877410291800826 1.9235403491125894929 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 3, 3] and d occurrences (as containment) of the composition, [5, 2, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 13 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 13 2 13 12 13 13 12 12 - 2 X1 X2 x - X2 x + X1 X2 x + X1 x + 2 X2 x - X1 x - X2 x 13 12 8 7 8 8 6 7 - x + x + X1 X2 x + X1 X2 x - X1 x - 2 X2 x - 2 X1 X2 x - 2 X1 x 7 8 5 6 6 4 5 3 4 + X2 x + 2 x + X1 X2 x + 3 X1 x - x - 2 X1 x - x + X1 x + 2 x 2 / 2 7 6 7 7 - 3 x + 3 x - 1) / ((-1 + x) (X1 X2 x + X1 X2 x - X1 x - X2 x / 5 6 7 5 4 5 3 4 3 - X1 X2 x - X1 x + x + 2 X1 x + X1 x - x - X1 x - x + x - 2 x + 1 )) and in Maple format -(X1*X2^2*x^13-2*X1*X2*x^13-X2^2*x^13+X1*X2*x^12+X1*x^13+2*X2*x^13-X1*x^12-X2*x ^12-x^13+x^12+X1*X2*x^8+X1*X2*x^7-X1*x^8-2*X2*x^8-2*X1*X2*x^6-2*X1*x^7+X2*x^7+2 *x^8+X1*X2*x^5+3*X1*x^6-x^6-2*X1*x^4-x^5+X1*x^3+2*x^4-3*x^2+3*x-1)/(-1+x)^2/(X1 *X2*x^7+X1*X2*x^6-X1*x^7-X2*x^7-X1*X2*x^5-X1*x^6+x^7+2*X1*x^5+X1*x^4-x^5-X1*x^3 -x^4+x^3-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 3, 3], are n 263 35 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [5, 2, 1], are n 111 57 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 2 35 57 ------------- 285 and in floating point 0.3134415640 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 1/2 2 7 285 ate normal pair with correlation, ------------- 285 1/2 1/2 2 7 285 341 i.e. , [[-------------, 0], [0, ---]] 285 285 ------------------------------------------------- Theorem Number, 54, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 3, 3], nor the composition, [6, 1, 1] Then infinity ----- \ n ) a(n) x = / ----- n = 0 14 13 12 11 10 9 7 6 4 3 2 x + 2 x + 2 x + 2 x + x + x - x - x + x - x - x + 2 x - 1 ------------------------------------------------------------------------- 7 6 5 4 2 2 (x + 2 x + 2 x + x + x + x - 1) (-1 + x) and in Maple format (x^14+2*x^13+2*x^12+2*x^11+x^10+x^9-x^7-x^6+x^4-x^3-x^2+2*x-1)/(x^7+2*x^6+2*x^5 +x^4+x^2+x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.67544294785551085893 1.9131864632213745587 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 3, 3] and d occurrences (as containment) of the composition, [6, 1, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 15 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 14 15 2 15 14 15 2 14 + X1 X2 x - 2 X1 X2 x - X2 x - 2 X1 X2 x + X1 x - X2 x 15 14 14 15 14 11 11 11 + 2 X2 x + X1 x + 2 X2 x - x - x + X1 X2 x - X1 x - X2 x 9 11 8 9 9 7 8 8 + X1 X2 x + x + X1 X2 x - X1 x - X2 x - 2 X1 X2 x - X1 x - X2 x 9 6 7 7 8 5 6 4 5 3 + x + X1 X2 x + X1 x + X2 x + x + X1 x - x - 2 X1 x - x + X1 x 4 2 / 2 8 7 8 + 2 x - 3 x + 3 x - 1) / ((-1 + x) (X1 X2 x + X1 X2 x - X1 x / 8 6 7 7 8 6 7 5 4 5 - X2 x - X1 X2 x - X1 x - X2 x + x + X1 x + x + X1 x + X1 x - x 3 4 3 - X1 x - x + x - 2 x + 1)) and in Maple format -(X1*X2^2*x^15+X1*X2^2*x^14-2*X1*X2*x^15-X2^2*x^15-2*X1*X2*x^14+X1*x^15-X2^2*x^ 14+2*X2*x^15+X1*x^14+2*X2*x^14-x^15-x^14+X1*X2*x^11-X1*x^11-X2*x^11+X1*X2*x^9+x ^11+X1*X2*x^8-X1*x^9-X2*x^9-2*X1*X2*x^7-X1*x^8-X2*x^8+x^9+X1*X2*x^6+X1*x^7+X2*x ^7+x^8+X1*x^5-x^6-2*X1*x^4-x^5+X1*x^3+2*x^4-3*x^2+3*x-1)/(-1+x)^2/(X1*X2*x^8+X1 *X2*x^7-X1*x^8-X2*x^8-X1*X2*x^6-X1*x^7-X2*x^7+x^8+X1*x^6+x^7+X1*x^5+X1*x^4-x^5- X1*x^3-x^4+x^3-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 3, 3], are n 263 35 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [6, 1, 1], are n 49 53 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 4 35 53 ------------- 1855 and in floating point 0.09287269204 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 4 1855 ate normal pair with correlation, --------- 1855 1/2 4 1855 1887 i.e. , [[---------, 0], [0, ----]] 1855 1855 ------------------------------------------------- Theorem Number, 55, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 4, 2], nor the composition, [1, 1, 6] Then infinity ----- 4 2 6 5 \ n (x + x - 2 x + 1) (x + x - 1) ) a(n) x = ------------------------------------- / 6 5 3 2 2 ----- (x + x + x + x + x - 1) (-1 + x) n = 0 and in Maple format (x^4+x^2-2*x+1)*(x^6+x^5-1)/(x^6+x^5+x^3+x^2+x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.68707893079649356212 1.9111834366854762350 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 4, 2] and d occurrences (as containment) of the composition, [1, 1, 6], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 10 9 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x + X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 10 10 8 9 9 10 7 8 - X1 x - X2 x + X1 X2 x - X1 x - X2 x + x - X1 X2 x - X1 x 8 9 6 7 7 8 7 5 6 4 - X2 x + x + X1 X2 x + X1 x + X2 x + x - x - X1 x - x + X1 x 5 4 2 / 7 6 7 + x - x - x + 2 x - 1) / ((-1 + x) (X1 X2 x - X1 X2 x - X1 x / 7 6 7 5 4 5 4 - X2 x + X1 x + x + X1 x - X1 x - x + x - 2 x + 1)) and in Maple format (X1*X2*x^10+X1*X2*x^9-X1*x^10-X2*x^10+X1*X2*x^8-X1*x^9-X2*x^9+x^10-X1*X2*x^7-X1 *x^8-X2*x^8+x^9+X1*X2*x^6+X1*x^7+X2*x^7+x^8-x^7-X1*x^5-x^6+X1*x^4+x^5-x^4-x^2+2 *x-1)/(-1+x)/(X1*X2*x^7-X1*X2*x^6-X1*x^7-X2*x^7+X1*x^6+x^7+X1*x^5-X1*x^4-x^5+x^ 4-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 4, 2], are n 183 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 1, 6], are n 49 53 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 8 3 53 ------------ 477 and in floating point 0.2114804230 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 8 159 ate normal pair with correlation, -------- 477 1/2 8 159 1559 i.e. , [[--------, 0], [0, ----]] 477 1431 ------------------------------------------------- Theorem Number, 56, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 4, 2], nor the composition, [1, 2, 5] Then infinity ----- 10 9 6 5 4 2 \ n x + x - x + x - x - x + 2 x - 1 ) a(n) x = -------------------------------------- / 9 8 5 4 ----- (-1 + x) (x + x - x + x - 2 x + 1) n = 0 and in Maple format (x^10+x^9-x^6+x^5-x^4-x^2+2*x-1)/(-1+x)/(x^9+x^8-x^5+x^4-2*x+1) The asymptotic expression for a(n) is, n 0.67229840142428698322 1.9159714162121856417 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 4, 2] and d occurrences (as containment) of the composition, [1, 2, 5], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 10 9 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x + X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 10 10 9 9 10 7 9 6 7 - X1 x - X2 x - X1 x - X2 x + x + X1 X2 x + x - X1 X2 x - X1 x 5 6 5 6 4 5 4 2 / + X1 X2 x + 2 X1 x - 2 X1 x - x + X1 x + x - x - x + 2 x - 1) / / 9 8 9 9 8 8 9 ((-1 + x) (X1 X2 x + X1 X2 x - X1 x - X2 x - X1 x - X2 x + x 6 8 5 6 5 4 5 4 + X1 X2 x + x - X1 X2 x - X1 x + 2 X1 x - X1 x - x + x - 2 x + 1)) and in Maple format (X1*X2*x^10+X1*X2*x^9-X1*x^10-X2*x^10-X1*x^9-X2*x^9+x^10+X1*X2*x^7+x^9-X1*X2*x^ 6-X1*x^7+X1*X2*x^5+2*X1*x^6-2*X1*x^5-x^6+X1*x^4+x^5-x^4-x^2+2*x-1)/(-1+x)/(X1* X2*x^9+X1*X2*x^8-X1*x^9-X2*x^9-X1*x^8-X2*x^8+x^9+X1*X2*x^6+x^8-X1*X2*x^5-X1*x^6 +2*X1*x^5-X1*x^4-x^5+x^4-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 4, 2], are n 183 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 2, 5], are n 111 57 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 14 19 -------- 171 and in floating point 0.3568689194 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 14 19 ate normal pair with correlation, -------- 171 1/2 14 19 1931 i.e. , [[--------, 0], [0, ----]] 171 1539 ------------------------------------------------- Theorem Number, 57, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 4, 2], nor the composition, [1, 3, 4] Then infinity ----- 8 7 6 4 \ n x + x + x + x - x + 1 ) a(n) x = ----------------------------------------- / 7 6 5 3 2 ----- (-1 + x) (x + x + x + x + x + x - 1) n = 0 and in Maple format (x^8+x^7+x^6+x^4-x+1)/(-1+x)/(x^7+x^6+x^5+x^3+x^2+x-1) The asymptotic expression for a(n) is, n 0.65748289688935283811 1.9221280043552247737 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 4, 2] and d occurrences (as containment) of the composition, [1, 3, 4], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 9 9 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x - X1 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 9 7 9 7 5 6 4 6 5 - X2 x + X1 X2 x + x - X1 x - X1 X2 x + X1 x + X1 X2 x - x + x 4 2 / 8 8 8 6 - x - x + 2 x - 1) / ((-1 + x) (X1 X2 x - X1 x - X2 x + X1 X2 x / 8 5 6 4 5 4 + x + X1 X2 x - X1 x - X1 X2 x - x + x - 2 x + 1)) and in Maple format (X1*X2*x^9-X1*x^9-X2*x^9+X1*X2*x^7+x^9-X1*x^7-X1*X2*x^5+X1*x^6+X1*X2*x^4-x^6+x^ 5-x^4-x^2+2*x-1)/(-1+x)/(X1*X2*x^8-X1*x^8-X2*x^8+X1*X2*x^6+x^8+X1*X2*x^5-X1*x^6 -X1*X2*x^4-x^5+x^4-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 4, 2], are n 183 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 3, 4], are n 135 65 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 22 3 65 ------------- 585 and in floating point 0.5251509079 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 22 195 ate normal pair with correlation, --------- 585 1/2 22 195 2723 i.e. , [[---------, 0], [0, ----]] 585 1755 ------------------------------------------------- Theorem Number, 58, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 4, 2], nor the composition, [1, 6, 1] Then infinity ----- 8 7 6 5 4 2 \ n x - x - x + x - x - x + 2 x - 1 ) a(n) x = ------------------------------------- / 6 5 3 2 2 ----- (x + x + x + x + x - 1) (-1 + x) n = 0 and in Maple format (x^8-x^7-x^6+x^5-x^4-x^2+2*x-1)/(x^6+x^5+x^3+x^2+x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.69790391550484586467 1.9111834366854762350 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 4, 2] and d occurrences (as containment) of the composition, [1, 6, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 8 7 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x - X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 8 8 6 7 7 8 7 5 6 4 - X1 x - X2 x + X1 X2 x + X1 x + X2 x + x - x - X1 x - x + X1 x 5 4 2 / 7 6 7 + x - x - x + 2 x - 1) / ((-1 + x) (X1 X2 x - X1 X2 x - X1 x / 7 6 7 5 4 5 4 - X2 x + X1 x + x + X1 x - X1 x - x + x - 2 x + 1)) and in Maple format (X1*X2*x^8-X1*X2*x^7-X1*x^8-X2*x^8+X1*X2*x^6+X1*x^7+X2*x^7+x^8-x^7-X1*x^5-x^6+ X1*x^4+x^5-x^4-x^2+2*x-1)/(-1+x)/(X1*X2*x^7-X1*X2*x^6-X1*x^7-X2*x^7+X1*x^6+x^7+ X1*x^5-X1*x^4-x^5+x^4-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 4, 2], are n 183 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 6, 1], are n 49 53 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 8 3 53 ------------ 477 and in floating point 0.2114804230 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 8 159 ate normal pair with correlation, -------- 477 1/2 8 159 1559 i.e. , [[--------, 0], [0, ----]] 477 1431 ------------------------------------------------- Theorem Number, 59, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 4, 2], nor the composition, [2, 1, 5] Then infinity ----- \ n ) a(n) x = / ----- n = 0 9 8 7 6 5 4 2 x + x + x - 2 x + x - x - x + 2 x - 1 ----------------------------------------------------------------- 12 11 10 8 7 6 5 4 (-1 + x) (x + 2 x + 2 x - x - x + x - x + x - 2 x + 1) and in Maple format (x^9+x^8+x^7-2*x^6+x^5-x^4-x^2+2*x-1)/(-1+x)/(x^12+2*x^11+2*x^10-x^8-x^7+x^6-x^ 5+x^4-2*x+1) The asymptotic expression for a(n) is, n 0.67448318694762719220 1.9143463083540318048 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 4, 2] and d occurrences (as containment) of the composition, [2, 1, 5], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 14 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 2 13 2 14 2 14 2 13 2 14 + X1 X2 x - 2 X1 X2 x - X1 X2 x - 2 X1 X2 x + X1 x 2 13 14 2 13 13 14 2 11 - X1 X2 x + 2 X1 X2 x + X1 x + 2 X1 X2 x - X1 x + X1 X2 x 13 11 11 9 8 9 9 - X1 x - 2 X1 X2 x + X1 x - X1 X2 x - X1 X2 x + X1 x + X2 x 7 8 8 9 6 7 7 8 5 - X1 X2 x + X1 x + X2 x - x + X1 X2 x + X1 x + X2 x - x - X1 X2 x 6 6 7 5 6 4 5 4 2 - 2 X1 x - X2 x - x + 2 X1 x + 2 x - X1 x - x + x + x - 2 x + 1) / 2 2 15 2 15 2 15 2 2 13 2 15 / (X1 X2 x - 2 X1 X2 x - X1 X2 x - X1 X2 x + X1 x / 15 2 13 2 13 15 2 12 2 12 + 2 X1 X2 x + X1 X2 x + X1 X2 x - X1 x - X1 X2 x + X1 X2 x 2 12 2 11 13 13 11 12 12 + X1 x - X1 X2 x - X1 x - X2 x + 2 X1 X2 x - X1 x - X2 x 13 10 11 12 9 10 10 + x - 2 X1 X2 x - X1 x + x - X1 X2 x + 2 X1 x + 2 X2 x 9 9 10 7 9 6 7 7 + X1 x + X2 x - 2 x + X1 X2 x - x - 2 X1 X2 x - X1 x - 2 X2 x 5 6 6 7 5 6 4 5 4 + X1 X2 x + 3 X1 x + X2 x + 2 x - 3 X1 x - 2 x + X1 x + 2 x - x 2 - 2 x + 3 x - 1) and in Maple format -(X1^2*X2^2*x^14+X1^2*X2^2*x^13-2*X1^2*X2*x^14-X1*X2^2*x^14-2*X1^2*X2*x^13+X1^2 *x^14-X1*X2^2*x^13+2*X1*X2*x^14+X1^2*x^13+2*X1*X2*x^13-X1*x^14+X1*X2^2*x^11-X1* x^13-2*X1*X2*x^11+X1*x^11-X1*X2*x^9-X1*X2*x^8+X1*x^9+X2*x^9-X1*X2*x^7+X1*x^8+X2 *x^8-x^9+X1*X2*x^6+X1*x^7+X2*x^7-x^8-X1*X2*x^5-2*X1*x^6-X2*x^6-x^7+2*X1*x^5+2*x ^6-X1*x^4-x^5+x^4+x^2-2*x+1)/(X1^2*X2^2*x^15-2*X1^2*X2*x^15-X1*X2^2*x^15-X1^2* X2^2*x^13+X1^2*x^15+2*X1*X2*x^15+X1^2*X2*x^13+X1*X2^2*x^13-X1*x^15-X1^2*X2*x^12 +X1*X2^2*x^12+X1^2*x^12-X1*X2^2*x^11-X1*x^13-X2*x^13+2*X1*X2*x^11-X1*x^12-X2*x^ 12+x^13-2*X1*X2*x^10-X1*x^11+x^12-X1*X2*x^9+2*X1*x^10+2*X2*x^10+X1*x^9+X2*x^9-2 *x^10+X1*X2*x^7-x^9-2*X1*X2*x^6-X1*x^7-2*X2*x^7+X1*X2*x^5+3*X1*x^6+X2*x^6+2*x^7 -3*X1*x^5-2*x^6+X1*x^4+2*x^5-x^4-2*x^2+3*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 4, 2], are n 183 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 1, 5], are n 113 57 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 4 19 ------- 57 and in floating point 0.3058876452 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 4 19 ate normal pair with correlation, ------- 57 1/2 4 19 203 i.e. , [[-------, 0], [0, ---]] 57 171 ------------------------------------------------- Theorem Number, 60, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 4, 2], nor the composition, [2, 2, 4] Then infinity ----- 9 8 6 5 4 2 \ n x + x - x + x - x - x + 2 x - 1 ) a(n) x = ----------------------------------------------- / 11 10 9 5 4 ----- (-1 + x) (x + 2 x + x - x + x - 2 x + 1) n = 0 and in Maple format (x^9+x^8-x^6+x^5-x^4-x^2+2*x-1)/(-1+x)/(x^11+2*x^10+x^9-x^5+x^4-2*x+1) The asymptotic expression for a(n) is, n 0.65906522740498320917 1.9197449317290998989 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 4, 2] and d occurrences (as containment) of the composition, [2, 2, 4], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 9 8 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x + X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 9 9 8 8 9 6 8 5 4 - X1 x - X2 x - X1 x - X2 x + x + X1 X2 x + x - X1 X2 x + X1 X2 x 6 5 4 2 / 2 12 2 11 2 12 - x + x - x - x + 2 x - 1) / (X1 X2 x + X1 X2 x - X1 x / 12 2 11 11 12 12 10 - 2 X1 X2 x - X1 x - 2 X1 X2 x + 2 X1 x + X2 x + X1 X2 x 11 11 12 9 10 10 11 9 + 2 X1 x + X2 x - x + X1 X2 x - X1 x - X2 x - x - X1 x 9 10 7 9 6 7 5 4 - X2 x + x + X1 X2 x + x - X1 X2 x - X1 x + 2 X1 X2 x - X1 X2 x 6 5 4 2 + x - 2 x + x + 2 x - 3 x + 1) and in Maple format -(X1*X2*x^9+X1*X2*x^8-X1*x^9-X2*x^9-X1*x^8-X2*x^8+x^9+X1*X2*x^6+x^8-X1*X2*x^5+ X1*X2*x^4-x^6+x^5-x^4-x^2+2*x-1)/(X1^2*X2*x^12+X1^2*X2*x^11-X1^2*x^12-2*X1*X2*x ^12-X1^2*x^11-2*X1*X2*x^11+2*X1*x^12+X2*x^12+X1*X2*x^10+2*X1*x^11+X2*x^11-x^12+ X1*X2*x^9-X1*x^10-X2*x^10-x^11-X1*x^9-X2*x^9+x^10+X1*X2*x^7+x^9-X1*X2*x^6-X1*x^ 7+2*X1*X2*x^5-X1*X2*x^4+x^6-2*x^5+x^4+2*x^2-3*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 4, 2], are n 183 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 2, 4], are n 75 69 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 20 23 -------- 207 and in floating point 0.4633653646 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 20 23 ate normal pair with correlation, -------- 207 1/2 20 23 2663 i.e. , [[--------, 0], [0, ----]] 207 1863 ------------------------------------------------- Theorem Number, 61, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 4, 2], nor the composition, [2, 3, 3] Then infinity ----- 7 6 4 3 2 \ n x - 2 x + x - x - x + 2 x - 1 ) a(n) x = ------------------------------------------------- / 10 7 6 5 4 3 ----- (-1 + x) (x - x + x - x - x + x - 2 x + 1) n = 0 and in Maple format (x^7-2*x^6+x^4-x^3-x^2+2*x-1)/(-1+x)/(x^10-x^7+x^6-x^5-x^4+x^3-2*x+1) The asymptotic expression for a(n) is, n 0.65532458744249045709 1.9223246520768555496 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 4, 2] and d occurrences (as containment) of the composition, [2, 3, 3], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 7 7 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x - X1 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 7 5 6 6 7 4 5 6 - X2 x - X1 X2 x + X1 x + X2 x + x + X1 X2 x + X2 x - 2 x 4 3 4 3 2 / 2 11 2 11 - 2 X2 x + X2 x + x - x - x + 2 x - 1) / (X1 X2 x - X1 x / 11 10 11 11 10 10 11 - 2 X1 X2 x + X1 X2 x + 2 X1 x + X2 x - X1 x - X2 x - x 8 10 7 8 8 7 7 8 + X1 X2 x + x - X1 X2 x - X1 x - X2 x + X1 x + 2 X2 x + x 5 6 6 7 4 5 6 4 + 2 X1 X2 x - X1 x - X2 x - 2 x - X1 X2 x - 2 X2 x + 2 x + 3 X2 x 3 4 3 2 - X2 x - 2 x + x + 2 x - 3 x + 1) and in Maple format -(X1*X2*x^7-X1*x^7-X2*x^7-X1*X2*x^5+X1*x^6+X2*x^6+x^7+X1*X2*x^4+X2*x^5-2*x^6-2* X2*x^4+X2*x^3+x^4-x^3-x^2+2*x-1)/(X1^2*X2*x^11-X1^2*x^11-2*X1*X2*x^11+X1*X2*x^ 10+2*X1*x^11+X2*x^11-X1*x^10-X2*x^10-x^11+X1*X2*x^8+x^10-X1*X2*x^7-X1*x^8-X2*x^ 8+X1*x^7+2*X2*x^7+x^8+2*X1*X2*x^5-X1*x^6-X2*x^6-2*x^7-X1*X2*x^4-2*X2*x^5+2*x^6+ 3*X2*x^4-X2*x^3-2*x^4+x^3+2*x^2-3*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 4, 2], are n 183 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 3, 3], are n 97 85 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 4 3 85 ------------ 153 and in floating point 0.4174828609 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 1/2 4 5 51 ate normal pair with correlation, ------------ 153 1/2 1/2 4 5 51 619 i.e. , [[------------, 0], [0, ---]] 153 459 ------------------------------------------------- Theorem Number, 62, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 4, 2], nor the composition, [2, 5, 1] Then infinity ----- \ n ) a(n) x = / ----- n = 0 12 11 10 7 6 5 4 3 2 x - x + x + x - 2 x + 2 x - x + x - 3 x + 3 x - 1 - ------------------------------------------------------------- 9 8 5 4 2 (x + x - x + x - 2 x + 1) (-1 + x) and in Maple format -(x^12-x^11+x^10+x^7-2*x^6+2*x^5-x^4+x^3-3*x^2+3*x-1)/(x^9+x^8-x^5+x^4-2*x+1)/( -1+x)^2 The asymptotic expression for a(n) is, n 0.67714921393955080398 1.9159714162121856417 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 4, 2] and d occurrences (as containment) of the composition, [2, 5, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 12 11 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x - X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 12 12 10 11 11 12 10 10 - X1 x - X2 x + X1 X2 x + X1 x + X2 x + x - X1 x - X2 x 11 8 10 7 8 6 7 - x - X1 X2 x + x + 2 X1 X2 x + X1 x - 2 X1 X2 x - 3 X1 x 5 6 7 5 6 4 5 4 3 2 + X1 X2 x + 4 X1 x + x - 3 X1 x - 2 x + X1 x + 2 x - x + x - 3 x / 2 9 8 9 9 8 + 3 x - 1) / ((-1 + x) (X1 X2 x + X1 X2 x - X1 x - X2 x - X1 x / 8 9 6 8 5 6 5 4 5 - X2 x + x + X1 X2 x + x - X1 X2 x - X1 x + 2 X1 x - X1 x - x 4 + x - 2 x + 1)) and in Maple format -(X1*X2*x^12-X1*X2*x^11-X1*x^12-X2*x^12+X1*X2*x^10+X1*x^11+X2*x^11+x^12-X1*x^10 -X2*x^10-x^11-X1*X2*x^8+x^10+2*X1*X2*x^7+X1*x^8-2*X1*X2*x^6-3*X1*x^7+X1*X2*x^5+ 4*X1*x^6+x^7-3*X1*x^5-2*x^6+X1*x^4+2*x^5-x^4+x^3-3*x^2+3*x-1)/(-1+x)^2/(X1*X2*x ^9+X1*X2*x^8-X1*x^9-X2*x^9-X1*x^8-X2*x^8+x^9+X1*X2*x^6+x^8-X1*X2*x^5-X1*x^6+2* X1*x^5-X1*x^4-x^5+x^4-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 4, 2], are n 183 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 5, 1], are n 111 57 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 14 19 -------- 171 and in floating point 0.3568689194 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 14 19 ate normal pair with correlation, -------- 171 1/2 14 19 1931 i.e. , [[--------, 0], [0, ----]] 171 1539 ------------------------------------------------- Theorem Number, 63, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 4, 2], nor the composition, [3, 1, 4] Then infinity ----- \ n ) a(n) x = / ----- n = 0 13 12 11 10 8 4 2 x + x + x - x - x + x + x - 2 x + 1 - ----------------------------------------------------------- 10 9 8 7 6 3 2 2 (x + 2 x + 3 x + 2 x + x + x + x + x - 1) (-1 + x) and in Maple format -(x^13+x^12+x^11-x^10-x^8+x^4+x^2-2*x+1)/(x^10+2*x^9+3*x^8+2*x^7+x^6+x^3+x^2+x-\ 1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.66396839102705721566 1.9175371779461324874 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 4, 2] and d occurrences (as containment) of the composition, [3, 1, 4], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 16 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 2 15 2 16 2 16 2 15 2 16 + X1 X2 x - 2 X1 X2 x - X1 X2 x - 2 X1 X2 x + X1 x 2 15 16 2 14 2 15 15 16 - X1 X2 x + 2 X1 X2 x - X1 X2 x + X1 x + 2 X1 X2 x - X1 x 2 2 12 2 14 14 15 2 2 11 - X1 X2 x + X1 x + 2 X1 X2 x - X1 x + X1 X2 x 2 12 2 12 14 14 2 2 10 2 11 + 2 X1 X2 x + X1 X2 x - 2 X1 x - X2 x - X1 X2 x - X1 X2 x 2 12 12 14 2 10 2 10 11 - X1 x - 2 X1 X2 x + x + X1 X2 x + X1 X2 x - 3 X1 X2 x 12 2 9 11 11 9 10 10 + X1 x - X1 X2 x + 3 X1 x + 2 X2 x + X1 X2 x - X1 x - X2 x 11 8 9 10 7 8 8 9 - 2 x + X1 X2 x + X2 x + x - X1 X2 x - X1 x - X2 x - x 6 7 8 5 6 4 5 5 + 2 X1 X2 x + X2 x + x - 2 X1 X2 x - 2 X2 x + X1 X2 x + X2 x + x 4 3 2 / 2 2 14 2 2 13 - x + x - 3 x + 3 x - 1) / ((-1 + x) (X1 X2 x + X1 X2 x / 2 14 2 14 2 13 2 14 2 13 - 2 X1 X2 x - X1 X2 x - 2 X1 X2 x + X1 x - X1 X2 x 14 2 2 11 2 12 2 13 2 12 + 2 X1 X2 x + X1 X2 x - X1 X2 x + X1 x + X1 X2 x 13 14 2 2 10 2 11 2 12 2 11 + 2 X1 X2 x - X1 x - X1 X2 x - 2 X1 X2 x + X1 x + X1 X2 x 13 2 10 2 11 2 10 11 12 - X1 x + X1 X2 x + X1 x + X1 X2 x - 2 X1 X2 x - X1 x 12 2 9 10 11 12 9 9 - X2 x - X1 X2 x - X1 X2 x + X1 x + x + X1 x + 2 X2 x 7 9 6 7 5 6 4 - X1 X2 x - 2 x + X1 X2 x + X1 x - 2 X1 X2 x - 2 X2 x + X1 X2 x 5 6 5 4 2 + X2 x + x + x - x - 2 x + 3 x - 1)) and in Maple format -(X1^2*X2^2*x^16+X1^2*X2^2*x^15-2*X1^2*X2*x^16-X1*X2^2*x^16-2*X1^2*X2*x^15+X1^2 *x^16-X1*X2^2*x^15+2*X1*X2*x^16-X1^2*X2*x^14+X1^2*x^15+2*X1*X2*x^15-X1*x^16-X1^ 2*X2^2*x^12+X1^2*x^14+2*X1*X2*x^14-X1*x^15+X1^2*X2^2*x^11+2*X1^2*X2*x^12+X1*X2^ 2*x^12-2*X1*x^14-X2*x^14-X1^2*X2^2*x^10-X1^2*X2*x^11-X1^2*x^12-2*X1*X2*x^12+x^ 14+X1^2*X2*x^10+X1*X2^2*x^10-3*X1*X2*x^11+X1*x^12-X1*X2^2*x^9+3*X1*x^11+2*X2*x^ 11+X1*X2*x^9-X1*x^10-X2*x^10-2*x^11+X1*X2*x^8+X2*x^9+x^10-X1*X2*x^7-X1*x^8-X2*x ^8-x^9+2*X1*X2*x^6+X2*x^7+x^8-2*X1*X2*x^5-2*X2*x^6+X1*X2*x^4+X2*x^5+x^5-x^4+x^3 -3*x^2+3*x-1)/(-1+x)/(X1^2*X2^2*x^14+X1^2*X2^2*x^13-2*X1^2*X2*x^14-X1*X2^2*x^14 -2*X1^2*X2*x^13+X1^2*x^14-X1*X2^2*x^13+2*X1*X2*x^14+X1^2*X2^2*x^11-X1^2*X2*x^12 +X1^2*x^13+X1*X2^2*x^12+2*X1*X2*x^13-X1*x^14-X1^2*X2^2*x^10-2*X1^2*X2*x^11+X1^2 *x^12+X1*X2^2*x^11-X1*x^13+X1^2*X2*x^10+X1^2*x^11+X1*X2^2*x^10-2*X1*X2*x^11-X1* x^12-X2*x^12-X1*X2^2*x^9-X1*X2*x^10+X1*x^11+x^12+X1*x^9+2*X2*x^9-X1*X2*x^7-2*x^ 9+X1*X2*x^6+X1*x^7-2*X1*X2*x^5-2*X2*x^6+X1*X2*x^4+X2*x^5+x^6+x^5-x^4-2*x^2+3*x-\ 1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 4, 2], are n 183 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 1, 4], are n 141 65 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 16 3 65 ------------- 585 and in floating point 0.3819279330 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 16 195 ate normal pair with correlation, --------- 585 1/2 16 195 2267 i.e. , [[---------, 0], [0, ----]] 585 1755 ------------------------------------------------- Theorem Number, 64, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 4, 2], nor the composition, [3, 2, 3] Then infinity ----- 14 11 9 8 5 4 3 2 \ n x + x - 2 x + x + x - x + x + x - 2 x + 1 ) a(n) x = - --------------------------------------------------- / 8 4 3 2 3 ----- (-1 + x) (x - x - x - x - x + 1) (x - x + 1) n = 0 and in Maple format -(x^14+x^11-2*x^9+x^8+x^5-x^4+x^3+x^2-2*x+1)/(-1+x)/(x^8-x^4-x^3-x^2-x+1)/(x^3- x+1) The asymptotic expression for a(n) is, n 0.65153311498131888405 1.9217220658969757404 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 4, 2] and d occurrences (as containment) of the composition, [3, 2, 3], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 15 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 15 2 15 2 2 13 2 15 2 14 - 2 X1 X2 x - 2 X1 X2 x - X1 X2 x + X1 x + X1 X2 x 15 2 15 2 13 2 13 14 + 4 X1 X2 x + X2 x + X1 X2 x + 2 X1 X2 x - 2 X1 X2 x 15 2 14 15 2 2 11 2 12 13 - 2 X1 x - X2 x - 2 X2 x + X1 X2 x - X1 X2 x - 2 X1 X2 x 14 2 13 14 15 2 2 10 2 11 + X1 x - X2 x + 2 X2 x + x - X1 X2 x - X1 X2 x 2 11 12 2 12 13 14 2 10 - 2 X1 X2 x + 2 X1 X2 x + X2 x + X2 x - x + X1 X2 x 2 10 11 12 2 11 12 2 9 + 4 X1 X2 x + X1 X2 x - X1 x + X2 x - 2 X2 x - 2 X1 X2 x 10 11 2 10 12 9 10 - 6 X1 X2 x + X1 x - 3 X2 x + x + 4 X1 X2 x + 2 X1 x 2 9 10 11 9 2 8 9 10 + 3 X2 x + 5 X2 x - x - 2 X1 x - X2 x - 6 X2 x - 2 x 7 8 9 6 7 8 5 - X1 X2 x + 2 X2 x + 3 x + 2 X1 X2 x + X2 x - x - 2 X1 X2 x 6 4 5 6 4 5 3 4 2 - 3 X2 x + X1 X2 x + 4 X2 x + x - 3 X2 x - 2 x + X2 x + 2 x - 3 x / 2 2 12 2 2 11 2 12 + 3 x - 1) / ((-1 + x) (X1 X2 x + X1 X2 x - 2 X1 X2 x / 2 12 2 2 10 2 11 2 12 12 - 2 X1 X2 x - X1 X2 x - 2 X1 X2 x + X1 x + 4 X1 X2 x 2 12 2 10 2 11 2 10 12 2 11 + X2 x + X1 X2 x + X1 x + 4 X1 X2 x - 2 X1 x - X2 x 12 2 9 10 2 10 11 12 - 2 X2 x - 2 X1 X2 x - 6 X1 X2 x - 2 X2 x + 2 X2 x + x 9 10 2 9 10 11 8 9 + 3 X1 X2 x + 2 X1 x + 3 X2 x + 3 X2 x - x - X1 X2 x - X1 x 2 8 9 10 7 8 8 9 6 - X2 x - 5 X2 x - x - X1 X2 x + X1 x + 3 X2 x + 2 x + X1 X2 x 7 8 5 6 4 5 6 4 + X1 x - 2 x - 2 X1 X2 x - 2 X2 x + X1 X2 x + 3 X2 x + x - 3 X2 x 5 3 4 3 2 - x + X2 x + 2 x - x - 2 x + 3 x - 1)) and in Maple format -(X1^2*X2^2*x^15-2*X1^2*X2*x^15-2*X1*X2^2*x^15-X1^2*X2^2*x^13+X1^2*x^15+X1*X2^2 *x^14+4*X1*X2*x^15+X2^2*x^15+X1^2*X2*x^13+2*X1*X2^2*x^13-2*X1*X2*x^14-2*X1*x^15 -X2^2*x^14-2*X2*x^15+X1^2*X2^2*x^11-X1*X2^2*x^12-2*X1*X2*x^13+X1*x^14-X2^2*x^13 +2*X2*x^14+x^15-X1^2*X2^2*x^10-X1^2*X2*x^11-2*X1*X2^2*x^11+2*X1*X2*x^12+X2^2*x^ 12+X2*x^13-x^14+X1^2*X2*x^10+4*X1*X2^2*x^10+X1*X2*x^11-X1*x^12+X2^2*x^11-2*X2*x ^12-2*X1*X2^2*x^9-6*X1*X2*x^10+X1*x^11-3*X2^2*x^10+x^12+4*X1*X2*x^9+2*X1*x^10+3 *X2^2*x^9+5*X2*x^10-x^11-2*X1*x^9-X2^2*x^8-6*X2*x^9-2*x^10-X1*X2*x^7+2*X2*x^8+3 *x^9+2*X1*X2*x^6+X2*x^7-x^8-2*X1*X2*x^5-3*X2*x^6+X1*X2*x^4+4*X2*x^5+x^6-3*X2*x^ 4-2*x^5+X2*x^3+2*x^4-3*x^2+3*x-1)/(-1+x)/(X1^2*X2^2*x^12+X1^2*X2^2*x^11-2*X1^2* X2*x^12-2*X1*X2^2*x^12-X1^2*X2^2*x^10-2*X1^2*X2*x^11+X1^2*x^12+4*X1*X2*x^12+X2^ 2*x^12+X1^2*X2*x^10+X1^2*x^11+4*X1*X2^2*x^10-2*X1*x^12-X2^2*x^11-2*X2*x^12-2*X1 *X2^2*x^9-6*X1*X2*x^10-2*X2^2*x^10+2*X2*x^11+x^12+3*X1*X2*x^9+2*X1*x^10+3*X2^2* x^9+3*X2*x^10-x^11-X1*X2*x^8-X1*x^9-X2^2*x^8-5*X2*x^9-x^10-X1*X2*x^7+X1*x^8+3* X2*x^8+2*x^9+X1*X2*x^6+X1*x^7-2*x^8-2*X1*X2*x^5-2*X2*x^6+X1*X2*x^4+3*X2*x^5+x^6 -3*X2*x^4-x^5+X2*x^3+2*x^4-x^3-2*x^2+3*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 4, 2], are n 183 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 2, 3], are n 89 77 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 20 3 77 ------------- 693 and in floating point 0.4386344635 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 20 231 ate normal pair with correlation, --------- 693 1/2 20 231 2879 i.e. , [[---------, 0], [0, ----]] 693 2079 ------------------------------------------------- Theorem Number, 65, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 4, 2], nor the composition, [3, 3, 2] Then infinity ----- 7 6 4 3 2 \ n x - 2 x + x - x - x + 2 x - 1 ) a(n) x = - ------------------------------------------- / 7 6 5 4 3 ----- (-1 + x) (x - x + x + x - x + 2 x - 1) n = 0 and in Maple format -(x^7-2*x^6+x^4-x^3-x^2+2*x-1)/(-1+x)/(x^7-x^6+x^5+x^4-x^3+2*x-1) The asymptotic expression for a(n) is, n 0.64732625001158940098 1.9249743097683141877 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 4, 2] and d occurrences (as containment) of the composition, [3, 3, 2], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 7 7 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x - X1 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 7 5 6 6 7 4 5 6 - X2 x - X1 X2 x + X1 x + X2 x + x + X1 X2 x + X2 x - 2 x 4 3 4 3 2 / 8 7 - 2 X2 x + X2 x + x - x - x + 2 x - 1) / (X1 X2 x - X1 X2 x / 8 8 7 7 8 5 6 6 7 - X1 x - X2 x + X1 x + 2 X2 x + x + 2 X1 X2 x - X1 x - X2 x - 2 x 4 5 6 4 3 4 3 2 - X1 X2 x - 2 X2 x + 2 x + 3 X2 x - X2 x - 2 x + x + 2 x - 3 x + 1 ) and in Maple format -(X1*X2*x^7-X1*x^7-X2*x^7-X1*X2*x^5+X1*x^6+X2*x^6+x^7+X1*X2*x^4+X2*x^5-2*x^6-2* X2*x^4+X2*x^3+x^4-x^3-x^2+2*x-1)/(X1*X2*x^8-X1*X2*x^7-X1*x^8-X2*x^8+X1*x^7+2*X2 *x^7+x^8+2*X1*X2*x^5-X1*x^6-X2*x^6-2*x^7-X1*X2*x^4-2*X2*x^5+2*x^6+3*X2*x^4-X2*x ^3-2*x^4+x^3+2*x^2-3*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 4, 2], are n 183 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 3, 2], are n 97 85 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 8 3 85 ------------ 255 and in floating point 0.5009794331 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 8 255 ate normal pair with correlation, -------- 255 1/2 8 255 383 i.e. , [[--------, 0], [0, ---]] 255 255 ------------------------------------------------- Theorem Number, 66, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 4, 2], nor the composition, [3, 4, 1] Then infinity ----- \ n ) a(n) x = / ----- n = 0 12 10 9 7 6 5 4 3 2 x - x + x + x - 2 x + 2 x - x + x - 3 x + 3 x - 1 - ------------------------------------------------------------ 7 6 5 3 2 3 (x + x + x + x + x + x - 1) (-1 + x) and in Maple format -(x^12-x^10+x^9+x^7-2*x^6+2*x^5-x^4+x^3-3*x^2+3*x-1)/(x^7+x^6+x^5+x^3+x^2+x-1)/ (-1+x)^3 The asymptotic expression for a(n) is, n 0.65557427343396184814 1.9221280043552247737 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 4, 2] and d occurrences (as containment) of the composition, [3, 4, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 12 12 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x - X1 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 12 10 12 9 10 10 8 9 - X2 x - X1 X2 x + x + X1 X2 x + X1 x + X2 x - X1 X2 x - X1 x 9 10 7 8 9 6 7 5 - X2 x - x + X1 X2 x + X1 x + x + X1 X2 x - 2 X1 x - 2 X1 X2 x 6 7 4 6 5 4 3 2 / + X1 x + x + X1 X2 x - 2 x + 2 x - x + x - 3 x + 3 x - 1) / ( / 2 8 8 8 6 8 5 6 (-1 + x) (X1 X2 x - X1 x - X2 x + X1 X2 x + x + X1 X2 x - X1 x 4 5 4 - X1 X2 x - x + x - 2 x + 1)) and in Maple format -(X1*X2*x^12-X1*x^12-X2*x^12-X1*X2*x^10+x^12+X1*X2*x^9+X1*x^10+X2*x^10-X1*X2*x^ 8-X1*x^9-X2*x^9-x^10+X1*X2*x^7+X1*x^8+x^9+X1*X2*x^6-2*X1*x^7-2*X1*X2*x^5+X1*x^6 +x^7+X1*X2*x^4-2*x^6+2*x^5-x^4+x^3-3*x^2+3*x-1)/(-1+x)^2/(X1*X2*x^8-X1*x^8-X2*x ^8+X1*X2*x^6+x^8+X1*X2*x^5-X1*x^6-X1*X2*x^4-x^5+x^4-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 4, 2], are n 183 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 4, 1], are n 135 65 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 22 3 65 ------------- 585 and in floating point 0.5251509079 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 22 195 ate normal pair with correlation, --------- 585 1/2 22 195 2723 i.e. , [[---------, 0], [0, ----]] 585 1755 ------------------------------------------------- Theorem Number, 67, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 4, 2], nor the composition, [4, 1, 3] Then infinity ----- 13 12 11 10 9 4 2 \ n x + x + x - x - x + x + x - 2 x + 1 ) a(n) x = ---------------------------------------------- / 7 6 4 ----- (-1 + x) (x + x - x + 2 x - 1) n = 0 and in Maple format (x^13+x^12+x^11-x^10-x^9+x^4+x^2-2*x+1)/(-1+x)/(x^7+x^6-x^4+2*x-1) The asymptotic expression for a(n) is, n 0.66581546041682687250 1.9168163613635124211 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 4, 2] and d occurrences (as containment) of the composition, [4, 1, 3], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 16 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 2 15 2 16 2 16 2 15 2 16 + X1 X2 x - 2 X1 X2 x - X1 X2 x - 2 X1 X2 x + X1 x 2 15 16 2 2 13 2 14 2 15 - X1 X2 x + 2 X1 X2 x - X1 X2 x - X1 X2 x + X1 x 15 16 2 13 2 14 2 13 14 + 2 X1 X2 x - X1 x + 2 X1 X2 x + X1 x + X1 X2 x + 2 X1 X2 x 15 2 2 11 2 13 13 14 14 - X1 x + X1 X2 x - X1 x - 2 X1 X2 x - 2 X1 x - X2 x 2 2 10 2 11 13 14 2 10 2 10 - X1 X2 x - X1 X2 x + X1 x + x + X1 X2 x + X1 X2 x 11 2 9 10 11 11 9 - 3 X1 X2 x - X1 X2 x - X1 X2 x + 3 X1 x + 2 X2 x + 3 X1 X2 x 11 9 9 7 9 6 7 - 2 x - 2 X1 x - X2 x - X1 X2 x + x + 2 X1 X2 x + X2 x 5 6 4 5 5 4 3 2 - 2 X1 X2 x - 2 X2 x + X1 X2 x + X2 x + x - x + x - 3 x + 3 x - 1) / 2 2 12 2 2 11 2 12 2 2 10 / ((-1 + x) (X1 X2 x + X1 X2 x - 2 X1 X2 x - X1 X2 x / 2 11 2 12 2 11 2 10 2 11 2 10 - 2 X1 X2 x + X1 x + X1 X2 x + X1 X2 x + X1 x + X1 X2 x 11 2 9 10 11 9 8 - 2 X1 X2 x - X1 X2 x - X1 X2 x + X1 x + 2 X1 X2 x - X1 X2 x 9 7 8 8 6 7 8 5 - X1 x - X1 X2 x + X1 x + X2 x + X1 X2 x + X1 x - x - 2 X1 X2 x 6 4 5 6 5 4 2 - 2 X2 x + X1 X2 x + X2 x + x + x - x - 2 x + 3 x - 1)) and in Maple format -(X1^2*X2^2*x^16+X1^2*X2^2*x^15-2*X1^2*X2*x^16-X1*X2^2*x^16-2*X1^2*X2*x^15+X1^2 *x^16-X1*X2^2*x^15+2*X1*X2*x^16-X1^2*X2^2*x^13-X1^2*X2*x^14+X1^2*x^15+2*X1*X2*x ^15-X1*x^16+2*X1^2*X2*x^13+X1^2*x^14+X1*X2^2*x^13+2*X1*X2*x^14-X1*x^15+X1^2*X2^ 2*x^11-X1^2*x^13-2*X1*X2*x^13-2*X1*x^14-X2*x^14-X1^2*X2^2*x^10-X1^2*X2*x^11+X1* x^13+x^14+X1^2*X2*x^10+X1*X2^2*x^10-3*X1*X2*x^11-X1*X2^2*x^9-X1*X2*x^10+3*X1*x^ 11+2*X2*x^11+3*X1*X2*x^9-2*x^11-2*X1*x^9-X2*x^9-X1*X2*x^7+x^9+2*X1*X2*x^6+X2*x^ 7-2*X1*X2*x^5-2*X2*x^6+X1*X2*x^4+X2*x^5+x^5-x^4+x^3-3*x^2+3*x-1)/(-1+x)/(X1^2* X2^2*x^12+X1^2*X2^2*x^11-2*X1^2*X2*x^12-X1^2*X2^2*x^10-2*X1^2*X2*x^11+X1^2*x^12 +X1*X2^2*x^11+X1^2*X2*x^10+X1^2*x^11+X1*X2^2*x^10-2*X1*X2*x^11-X1*X2^2*x^9-X1* X2*x^10+X1*x^11+2*X1*X2*x^9-X1*X2*x^8-X1*x^9-X1*X2*x^7+X1*x^8+X2*x^8+X1*X2*x^6+ X1*x^7-x^8-2*X1*X2*x^5-2*X2*x^6+X1*X2*x^4+X2*x^5+x^6+x^5-x^4-2*x^2+3*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 4, 2], are n 183 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 1, 3], are n 141 65 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 16 3 65 ------------- 585 and in floating point 0.3819279330 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 16 195 ate normal pair with correlation, --------- 585 1/2 16 195 2267 i.e. , [[---------, 0], [0, ----]] 585 1755 ------------------------------------------------- Theorem Number, 68, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 4, 2], nor the composition, [4, 2, 2] Then infinity ----- 9 8 6 5 4 2 \ n x + x - x + x - x - x + 2 x - 1 ) a(n) x = - ------------------------------------- / 5 4 ----- (-1 + x) (x - x + 2 x - 1) n = 0 and in Maple format -(x^9+x^8-x^6+x^5-x^4-x^2+2*x-1)/(-1+x)/(x^5-x^4+2*x-1) The asymptotic expression for a(n) is, n 0.61915988369516142914 1.9331849818995204468 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 4, 2] and d occurrences (as containment) of the composition, [4, 2, 2], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 9 8 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x + X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 9 9 8 8 9 6 8 5 4 - X1 x - X2 x - X1 x - X2 x + x + X1 X2 x + x - X1 X2 x + X1 X2 x 6 5 4 2 / 7 6 7 - x + x - x - x + 2 x - 1) / (X1 X2 x - X1 X2 x - X1 x / 5 4 6 5 4 2 + 2 X1 X2 x - X1 X2 x + x - 2 x + x + 2 x - 3 x + 1) and in Maple format -(X1*X2*x^9+X1*X2*x^8-X1*x^9-X2*x^9-X1*x^8-X2*x^8+x^9+X1*X2*x^6+x^8-X1*X2*x^5+ X1*X2*x^4-x^6+x^5-x^4-x^2+2*x-1)/(X1*X2*x^7-X1*X2*x^6-X1*x^7+2*X1*X2*x^5-X1*X2* x^4+x^6-2*x^5+x^4+2*x^2-3*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 4, 2], are n 183 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 2, 2], are n 75 69 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 32 23 -------- 207 and in floating point 0.7413845834 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 32 23 ate normal pair with correlation, -------- 207 1/2 32 23 3911 i.e. , [[--------, 0], [0, ----]] 207 1863 ------------------------------------------------- Theorem Number, 69, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 4, 2], nor the composition, [4, 3, 1] Then infinity ----- \ n ) a(n) x = / ----- n = 0 3 2 9 8 6 5 4 3 2 (x + x - 1) (x - x + x - x + 2 x - 3 x + 4 x - 3 x + 1) ---------------------------------------------------------------- 5 4 2 (x - x + 2 x - 1) (-1 + x) and in Maple format (x^3+x^2-1)*(x^9-x^8+x^6-x^5+2*x^4-3*x^3+4*x^2-3*x+1)/(x^5-x^4+2*x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.61747222077046425935 1.9331849818995204468 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 4, 2] and d occurrences (as containment) of the composition, [4, 3, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 12 12 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x - X1 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 12 10 12 10 10 10 7 8 - X2 x - X1 X2 x + x + X1 x + X2 x - x + X1 X2 x - X2 x 6 7 8 5 6 7 4 6 + X1 X2 x - 2 X1 x + x - 2 X1 X2 x + X1 x + x + X1 X2 x - 2 x 5 4 3 2 / 2 + 2 x - x + x - 3 x + 3 x - 1) / ((-1 + x) / 6 5 6 4 5 4 (X1 X2 x + X1 X2 x - X1 x - X1 X2 x - x + x - 2 x + 1)) and in Maple format -(X1*X2*x^12-X1*x^12-X2*x^12-X1*X2*x^10+x^12+X1*x^10+X2*x^10-x^10+X1*X2*x^7-X2* x^8+X1*X2*x^6-2*X1*x^7+x^8-2*X1*X2*x^5+X1*x^6+x^7+X1*X2*x^4-2*x^6+2*x^5-x^4+x^3 -3*x^2+3*x-1)/(-1+x)^2/(X1*X2*x^6+X1*X2*x^5-X1*x^6-X1*X2*x^4-x^5+x^4-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 4, 2], are n 183 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 3, 1], are n 135 65 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 2 3 65 ------------ 39 and in floating point 0.7161148743 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 1/2 2 5 39 ate normal pair with correlation, ------------ 39 1/2 1/2 2 5 39 79 i.e. , [[------------, 0], [0, --]] 39 39 ------------------------------------------------- Theorem Number, 70, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 4, 2], nor the composition, [5, 1, 2] Then infinity ----- 10 9 7 6 5 4 2 \ n x + x + x - 2 x + x - x - x + 2 x - 1 ) a(n) x = - --------------------------------------------- / 7 6 5 4 ----- (-1 + x) (x - x + x - x + 2 x - 1) n = 0 and in Maple format -(x^10+x^9+x^7-2*x^6+x^5-x^4-x^2+2*x-1)/(-1+x)/(x^7-x^6+x^5-x^4+2*x-1) The asymptotic expression for a(n) is, n 0.67750027158103274656 1.9132221246804735080 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 4, 2] and d occurrences (as containment) of the composition, [5, 1, 2], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 11 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 11 10 11 9 10 10 9 - 2 X1 X2 x - X1 X2 x + X1 x - X1 X2 x + X1 x + X2 x + X1 x 9 10 7 9 6 7 7 5 + X2 x - x - X1 X2 x - x + X1 X2 x + X1 x + X2 x - X1 X2 x 6 6 7 5 6 4 5 4 2 - 2 X1 x - X2 x - x + 2 X1 x + 2 x - X1 x - x + x + x - 2 x + 1) / 2 12 2 11 12 11 12 11 / (X1 X2 x - X1 X2 x - 2 X1 X2 x + 2 X1 X2 x + X1 x - X1 x / 8 7 8 8 6 7 7 8 - X1 X2 x + X1 X2 x + X1 x + X2 x - 2 X1 X2 x - X1 x - 2 X2 x - x 5 6 6 7 5 6 4 5 4 + X1 X2 x + 3 X1 x + X2 x + 2 x - 3 X1 x - 2 x + X1 x + 2 x - x 2 - 2 x + 3 x - 1) and in Maple format -(X1*X2^2*x^11-2*X1*X2*x^11-X1*X2*x^10+X1*x^11-X1*X2*x^9+X1*x^10+X2*x^10+X1*x^9 +X2*x^9-x^10-X1*X2*x^7-x^9+X1*X2*x^6+X1*x^7+X2*x^7-X1*X2*x^5-2*X1*x^6-X2*x^6-x^ 7+2*X1*x^5+2*x^6-X1*x^4-x^5+x^4+x^2-2*x+1)/(X1*X2^2*x^12-X1*X2^2*x^11-2*X1*X2*x ^12+2*X1*X2*x^11+X1*x^12-X1*x^11-X1*X2*x^8+X1*X2*x^7+X1*x^8+X2*x^8-2*X1*X2*x^6- X1*x^7-2*X2*x^7-x^8+X1*X2*x^5+3*X1*x^6+X2*x^6+2*x^7-3*X1*x^5-2*x^6+X1*x^4+2*x^5 -x^4-2*x^2+3*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 4, 2], are n 183 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [5, 1, 2], are n 113 57 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 4 19 ------- 57 and in floating point 0.3058876452 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 4 19 ate normal pair with correlation, ------- 57 1/2 4 19 203 i.e. , [[-------, 0], [0, ---]] 57 171 ------------------------------------------------- Theorem Number, 71, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 4, 2], nor the composition, [5, 2, 1] Then infinity ----- 12 11 8 7 6 5 4 3 2 \ n x - x + x + x - 2 x + 2 x - x + x - 3 x + 3 x - 1 ) a(n) x = ------------------------------------------------------------ / 5 4 2 ----- (x - x + 2 x - 1) (-1 + x) n = 0 and in Maple format (x^12-x^11+x^8+x^7-2*x^6+2*x^5-x^4+x^3-3*x^2+3*x-1)/(x^5-x^4+2*x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.61442764446410597890 1.9331849818995204468 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 4, 2] and d occurrences (as containment) of the composition, [5, 2, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 12 11 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x - X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 12 12 11 11 12 11 7 8 - X1 x - X2 x + X1 x + X2 x + x - x + 2 X1 X2 x - X2 x 6 7 8 5 6 7 5 6 - 2 X1 X2 x - 3 X1 x + x + X1 X2 x + 4 X1 x + x - 3 X1 x - 2 x 4 5 4 3 2 / 2 + X1 x + 2 x - x + x - 3 x + 3 x - 1) / ((-1 + x) / 6 5 6 5 4 5 4 (X1 X2 x - X1 X2 x - X1 x + 2 X1 x - X1 x - x + x - 2 x + 1)) and in Maple format -(X1*X2*x^12-X1*X2*x^11-X1*x^12-X2*x^12+X1*x^11+X2*x^11+x^12-x^11+2*X1*X2*x^7- X2*x^8-2*X1*X2*x^6-3*X1*x^7+x^8+X1*X2*x^5+4*X1*x^6+x^7-3*X1*x^5-2*x^6+X1*x^4+2* x^5-x^4+x^3-3*x^2+3*x-1)/(-1+x)^2/(X1*X2*x^6-X1*X2*x^5-X1*x^6+2*X1*x^5-X1*x^4-x ^5+x^4-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 4, 2], are n 183 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [5, 2, 1], are n 111 57 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 26 19 -------- 171 and in floating point 0.6627565645 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 26 19 ate normal pair with correlation, -------- 171 1/2 26 19 2891 i.e. , [[--------, 0], [0, ----]] 171 1539 ------------------------------------------------- Theorem Number, 72, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 4, 2], nor the composition, [6, 1, 1] Then infinity ----- \ n ) a(n) x = / ----- n = 0 2 11 10 9 8 5 4 2 (x - x + 1) (x + 2 x + 2 x + x - x - 2 x + x + x - 1) --------------------------------------------------------------- 6 5 3 2 2 (x + x + x + x + x - 1) (-1 + x) and in Maple format (x^2-x+1)*(x^11+2*x^10+2*x^9+x^8-x^5-2*x^4+x^2+x-1)/(x^6+x^5+x^3+x^2+x-1)/(-1+x )^2 The asymptotic expression for a(n) is, n 0.68358264996735542565 1.9111834366854762350 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 4, 2] and d occurrences (as containment) of the composition, [6, 1, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 14 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 14 2 14 14 14 14 8 - 2 X1 X2 x - X2 x + X1 x + 2 X2 x - x + 2 X1 X2 x 7 8 8 6 7 7 8 6 - 2 X1 X2 x - 2 X1 x - 2 X2 x + X1 X2 x + X1 x + X2 x + 2 x + X1 x 5 6 4 5 4 3 2 / 2 - 2 X1 x - 2 x + X1 x + 2 x - x + x - 3 x + 3 x - 1) / ((-1 + x) / 7 6 7 7 6 7 5 4 5 4 (X1 X2 x - X1 X2 x - X1 x - X2 x + X1 x + x + X1 x - X1 x - x + x - 2 x + 1)) and in Maple format -(X1*X2^2*x^14-2*X1*X2*x^14-X2^2*x^14+X1*x^14+2*X2*x^14-x^14+2*X1*X2*x^8-2*X1* X2*x^7-2*X1*x^8-2*X2*x^8+X1*X2*x^6+X1*x^7+X2*x^7+2*x^8+X1*x^6-2*X1*x^5-2*x^6+X1 *x^4+2*x^5-x^4+x^3-3*x^2+3*x-1)/(-1+x)^2/(X1*X2*x^7-X1*X2*x^6-X1*x^7-X2*x^7+X1* x^6+x^7+X1*x^5-X1*x^4-x^5+x^4-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 4, 2], are n 183 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [6, 1, 1], are n 49 53 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 8 3 53 ------------ 477 and in floating point 0.2114804230 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 8 159 ate normal pair with correlation, -------- 477 1/2 8 159 1559 i.e. , [[--------, 0], [0, ----]] 477 1431 ------------------------------------------------- Theorem Number, 73, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 5, 1], nor the composition, [1, 1, 6] Then infinity ----- 11 10 9 8 5 2 \ n x + x + x + x - x - x + 2 x - 1 ) a(n) x = --------------------------------------- / 4 3 2 2 ----- (x + x + x + x - 1) (-1 + x) n = 0 and in Maple format (x^11+x^10+x^9+x^8-x^5-x^2+2*x-1)/(x^4+x^3+x^2+x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.63351683346155297547 1.9275619754829253043 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 5, 1] and d occurrences (as containment) of the composition, [1, 1, 6], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 11 10 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x + X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 11 11 9 10 10 11 8 9 - X1 x - X2 x + X1 X2 x - X1 x - X2 x + x + X1 X2 x - X1 x 9 10 8 8 9 6 8 6 5 5 - X2 x + x - X1 x - X2 x + x + X1 X2 x + x - X1 x + X1 x - x 2 / 6 6 5 5 - x + 2 x - 1) / ((-1 + x) (X1 X2 x - X1 x + X1 x - x + 2 x - 1)) / and in Maple format -(X1*X2*x^11+X1*X2*x^10-X1*x^11-X2*x^11+X1*X2*x^9-X1*x^10-X2*x^10+x^11+X1*X2*x^ 8-X1*x^9-X2*x^9+x^10-X1*x^8-X2*x^8+x^9+X1*X2*x^6+x^8-X1*x^6+X1*x^5-x^5-x^2+2*x-\ 1)/(-1+x)/(X1*X2*x^6-X1*x^6+X1*x^5-x^5+2*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 5, 1], are n 139 23 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 1, 6], are n 49 53 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 22 23 53 -------------- 1219 and in floating point 0.6301164650 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 22 1219 ate normal pair with correlation, ---------- 1219 1/2 22 1219 2187 i.e. , [[----------, 0], [0, ----]] 1219 1219 ------------------------------------------------- Theorem Number, 74, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 5, 1], nor the composition, [1, 2, 5] Then infinity ----- 10 9 8 5 2 \ n x + x + x - x - x + 2 x - 1 ) a(n) x = --------------------------------- / 4 3 2 2 ----- (x + x + x + x - 1) (-1 + x) n = 0 and in Maple format (x^10+x^9+x^8-x^5-x^2+2*x-1)/(x^4+x^3+x^2+x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.63530875266095142903 1.9275619754829253043 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 5, 1] and d occurrences (as containment) of the composition, [1, 2, 5], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 10 9 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x + X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 10 10 8 9 9 10 7 8 - X1 x - X2 x + X1 X2 x - X1 x - X2 x + x + X1 X2 x - X1 x 8 9 6 7 8 5 6 5 2 - X2 x + x - X1 X2 x - X1 x + x + X1 X2 x + X1 x - x - x + 2 x / 6 5 6 5 - 1) / ((-1 + x) (X1 X2 x - X1 X2 x - X1 x + x - 2 x + 1)) / and in Maple format (X1*X2*x^10+X1*X2*x^9-X1*x^10-X2*x^10+X1*X2*x^8-X1*x^9-X2*x^9+x^10+X1*X2*x^7-X1 *x^8-X2*x^8+x^9-X1*X2*x^6-X1*x^7+x^8+X1*X2*x^5+X1*x^6-x^5-x^2+2*x-1)/(-1+x)/(X1 *X2*x^6-X1*X2*x^5-X1*x^6+x^5-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 5, 1], are n 139 23 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 2, 5], are n 111 57 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 8 23 57 ------------- 437 and in floating point 0.6628418121 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 1/2 8 3 437 ate normal pair with correlation, ------------- 437 1/2 1/2 8 3 437 821 i.e. , [[-------------, 0], [0, ---]] 437 437 ------------------------------------------------- Theorem Number, 75, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 5, 1], nor the composition, [1, 3, 4] Then infinity ----- 8 7 6 4 \ n x + 2 x + x + x - x + 1 ) a(n) x = ------------------------------------ / 6 5 3 2 ----- (-1 + x) (x + x + x + x + x - 1) n = 0 and in Maple format (x^8+2*x^7+x^6+x^4-x+1)/(-1+x)/(x^6+x^5+x^3+x^2+x-1) The asymptotic expression for a(n) is, n 0.69079734461128122568 1.9111834366854762350 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 5, 1] and d occurrences (as containment) of the composition, [1, 3, 4], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 9 8 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x + X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 9 9 8 8 9 6 7 8 5 - X1 x - X2 x - X1 x - X2 x + x - X1 X2 x + X2 x + x + X1 X2 x 6 6 7 5 6 4 5 4 2 / + X1 x + X2 x - x - 2 X2 x - x + X2 x + x - x - x + 2 x - 1) / / 7 6 7 7 5 6 7 ((-1 + x) (X1 X2 x + X1 X2 x - X1 x - X2 x - X1 X2 x - X1 x + x 5 4 5 4 + 2 X2 x - X2 x - x + x - 2 x + 1)) and in Maple format (X1*X2*x^9+X1*X2*x^8-X1*x^9-X2*x^9-X1*x^8-X2*x^8+x^9-X1*X2*x^6+X2*x^7+x^8+X1*X2 *x^5+X1*x^6+X2*x^6-x^7-2*X2*x^5-x^6+X2*x^4+x^5-x^4-x^2+2*x-1)/(-1+x)/(X1*X2*x^7 +X1*X2*x^6-X1*x^7-X2*x^7-X1*X2*x^5-X1*x^6+x^7+2*X2*x^5-X2*x^4-x^5+x^4-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 5, 1], are n 139 23 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 3, 4], are n 135 65 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 12 23 65 -------------- 1495 and in floating point 0.3103563600 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 12 1495 ate normal pair with correlation, ---------- 1495 1/2 12 1495 1783 i.e. , [[----------, 0], [0, ----]] 1495 1495 ------------------------------------------------- Theorem Number, 76, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 5, 1], nor the composition, [1, 4, 3] Then infinity ----- 8 7 6 5 4 2 \ n x - x - x + x - x - x + 2 x - 1 ) a(n) x = ------------------------------------- / 6 5 3 2 2 ----- (x + x + x + x + x - 1) (-1 + x) n = 0 and in Maple format (x^8-x^7-x^6+x^5-x^4-x^2+2*x-1)/(x^6+x^5+x^3+x^2+x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.69790391550484586467 1.9111834366854762350 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 5, 1] and d occurrences (as containment) of the composition, [1, 4, 3], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 8 8 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x - X1 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 8 6 7 8 5 6 6 7 5 - X2 x - X1 X2 x + X2 x + x + X1 X2 x + X1 x + X2 x - x - 2 X2 x 6 4 5 4 2 / 7 6 - x + X2 x + x - x - x + 2 x - 1) / ((-1 + x) (X1 X2 x + X1 X2 x / 7 7 5 6 7 5 4 5 4 - X1 x - X2 x - X1 X2 x - X1 x + x + 2 X2 x - X2 x - x + x - 2 x + 1)) and in Maple format (X1*X2*x^8-X1*x^8-X2*x^8-X1*X2*x^6+X2*x^7+x^8+X1*X2*x^5+X1*x^6+X2*x^6-x^7-2*X2* x^5-x^6+X2*x^4+x^5-x^4-x^2+2*x-1)/(-1+x)/(X1*X2*x^7+X1*X2*x^6-X1*x^7-X2*x^7-X1* X2*x^5-X1*x^6+x^7+2*X2*x^5-X2*x^4-x^5+x^4-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 5, 1], are n 139 23 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 4, 3], are n 135 65 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 12 23 65 -------------- 1495 and in floating point 0.3103563600 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 12 1495 ate normal pair with correlation, ---------- 1495 1/2 12 1495 1783 i.e. , [[----------, 0], [0, ----]] 1495 1495 ------------------------------------------------- Theorem Number, 77, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 5, 1], nor the composition, [2, 1, 5] Then infinity ----- \ n ) a(n) x = / ----- n = 0 15 14 13 12 11 10 9 8 5 2 x + x + x + x - x - x - x - x + x + x - 2 x + 1 - --------------------------------------------------------------- 4 3 2 2 (x + x + x + x - 1) (-1 + x) and in Maple format -(x^15+x^14+x^13+x^12-x^11-x^10-x^9-x^8+x^5+x^2-2*x+1)/(x^4+x^3+x^2+x-1)/(-1+x) ^2 The asymptotic expression for a(n) is, n 0.63530875266095142903 1.9275619754829253043 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 5, 1] and d occurrences (as containment) of the composition, [2, 1, 5], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 17 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 17 2 17 2 16 2 17 17 2 16 - 2 X1 X2 x - X1 X2 x - X1 X2 x + X1 x + 2 X1 X2 x + X1 x 16 17 2 2 13 16 16 2 2 12 + 2 X1 X2 x - X1 x - X1 X2 x - 2 X1 x - X2 x + X1 X2 x 2 13 2 13 16 2 2 11 2 12 2 13 + 2 X1 X2 x + X1 X2 x + x - X1 X2 x - X1 X2 x - X1 x 13 2 11 12 13 11 12 - 2 X1 X2 x + X1 X2 x - 3 X1 X2 x + X1 x + X1 X2 x + 3 X1 x 12 11 12 8 8 8 6 8 + 2 X2 x - X1 x - 2 x + X1 X2 x - X1 x - X2 x - X1 X2 x + x 5 6 5 3 2 / + X1 X2 x + x - x + x - 3 x + 3 x - 1) / ((-1 + x) / 6 5 6 5 6 6 (X1 X2 x - X1 X2 x - X1 x + x - 2 x + 1) (X1 X2 x - X1 x + x - 1)) and in Maple format -(X1^2*X2^2*x^17-2*X1^2*X2*x^17-X1*X2^2*x^17-X1^2*X2*x^16+X1^2*x^17+2*X1*X2*x^ 17+X1^2*x^16+2*X1*X2*x^16-X1*x^17-X1^2*X2^2*x^13-2*X1*x^16-X2*x^16+X1^2*X2^2*x^ 12+2*X1^2*X2*x^13+X1*X2^2*x^13+x^16-X1^2*X2^2*x^11-X1^2*X2*x^12-X1^2*x^13-2*X1* X2*x^13+X1^2*X2*x^11-3*X1*X2*x^12+X1*x^13+X1*X2*x^11+3*X1*x^12+2*X2*x^12-X1*x^ 11-2*x^12+X1*X2*x^8-X1*x^8-X2*x^8-X1*X2*x^6+x^8+X1*X2*x^5+x^6-x^5+x^3-3*x^2+3*x -1)/(-1+x)/(X1*X2*x^6-X1*X2*x^5-X1*x^6+x^5-2*x+1)/(X1*X2*x^6-X1*x^6+x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 5, 1], are n 139 23 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 1, 5], are n 113 57 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 8 23 57 ------------- 437 and in floating point 0.6628418121 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 1/2 8 3 437 ate normal pair with correlation, ------------- 437 1/2 1/2 8 3 437 821 i.e. , [[-------------, 0], [0, ---]] 437 437 ------------------------------------------------- Theorem Number, 78, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 5, 1], nor the composition, [2, 2, 4] Then infinity ----- \ n ) a(n) x = / ----- n = 0 14 13 12 11 9 7 6 5 4 2 x + x + x - x - x - x + 2 x - x + x + x - 2 x + 1 - --------------------------------------------------------------- 9 8 7 5 3 2 2 (x + x + x + x + x + x + x - 1) (-1 + x) and in Maple format -(x^14+x^13+x^12-x^11-x^9-x^7+2*x^6-x^5+x^4+x^2-2*x+1)/(x^9+x^8+x^7+x^5+x^3+x^2 +x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.69239526746995424995 1.9099139227361088013 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 5, 1] and d occurrences (as containment) of the composition, [2, 2, 4], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 15 2 15 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x - X1 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 15 15 15 2 12 15 2 11 - 2 X1 X2 x + 2 X1 x + X2 x - X1 X2 x - x + X1 X2 x 2 12 12 2 11 11 12 12 + X1 x + 3 X1 X2 x - X1 x - 2 X1 X2 x - 3 X1 x - 2 X2 x 10 11 11 12 9 10 10 11 + X1 X2 x + 2 X1 x + X2 x + 2 x - X1 X2 x - X1 x - X2 x - x 8 9 9 10 7 8 8 9 + X1 X2 x + X1 x + X2 x + x - 2 X1 X2 x - X1 x - X2 x - x 6 7 7 8 5 6 6 7 + 2 X1 X2 x + 2 X1 x + 3 X2 x + x - X1 X2 x - X1 x - 4 X2 x - 3 x 5 6 4 5 4 3 2 / + 3 X2 x + 3 x - X2 x - 2 x + x - x + 3 x - 3 x + 1) / ((-1 + x) / 2 11 2 11 11 10 11 11 (X1 X2 x - X1 x - 2 X1 X2 x + X1 X2 x + 2 X1 x + X2 x 10 10 11 8 10 7 8 8 - X1 x - X2 x - x + X1 X2 x + x - X1 X2 x - X1 x - X2 x 6 7 7 8 5 6 6 7 + 2 X1 X2 x + X1 x + 2 X2 x + x - X1 X2 x - X1 x - 3 X2 x - 2 x 5 6 4 5 4 2 + 3 X2 x + 2 x - X2 x - 2 x + x + 2 x - 3 x + 1)) and in Maple format -(X1^2*X2*x^15-X1^2*x^15-2*X1*X2*x^15+2*X1*x^15+X2*x^15-X1^2*X2*x^12-x^15+X1^2* X2*x^11+X1^2*x^12+3*X1*X2*x^12-X1^2*x^11-2*X1*X2*x^11-3*X1*x^12-2*X2*x^12+X1*X2 *x^10+2*X1*x^11+X2*x^11+2*x^12-X1*X2*x^9-X1*x^10-X2*x^10-x^11+X1*X2*x^8+X1*x^9+ X2*x^9+x^10-2*X1*X2*x^7-X1*x^8-X2*x^8-x^9+2*X1*X2*x^6+2*X1*x^7+3*X2*x^7+x^8-X1* X2*x^5-X1*x^6-4*X2*x^6-3*x^7+3*X2*x^5+3*x^6-X2*x^4-2*x^5+x^4-x^3+3*x^2-3*x+1)/( -1+x)/(X1^2*X2*x^11-X1^2*x^11-2*X1*X2*x^11+X1*X2*x^10+2*X1*x^11+X2*x^11-X1*x^10 -X2*x^10-x^11+X1*X2*x^8+x^10-X1*X2*x^7-X1*x^8-X2*x^8+2*X1*X2*x^6+X1*x^7+2*X2*x^ 7+x^8-X1*X2*x^5-X1*x^6-3*X2*x^6-2*x^7+3*X2*x^5+2*x^6-X2*x^4-2*x^5+x^4+2*x^2-3*x +1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 5, 1], are n 139 23 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 2, 4], are n 75 69 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 10 3 ------- 69 and in floating point 0.2510218562 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 10 3 ate normal pair with correlation, ------- 69 1/2 10 3 1787 i.e. , [[-------, 0], [0, ----]] 69 1587 ------------------------------------------------- Theorem Number, 79, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 5, 1], nor the composition, [2, 3, 3] Then infinity ----- 13 12 11 9 8 6 4 3 2 \ n x + x - x + x - x + 2 x - x + x + x - 2 x + 1 ) a(n) x = - --------------------------------------------------------- / 9 8 7 6 5 4 2 2 ----- (x + 2 x + x + x + 2 x + x + x + x - 1) (-1 + x) n = 0 and in Maple format -(x^13+x^12-x^11+x^9-x^8+2*x^6-x^4+x^3+x^2-2*x+1)/(x^9+2*x^8+x^7+x^6+2*x^5+x^4+ x^2+x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.70464553543661030161 1.9073680513412163549 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 5, 1] and d occurrences (as containment) of the composition, [2, 3, 3], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 14 2 14 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x - X1 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 14 2 12 14 14 2 11 2 12 - 2 X1 X2 x - X1 X2 x + 2 X1 x + X2 x + X1 X2 x + X1 x 12 14 2 11 11 12 12 + 3 X1 X2 x - x - X1 x - 2 X1 X2 x - 3 X1 x - 2 X2 x 10 11 11 12 9 10 10 - X1 X2 x + 2 X1 x + X2 x + 2 x + 2 X1 X2 x + X1 x + X2 x 11 8 9 9 10 7 8 8 - x - X1 X2 x - 2 X1 x - 2 X2 x - x - X1 X2 x + X1 x + X2 x 9 6 7 7 8 5 6 6 + 2 x + 2 X1 X2 x + X1 x + 2 X2 x - x - X1 X2 x - X1 x - 3 X2 x 7 6 4 5 3 4 2 / - 2 x + 2 x + 2 X2 x + x - X2 x - 2 x + 3 x - 3 x + 1) / ( / 2 11 2 11 11 11 11 (-1 + x) (X1 X2 x - X1 x - 2 X1 X2 x + 2 X1 x + X2 x 9 11 8 9 9 8 8 9 + 2 X1 X2 x - x - X1 X2 x - 2 X1 x - 2 X2 x + X1 x + X2 x + 2 x 6 7 8 5 6 6 7 5 6 + 2 X1 X2 x + X2 x - x - X1 X2 x - X1 x - 3 X2 x - x + X2 x + 2 x 4 3 4 3 2 + 2 X2 x - X2 x - 2 x + x + 2 x - 3 x + 1)) and in Maple format -(X1^2*X2*x^14-X1^2*x^14-2*X1*X2*x^14-X1^2*X2*x^12+2*X1*x^14+X2*x^14+X1^2*X2*x^ 11+X1^2*x^12+3*X1*X2*x^12-x^14-X1^2*x^11-2*X1*X2*x^11-3*X1*x^12-2*X2*x^12-X1*X2 *x^10+2*X1*x^11+X2*x^11+2*x^12+2*X1*X2*x^9+X1*x^10+X2*x^10-x^11-X1*X2*x^8-2*X1* x^9-2*X2*x^9-x^10-X1*X2*x^7+X1*x^8+X2*x^8+2*x^9+2*X1*X2*x^6+X1*x^7+2*X2*x^7-x^8 -X1*X2*x^5-X1*x^6-3*X2*x^6-2*x^7+2*x^6+2*X2*x^4+x^5-X2*x^3-2*x^4+3*x^2-3*x+1)/( -1+x)/(X1^2*X2*x^11-X1^2*x^11-2*X1*X2*x^11+2*X1*x^11+X2*x^11+2*X1*X2*x^9-x^11- X1*X2*x^8-2*X1*x^9-2*X2*x^9+X1*x^8+X2*x^8+2*x^9+2*X1*X2*x^6+X2*x^7-x^8-X1*X2*x^ 5-X1*x^6-3*X2*x^6-x^7+X2*x^5+2*x^6+2*X2*x^4-X2*x^3-2*x^4+x^3+2*x^2-3*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 5, 1], are n 139 23 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 3, 3], are n 97 85 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 6 23 85 ------------- 1955 and in floating point 0.1356993819 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 6 1955 ate normal pair with correlation, --------- 1955 1/2 6 1955 2027 i.e. , [[---------, 0], [0, ----]] 1955 1955 ------------------------------------------------- Theorem Number, 80, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 5, 1], nor the composition, [2, 4, 2] Then infinity ----- 12 11 10 7 6 5 4 2 \ n x - x + x - x + 2 x - x + x + x - 2 x + 1 ) a(n) x = - ---------------------------------------------------- / 9 8 7 5 3 2 2 ----- (x + x + x + x + x + x + x - 1) (-1 + x) n = 0 and in Maple format -(x^12-x^11+x^10-x^7+2*x^6-x^5+x^4+x^2-2*x+1)/(x^9+x^8+x^7+x^5+x^3+x^2+x-1)/(-1 +x)^2 The asymptotic expression for a(n) is, n 0.70233752396058589616 1.9099139227361088013 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 5, 1] and d occurrences (as containment) of the composition, [2, 4, 2], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 13 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 12 2 13 13 2 11 2 12 12 - X1 X2 x - X1 x - 2 X1 X2 x + X1 X2 x + X1 x + 3 X1 X2 x 13 13 2 11 11 12 12 13 + 2 X1 x + X2 x - X1 x - 3 X1 X2 x - 3 X1 x - 2 X2 x - x 10 11 11 12 10 10 11 + X1 X2 x + 3 X1 x + 2 X2 x + 2 x - X1 x - X2 x - 2 x 8 10 7 8 8 6 7 + X1 X2 x + x - 2 X1 X2 x - X1 x - X2 x + 2 X1 X2 x + 2 X1 x 7 8 5 6 6 7 5 6 + 3 X2 x + x - X1 X2 x - X1 x - 4 X2 x - 3 x + 3 X2 x + 3 x 4 5 4 3 2 / 2 11 - X2 x - 2 x + x - x + 3 x - 3 x + 1) / ((-1 + x) (X1 X2 x / 2 11 11 10 11 11 10 10 - X1 x - 2 X1 X2 x + X1 X2 x + 2 X1 x + X2 x - X1 x - X2 x 11 8 10 7 8 8 6 7 - x + X1 X2 x + x - X1 X2 x - X1 x - X2 x + 2 X1 X2 x + X1 x 7 8 5 6 6 7 5 6 + 2 X2 x + x - X1 X2 x - X1 x - 3 X2 x - 2 x + 3 X2 x + 2 x 4 5 4 2 - X2 x - 2 x + x + 2 x - 3 x + 1)) and in Maple format -(X1^2*X2*x^13-X1^2*X2*x^12-X1^2*x^13-2*X1*X2*x^13+X1^2*X2*x^11+X1^2*x^12+3*X1* X2*x^12+2*X1*x^13+X2*x^13-X1^2*x^11-3*X1*X2*x^11-3*X1*x^12-2*X2*x^12-x^13+X1*X2 *x^10+3*X1*x^11+2*X2*x^11+2*x^12-X1*x^10-X2*x^10-2*x^11+X1*X2*x^8+x^10-2*X1*X2* x^7-X1*x^8-X2*x^8+2*X1*X2*x^6+2*X1*x^7+3*X2*x^7+x^8-X1*X2*x^5-X1*x^6-4*X2*x^6-3 *x^7+3*X2*x^5+3*x^6-X2*x^4-2*x^5+x^4-x^3+3*x^2-3*x+1)/(-1+x)/(X1^2*X2*x^11-X1^2 *x^11-2*X1*X2*x^11+X1*X2*x^10+2*X1*x^11+X2*x^11-X1*x^10-X2*x^10-x^11+X1*X2*x^8+ x^10-X1*X2*x^7-X1*x^8-X2*x^8+2*X1*X2*x^6+X1*x^7+2*X2*x^7+x^8-X1*X2*x^5-X1*x^6-3 *X2*x^6-2*x^7+3*X2*x^5+2*x^6-X2*x^4-2*x^5+x^4+2*x^2-3*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 5, 1], are n 139 23 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 4, 2], are n 75 69 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 10 3 ------- 69 and in floating point 0.2510218562 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 10 3 ate normal pair with correlation, ------- 69 1/2 10 3 1787 i.e. , [[-------, 0], [0, ----]] 69 1587 ------------------------------------------------- Theorem Number, 81, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 5, 1], nor the composition, [3, 1, 4] Then infinity ----- \ n ) a(n) x = / ----- n = 0 17 16 15 14 13 12 11 10 4 2 x + 2 x + 2 x + x + x + 2 x - x - x + x + x - 2 x + 1 - ----------------------------------------------------------------------- 6 5 3 2 6 5 2 (x + x + x + x + x - 1) (x + x + 1) (-1 + x) and in Maple format -(x^17+2*x^16+2*x^15+x^14+x^13+2*x^12-x^11-x^10+x^4+x^2-2*x+1)/(x^6+x^5+x^3+x^2 +x-1)/(x^6+x^5+1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.68733873783470785714 1.9111834366854762350 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 5, 1] and d occurrences (as containment) of the composition, [3, 1, 4], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 18 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 2 17 2 18 2 18 2 2 16 2 17 + 2 X1 X2 x - 2 X1 X2 x - 2 X1 X2 x + X1 X2 x - 4 X1 X2 x 2 18 2 17 18 2 18 2 2 15 + X1 x - 3 X1 X2 x + 4 X1 X2 x + X2 x - X1 X2 x 2 16 2 17 17 18 2 17 18 - 3 X1 X2 x + 2 X1 x + 6 X1 X2 x - 2 X1 x + X2 x - 2 X2 x 2 2 14 2 15 2 16 2 15 16 - X1 X2 x + X1 X2 x + 2 X1 x + 3 X1 X2 x + 2 X1 X2 x 17 2 16 17 18 2 2 13 2 14 - 3 X1 x - X2 x - 2 X2 x + x + X1 X2 x + 2 X1 X2 x 2 14 15 16 2 15 16 17 + X1 X2 x - 4 X1 X2 x - 2 X1 x - 2 X2 x + X2 x + x 2 13 2 14 2 13 14 15 15 - X1 X2 x - X1 x - 3 X1 X2 x - 2 X1 X2 x + X1 x + 3 X2 x 2 2 11 2 12 13 14 2 13 15 - X1 X2 x + 2 X1 X2 x + 4 X1 X2 x + X1 x + 2 X2 x - x 2 11 2 11 12 13 2 12 13 + X1 X2 x + 2 X1 X2 x - 5 X1 X2 x - X1 x - 2 X2 x - 3 X2 x 2 10 11 12 12 13 10 - 2 X1 X2 x - 3 X1 X2 x + 3 X1 x + 5 X2 x + x + 3 X1 X2 x 11 2 10 12 9 10 2 9 10 + X1 x + 2 X2 x - 3 x + X1 X2 x - X1 x - X2 x - 3 X2 x 9 9 10 6 5 6 5 4 5 - X1 x + X2 x + x - X1 X2 x + X1 X2 x + X2 x - 2 X2 x + X2 x + x 4 3 2 / - x + x - 3 x + 3 x - 1) / ((-1 + x) / 7 6 7 7 6 7 5 5 (X1 X2 x + X1 X2 x - X1 x - X2 x - X1 x + x + X2 x - x + x - 1) ( 7 6 7 7 5 6 7 5 X1 X2 x + X1 X2 x - X1 x - X2 x - X1 X2 x - X1 x + x + 2 X2 x 4 5 4 - X2 x - x + x - 2 x + 1)) and in Maple format -(X1^2*X2^2*x^18+2*X1^2*X2^2*x^17-2*X1^2*X2*x^18-2*X1*X2^2*x^18+X1^2*X2^2*x^16-\ 4*X1^2*X2*x^17+X1^2*x^18-3*X1*X2^2*x^17+4*X1*X2*x^18+X2^2*x^18-X1^2*X2^2*x^15-3 *X1^2*X2*x^16+2*X1^2*x^17+6*X1*X2*x^17-2*X1*x^18+X2^2*x^17-2*X2*x^18-X1^2*X2^2* x^14+X1^2*X2*x^15+2*X1^2*x^16+3*X1*X2^2*x^15+2*X1*X2*x^16-3*X1*x^17-X2^2*x^16-2 *X2*x^17+x^18+X1^2*X2^2*x^13+2*X1^2*X2*x^14+X1*X2^2*x^14-4*X1*X2*x^15-2*X1*x^16 -2*X2^2*x^15+X2*x^16+x^17-X1^2*X2*x^13-X1^2*x^14-3*X1*X2^2*x^13-2*X1*X2*x^14+X1 *x^15+3*X2*x^15-X1^2*X2^2*x^11+2*X1*X2^2*x^12+4*X1*X2*x^13+X1*x^14+2*X2^2*x^13- x^15+X1^2*X2*x^11+2*X1*X2^2*x^11-5*X1*X2*x^12-X1*x^13-2*X2^2*x^12-3*X2*x^13-2* X1*X2^2*x^10-3*X1*X2*x^11+3*X1*x^12+5*X2*x^12+x^13+3*X1*X2*x^10+X1*x^11+2*X2^2* x^10-3*x^12+X1*X2*x^9-X1*x^10-X2^2*x^9-3*X2*x^10-X1*x^9+X2*x^9+x^10-X1*X2*x^6+ X1*X2*x^5+X2*x^6-2*X2*x^5+X2*x^4+x^5-x^4+x^3-3*x^2+3*x-1)/(-1+x)/(X1*X2*x^7+X1* X2*x^6-X1*x^7-X2*x^7-X1*x^6+x^7+X2*x^5-x^5+x-1)/(X1*X2*x^7+X1*X2*x^6-X1*x^7-X2* x^7-X1*X2*x^5-X1*x^6+x^7+2*X2*x^5-X2*x^4-x^5+x^4-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 5, 1], are n 139 23 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 1, 4], are n 141 65 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 12 23 65 -------------- 1495 and in floating point 0.3103563600 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 12 1495 ate normal pair with correlation, ---------- 1495 1/2 12 1495 1783 i.e. , [[----------, 0], [0, ----]] 1495 1495 ------------------------------------------------- Theorem Number, 82, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 5, 1], nor the composition, [3, 2, 3] Then infinity ----- \ n 16 15 14 13 12 10 9 8 7 ) a(n) x = - (x + 2 x + x + x + 2 x - 2 x + x + x - x / ----- n = 0 6 5 4 3 2 / + x + x - x + x + x - 2 x + 1) / ( / 12 11 10 9 8 7 6 5 4 2 (x + 3 x + 3 x + 2 x + 3 x + 3 x + x + x + x + x + x - 1) 2 (-1 + x) ) and in Maple format -(x^16+2*x^15+x^14+x^13+2*x^12-2*x^10+x^9+x^8-x^7+x^6+x^5-x^4+x^3+x^2-2*x+1)/(x ^12+3*x^11+3*x^10+2*x^9+3*x^8+3*x^7+x^6+x^5+x^4+x^2+x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.70533351701574926057 1.9060446822436648581 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 5, 1] and d occurrences (as containment) of the composition, [3, 2, 3], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 17 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 2 16 2 17 2 17 2 2 15 2 16 + X1 X2 x - 2 X1 X2 x - 2 X1 X2 x - X1 X2 x - 2 X1 X2 x 2 17 2 16 17 2 17 2 15 2 16 + X1 x - 2 X1 X2 x + 4 X1 X2 x + X2 x + X1 X2 x + X1 x 2 15 16 17 2 16 17 2 2 13 + 3 X1 X2 x + 4 X1 X2 x - 2 X1 x + X2 x - 2 X2 x + X1 X2 x 2 14 2 14 15 16 2 15 16 - X1 X2 x + X1 X2 x - 4 X1 X2 x - 2 X1 x - 2 X2 x - 2 X2 x 17 2 2 12 2 13 2 14 2 13 15 + x - X1 X2 x - X1 X2 x + X1 x - 3 X1 X2 x + X1 x 2 14 15 16 2 12 2 12 13 - X2 x + 3 X2 x + x + 2 X1 X2 x + 2 X1 X2 x + 4 X1 X2 x 14 2 13 14 15 2 11 2 12 2 11 - X1 x + 2 X2 x + X2 x - x - X1 X2 x - X1 x + 2 X1 X2 x 12 13 2 12 13 2 11 2 10 - 5 X1 X2 x - X1 x - X2 x - 3 X2 x + X1 x - 2 X1 X2 x 11 12 2 11 12 13 10 - 3 X1 X2 x + 3 X1 x - 2 X2 x + 3 X2 x + x + 5 X1 X2 x 11 2 10 11 12 9 10 2 9 + X1 x + 2 X2 x + 4 X2 x - 2 x - X1 X2 x - 3 X1 x + X2 x 10 11 8 9 2 8 9 10 - 5 X2 x - 2 x - X1 X2 x + X1 x - X2 x - X2 x + 3 x 7 8 8 6 7 7 8 + 2 X1 X2 x + X1 x + 3 X2 x - 2 X1 X2 x - 2 X1 x - 2 X2 x - 2 x 5 6 6 7 5 4 5 3 4 + X1 X2 x + X1 x + X2 x + 2 x + X2 x - 2 X2 x - 2 x + X2 x + 2 x 2 / 2 2 14 2 2 13 2 14 - 3 x + 3 x - 1) / ((-1 + x) (X1 X2 x + X1 X2 x - 2 X1 X2 x / 2 14 2 2 12 2 13 2 14 2 13 - 2 X1 X2 x - X1 X2 x - 2 X1 X2 x + X1 x - 2 X1 X2 x 14 2 14 2 12 2 13 2 12 + 4 X1 X2 x + X2 x + X1 X2 x + X1 x + 4 X1 X2 x 13 14 2 13 14 2 11 2 11 + 4 X1 X2 x - 2 X1 x + X2 x - 2 X2 x - X1 X2 x + 2 X1 X2 x 12 13 2 12 13 14 2 11 - 6 X1 X2 x - 2 X1 x - 3 X2 x - 2 X2 x + x + X1 x 2 10 11 12 2 11 12 13 - 2 X1 X2 x - 2 X1 X2 x + 2 X1 x - 2 X2 x + 5 X2 x + x 10 2 10 11 12 9 10 + 3 X1 X2 x + 3 X2 x + 3 X2 x - 2 x - 2 X1 X2 x - X1 x 2 9 10 11 8 9 2 8 10 7 + X2 x - 5 X2 x - x - X1 X2 x + 2 X1 x - X2 x + 2 x + X1 X2 x 8 8 9 6 7 7 8 5 + X1 x + 3 X2 x - x - 2 X1 X2 x - X1 x - 2 X2 x - 2 x + X1 X2 x 6 6 7 4 5 3 4 3 2 + X1 x + X2 x + 2 x - 2 X2 x - x + X2 x + 2 x - x - 2 x + 3 x - 1 )) and in Maple format -(X1^2*X2^2*x^17+X1^2*X2^2*x^16-2*X1^2*X2*x^17-2*X1*X2^2*x^17-X1^2*X2^2*x^15-2* X1^2*X2*x^16+X1^2*x^17-2*X1*X2^2*x^16+4*X1*X2*x^17+X2^2*x^17+X1^2*X2*x^15+X1^2* x^16+3*X1*X2^2*x^15+4*X1*X2*x^16-2*X1*x^17+X2^2*x^16-2*X2*x^17+X1^2*X2^2*x^13- X1^2*X2*x^14+X1*X2^2*x^14-4*X1*X2*x^15-2*X1*x^16-2*X2^2*x^15-2*X2*x^16+x^17-X1^ 2*X2^2*x^12-X1^2*X2*x^13+X1^2*x^14-3*X1*X2^2*x^13+X1*x^15-X2^2*x^14+3*X2*x^15+x ^16+2*X1^2*X2*x^12+2*X1*X2^2*x^12+4*X1*X2*x^13-X1*x^14+2*X2^2*x^13+X2*x^14-x^15 -X1^2*X2*x^11-X1^2*x^12+2*X1*X2^2*x^11-5*X1*X2*x^12-X1*x^13-X2^2*x^12-3*X2*x^13 +X1^2*x^11-2*X1*X2^2*x^10-3*X1*X2*x^11+3*X1*x^12-2*X2^2*x^11+3*X2*x^12+x^13+5* X1*X2*x^10+X1*x^11+2*X2^2*x^10+4*X2*x^11-2*x^12-X1*X2*x^9-3*X1*x^10+X2^2*x^9-5* X2*x^10-2*x^11-X1*X2*x^8+X1*x^9-X2^2*x^8-X2*x^9+3*x^10+2*X1*X2*x^7+X1*x^8+3*X2* x^8-2*X1*X2*x^6-2*X1*x^7-2*X2*x^7-2*x^8+X1*X2*x^5+X1*x^6+X2*x^6+2*x^7+X2*x^5-2* X2*x^4-2*x^5+X2*x^3+2*x^4-3*x^2+3*x-1)/(-1+x)/(X1^2*X2^2*x^14+X1^2*X2^2*x^13-2* X1^2*X2*x^14-2*X1*X2^2*x^14-X1^2*X2^2*x^12-2*X1^2*X2*x^13+X1^2*x^14-2*X1*X2^2*x ^13+4*X1*X2*x^14+X2^2*x^14+X1^2*X2*x^12+X1^2*x^13+4*X1*X2^2*x^12+4*X1*X2*x^13-2 *X1*x^14+X2^2*x^13-2*X2*x^14-X1^2*X2*x^11+2*X1*X2^2*x^11-6*X1*X2*x^12-2*X1*x^13 -3*X2^2*x^12-2*X2*x^13+x^14+X1^2*x^11-2*X1*X2^2*x^10-2*X1*X2*x^11+2*X1*x^12-2* X2^2*x^11+5*X2*x^12+x^13+3*X1*X2*x^10+3*X2^2*x^10+3*X2*x^11-2*x^12-2*X1*X2*x^9- X1*x^10+X2^2*x^9-5*X2*x^10-x^11-X1*X2*x^8+2*X1*x^9-X2^2*x^8+2*x^10+X1*X2*x^7+X1 *x^8+3*X2*x^8-x^9-2*X1*X2*x^6-X1*x^7-2*X2*x^7-2*x^8+X1*X2*x^5+X1*x^6+X2*x^6+2*x ^7-2*X2*x^4-x^5+X2*x^3+2*x^4-x^3-2*x^2+3*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 5, 1], are n 139 23 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 2, 3], are n 89 77 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 6 23 77 ------------- 1771 and in floating point 0.1425745369 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 6 1771 ate normal pair with correlation, --------- 1771 1/2 6 1771 1843 i.e. , [[---------, 0], [0, ----]] 1771 1771 ------------------------------------------------- Theorem Number, 83, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 5, 1], nor the composition, [3, 3, 2] Then infinity ----- 13 12 11 9 8 6 4 3 2 \ n x + x - x + x - x + 2 x - x + x + x - 2 x + 1 ) a(n) x = - --------------------------------------------------------- / 9 8 7 6 5 4 2 2 ----- (x + 2 x + x + x + 2 x + x + x + x - 1) (-1 + x) n = 0 and in Maple format -(x^13+x^12-x^11+x^9-x^8+2*x^6-x^4+x^3+x^2-2*x+1)/(x^9+2*x^8+x^7+x^6+2*x^5+x^4+ x^2+x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.70464553543661030161 1.9073680513412163549 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 5, 1] and d occurrences (as containment) of the composition, [3, 3, 2], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 14 2 14 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x - X1 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 14 2 12 14 14 2 11 2 12 - 2 X1 X2 x - X1 X2 x + 2 X1 x + X2 x + X1 X2 x + X1 x 12 14 2 11 11 12 12 + 3 X1 X2 x - x - X1 x - 2 X1 X2 x - 3 X1 x - 2 X2 x 10 11 11 12 9 10 10 - X1 X2 x + 2 X1 x + X2 x + 2 x + 2 X1 X2 x + X1 x + X2 x 11 8 9 9 10 7 8 8 - x - X1 X2 x - 2 X1 x - 2 X2 x - x - X1 X2 x + X1 x + X2 x 9 6 7 7 8 5 6 6 + 2 x + 2 X1 X2 x + X1 x + 2 X2 x - x - X1 X2 x - X1 x - 3 X2 x 7 6 4 5 3 4 2 / - 2 x + 2 x + 2 X2 x + x - X2 x - 2 x + 3 x - 3 x + 1) / ( / 2 11 2 11 11 11 11 (-1 + x) (X1 X2 x - X1 x - 2 X1 X2 x + 2 X1 x + X2 x 9 11 8 9 9 8 8 9 + 2 X1 X2 x - x - X1 X2 x - 2 X1 x - 2 X2 x + X1 x + X2 x + 2 x 6 7 8 5 6 6 7 5 6 + 2 X1 X2 x + X2 x - x - X1 X2 x - X1 x - 3 X2 x - x + X2 x + 2 x 4 3 4 3 2 + 2 X2 x - X2 x - 2 x + x + 2 x - 3 x + 1)) and in Maple format -(X1^2*X2*x^14-X1^2*x^14-2*X1*X2*x^14-X1^2*X2*x^12+2*X1*x^14+X2*x^14+X1^2*X2*x^ 11+X1^2*x^12+3*X1*X2*x^12-x^14-X1^2*x^11-2*X1*X2*x^11-3*X1*x^12-2*X2*x^12-X1*X2 *x^10+2*X1*x^11+X2*x^11+2*x^12+2*X1*X2*x^9+X1*x^10+X2*x^10-x^11-X1*X2*x^8-2*X1* x^9-2*X2*x^9-x^10-X1*X2*x^7+X1*x^8+X2*x^8+2*x^9+2*X1*X2*x^6+X1*x^7+2*X2*x^7-x^8 -X1*X2*x^5-X1*x^6-3*X2*x^6-2*x^7+2*x^6+2*X2*x^4+x^5-X2*x^3-2*x^4+3*x^2-3*x+1)/( -1+x)/(X1^2*X2*x^11-X1^2*x^11-2*X1*X2*x^11+2*X1*x^11+X2*x^11+2*X1*X2*x^9-x^11- X1*X2*x^8-2*X1*x^9-2*X2*x^9+X1*x^8+X2*x^8+2*x^9+2*X1*X2*x^6+X2*x^7-x^8-X1*X2*x^ 5-X1*x^6-3*X2*x^6-x^7+X2*x^5+2*x^6+2*X2*x^4-X2*x^3-2*x^4+x^3+2*x^2-3*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 5, 1], are n 139 23 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 3, 2], are n 97 85 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 6 23 85 ------------- 1955 and in floating point 0.1356993819 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 6 1955 ate normal pair with correlation, --------- 1955 1/2 6 1955 2027 i.e. , [[---------, 0], [0, ----]] 1955 1955 ------------------------------------------------- Theorem Number, 84, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 5, 1], nor the composition, [3, 4, 1] Then infinity ----- 8 7 6 5 4 2 \ n x - x - x + x - x - x + 2 x - 1 ) a(n) x = ------------------------------------- / 6 5 3 2 2 ----- (x + x + x + x + x - 1) (-1 + x) n = 0 and in Maple format (x^8-x^7-x^6+x^5-x^4-x^2+2*x-1)/(x^6+x^5+x^3+x^2+x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.69790391550484586467 1.9111834366854762350 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 5, 1] and d occurrences (as containment) of the composition, [3, 4, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 8 8 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x - X1 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 8 6 7 8 5 6 6 7 5 - X2 x - X1 X2 x + X2 x + x + X1 X2 x + X1 x + X2 x - x - 2 X2 x 6 4 5 4 2 / 7 6 - x + X2 x + x - x - x + 2 x - 1) / ((-1 + x) (X1 X2 x + X1 X2 x / 7 7 5 6 7 5 4 5 4 - X1 x - X2 x - X1 X2 x - X1 x + x + 2 X2 x - X2 x - x + x - 2 x + 1)) and in Maple format (X1*X2*x^8-X1*x^8-X2*x^8-X1*X2*x^6+X2*x^7+x^8+X1*X2*x^5+X1*x^6+X2*x^6-x^7-2*X2* x^5-x^6+X2*x^4+x^5-x^4-x^2+2*x-1)/(-1+x)/(X1*X2*x^7+X1*X2*x^6-X1*x^7-X2*x^7-X1* X2*x^5-X1*x^6+x^7+2*X2*x^5-X2*x^4-x^5+x^4-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 5, 1], are n 139 23 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 4, 1], are n 135 65 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 12 23 65 -------------- 1495 and in floating point 0.3103563600 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 12 1495 ate normal pair with correlation, ---------- 1495 1/2 12 1495 1783 i.e. , [[----------, 0], [0, ----]] 1495 1495 ------------------------------------------------- Theorem Number, 85, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 5, 1], nor the composition, [4, 1, 3] Then infinity ----- \ n ) a(n) x = / ----- n = 0 17 16 15 14 13 12 11 10 4 2 x + 2 x + 2 x + x + x + 2 x - x - x + x + x - 2 x + 1 - ----------------------------------------------------------------------- 6 5 3 2 6 5 2 (x + x + x + x + x - 1) (x + x + 1) (-1 + x) and in Maple format -(x^17+2*x^16+2*x^15+x^14+x^13+2*x^12-x^11-x^10+x^4+x^2-2*x+1)/(x^6+x^5+x^3+x^2 +x-1)/(x^6+x^5+1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.68733873783470785714 1.9111834366854762350 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 5, 1] and d occurrences (as containment) of the composition, [4, 1, 3], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 18 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 2 17 2 18 2 18 2 2 16 2 17 + 2 X1 X2 x - 2 X1 X2 x - 2 X1 X2 x + X1 X2 x - 4 X1 X2 x 2 18 2 17 18 2 18 2 2 15 + X1 x - 3 X1 X2 x + 4 X1 X2 x + X2 x - X1 X2 x 2 16 2 17 17 18 2 17 18 - 3 X1 X2 x + 2 X1 x + 6 X1 X2 x - 2 X1 x + X2 x - 2 X2 x 2 2 14 2 15 2 16 2 15 16 - X1 X2 x + X1 X2 x + 2 X1 x + 3 X1 X2 x + 2 X1 X2 x 17 2 16 17 18 2 2 13 2 14 - 3 X1 x - X2 x - 2 X2 x + x + X1 X2 x + 2 X1 X2 x 2 14 15 16 2 15 16 17 + X1 X2 x - 4 X1 X2 x - 2 X1 x - 2 X2 x + X2 x + x 2 13 2 14 2 13 14 15 15 - X1 X2 x - X1 x - 3 X1 X2 x - 2 X1 X2 x + X1 x + 3 X2 x 2 2 11 2 12 13 14 2 13 15 - X1 X2 x + 2 X1 X2 x + 4 X1 X2 x + X1 x + 2 X2 x - x 2 11 2 11 12 13 2 12 13 + X1 X2 x + 2 X1 X2 x - 5 X1 X2 x - X1 x - 2 X2 x - 3 X2 x 2 10 11 12 12 13 10 - 2 X1 X2 x - 3 X1 X2 x + 3 X1 x + 5 X2 x + x + 3 X1 X2 x 11 2 10 12 9 10 2 9 10 + X1 x + 2 X2 x - 3 x + X1 X2 x - X1 x - X2 x - 3 X2 x 9 9 10 6 5 6 5 4 5 - X1 x + X2 x + x - X1 X2 x + X1 X2 x + X2 x - 2 X2 x + X2 x + x 4 3 2 / - x + x - 3 x + 3 x - 1) / ((-1 + x) / 7 6 7 7 6 7 5 5 (X1 X2 x + X1 X2 x - X1 x - X2 x - X1 x + x + X2 x - x + x - 1) ( 7 6 7 7 5 6 7 5 X1 X2 x + X1 X2 x - X1 x - X2 x - X1 X2 x - X1 x + x + 2 X2 x 4 5 4 - X2 x - x + x - 2 x + 1)) and in Maple format -(X1^2*X2^2*x^18+2*X1^2*X2^2*x^17-2*X1^2*X2*x^18-2*X1*X2^2*x^18+X1^2*X2^2*x^16-\ 4*X1^2*X2*x^17+X1^2*x^18-3*X1*X2^2*x^17+4*X1*X2*x^18+X2^2*x^18-X1^2*X2^2*x^15-3 *X1^2*X2*x^16+2*X1^2*x^17+6*X1*X2*x^17-2*X1*x^18+X2^2*x^17-2*X2*x^18-X1^2*X2^2* x^14+X1^2*X2*x^15+2*X1^2*x^16+3*X1*X2^2*x^15+2*X1*X2*x^16-3*X1*x^17-X2^2*x^16-2 *X2*x^17+x^18+X1^2*X2^2*x^13+2*X1^2*X2*x^14+X1*X2^2*x^14-4*X1*X2*x^15-2*X1*x^16 -2*X2^2*x^15+X2*x^16+x^17-X1^2*X2*x^13-X1^2*x^14-3*X1*X2^2*x^13-2*X1*X2*x^14+X1 *x^15+3*X2*x^15-X1^2*X2^2*x^11+2*X1*X2^2*x^12+4*X1*X2*x^13+X1*x^14+2*X2^2*x^13- x^15+X1^2*X2*x^11+2*X1*X2^2*x^11-5*X1*X2*x^12-X1*x^13-2*X2^2*x^12-3*X2*x^13-2* X1*X2^2*x^10-3*X1*X2*x^11+3*X1*x^12+5*X2*x^12+x^13+3*X1*X2*x^10+X1*x^11+2*X2^2* x^10-3*x^12+X1*X2*x^9-X1*x^10-X2^2*x^9-3*X2*x^10-X1*x^9+X2*x^9+x^10-X1*X2*x^6+ X1*X2*x^5+X2*x^6-2*X2*x^5+X2*x^4+x^5-x^4+x^3-3*x^2+3*x-1)/(-1+x)/(X1*X2*x^7+X1* X2*x^6-X1*x^7-X2*x^7-X1*x^6+x^7+X2*x^5-x^5+x-1)/(X1*X2*x^7+X1*X2*x^6-X1*x^7-X2* x^7-X1*X2*x^5-X1*x^6+x^7+2*X2*x^5-X2*x^4-x^5+x^4-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 5, 1], are n 139 23 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 1, 3], are n 141 65 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 12 23 65 -------------- 1495 and in floating point 0.3103563600 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 12 1495 ate normal pair with correlation, ---------- 1495 1/2 12 1495 1783 i.e. , [[----------, 0], [0, ----]] 1495 1495 ------------------------------------------------- Theorem Number, 86, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 5, 1], nor the composition, [4, 2, 2] Then infinity ----- \ n ) a(n) x = / ----- n = 0 14 13 12 11 9 7 6 5 4 2 x + x + x - x - x - x + 2 x - x + x + x - 2 x + 1 - --------------------------------------------------------------- 9 8 7 5 3 2 2 (x + x + x + x + x + x + x - 1) (-1 + x) and in Maple format -(x^14+x^13+x^12-x^11-x^9-x^7+2*x^6-x^5+x^4+x^2-2*x+1)/(x^9+x^8+x^7+x^5+x^3+x^2 +x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.69239526746995424995 1.9099139227361088013 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 5, 1] and d occurrences (as containment) of the composition, [4, 2, 2], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 15 2 15 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x - X1 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 15 15 15 2 12 15 2 11 - 2 X1 X2 x + 2 X1 x + X2 x - X1 X2 x - x + X1 X2 x 2 12 12 2 11 11 12 12 + X1 x + 3 X1 X2 x - X1 x - 2 X1 X2 x - 3 X1 x - 2 X2 x 10 11 11 12 9 10 10 11 + X1 X2 x + 2 X1 x + X2 x + 2 x - X1 X2 x - X1 x - X2 x - x 8 9 9 10 7 8 8 9 + X1 X2 x + X1 x + X2 x + x - 2 X1 X2 x - X1 x - X2 x - x 6 7 7 8 5 6 6 7 + 2 X1 X2 x + 2 X1 x + 3 X2 x + x - X1 X2 x - X1 x - 4 X2 x - 3 x 5 6 4 5 4 3 2 / + 3 X2 x + 3 x - X2 x - 2 x + x - x + 3 x - 3 x + 1) / ((-1 + x) / 2 11 2 11 11 10 11 11 (X1 X2 x - X1 x - 2 X1 X2 x + X1 X2 x + 2 X1 x + X2 x 10 10 11 8 10 7 8 8 - X1 x - X2 x - x + X1 X2 x + x - X1 X2 x - X1 x - X2 x 6 7 7 8 5 6 6 7 + 2 X1 X2 x + X1 x + 2 X2 x + x - X1 X2 x - X1 x - 3 X2 x - 2 x 5 6 4 5 4 2 + 3 X2 x + 2 x - X2 x - 2 x + x + 2 x - 3 x + 1)) and in Maple format -(X1^2*X2*x^15-X1^2*x^15-2*X1*X2*x^15+2*X1*x^15+X2*x^15-X1^2*X2*x^12-x^15+X1^2* X2*x^11+X1^2*x^12+3*X1*X2*x^12-X1^2*x^11-2*X1*X2*x^11-3*X1*x^12-2*X2*x^12+X1*X2 *x^10+2*X1*x^11+X2*x^11+2*x^12-X1*X2*x^9-X1*x^10-X2*x^10-x^11+X1*X2*x^8+X1*x^9+ X2*x^9+x^10-2*X1*X2*x^7-X1*x^8-X2*x^8-x^9+2*X1*X2*x^6+2*X1*x^7+3*X2*x^7+x^8-X1* X2*x^5-X1*x^6-4*X2*x^6-3*x^7+3*X2*x^5+3*x^6-X2*x^4-2*x^5+x^4-x^3+3*x^2-3*x+1)/( -1+x)/(X1^2*X2*x^11-X1^2*x^11-2*X1*X2*x^11+X1*X2*x^10+2*X1*x^11+X2*x^11-X1*x^10 -X2*x^10-x^11+X1*X2*x^8+x^10-X1*X2*x^7-X1*x^8-X2*x^8+2*X1*X2*x^6+X1*x^7+2*X2*x^ 7+x^8-X1*X2*x^5-X1*x^6-3*X2*x^6-2*x^7+3*X2*x^5+2*x^6-X2*x^4-2*x^5+x^4+2*x^2-3*x +1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 5, 1], are n 139 23 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 2, 2], are n 75 69 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 10 3 ------- 69 and in floating point 0.2510218562 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 10 3 ate normal pair with correlation, ------- 69 1/2 10 3 1787 i.e. , [[-------, 0], [0, ----]] 69 1587 ------------------------------------------------- Theorem Number, 87, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 5, 1], nor the composition, [4, 3, 1] Then infinity ----- 8 7 6 4 \ n x + 2 x + x + x - x + 1 ) a(n) x = ------------------------------------ / 6 5 3 2 ----- (-1 + x) (x + x + x + x + x - 1) n = 0 and in Maple format (x^8+2*x^7+x^6+x^4-x+1)/(-1+x)/(x^6+x^5+x^3+x^2+x-1) The asymptotic expression for a(n) is, n 0.69079734461128122568 1.9111834366854762350 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 5, 1] and d occurrences (as containment) of the composition, [4, 3, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 9 8 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x + X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 9 9 8 8 9 6 7 8 5 - X1 x - X2 x - X1 x - X2 x + x - X1 X2 x + X2 x + x + X1 X2 x 6 6 7 5 6 4 5 4 2 / + X1 x + X2 x - x - 2 X2 x - x + X2 x + x - x - x + 2 x - 1) / / 7 6 7 7 5 6 7 ((-1 + x) (X1 X2 x + X1 X2 x - X1 x - X2 x - X1 X2 x - X1 x + x 5 4 5 4 + 2 X2 x - X2 x - x + x - 2 x + 1)) and in Maple format (X1*X2*x^9+X1*X2*x^8-X1*x^9-X2*x^9-X1*x^8-X2*x^8+x^9-X1*X2*x^6+X2*x^7+x^8+X1*X2 *x^5+X1*x^6+X2*x^6-x^7-2*X2*x^5-x^6+X2*x^4+x^5-x^4-x^2+2*x-1)/(-1+x)/(X1*X2*x^7 +X1*X2*x^6-X1*x^7-X2*x^7-X1*X2*x^5-X1*x^6+x^7+2*X2*x^5-X2*x^4-x^5+x^4-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 5, 1], are n 139 23 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 3, 1], are n 135 65 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 12 23 65 -------------- 1495 and in floating point 0.3103563600 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 12 1495 ate normal pair with correlation, ---------- 1495 1/2 12 1495 1783 i.e. , [[----------, 0], [0, ----]] 1495 1495 ------------------------------------------------- Theorem Number, 88, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 5, 1], nor the composition, [5, 1, 2] Then infinity ----- \ n ) a(n) x = / ----- n = 0 15 14 13 12 11 10 9 8 5 2 x + x + x + x - x - x - x - x + x + x - 2 x + 1 - --------------------------------------------------------------- 4 3 2 2 (x + x + x + x - 1) (-1 + x) and in Maple format -(x^15+x^14+x^13+x^12-x^11-x^10-x^9-x^8+x^5+x^2-2*x+1)/(x^4+x^3+x^2+x-1)/(-1+x) ^2 The asymptotic expression for a(n) is, n 0.63530875266095142903 1.9275619754829253043 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 5, 1] and d occurrences (as containment) of the composition, [5, 1, 2], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 17 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 17 2 17 2 16 2 17 17 2 16 - 2 X1 X2 x - X1 X2 x - X1 X2 x + X1 x + 2 X1 X2 x + X1 x 16 17 2 2 13 16 16 2 2 12 + 2 X1 X2 x - X1 x - X1 X2 x - 2 X1 x - X2 x + X1 X2 x 2 13 2 13 16 2 2 11 2 12 2 13 + 2 X1 X2 x + X1 X2 x + x - X1 X2 x - X1 X2 x - X1 x 13 2 11 12 13 11 12 - 2 X1 X2 x + X1 X2 x - 3 X1 X2 x + X1 x + X1 X2 x + 3 X1 x 12 11 12 8 8 8 6 8 + 2 X2 x - X1 x - 2 x + X1 X2 x - X1 x - X2 x - X1 X2 x + x 5 6 5 3 2 / + X1 X2 x + x - x + x - 3 x + 3 x - 1) / ((-1 + x) / 6 6 6 5 6 5 (X1 X2 x - X1 x + x - 1) (X1 X2 x - X1 X2 x - X1 x + x - 2 x + 1)) and in Maple format -(X1^2*X2^2*x^17-2*X1^2*X2*x^17-X1*X2^2*x^17-X1^2*X2*x^16+X1^2*x^17+2*X1*X2*x^ 17+X1^2*x^16+2*X1*X2*x^16-X1*x^17-X1^2*X2^2*x^13-2*X1*x^16-X2*x^16+X1^2*X2^2*x^ 12+2*X1^2*X2*x^13+X1*X2^2*x^13+x^16-X1^2*X2^2*x^11-X1^2*X2*x^12-X1^2*x^13-2*X1* X2*x^13+X1^2*X2*x^11-3*X1*X2*x^12+X1*x^13+X1*X2*x^11+3*X1*x^12+2*X2*x^12-X1*x^ 11-2*x^12+X1*X2*x^8-X1*x^8-X2*x^8-X1*X2*x^6+x^8+X1*X2*x^5+x^6-x^5+x^3-3*x^2+3*x -1)/(-1+x)/(X1*X2*x^6-X1*x^6+x-1)/(X1*X2*x^6-X1*X2*x^5-X1*x^6+x^5-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 5, 1], are n 139 23 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [5, 1, 2], are n 113 57 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 8 23 57 ------------- 437 and in floating point 0.6628418121 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 1/2 8 3 437 ate normal pair with correlation, ------------- 437 1/2 1/2 8 3 437 821 i.e. , [[-------------, 0], [0, ---]] 437 437 ------------------------------------------------- Theorem Number, 89, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 5, 1], nor the composition, [5, 2, 1] Then infinity ----- 10 9 8 5 2 \ n x + x + x - x - x + 2 x - 1 ) a(n) x = --------------------------------- / 4 3 2 2 ----- (x + x + x + x - 1) (-1 + x) n = 0 and in Maple format (x^10+x^9+x^8-x^5-x^2+2*x-1)/(x^4+x^3+x^2+x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.63530875266095142903 1.9275619754829253043 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 5, 1] and d occurrences (as containment) of the composition, [5, 2, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 10 9 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x + X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 10 10 8 9 9 10 7 8 - X1 x - X2 x + X1 X2 x - X1 x - X2 x + x + X1 X2 x - X1 x 8 9 6 7 8 5 6 5 2 - X2 x + x - X1 X2 x - X1 x + x + X1 X2 x + X1 x - x - x + 2 x / 6 5 6 5 - 1) / ((-1 + x) (X1 X2 x - X1 X2 x - X1 x + x - 2 x + 1)) / and in Maple format (X1*X2*x^10+X1*X2*x^9-X1*x^10-X2*x^10+X1*X2*x^8-X1*x^9-X2*x^9+x^10+X1*X2*x^7-X1 *x^8-X2*x^8+x^9-X1*X2*x^6-X1*x^7+x^8+X1*X2*x^5+X1*x^6-x^5-x^2+2*x-1)/(-1+x)/(X1 *X2*x^6-X1*X2*x^5-X1*x^6+x^5-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 5, 1], are n 139 23 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [5, 2, 1], are n 111 57 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 8 23 57 ------------- 437 and in floating point 0.6628418121 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 1/2 8 3 437 ate normal pair with correlation, ------------- 437 1/2 1/2 8 3 437 821 i.e. , [[-------------, 0], [0, ---]] 437 437 ------------------------------------------------- Theorem Number, 90, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [1, 5, 1], nor the composition, [6, 1, 1] Then infinity ----- 11 10 9 8 5 2 \ n x + x + x + x - x - x + 2 x - 1 ) a(n) x = --------------------------------------- / 4 3 2 2 ----- (x + x + x + x - 1) (-1 + x) n = 0 and in Maple format (x^11+x^10+x^9+x^8-x^5-x^2+2*x-1)/(x^4+x^3+x^2+x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.63351683346155297547 1.9275619754829253043 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 5, 1] and d occurrences (as containment) of the composition, [6, 1, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 11 10 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x + X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 11 11 9 10 10 11 8 9 - X1 x - X2 x + X1 X2 x - X1 x - X2 x + x + X1 X2 x - X1 x 9 10 8 8 9 6 8 6 5 5 - X2 x + x - X1 x - X2 x + x + X1 X2 x + x - X1 x + X1 x - x 2 / 6 6 5 5 - x + 2 x - 1) / ((-1 + x) (X1 X2 x - X1 x + X1 x - x + 2 x - 1)) / and in Maple format -(X1*X2*x^11+X1*X2*x^10-X1*x^11-X2*x^11+X1*X2*x^9-X1*x^10-X2*x^10+x^11+X1*X2*x^ 8-X1*x^9-X2*x^9+x^10-X1*x^8-X2*x^8+x^9+X1*X2*x^6+x^8-X1*x^6+X1*x^5-x^5-x^2+2*x-\ 1)/(-1+x)/(X1*X2*x^6-X1*x^6+X1*x^5-x^5+2*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 5, 1], are n 139 23 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [6, 1, 1], are n 49 53 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 22 23 53 -------------- 1219 and in floating point 0.6301164650 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 22 1219 ate normal pair with correlation, ---------- 1219 1/2 22 1219 2187 i.e. , [[----------, 0], [0, ----]] 1219 1219 ------------------------------------------------- Theorem Number, 91, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 1, 4], nor the composition, [1, 1, 6] Then infinity ----- 11 9 4 2 \ n x - x - x - x + 2 x - 1 ) a(n) x = --------------------------------------------------- / 6 5 3 2 6 5 2 ----- (x + x + x + x + x - 1) (x + x + 1) (-1 + x) n = 0 and in Maple format (x^11-x^9-x^4-x^2+2*x-1)/(x^6+x^5+x^3+x^2+x-1)/(x^6+x^5+1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.69440972834504952286 1.9111834366854762350 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 1, 4] and d occurrences (as containment) of the composition, [1, 1, 6], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 13 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 13 2 13 2 13 13 2 11 - 2 X1 X2 x - X1 X2 x + X1 x + 2 X1 X2 x + 2 X1 X2 x 13 2 11 11 11 11 2 9 11 - X1 x - 2 X1 x - 3 X1 X2 x + 3 X1 x + X2 x + X1 x - x 9 9 6 6 4 4 2 / - 2 X1 x + x - X1 X2 x + X1 x - X1 x + x + x - 2 x + 1) / (( / 7 6 7 7 6 7 5 4 5 4 X1 X2 x - X1 X2 x - X1 x - X2 x + X1 x + x + X1 x - X1 x - x + x 7 7 7 7 5 5 - 2 x + 1) (X1 X2 x - X1 x - X2 x + x + X1 x - x + x - 1)) and in Maple format -(X1^2*X2^2*x^13-2*X1^2*X2*x^13-X1*X2^2*x^13+X1^2*x^13+2*X1*X2*x^13+2*X1^2*X2*x ^11-X1*x^13-2*X1^2*x^11-3*X1*X2*x^11+3*X1*x^11+X2*x^11+X1^2*x^9-x^11-2*X1*x^9+x ^9-X1*X2*x^6+X1*x^6-X1*x^4+x^4+x^2-2*x+1)/(X1*X2*x^7-X1*X2*x^6-X1*x^7-X2*x^7+X1 *x^6+x^7+X1*x^5-X1*x^4-x^5+x^4-2*x+1)/(X1*X2*x^7-X1*x^7-X2*x^7+x^7+X1*x^5-x^5+x -1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 1, 4], are n 191 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 1, 6], are n 49 53 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 8 3 53 ------------ 477 and in floating point 0.2114804230 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 8 159 ate normal pair with correlation, -------- 477 1/2 8 159 1559 i.e. , [[--------, 0], [0, ----]] 477 1431 ------------------------------------------------- Theorem Number, 92, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 1, 4], nor the composition, [1, 2, 5] Then infinity ----- 10 9 4 2 \ n x - x - x - x + 2 x - 1 ) a(n) x = - -------------------------------------------- / 10 9 7 6 4 ----- (-1 + x) (x - x + x + x - x + 2 x - 1) n = 0 and in Maple format -(x^10-x^9-x^4-x^2+2*x-1)/(-1+x)/(x^10-x^9+x^7+x^6-x^4+2*x-1) The asymptotic expression for a(n) is, n 0.68264941498911845581 1.9140011778740215244 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 1, 4] and d occurrences (as containment) of the composition, [1, 2, 5], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 10 2 10 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x - X1 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 10 2 9 10 10 9 10 7 9 - 2 X1 X2 x + X1 x + 2 X1 x + X2 x - 2 X1 x - x - X1 X2 x + x 6 7 5 6 5 4 4 2 + X1 X2 x + X1 x - X1 X2 x - X1 x + X1 x - X1 x + x + x - 2 x + 1) / 2 11 2 10 2 11 11 2 10 / (X1 X2 x - X1 X2 x - X1 x - 2 X1 X2 x + 2 X1 x / 10 11 11 2 9 10 10 11 + 2 X1 X2 x + 2 X1 x + X2 x - X1 x - 4 X1 x - X2 x - x 8 9 10 7 8 8 9 6 - X1 X2 x + 2 X1 x + 2 x + X1 X2 x + X1 x + X2 x - x - 2 X1 X2 x 7 8 5 6 5 6 4 5 4 2 - X1 x - x + X1 X2 x + X1 x - 2 X1 x + x + X1 x + x - x - 2 x + 3 x - 1) and in Maple format -(X1^2*X2*x^10-X1^2*x^10-2*X1*X2*x^10+X1^2*x^9+2*X1*x^10+X2*x^10-2*X1*x^9-x^10- X1*X2*x^7+x^9+X1*X2*x^6+X1*x^7-X1*X2*x^5-X1*x^6+X1*x^5-X1*x^4+x^4+x^2-2*x+1)/( X1^2*X2*x^11-X1^2*X2*x^10-X1^2*x^11-2*X1*X2*x^11+2*X1^2*x^10+2*X1*X2*x^10+2*X1* x^11+X2*x^11-X1^2*x^9-4*X1*x^10-X2*x^10-x^11-X1*X2*x^8+2*X1*x^9+2*x^10+X1*X2*x^ 7+X1*x^8+X2*x^8-x^9-2*X1*X2*x^6-X1*x^7-x^8+X1*X2*x^5+X1*x^6-2*X1*x^5+x^6+X1*x^4 +x^5-x^4-2*x^2+3*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 1, 4], are n 191 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 2, 5], are n 111 57 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 4 19 ------- 57 and in floating point 0.3058876452 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 4 19 ate normal pair with correlation, ------- 57 1/2 4 19 203 i.e. , [[-------, 0], [0, ---]] 57 171 ------------------------------------------------- Theorem Number, 93, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 1, 4], nor the composition, [1, 3, 4] Then infinity ----- 4 2 \ n x + x - 2 x + 1 ) a(n) x = --------------------------------- / 7 6 4 ----- (-1 + x) (x + x - x + 2 x - 1) n = 0 and in Maple format (x^4+x^2-2*x+1)/(-1+x)/(x^7+x^6-x^4+2*x-1) The asymptotic expression for a(n) is, n 0.67238389941451345857 1.9168163613635124211 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 1, 4] and d occurrences (as containment) of the composition, [1, 3, 4], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 9 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 9 9 7 7 5 4 5 4 - 2 X1 X2 x + X2 x - X1 X2 x + X1 x + X1 X2 x - X1 X2 x - X1 x + x 2 / 2 10 2 9 10 9 + x - 2 x + 1) / (X1 X2 x - X1 X2 x - 2 X1 X2 x + 2 X1 X2 x / 10 8 9 7 8 8 7 8 + X2 x - X1 X2 x - X2 x + X1 X2 x + X1 x + X2 x - X1 x - x 5 6 4 5 6 5 4 2 - 2 X1 X2 x - X1 x + X1 X2 x + X1 x + x + x - x - 2 x + 3 x - 1) and in Maple format -(X1^2*X2*x^9-2*X1*X2*x^9+X2*x^9-X1*X2*x^7+X1*x^7+X1*X2*x^5-X1*X2*x^4-X1*x^5+x^ 4+x^2-2*x+1)/(X1^2*X2*x^10-X1^2*X2*x^9-2*X1*X2*x^10+2*X1*X2*x^9+X2*x^10-X1*X2*x ^8-X2*x^9+X1*X2*x^7+X1*x^8+X2*x^8-X1*x^7-x^8-2*X1*X2*x^5-X1*x^6+X1*X2*x^4+X1*x^ 5+x^6+x^5-x^4-2*x^2+3*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 1, 4], are n 191 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 3, 4], are n 135 65 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 16 3 65 ------------- 585 and in floating point 0.3819279330 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 16 195 ate normal pair with correlation, --------- 585 1/2 16 195 2267 i.e. , [[---------, 0], [0, ----]] 585 1755 ------------------------------------------------- Theorem Number, 94, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 1, 4], nor the composition, [1, 4, 3] Then infinity ----- 7 6 5 4 \ n x + x + x + x - x + 1 ) a(n) x = ---------------------------------------------------------- / 10 9 8 7 6 3 2 ----- (-1 + x) (x + 2 x + 3 x + 2 x + x + x + x + x - 1) n = 0 and in Maple format (x^7+x^6+x^5+x^4-x+1)/(-1+x)/(x^10+2*x^9+3*x^8+2*x^7+x^6+x^3+x^2+x-1) The asymptotic expression for a(n) is, n 0.66418334929247170970 1.9175371779461324874 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 1, 4] and d occurrences (as containment) of the composition, [1, 4, 3], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 12 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 12 2 12 12 2 12 12 2 9 - X1 X2 x - 2 X1 X2 x + 2 X1 X2 x + X2 x - X2 x + X1 X2 x 9 8 9 7 8 8 7 8 - 2 X1 X2 x - X1 X2 x + X2 x - X1 X2 x + X1 x + X2 x + X1 x - x 5 4 5 4 2 / 2 2 13 + X1 X2 x - X1 X2 x - X1 x + x + x - 2 x + 1) / (X1 X2 x / 2 2 12 2 13 2 13 2 12 2 12 - X1 X2 x - X1 X2 x - 2 X1 X2 x + X1 X2 x + X1 X2 x 13 2 13 13 2 10 12 12 + 2 X1 X2 x + X2 x - X2 x + X1 X2 x - X1 x - X2 x 2 9 10 12 10 9 9 7 - X1 X2 x - 2 X1 X2 x + x + X2 x + 2 X1 x + X2 x + X1 X2 x 9 7 5 6 4 5 6 5 4 - 2 x - X1 x - 2 X1 X2 x - X1 x + X1 X2 x + X1 x + x + x - x 2 - 2 x + 3 x - 1) and in Maple format -(X1^2*X2^2*x^12-X1^2*X2*x^12-2*X1*X2^2*x^12+2*X1*X2*x^12+X2^2*x^12-X2*x^12+X1^ 2*X2*x^9-2*X1*X2*x^9-X1*X2*x^8+X2*x^9-X1*X2*x^7+X1*x^8+X2*x^8+X1*x^7-x^8+X1*X2* x^5-X1*X2*x^4-X1*x^5+x^4+x^2-2*x+1)/(X1^2*X2^2*x^13-X1^2*X2^2*x^12-X1^2*X2*x^13 -2*X1*X2^2*x^13+X1^2*X2*x^12+X1*X2^2*x^12+2*X1*X2*x^13+X2^2*x^13-X2*x^13+X1^2* X2*x^10-X1*x^12-X2*x^12-X1^2*X2*x^9-2*X1*X2*x^10+x^12+X2*x^10+2*X1*x^9+X2*x^9+ X1*X2*x^7-2*x^9-X1*x^7-2*X1*X2*x^5-X1*x^6+X1*X2*x^4+X1*x^5+x^6+x^5-x^4-2*x^2+3* x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 1, 4], are n 191 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 4, 3], are n 135 65 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 16 3 65 ------------- 585 and in floating point 0.3819279330 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 16 195 ate normal pair with correlation, --------- 585 1/2 16 195 2267 i.e. , [[---------, 0], [0, ----]] 585 1755 ------------------------------------------------- Theorem Number, 95, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 1, 4], nor the composition, [1, 5, 2] Then infinity ----- \ n 14 12 10 8 4 2 / ) a(n) x = - (x - x - x - x + x + x - 2 x + 1) / ((-1 + x) / / ----- n = 0 14 13 12 11 10 9 8 7 6 4 (x + x + x + 2 x + x + x - x - x - x + x - 2 x + 1)) and in Maple format -(x^14-x^12-x^10-x^8+x^4+x^2-2*x+1)/(-1+x)/(x^14+x^13+x^12+2*x^11+x^10+x^9-x^8- x^7-x^6+x^4-2*x+1) The asymptotic expression for a(n) is, n 0.67221179878675203708 1.9144061473285935513 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 1, 4] and d occurrences (as containment) of the composition, [1, 5, 2], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 14 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 2 13 2 14 2 14 2 13 2 14 + X1 X2 x - 2 X1 X2 x - 2 X1 X2 x - X1 X2 x + X1 x 2 13 14 2 14 2 12 13 - 2 X1 X2 x + 4 X1 X2 x + X2 x + X1 X2 x + 2 X1 X2 x 14 2 13 14 2 12 12 13 14 - 2 X1 x + X2 x - 2 X2 x - X1 x - 2 X1 X2 x - X2 x + x 2 10 12 12 2 10 10 12 2 9 + X1 X2 x + 2 X1 x + X2 x - X1 x - 2 X1 X2 x - x + X1 x 9 10 10 8 9 9 10 7 - X1 X2 x + 2 X1 x + X2 x - X1 X2 x - X1 x + X2 x - x - X1 X2 x 8 8 6 7 8 5 6 5 4 + X1 x + X2 x + X1 X2 x + X1 x - x - X1 X2 x - X1 x + X1 x - X1 x 4 2 / 2 2 15 2 15 2 15 + x + x - 2 x + 1) / (X1 X2 x - 2 X1 X2 x - 2 X1 X2 x / 2 2 13 2 14 2 15 2 14 15 2 15 - X1 X2 x + X1 X2 x + X1 x - X1 X2 x + 4 X1 X2 x + X2 x 2 13 2 14 2 13 15 2 14 15 + 2 X1 X2 x - X1 x + X1 X2 x - 2 X1 x + X2 x - 2 X2 x 2 12 2 13 13 14 14 15 2 11 - X1 X2 x - X1 x - 2 X1 X2 x + X1 x - X2 x + x + X1 X2 x 2 12 12 13 2 10 2 11 11 + X1 x + 2 X1 X2 x + X1 x - X1 X2 x - X1 x - 2 X1 X2 x 12 12 2 10 11 11 12 2 9 - 2 X1 x - X2 x + 2 X1 x + 2 X1 x + X2 x + x - X1 x 9 10 10 11 9 9 7 9 - X1 X2 x - 2 X1 x + X2 x - x + 3 X1 x + X2 x + X1 X2 x - 2 x 6 7 5 6 5 6 4 5 4 - 2 X1 X2 x - X1 x + X1 X2 x + X1 x - 2 X1 x + x + X1 x + x - x 2 - 2 x + 3 x - 1) and in Maple format -(X1^2*X2^2*x^14+X1^2*X2^2*x^13-2*X1^2*X2*x^14-2*X1*X2^2*x^14-X1^2*X2*x^13+X1^2 *x^14-2*X1*X2^2*x^13+4*X1*X2*x^14+X2^2*x^14+X1^2*X2*x^12+2*X1*X2*x^13-2*X1*x^14 +X2^2*x^13-2*X2*x^14-X1^2*x^12-2*X1*X2*x^12-X2*x^13+x^14+X1^2*X2*x^10+2*X1*x^12 +X2*x^12-X1^2*x^10-2*X1*X2*x^10-x^12+X1^2*x^9-X1*X2*x^9+2*X1*x^10+X2*x^10-X1*X2 *x^8-X1*x^9+X2*x^9-x^10-X1*X2*x^7+X1*x^8+X2*x^8+X1*X2*x^6+X1*x^7-x^8-X1*X2*x^5- X1*x^6+X1*x^5-X1*x^4+x^4+x^2-2*x+1)/(X1^2*X2^2*x^15-2*X1^2*X2*x^15-2*X1*X2^2*x^ 15-X1^2*X2^2*x^13+X1^2*X2*x^14+X1^2*x^15-X1*X2^2*x^14+4*X1*X2*x^15+X2^2*x^15+2* X1^2*X2*x^13-X1^2*x^14+X1*X2^2*x^13-2*X1*x^15+X2^2*x^14-2*X2*x^15-X1^2*X2*x^12- X1^2*x^13-2*X1*X2*x^13+X1*x^14-X2*x^14+x^15+X1^2*X2*x^11+X1^2*x^12+2*X1*X2*x^12 +X1*x^13-X1^2*X2*x^10-X1^2*x^11-2*X1*X2*x^11-2*X1*x^12-X2*x^12+2*X1^2*x^10+2*X1 *x^11+X2*x^11+x^12-X1^2*x^9-X1*X2*x^9-2*X1*x^10+X2*x^10-x^11+3*X1*x^9+X2*x^9+X1 *X2*x^7-2*x^9-2*X1*X2*x^6-X1*x^7+X1*X2*x^5+X1*x^6-2*X1*x^5+x^6+X1*x^4+x^5-x^4-2 *x^2+3*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 1, 4], are n 191 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 5, 2], are n 111 57 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 4 19 ------- 57 and in floating point 0.3058876452 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 4 19 ate normal pair with correlation, ------- 57 1/2 4 19 203 i.e. , [[-------, 0], [0, ---]] 57 171 ------------------------------------------------- Theorem Number, 96, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 1, 4], nor the composition, [1, 6, 1] Then infinity ----- 17 16 15 12 11 10 4 2 \ n x + 2 x + x + x - 2 x - x + x + x - 2 x + 1 ) a(n) x = - --------------------------------------------------------- / 6 5 3 2 6 5 2 ----- (x + x + x + x + x - 1) (x + x + 1) (-1 + x) n = 0 and in Maple format -(x^17+2*x^16+x^15+x^12-2*x^11-x^10+x^4+x^2-2*x+1)/(x^6+x^5+x^3+x^2+x-1)/(x^6+x ^5+1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.68363894095775044155 1.9111834366854762350 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 1, 4] and d occurrences (as containment) of the composition, [1, 6, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 18 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 18 2 18 2 18 2 17 18 - 2 X1 X2 x - 2 X1 X2 x + X1 x - X1 X2 x + 4 X1 X2 x 2 18 2 2 15 2 16 17 18 2 17 + X2 x - X1 X2 x + X1 X2 x + 2 X1 X2 x - 2 X1 x + X2 x 18 2 2 14 2 15 2 16 2 15 - 2 X2 x + X1 X2 x + 2 X1 X2 x - X1 x + 2 X1 X2 x 16 17 17 18 2 2 13 2 14 - 2 X1 X2 x - X1 x - 2 X2 x + x - X1 X2 x - 2 X1 X2 x 2 15 2 14 15 16 2 15 16 17 - X1 x - X1 X2 x - 4 X1 X2 x + 2 X1 x - X2 x + X2 x + x 2 13 2 14 2 13 14 15 15 + X1 X2 x + X1 x + X1 X2 x + 2 X1 X2 x + 2 X1 x + 2 X2 x 16 2 12 14 15 2 11 2 12 - x + 2 X1 X2 x - X1 x - x - 2 X1 X2 x - 2 X1 x 12 13 13 2 11 11 12 - 5 X1 X2 x - X1 x - X2 x + 2 X1 x + 3 X1 X2 x + 5 X1 x 12 13 2 10 11 11 12 2 9 9 + 3 X2 x + x + X1 x - 3 X1 x - X2 x - 3 x - X1 x + X1 X2 x 10 11 9 9 10 7 6 7 - 2 X1 x + x + X1 x - X2 x + x - X1 X2 x + X1 X2 x + X1 x 6 5 4 5 4 3 2 / - X1 x - X1 x + X1 x + x - x + x - 3 x + 3 x - 1) / ((-1 + x) ( / 7 6 7 7 6 7 5 4 5 4 X1 X2 x - X1 X2 x - X1 x - X2 x + X1 x + x + X1 x - X1 x - x + x 7 7 7 7 5 5 - 2 x + 1) (X1 X2 x - X1 x - X2 x + x + X1 x - x + x - 1)) and in Maple format -(X1^2*X2^2*x^18-2*X1^2*X2*x^18-2*X1*X2^2*x^18+X1^2*x^18-X1*X2^2*x^17+4*X1*X2*x ^18+X2^2*x^18-X1^2*X2^2*x^15+X1^2*X2*x^16+2*X1*X2*x^17-2*X1*x^18+X2^2*x^17-2*X2 *x^18+X1^2*X2^2*x^14+2*X1^2*X2*x^15-X1^2*x^16+2*X1*X2^2*x^15-2*X1*X2*x^16-X1*x^ 17-2*X2*x^17+x^18-X1^2*X2^2*x^13-2*X1^2*X2*x^14-X1^2*x^15-X1*X2^2*x^14-4*X1*X2* x^15+2*X1*x^16-X2^2*x^15+X2*x^16+x^17+X1^2*X2*x^13+X1^2*x^14+X1*X2^2*x^13+2*X1* X2*x^14+2*X1*x^15+2*X2*x^15-x^16+2*X1^2*X2*x^12-X1*x^14-x^15-2*X1^2*X2*x^11-2* X1^2*x^12-5*X1*X2*x^12-X1*x^13-X2*x^13+2*X1^2*x^11+3*X1*X2*x^11+5*X1*x^12+3*X2* x^12+x^13+X1^2*x^10-3*X1*x^11-X2*x^11-3*x^12-X1^2*x^9+X1*X2*x^9-2*X1*x^10+x^11+ X1*x^9-X2*x^9+x^10-X1*X2*x^7+X1*X2*x^6+X1*x^7-X1*x^6-X1*x^5+X1*x^4+x^5-x^4+x^3-\ 3*x^2+3*x-1)/(-1+x)/(X1*X2*x^7-X1*X2*x^6-X1*x^7-X2*x^7+X1*x^6+x^7+X1*x^5-X1*x^4 -x^5+x^4-2*x+1)/(X1*X2*x^7-X1*x^7-X2*x^7+x^7+X1*x^5-x^5+x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 1, 4], are n 191 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 6, 1], are n 49 53 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 8 3 53 ------------ 477 and in floating point 0.2114804230 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 8 159 ate normal pair with correlation, -------- 477 1/2 8 159 1559 i.e. , [[--------, 0], [0, ----]] 477 1431 ------------------------------------------------- Theorem Number, 97, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 1, 4], nor the composition, [2, 3, 3] Then infinity ----- 8 7 6 5 4 3 2 \ n x - x + x + x - x + x + x - 2 x + 1 ) a(n) x = - -------------------------------------------------- / 9 8 7 5 4 3 2 ----- x - 2 x + 2 x - x + 2 x - x - 2 x + 3 x - 1 n = 0 and in Maple format -(x^8-x^7+x^6+x^5-x^4+x^3+x^2-2*x+1)/(x^9-2*x^8+2*x^7-x^5+2*x^4-x^3-2*x^2+3*x-1 ) The asymptotic expression for a(n) is, n 0.65556997991907913494 1.9211122916754298187 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 1, 4] and d occurrences (as containment) of the composition, [2, 3, 3], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 9 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 9 8 9 7 8 8 7 7 - 2 X1 X2 x + X1 X2 x + X2 x - X1 X2 x - X1 x - X2 x + X1 x + X2 x 8 5 6 7 4 5 5 6 4 + x + X1 X2 x - X2 x - x - X1 X2 x - X1 x - X2 x + x + 2 X2 x 5 3 4 3 2 / 2 10 2 9 + x - X2 x - x + x + x - 2 x + 1) / (X1 X2 x - X1 X2 x / 10 9 10 8 9 9 - 2 X1 X2 x + 3 X1 X2 x + X2 x - 2 X1 X2 x - X1 x - 2 X2 x 7 8 8 9 7 7 8 5 + X1 X2 x + 2 X1 x + 2 X2 x + x - X1 x - 2 X2 x - 2 x - 2 X1 X2 x 6 6 7 4 5 5 4 5 3 - X1 x + X2 x + 2 x + X1 X2 x + X1 x + 2 X2 x - 3 X2 x - x + X2 x 4 3 2 + 2 x - x - 2 x + 3 x - 1) and in Maple format -(X1^2*X2*x^9-2*X1*X2*x^9+X1*X2*x^8+X2*x^9-X1*X2*x^7-X1*x^8-X2*x^8+X1*x^7+X2*x^ 7+x^8+X1*X2*x^5-X2*x^6-x^7-X1*X2*x^4-X1*x^5-X2*x^5+x^6+2*X2*x^4+x^5-X2*x^3-x^4+ x^3+x^2-2*x+1)/(X1^2*X2*x^10-X1^2*X2*x^9-2*X1*X2*x^10+3*X1*X2*x^9+X2*x^10-2*X1* X2*x^8-X1*x^9-2*X2*x^9+X1*X2*x^7+2*X1*x^8+2*X2*x^8+x^9-X1*x^7-2*X2*x^7-2*x^8-2* X1*X2*x^5-X1*x^6+X2*x^6+2*x^7+X1*X2*x^4+X1*x^5+2*X2*x^5-3*X2*x^4-x^5+X2*x^3+2*x ^4-x^3-2*x^2+3*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 1, 4], are n 191 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 3, 3], are n 97 85 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 2 3 85 ------------ 85 and in floating point 0.3757345747 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 1/2 2 3 85 ate normal pair with correlation, ------------ 85 1/2 1/2 2 3 85 109 i.e. , [[------------, 0], [0, ---]] 85 85 ------------------------------------------------- Theorem Number, 98, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 1, 4], nor the composition, [2, 4, 2] Then infinity ----- 12 11 8 4 2 \ n x + x + x - x - x + 2 x - 1 ) a(n) x = - ------------------------------------------------ / 13 11 9 6 5 4 2 ----- x - x + 2 x - x - x + x + 2 x - 3 x + 1 n = 0 and in Maple format -(x^12+x^11+x^8-x^4-x^2+2*x-1)/(x^13-x^11+2*x^9-x^6-x^5+x^4+2*x^2-3*x+1) The asymptotic expression for a(n) is, n 0.66010691781921974090 1.9182218813446505063 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 1, 4] and d occurrences (as containment) of the composition, [2, 4, 2], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 12 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 11 2 12 12 2 11 11 12 + X1 X2 x - X1 x - 2 X1 X2 x - X1 x - 2 X1 X2 x + 2 X1 x 12 2 9 11 11 12 9 11 + X2 x + X1 X2 x + 2 X1 x + X2 x - x - 2 X1 X2 x - x 8 9 8 8 6 8 5 6 - X1 X2 x + X2 x + X1 x + X2 x - X1 X2 x - x + X1 X2 x + X1 x 4 5 4 2 / 2 13 2 13 - X1 X2 x - X1 x + x + x - 2 x + 1) / (X1 X2 x - X1 x / 13 2 11 13 13 2 10 2 11 - 2 X1 X2 x - X1 X2 x + 2 X1 x + X2 x + X1 X2 x + X1 x 11 13 2 9 10 11 11 10 + 2 X1 X2 x - x - X1 X2 x - 2 X1 X2 x - 2 X1 x - X2 x + X2 x 11 9 9 7 9 6 7 5 + x + 2 X1 x + X2 x - X1 X2 x - 2 x + X1 X2 x + X1 x - 2 X1 X2 x 6 4 5 6 5 4 2 - 2 X1 x + X1 X2 x + X1 x + x + x - x - 2 x + 3 x - 1) and in Maple format -(X1^2*X2*x^12+X1^2*X2*x^11-X1^2*x^12-2*X1*X2*x^12-X1^2*x^11-2*X1*X2*x^11+2*X1* x^12+X2*x^12+X1^2*X2*x^9+2*X1*x^11+X2*x^11-x^12-2*X1*X2*x^9-x^11-X1*X2*x^8+X2*x ^9+X1*x^8+X2*x^8-X1*X2*x^6-x^8+X1*X2*x^5+X1*x^6-X1*X2*x^4-X1*x^5+x^4+x^2-2*x+1) /(X1^2*X2*x^13-X1^2*x^13-2*X1*X2*x^13-X1^2*X2*x^11+2*X1*x^13+X2*x^13+X1^2*X2*x^ 10+X1^2*x^11+2*X1*X2*x^11-x^13-X1^2*X2*x^9-2*X1*X2*x^10-2*X1*x^11-X2*x^11+X2*x^ 10+x^11+2*X1*x^9+X2*x^9-X1*X2*x^7-2*x^9+X1*X2*x^6+X1*x^7-2*X1*X2*x^5-2*X1*x^6+ X1*X2*x^4+X1*x^5+x^6+x^5-x^4-2*x^2+3*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 1, 4], are n 191 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 4, 2], are n 75 69 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 2 23 ------- 23 and in floating point 0.4170288281 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 2 23 ate normal pair with correlation, ------- 23 1/2 2 23 31 i.e. , [[-------, 0], [0, --]] 23 23 ------------------------------------------------- Theorem Number, 99, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 1, 4], nor the composition, [2, 5, 1] Then infinity ----- 2 8 7 5 4 2 \ n (x - x + 1) (2 x + 2 x - 2 x - 2 x + x + x - 1) ) a(n) x = - ----------------------------------------------------- / 10 9 7 6 4 ----- (-1 + x) (x - x + x + x - x + 2 x - 1) n = 0 and in Maple format -(x^2-x+1)*(2*x^8+2*x^7-2*x^5-2*x^4+x^2+x-1)/(-1+x)/(x^10-x^9+x^7+x^6-x^4+2*x-1 ) The asymptotic expression for a(n) is, n 0.67273009850846616931 1.9140011778740215244 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 1, 4] and d occurrences (as containment) of the composition, [2, 5, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 10 2 10 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x - X1 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 10 2 9 9 10 10 9 9 - 3 X1 X2 x + X1 x - X1 X2 x + 3 X1 x + 2 X2 x - X1 x + X2 x 10 7 6 7 5 6 5 4 - 2 x - X1 X2 x + X1 X2 x + X1 x - X1 X2 x - X1 x + X1 x - X1 x 4 2 / 2 11 2 10 2 11 11 + x + x - 2 x + 1) / (X1 X2 x - X1 X2 x - X1 x - 2 X1 X2 x / 2 10 10 11 11 2 9 10 10 + 2 X1 x + 2 X1 X2 x + 2 X1 x + X2 x - X1 x - 4 X1 x - X2 x 11 8 9 10 7 8 8 9 - x - X1 X2 x + 2 X1 x + 2 x + X1 X2 x + X1 x + X2 x - x 6 7 8 5 6 5 6 4 5 - 2 X1 X2 x - X1 x - x + X1 X2 x + X1 x - 2 X1 x + x + X1 x + x 4 2 - x - 2 x + 3 x - 1) and in Maple format -(X1^2*X2*x^10-X1^2*x^10-3*X1*X2*x^10+X1^2*x^9-X1*X2*x^9+3*X1*x^10+2*X2*x^10-X1 *x^9+X2*x^9-2*x^10-X1*X2*x^7+X1*X2*x^6+X1*x^7-X1*X2*x^5-X1*x^6+X1*x^5-X1*x^4+x^ 4+x^2-2*x+1)/(X1^2*X2*x^11-X1^2*X2*x^10-X1^2*x^11-2*X1*X2*x^11+2*X1^2*x^10+2*X1 *X2*x^10+2*X1*x^11+X2*x^11-X1^2*x^9-4*X1*x^10-X2*x^10-x^11-X1*X2*x^8+2*X1*x^9+2 *x^10+X1*X2*x^7+X1*x^8+X2*x^8-x^9-2*X1*X2*x^6-X1*x^7-x^8+X1*X2*x^5+X1*x^6-2*X1* x^5+x^6+X1*x^4+x^5-x^4-2*x^2+3*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 1, 4], are n 191 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 5, 1], are n 111 57 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 4 19 ------- 57 and in floating point 0.3058876452 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 4 19 ate normal pair with correlation, ------- 57 1/2 4 19 203 i.e. , [[-------, 0], [0, ---]] 57 171 ------------------------------------------------- Theorem Number, 100, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 1, 4], nor the composition, [3, 2, 3] Then infinity ----- \ n ) a(n) x = / ----- n = 0 9 8 6 5 4 3 2 x - 2 x + x - 2 x + x - x - x + 2 x - 1 -------------------------------------------------------------------- 12 11 9 8 6 5 4 3 2 x - 2 x + 3 x - 3 x + 3 x - 2 x + 2 x - x - 2 x + 3 x - 1 and in Maple format (x^9-2*x^8+x^6-2*x^5+x^4-x^3-x^2+2*x-1)/(x^12-2*x^11+3*x^9-3*x^8+3*x^6-2*x^5+2* x^4-x^3-2*x^2+3*x-1) The asymptotic expression for a(n) is, n 0.65537287717026354174 1.9205873909649774992 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 1, 4] and d occurrences (as containment) of the composition, [3, 2, 3], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 10 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 10 2 10 2 9 2 9 10 - X1 X2 x - 2 X1 X2 x + X1 X2 x + 2 X1 X2 x + 2 X1 X2 x 2 10 9 2 9 10 8 9 2 8 + X2 x - 5 X1 X2 x - 2 X2 x - X2 x + X1 X2 x + X1 x + X2 x 9 8 8 9 6 8 5 6 + 4 X2 x - X1 x - 3 X2 x - x - X1 X2 x + 2 x + X1 X2 x + X1 x 6 4 5 5 6 4 5 3 4 + X2 x - X1 X2 x - X1 x - 2 X2 x - x + 2 X2 x + 2 x - X2 x - x 3 2 / 2 2 12 2 2 11 2 12 + x + x - 2 x + 1) / (X1 X2 x + X1 X2 x - 2 X1 X2 x / 2 12 2 2 10 2 11 2 12 12 - 2 X1 X2 x - X1 X2 x - X1 X2 x + X1 x + 4 X1 X2 x 2 12 2 10 2 10 11 12 2 11 + X2 x + 2 X1 X2 x + 4 X1 X2 x - 2 X1 X2 x - 2 X1 x - X2 x 12 2 9 2 9 10 11 2 10 - 2 X2 x - X1 X2 x - 2 X1 X2 x - 7 X1 X2 x + 2 X1 x - 2 X2 x 11 12 9 10 2 9 10 11 + 3 X2 x + x + 6 X1 X2 x + X1 x + 3 X2 x + 3 X2 x - 2 x 8 9 2 8 9 7 8 8 - 2 X1 X2 x - 2 X1 x - X2 x - 7 X2 x - X1 X2 x + 2 X1 x + 4 X2 x 9 6 7 8 5 6 6 + 3 x + X1 X2 x + X1 x - 3 x - 2 X1 X2 x - 2 X1 x - 2 X2 x 4 5 5 6 4 5 3 4 3 + X1 X2 x + X1 x + 3 X2 x + 3 x - 3 X2 x - 2 x + X2 x + 2 x - x 2 - 2 x + 3 x - 1) and in Maple format -(X1^2*X2^2*x^10-X1^2*X2*x^10-2*X1*X2^2*x^10+X1^2*X2*x^9+2*X1*X2^2*x^9+2*X1*X2* x^10+X2^2*x^10-5*X1*X2*x^9-2*X2^2*x^9-X2*x^10+X1*X2*x^8+X1*x^9+X2^2*x^8+4*X2*x^ 9-X1*x^8-3*X2*x^8-x^9-X1*X2*x^6+2*x^8+X1*X2*x^5+X1*x^6+X2*x^6-X1*X2*x^4-X1*x^5-\ 2*X2*x^5-x^6+2*X2*x^4+2*x^5-X2*x^3-x^4+x^3+x^2-2*x+1)/(X1^2*X2^2*x^12+X1^2*X2^2 *x^11-2*X1^2*X2*x^12-2*X1*X2^2*x^12-X1^2*X2^2*x^10-X1^2*X2*x^11+X1^2*x^12+4*X1* X2*x^12+X2^2*x^12+2*X1^2*X2*x^10+4*X1*X2^2*x^10-2*X1*X2*x^11-2*X1*x^12-X2^2*x^ 11-2*X2*x^12-X1^2*X2*x^9-2*X1*X2^2*x^9-7*X1*X2*x^10+2*X1*x^11-2*X2^2*x^10+3*X2* x^11+x^12+6*X1*X2*x^9+X1*x^10+3*X2^2*x^9+3*X2*x^10-2*x^11-2*X1*X2*x^8-2*X1*x^9- X2^2*x^8-7*X2*x^9-X1*X2*x^7+2*X1*x^8+4*X2*x^8+3*x^9+X1*X2*x^6+X1*x^7-3*x^8-2*X1 *X2*x^5-2*X1*x^6-2*X2*x^6+X1*X2*x^4+X1*x^5+3*X2*x^5+3*x^6-3*X2*x^4-2*x^5+X2*x^3 +2*x^4-x^3-2*x^2+3*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 1, 4], are n 191 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 2, 3], are n 89 77 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 2 3 77 ------------ 77 and in floating point 0.3947710171 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 1/2 2 3 77 ate normal pair with correlation, ------------ 77 1/2 1/2 2 3 77 101 i.e. , [[------------, 0], [0, ---]] 77 77 ------------------------------------------------- Theorem Number, 101, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 1, 4], nor the composition, [3, 3, 2] Then infinity ----- \ n ) a(n) x = - ( / ----- n = 0 14 11 10 9 8 7 6 5 4 3 2 / x + x + x + x - x + x - x - x + x - x - x + 2 x - 1) / ( / 15 14 12 11 9 8 7 5 4 3 2 x - x + x + x - x + 2 x - 2 x + x - 2 x + x + 2 x - 3 x + 1) and in Maple format -(x^14+x^11+x^10+x^9-x^8+x^7-x^6-x^5+x^4-x^3-x^2+2*x-1)/(x^15-x^14+x^12+x^11-x^ 9+2*x^8-2*x^7+x^5-2*x^4+x^3+2*x^2-3*x+1) The asymptotic expression for a(n) is, n 0.66099415014764611466 1.9170546374446963081 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 1, 4] and d occurrences (as containment) of the composition, [3, 3, 2], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 3 15 3 15 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x - X1 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 15 2 14 2 15 15 2 14 - 3 X1 X2 x + X1 X2 x + 3 X1 x + 3 X1 X2 x - X1 x 14 15 15 2 12 14 14 15 - 2 X1 X2 x - 3 X1 x - X2 x - X1 X2 x + 2 X1 x + X2 x + x 2 12 12 14 2 10 12 12 2 9 + X1 x + 2 X1 X2 x - x - X1 X2 x - 2 X1 x - X2 x + X1 X2 x 10 12 9 10 8 9 9 + 2 X1 X2 x + x - 4 X1 X2 x - X2 x + 2 X1 X2 x + 2 X1 x + 3 X2 x 7 8 8 9 6 7 7 8 - X1 X2 x - 2 X1 x - 2 X2 x - 2 x - X1 X2 x + X1 x + 2 X2 x + 2 x 5 6 7 4 5 5 4 5 + 2 X1 X2 x + X1 x - 2 x - X1 X2 x - X1 x - 3 X2 x + 3 X2 x + 2 x 3 4 2 / 3 16 3 15 3 16 - X2 x - 2 x + 3 x - 3 x + 1) / (X1 X2 x - X1 X2 x - X1 x / 2 16 3 15 2 15 2 16 16 - 3 X1 X2 x + X1 x + 4 X1 X2 x + 3 X1 x + 3 X1 X2 x 2 14 2 15 15 16 16 2 13 - X1 X2 x - 4 X1 x - 5 X1 X2 x - 3 X1 x - X2 x - X1 X2 x 2 14 14 15 15 16 2 13 + X1 x + 2 X1 X2 x + 5 X1 x + 2 X2 x + x + X1 x 13 14 14 15 2 11 13 13 + 2 X1 X2 x - 2 X1 x - X2 x - 2 x - X1 X2 x - 2 X1 x - X2 x 14 2 10 11 13 2 9 10 11 + x + 2 X1 X2 x + X1 X2 x + x - X1 X2 x - 5 X1 X2 x + X1 x 9 10 10 11 8 9 9 + 5 X1 X2 x + X1 x + 3 X2 x - x - 3 X1 X2 x - 3 X1 x - 4 X2 x 10 7 8 8 9 6 7 8 - x + X1 X2 x + 3 X1 x + 4 X2 x + 3 x + 2 X1 X2 x - 3 X2 x - 4 x 5 6 6 7 4 5 5 6 - 3 X1 X2 x - 2 X1 x - X2 x + 2 x + X1 X2 x + X1 x + 5 X2 x + x 4 5 3 4 3 2 - 4 X2 x - 3 x + X2 x + 3 x + x - 5 x + 4 x - 1) and in Maple format -(X1^3*X2*x^15-X1^3*x^15-3*X1^2*X2*x^15+X1^2*X2*x^14+3*X1^2*x^15+3*X1*X2*x^15- X1^2*x^14-2*X1*X2*x^14-3*X1*x^15-X2*x^15-X1^2*X2*x^12+2*X1*x^14+X2*x^14+x^15+X1 ^2*x^12+2*X1*X2*x^12-x^14-X1^2*X2*x^10-2*X1*x^12-X2*x^12+X1^2*X2*x^9+2*X1*X2*x^ 10+x^12-4*X1*X2*x^9-X2*x^10+2*X1*X2*x^8+2*X1*x^9+3*X2*x^9-X1*X2*x^7-2*X1*x^8-2* X2*x^8-2*x^9-X1*X2*x^6+X1*x^7+2*X2*x^7+2*x^8+2*X1*X2*x^5+X1*x^6-2*x^7-X1*X2*x^4 -X1*x^5-3*X2*x^5+3*X2*x^4+2*x^5-X2*x^3-2*x^4+3*x^2-3*x+1)/(X1^3*X2*x^16-X1^3*X2 *x^15-X1^3*x^16-3*X1^2*X2*x^16+X1^3*x^15+4*X1^2*X2*x^15+3*X1^2*x^16+3*X1*X2*x^ 16-X1^2*X2*x^14-4*X1^2*x^15-5*X1*X2*x^15-3*X1*x^16-X2*x^16-X1^2*X2*x^13+X1^2*x^ 14+2*X1*X2*x^14+5*X1*x^15+2*X2*x^15+x^16+X1^2*x^13+2*X1*X2*x^13-2*X1*x^14-X2*x^ 14-2*x^15-X1^2*X2*x^11-2*X1*x^13-X2*x^13+x^14+2*X1^2*X2*x^10+X1*X2*x^11+x^13-X1 ^2*X2*x^9-5*X1*X2*x^10+X1*x^11+5*X1*X2*x^9+X1*x^10+3*X2*x^10-x^11-3*X1*X2*x^8-3 *X1*x^9-4*X2*x^9-x^10+X1*X2*x^7+3*X1*x^8+4*X2*x^8+3*x^9+2*X1*X2*x^6-3*X2*x^7-4* x^8-3*X1*X2*x^5-2*X1*x^6-X2*x^6+2*x^7+X1*X2*x^4+X1*x^5+5*X2*x^5+x^6-4*X2*x^4-3* x^5+X2*x^3+3*x^4+x^3-5*x^2+4*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 1, 4], are n 191 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 3, 2], are n 97 85 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 14 3 85 ------------- 765 and in floating point 0.2922380026 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 14 255 ate normal pair with correlation, --------- 765 1/2 14 255 2687 i.e. , [[---------, 0], [0, ----]] 765 2295 ------------------------------------------------- Theorem Number, 102, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 1, 4], nor the composition, [3, 4, 1] Then infinity ----- 15 12 9 5 4 3 2 \ n x - x + x + x - x + x - 3 x + 3 x - 1 ) a(n) x = ---------------------------------------------- / 7 6 4 2 ----- (x + x - x + 2 x - 1) (-1 + x) n = 0 and in Maple format (x^15-x^12+x^9+x^5-x^4+x^3-3*x^2+3*x-1)/(x^7+x^6-x^4+2*x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.66071460672041514761 1.9168163613635124211 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 1, 4] and d occurrences (as containment) of the composition, [3, 4, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 15 2 15 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x - X1 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 15 15 15 2 12 15 2 12 - 2 X1 X2 x + 2 X1 x + X2 x - X1 X2 x - x + X1 x 12 2 10 12 12 2 9 10 + 2 X1 X2 x - X1 X2 x - 2 X1 x - X2 x + X1 X2 x + 2 X1 X2 x 12 9 10 8 9 9 7 + x - 3 X1 X2 x - X2 x + X1 X2 x + X1 x + 2 X2 x - X1 X2 x 8 9 6 7 5 6 4 5 - X1 x - x - X1 X2 x + X1 x + 2 X1 X2 x + X1 x - X1 X2 x - X1 x 5 4 3 2 / 2 10 2 9 - x + x - x + 3 x - 3 x + 1) / ((-1 + x) (X1 X2 x - X1 X2 x / 10 9 10 8 9 7 8 - 2 X1 X2 x + 2 X1 X2 x + X2 x - X1 X2 x - X2 x + X1 X2 x + X1 x 8 7 8 5 6 4 5 6 5 + X2 x - X1 x - x - 2 X1 X2 x - X1 x + X1 X2 x + X1 x + x + x 4 2 - x - 2 x + 3 x - 1)) and in Maple format (X1^2*X2*x^15-X1^2*x^15-2*X1*X2*x^15+2*X1*x^15+X2*x^15-X1^2*X2*x^12-x^15+X1^2*x ^12+2*X1*X2*x^12-X1^2*X2*x^10-2*X1*x^12-X2*x^12+X1^2*X2*x^9+2*X1*X2*x^10+x^12-3 *X1*X2*x^9-X2*x^10+X1*X2*x^8+X1*x^9+2*X2*x^9-X1*X2*x^7-X1*x^8-x^9-X1*X2*x^6+X1* x^7+2*X1*X2*x^5+X1*x^6-X1*X2*x^4-X1*x^5-x^5+x^4-x^3+3*x^2-3*x+1)/(-1+x)/(X1^2* X2*x^10-X1^2*X2*x^9-2*X1*X2*x^10+2*X1*X2*x^9+X2*x^10-X1*X2*x^8-X2*x^9+X1*X2*x^7 +X1*x^8+X2*x^8-X1*x^7-x^8-2*X1*X2*x^5-X1*x^6+X1*X2*x^4+X1*x^5+x^6+x^5-x^4-2*x^2 +3*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 1, 4], are n 191 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 4, 1], are n 135 65 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 16 3 65 ------------- 585 and in floating point 0.3819279330 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 16 195 ate normal pair with correlation, --------- 585 1/2 16 195 2267 i.e. , [[---------, 0], [0, ----]] 585 1755 ------------------------------------------------- Theorem Number, 103, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 1, 4], nor the composition, [4, 1, 3] Then infinity ----- \ n ) a(n) x = - ( / ----- n = 0 22 21 20 19 13 12 11 10 9 4 2 x + 2 x + 2 x + x + x + x + x + x + x - x - x + 2 x - 1) / 7 6 5 3 2 / ((x + x + x + x + x + x - 1) / 16 12 9 6 5 2 (x - x + x + x - x - x + 2 x - 1)) and in Maple format -(x^22+2*x^21+2*x^20+x^19+x^13+x^12+x^11+x^10+x^9-x^4-x^2+2*x-1)/(x^7+x^6+x^5+x ^3+x^2+x-1)/(x^16-x^12+x^9+x^6-x^5-x^2+2*x-1) The asymptotic expression for a(n) is, n 0.64880167659124523698 1.9221280043552247737 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 1, 4] and d occurrences (as containment) of the composition, [4, 1, 3], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 3 3 23 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 3 3 22 3 2 23 2 3 23 3 3 21 + X1 X2 x - 3 X1 X2 x - 3 X1 X2 x + X1 X2 x 3 2 22 3 23 2 3 22 2 2 23 - 3 X1 X2 x + 3 X1 X2 x - 3 X1 X2 x + 9 X1 X2 x 3 23 3 3 20 3 2 21 3 22 3 23 + 3 X1 X2 x + X1 X2 x - 3 X1 X2 x + 3 X1 X2 x - X1 x 2 3 21 2 2 22 2 23 3 22 - 2 X1 X2 x + 9 X1 X2 x - 9 X1 X2 x + 3 X1 X2 x 2 23 3 23 3 3 19 3 2 20 3 21 - 9 X1 X2 x - X2 x + X1 X2 x - 2 X1 X2 x + 3 X1 X2 x 3 22 2 3 20 2 2 21 2 22 2 23 - X1 x - 2 X1 X2 x + 6 X1 X2 x - 9 X1 X2 x + 3 X1 x 3 21 2 22 23 3 22 2 23 + X1 X2 x - 9 X1 X2 x + 9 X1 X2 x - X2 x + 3 X2 x 3 2 19 3 20 3 21 2 3 19 2 2 20 - 2 X1 X2 x + X1 X2 x - X1 x - 2 X1 X2 x + 3 X1 X2 x 2 21 2 22 3 20 2 21 22 - 6 X1 X2 x + 3 X1 x + X1 X2 x - 3 X1 X2 x + 9 X1 X2 x 23 2 22 23 3 19 2 2 19 2 21 - 3 X1 x + 3 X2 x - 3 X2 x + X1 X2 x + 3 X1 X2 x + 2 X1 x 3 19 21 22 22 23 3 3 16 + X1 X2 x + 3 X1 X2 x - 3 X1 x - 3 X2 x + x + X1 X2 x 2 20 20 21 2 20 22 3 2 16 - X1 x - 3 X1 X2 x - X1 x - X2 x + x - X1 X2 x 2 3 16 2 19 19 20 2 19 20 - X1 X2 x - X1 x - 3 X1 X2 x + 2 X1 x - X2 x + 2 X2 x 2 2 16 19 19 20 2 16 2 16 - 2 X1 X2 x + 2 X1 x + 2 X2 x - x + 4 X1 X2 x + 4 X1 X2 x 19 2 16 16 2 16 2 2 13 2 14 - x - X1 x - 5 X1 X2 x - X2 x + 2 X1 X2 x - X1 X2 x 16 16 2 13 2 13 14 2 14 + X1 x + X2 x - 4 X1 X2 x - 2 X1 X2 x + 2 X1 X2 x + X2 x 2 2 11 2 13 13 14 14 2 11 - X1 X2 x + 2 X1 x + 4 X1 X2 x - X1 x - 2 X2 x + X1 X2 x 13 14 2 2 9 2 10 11 10 - 2 X1 x + x + X1 X2 x - X1 X2 x + X1 X2 x + 2 X1 X2 x 11 9 10 9 9 9 5 - X1 x - 4 X1 X2 x - X2 x + 2 X1 x + 2 X2 x - x + X1 X2 x 4 5 4 3 2 / 8 8 8 - X1 X2 x - x + x - x + 3 x - 3 x + 1) / ((X1 X2 x - X1 x - X2 x / 6 8 5 6 4 5 4 + X1 X2 x + x + X1 X2 x - X1 x - X1 X2 x - x + x - 2 x + 1) ( 2 2 16 2 16 2 16 2 16 16 2 16 X1 X2 x - 2 X1 X2 x - 2 X1 X2 x + X1 x + 4 X1 X2 x + X2 x 16 16 2 2 12 16 2 12 2 12 - 2 X1 x - 2 X2 x + X1 X2 x + x - X1 X2 x - X1 X2 x 12 12 12 9 9 9 7 9 7 + X1 x + X2 x - x + X1 X2 x - X1 x - X2 x - X1 X2 x + x + X1 x 5 6 6 5 2 + X1 X2 x - X1 x + x - x - x + 2 x - 1)) and in Maple format -(X1^3*X2^3*x^23+X1^3*X2^3*x^22-3*X1^3*X2^2*x^23-3*X1^2*X2^3*x^23+X1^3*X2^3*x^ 21-3*X1^3*X2^2*x^22+3*X1^3*X2*x^23-3*X1^2*X2^3*x^22+9*X1^2*X2^2*x^23+3*X1*X2^3* x^23+X1^3*X2^3*x^20-3*X1^3*X2^2*x^21+3*X1^3*X2*x^22-X1^3*x^23-2*X1^2*X2^3*x^21+ 9*X1^2*X2^2*x^22-9*X1^2*X2*x^23+3*X1*X2^3*x^22-9*X1*X2^2*x^23-X2^3*x^23+X1^3*X2 ^3*x^19-2*X1^3*X2^2*x^20+3*X1^3*X2*x^21-X1^3*x^22-2*X1^2*X2^3*x^20+6*X1^2*X2^2* x^21-9*X1^2*X2*x^22+3*X1^2*x^23+X1*X2^3*x^21-9*X1*X2^2*x^22+9*X1*X2*x^23-X2^3*x ^22+3*X2^2*x^23-2*X1^3*X2^2*x^19+X1^3*X2*x^20-X1^3*x^21-2*X1^2*X2^3*x^19+3*X1^2 *X2^2*x^20-6*X1^2*X2*x^21+3*X1^2*x^22+X1*X2^3*x^20-3*X1*X2^2*x^21+9*X1*X2*x^22-\ 3*X1*x^23+3*X2^2*x^22-3*X2*x^23+X1^3*X2*x^19+3*X1^2*X2^2*x^19+2*X1^2*x^21+X1*X2 ^3*x^19+3*X1*X2*x^21-3*X1*x^22-3*X2*x^22+x^23+X1^3*X2^3*x^16-X1^2*x^20-3*X1*X2* x^20-X1*x^21-X2^2*x^20+x^22-X1^3*X2^2*x^16-X1^2*X2^3*x^16-X1^2*x^19-3*X1*X2*x^ 19+2*X1*x^20-X2^2*x^19+2*X2*x^20-2*X1^2*X2^2*x^16+2*X1*x^19+2*X2*x^19-x^20+4*X1 ^2*X2*x^16+4*X1*X2^2*x^16-x^19-X1^2*x^16-5*X1*X2*x^16-X2^2*x^16+2*X1^2*X2^2*x^ 13-X1*X2^2*x^14+X1*x^16+X2*x^16-4*X1^2*X2*x^13-2*X1*X2^2*x^13+2*X1*X2*x^14+X2^2 *x^14-X1^2*X2^2*x^11+2*X1^2*x^13+4*X1*X2*x^13-X1*x^14-2*X2*x^14+X1^2*X2*x^11-2* X1*x^13+x^14+X1^2*X2^2*x^9-X1^2*X2*x^10+X1*X2*x^11+2*X1*X2*x^10-X1*x^11-4*X1*X2 *x^9-X2*x^10+2*X1*x^9+2*X2*x^9-x^9+X1*X2*x^5-X1*X2*x^4-x^5+x^4-x^3+3*x^2-3*x+1) /(X1*X2*x^8-X1*x^8-X2*x^8+X1*X2*x^6+x^8+X1*X2*x^5-X1*x^6-X1*X2*x^4-x^5+x^4-2*x+ 1)/(X1^2*X2^2*x^16-2*X1^2*X2*x^16-2*X1*X2^2*x^16+X1^2*x^16+4*X1*X2*x^16+X2^2*x^ 16-2*X1*x^16-2*X2*x^16+X1^2*X2^2*x^12+x^16-X1^2*X2*x^12-X1*X2^2*x^12+X1*x^12+X2 *x^12-x^12+X1*X2*x^9-X1*x^9-X2*x^9-X1*X2*x^7+x^9+X1*x^7+X1*X2*x^5-X1*x^6+x^6-x^ 5-x^2+2*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 1, 4], are n 191 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 1, 3], are n 141 65 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 22 3 65 ------------- 585 and in floating point 0.5251509079 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 22 195 ate normal pair with correlation, --------- 585 1/2 22 195 2723 i.e. , [[---------, 0], [0, ----]] 585 1755 ------------------------------------------------- Theorem Number, 104, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 1, 4], nor the composition, [4, 2, 2] Then infinity ----- \ n ) a(n) x = / ----- n = 0 14 13 11 10 9 8 4 2 x + 2 x - 2 x - 2 x - x - x + x + x - 2 x + 1 - ---------------------------------------------------------------- 15 14 12 11 9 6 5 4 2 x + 2 x - 2 x - 2 x - x + x + x - x - 2 x + 3 x - 1 and in Maple format -(x^14+2*x^13-2*x^11-2*x^10-x^9-x^8+x^4+x^2-2*x+1)/(x^15+2*x^14-2*x^12-2*x^11-x ^9+x^6+x^5-x^4-2*x^2+3*x-1) The asymptotic expression for a(n) is, n 0.65170350411535370406 1.9191057201085461028 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 1, 4] and d occurrences (as containment) of the composition, [4, 2, 2], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 3 2 16 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 3 2 15 3 16 2 2 16 3 15 2 2 15 + X1 X2 x - X1 X2 x - 3 X1 X2 x - X1 X2 x - 4 X1 X2 x 2 16 2 16 2 15 2 15 16 + 3 X1 X2 x + 3 X1 X2 x + 5 X1 X2 x + 5 X1 X2 x - 3 X1 X2 x 2 16 2 2 13 2 14 2 15 15 2 15 - X2 x + X1 X2 x + X1 X2 x - X1 x - 7 X1 X2 x - 2 X2 x 16 2 13 2 14 2 13 14 15 + X2 x - 2 X1 X2 x - X1 x - 3 X1 X2 x - 2 X1 X2 x + 2 X1 x 15 2 12 2 13 2 12 13 14 + 3 X2 x - X1 X2 x + X1 x - X1 X2 x + 6 X1 X2 x + 2 X1 x 2 13 14 15 2 12 12 13 2 12 + 2 X2 x + X2 x - x + X1 x + 4 X1 X2 x - 3 X1 x + X2 x 13 14 2 10 12 12 13 2 9 - 4 X2 x - x - X1 X2 x - 3 X1 x - 3 X2 x + 2 x + X1 X2 x 10 12 9 10 8 9 10 + X1 X2 x + 2 x - 2 X1 X2 x + X1 x - X1 X2 x + X2 x - x 7 8 8 6 7 8 5 + X1 X2 x + X1 x + X2 x - 2 X1 X2 x - X1 x - x + 2 X1 X2 x 6 4 5 5 4 3 2 / + 2 X1 x - X1 X2 x - X1 x - x + x - x + 3 x - 3 x + 1) / ( / 3 2 17 3 17 2 2 17 3 2 15 2 2 16 X1 X2 x - X1 X2 x - 3 X1 X2 x - X1 X2 x - X1 X2 x 2 17 2 17 3 15 2 2 15 2 16 + 3 X1 X2 x + 3 X1 X2 x + X1 X2 x + 3 X1 X2 x + 2 X1 X2 x 2 16 17 2 17 2 15 2 16 + 2 X1 X2 x - 3 X1 X2 x - X2 x - 2 X1 X2 x - X1 x 2 15 16 2 16 17 2 2 13 2 14 - 3 X1 X2 x - 4 X1 X2 x - X2 x + X2 x - X1 X2 x - X1 X2 x 2 15 2 14 15 16 2 15 16 - X1 x - X1 X2 x + X1 X2 x + 2 X1 x + X2 x + 2 X2 x 2 14 2 13 14 15 2 14 16 2 13 + X1 x + X1 X2 x + 4 X1 X2 x + 2 X1 x + X2 x - x + X1 x 13 14 14 15 2 11 13 + 2 X1 X2 x - 3 X1 x - 3 X2 x - x - X1 X2 x - 3 X1 x 13 14 2 10 13 2 9 10 - 2 X2 x + 2 x + 2 X1 X2 x + 2 x - X1 X2 x - 3 X1 X2 x 11 11 9 10 10 11 8 + 2 X1 x + X2 x + X1 X2 x - X1 x + X2 x - 2 x + X1 X2 x 9 10 7 8 9 6 7 + X1 x + x - 2 X1 X2 x - X1 x - x + 3 X1 X2 x + 3 X1 x 5 6 7 4 5 5 4 3 2 - 3 X1 X2 x - 3 X1 x - x + X1 X2 x + X1 x + 2 x - x + 2 x - 5 x + 4 x - 1) and in Maple format -(X1^3*X2^2*x^16+X1^3*X2^2*x^15-X1^3*X2*x^16-3*X1^2*X2^2*x^16-X1^3*X2*x^15-4*X1 ^2*X2^2*x^15+3*X1^2*X2*x^16+3*X1*X2^2*x^16+5*X1^2*X2*x^15+5*X1*X2^2*x^15-3*X1* X2*x^16-X2^2*x^16+X1^2*X2^2*x^13+X1^2*X2*x^14-X1^2*x^15-7*X1*X2*x^15-2*X2^2*x^ 15+X2*x^16-2*X1^2*X2*x^13-X1^2*x^14-3*X1*X2^2*x^13-2*X1*X2*x^14+2*X1*x^15+3*X2* x^15-X1^2*X2*x^12+X1^2*x^13-X1*X2^2*x^12+6*X1*X2*x^13+2*X1*x^14+2*X2^2*x^13+X2* x^14-x^15+X1^2*x^12+4*X1*X2*x^12-3*X1*x^13+X2^2*x^12-4*X2*x^13-x^14-X1^2*X2*x^ 10-3*X1*x^12-3*X2*x^12+2*x^13+X1^2*X2*x^9+X1*X2*x^10+2*x^12-2*X1*X2*x^9+X1*x^10 -X1*X2*x^8+X2*x^9-x^10+X1*X2*x^7+X1*x^8+X2*x^8-2*X1*X2*x^6-X1*x^7-x^8+2*X1*X2*x ^5+2*X1*x^6-X1*X2*x^4-X1*x^5-x^5+x^4-x^3+3*x^2-3*x+1)/(X1^3*X2^2*x^17-X1^3*X2*x ^17-3*X1^2*X2^2*x^17-X1^3*X2^2*x^15-X1^2*X2^2*x^16+3*X1^2*X2*x^17+3*X1*X2^2*x^ 17+X1^3*X2*x^15+3*X1^2*X2^2*x^15+2*X1^2*X2*x^16+2*X1*X2^2*x^16-3*X1*X2*x^17-X2^ 2*x^17-2*X1^2*X2*x^15-X1^2*x^16-3*X1*X2^2*x^15-4*X1*X2*x^16-X2^2*x^16+X2*x^17- X1^2*X2^2*x^13-X1^2*X2*x^14-X1^2*x^15-X1*X2^2*x^14+X1*X2*x^15+2*X1*x^16+X2^2*x^ 15+2*X2*x^16+X1^2*x^14+X1*X2^2*x^13+4*X1*X2*x^14+2*X1*x^15+X2^2*x^14-x^16+X1^2* x^13+2*X1*X2*x^13-3*X1*x^14-3*X2*x^14-x^15-X1^2*X2*x^11-3*X1*x^13-2*X2*x^13+2*x ^14+2*X1^2*X2*x^10+2*x^13-X1^2*X2*x^9-3*X1*X2*x^10+2*X1*x^11+X2*x^11+X1*X2*x^9- X1*x^10+X2*x^10-2*x^11+X1*X2*x^8+X1*x^9+x^10-2*X1*X2*x^7-X1*x^8-x^9+3*X1*X2*x^6 +3*X1*x^7-3*X1*X2*x^5-3*X1*x^6-x^7+X1*X2*x^4+X1*x^5+2*x^5-x^4+2*x^3-5*x^2+4*x-1 ) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 1, 4], are n 191 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 2, 2], are n 75 69 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 2 23 ------- 23 and in floating point 0.4170288281 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 2 23 ate normal pair with correlation, ------- 23 1/2 2 23 31 i.e. , [[-------, 0], [0, --]] 23 23 ------------------------------------------------- Theorem Number, 105, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 1, 4], nor the composition, [4, 3, 1] Then infinity ----- \ n 18 17 16 15 14 13 12 11 10 ) a(n) x = (x + 2 x + 2 x + x - x - x + x + 2 x + 2 x / ----- n = 0 8 4 2 / + x - x - x + 2 x - 1) / ( / 10 9 8 7 6 3 2 2 (x + 2 x + 3 x + 2 x + x + x + x + x - 1) (-1 + x) ) and in Maple format (x^18+2*x^17+2*x^16+x^15-x^14-x^13+x^12+2*x^11+2*x^10+x^8-x^4-x^2+2*x-1)/(x^10+ 2*x^9+3*x^8+2*x^7+x^6+x^3+x^2+x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.65353748308844606444 1.9175371779461324874 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 1, 4] and d occurrences (as containment) of the composition, [4, 3, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 3 2 19 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 3 2 18 3 19 2 2 19 3 18 3 19 + X1 X2 x - 2 X1 X2 x - 3 X1 X2 x - 2 X1 X2 x + X1 x 2 2 18 2 19 2 19 3 18 2 18 - 3 X1 X2 x + 6 X1 X2 x + 3 X1 X2 x + X1 x + 6 X1 X2 x 2 19 2 18 19 2 19 2 2 16 - 3 X1 x + 3 X1 X2 x - 6 X1 X2 x - X2 x + X1 X2 x 2 18 18 19 2 18 19 2 2 15 - 3 X1 x - 6 X1 X2 x + 3 X1 x - X2 x + 2 X2 x + X1 X2 x 2 16 2 16 18 18 19 2 2 14 - 2 X1 X2 x - 2 X1 X2 x + 3 X1 x + 2 X2 x - x - X1 X2 x 2 15 2 16 2 15 16 2 16 18 - 3 X1 X2 x + X1 x - 2 X1 X2 x + 4 X1 X2 x + X2 x - x 2 2 13 2 14 2 15 2 14 15 + X1 X2 x + X1 X2 x + 2 X1 x + 2 X1 X2 x + 6 X1 X2 x 16 2 15 16 2 2 12 2 13 14 - 2 X1 x + X2 x - 2 X2 x - X1 X2 x - X1 X2 x - 2 X1 X2 x 15 2 14 15 16 2 12 2 13 - 4 X1 x - X2 x - 3 X2 x + x + 2 X1 X2 x - X1 x 2 12 13 14 15 2 12 12 + 2 X1 X2 x - 2 X1 X2 x + X2 x + 2 x - X1 x - 4 X1 X2 x 13 2 12 13 2 10 12 12 13 + 3 X1 x - X2 x + 2 X2 x + X1 X2 x + 2 X1 x + 2 X2 x - 2 x 2 9 12 9 10 10 9 10 - X1 X2 x - x + X1 X2 x - 2 X1 x - X2 x + X1 x + 2 x 7 8 9 6 7 8 5 6 + X1 X2 x - X2 x - x + X1 X2 x - X1 x + x - 2 X1 X2 x - X1 x 4 5 5 4 3 2 / + X1 X2 x + X1 x + x - x + x - 3 x + 3 x - 1) / ((-1 + x) ( / 2 2 13 2 2 12 2 13 2 13 2 12 X1 X2 x - X1 X2 x - X1 X2 x - 2 X1 X2 x + X1 X2 x 2 12 13 2 13 13 2 10 12 + X1 X2 x + 2 X1 X2 x + X2 x - X2 x + X1 X2 x - X1 x 12 2 9 10 12 10 9 9 - X2 x - X1 X2 x - 2 X1 X2 x + x + X2 x + 2 X1 x + X2 x 7 9 7 5 6 4 5 6 + X1 X2 x - 2 x - X1 x - 2 X1 X2 x - X1 x + X1 X2 x + X1 x + x 5 4 2 + x - x - 2 x + 3 x - 1)) and in Maple format -(X1^3*X2^2*x^19+X1^3*X2^2*x^18-2*X1^3*X2*x^19-3*X1^2*X2^2*x^19-2*X1^3*X2*x^18+ X1^3*x^19-3*X1^2*X2^2*x^18+6*X1^2*X2*x^19+3*X1*X2^2*x^19+X1^3*x^18+6*X1^2*X2*x^ 18-3*X1^2*x^19+3*X1*X2^2*x^18-6*X1*X2*x^19-X2^2*x^19+X1^2*X2^2*x^16-3*X1^2*x^18 -6*X1*X2*x^18+3*X1*x^19-X2^2*x^18+2*X2*x^19+X1^2*X2^2*x^15-2*X1^2*X2*x^16-2*X1* X2^2*x^16+3*X1*x^18+2*X2*x^18-x^19-X1^2*X2^2*x^14-3*X1^2*X2*x^15+X1^2*x^16-2*X1 *X2^2*x^15+4*X1*X2*x^16+X2^2*x^16-x^18+X1^2*X2^2*x^13+X1^2*X2*x^14+2*X1^2*x^15+ 2*X1*X2^2*x^14+6*X1*X2*x^15-2*X1*x^16+X2^2*x^15-2*X2*x^16-X1^2*X2^2*x^12-X1*X2^ 2*x^13-2*X1*X2*x^14-4*X1*x^15-X2^2*x^14-3*X2*x^15+x^16+2*X1^2*X2*x^12-X1^2*x^13 +2*X1*X2^2*x^12-2*X1*X2*x^13+X2*x^14+2*x^15-X1^2*x^12-4*X1*X2*x^12+3*X1*x^13-X2 ^2*x^12+2*X2*x^13+X1^2*X2*x^10+2*X1*x^12+2*X2*x^12-2*x^13-X1^2*X2*x^9-x^12+X1* X2*x^9-2*X1*x^10-X2*x^10+X1*x^9+2*x^10+X1*X2*x^7-X2*x^8-x^9+X1*X2*x^6-X1*x^7+x^ 8-2*X1*X2*x^5-X1*x^6+X1*X2*x^4+X1*x^5+x^5-x^4+x^3-3*x^2+3*x-1)/(-1+x)/(X1^2*X2^ 2*x^13-X1^2*X2^2*x^12-X1^2*X2*x^13-2*X1*X2^2*x^13+X1^2*X2*x^12+X1*X2^2*x^12+2* X1*X2*x^13+X2^2*x^13-X2*x^13+X1^2*X2*x^10-X1*x^12-X2*x^12-X1^2*X2*x^9-2*X1*X2*x ^10+x^12+X2*x^10+2*X1*x^9+X2*x^9+X1*X2*x^7-2*x^9-X1*x^7-2*X1*X2*x^5-X1*x^6+X1* X2*x^4+X1*x^5+x^6+x^5-x^4-2*x^2+3*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 1, 4], are n 191 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 3, 1], are n 135 65 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 16 3 65 ------------- 585 and in floating point 0.3819279330 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 16 195 ate normal pair with correlation, --------- 585 1/2 16 195 2267 i.e. , [[---------, 0], [0, ----]] 585 1755 ------------------------------------------------- Theorem Number, 106, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 1, 4], nor the composition, [5, 1, 2] Then infinity ----- \ n 26 25 24 23 21 20 19 18 ) a(n) x = - (x + 4 x + 6 x + 4 x - 3 x - 3 x - x - x / ----- n = 0 17 16 15 14 11 10 9 5 4 3 2 - x + x + 2 x + x + x - 2 x - x - x + x - x + 3 x - 3 x / + 1) / ( / 18 17 16 13 12 10 9 6 5 2 (x + 2 x + x - x - x + x + x + x - x - x + 2 x - 1) 9 8 5 4 (x + x - x + x - 2 x + 1)) and in Maple format -(x^26+4*x^25+6*x^24+4*x^23-3*x^21-3*x^20-x^19-x^18-x^17+x^16+2*x^15+x^14+x^11-\ 2*x^10-x^9-x^5+x^4-x^3+3*x^2-3*x+1)/(x^18+2*x^17+x^16-x^13-x^12+x^10+x^9+x^6-x^ 5-x^2+2*x-1)/(x^9+x^8-x^5+x^4-2*x+1) The asymptotic expression for a(n) is, n 0.66036270785748552732 1.9159714162121856417 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 1, 4] and d occurrences (as containment) of the composition, [5, 1, 2], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 3 3 26 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 3 3 25 3 2 26 2 3 26 3 3 24 + 4 X1 X2 x - 3 X1 X2 x - 3 X1 X2 x + 6 X1 X2 x 3 2 25 3 26 2 3 25 2 2 26 - 12 X1 X2 x + 3 X1 X2 x - 12 X1 X2 x + 9 X1 X2 x 3 26 3 3 23 3 2 24 3 25 3 26 + 3 X1 X2 x + 5 X1 X2 x - 18 X1 X2 x + 12 X1 X2 x - X1 x 2 3 24 2 2 25 2 26 3 25 - 18 X1 X2 x + 36 X1 X2 x - 9 X1 X2 x + 12 X1 X2 x 2 26 3 26 3 3 22 3 2 23 3 24 - 9 X1 X2 x - X2 x + 4 X1 X2 x - 15 X1 X2 x + 18 X1 X2 x 3 25 2 3 23 2 2 24 2 25 2 26 - 4 X1 x - 14 X1 X2 x + 54 X1 X2 x - 36 X1 X2 x + 3 X1 x 3 24 2 25 26 3 25 2 26 + 18 X1 X2 x - 36 X1 X2 x + 9 X1 X2 x - 4 X2 x + 3 X2 x 3 3 21 3 2 22 3 23 3 24 + 3 X1 X2 x - 11 X1 X2 x + 15 X1 X2 x - 6 X1 x 2 3 22 2 2 23 2 24 2 25 - 9 X1 X2 x + 42 X1 X2 x - 54 X1 X2 x + 12 X1 x 3 23 2 24 25 26 3 24 + 13 X1 X2 x - 54 X1 X2 x + 36 X1 X2 x - 3 X1 x - 6 X2 x 2 25 26 3 3 20 3 2 21 3 22 + 12 X2 x - 3 X2 x + X1 X2 x - 6 X1 X2 x + 10 X1 X2 x 3 23 2 3 21 2 2 22 2 23 2 24 - 5 X1 x - 6 X1 X2 x + 24 X1 X2 x - 42 X1 X2 x + 18 X1 x 3 22 2 23 24 25 3 23 + 6 X1 X2 x - 39 X1 X2 x + 54 X1 X2 x - 12 X1 x - 4 X2 x 2 24 25 26 3 3 19 3 21 3 22 + 18 X2 x - 12 X2 x + x + X1 X2 x + 3 X1 X2 x - 3 X1 x 2 3 20 2 2 21 2 22 2 23 - 2 X1 X2 x + 9 X1 X2 x - 21 X1 X2 x + 14 X1 x 3 21 2 22 23 24 3 22 + 3 X1 X2 x - 15 X1 X2 x + 39 X1 X2 x - 18 X1 x - X2 x 2 23 24 25 3 3 18 3 2 19 + 12 X2 x - 18 X2 x + 4 x + X1 X2 x - 2 X1 X2 x 3 20 2 3 19 2 2 20 2 22 3 20 - 3 X1 X2 x - X1 X2 x - 3 X1 X2 x + 6 X1 x + X1 X2 x 22 23 2 22 23 24 3 2 18 + 12 X1 X2 x - 13 X1 x + 2 X2 x - 12 X2 x + 6 x - X1 X2 x 3 19 3 20 2 3 18 2 20 2 21 + X1 X2 x + 2 X1 x - X1 X2 x + 12 X1 X2 x - 3 X1 x 2 20 21 22 2 21 22 23 + 6 X1 X2 x - 9 X1 X2 x - 3 X1 x - 3 X2 x - X2 x + 4 x 3 2 17 3 18 2 2 18 2 19 2 20 + 2 X1 X2 x - X1 X2 x - 2 X1 X2 x + 3 X1 X2 x - 7 X1 x 2 19 20 21 2 20 21 + 3 X1 X2 x - 15 X1 X2 x + 6 X1 x - 3 X2 x + 6 X2 x 3 17 3 18 2 2 17 2 18 2 19 - 3 X1 X2 x + X1 x - 6 X1 X2 x + 7 X1 X2 x - 2 X1 x 2 18 19 20 2 19 20 21 + 4 X1 X2 x - 6 X1 X2 x + 8 X1 x - X2 x + 6 X2 x - 3 x 3 16 3 17 2 17 2 18 2 17 + X1 X2 x + X1 x + 9 X1 X2 x - 4 X1 x + 6 X1 X2 x 18 19 2 18 19 20 3 16 - 8 X1 X2 x + 3 X1 x - X2 x + 2 X2 x - 3 x - X1 x 2 2 15 2 16 2 17 17 18 + 2 X1 X2 x - 3 X1 X2 x - 3 X1 x - 9 X1 X2 x + 4 X1 x 2 17 18 19 2 2 14 2 15 2 16 - 2 X2 x + 2 X2 x - x + X1 X2 x - 4 X1 X2 x + 3 X1 x 2 15 16 17 17 18 2 14 - 4 X1 X2 x + 3 X1 X2 x + 3 X1 x + 3 X2 x - x - 2 X1 X2 x 2 15 2 14 15 16 2 15 16 + 2 X1 x - 2 X1 X2 x + 8 X1 X2 x - 3 X1 x + 2 X2 x - X2 x 17 2 2 12 2 14 14 15 2 14 - x - X1 X2 x + X1 x + 4 X1 X2 x - 4 X1 x + X2 x 15 16 2 2 11 2 12 14 14 - 4 X2 x + x + X1 X2 x + 2 X1 X2 x - 2 X1 x - 2 X2 x 15 2 11 2 12 14 2 10 2 11 + 2 x - 4 X1 X2 x - X1 x + x + 2 X1 X2 x + 3 X1 x 11 2 10 10 11 11 2 9 + 3 X1 X2 x - 3 X1 x - 3 X1 X2 x - 3 X1 x - X2 x + X1 x 9 10 10 11 9 10 9 6 - 2 X1 X2 x + 5 X1 x + X2 x + x + 2 X2 x - 2 x - x + X1 X2 x 5 6 5 4 5 4 3 2 / - X1 X2 x - X1 x + 2 X1 x - X1 x - x + x - x + 3 x - 3 x + 1) / / 9 8 9 9 8 8 9 6 8 ((X1 X2 x + X1 X2 x - X1 x - X2 x - X1 x - X2 x + x + X1 X2 x + x 5 6 5 4 5 4 2 2 18 - X1 X2 x - X1 x + 2 X1 x - X1 x - x + x - 2 x + 1) (X1 X2 x 2 2 17 2 18 2 18 2 2 16 2 17 + 2 X1 X2 x - 2 X1 X2 x - 2 X1 X2 x + X1 X2 x - 4 X1 X2 x 2 18 2 17 18 2 18 2 16 + X1 x - 4 X1 X2 x + 4 X1 X2 x + X2 x - 2 X1 X2 x 2 17 2 16 17 18 2 17 18 + 2 X1 x - 2 X1 X2 x + 8 X1 X2 x - 2 X1 x + 2 X2 x - 2 X2 x 2 2 14 2 16 16 17 2 16 17 + X1 X2 x + X1 x + 4 X1 X2 x - 4 X1 x + X2 x - 4 X2 x 18 2 2 13 2 14 2 14 16 16 + x + X1 X2 x - 2 X1 X2 x - X1 X2 x - 2 X1 x - 2 X2 x 17 2 13 2 14 2 13 14 16 + 2 x - X1 X2 x + X1 x - X1 X2 x + 2 X1 X2 x + x 2 12 14 2 12 12 13 13 12 + X1 X2 x - X1 x - X1 x - 2 X1 X2 x + X1 x + X2 x + 2 X1 x 12 13 10 12 9 10 10 9 + X2 x - x + X1 X2 x - x + X1 X2 x - X1 x - X2 x - X1 x 9 10 7 9 6 7 6 5 6 - X2 x + x - X1 X2 x + x + X1 X2 x + X1 x - 2 X1 x + X1 x + x 5 2 - x - x + 2 x - 1)) and in Maple format -(X1^3*X2^3*x^26+4*X1^3*X2^3*x^25-3*X1^3*X2^2*x^26-3*X1^2*X2^3*x^26+6*X1^3*X2^3 *x^24-12*X1^3*X2^2*x^25+3*X1^3*X2*x^26-12*X1^2*X2^3*x^25+9*X1^2*X2^2*x^26+3*X1* X2^3*x^26+5*X1^3*X2^3*x^23-18*X1^3*X2^2*x^24+12*X1^3*X2*x^25-X1^3*x^26-18*X1^2* X2^3*x^24+36*X1^2*X2^2*x^25-9*X1^2*X2*x^26+12*X1*X2^3*x^25-9*X1*X2^2*x^26-X2^3* x^26+4*X1^3*X2^3*x^22-15*X1^3*X2^2*x^23+18*X1^3*X2*x^24-4*X1^3*x^25-14*X1^2*X2^ 3*x^23+54*X1^2*X2^2*x^24-36*X1^2*X2*x^25+3*X1^2*x^26+18*X1*X2^3*x^24-36*X1*X2^2 *x^25+9*X1*X2*x^26-4*X2^3*x^25+3*X2^2*x^26+3*X1^3*X2^3*x^21-11*X1^3*X2^2*x^22+ 15*X1^3*X2*x^23-6*X1^3*x^24-9*X1^2*X2^3*x^22+42*X1^2*X2^2*x^23-54*X1^2*X2*x^24+ 12*X1^2*x^25+13*X1*X2^3*x^23-54*X1*X2^2*x^24+36*X1*X2*x^25-3*X1*x^26-6*X2^3*x^ 24+12*X2^2*x^25-3*X2*x^26+X1^3*X2^3*x^20-6*X1^3*X2^2*x^21+10*X1^3*X2*x^22-5*X1^ 3*x^23-6*X1^2*X2^3*x^21+24*X1^2*X2^2*x^22-42*X1^2*X2*x^23+18*X1^2*x^24+6*X1*X2^ 3*x^22-39*X1*X2^2*x^23+54*X1*X2*x^24-12*X1*x^25-4*X2^3*x^23+18*X2^2*x^24-12*X2* x^25+x^26+X1^3*X2^3*x^19+3*X1^3*X2*x^21-3*X1^3*x^22-2*X1^2*X2^3*x^20+9*X1^2*X2^ 2*x^21-21*X1^2*X2*x^22+14*X1^2*x^23+3*X1*X2^3*x^21-15*X1*X2^2*x^22+39*X1*X2*x^ 23-18*X1*x^24-X2^3*x^22+12*X2^2*x^23-18*X2*x^24+4*x^25+X1^3*X2^3*x^18-2*X1^3*X2 ^2*x^19-3*X1^3*X2*x^20-X1^2*X2^3*x^19-3*X1^2*X2^2*x^20+6*X1^2*x^22+X1*X2^3*x^20 +12*X1*X2*x^22-13*X1*x^23+2*X2^2*x^22-12*X2*x^23+6*x^24-X1^3*X2^2*x^18+X1^3*X2* x^19+2*X1^3*x^20-X1^2*X2^3*x^18+12*X1^2*X2*x^20-3*X1^2*x^21+6*X1*X2^2*x^20-9*X1 *X2*x^21-3*X1*x^22-3*X2^2*x^21-X2*x^22+4*x^23+2*X1^3*X2^2*x^17-X1^3*X2*x^18-2* X1^2*X2^2*x^18+3*X1^2*X2*x^19-7*X1^2*x^20+3*X1*X2^2*x^19-15*X1*X2*x^20+6*X1*x^ 21-3*X2^2*x^20+6*X2*x^21-3*X1^3*X2*x^17+X1^3*x^18-6*X1^2*X2^2*x^17+7*X1^2*X2*x^ 18-2*X1^2*x^19+4*X1*X2^2*x^18-6*X1*X2*x^19+8*X1*x^20-X2^2*x^19+6*X2*x^20-3*x^21 +X1^3*X2*x^16+X1^3*x^17+9*X1^2*X2*x^17-4*X1^2*x^18+6*X1*X2^2*x^17-8*X1*X2*x^18+ 3*X1*x^19-X2^2*x^18+2*X2*x^19-3*x^20-X1^3*x^16+2*X1^2*X2^2*x^15-3*X1^2*X2*x^16-\ 3*X1^2*x^17-9*X1*X2*x^17+4*X1*x^18-2*X2^2*x^17+2*X2*x^18-x^19+X1^2*X2^2*x^14-4* X1^2*X2*x^15+3*X1^2*x^16-4*X1*X2^2*x^15+3*X1*X2*x^16+3*X1*x^17+3*X2*x^17-x^18-2 *X1^2*X2*x^14+2*X1^2*x^15-2*X1*X2^2*x^14+8*X1*X2*x^15-3*X1*x^16+2*X2^2*x^15-X2* x^16-x^17-X1^2*X2^2*x^12+X1^2*x^14+4*X1*X2*x^14-4*X1*x^15+X2^2*x^14-4*X2*x^15+x ^16+X1^2*X2^2*x^11+2*X1^2*X2*x^12-2*X1*x^14-2*X2*x^14+2*x^15-4*X1^2*X2*x^11-X1^ 2*x^12+x^14+2*X1^2*X2*x^10+3*X1^2*x^11+3*X1*X2*x^11-3*X1^2*x^10-3*X1*X2*x^10-3* X1*x^11-X2*x^11+X1^2*x^9-2*X1*X2*x^9+5*X1*x^10+X2*x^10+x^11+2*X2*x^9-2*x^10-x^9 +X1*X2*x^6-X1*X2*x^5-X1*x^6+2*X1*x^5-X1*x^4-x^5+x^4-x^3+3*x^2-3*x+1)/(X1*X2*x^9 +X1*X2*x^8-X1*x^9-X2*x^9-X1*x^8-X2*x^8+x^9+X1*X2*x^6+x^8-X1*X2*x^5-X1*x^6+2*X1* x^5-X1*x^4-x^5+x^4-2*x+1)/(X1^2*X2^2*x^18+2*X1^2*X2^2*x^17-2*X1^2*X2*x^18-2*X1* X2^2*x^18+X1^2*X2^2*x^16-4*X1^2*X2*x^17+X1^2*x^18-4*X1*X2^2*x^17+4*X1*X2*x^18+ X2^2*x^18-2*X1^2*X2*x^16+2*X1^2*x^17-2*X1*X2^2*x^16+8*X1*X2*x^17-2*X1*x^18+2*X2 ^2*x^17-2*X2*x^18+X1^2*X2^2*x^14+X1^2*x^16+4*X1*X2*x^16-4*X1*x^17+X2^2*x^16-4* X2*x^17+x^18+X1^2*X2^2*x^13-2*X1^2*X2*x^14-X1*X2^2*x^14-2*X1*x^16-2*X2*x^16+2*x ^17-X1^2*X2*x^13+X1^2*x^14-X1*X2^2*x^13+2*X1*X2*x^14+x^16+X1^2*X2*x^12-X1*x^14- X1^2*x^12-2*X1*X2*x^12+X1*x^13+X2*x^13+2*X1*x^12+X2*x^12-x^13+X1*X2*x^10-x^12+ X1*X2*x^9-X1*x^10-X2*x^10-X1*x^9-X2*x^9+x^10-X1*X2*x^7+x^9+X1*X2*x^6+X1*x^7-2* X1*x^6+X1*x^5+x^6-x^5-x^2+2*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 1, 4], are n 191 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [5, 1, 2], are n 113 57 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 14 19 -------- 171 and in floating point 0.3568689194 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 14 19 ate normal pair with correlation, -------- 171 1/2 14 19 1931 i.e. , [[--------, 0], [0, ----]] 171 1539 ------------------------------------------------- Theorem Number, 107, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 1, 4], nor the composition, [5, 2, 1] Then infinity ----- \ n 21 20 19 18 17 14 13 12 ) a(n) x = (x + 2 x + x - x - 2 x + 2 x + 2 x - x / ----- n = 0 10 9 8 5 4 3 2 / - 2 x + x - x - x + x - x + 3 x - 3 x + 1) / ( / 14 13 12 11 10 9 8 7 6 4 (x + x + x + 2 x + x + x - x - x - x + x - 2 x + 1) 2 (-1 + x) ) and in Maple format (x^21+2*x^20+x^19-x^18-2*x^17+2*x^14+2*x^13-x^12-2*x^10+x^9-x^8-x^5+x^4-x^3+3*x ^2-3*x+1)/(x^14+x^13+x^12+2*x^11+x^10+x^9-x^8-x^7-x^6+x^4-2*x+1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.66293686307441587764 1.9144061473285935513 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 1, 4] and d occurrences (as containment) of the composition, [5, 2, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 3 2 21 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 3 2 20 3 21 2 2 21 3 2 19 + 2 X1 X2 x - 2 X1 X2 x - 3 X1 X2 x + X1 X2 x 3 20 3 21 2 2 20 2 21 2 21 - 4 X1 X2 x + X1 x - 6 X1 X2 x + 6 X1 X2 x + 3 X1 X2 x 3 19 3 20 2 2 19 2 20 2 21 - 2 X1 X2 x + 2 X1 x - 3 X1 X2 x + 12 X1 X2 x - 3 X1 x 2 20 21 2 21 3 19 2 2 18 + 6 X1 X2 x - 6 X1 X2 x - X2 x + X1 x + X1 X2 x 2 19 2 20 2 19 20 21 + 6 X1 X2 x - 6 X1 x + 3 X1 X2 x - 12 X1 X2 x + 3 X1 x 2 20 21 2 2 17 2 18 2 19 - 2 X2 x + 2 X2 x + 2 X1 X2 x - 2 X1 X2 x - 3 X1 x 2 18 19 20 2 19 20 21 - 2 X1 X2 x - 6 X1 X2 x + 6 X1 x - X2 x + 4 X2 x - x 2 17 2 18 2 17 18 19 2 18 - 4 X1 X2 x + X1 x - 4 X1 X2 x + 4 X1 X2 x + 3 X1 x + X2 x 19 20 2 17 17 18 2 17 + 2 X2 x - 2 x + 2 X1 x + 8 X1 X2 x - 2 X1 x + 2 X2 x 18 19 2 15 2 15 17 17 18 - 2 X2 x - x - X1 X2 x + X1 X2 x - 4 X1 x - 4 X2 x + x 2 2 13 2 14 2 15 2 14 2 15 17 - X1 X2 x + X1 X2 x + X1 x + X1 X2 x - X2 x + 2 x 2 13 2 14 2 13 14 15 2 14 + 3 X1 X2 x - X1 x + 2 X1 X2 x - 4 X1 X2 x - X1 x - X2 x 15 2 12 2 13 13 14 2 13 + X2 x - X1 X2 x - 2 X1 x - 6 X1 X2 x + 3 X1 x - X2 x 14 2 11 2 12 12 13 13 + 3 X2 x + X1 X2 x + X1 x + 2 X1 X2 x + 4 X1 x + 3 X2 x 14 2 10 2 11 11 12 12 13 - 2 x - X1 X2 x - X1 x - X1 X2 x - 2 X1 x - X2 x - 2 x 2 10 10 11 12 2 9 10 10 + 2 X1 x + 2 X1 X2 x + X1 x + x - X1 x - 4 X1 x - X2 x 9 10 7 8 9 6 7 8 + 2 X1 x + 2 x + 2 X1 X2 x - X2 x - x - 2 X1 X2 x - 2 X1 x + x 5 6 5 4 5 4 3 2 + X1 X2 x + 2 X1 x - 2 X1 x + X1 x + x - x + x - 3 x + 3 x - 1) / 2 2 15 2 15 2 15 2 2 13 / ((-1 + x) (X1 X2 x - 2 X1 X2 x - 2 X1 X2 x - X1 X2 x / 2 14 2 15 2 14 15 2 15 2 13 + X1 X2 x + X1 x - X1 X2 x + 4 X1 X2 x + X2 x + 2 X1 X2 x 2 14 2 13 15 2 14 15 2 12 - X1 x + X1 X2 x - 2 X1 x + X2 x - 2 X2 x - X1 X2 x 2 13 13 14 14 15 2 11 2 12 - X1 x - 2 X1 X2 x + X1 x - X2 x + x + X1 X2 x + X1 x 12 13 2 10 2 11 11 12 + 2 X1 X2 x + X1 x - X1 X2 x - X1 x - 2 X1 X2 x - 2 X1 x 12 2 10 11 11 12 2 9 9 - X2 x + 2 X1 x + 2 X1 x + X2 x + x - X1 x - X1 X2 x 10 10 11 9 9 7 9 6 - 2 X1 x + X2 x - x + 3 X1 x + X2 x + X1 X2 x - 2 x - 2 X1 X2 x 7 5 6 5 6 4 5 4 2 - X1 x + X1 X2 x + X1 x - 2 X1 x + x + X1 x + x - x - 2 x + 3 x - 1)) and in Maple format -(X1^3*X2^2*x^21+2*X1^3*X2^2*x^20-2*X1^3*X2*x^21-3*X1^2*X2^2*x^21+X1^3*X2^2*x^ 19-4*X1^3*X2*x^20+X1^3*x^21-6*X1^2*X2^2*x^20+6*X1^2*X2*x^21+3*X1*X2^2*x^21-2*X1 ^3*X2*x^19+2*X1^3*x^20-3*X1^2*X2^2*x^19+12*X1^2*X2*x^20-3*X1^2*x^21+6*X1*X2^2*x ^20-6*X1*X2*x^21-X2^2*x^21+X1^3*x^19+X1^2*X2^2*x^18+6*X1^2*X2*x^19-6*X1^2*x^20+ 3*X1*X2^2*x^19-12*X1*X2*x^20+3*X1*x^21-2*X2^2*x^20+2*X2*x^21+2*X1^2*X2^2*x^17-2 *X1^2*X2*x^18-3*X1^2*x^19-2*X1*X2^2*x^18-6*X1*X2*x^19+6*X1*x^20-X2^2*x^19+4*X2* x^20-x^21-4*X1^2*X2*x^17+X1^2*x^18-4*X1*X2^2*x^17+4*X1*X2*x^18+3*X1*x^19+X2^2*x ^18+2*X2*x^19-2*x^20+2*X1^2*x^17+8*X1*X2*x^17-2*X1*x^18+2*X2^2*x^17-2*X2*x^18-x ^19-X1^2*X2*x^15+X1*X2^2*x^15-4*X1*x^17-4*X2*x^17+x^18-X1^2*X2^2*x^13+X1^2*X2*x ^14+X1^2*x^15+X1*X2^2*x^14-X2^2*x^15+2*x^17+3*X1^2*X2*x^13-X1^2*x^14+2*X1*X2^2* x^13-4*X1*X2*x^14-X1*x^15-X2^2*x^14+X2*x^15-X1^2*X2*x^12-2*X1^2*x^13-6*X1*X2*x^ 13+3*X1*x^14-X2^2*x^13+3*X2*x^14+X1^2*X2*x^11+X1^2*x^12+2*X1*X2*x^12+4*X1*x^13+ 3*X2*x^13-2*x^14-X1^2*X2*x^10-X1^2*x^11-X1*X2*x^11-2*X1*x^12-X2*x^12-2*x^13+2* X1^2*x^10+2*X1*X2*x^10+X1*x^11+x^12-X1^2*x^9-4*X1*x^10-X2*x^10+2*X1*x^9+2*x^10+ 2*X1*X2*x^7-X2*x^8-x^9-2*X1*X2*x^6-2*X1*x^7+x^8+X1*X2*x^5+2*X1*x^6-2*X1*x^5+X1* x^4+x^5-x^4+x^3-3*x^2+3*x-1)/(-1+x)/(X1^2*X2^2*x^15-2*X1^2*X2*x^15-2*X1*X2^2*x^ 15-X1^2*X2^2*x^13+X1^2*X2*x^14+X1^2*x^15-X1*X2^2*x^14+4*X1*X2*x^15+X2^2*x^15+2* X1^2*X2*x^13-X1^2*x^14+X1*X2^2*x^13-2*X1*x^15+X2^2*x^14-2*X2*x^15-X1^2*X2*x^12- X1^2*x^13-2*X1*X2*x^13+X1*x^14-X2*x^14+x^15+X1^2*X2*x^11+X1^2*x^12+2*X1*X2*x^12 +X1*x^13-X1^2*X2*x^10-X1^2*x^11-2*X1*X2*x^11-2*X1*x^12-X2*x^12+2*X1^2*x^10+2*X1 *x^11+X2*x^11+x^12-X1^2*x^9-X1*X2*x^9-2*X1*x^10+X2*x^10-x^11+3*X1*x^9+X2*x^9+X1 *X2*x^7-2*x^9-2*X1*X2*x^6-X1*x^7+X1*X2*x^5+X1*x^6-2*X1*x^5+x^6+X1*x^4+x^5-x^4-2 *x^2+3*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 1, 4], are n 191 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [5, 2, 1], are n 111 57 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 4 19 ------- 57 and in floating point 0.3058876452 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 4 19 ate normal pair with correlation, ------- 57 1/2 4 19 203 i.e. , [[-------, 0], [0, ---]] 57 171 ------------------------------------------------- Theorem Number, 108, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 1, 4], nor the composition, [6, 1, 1] Then infinity ----- \ n 18 17 16 14 13 12 11 ) a(n) x = - (x + 2 x + 2 x - 3 x - 3 x - 2 x - 3 x / ----- n = 0 10 4 2 / 6 5 3 2 - 2 x + x + x - 2 x + 1) / ((x + x + x + x + x - 1) / 6 5 2 (x + x + 1) (-1 + x) ) and in Maple format -(x^18+2*x^17+2*x^16-3*x^14-3*x^13-2*x^12-3*x^11-2*x^10+x^4+x^2-2*x+1)/(x^6+x^5 +x^3+x^2+x-1)/(x^6+x^5+1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.67303521612186301918 1.9111834366854762350 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 1, 4] and d occurrences (as containment) of the composition, [6, 1, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 19 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 2 18 2 19 2 19 2 18 2 19 + X1 X2 x - 2 X1 X2 x - 2 X1 X2 x - 2 X1 X2 x + X1 x 2 18 19 2 19 2 2 16 2 18 - 2 X1 X2 x + 4 X1 X2 x + X2 x - X1 X2 x + X1 x 18 19 2 18 19 2 2 15 2 16 + 4 X1 X2 x - 2 X1 x + X2 x - 2 X2 x - X1 X2 x + 2 X1 X2 x 2 16 18 18 19 2 2 14 2 15 + 3 X1 X2 x - 2 X1 x - 2 X2 x + x + X1 X2 x + 3 X1 X2 x 2 16 2 15 16 2 16 18 2 2 13 - X1 x + 3 X1 X2 x - 6 X1 X2 x - 2 X2 x + x - X1 X2 x 2 14 2 15 2 14 15 16 - 2 X1 X2 x - 2 X1 x - X1 X2 x - 8 X1 X2 x + 3 X1 x 2 15 16 2 13 2 14 2 13 14 - 2 X2 x + 4 X2 x + X1 X2 x + X1 x + X1 X2 x + 2 X1 X2 x 15 15 16 2 12 14 15 + 5 X1 x + 5 X2 x - 2 x + 2 X1 X2 x - X1 x - 3 x 2 11 2 12 12 13 13 2 11 - 2 X1 X2 x - 2 X1 x - 3 X1 X2 x - X1 x - X2 x + 2 X1 x 11 12 12 13 2 10 10 11 + 3 X1 X2 x + 3 X1 x + X2 x + x + X1 x + X1 X2 x - 3 X1 x 11 12 2 9 9 10 10 11 9 - X2 x - x - X1 x + X1 X2 x - 3 X1 x - X2 x + x + X1 x 9 10 7 6 7 6 5 4 5 - X2 x + 2 x - X1 X2 x + X1 X2 x + X1 x - X1 x - X1 x + X1 x + x 4 3 2 / - x + x - 3 x + 3 x - 1) / ((-1 + x) / 7 7 7 7 5 5 7 6 (X1 X2 x - X1 x - X2 x + x + X1 x - x + x - 1) (X1 X2 x - X1 X2 x 7 7 6 7 5 4 5 4 - X1 x - X2 x + X1 x + x + X1 x - X1 x - x + x - 2 x + 1)) and in Maple format -(X1^2*X2^2*x^19+X1^2*X2^2*x^18-2*X1^2*X2*x^19-2*X1*X2^2*x^19-2*X1^2*X2*x^18+X1 ^2*x^19-2*X1*X2^2*x^18+4*X1*X2*x^19+X2^2*x^19-X1^2*X2^2*x^16+X1^2*x^18+4*X1*X2* x^18-2*X1*x^19+X2^2*x^18-2*X2*x^19-X1^2*X2^2*x^15+2*X1^2*X2*x^16+3*X1*X2^2*x^16 -2*X1*x^18-2*X2*x^18+x^19+X1^2*X2^2*x^14+3*X1^2*X2*x^15-X1^2*x^16+3*X1*X2^2*x^ 15-6*X1*X2*x^16-2*X2^2*x^16+x^18-X1^2*X2^2*x^13-2*X1^2*X2*x^14-2*X1^2*x^15-X1* X2^2*x^14-8*X1*X2*x^15+3*X1*x^16-2*X2^2*x^15+4*X2*x^16+X1^2*X2*x^13+X1^2*x^14+ X1*X2^2*x^13+2*X1*X2*x^14+5*X1*x^15+5*X2*x^15-2*x^16+2*X1^2*X2*x^12-X1*x^14-3*x ^15-2*X1^2*X2*x^11-2*X1^2*x^12-3*X1*X2*x^12-X1*x^13-X2*x^13+2*X1^2*x^11+3*X1*X2 *x^11+3*X1*x^12+X2*x^12+x^13+X1^2*x^10+X1*X2*x^10-3*X1*x^11-X2*x^11-x^12-X1^2*x ^9+X1*X2*x^9-3*X1*x^10-X2*x^10+x^11+X1*x^9-X2*x^9+2*x^10-X1*X2*x^7+X1*X2*x^6+X1 *x^7-X1*x^6-X1*x^5+X1*x^4+x^5-x^4+x^3-3*x^2+3*x-1)/(-1+x)/(X1*X2*x^7-X1*x^7-X2* x^7+x^7+X1*x^5-x^5+x-1)/(X1*X2*x^7-X1*X2*x^6-X1*x^7-X2*x^7+X1*x^6+x^7+X1*x^5-X1 *x^4-x^5+x^4-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 1, 4], are n 191 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [6, 1, 1], are n 49 53 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 8 3 53 ------------ 477 and in floating point 0.2114804230 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 8 159 ate normal pair with correlation, -------- 477 1/2 8 159 1559 i.e. , [[--------, 0], [0, ----]] 477 1431 ------------------------------------------------- Theorem Number, 109, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 2, 3], nor the composition, [1, 1, 6] Then infinity ----- \ n ) a(n) x = / ----- n = 0 13 10 8 7 5 4 3 2 x - x + x - x - x + x - x - x + 2 x - 1 ---------------------------------------------------------------------- 10 9 8 7 6 5 3 2 2 (x + 1) (x + x + 2 x + x + x + x + x - x + 2 x - 1) (-1 + x) and in Maple format (x^13-x^10+x^8-x^7-x^5+x^4-x^3-x^2+2*x-1)/(x+1)/(x^10+x^9+2*x^8+x^7+x^6+x^5+x^3 -x^2+2*x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.67919107669695186122 1.9139965407423997306 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 2, 3] and d occurrences (as containment) of the composition, [1, 1, 6], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 13 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 13 2 13 13 13 13 10 10 - 2 X1 X2 x - X2 x + X1 x + 2 X2 x - x + X1 X2 x - X1 x 10 8 10 7 8 8 6 7 - X2 x - X1 X2 x + x + X1 X2 x + X1 x + X2 x - X1 X2 x - X1 x 7 8 6 7 5 4 5 3 4 3 2 - X2 x - x + X1 x + x - X1 x + X1 x + x - X1 x - x + x + x / 2 13 13 2 13 13 13 - 2 x + 1) / (X1 X2 x - 2 X1 X2 x - X2 x + X1 x + 2 X2 x / 13 10 9 10 10 8 9 9 - x + X1 X2 x + X1 X2 x - X1 x - X2 x - X1 X2 x - X1 x - X2 x 10 7 8 8 9 6 7 7 8 + x + 2 X1 X2 x + X1 x + X2 x + x - X1 X2 x - 2 X1 x - X2 x - x 6 7 5 6 4 5 3 4 3 2 + 2 X1 x + x - X1 x - x + 2 X1 x + x - X1 x - 2 x + x + 2 x - 3 x + 1) and in Maple format (X1*X2^2*x^13-2*X1*X2*x^13-X2^2*x^13+X1*x^13+2*X2*x^13-x^13+X1*X2*x^10-X1*x^10- X2*x^10-X1*X2*x^8+x^10+X1*X2*x^7+X1*x^8+X2*x^8-X1*X2*x^6-X1*x^7-X2*x^7-x^8+X1*x ^6+x^7-X1*x^5+X1*x^4+x^5-X1*x^3-x^4+x^3+x^2-2*x+1)/(X1*X2^2*x^13-2*X1*X2*x^13- X2^2*x^13+X1*x^13+2*X2*x^13-x^13+X1*X2*x^10+X1*X2*x^9-X1*x^10-X2*x^10-X1*X2*x^8 -X1*x^9-X2*x^9+x^10+2*X1*X2*x^7+X1*x^8+X2*x^8+x^9-X1*X2*x^6-2*X1*x^7-X2*x^7-x^8 +2*X1*x^6+x^7-X1*x^5-x^6+2*X1*x^4+x^5-X1*x^3-2*x^4+x^3+2*x^2-3*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 2, 3], are n 315 39 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 1, 6], are n 49 53 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 2 39 53 ------------- 2067 and in floating point 0.04399058701 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 2 2067 ate normal pair with correlation, --------- 2067 1/2 2 2067 2075 i.e. , [[---------, 0], [0, ----]] 2067 2067 ------------------------------------------------- Theorem Number, 110, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 2, 3], nor the composition, [1, 2, 5] Then infinity ----- 5 4 3 2 \ n x - x + x + x - 2 x + 1 ) a(n) x = -------------------------------------------- / 3 4 3 2 ----- (-1 + x) (x - x + 1) (x + x + x + x - 1) n = 0 and in Maple format (x^5-x^4+x^3+x^2-2*x+1)/(-1+x)/(x^3-x+1)/(x^4+x^3+x^2+x-1) The asymptotic expression for a(n) is, n 0.63758295460268560269 1.9275619754829253043 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 2, 3] and d occurrences (as containment) of the composition, [1, 2, 5], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 7 6 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x - X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 7 5 6 4 5 3 4 3 2 - X1 x + X1 X2 x + X1 x - X1 x - x + X1 x + x - x - x + 2 x - 1) / 8 7 8 8 6 7 8 / (X1 X2 x - X1 X2 x - X1 x - X2 x + 2 X1 X2 x + X1 x + x / 5 6 6 4 5 3 4 3 2 - X1 X2 x - X1 x - x + 2 X1 x + x - X1 x - 2 x + x + 2 x - 3 x + 1) and in Maple format -(X1*X2*x^7-X1*X2*x^6-X1*x^7+X1*X2*x^5+X1*x^6-X1*x^4-x^5+X1*x^3+x^4-x^3-x^2+2*x -1)/(X1*X2*x^8-X1*X2*x^7-X1*x^8-X2*x^8+2*X1*X2*x^6+X1*x^7+x^8-X1*X2*x^5-X1*x^6- x^6+2*X1*x^4+x^5-X1*x^3-2*x^4+x^3+2*x^2-3*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 2, 3], are n 315 39 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 2, 5], are n 111 57 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 16 13 19 -------------- 741 and in floating point 0.3393518736 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 16 247 ate normal pair with correlation, --------- 741 1/2 16 247 2735 i.e. , [[---------, 0], [0, ----]] 741 2223 ------------------------------------------------- Theorem Number, 111, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 2, 3], nor the composition, [1, 3, 4] Then infinity ----- 5 4 3 2 \ n x - x + x + x - 2 x + 1 ) a(n) x = -------------------------------------------- / 3 4 3 2 ----- (-1 + x) (x - x + 1) (x + x + x + x - 1) n = 0 and in Maple format (x^5-x^4+x^3+x^2-2*x+1)/(-1+x)/(x^3-x+1)/(x^4+x^3+x^2+x-1) The asymptotic expression for a(n) is, n 0.63758295460268560269 1.9275619754829253043 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 2, 3] and d occurrences (as containment) of the composition, [1, 3, 4], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 7 7 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x - X1 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 5 4 5 4 5 3 4 3 2 - X1 X2 x + X1 X2 x + 2 X1 x - 2 X1 x - x + X1 x + x - x - x / 8 7 8 8 7 8 + 2 x - 1) / (X1 X2 x - X1 X2 x - X1 x - X2 x + X1 x + x / 5 6 4 5 6 4 5 3 + 2 X1 X2 x + X1 x - X1 X2 x - 3 X1 x - x + 3 X1 x + x - X1 x 4 3 2 - 2 x + x + 2 x - 3 x + 1) and in Maple format -(X1*X2*x^7-X1*x^7-X1*X2*x^5+X1*X2*x^4+2*X1*x^5-2*X1*x^4-x^5+X1*x^3+x^4-x^3-x^2 +2*x-1)/(X1*X2*x^8-X1*X2*x^7-X1*x^8-X2*x^8+X1*x^7+x^8+2*X1*X2*x^5+X1*x^6-X1*X2* x^4-3*X1*x^5-x^6+3*X1*x^4+x^5-X1*x^3-2*x^4+x^3+2*x^2-3*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 2, 3], are n 315 39 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 3, 4], are n 135 65 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 4 3 5 ----------- 39 and in floating point 0.3972290613 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 1/2 4 3 5 ate normal pair with correlation, ----------- 39 1/2 1/2 4 3 5 667 i.e. , [[-----------, 0], [0, ---]] 39 507 ------------------------------------------------- Theorem Number, 112, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 2, 3], nor the composition, [1, 4, 3] Then infinity ----- 5 4 3 2 \ n x - x + x + x - 2 x + 1 ) a(n) x = -------------------------------------------- / 3 4 3 2 ----- (-1 + x) (x - x + 1) (x + x + x + x - 1) n = 0 and in Maple format (x^5-x^4+x^3+x^2-2*x+1)/(-1+x)/(x^3-x+1)/(x^4+x^3+x^2+x-1) The asymptotic expression for a(n) is, n 0.63758295460268560269 1.9275619754829253043 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 2, 3] and d occurrences (as containment) of the composition, [1, 4, 3], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 7 7 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x - X1 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 5 4 5 4 5 3 4 3 2 - X1 X2 x + X1 X2 x + 2 X1 x - 2 X1 x - x + X1 x + x - x - x / 8 7 8 8 7 8 + 2 x - 1) / (X1 X2 x - X1 X2 x - X1 x - X2 x + X1 x + x / 5 6 4 5 6 4 5 3 + 2 X1 X2 x + X1 x - X1 X2 x - 3 X1 x - x + 3 X1 x + x - X1 x 4 3 2 - 2 x + x + 2 x - 3 x + 1) and in Maple format -(X1*X2*x^7-X1*x^7-X1*X2*x^5+X1*X2*x^4+2*X1*x^5-2*X1*x^4-x^5+X1*x^3+x^4-x^3-x^2 +2*x-1)/(X1*X2*x^8-X1*X2*x^7-X1*x^8-X2*x^8+X1*x^7+x^8+2*X1*X2*x^5+X1*x^6-X1*X2* x^4-3*X1*x^5-x^6+3*X1*x^4+x^5-X1*x^3-2*x^4+x^3+2*x^2-3*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 2, 3], are n 315 39 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 4, 3], are n 135 65 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 4 3 5 ----------- 39 and in floating point 0.3972290613 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 1/2 4 3 5 ate normal pair with correlation, ----------- 39 1/2 1/2 4 3 5 667 i.e. , [[-----------, 0], [0, ---]] 39 507 ------------------------------------------------- Theorem Number, 113, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 2, 3], nor the composition, [1, 5, 2] Then infinity ----- \ n ) a(n) x = / ----- n = 0 8 7 6 5 3 x + x + x + x + x - x + 1 ------------------------------------------------------------------------- 11 10 9 8 7 6 5 4 2 (-1 + x) (x + 2 x + 3 x + 3 x + 3 x + 2 x + x + x + x + x - 1) and in Maple format (x^8+x^7+x^6+x^5+x^3-x+1)/(-1+x)/(x^11+2*x^10+3*x^9+3*x^8+3*x^7+2*x^6+x^5+x^4+x ^2+x-1) The asymptotic expression for a(n) is, n 0.64629602075846469709 1.9230256556702387387 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 2, 3] and d occurrences (as containment) of the composition, [1, 5, 2], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 9 9 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x - X1 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 9 7 9 6 7 5 6 4 5 - X2 x + X1 X2 x + x - X1 X2 x - X1 x + X1 X2 x + X1 x - X1 x - x 3 4 3 2 / 2 13 13 2 13 + X1 x + x - x - x + 2 x - 1) / (X1 X2 x - 2 X1 X2 x - X2 x / 13 13 13 10 10 10 8 10 + X1 x + 2 X2 x - x + X1 X2 x - X1 x - X2 x + X1 X2 x + x 7 8 8 6 7 8 5 6 - X1 X2 x - X1 x - X2 x + 2 X1 X2 x + X1 x + x - X1 X2 x - X1 x 6 4 5 3 4 3 2 - x + 2 X1 x + x - X1 x - 2 x + x + 2 x - 3 x + 1) and in Maple format -(X1*X2*x^9-X1*x^9-X2*x^9+X1*X2*x^7+x^9-X1*X2*x^6-X1*x^7+X1*X2*x^5+X1*x^6-X1*x^ 4-x^5+X1*x^3+x^4-x^3-x^2+2*x-1)/(X1*X2^2*x^13-2*X1*X2*x^13-X2^2*x^13+X1*x^13+2* X2*x^13-x^13+X1*X2*x^10-X1*x^10-X2*x^10+X1*X2*x^8+x^10-X1*X2*x^7-X1*x^8-X2*x^8+ 2*X1*X2*x^6+X1*x^7+x^8-X1*X2*x^5-X1*x^6-x^6+2*X1*x^4+x^5-X1*x^3-2*x^4+x^3+2*x^2 -3*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 2, 3], are n 315 39 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 5, 2], are n 111 57 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 4 13 19 ------------- 247 and in floating point 0.2545139052 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 4 247 ate normal pair with correlation, -------- 247 1/2 4 247 279 i.e. , [[--------, 0], [0, ---]] 247 247 ------------------------------------------------- Theorem Number, 114, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 2, 3], nor the composition, [1, 6, 1] Then infinity ----- \ n ) a(n) x = / ----- n = 0 2 12 11 8 4 2 (x - x + 1) (x + x - x + x + x - 2 x + 1) - -------------------------------------------------------------- 10 9 8 7 6 5 3 2 2 (x + x + 2 x + x + x + x + x - x + 2 x - 1) (-1 + x) and in Maple format -(x^2-x+1)*(x^12+x^11-x^8+x^4+x^2-2*x+1)/(x^10+x^9+2*x^8+x^7+x^6+x^5+x^3-x^2+2* x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.67777121983368566754 1.9139965407423997306 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 2, 3] and d occurrences (as containment) of the composition, [1, 6, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 16 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 16 2 16 2 14 16 16 2 13 - 2 X1 X2 x - X2 x - X1 X2 x + X1 x + 2 X2 x + X1 X2 x 14 2 14 16 13 14 2 13 14 + 2 X1 X2 x + X2 x - x - 2 X1 X2 x - X1 x - X2 x - 2 X2 x 12 13 13 14 12 12 13 12 + X1 X2 x + X1 x + 2 X2 x + x - X1 x - X2 x - x + x 9 8 9 9 7 8 8 + X1 X2 x - 2 X1 X2 x - X1 x - X2 x + 2 X1 X2 x + 2 X1 x + 2 X2 x 9 6 7 7 8 6 7 5 6 + x - X1 X2 x - 2 X1 x - X2 x - 2 x + 2 X1 x + x - 2 X1 x - x 4 5 3 4 2 / + 2 X1 x + 2 x - X1 x - 2 x + 3 x - 3 x + 1) / ((-1 + x) ( / 2 13 13 2 13 13 13 13 10 X1 X2 x - 2 X1 X2 x - X2 x + X1 x + 2 X2 x - x + X1 X2 x 9 10 10 8 9 9 10 7 + X1 X2 x - X1 x - X2 x - X1 X2 x - X1 x - X2 x + x + 2 X1 X2 x 8 8 9 6 7 7 8 6 7 + X1 x + X2 x + x - X1 X2 x - 2 X1 x - X2 x - x + 2 X1 x + x 5 6 4 5 3 4 3 2 - X1 x - x + 2 X1 x + x - X1 x - 2 x + x + 2 x - 3 x + 1)) and in Maple format -(X1*X2^2*x^16-2*X1*X2*x^16-X2^2*x^16-X1*X2^2*x^14+X1*x^16+2*X2*x^16+X1*X2^2*x^ 13+2*X1*X2*x^14+X2^2*x^14-x^16-2*X1*X2*x^13-X1*x^14-X2^2*x^13-2*X2*x^14+X1*X2*x ^12+X1*x^13+2*X2*x^13+x^14-X1*x^12-X2*x^12-x^13+x^12+X1*X2*x^9-2*X1*X2*x^8-X1*x ^9-X2*x^9+2*X1*X2*x^7+2*X1*x^8+2*X2*x^8+x^9-X1*X2*x^6-2*X1*x^7-X2*x^7-2*x^8+2* X1*x^6+x^7-2*X1*x^5-x^6+2*X1*x^4+2*x^5-X1*x^3-2*x^4+3*x^2-3*x+1)/(-1+x)/(X1*X2^ 2*x^13-2*X1*X2*x^13-X2^2*x^13+X1*x^13+2*X2*x^13-x^13+X1*X2*x^10+X1*X2*x^9-X1*x^ 10-X2*x^10-X1*X2*x^8-X1*x^9-X2*x^9+x^10+2*X1*X2*x^7+X1*x^8+X2*x^8+x^9-X1*X2*x^6 -2*X1*x^7-X2*x^7-x^8+2*X1*x^6+x^7-X1*x^5-x^6+2*X1*x^4+x^5-X1*x^3-2*x^4+x^3+2*x^ 2-3*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 2, 3], are n 315 39 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 6, 1], are n 49 53 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 2 39 53 ------------- 2067 and in floating point 0.04399058701 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 2 2067 ate normal pair with correlation, --------- 2067 1/2 2 2067 2075 i.e. , [[---------, 0], [0, ----]] 2067 2067 ------------------------------------------------- Theorem Number, 115, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 2, 3], nor the composition, [2, 1, 5] Then infinity ----- 9 7 6 5 4 3 2 \ n x - x + x + x - x + x + x - 2 x + 1 ) a(n) x = - ------------------------------------------------------ / 10 9 8 7 5 4 3 2 ----- x - x - x + 2 x - x + 2 x - x - 2 x + 3 x - 1 n = 0 and in Maple format -(x^9-x^7+x^6+x^5-x^4+x^3+x^2-2*x+1)/(x^10-x^9-x^8+2*x^7-x^5+2*x^4-x^3-2*x^2+3* x-1) The asymptotic expression for a(n) is, n 0.64281047729344093126 1.9255092871632283681 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 2, 3] and d occurrences (as containment) of the composition, [2, 1, 5], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 11 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 11 11 9 9 9 7 9 - 2 X1 X2 x + X1 x + X1 X2 x - X1 x - X2 x - X1 X2 x + x 6 7 7 5 6 6 7 6 4 + X1 X2 x + X1 x + X2 x - X1 X2 x - X1 x - X2 x - x + x + X1 x 5 3 4 3 2 / 2 12 2 11 + x - X1 x - x + x + x - 2 x + 1) / (X1 X2 x - X1 X2 x / 12 11 12 10 11 9 - 2 X1 X2 x + 2 X1 X2 x + X1 x + X1 X2 x - X1 x - X1 X2 x 10 10 8 9 9 10 7 8 - X1 x - X2 x - X1 X2 x + X1 x + X2 x + x + X1 X2 x + X1 x 8 9 6 7 7 8 5 6 + X2 x - x - 2 X1 X2 x - X1 x - 2 X2 x - x + X1 X2 x + X1 x 6 7 4 5 3 4 3 2 + X2 x + 2 x - 2 X1 x - x + X1 x + 2 x - x - 2 x + 3 x - 1) and in Maple format -(X1*X2^2*x^11-2*X1*X2*x^11+X1*x^11+X1*X2*x^9-X1*x^9-X2*x^9-X1*X2*x^7+x^9+X1*X2 *x^6+X1*x^7+X2*x^7-X1*X2*x^5-X1*x^6-X2*x^6-x^7+x^6+X1*x^4+x^5-X1*x^3-x^4+x^3+x^ 2-2*x+1)/(X1*X2^2*x^12-X1*X2^2*x^11-2*X1*X2*x^12+2*X1*X2*x^11+X1*x^12+X1*X2*x^ 10-X1*x^11-X1*X2*x^9-X1*x^10-X2*x^10-X1*X2*x^8+X1*x^9+X2*x^9+x^10+X1*X2*x^7+X1* x^8+X2*x^8-x^9-2*X1*X2*x^6-X1*x^7-2*X2*x^7-x^8+X1*X2*x^5+X1*x^6+X2*x^6+2*x^7-2* X1*x^4-x^5+X1*x^3+2*x^4-x^3-2*x^2+3*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 2, 3], are n 315 39 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 1, 5], are n 113 57 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 14 13 19 -------------- 741 and in floating point 0.2969328894 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 14 247 ate normal pair with correlation, --------- 741 1/2 14 247 2615 i.e. , [[---------, 0], [0, ----]] 741 2223 ------------------------------------------------- Theorem Number, 116, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 2, 3], nor the composition, [2, 4, 2] Then infinity ----- 6 4 3 2 \ n x + x - x + 2 x - 2 x + 1 ) a(n) x = - ------------------------------------ / 7 5 4 3 2 ----- x + x - x + 2 x - 3 x + 3 x - 1 n = 0 and in Maple format -(x^6+x^4-x^3+2*x^2-2*x+1)/(x^7+x^5-x^4+2*x^3-3*x^2+3*x-1) The asymptotic expression for a(n) is, n 0.60945823930910114296 1.9365136360000287543 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 2, 3] and d occurrences (as containment) of the composition, [2, 4, 2], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 8 8 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x - X1 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 8 6 8 5 6 4 5 4 - X2 x + X1 X2 x + x - X1 X2 x - X1 x + X1 X2 x + 2 X1 x - 2 X1 x 5 3 4 3 2 / 9 9 9 - x + X1 x + x - x - x + 2 x - 1) / (X1 X2 x - X1 x - X2 x / 7 9 6 7 5 6 4 + X1 X2 x + x - X1 X2 x - X1 x + 2 X1 X2 x + 2 X1 x - X1 X2 x 5 6 4 5 3 4 3 2 - 3 X1 x - x + 3 X1 x + x - X1 x - 2 x + x + 2 x - 3 x + 1) and in Maple format -(X1*X2*x^8-X1*x^8-X2*x^8+X1*X2*x^6+x^8-X1*X2*x^5-X1*x^6+X1*X2*x^4+2*X1*x^5-2* X1*x^4-x^5+X1*x^3+x^4-x^3-x^2+2*x-1)/(X1*X2*x^9-X1*x^9-X2*x^9+X1*X2*x^7+x^9-X1* X2*x^6-X1*x^7+2*X1*X2*x^5+2*X1*x^6-X1*X2*x^4-3*X1*x^5-x^6+3*X1*x^4+x^5-X1*x^3-2 *x^4+x^3+2*x^2-3*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 2, 3], are n 315 39 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 4, 2], are n 75 69 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 10 13 23 -------------- 299 and in floating point 0.5783149317 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 10 299 ate normal pair with correlation, --------- 299 1/2 10 299 499 i.e. , [[---------, 0], [0, ---]] 299 299 ------------------------------------------------- Theorem Number, 117, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 2, 3], nor the composition, [2, 5, 1] Then infinity ----- 8 7 6 5 3 \ n x + x + x + x + x - x + 1 ) a(n) x = - ----------------------------------- / 3 4 3 2 ----- (x - x + 1) (x + x + x + x - 1) n = 0 and in Maple format -(x^8+x^7+x^6+x^5+x^3-x+1)/(x^3-x+1)/(x^4+x^3+x^2+x-1) The asymptotic expression for a(n) is, n 0.63242246488998289333 1.9275619754829253043 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 2, 3] and d occurrences (as containment) of the composition, [2, 5, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 9 9 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x - X1 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 9 7 9 6 7 5 6 4 5 - X2 x + X1 X2 x + x - X1 X2 x - X1 x + X1 X2 x + X1 x - X1 x - x 3 4 3 2 / 8 7 8 8 + X1 x + x - x - x + 2 x - 1) / (X1 X2 x - X1 X2 x - X1 x - X2 x / 6 7 8 5 6 6 4 5 3 + 2 X1 X2 x + X1 x + x - X1 X2 x - X1 x - x + 2 X1 x + x - X1 x 4 3 2 - 2 x + x + 2 x - 3 x + 1) and in Maple format -(X1*X2*x^9-X1*x^9-X2*x^9+X1*X2*x^7+x^9-X1*X2*x^6-X1*x^7+X1*X2*x^5+X1*x^6-X1*x^ 4-x^5+X1*x^3+x^4-x^3-x^2+2*x-1)/(X1*X2*x^8-X1*X2*x^7-X1*x^8-X2*x^8+2*X1*X2*x^6+ X1*x^7+x^8-X1*X2*x^5-X1*x^6-x^6+2*X1*x^4+x^5-X1*x^3-2*x^4+x^3+2*x^2-3*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 2, 3], are n 315 39 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 5, 1], are n 111 57 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 16 13 19 -------------- 741 and in floating point 0.3393518736 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 16 247 ate normal pair with correlation, --------- 741 1/2 16 247 2735 i.e. , [[---------, 0], [0, ----]] 741 2223 ------------------------------------------------- Theorem Number, 118, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 2, 3], nor the composition, [3, 1, 4] Then infinity ----- 7 6 4 3 2 \ n x - x + 2 x - 3 x + 4 x - 3 x + 1 ) a(n) x = - ---------------------------------------------------- / 9 7 6 5 4 3 2 ----- x - 2 x + 2 x + x - 3 x + 5 x - 6 x + 4 x - 1 n = 0 and in Maple format -(x^7-x^6+2*x^4-3*x^3+4*x^2-3*x+1)/(x^9-2*x^7+2*x^6+x^5-3*x^4+5*x^3-6*x^2+4*x-1 ) The asymptotic expression for a(n) is, n 0.63104184045431048184 1.9285748396761772314 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 2, 3] and d occurrences (as containment) of the composition, [3, 1, 4], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 10 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 10 2 9 2 10 2 9 2 9 9 - 2 X1 X2 x + X1 X2 x + X1 x + X1 X2 x - X1 x - 3 X1 X2 x 8 9 8 8 6 8 5 6 + X1 X2 x + 2 X1 x - X1 x - X2 x - X1 X2 x + x + X1 X2 x + X1 x 6 4 5 5 6 4 5 3 4 + X2 x - X1 X2 x - 2 X1 x - X2 x - x + 2 X1 x + 2 x - X1 x - x 3 2 / 2 2 12 2 2 11 2 12 + x + x - 2 x + 1) / (X1 X2 x + X1 X2 x - 2 X1 X2 x / 2 2 10 2 11 2 12 2 11 2 10 - X1 X2 x - 2 X1 X2 x + X1 x + X1 X2 x + 3 X1 X2 x 2 11 2 10 11 2 9 2 10 2 9 + X1 x + X1 X2 x - 2 X1 X2 x - X1 X2 x - 2 X1 x - X1 X2 x 10 11 2 9 9 10 10 - 2 X1 X2 x + X1 x + X1 x + 4 X1 X2 x + X1 x - X2 x 8 9 9 10 7 8 8 9 - 2 X1 X2 x - 3 X1 x - X2 x + x - X1 X2 x + 2 X1 x + 2 X2 x + x 6 7 8 5 6 6 4 + X1 X2 x + X1 x - 2 x - 2 X1 X2 x - 2 X1 x - 2 X2 x + X1 X2 x 5 5 6 4 5 3 4 3 2 + 3 X1 x + X2 x + 3 x - 3 X1 x - 2 x + X1 x + 2 x - x - 2 x + 3 x - 1) and in Maple format -(X1^2*X2^2*x^10-2*X1^2*X2*x^10+X1^2*X2*x^9+X1^2*x^10+X1*X2^2*x^9-X1^2*x^9-3*X1 *X2*x^9+X1*X2*x^8+2*X1*x^9-X1*x^8-X2*x^8-X1*X2*x^6+x^8+X1*X2*x^5+X1*x^6+X2*x^6- X1*X2*x^4-2*X1*x^5-X2*x^5-x^6+2*X1*x^4+2*x^5-X1*x^3-x^4+x^3+x^2-2*x+1)/(X1^2*X2 ^2*x^12+X1^2*X2^2*x^11-2*X1^2*X2*x^12-X1^2*X2^2*x^10-2*X1^2*X2*x^11+X1^2*x^12+ X1*X2^2*x^11+3*X1^2*X2*x^10+X1^2*x^11+X1*X2^2*x^10-2*X1*X2*x^11-X1^2*X2*x^9-2* X1^2*x^10-X1*X2^2*x^9-2*X1*X2*x^10+X1*x^11+X1^2*x^9+4*X1*X2*x^9+X1*x^10-X2*x^10 -2*X1*X2*x^8-3*X1*x^9-X2*x^9+x^10-X1*X2*x^7+2*X1*x^8+2*X2*x^8+x^9+X1*X2*x^6+X1* x^7-2*x^8-2*X1*X2*x^5-2*X1*x^6-2*X2*x^6+X1*X2*x^4+3*X1*x^5+X2*x^5+3*x^6-3*X1*x^ 4-2*x^5+X1*x^3+2*x^4-x^3-2*x^2+3*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 2, 3], are n 315 39 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 1, 4], are n 141 65 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 6 3 5 ----------- 65 and in floating point 0.3575061550 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 1/2 6 3 5 ate normal pair with correlation, ----------- 65 1/2 1/2 6 3 5 1061 i.e. , [[-----------, 0], [0, ----]] 65 845 ------------------------------------------------- Theorem Number, 119, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 2, 3], nor the composition, [3, 3, 2] Then infinity ----- 6 4 3 2 \ n x + x - x + 2 x - 2 x + 1 ) a(n) x = - ------------------------------------ / 7 5 4 3 2 ----- x + x - x + 2 x - 3 x + 3 x - 1 n = 0 and in Maple format -(x^6+x^4-x^3+2*x^2-2*x+1)/(x^7+x^5-x^4+2*x^3-3*x^2+3*x-1) The asymptotic expression for a(n) is, n 0.60945823930910114296 1.9365136360000287543 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 2, 3] and d occurrences (as containment) of the composition, [3, 3, 2], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 10 2 10 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x - X1 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 10 9 10 8 9 9 7 - X1 X2 x - X1 X2 x + X1 x + X1 X2 x + X1 x + X2 x - X1 X2 x 8 8 9 6 7 8 5 6 - X1 x - X2 x - x + X1 X2 x + X1 x + x + X1 X2 x - 2 X1 x 4 5 6 3 5 4 2 / - 2 X1 X2 x + X1 x + x + X1 X2 x - 2 x + 2 x - 3 x + 3 x - 1) / ( / 2 11 2 11 11 10 11 9 10 X1 X2 x - X1 x - X1 X2 x - X1 X2 x + X1 x + X1 X2 x + X1 x 10 8 9 9 10 7 8 9 + X2 x - X1 X2 x - X1 x - X2 x - x + 2 X1 X2 x + X1 x + x 6 7 5 6 7 4 5 - X1 X2 x - 3 X1 x - 2 X1 X2 x + 3 X1 x + x + 3 X1 X2 x - X1 x 6 3 5 4 3 2 - 2 x - X1 X2 x + 3 x - 3 x - x + 5 x - 4 x + 1) and in Maple format -(X1^2*X2*x^10-X1^2*x^10-X1*X2*x^10-X1*X2*x^9+X1*x^10+X1*X2*x^8+X1*x^9+X2*x^9- X1*X2*x^7-X1*x^8-X2*x^8-x^9+X1*X2*x^6+X1*x^7+x^8+X1*X2*x^5-2*X1*x^6-2*X1*X2*x^4 +X1*x^5+x^6+X1*X2*x^3-2*x^5+2*x^4-3*x^2+3*x-1)/(X1^2*X2*x^11-X1^2*x^11-X1*X2*x^ 11-X1*X2*x^10+X1*x^11+X1*X2*x^9+X1*x^10+X2*x^10-X1*X2*x^8-X1*x^9-X2*x^9-x^10+2* X1*X2*x^7+X1*x^8+x^9-X1*X2*x^6-3*X1*x^7-2*X1*X2*x^5+3*X1*x^6+x^7+3*X1*X2*x^4-X1 *x^5-2*x^6-X1*X2*x^3+3*x^5-3*x^4-x^3+5*x^2-4*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 2, 3], are n 315 39 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 3, 2], are n 97 85 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 2 39 85 ------------- 195 and in floating point 0.5905234533 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 1/2 2 17 195 ate normal pair with correlation, -------------- 195 1/2 1/2 2 17 195 331 i.e. , [[--------------, 0], [0, ---]] 195 195 ------------------------------------------------- Theorem Number, 120, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 2, 3], nor the composition, [3, 4, 1] Then infinity ----- 8 7 6 5 3 \ n x + x + x + x + x - x + 1 ) a(n) x = - ----------------------------------- / 3 4 3 2 ----- (x - x + 1) (x + x + x + x - 1) n = 0 and in Maple format -(x^8+x^7+x^6+x^5+x^3-x+1)/(x^3-x+1)/(x^4+x^3+x^2+x-1) The asymptotic expression for a(n) is, n 0.63242246488998289333 1.9275619754829253043 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 2, 3] and d occurrences (as containment) of the composition, [3, 4, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 11 2 11 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x - X1 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 11 10 11 9 10 10 8 - X1 X2 x - X1 X2 x + X1 x + X1 X2 x + X1 x + X2 x - X1 X2 x 9 9 10 7 8 9 6 7 - X1 x - X2 x - x + X1 X2 x + X1 x + x + X1 X2 x - X1 x 5 6 4 5 6 4 5 3 - 2 X1 X2 x - 2 X1 x + X1 X2 x + 4 X1 x + x - 3 X1 x - 2 x + X1 x 4 2 / 8 7 8 + 2 x - 3 x + 3 x - 1) / ((-1 + x) (X1 X2 x - X1 X2 x - X1 x / 8 7 8 5 6 4 5 6 - X2 x + X1 x + x + 2 X1 X2 x + X1 x - X1 X2 x - 3 X1 x - x 4 5 3 4 3 2 + 3 X1 x + x - X1 x - 2 x + x + 2 x - 3 x + 1)) and in Maple format (X1^2*X2*x^11-X1^2*x^11-X1*X2*x^11-X1*X2*x^10+X1*x^11+X1*X2*x^9+X1*x^10+X2*x^10 -X1*X2*x^8-X1*x^9-X2*x^9-x^10+X1*X2*x^7+X1*x^8+x^9+X1*X2*x^6-X1*x^7-2*X1*X2*x^5 -2*X1*x^6+X1*X2*x^4+4*X1*x^5+x^6-3*X1*x^4-2*x^5+X1*x^3+2*x^4-3*x^2+3*x-1)/(-1+x )/(X1*X2*x^8-X1*X2*x^7-X1*x^8-X2*x^8+X1*x^7+x^8+2*X1*X2*x^5+X1*x^6-X1*X2*x^4-3* X1*x^5-x^6+3*X1*x^4+x^5-X1*x^3-2*x^4+x^3+2*x^2-3*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 2, 3], are n 315 39 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 4, 1], are n 135 65 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 4 3 5 ----------- 39 and in floating point 0.3972290613 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 1/2 4 3 5 ate normal pair with correlation, ----------- 39 1/2 1/2 4 3 5 667 i.e. , [[-----------, 0], [0, ---]] 39 507 ------------------------------------------------- Theorem Number, 121, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 2, 3], nor the composition, [4, 1, 3] Then infinity ----- 7 6 4 3 2 \ n x - x + 2 x - 3 x + 4 x - 3 x + 1 ) a(n) x = - ---------------------------------------------------- / 9 7 6 5 4 3 2 ----- x - 2 x + 2 x + x - 3 x + 5 x - 6 x + 4 x - 1 n = 0 and in Maple format -(x^7-x^6+2*x^4-3*x^3+4*x^2-3*x+1)/(x^9-2*x^7+2*x^6+x^5-3*x^4+5*x^3-6*x^2+4*x-1 ) The asymptotic expression for a(n) is, n 0.63104184045431048184 1.9285748396761772314 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 2, 3] and d occurrences (as containment) of the composition, [4, 1, 3], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 10 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 10 2 9 2 10 2 9 2 9 9 - 2 X1 X2 x + X1 X2 x + X1 x + X1 X2 x - X1 x - 3 X1 X2 x 8 9 8 8 6 8 5 6 + X1 X2 x + 2 X1 x - X1 x - X2 x - X1 X2 x + x + X1 X2 x + X1 x 6 4 5 5 6 4 5 3 4 + X2 x - X1 X2 x - 2 X1 x - X2 x - x + 2 X1 x + 2 x - X1 x - x 3 2 / 2 2 12 2 2 11 2 12 + x + x - 2 x + 1) / (X1 X2 x + X1 X2 x - 2 X1 X2 x / 2 2 10 2 11 2 12 2 11 2 10 - X1 X2 x - 2 X1 X2 x + X1 x + X1 X2 x + 3 X1 X2 x 2 11 2 10 11 2 9 2 10 2 9 + X1 x + X1 X2 x - 2 X1 X2 x - X1 X2 x - 2 X1 x - X1 X2 x 10 11 2 9 9 10 10 - 2 X1 X2 x + X1 x + X1 x + 4 X1 X2 x + X1 x - X2 x 8 9 9 10 7 8 8 9 - 2 X1 X2 x - 3 X1 x - X2 x + x - X1 X2 x + 2 X1 x + 2 X2 x + x 6 7 8 5 6 6 4 + X1 X2 x + X1 x - 2 x - 2 X1 X2 x - 2 X1 x - 2 X2 x + X1 X2 x 5 5 6 4 5 3 4 3 2 + 3 X1 x + X2 x + 3 x - 3 X1 x - 2 x + X1 x + 2 x - x - 2 x + 3 x - 1) and in Maple format -(X1^2*X2^2*x^10-2*X1^2*X2*x^10+X1^2*X2*x^9+X1^2*x^10+X1*X2^2*x^9-X1^2*x^9-3*X1 *X2*x^9+X1*X2*x^8+2*X1*x^9-X1*x^8-X2*x^8-X1*X2*x^6+x^8+X1*X2*x^5+X1*x^6+X2*x^6- X1*X2*x^4-2*X1*x^5-X2*x^5-x^6+2*X1*x^4+2*x^5-X1*x^3-x^4+x^3+x^2-2*x+1)/(X1^2*X2 ^2*x^12+X1^2*X2^2*x^11-2*X1^2*X2*x^12-X1^2*X2^2*x^10-2*X1^2*X2*x^11+X1^2*x^12+ X1*X2^2*x^11+3*X1^2*X2*x^10+X1^2*x^11+X1*X2^2*x^10-2*X1*X2*x^11-X1^2*X2*x^9-2* X1^2*x^10-X1*X2^2*x^9-2*X1*X2*x^10+X1*x^11+X1^2*x^9+4*X1*X2*x^9+X1*x^10-X2*x^10 -2*X1*X2*x^8-3*X1*x^9-X2*x^9+x^10-X1*X2*x^7+2*X1*x^8+2*X2*x^8+x^9+X1*X2*x^6+X1* x^7-2*x^8-2*X1*X2*x^5-2*X1*x^6-2*X2*x^6+X1*X2*x^4+3*X1*x^5+X2*x^5+3*x^6-3*X1*x^ 4-2*x^5+X1*x^3+2*x^4-x^3-2*x^2+3*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 2, 3], are n 315 39 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 1, 3], are n 141 65 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 6 3 5 ----------- 65 and in floating point 0.3575061550 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 1/2 6 3 5 ate normal pair with correlation, ----------- 65 1/2 1/2 6 3 5 1061 i.e. , [[-----------, 0], [0, ----]] 65 845 ------------------------------------------------- Theorem Number, 122, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 2, 3], nor the composition, [4, 2, 2] Then infinity ----- 10 6 4 3 2 \ n x - x - x + x - 2 x + 2 x - 1 ) a(n) x = - ------------------------------------------ / 11 7 5 4 3 2 ----- x - x - x + x - 2 x + 3 x - 3 x + 1 n = 0 and in Maple format -(x^10-x^6-x^4+x^3-2*x^2+2*x-1)/(x^11-x^7-x^5+x^4-2*x^3+3*x^2-3*x+1) The asymptotic expression for a(n) is, n 0.61318603545226866829 1.9346824023256656451 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 2, 3] and d occurrences (as containment) of the composition, [4, 2, 2], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 13 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 13 2 13 2 13 2 12 13 - 2 X1 X2 x - 2 X1 X2 x + X1 x + X1 X2 x + 4 X1 X2 x 2 13 2 11 12 13 2 12 13 + X2 x + X1 X2 x - 2 X1 X2 x - 2 X1 x - X2 x - 2 X2 x 2 11 11 12 12 13 10 11 - X1 x - 2 X1 X2 x + X1 x + 2 X2 x + x + X1 X2 x + 2 X1 x 11 12 9 10 10 11 8 9 + X2 x - x - X1 X2 x - X1 x - X2 x - x + X1 X2 x + X1 x 9 10 7 8 8 9 6 7 8 + X2 x + x - X1 X2 x - X1 x - X2 x - x + 2 X1 X2 x + X1 x + x 5 6 4 5 6 4 5 3 - 2 X1 X2 x - 3 X1 x + X1 X2 x + 4 X1 x + x - 3 X1 x - 2 x + X1 x 4 2 / 2 2 14 2 14 2 14 + 2 x - 3 x + 3 x - 1) / (X1 X2 x - 2 X1 X2 x - 2 X1 X2 x / 2 14 2 13 14 2 14 2 12 13 + X1 x + X1 X2 x + 4 X1 X2 x + X2 x + X1 X2 x - 2 X1 X2 x 14 2 13 14 2 12 12 13 - 2 X1 x - X2 x - 2 X2 x - X1 x - 2 X1 X2 x + X1 x 13 14 11 12 12 13 10 + 2 X2 x + x + X1 X2 x + 2 X1 x + X2 x - x - X1 X2 x 11 11 12 9 10 10 11 8 - X1 x - X2 x - x + X1 X2 x + X1 x + X2 x + x - X1 X2 x 9 9 10 7 8 9 6 7 - X1 x - X2 x - x + 2 X1 X2 x + X1 x + x - 3 X1 X2 x - 3 X1 x 5 6 7 4 5 6 4 5 + 3 X1 X2 x + 5 X1 x + x - X1 X2 x - 6 X1 x - 2 x + 4 X1 x + 3 x 3 4 3 2 - X1 x - 3 x - x + 5 x - 4 x + 1) and in Maple format -(X1^2*X2^2*x^13-2*X1^2*X2*x^13-2*X1*X2^2*x^13+X1^2*x^13+X1*X2^2*x^12+4*X1*X2*x ^13+X2^2*x^13+X1^2*X2*x^11-2*X1*X2*x^12-2*X1*x^13-X2^2*x^12-2*X2*x^13-X1^2*x^11 -2*X1*X2*x^11+X1*x^12+2*X2*x^12+x^13+X1*X2*x^10+2*X1*x^11+X2*x^11-x^12-X1*X2*x^ 9-X1*x^10-X2*x^10-x^11+X1*X2*x^8+X1*x^9+X2*x^9+x^10-X1*X2*x^7-X1*x^8-X2*x^8-x^9 +2*X1*X2*x^6+X1*x^7+x^8-2*X1*X2*x^5-3*X1*x^6+X1*X2*x^4+4*X1*x^5+x^6-3*X1*x^4-2* x^5+X1*x^3+2*x^4-3*x^2+3*x-1)/(X1^2*X2^2*x^14-2*X1^2*X2*x^14-2*X1*X2^2*x^14+X1^ 2*x^14+X1*X2^2*x^13+4*X1*X2*x^14+X2^2*x^14+X1^2*X2*x^12-2*X1*X2*x^13-2*X1*x^14- X2^2*x^13-2*X2*x^14-X1^2*x^12-2*X1*X2*x^12+X1*x^13+2*X2*x^13+x^14+X1*X2*x^11+2* X1*x^12+X2*x^12-x^13-X1*X2*x^10-X1*x^11-X2*x^11-x^12+X1*X2*x^9+X1*x^10+X2*x^10+ x^11-X1*X2*x^8-X1*x^9-X2*x^9-x^10+2*X1*X2*x^7+X1*x^8+x^9-3*X1*X2*x^6-3*X1*x^7+3 *X1*X2*x^5+5*X1*x^6+x^7-X1*X2*x^4-6*X1*x^5-2*x^6+4*X1*x^4+3*x^5-X1*x^3-3*x^4-x^ 3+5*x^2-4*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 2, 3], are n 315 39 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 2, 2], are n 75 69 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 2 13 23 ------------- 69 and in floating point 0.5012062743 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 1/2 2 13 23 ate normal pair with correlation, ------------- 69 1/2 1/2 2 13 23 311 i.e. , [[-------------, 0], [0, ---]] 69 207 ------------------------------------------------- Theorem Number, 123, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 2, 3], nor the composition, [4, 3, 1] Then infinity ----- 8 7 6 5 3 \ n x + x + x + x + x - x + 1 ) a(n) x = - ----------------------------------- / 3 4 3 2 ----- (x - x + 1) (x + x + x + x - 1) n = 0 and in Maple format -(x^8+x^7+x^6+x^5+x^3-x+1)/(x^3-x+1)/(x^4+x^3+x^2+x-1) The asymptotic expression for a(n) is, n 0.63242246488998289333 1.9275619754829253043 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 2, 3] and d occurrences (as containment) of the composition, [4, 3, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 11 2 11 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x - X1 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 11 10 11 9 10 10 8 - X1 X2 x - X1 X2 x + X1 x + X1 X2 x + X1 x + X2 x - X1 X2 x 9 9 10 7 8 9 6 7 - X1 x - X2 x - x + X1 X2 x + X1 x + x + X1 X2 x - X1 x 5 6 4 5 6 4 5 3 - 2 X1 X2 x - 2 X1 x + X1 X2 x + 4 X1 x + x - 3 X1 x - 2 x + X1 x 4 2 / 8 7 8 + 2 x - 3 x + 3 x - 1) / ((-1 + x) (X1 X2 x - X1 X2 x - X1 x / 8 7 8 5 6 4 5 6 - X2 x + X1 x + x + 2 X1 X2 x + X1 x - X1 X2 x - 3 X1 x - x 4 5 3 4 3 2 + 3 X1 x + x - X1 x - 2 x + x + 2 x - 3 x + 1)) and in Maple format (X1^2*X2*x^11-X1^2*x^11-X1*X2*x^11-X1*X2*x^10+X1*x^11+X1*X2*x^9+X1*x^10+X2*x^10 -X1*X2*x^8-X1*x^9-X2*x^9-x^10+X1*X2*x^7+X1*x^8+x^9+X1*X2*x^6-X1*x^7-2*X1*X2*x^5 -2*X1*x^6+X1*X2*x^4+4*X1*x^5+x^6-3*X1*x^4-2*x^5+X1*x^3+2*x^4-3*x^2+3*x-1)/(-1+x )/(X1*X2*x^8-X1*X2*x^7-X1*x^8-X2*x^8+X1*x^7+x^8+2*X1*X2*x^5+X1*x^6-X1*X2*x^4-3* X1*x^5-x^6+3*X1*x^4+x^5-X1*x^3-2*x^4+x^3+2*x^2-3*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 2, 3], are n 315 39 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 3, 1], are n 135 65 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 4 3 5 ----------- 39 and in floating point 0.3972290613 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 1/2 4 3 5 ate normal pair with correlation, ----------- 39 1/2 1/2 4 3 5 667 i.e. , [[-----------, 0], [0, ---]] 39 507 ------------------------------------------------- Theorem Number, 124, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 2, 3], nor the composition, [5, 1, 2] Then infinity ----- \ n ) a(n) x = / ----- n = 0 13 12 11 9 8 7 6 5 3 x + x + x - x - x - x - 2 x - x - x + x - 1 - ---------------------------------------------------------------- 2 12 11 9 8 7 5 3 2 (x + 1) (x + x - x - x + x - 3 x + 3 x - x - 2 x + 1) and in Maple format -(x^13+x^12+x^11-x^9-x^8-x^7-2*x^6-x^5-x^3+x-1)/(x^2+1)/(x^12+x^11-x^9-x^8+x^7-\ 3*x^5+3*x^3-x^2-2*x+1) The asymptotic expression for a(n) is, n 0.64772484905004026630 1.9216163409913324968 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 2, 3] and d occurrences (as containment) of the composition, [5, 1, 2], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 3 18 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 2 18 3 18 2 18 2 18 2 18 - 3 X1 X2 x - X1 X2 x + 3 X1 X2 x + 3 X1 X2 x - X1 x 18 18 2 15 2 14 15 2 15 - 3 X1 X2 x + X1 x + X1 X2 x - X1 X2 x - 2 X1 X2 x - X2 x 14 15 2 14 15 2 12 2 12 + 2 X1 X2 x + X1 x + X2 x + 2 X2 x - X1 X2 x - X1 X2 x 14 14 15 2 12 2 11 12 14 - X1 x - 2 X2 x - x + X1 x + X1 X2 x + 4 X1 X2 x + x 11 12 12 10 11 12 10 - 2 X1 X2 x - 3 X1 x - X2 x - X1 X2 x + X1 x + x + X1 x 10 8 10 7 8 8 6 + X2 x + X1 X2 x - x - 2 X1 X2 x - X1 x - X2 x + 2 X1 X2 x 7 7 8 5 6 6 7 5 + 2 X1 x + 2 X2 x + x - X1 X2 x - X1 x - X2 x - 2 x - X1 x 4 5 3 4 2 / 2 3 19 + 2 X1 x + 2 x - X1 x - 2 x + 3 x - 3 x + 1) / (X1 X2 x / 2 3 18 2 2 19 3 19 2 2 18 2 19 - X1 X2 x - 3 X1 X2 x - X1 X2 x + 3 X1 X2 x + 3 X1 X2 x 3 18 2 19 2 18 2 19 2 18 + X1 X2 x + 3 X1 X2 x - 3 X1 X2 x - X1 x - 3 X1 X2 x 19 2 18 18 19 2 2 15 2 16 - 3 X1 X2 x + X1 x + 3 X1 X2 x + X1 x - X1 X2 x + X1 X2 x 18 2 15 16 2 16 2 15 16 - X1 x + 2 X1 X2 x - 2 X1 X2 x - X2 x - X1 x + X1 x 2 15 16 2 13 2 13 15 16 2 13 + X2 x + 2 X2 x - X1 X2 x - X1 X2 x - 2 X2 x - x + X1 x 2 12 13 15 2 11 12 13 + 2 X1 X2 x + 4 X1 X2 x + x - X1 X2 x - 4 X1 X2 x - 3 X1 x 13 12 13 10 11 11 10 - X2 x + 2 X1 x + x + 2 X1 X2 x + X1 x + 2 X2 x - 2 X1 x 10 11 8 10 7 8 8 - 2 X2 x - 2 x - 2 X1 X2 x + 2 x + 3 X1 X2 x + 2 X1 x + 3 X2 x 6 7 7 8 5 6 6 7 - 3 X1 X2 x - 2 X1 x - 3 X2 x - 3 x + X1 X2 x + X1 x + X2 x + 2 x 5 6 4 5 3 4 3 2 + 2 X1 x + x - 3 X1 x - 3 x + X1 x + 3 x + x - 5 x + 4 x - 1) and in Maple format -(X1^2*X2^3*x^18-3*X1^2*X2^2*x^18-X1*X2^3*x^18+3*X1^2*X2*x^18+3*X1*X2^2*x^18-X1 ^2*x^18-3*X1*X2*x^18+X1*x^18+X1*X2^2*x^15-X1*X2^2*x^14-2*X1*X2*x^15-X2^2*x^15+2 *X1*X2*x^14+X1*x^15+X2^2*x^14+2*X2*x^15-X1^2*X2*x^12-X1*X2^2*x^12-X1*x^14-2*X2* x^14-x^15+X1^2*x^12+X1*X2^2*x^11+4*X1*X2*x^12+x^14-2*X1*X2*x^11-3*X1*x^12-X2*x^ 12-X1*X2*x^10+X1*x^11+x^12+X1*x^10+X2*x^10+X1*X2*x^8-x^10-2*X1*X2*x^7-X1*x^8-X2 *x^8+2*X1*X2*x^6+2*X1*x^7+2*X2*x^7+x^8-X1*X2*x^5-X1*x^6-X2*x^6-2*x^7-X1*x^5+2* X1*x^4+2*x^5-X1*x^3-2*x^4+3*x^2-3*x+1)/(X1^2*X2^3*x^19-X1^2*X2^3*x^18-3*X1^2*X2 ^2*x^19-X1*X2^3*x^19+3*X1^2*X2^2*x^18+3*X1^2*X2*x^19+X1*X2^3*x^18+3*X1*X2^2*x^ 19-3*X1^2*X2*x^18-X1^2*x^19-3*X1*X2^2*x^18-3*X1*X2*x^19+X1^2*x^18+3*X1*X2*x^18+ X1*x^19-X1^2*X2^2*x^15+X1*X2^2*x^16-X1*x^18+2*X1^2*X2*x^15-2*X1*X2*x^16-X2^2*x^ 16-X1^2*x^15+X1*x^16+X2^2*x^15+2*X2*x^16-X1^2*X2*x^13-X1*X2^2*x^13-2*X2*x^15-x^ 16+X1^2*x^13+2*X1*X2^2*x^12+4*X1*X2*x^13+x^15-X1*X2^2*x^11-4*X1*X2*x^12-3*X1*x^ 13-X2*x^13+2*X1*x^12+x^13+2*X1*X2*x^10+X1*x^11+2*X2*x^11-2*X1*x^10-2*X2*x^10-2* x^11-2*X1*X2*x^8+2*x^10+3*X1*X2*x^7+2*X1*x^8+3*X2*x^8-3*X1*X2*x^6-2*X1*x^7-3*X2 *x^7-3*x^8+X1*X2*x^5+X1*x^6+X2*x^6+2*x^7+2*X1*x^5+x^6-3*X1*x^4-3*x^5+X1*x^3+3*x ^4+x^3-5*x^2+4*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 2, 3], are n 315 39 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [5, 1, 2], are n 113 57 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 10 13 19 -------------- 741 and in floating point 0.2120949210 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 10 247 ate normal pair with correlation, --------- 741 1/2 10 247 2423 i.e. , [[---------, 0], [0, ----]] 741 2223 ------------------------------------------------- Theorem Number, 125, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 2, 3], nor the composition, [5, 2, 1] Then infinity ----- \ n ) a(n) x = / ----- n = 0 13 12 11 8 7 6 5 3 x + x + x - x - x - x - x - x + x - 1 - ------------------------------------------------------------------------- 11 10 9 8 7 6 5 4 2 (-1 + x) (x + 2 x + 3 x + 3 x + 3 x + 2 x + x + x + x + x - 1) and in Maple format -(x^13+x^12+x^11-x^8-x^7-x^6-x^5-x^3+x-1)/(-1+x)/(x^11+2*x^10+3*x^9+3*x^8+3*x^7 +2*x^6+x^5+x^4+x^2+x-1) The asymptotic expression for a(n) is, n 0.64504238443304350890 1.9230256556702387387 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 2, 3] and d occurrences (as containment) of the composition, [5, 2, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 15 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 15 2 15 2 15 2 14 15 - 2 X1 X2 x - 2 X1 X2 x + X1 x + X1 X2 x + 4 X1 X2 x 2 15 14 15 2 14 15 2 12 + X2 x - 2 X1 X2 x - 2 X1 x - X2 x - 2 X2 x + X1 X2 x 14 14 15 2 12 12 14 11 + X1 x + 2 X2 x + x - X1 x - 2 X1 X2 x - x + X1 X2 x 12 12 10 11 11 12 9 + 2 X1 x + X2 x - X1 X2 x - X1 x - X2 x - x + X1 X2 x 10 10 11 8 9 9 10 7 + X1 x + X2 x + x - X1 X2 x - X1 x - X2 x - x + 2 X1 X2 x 8 9 6 7 5 6 5 6 + X1 x + x - 2 X1 X2 x - 2 X1 x + X1 X2 x + X1 x + X1 x + x 4 5 3 4 2 / - 2 X1 x - 2 x + X1 x + 2 x - 3 x + 3 x - 1) / ((-1 + x) ( / 2 13 13 2 13 13 13 13 10 X1 X2 x - 2 X1 X2 x - X2 x + X1 x + 2 X2 x - x + X1 X2 x 10 10 8 10 7 8 8 6 - X1 x - X2 x + X1 X2 x + x - X1 X2 x - X1 x - X2 x + 2 X1 X2 x 7 8 5 6 6 4 5 3 4 3 + X1 x + x - X1 X2 x - X1 x - x + 2 X1 x + x - X1 x - 2 x + x 2 + 2 x - 3 x + 1)) and in Maple format (X1^2*X2^2*x^15-2*X1^2*X2*x^15-2*X1*X2^2*x^15+X1^2*x^15+X1*X2^2*x^14+4*X1*X2*x^ 15+X2^2*x^15-2*X1*X2*x^14-2*X1*x^15-X2^2*x^14-2*X2*x^15+X1^2*X2*x^12+X1*x^14+2* X2*x^14+x^15-X1^2*x^12-2*X1*X2*x^12-x^14+X1*X2*x^11+2*X1*x^12+X2*x^12-X1*X2*x^ 10-X1*x^11-X2*x^11-x^12+X1*X2*x^9+X1*x^10+X2*x^10+x^11-X1*X2*x^8-X1*x^9-X2*x^9- x^10+2*X1*X2*x^7+X1*x^8+x^9-2*X1*X2*x^6-2*X1*x^7+X1*X2*x^5+X1*x^6+X1*x^5+x^6-2* X1*x^4-2*x^5+X1*x^3+2*x^4-3*x^2+3*x-1)/(-1+x)/(X1*X2^2*x^13-2*X1*X2*x^13-X2^2*x ^13+X1*x^13+2*X2*x^13-x^13+X1*X2*x^10-X1*x^10-X2*x^10+X1*X2*x^8+x^10-X1*X2*x^7- X1*x^8-X2*x^8+2*X1*X2*x^6+X1*x^7+x^8-X1*X2*x^5-X1*x^6-x^6+2*X1*x^4+x^5-X1*x^3-2 *x^4+x^3+2*x^2-3*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 2, 3], are n 315 39 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [5, 2, 1], are n 111 57 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 4 13 19 ------------- 247 and in floating point 0.2545139052 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 4 247 ate normal pair with correlation, -------- 247 1/2 4 247 279 i.e. , [[--------, 0], [0, ---]] 247 247 ------------------------------------------------- Theorem Number, 126, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 2, 3], nor the composition, [6, 1, 1] Then infinity ----- \ n ) a(n) x = / ----- n = 0 14 13 12 9 7 6 4 3 2 x + x + x - x - x - x - x + x - 2 x + 2 x - 1 - ------------------------------------------------------------- 10 9 8 7 6 5 3 2 (-1 + x) (x + x + 2 x + x + x + x + x - x + 2 x - 1) and in Maple format -(x^14+x^13+x^12-x^9-x^7-x^6-x^4+x^3-2*x^2+2*x-1)/(-1+x)/(x^10+x^9+2*x^8+x^7+x^ 6+x^5+x^3-x^2+2*x-1) The asymptotic expression for a(n) is, n 0.67395690992279832777 1.9139965407423997306 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 2, 3] and d occurrences (as containment) of the composition, [6, 1, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 17 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 17 2 17 2 17 17 2 17 - 2 X1 X2 x - 2 X1 X2 x + X1 x + 4 X1 X2 x + X2 x 2 15 17 17 2 14 15 2 15 + X1 X2 x - 2 X1 x - 2 X2 x + X1 X2 x - 2 X1 X2 x - X2 x 17 2 13 2 13 14 15 2 14 + x + X1 X2 x - X1 X2 x - 2 X1 X2 x + X1 x - X2 x 15 2 13 14 2 13 14 15 13 13 + 2 X2 x - X1 x + X1 x + X2 x + 2 X2 x - x + X1 x - X2 x 14 11 11 11 9 11 8 9 - x + X1 X2 x - X1 x - X2 x - X1 X2 x + x + 2 X1 X2 x + X1 x 9 7 8 8 9 6 7 7 + X2 x - 2 X1 X2 x - 2 X1 x - 2 X2 x - x + X1 X2 x + 2 X1 x + X2 x 8 6 7 5 6 4 5 3 4 + 2 x - 2 X1 x - x + 2 X1 x + x - 2 X1 x - 2 x + X1 x + 2 x 2 / 2 13 13 2 13 - 3 x + 3 x - 1) / ((-1 + x) (X1 X2 x - 2 X1 X2 x - X2 x / 13 13 13 10 9 10 10 + X1 x + 2 X2 x - x + X1 X2 x + X1 X2 x - X1 x - X2 x 8 9 9 10 7 8 8 9 - X1 X2 x - X1 x - X2 x + x + 2 X1 X2 x + X1 x + X2 x + x 6 7 7 8 6 7 5 6 4 - X1 X2 x - 2 X1 x - X2 x - x + 2 X1 x + x - X1 x - x + 2 X1 x 5 3 4 3 2 + x - X1 x - 2 x + x + 2 x - 3 x + 1)) and in Maple format (X1^2*X2^2*x^17-2*X1^2*X2*x^17-2*X1*X2^2*x^17+X1^2*x^17+4*X1*X2*x^17+X2^2*x^17+ X1*X2^2*x^15-2*X1*x^17-2*X2*x^17+X1*X2^2*x^14-2*X1*X2*x^15-X2^2*x^15+x^17+X1^2* X2*x^13-X1*X2^2*x^13-2*X1*X2*x^14+X1*x^15-X2^2*x^14+2*X2*x^15-X1^2*x^13+X1*x^14 +X2^2*x^13+2*X2*x^14-x^15+X1*x^13-X2*x^13-x^14+X1*X2*x^11-X1*x^11-X2*x^11-X1*X2 *x^9+x^11+2*X1*X2*x^8+X1*x^9+X2*x^9-2*X1*X2*x^7-2*X1*x^8-2*X2*x^8-x^9+X1*X2*x^6 +2*X1*x^7+X2*x^7+2*x^8-2*X1*x^6-x^7+2*X1*x^5+x^6-2*X1*x^4-2*x^5+X1*x^3+2*x^4-3* x^2+3*x-1)/(-1+x)/(X1*X2^2*x^13-2*X1*X2*x^13-X2^2*x^13+X1*x^13+2*X2*x^13-x^13+ X1*X2*x^10+X1*X2*x^9-X1*x^10-X2*x^10-X1*X2*x^8-X1*x^9-X2*x^9+x^10+2*X1*X2*x^7+ X1*x^8+X2*x^8+x^9-X1*X2*x^6-2*X1*x^7-X2*x^7-x^8+2*X1*x^6+x^7-X1*x^5-x^6+2*X1*x^ 4+x^5-X1*x^3-2*x^4+x^3+2*x^2-3*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 2, 3], are n 315 39 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [6, 1, 1], are n 49 53 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 2 39 53 ------------- 2067 and in floating point 0.04399058701 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 2 2067 ate normal pair with correlation, --------- 2067 1/2 2 2067 2075 i.e. , [[---------, 0], [0, ----]] 2067 2067 ------------------------------------------------- Theorem Number, 127, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 3, 2], nor the composition, [1, 1, 6] Then infinity ----- \ n ) a(n) x = / ----- n = 0 11 9 7 6 4 3 2 x + x + x + x + x - x + 2 x - 2 x + 1 ------------------------------------------------------------- 10 9 8 7 6 5 3 2 (-1 + x) (x + x + 2 x + x + x + x + x - x + 2 x - 1) and in Maple format (x^11+x^9+x^7+x^6+x^4-x^3+2*x^2-2*x+1)/(-1+x)/(x^10+x^9+2*x^8+x^7+x^6+x^5+x^3-x ^2+2*x-1) The asymptotic expression for a(n) is, n 0.67617629498732100894 1.9139965407423997306 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 3, 2] and d occurrences (as containment) of the composition, [1, 1, 6], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 13 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 13 2 13 13 13 13 8 7 - 2 X1 X2 x - X2 x + X1 x + 2 X2 x - x - X1 X2 x + X1 X2 x 8 8 6 7 7 8 6 7 5 + X1 x + X2 x - X1 X2 x - X1 x - X2 x - x + X1 x + x - X1 x 4 5 3 4 3 2 / 2 13 + X1 x + x - X1 x - x + x + x - 2 x + 1) / (X1 X2 x / 13 2 13 13 13 13 10 9 - 2 X1 X2 x - X2 x + X1 x + 2 X2 x - x + X1 X2 x + X1 X2 x 10 10 8 9 9 10 7 8 - X1 x - X2 x - X1 X2 x - X1 x - X2 x + x + 2 X1 X2 x + X1 x 8 9 6 7 7 8 6 7 5 6 + X2 x + x - X1 X2 x - 2 X1 x - X2 x - x + 2 X1 x + x - X1 x - x 4 5 3 4 3 2 + 2 X1 x + x - X1 x - 2 x + x + 2 x - 3 x + 1) and in Maple format (X1*X2^2*x^13-2*X1*X2*x^13-X2^2*x^13+X1*x^13+2*X2*x^13-x^13-X1*X2*x^8+X1*X2*x^7 +X1*x^8+X2*x^8-X1*X2*x^6-X1*x^7-X2*x^7-x^8+X1*x^6+x^7-X1*x^5+X1*x^4+x^5-X1*x^3- x^4+x^3+x^2-2*x+1)/(X1*X2^2*x^13-2*X1*X2*x^13-X2^2*x^13+X1*x^13+2*X2*x^13-x^13+ X1*X2*x^10+X1*X2*x^9-X1*x^10-X2*x^10-X1*X2*x^8-X1*x^9-X2*x^9+x^10+2*X1*X2*x^7+ X1*x^8+X2*x^8+x^9-X1*X2*x^6-2*X1*x^7-X2*x^7-x^8+2*X1*x^6+x^7-X1*x^5-x^6+2*X1*x^ 4+x^5-X1*x^3-2*x^4+x^3+2*x^2-3*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 3, 2], are n 315 39 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 1, 6], are n 49 53 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 2 39 53 ------------- 2067 and in floating point 0.04399058701 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 2 2067 ate normal pair with correlation, --------- 2067 1/2 2 2067 2075 i.e. , [[---------, 0], [0, ----]] 2067 2067 ------------------------------------------------- Theorem Number, 128, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 3, 2], nor the composition, [1, 2, 5] Then infinity ----- 8 7 6 5 3 \ n x + x + x + x + x - x + 1 ) a(n) x = - ----------------------------------- / 3 4 3 2 ----- (x - x + 1) (x + x + x + x - 1) n = 0 and in Maple format -(x^8+x^7+x^6+x^5+x^3-x+1)/(x^3-x+1)/(x^4+x^3+x^2+x-1) The asymptotic expression for a(n) is, n 0.63242246488998289333 1.9275619754829253043 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 3, 2] and d occurrences (as containment) of the composition, [1, 2, 5], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 9 9 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x - X1 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 9 7 9 6 7 5 6 4 5 - X2 x + X1 X2 x + x - X1 X2 x - X1 x + X1 X2 x + X1 x - X1 x - x 3 4 3 2 / 8 7 8 8 + X1 x + x - x - x + 2 x - 1) / (X1 X2 x - X1 X2 x - X1 x - X2 x / 6 7 8 5 6 6 4 5 3 + 2 X1 X2 x + X1 x + x - X1 X2 x - X1 x - x + 2 X1 x + x - X1 x 4 3 2 - 2 x + x + 2 x - 3 x + 1) and in Maple format -(X1*X2*x^9-X1*x^9-X2*x^9+X1*X2*x^7+x^9-X1*X2*x^6-X1*x^7+X1*X2*x^5+X1*x^6-X1*x^ 4-x^5+X1*x^3+x^4-x^3-x^2+2*x-1)/(X1*X2*x^8-X1*X2*x^7-X1*x^8-X2*x^8+2*X1*X2*x^6+ X1*x^7+x^8-X1*X2*x^5-X1*x^6-x^6+2*X1*x^4+x^5-X1*x^3-2*x^4+x^3+2*x^2-3*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 3, 2], are n 315 39 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 2, 5], are n 111 57 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 16 13 19 -------------- 741 and in floating point 0.3393518736 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 16 247 ate normal pair with correlation, --------- 741 1/2 16 247 2735 i.e. , [[---------, 0], [0, ----]] 741 2223 ------------------------------------------------- Theorem Number, 129, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 3, 2], nor the composition, [1, 3, 4] Then infinity ----- 5 4 3 2 \ n x - x + x + x - 2 x + 1 ) a(n) x = - ----------------------------------------------- / 10 8 6 5 4 3 2 ----- x - x + x - x + 2 x - x - 2 x + 3 x - 1 n = 0 and in Maple format -(x^5-x^4+x^3+x^2-2*x+1)/(x^10-x^8+x^6-x^5+2*x^4-x^3-2*x^2+3*x-1) The asymptotic expression for a(n) is, n 0.62193578086462630009 1.9325948503424936840 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 3, 2] and d occurrences (as containment) of the composition, [1, 3, 4], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 7 7 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x - X1 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 5 4 5 4 5 3 4 3 2 - X1 X2 x + X1 X2 x + 2 X1 x - 2 X1 x - x + X1 x + x - x - x / 10 10 10 8 10 7 + 2 x - 1) / (X1 X2 x - X1 x - X2 x - X1 X2 x + x + X1 X2 x / 8 8 7 8 5 6 4 5 + X1 x + X2 x - X1 x - x - 2 X1 X2 x - X1 x + X1 X2 x + 3 X1 x 6 4 5 3 4 3 2 + x - 3 X1 x - x + X1 x + 2 x - x - 2 x + 3 x - 1) and in Maple format (X1*X2*x^7-X1*x^7-X1*X2*x^5+X1*X2*x^4+2*X1*x^5-2*X1*x^4-x^5+X1*x^3+x^4-x^3-x^2+ 2*x-1)/(X1*X2*x^10-X1*x^10-X2*x^10-X1*X2*x^8+x^10+X1*X2*x^7+X1*x^8+X2*x^8-X1*x^ 7-x^8-2*X1*X2*x^5-X1*x^6+X1*X2*x^4+3*X1*x^5+x^6-3*X1*x^4-x^5+X1*x^3+2*x^4-x^3-2 *x^2+3*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 3, 2], are n 315 39 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 3, 4], are n 135 65 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 8 3 5 ----------- 65 and in floating point 0.4766748734 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 1/2 8 3 5 ate normal pair with correlation, ----------- 65 1/2 1/2 8 3 5 1229 i.e. , [[-----------, 0], [0, ----]] 65 845 ------------------------------------------------- Theorem Number, 130, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 3, 2], nor the composition, [1, 4, 3] Then infinity ----- 5 4 3 2 \ n x - x + x + x - 2 x + 1 ) a(n) x = - ----------------------------------------------- / 10 8 6 5 4 3 2 ----- x - x + x - x + 2 x - x - 2 x + 3 x - 1 n = 0 and in Maple format -(x^5-x^4+x^3+x^2-2*x+1)/(x^10-x^8+x^6-x^5+2*x^4-x^3-2*x^2+3*x-1) The asymptotic expression for a(n) is, n 0.62193578086462630009 1.9325948503424936840 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 3, 2] and d occurrences (as containment) of the composition, [1, 4, 3], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 7 7 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x - X1 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 5 4 5 4 5 3 4 3 2 - X1 X2 x + X1 X2 x + 2 X1 x - 2 X1 x - x + X1 x + x - x - x / 10 10 10 8 10 7 + 2 x - 1) / (X1 X2 x - X1 x - X2 x - X1 X2 x + x + X1 X2 x / 8 8 7 8 5 6 4 5 + X1 x + X2 x - X1 x - x - 2 X1 X2 x - X1 x + X1 X2 x + 3 X1 x 6 4 5 3 4 3 2 + x - 3 X1 x - x + X1 x + 2 x - x - 2 x + 3 x - 1) and in Maple format (X1*X2*x^7-X1*x^7-X1*X2*x^5+X1*X2*x^4+2*X1*x^5-2*X1*x^4-x^5+X1*x^3+x^4-x^3-x^2+ 2*x-1)/(X1*X2*x^10-X1*x^10-X2*x^10-X1*X2*x^8+x^10+X1*X2*x^7+X1*x^8+X2*x^8-X1*x^ 7-x^8-2*X1*X2*x^5-X1*x^6+X1*X2*x^4+3*X1*x^5+x^6-3*X1*x^4-x^5+X1*x^3+2*x^4-x^3-2 *x^2+3*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 3, 2], are n 315 39 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 4, 3], are n 135 65 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 8 3 5 ----------- 65 and in floating point 0.4766748734 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 1/2 8 3 5 ate normal pair with correlation, ----------- 65 1/2 1/2 8 3 5 1229 i.e. , [[-----------, 0], [0, ----]] 65 845 ------------------------------------------------- Theorem Number, 131, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 3, 2], nor the composition, [1, 5, 2] Then infinity ----- 5 4 3 2 \ n x - x + x + x - 2 x + 1 ) a(n) x = -------------------------------------------- / 3 4 3 2 ----- (-1 + x) (x - x + 1) (x + x + x + x - 1) n = 0 and in Maple format (x^5-x^4+x^3+x^2-2*x+1)/(-1+x)/(x^3-x+1)/(x^4+x^3+x^2+x-1) The asymptotic expression for a(n) is, n 0.63758295460268560269 1.9275619754829253043 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 3, 2] and d occurrences (as containment) of the composition, [1, 5, 2], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 7 6 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x - X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 7 5 6 4 5 3 4 3 2 - X1 x + X1 X2 x + X1 x - X1 x - x + X1 x + x - x - x + 2 x - 1) / 8 7 8 8 6 7 8 / (X1 X2 x - X1 X2 x - X1 x - X2 x + 2 X1 X2 x + X1 x + x / 5 6 6 4 5 3 4 3 2 - X1 X2 x - X1 x - x + 2 X1 x + x - X1 x - 2 x + x + 2 x - 3 x + 1) and in Maple format -(X1*X2*x^7-X1*X2*x^6-X1*x^7+X1*X2*x^5+X1*x^6-X1*x^4-x^5+X1*x^3+x^4-x^3-x^2+2*x -1)/(X1*X2*x^8-X1*X2*x^7-X1*x^8-X2*x^8+2*X1*X2*x^6+X1*x^7+x^8-X1*X2*x^5-X1*x^6- x^6+2*X1*x^4+x^5-X1*x^3-2*x^4+x^3+2*x^2-3*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 3, 2], are n 315 39 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 5, 2], are n 111 57 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 16 13 19 -------------- 741 and in floating point 0.3393518736 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 16 247 ate normal pair with correlation, --------- 741 1/2 16 247 2735 i.e. , [[---------, 0], [0, ----]] 741 2223 ------------------------------------------------- Theorem Number, 132, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 3, 2], nor the composition, [1, 6, 1] Then infinity ----- \ n ) a(n) x = / ----- n = 0 14 12 10 8 7 5 4 3 2 x + x + x - x + x + x - x + x + x - 2 x + 1 - ---------------------------------------------------------------------- 10 9 8 7 6 5 3 2 2 (x + 1) (x + x + 2 x + x + x + x + x - x + 2 x - 1) (-1 + x) and in Maple format -(x^14+x^12+x^10-x^8+x^7+x^5-x^4+x^3+x^2-2*x+1)/(x+1)/(x^10+x^9+2*x^8+x^7+x^6+x ^5+x^3-x^2+2*x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.68066863342843301004 1.9139965407423997306 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 3, 2] and d occurrences (as containment) of the composition, [1, 6, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 15 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 14 15 2 15 2 13 14 15 - X1 X2 x - 2 X1 X2 x - X2 x + X1 X2 x + 2 X1 X2 x + X1 x 2 14 15 13 14 2 13 14 15 + X2 x + 2 X2 x - 2 X1 X2 x - X1 x - X2 x - 2 X2 x - x 12 13 13 14 11 12 12 13 + X1 X2 x + X1 x + 2 X2 x + x - X1 X2 x - X1 x - X2 x - x 10 11 11 12 9 10 10 11 + X1 X2 x + X1 x + X2 x + x + X1 X2 x - X1 x - X2 x - x 8 9 9 10 7 8 8 9 - 2 X1 X2 x - X1 x - X2 x + x + 2 X1 X2 x + 2 X1 x + 2 X2 x + x 6 7 7 8 6 7 5 6 - X1 X2 x - 2 X1 x - X2 x - 2 x + 2 X1 x + x - 2 X1 x - x 4 5 3 4 2 / + 2 X1 x + 2 x - X1 x - 2 x + 3 x - 3 x + 1) / ((-1 + x) ( / 2 13 13 2 13 13 13 13 10 X1 X2 x - 2 X1 X2 x - X2 x + X1 x + 2 X2 x - x + X1 X2 x 9 10 10 8 9 9 10 7 + X1 X2 x - X1 x - X2 x - X1 X2 x - X1 x - X2 x + x + 2 X1 X2 x 8 8 9 6 7 7 8 6 7 + X1 x + X2 x + x - X1 X2 x - 2 X1 x - X2 x - x + 2 X1 x + x 5 6 4 5 3 4 3 2 - X1 x - x + 2 X1 x + x - X1 x - 2 x + x + 2 x - 3 x + 1)) and in Maple format -(X1*X2^2*x^15-X1*X2^2*x^14-2*X1*X2*x^15-X2^2*x^15+X1*X2^2*x^13+2*X1*X2*x^14+X1 *x^15+X2^2*x^14+2*X2*x^15-2*X1*X2*x^13-X1*x^14-X2^2*x^13-2*X2*x^14-x^15+X1*X2*x ^12+X1*x^13+2*X2*x^13+x^14-X1*X2*x^11-X1*x^12-X2*x^12-x^13+X1*X2*x^10+X1*x^11+ X2*x^11+x^12+X1*X2*x^9-X1*x^10-X2*x^10-x^11-2*X1*X2*x^8-X1*x^9-X2*x^9+x^10+2*X1 *X2*x^7+2*X1*x^8+2*X2*x^8+x^9-X1*X2*x^6-2*X1*x^7-X2*x^7-2*x^8+2*X1*x^6+x^7-2*X1 *x^5-x^6+2*X1*x^4+2*x^5-X1*x^3-2*x^4+3*x^2-3*x+1)/(-1+x)/(X1*X2^2*x^13-2*X1*X2* x^13-X2^2*x^13+X1*x^13+2*X2*x^13-x^13+X1*X2*x^10+X1*X2*x^9-X1*x^10-X2*x^10-X1* X2*x^8-X1*x^9-X2*x^9+x^10+2*X1*X2*x^7+X1*x^8+X2*x^8+x^9-X1*X2*x^6-2*X1*x^7-X2*x ^7-x^8+2*X1*x^6+x^7-X1*x^5-x^6+2*X1*x^4+x^5-X1*x^3-2*x^4+x^3+2*x^2-3*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 3, 2], are n 315 39 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 6, 1], are n 49 53 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 2 39 53 ------------- 2067 and in floating point 0.04399058701 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 2 2067 ate normal pair with correlation, --------- 2067 1/2 2 2067 2075 i.e. , [[---------, 0], [0, ----]] 2067 2067 ------------------------------------------------- Theorem Number, 133, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 3, 2], nor the composition, [2, 1, 5] Then infinity ----- 12 11 10 9 8 7 6 5 3 \ n x + x + x + x + x + x + 2 x + x + x - x + 1 ) a(n) x = - ------------------------------------------------------- / 13 8 7 4 3 ----- x + x + 2 x + x - x + 2 x - 1 n = 0 and in Maple format -(x^12+x^11+x^10+x^9+x^8+x^7+2*x^6+x^5+x^3-x+1)/(x^13+x^8+2*x^7+x^4-x^3+2*x-1) The asymptotic expression for a(n) is, n 0.65262822115524216049 1.9206669544913373883 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 3, 2] and d occurrences (as containment) of the composition, [2, 1, 5], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 13 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 13 2 13 2 11 13 13 11 - 2 X1 X2 x - X2 x + X1 X2 x + X1 x + 2 X2 x - 2 X1 X2 x 13 11 7 6 7 7 5 6 - x + X1 x - X1 X2 x + X1 X2 x + X1 x + X2 x - X1 X2 x - X1 x 6 7 6 4 5 3 4 3 2 / - X2 x - x + x + X1 x + x - X1 x - x + x + x - 2 x + 1) / ( / 2 14 2 13 14 2 14 2 12 13 X1 X2 x - X1 X2 x - 2 X1 X2 x - X2 x + X1 X2 x + 2 X1 X2 x 14 2 13 14 2 11 12 13 + X1 x + X2 x + 2 X2 x - X1 X2 x - 2 X1 X2 x - X1 x 13 14 11 12 13 11 9 - 2 X2 x - x + 2 X1 X2 x + X1 x + x - X1 x - X1 X2 x 8 9 9 7 8 8 9 6 - X1 X2 x + X1 x + X2 x + X1 X2 x + X1 x + X2 x - x - 2 X1 X2 x 7 7 8 5 6 6 7 4 5 - X1 x - 2 X2 x - x + X1 X2 x + X1 x + X2 x + 2 x - 2 X1 x - x 3 4 3 2 + X1 x + 2 x - x - 2 x + 3 x - 1) and in Maple format -(X1*X2^2*x^13-2*X1*X2*x^13-X2^2*x^13+X1*X2^2*x^11+X1*x^13+2*X2*x^13-2*X1*X2*x^ 11-x^13+X1*x^11-X1*X2*x^7+X1*X2*x^6+X1*x^7+X2*x^7-X1*X2*x^5-X1*x^6-X2*x^6-x^7+x ^6+X1*x^4+x^5-X1*x^3-x^4+x^3+x^2-2*x+1)/(X1*X2^2*x^14-X1*X2^2*x^13-2*X1*X2*x^14 -X2^2*x^14+X1*X2^2*x^12+2*X1*X2*x^13+X1*x^14+X2^2*x^13+2*X2*x^14-X1*X2^2*x^11-2 *X1*X2*x^12-X1*x^13-2*X2*x^13-x^14+2*X1*X2*x^11+X1*x^12+x^13-X1*x^11-X1*X2*x^9- X1*X2*x^8+X1*x^9+X2*x^9+X1*X2*x^7+X1*x^8+X2*x^8-x^9-2*X1*X2*x^6-X1*x^7-2*X2*x^7 -x^8+X1*X2*x^5+X1*x^6+X2*x^6+2*x^7-2*X1*x^4-x^5+X1*x^3+2*x^4-x^3-2*x^2+3*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 3, 2], are n 315 39 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 1, 5], are n 113 57 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 10 13 19 -------------- 741 and in floating point 0.2120949210 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 10 247 ate normal pair with correlation, --------- 741 1/2 10 247 2423 i.e. , [[---------, 0], [0, ----]] 741 2223 ------------------------------------------------- Theorem Number, 134, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 3, 2], nor the composition, [2, 2, 4] Then infinity ----- 6 4 3 2 \ n x + x - x + 2 x - 2 x + 1 ) a(n) x = - ------------------------------------ / 7 5 4 3 2 ----- x + x - x + 2 x - 3 x + 3 x - 1 n = 0 and in Maple format -(x^6+x^4-x^3+2*x^2-2*x+1)/(x^7+x^5-x^4+2*x^3-3*x^2+3*x-1) The asymptotic expression for a(n) is, n 0.60945823930910114296 1.9365136360000287543 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 3, 2] and d occurrences (as containment) of the composition, [2, 2, 4], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 8 8 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x - X1 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 8 6 8 5 6 4 5 4 - X2 x + X1 X2 x + x - X1 X2 x - X1 x + X1 X2 x + 2 X1 x - 2 X1 x 5 3 4 3 2 / 9 9 9 - x + X1 x + x - x - x + 2 x - 1) / (X1 X2 x - X1 x - X2 x / 7 9 6 7 5 6 4 + X1 X2 x + x - X1 X2 x - X1 x + 2 X1 X2 x + 2 X1 x - X1 X2 x 5 6 4 5 3 4 3 2 - 3 X1 x - x + 3 X1 x + x - X1 x - 2 x + x + 2 x - 3 x + 1) and in Maple format -(X1*X2*x^8-X1*x^8-X2*x^8+X1*X2*x^6+x^8-X1*X2*x^5-X1*x^6+X1*X2*x^4+2*X1*x^5-2* X1*x^4-x^5+X1*x^3+x^4-x^3-x^2+2*x-1)/(X1*X2*x^9-X1*x^9-X2*x^9+X1*X2*x^7+x^9-X1* X2*x^6-X1*x^7+2*X1*X2*x^5+2*X1*x^6-X1*X2*x^4-3*X1*x^5-x^6+3*X1*x^4+x^5-X1*x^3-2 *x^4+x^3+2*x^2-3*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 3, 2], are n 315 39 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 2, 4], are n 75 69 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 10 13 23 -------------- 299 and in floating point 0.5783149317 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 10 299 ate normal pair with correlation, --------- 299 1/2 10 299 499 i.e. , [[---------, 0], [0, ---]] 299 299 ------------------------------------------------- Theorem Number, 135, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 3, 2], nor the composition, [2, 5, 1] Then infinity ----- 5 4 3 2 \ n x - x + x + x - 2 x + 1 ) a(n) x = -------------------------------------------- / 3 4 3 2 ----- (-1 + x) (x - x + 1) (x + x + x + x - 1) n = 0 and in Maple format (x^5-x^4+x^3+x^2-2*x+1)/(-1+x)/(x^3-x+1)/(x^4+x^3+x^2+x-1) The asymptotic expression for a(n) is, n 0.63758295460268560269 1.9275619754829253043 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 3, 2] and d occurrences (as containment) of the composition, [2, 5, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 7 6 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x - X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 7 5 6 4 5 3 4 3 2 - X1 x + X1 X2 x + X1 x - X1 x - x + X1 x + x - x - x + 2 x - 1) / 8 7 8 8 6 7 8 / (X1 X2 x - X1 X2 x - X1 x - X2 x + 2 X1 X2 x + X1 x + x / 5 6 6 4 5 3 4 3 2 - X1 X2 x - X1 x - x + 2 X1 x + x - X1 x - 2 x + x + 2 x - 3 x + 1) and in Maple format -(X1*X2*x^7-X1*X2*x^6-X1*x^7+X1*X2*x^5+X1*x^6-X1*x^4-x^5+X1*x^3+x^4-x^3-x^2+2*x -1)/(X1*X2*x^8-X1*X2*x^7-X1*x^8-X2*x^8+2*X1*X2*x^6+X1*x^7+x^8-X1*X2*x^5-X1*x^6- x^6+2*X1*x^4+x^5-X1*x^3-2*x^4+x^3+2*x^2-3*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 3, 2], are n 315 39 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 5, 1], are n 111 57 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 16 13 19 -------------- 741 and in floating point 0.3393518736 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 16 247 ate normal pair with correlation, --------- 741 1/2 16 247 2735 i.e. , [[---------, 0], [0, ----]] 741 2223 ------------------------------------------------- Theorem Number, 136, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 3, 2], nor the composition, [3, 1, 4] Then infinity ----- \ n ) a(n) x = / ----- n = 0 10 9 8 6 5 4 3 2 x - x + x - x + 2 x - x + x + x - 2 x + 1 ------------------------------------------------------------------------ 13 11 10 9 8 6 5 4 3 2 x - 2 x + x - x + 2 x - 3 x + 2 x - 2 x + x + 2 x - 3 x + 1 and in Maple format (x^10-x^9+x^8-x^6+2*x^5-x^4+x^3+x^2-2*x+1)/(x^13-2*x^11+x^10-x^9+2*x^8-3*x^6+2* x^5-2*x^4+x^3+2*x^2-3*x+1) The asymptotic expression for a(n) is, n 0.64411604859512176136 1.9232078865547166800 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 3, 2] and d occurrences (as containment) of the composition, [3, 1, 4], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 3 15 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 2 15 3 15 2 2 14 2 15 2 15 - 2 X1 X2 x - X1 X2 x + X1 X2 x + X1 X2 x + 2 X1 X2 x 2 2 13 2 14 2 14 15 2 13 - X1 X2 x - 2 X1 X2 x - X1 X2 x - X1 X2 x + 2 X1 X2 x 2 14 2 13 14 2 2 11 2 12 2 13 + X1 x + X1 X2 x + 2 X1 X2 x + X1 X2 x - X1 X2 x - X1 x 2 12 13 14 2 2 10 2 11 2 12 + X1 X2 x - 2 X1 X2 x - X1 x - X1 X2 x - 2 X1 X2 x + X1 x 2 11 13 2 12 2 10 2 11 2 10 - X1 X2 x + X1 x - X2 x + 3 X1 X2 x + X1 x + X1 X2 x 11 12 2 11 12 2 9 2 10 + 2 X1 X2 x - X1 x + X2 x + X2 x - X1 X2 x - 2 X1 x 2 9 10 11 11 2 9 9 - X1 X2 x - 5 X1 X2 x - X1 x - 2 X2 x + X1 x + 5 X1 X2 x 10 10 11 8 9 9 10 + 4 X1 x + 2 X2 x + x - X1 X2 x - 4 X1 x - 2 X2 x - 2 x 7 8 8 9 6 7 7 8 - X1 X2 x + X1 x + X2 x + 2 x + 2 X1 X2 x + X1 x + X2 x - x 5 6 6 7 4 5 5 6 - 2 X1 X2 x - 3 X1 x - 2 X2 x - x + X1 X2 x + 4 X1 x + X2 x + 3 x 4 5 3 4 2 / 2 3 17 - 3 X1 x - 3 x + X1 x + 2 x - 3 x + 3 x - 1) / (X1 X2 x / 2 3 16 2 2 17 3 17 2 3 15 2 2 16 + X1 X2 x - 3 X1 X2 x - X1 X2 x - X1 X2 x - 2 X1 X2 x 2 17 3 16 2 17 2 2 15 2 16 + 3 X1 X2 x - X1 X2 x + 3 X1 X2 x + 4 X1 X2 x + X1 X2 x 2 17 3 15 2 16 17 2 2 14 - X1 x + X1 X2 x + 2 X1 X2 x - 3 X1 X2 x - 2 X1 X2 x 2 15 2 15 16 17 2 2 13 - 5 X1 X2 x - 4 X1 X2 x - X1 X2 x + X1 x + X1 X2 x 2 14 2 15 2 14 15 2 13 + 4 X1 X2 x + 2 X1 x + 3 X1 X2 x + 5 X1 X2 x - 3 X1 X2 x 2 14 14 15 2 14 2 2 11 2 13 - 2 X1 x - 6 X1 X2 x - 2 X1 x - X2 x - 2 X1 X2 x + 2 X1 x 2 12 13 14 14 2 2 10 - X1 X2 x + 2 X1 X2 x + 3 X1 x + 2 X2 x + X1 X2 x 2 11 2 11 12 13 2 12 13 + 5 X1 X2 x + X1 X2 x + 2 X1 X2 x - 2 X1 x + 2 X2 x - X2 x 14 2 10 2 11 2 10 11 12 - x - 4 X1 X2 x - 3 X1 x - 2 X1 X2 x - 5 X1 X2 x - X1 x 2 11 12 13 2 9 2 10 2 9 - X2 x - 4 X2 x + x + X1 X2 x + 3 X1 x + X1 X2 x 10 11 11 12 2 9 9 + 8 X1 X2 x + 4 X1 x + 4 X2 x + 2 x - X1 x - 6 X1 X2 x 10 10 11 8 9 9 10 - 6 X1 x - 2 X2 x - 3 x + X1 X2 x + 5 X1 x + 3 X2 x + 2 x 7 8 8 9 6 7 7 + 2 X1 X2 x - X1 x - 2 X2 x - 3 x - 3 X1 X2 x - 3 X1 x - 2 X2 x 8 5 6 6 7 4 5 + 2 x + 3 X1 X2 x + 5 X1 x + 3 X2 x + 3 x - X1 X2 x - 6 X1 x 5 6 4 5 3 4 3 2 - X2 x - 5 x + 4 X1 x + 4 x - X1 x - 3 x - x + 5 x - 4 x + 1) and in Maple format -(X1^2*X2^3*x^15-2*X1^2*X2^2*x^15-X1*X2^3*x^15+X1^2*X2^2*x^14+X1^2*X2*x^15+2*X1 *X2^2*x^15-X1^2*X2^2*x^13-2*X1^2*X2*x^14-X1*X2^2*x^14-X1*X2*x^15+2*X1^2*X2*x^13 +X1^2*x^14+X1*X2^2*x^13+2*X1*X2*x^14+X1^2*X2^2*x^11-X1^2*X2*x^12-X1^2*x^13+X1* X2^2*x^12-2*X1*X2*x^13-X1*x^14-X1^2*X2^2*x^10-2*X1^2*X2*x^11+X1^2*x^12-X1*X2^2* x^11+X1*x^13-X2^2*x^12+3*X1^2*X2*x^10+X1^2*x^11+X1*X2^2*x^10+2*X1*X2*x^11-X1*x^ 12+X2^2*x^11+X2*x^12-X1^2*X2*x^9-2*X1^2*x^10-X1*X2^2*x^9-5*X1*X2*x^10-X1*x^11-2 *X2*x^11+X1^2*x^9+5*X1*X2*x^9+4*X1*x^10+2*X2*x^10+x^11-X1*X2*x^8-4*X1*x^9-2*X2* x^9-2*x^10-X1*X2*x^7+X1*x^8+X2*x^8+2*x^9+2*X1*X2*x^6+X1*x^7+X2*x^7-x^8-2*X1*X2* x^5-3*X1*x^6-2*X2*x^6-x^7+X1*X2*x^4+4*X1*x^5+X2*x^5+3*x^6-3*X1*x^4-3*x^5+X1*x^3 +2*x^4-3*x^2+3*x-1)/(X1^2*X2^3*x^17+X1^2*X2^3*x^16-3*X1^2*X2^2*x^17-X1*X2^3*x^ 17-X1^2*X2^3*x^15-2*X1^2*X2^2*x^16+3*X1^2*X2*x^17-X1*X2^3*x^16+3*X1*X2^2*x^17+4 *X1^2*X2^2*x^15+X1^2*X2*x^16-X1^2*x^17+X1*X2^3*x^15+2*X1*X2^2*x^16-3*X1*X2*x^17 -2*X1^2*X2^2*x^14-5*X1^2*X2*x^15-4*X1*X2^2*x^15-X1*X2*x^16+X1*x^17+X1^2*X2^2*x^ 13+4*X1^2*X2*x^14+2*X1^2*x^15+3*X1*X2^2*x^14+5*X1*X2*x^15-3*X1^2*X2*x^13-2*X1^2 *x^14-6*X1*X2*x^14-2*X1*x^15-X2^2*x^14-2*X1^2*X2^2*x^11+2*X1^2*x^13-X1*X2^2*x^ 12+2*X1*X2*x^13+3*X1*x^14+2*X2*x^14+X1^2*X2^2*x^10+5*X1^2*X2*x^11+X1*X2^2*x^11+ 2*X1*X2*x^12-2*X1*x^13+2*X2^2*x^12-X2*x^13-x^14-4*X1^2*X2*x^10-3*X1^2*x^11-2*X1 *X2^2*x^10-5*X1*X2*x^11-X1*x^12-X2^2*x^11-4*X2*x^12+x^13+X1^2*X2*x^9+3*X1^2*x^ 10+X1*X2^2*x^9+8*X1*X2*x^10+4*X1*x^11+4*X2*x^11+2*x^12-X1^2*x^9-6*X1*X2*x^9-6* X1*x^10-2*X2*x^10-3*x^11+X1*X2*x^8+5*X1*x^9+3*X2*x^9+2*x^10+2*X1*X2*x^7-X1*x^8-\ 2*X2*x^8-3*x^9-3*X1*X2*x^6-3*X1*x^7-2*X2*x^7+2*x^8+3*X1*X2*x^5+5*X1*x^6+3*X2*x^ 6+3*x^7-X1*X2*x^4-6*X1*x^5-X2*x^5-5*x^6+4*X1*x^4+4*x^5-X1*x^3-3*x^4-x^3+5*x^2-4 *x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 3, 2], are n 315 39 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 1, 4], are n 141 65 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 14 3 5 ------------ 195 and in floating point 0.2780603428 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 14 15 ate normal pair with correlation, -------- 195 1/2 14 15 2927 i.e. , [[--------, 0], [0, ----]] 195 2535 ------------------------------------------------- Theorem Number, 137, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 3, 2], nor the composition, [3, 2, 3] Then infinity ----- 10 8 5 4 3 2 \ n x - x + x - x + x + x - 2 x + 1 ) a(n) x = - -------------------------------------- / 3 8 5 ----- (x - x + 1) (x - x + 2 x - 1) n = 0 and in Maple format -(x^10-x^8+x^5-x^4+x^3+x^2-2*x+1)/(x^3-x+1)/(x^8-x^5+2*x-1) The asymptotic expression for a(n) is, n 0.60406319431018317656 1.9389552798581449825 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 3, 2] and d occurrences (as containment) of the composition, [3, 2, 3], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 13 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 13 2 13 2 2 11 2 13 13 - 2 X1 X2 x - X1 X2 x - X1 X2 x + X1 x + 2 X1 X2 x 2 11 2 11 13 2 2 9 2 2 8 2 9 + X1 X2 x + X1 X2 x - X1 x + X1 X2 x - X1 X2 x - X1 X2 x 10 11 11 2 8 9 10 10 - X1 X2 x - X1 x - X2 x + X1 X2 x - 2 X1 X2 x + X1 x + X2 x 11 8 9 9 10 8 8 9 + x + 2 X1 X2 x + 2 X1 x + X2 x - x - 2 X1 x - X2 x - x 6 8 5 4 6 3 5 4 - X1 X2 x + x + 2 X1 X2 x - 2 X1 X2 x + x + X1 X2 x - 2 x + 2 x 2 / 2 2 14 2 14 2 14 - 3 x + 3 x - 1) / (X1 X2 x - 2 X1 X2 x - X1 X2 x / 2 2 12 2 14 14 2 2 11 2 12 - X1 X2 x + X1 x + 2 X1 X2 x + X1 X2 x + X1 X2 x 2 12 14 2 11 2 2 9 2 10 2 11 + X1 X2 x - X1 x - 2 X1 X2 x - 2 X1 X2 x + X1 X2 x + X1 x 11 12 12 2 2 8 2 9 2 10 - X1 X2 x - X1 x - X2 x + X1 X2 x + 2 X1 X2 x - X1 x 10 11 11 12 2 8 9 10 - 2 X1 X2 x + X1 x + X2 x + x - X1 X2 x + 3 X1 X2 x + 2 X1 x 10 11 8 9 9 10 8 9 + X2 x - x - 2 X1 X2 x - 3 X1 x - X2 x - x + 2 X1 x + x 6 7 5 7 4 6 3 + 2 X1 X2 x - X1 x - 3 X1 X2 x + x + 3 X1 X2 x - 2 x - X1 X2 x 5 4 3 2 + 3 x - 3 x - x + 5 x - 4 x + 1) and in Maple format -(X1^2*X2^2*x^13-2*X1^2*X2*x^13-X1*X2^2*x^13-X1^2*X2^2*x^11+X1^2*x^13+2*X1*X2*x ^13+X1^2*X2*x^11+X1*X2^2*x^11-X1*x^13+X1^2*X2^2*x^9-X1^2*X2^2*x^8-X1^2*X2*x^9- X1*X2*x^10-X1*x^11-X2*x^11+X1^2*X2*x^8-2*X1*X2*x^9+X1*x^10+X2*x^10+x^11+2*X1*X2 *x^8+2*X1*x^9+X2*x^9-x^10-2*X1*x^8-X2*x^8-x^9-X1*X2*x^6+x^8+2*X1*X2*x^5-2*X1*X2 *x^4+x^6+X1*X2*x^3-2*x^5+2*x^4-3*x^2+3*x-1)/(X1^2*X2^2*x^14-2*X1^2*X2*x^14-X1* X2^2*x^14-X1^2*X2^2*x^12+X1^2*x^14+2*X1*X2*x^14+X1^2*X2^2*x^11+X1^2*X2*x^12+X1* X2^2*x^12-X1*x^14-2*X1^2*X2*x^11-2*X1^2*X2^2*x^9+X1^2*X2*x^10+X1^2*x^11-X1*X2*x ^11-X1*x^12-X2*x^12+X1^2*X2^2*x^8+2*X1^2*X2*x^9-X1^2*x^10-2*X1*X2*x^10+X1*x^11+ X2*x^11+x^12-X1^2*X2*x^8+3*X1*X2*x^9+2*X1*x^10+X2*x^10-x^11-2*X1*X2*x^8-3*X1*x^ 9-X2*x^9-x^10+2*X1*x^8+x^9+2*X1*X2*x^6-X1*x^7-3*X1*X2*x^5+x^7+3*X1*X2*x^4-2*x^6 -X1*X2*x^3+3*x^5-3*x^4-x^3+5*x^2-4*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 3, 2], are n 315 39 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 2, 3], are n 89 77 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 38 39 77 -------------- 3003 and in floating point 0.6934352749 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 38 3003 ate normal pair with correlation, ---------- 3003 1/2 38 3003 5891 i.e. , [[----------, 0], [0, ----]] 3003 3003 ------------------------------------------------- Theorem Number, 138, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 3, 2], nor the composition, [3, 4, 1] Then infinity ----- 5 4 3 2 \ n x - x + x + x - 2 x + 1 ) a(n) x = - ----------------------------------------------- / 10 8 6 5 4 3 2 ----- x - x + x - x + 2 x - x - 2 x + 3 x - 1 n = 0 and in Maple format -(x^5-x^4+x^3+x^2-2*x+1)/(x^10-x^8+x^6-x^5+2*x^4-x^3-2*x^2+3*x-1) The asymptotic expression for a(n) is, n 0.62193578086462630009 1.9325948503424936840 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 3, 2] and d occurrences (as containment) of the composition, [3, 4, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 7 7 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x - X1 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 5 4 5 4 5 3 4 3 2 - X1 X2 x + X1 X2 x + 2 X1 x - 2 X1 x - x + X1 x + x - x - x / 10 10 10 8 10 7 + 2 x - 1) / (X1 X2 x - X1 x - X2 x - X1 X2 x + x + X1 X2 x / 8 8 7 8 5 6 4 5 + X1 x + X2 x - X1 x - x - 2 X1 X2 x - X1 x + X1 X2 x + 3 X1 x 6 4 5 3 4 3 2 + x - 3 X1 x - x + X1 x + 2 x - x - 2 x + 3 x - 1) and in Maple format (X1*X2*x^7-X1*x^7-X1*X2*x^5+X1*X2*x^4+2*X1*x^5-2*X1*x^4-x^5+X1*x^3+x^4-x^3-x^2+ 2*x-1)/(X1*X2*x^10-X1*x^10-X2*x^10-X1*X2*x^8+x^10+X1*X2*x^7+X1*x^8+X2*x^8-X1*x^ 7-x^8-2*X1*X2*x^5-X1*x^6+X1*X2*x^4+3*X1*x^5+x^6-3*X1*x^4-x^5+X1*x^3+2*x^4-x^3-2 *x^2+3*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 3, 2], are n 315 39 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 4, 1], are n 135 65 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 8 3 5 ----------- 65 and in floating point 0.4766748734 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 1/2 8 3 5 ate normal pair with correlation, ----------- 65 1/2 1/2 8 3 5 1229 i.e. , [[-----------, 0], [0, ----]] 65 845 ------------------------------------------------- Theorem Number, 139, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 3, 2], nor the composition, [4, 1, 3] Then infinity ----- \ n ) a(n) x = / ----- n = 0 10 9 8 6 5 4 3 2 x - x + x - x + 2 x - x + x + x - 2 x + 1 ------------------------------------------------------------------------ 13 11 10 9 8 6 5 4 3 2 x - 2 x + x - x + 2 x - 3 x + 2 x - 2 x + x + 2 x - 3 x + 1 and in Maple format (x^10-x^9+x^8-x^6+2*x^5-x^4+x^3+x^2-2*x+1)/(x^13-2*x^11+x^10-x^9+2*x^8-3*x^6+2* x^5-2*x^4+x^3+2*x^2-3*x+1) The asymptotic expression for a(n) is, n 0.64411604859512176136 1.9232078865547166800 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 3, 2] and d occurrences (as containment) of the composition, [4, 1, 3], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 3 15 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 2 15 3 15 2 2 14 2 15 2 15 - 2 X1 X2 x - X1 X2 x + X1 X2 x + X1 X2 x + 2 X1 X2 x 2 2 13 2 14 2 14 15 2 13 - X1 X2 x - 2 X1 X2 x - X1 X2 x - X1 X2 x + 2 X1 X2 x 2 14 2 13 14 2 2 11 2 12 2 13 + X1 x + X1 X2 x + 2 X1 X2 x + X1 X2 x - X1 X2 x - X1 x 2 12 13 14 2 2 10 2 11 2 12 + X1 X2 x - 2 X1 X2 x - X1 x - X1 X2 x - 2 X1 X2 x + X1 x 2 11 13 2 12 2 10 2 11 2 10 - X1 X2 x + X1 x - X2 x + 3 X1 X2 x + X1 x + X1 X2 x 11 12 2 11 12 2 9 2 10 + 2 X1 X2 x - X1 x + X2 x + X2 x - X1 X2 x - 2 X1 x 2 9 10 11 11 2 9 9 - X1 X2 x - 5 X1 X2 x - X1 x - 2 X2 x + X1 x + 5 X1 X2 x 10 10 11 8 9 9 10 + 4 X1 x + 2 X2 x + x - X1 X2 x - 4 X1 x - 2 X2 x - 2 x 7 8 8 9 6 7 7 8 - X1 X2 x + X1 x + X2 x + 2 x + 2 X1 X2 x + X1 x + X2 x - x 5 6 6 7 4 5 5 6 - 2 X1 X2 x - 3 X1 x - 2 X2 x - x + X1 X2 x + 4 X1 x + X2 x + 3 x 4 5 3 4 2 / 2 3 17 - 3 X1 x - 3 x + X1 x + 2 x - 3 x + 3 x - 1) / (X1 X2 x / 2 3 16 2 2 17 3 17 2 3 15 2 2 16 + X1 X2 x - 3 X1 X2 x - X1 X2 x - X1 X2 x - 2 X1 X2 x 2 17 3 16 2 17 2 2 15 2 16 + 3 X1 X2 x - X1 X2 x + 3 X1 X2 x + 4 X1 X2 x + X1 X2 x 2 17 3 15 2 16 17 2 2 14 - X1 x + X1 X2 x + 2 X1 X2 x - 3 X1 X2 x - 2 X1 X2 x 2 15 2 15 16 17 2 2 13 - 5 X1 X2 x - 4 X1 X2 x - X1 X2 x + X1 x + X1 X2 x 2 14 2 15 2 14 15 2 13 + 4 X1 X2 x + 2 X1 x + 3 X1 X2 x + 5 X1 X2 x - 3 X1 X2 x 2 14 14 15 2 14 2 2 11 2 13 - 2 X1 x - 6 X1 X2 x - 2 X1 x - X2 x - 2 X1 X2 x + 2 X1 x 2 12 13 14 14 2 2 10 - X1 X2 x + 2 X1 X2 x + 3 X1 x + 2 X2 x + X1 X2 x 2 11 2 11 12 13 2 12 13 + 5 X1 X2 x + X1 X2 x + 2 X1 X2 x - 2 X1 x + 2 X2 x - X2 x 14 2 10 2 11 2 10 11 12 - x - 4 X1 X2 x - 3 X1 x - 2 X1 X2 x - 5 X1 X2 x - X1 x 2 11 12 13 2 9 2 10 2 9 - X2 x - 4 X2 x + x + X1 X2 x + 3 X1 x + X1 X2 x 10 11 11 12 2 9 9 + 8 X1 X2 x + 4 X1 x + 4 X2 x + 2 x - X1 x - 6 X1 X2 x 10 10 11 8 9 9 10 - 6 X1 x - 2 X2 x - 3 x + X1 X2 x + 5 X1 x + 3 X2 x + 2 x 7 8 8 9 6 7 7 + 2 X1 X2 x - X1 x - 2 X2 x - 3 x - 3 X1 X2 x - 3 X1 x - 2 X2 x 8 5 6 6 7 4 5 + 2 x + 3 X1 X2 x + 5 X1 x + 3 X2 x + 3 x - X1 X2 x - 6 X1 x 5 6 4 5 3 4 3 2 - X2 x - 5 x + 4 X1 x + 4 x - X1 x - 3 x - x + 5 x - 4 x + 1) and in Maple format -(X1^2*X2^3*x^15-2*X1^2*X2^2*x^15-X1*X2^3*x^15+X1^2*X2^2*x^14+X1^2*X2*x^15+2*X1 *X2^2*x^15-X1^2*X2^2*x^13-2*X1^2*X2*x^14-X1*X2^2*x^14-X1*X2*x^15+2*X1^2*X2*x^13 +X1^2*x^14+X1*X2^2*x^13+2*X1*X2*x^14+X1^2*X2^2*x^11-X1^2*X2*x^12-X1^2*x^13+X1* X2^2*x^12-2*X1*X2*x^13-X1*x^14-X1^2*X2^2*x^10-2*X1^2*X2*x^11+X1^2*x^12-X1*X2^2* x^11+X1*x^13-X2^2*x^12+3*X1^2*X2*x^10+X1^2*x^11+X1*X2^2*x^10+2*X1*X2*x^11-X1*x^ 12+X2^2*x^11+X2*x^12-X1^2*X2*x^9-2*X1^2*x^10-X1*X2^2*x^9-5*X1*X2*x^10-X1*x^11-2 *X2*x^11+X1^2*x^9+5*X1*X2*x^9+4*X1*x^10+2*X2*x^10+x^11-X1*X2*x^8-4*X1*x^9-2*X2* x^9-2*x^10-X1*X2*x^7+X1*x^8+X2*x^8+2*x^9+2*X1*X2*x^6+X1*x^7+X2*x^7-x^8-2*X1*X2* x^5-3*X1*x^6-2*X2*x^6-x^7+X1*X2*x^4+4*X1*x^5+X2*x^5+3*x^6-3*X1*x^4-3*x^5+X1*x^3 +2*x^4-3*x^2+3*x-1)/(X1^2*X2^3*x^17+X1^2*X2^3*x^16-3*X1^2*X2^2*x^17-X1*X2^3*x^ 17-X1^2*X2^3*x^15-2*X1^2*X2^2*x^16+3*X1^2*X2*x^17-X1*X2^3*x^16+3*X1*X2^2*x^17+4 *X1^2*X2^2*x^15+X1^2*X2*x^16-X1^2*x^17+X1*X2^3*x^15+2*X1*X2^2*x^16-3*X1*X2*x^17 -2*X1^2*X2^2*x^14-5*X1^2*X2*x^15-4*X1*X2^2*x^15-X1*X2*x^16+X1*x^17+X1^2*X2^2*x^ 13+4*X1^2*X2*x^14+2*X1^2*x^15+3*X1*X2^2*x^14+5*X1*X2*x^15-3*X1^2*X2*x^13-2*X1^2 *x^14-6*X1*X2*x^14-2*X1*x^15-X2^2*x^14-2*X1^2*X2^2*x^11+2*X1^2*x^13-X1*X2^2*x^ 12+2*X1*X2*x^13+3*X1*x^14+2*X2*x^14+X1^2*X2^2*x^10+5*X1^2*X2*x^11+X1*X2^2*x^11+ 2*X1*X2*x^12-2*X1*x^13+2*X2^2*x^12-X2*x^13-x^14-4*X1^2*X2*x^10-3*X1^2*x^11-2*X1 *X2^2*x^10-5*X1*X2*x^11-X1*x^12-X2^2*x^11-4*X2*x^12+x^13+X1^2*X2*x^9+3*X1^2*x^ 10+X1*X2^2*x^9+8*X1*X2*x^10+4*X1*x^11+4*X2*x^11+2*x^12-X1^2*x^9-6*X1*X2*x^9-6* X1*x^10-2*X2*x^10-3*x^11+X1*X2*x^8+5*X1*x^9+3*X2*x^9+2*x^10+2*X1*X2*x^7-X1*x^8-\ 2*X2*x^8-3*x^9-3*X1*X2*x^6-3*X1*x^7-2*X2*x^7+2*x^8+3*X1*X2*x^5+5*X1*x^6+3*X2*x^ 6+3*x^7-X1*X2*x^4-6*X1*x^5-X2*x^5-5*x^6+4*X1*x^4+4*x^5-X1*x^3-3*x^4-x^3+5*x^2-4 *x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 3, 2], are n 315 39 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 1, 3], are n 141 65 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 14 3 5 ------------ 195 and in floating point 0.2780603428 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 14 15 ate normal pair with correlation, -------- 195 1/2 14 15 2927 i.e. , [[--------, 0], [0, ----]] 195 2535 ------------------------------------------------- Theorem Number, 140, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 3, 2], nor the composition, [4, 2, 2] Then infinity ----- 6 4 3 2 \ n x + x - x + 2 x - 2 x + 1 ) a(n) x = - ------------------------------------ / 7 5 4 3 2 ----- x + x - x + 2 x - 3 x + 3 x - 1 n = 0 and in Maple format -(x^6+x^4-x^3+2*x^2-2*x+1)/(x^7+x^5-x^4+2*x^3-3*x^2+3*x-1) The asymptotic expression for a(n) is, n 0.60945823930910114296 1.9365136360000287543 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 3, 2] and d occurrences (as containment) of the composition, [4, 2, 2], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 8 8 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x - X1 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 8 6 8 5 6 4 5 4 - X2 x + X1 X2 x + x - X1 X2 x - X1 x + X1 X2 x + 2 X1 x - 2 X1 x 5 3 4 3 2 / 9 9 9 - x + X1 x + x - x - x + 2 x - 1) / (X1 X2 x - X1 x - X2 x / 7 9 6 7 5 6 4 + X1 X2 x + x - X1 X2 x - X1 x + 2 X1 X2 x + 2 X1 x - X1 X2 x 5 6 4 5 3 4 3 2 - 3 X1 x - x + 3 X1 x + x - X1 x - 2 x + x + 2 x - 3 x + 1) and in Maple format -(X1*X2*x^8-X1*x^8-X2*x^8+X1*X2*x^6+x^8-X1*X2*x^5-X1*x^6+X1*X2*x^4+2*X1*x^5-2* X1*x^4-x^5+X1*x^3+x^4-x^3-x^2+2*x-1)/(X1*X2*x^9-X1*x^9-X2*x^9+X1*X2*x^7+x^9-X1* X2*x^6-X1*x^7+2*X1*X2*x^5+2*X1*x^6-X1*X2*x^4-3*X1*x^5-x^6+3*X1*x^4+x^5-X1*x^3-2 *x^4+x^3+2*x^2-3*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 3, 2], are n 315 39 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 2, 2], are n 75 69 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 10 13 23 -------------- 299 and in floating point 0.5783149317 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 10 299 ate normal pair with correlation, --------- 299 1/2 10 299 499 i.e. , [[---------, 0], [0, ---]] 299 299 ------------------------------------------------- Theorem Number, 141, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 3, 2], nor the composition, [4, 3, 1] Then infinity ----- 5 4 3 2 \ n x - x + x + x - 2 x + 1 ) a(n) x = - ----------------------------------------------- / 10 8 6 5 4 3 2 ----- x - x + x - x + 2 x - x - 2 x + 3 x - 1 n = 0 and in Maple format -(x^5-x^4+x^3+x^2-2*x+1)/(x^10-x^8+x^6-x^5+2*x^4-x^3-2*x^2+3*x-1) The asymptotic expression for a(n) is, n 0.62193578086462630009 1.9325948503424936840 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 3, 2] and d occurrences (as containment) of the composition, [4, 3, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 7 7 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x - X1 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 5 4 5 4 5 3 4 3 2 - X1 X2 x + X1 X2 x + 2 X1 x - 2 X1 x - x + X1 x + x - x - x / 10 10 10 8 10 7 + 2 x - 1) / (X1 X2 x - X1 x - X2 x - X1 X2 x + x + X1 X2 x / 8 8 7 8 5 6 4 5 + X1 x + X2 x - X1 x - x - 2 X1 X2 x - X1 x + X1 X2 x + 3 X1 x 6 4 5 3 4 3 2 + x - 3 X1 x - x + X1 x + 2 x - x - 2 x + 3 x - 1) and in Maple format (X1*X2*x^7-X1*x^7-X1*X2*x^5+X1*X2*x^4+2*X1*x^5-2*X1*x^4-x^5+X1*x^3+x^4-x^3-x^2+ 2*x-1)/(X1*X2*x^10-X1*x^10-X2*x^10-X1*X2*x^8+x^10+X1*X2*x^7+X1*x^8+X2*x^8-X1*x^ 7-x^8-2*X1*X2*x^5-X1*x^6+X1*X2*x^4+3*X1*x^5+x^6-3*X1*x^4-x^5+X1*x^3+2*x^4-x^3-2 *x^2+3*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 3, 2], are n 315 39 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 3, 1], are n 135 65 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 8 3 5 ----------- 65 and in floating point 0.4766748734 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 1/2 8 3 5 ate normal pair with correlation, ----------- 65 1/2 1/2 8 3 5 1229 i.e. , [[-----------, 0], [0, ----]] 65 845 ------------------------------------------------- Theorem Number, 142, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 3, 2], nor the composition, [5, 1, 2] Then infinity ----- 12 11 10 9 8 7 6 5 3 \ n x + x + x + x + x + x + 2 x + x + x - x + 1 ) a(n) x = - ------------------------------------------------------- / 13 8 7 4 3 ----- x + x + 2 x + x - x + 2 x - 1 n = 0 and in Maple format -(x^12+x^11+x^10+x^9+x^8+x^7+2*x^6+x^5+x^3-x+1)/(x^13+x^8+2*x^7+x^4-x^3+2*x-1) The asymptotic expression for a(n) is, n 0.65262822115524216049 1.9206669544913373883 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 3, 2] and d occurrences (as containment) of the composition, [5, 1, 2], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 13 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 13 2 13 2 11 13 13 11 - 2 X1 X2 x - X2 x + X1 X2 x + X1 x + 2 X2 x - 2 X1 X2 x 13 11 7 6 7 7 5 6 - x + X1 x - X1 X2 x + X1 X2 x + X1 x + X2 x - X1 X2 x - X1 x 6 7 6 4 5 3 4 3 2 / - X2 x - x + x + X1 x + x - X1 x - x + x + x - 2 x + 1) / ( / 2 14 2 13 14 2 14 2 12 13 X1 X2 x - X1 X2 x - 2 X1 X2 x - X2 x + X1 X2 x + 2 X1 X2 x 14 2 13 14 2 11 12 13 + X1 x + X2 x + 2 X2 x - X1 X2 x - 2 X1 X2 x - X1 x 13 14 11 12 13 11 9 - 2 X2 x - x + 2 X1 X2 x + X1 x + x - X1 x - X1 X2 x 8 9 9 7 8 8 9 6 - X1 X2 x + X1 x + X2 x + X1 X2 x + X1 x + X2 x - x - 2 X1 X2 x 7 7 8 5 6 6 7 4 5 - X1 x - 2 X2 x - x + X1 X2 x + X1 x + X2 x + 2 x - 2 X1 x - x 3 4 3 2 + X1 x + 2 x - x - 2 x + 3 x - 1) and in Maple format -(X1*X2^2*x^13-2*X1*X2*x^13-X2^2*x^13+X1*X2^2*x^11+X1*x^13+2*X2*x^13-2*X1*X2*x^ 11-x^13+X1*x^11-X1*X2*x^7+X1*X2*x^6+X1*x^7+X2*x^7-X1*X2*x^5-X1*x^6-X2*x^6-x^7+x ^6+X1*x^4+x^5-X1*x^3-x^4+x^3+x^2-2*x+1)/(X1*X2^2*x^14-X1*X2^2*x^13-2*X1*X2*x^14 -X2^2*x^14+X1*X2^2*x^12+2*X1*X2*x^13+X1*x^14+X2^2*x^13+2*X2*x^14-X1*X2^2*x^11-2 *X1*X2*x^12-X1*x^13-2*X2*x^13-x^14+2*X1*X2*x^11+X1*x^12+x^13-X1*x^11-X1*X2*x^9- X1*X2*x^8+X1*x^9+X2*x^9+X1*X2*x^7+X1*x^8+X2*x^8-x^9-2*X1*X2*x^6-X1*x^7-2*X2*x^7 -x^8+X1*X2*x^5+X1*x^6+X2*x^6+2*x^7-2*X1*x^4-x^5+X1*x^3+2*x^4-x^3-2*x^2+3*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 3, 2], are n 315 39 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [5, 1, 2], are n 113 57 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 10 13 19 -------------- 741 and in floating point 0.2120949210 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 10 247 ate normal pair with correlation, --------- 741 1/2 10 247 2423 i.e. , [[---------, 0], [0, ----]] 741 2223 ------------------------------------------------- Theorem Number, 143, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 3, 2], nor the composition, [5, 2, 1] Then infinity ----- 8 7 6 5 3 \ n x + x + x + x + x - x + 1 ) a(n) x = - ----------------------------------- / 3 4 3 2 ----- (x - x + 1) (x + x + x + x - 1) n = 0 and in Maple format -(x^8+x^7+x^6+x^5+x^3-x+1)/(x^3-x+1)/(x^4+x^3+x^2+x-1) The asymptotic expression for a(n) is, n 0.63242246488998289333 1.9275619754829253043 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 3, 2] and d occurrences (as containment) of the composition, [5, 2, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 9 9 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x - X1 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 9 7 9 6 7 5 6 4 5 - X2 x + X1 X2 x + x - X1 X2 x - X1 x + X1 X2 x + X1 x - X1 x - x 3 4 3 2 / 8 7 8 8 + X1 x + x - x - x + 2 x - 1) / (X1 X2 x - X1 X2 x - X1 x - X2 x / 6 7 8 5 6 6 4 5 3 + 2 X1 X2 x + X1 x + x - X1 X2 x - X1 x - x + 2 X1 x + x - X1 x 4 3 2 - 2 x + x + 2 x - 3 x + 1) and in Maple format -(X1*X2*x^9-X1*x^9-X2*x^9+X1*X2*x^7+x^9-X1*X2*x^6-X1*x^7+X1*X2*x^5+X1*x^6-X1*x^ 4-x^5+X1*x^3+x^4-x^3-x^2+2*x-1)/(X1*X2*x^8-X1*X2*x^7-X1*x^8-X2*x^8+2*X1*X2*x^6+ X1*x^7+x^8-X1*X2*x^5-X1*x^6-x^6+2*X1*x^4+x^5-X1*x^3-2*x^4+x^3+2*x^2-3*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 3, 2], are n 315 39 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [5, 2, 1], are n 111 57 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 16 13 19 -------------- 741 and in floating point 0.3393518736 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 16 247 ate normal pair with correlation, --------- 741 1/2 16 247 2735 i.e. , [[---------, 0], [0, ----]] 741 2223 ------------------------------------------------- Theorem Number, 144, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 3, 2], nor the composition, [6, 1, 1] Then infinity ----- \ n ) a(n) x = / ----- n = 0 11 9 7 6 4 3 2 x + x + x + x + x - x + 2 x - 2 x + 1 ------------------------------------------------------------- 10 9 8 7 6 5 3 2 (-1 + x) (x + x + 2 x + x + x + x + x - x + 2 x - 1) and in Maple format (x^11+x^9+x^7+x^6+x^4-x^3+2*x^2-2*x+1)/(-1+x)/(x^10+x^9+2*x^8+x^7+x^6+x^5+x^3-x ^2+2*x-1) The asymptotic expression for a(n) is, n 0.67617629498732100894 1.9139965407423997306 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 3, 2] and d occurrences (as containment) of the composition, [6, 1, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 13 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 13 2 13 13 13 13 8 7 - 2 X1 X2 x - X2 x + X1 x + 2 X2 x - x - X1 X2 x + X1 X2 x 8 8 6 7 7 8 6 7 5 + X1 x + X2 x - X1 X2 x - X1 x - X2 x - x + X1 x + x - X1 x 4 5 3 4 3 2 / 2 13 + X1 x + x - X1 x - x + x + x - 2 x + 1) / (X1 X2 x / 13 2 13 13 13 13 10 9 - 2 X1 X2 x - X2 x + X1 x + 2 X2 x - x + X1 X2 x + X1 X2 x 10 10 8 9 9 10 7 8 - X1 x - X2 x - X1 X2 x - X1 x - X2 x + x + 2 X1 X2 x + X1 x 8 9 6 7 7 8 6 7 5 6 + X2 x + x - X1 X2 x - 2 X1 x - X2 x - x + 2 X1 x + x - X1 x - x 4 5 3 4 3 2 + 2 X1 x + x - X1 x - 2 x + x + 2 x - 3 x + 1) and in Maple format (X1*X2^2*x^13-2*X1*X2*x^13-X2^2*x^13+X1*x^13+2*X2*x^13-x^13-X1*X2*x^8+X1*X2*x^7 +X1*x^8+X2*x^8-X1*X2*x^6-X1*x^7-X2*x^7-x^8+X1*x^6+x^7-X1*x^5+X1*x^4+x^5-X1*x^3- x^4+x^3+x^2-2*x+1)/(X1*X2^2*x^13-2*X1*X2*x^13-X2^2*x^13+X1*x^13+2*X2*x^13-x^13+ X1*X2*x^10+X1*X2*x^9-X1*x^10-X2*x^10-X1*X2*x^8-X1*x^9-X2*x^9+x^10+2*X1*X2*x^7+ X1*x^8+X2*x^8+x^9-X1*X2*x^6-2*X1*x^7-X2*x^7-x^8+2*X1*x^6+x^7-X1*x^5-x^6+2*X1*x^ 4+x^5-X1*x^3-2*x^4+x^3+2*x^2-3*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 3, 2], are n 315 39 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [6, 1, 1], are n 49 53 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 2 39 53 ------------- 2067 and in floating point 0.04399058701 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 2 2067 ate normal pair with correlation, --------- 2067 1/2 2 2067 2075 i.e. , [[---------, 0], [0, ----]] 2067 2067 ------------------------------------------------- Theorem Number, 145, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 4, 1], nor the composition, [1, 1, 6] Then infinity ----- \ n ) a(n) x = / ----- n = 0 2 11 10 9 8 5 4 2 (x - x + 1) (x + 2 x + 2 x + x - x - 2 x + x + x - 1) --------------------------------------------------------------- 6 5 3 2 2 (x + x + x + x + x - 1) (-1 + x) and in Maple format (x^2-x+1)*(x^11+2*x^10+2*x^9+x^8-x^5-2*x^4+x^2+x-1)/(x^6+x^5+x^3+x^2+x-1)/(-1+x )^2 The asymptotic expression for a(n) is, n 0.68358264996735542565 1.9111834366854762350 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 4, 1] and d occurrences (as containment) of the composition, [1, 1, 6], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 14 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 14 2 14 14 14 14 8 - 2 X1 X2 x - X2 x + X1 x + 2 X2 x - x + 2 X1 X2 x 7 8 8 6 7 7 8 6 - 2 X1 X2 x - 2 X1 x - 2 X2 x + X1 X2 x + X1 x + X2 x + 2 x + X1 x 5 6 4 5 4 3 2 / 2 - 2 X1 x - 2 x + X1 x + 2 x - x + x - 3 x + 3 x - 1) / ((-1 + x) / 7 6 7 7 6 7 5 4 5 4 (X1 X2 x - X1 X2 x - X1 x - X2 x + X1 x + x + X1 x - X1 x - x + x - 2 x + 1)) and in Maple format -(X1*X2^2*x^14-2*X1*X2*x^14-X2^2*x^14+X1*x^14+2*X2*x^14-x^14+2*X1*X2*x^8-2*X1* X2*x^7-2*X1*x^8-2*X2*x^8+X1*X2*x^6+X1*x^7+X2*x^7+2*x^8+X1*x^6-2*X1*x^5-2*x^6+X1 *x^4+2*x^5-x^4+x^3-3*x^2+3*x-1)/(-1+x)^2/(X1*X2*x^7-X1*X2*x^6-X1*x^7-X2*x^7+X1* x^6+x^7+X1*x^5-X1*x^4-x^5+x^4-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 4, 1], are n 183 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 1, 6], are n 49 53 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 8 3 53 ------------ 477 and in floating point 0.2114804230 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 8 159 ate normal pair with correlation, -------- 477 1/2 8 159 1559 i.e. , [[--------, 0], [0, ----]] 477 1431 ------------------------------------------------- Theorem Number, 146, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 4, 1], nor the composition, [1, 2, 5] Then infinity ----- 12 11 8 7 6 5 4 3 2 \ n x - x + x + x - 2 x + 2 x - x + x - 3 x + 3 x - 1 ) a(n) x = ------------------------------------------------------------ / 5 4 2 ----- (x - x + 2 x - 1) (-1 + x) n = 0 and in Maple format (x^12-x^11+x^8+x^7-2*x^6+2*x^5-x^4+x^3-3*x^2+3*x-1)/(x^5-x^4+2*x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.61442764446410597890 1.9331849818995204468 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 4, 1] and d occurrences (as containment) of the composition, [1, 2, 5], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 12 11 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x - X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 12 12 11 11 12 11 7 8 - X1 x - X2 x + X1 x + X2 x + x - x + 2 X1 X2 x - X2 x 6 7 8 5 6 7 5 6 - 2 X1 X2 x - 3 X1 x + x + X1 X2 x + 4 X1 x + x - 3 X1 x - 2 x 4 5 4 3 2 / 2 + X1 x + 2 x - x + x - 3 x + 3 x - 1) / ((-1 + x) / 6 5 6 5 4 5 4 (X1 X2 x - X1 X2 x - X1 x + 2 X1 x - X1 x - x + x - 2 x + 1)) and in Maple format -(X1*X2*x^12-X1*X2*x^11-X1*x^12-X2*x^12+X1*x^11+X2*x^11+x^12-x^11+2*X1*X2*x^7- X2*x^8-2*X1*X2*x^6-3*X1*x^7+x^8+X1*X2*x^5+4*X1*x^6+x^7-3*X1*x^5-2*x^6+X1*x^4+2* x^5-x^4+x^3-3*x^2+3*x-1)/(-1+x)^2/(X1*X2*x^6-X1*X2*x^5-X1*x^6+2*X1*x^5-X1*x^4-x ^5+x^4-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 4, 1], are n 183 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 2, 5], are n 111 57 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 26 19 -------- 171 and in floating point 0.6627565645 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 26 19 ate normal pair with correlation, -------- 171 1/2 26 19 2891 i.e. , [[--------, 0], [0, ----]] 171 1539 ------------------------------------------------- Theorem Number, 147, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 4, 1], nor the composition, [1, 3, 4] Then infinity ----- \ n ) a(n) x = / ----- n = 0 3 2 9 8 6 5 4 3 2 (x + x - 1) (x - x + x - x + 2 x - 3 x + 4 x - 3 x + 1) ---------------------------------------------------------------- 5 4 2 (x - x + 2 x - 1) (-1 + x) and in Maple format (x^3+x^2-1)*(x^9-x^8+x^6-x^5+2*x^4-3*x^3+4*x^2-3*x+1)/(x^5-x^4+2*x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.61747222077046425935 1.9331849818995204468 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 4, 1] and d occurrences (as containment) of the composition, [1, 3, 4], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 12 12 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x - X1 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 12 10 12 10 10 10 7 8 - X2 x - X1 X2 x + x + X1 x + X2 x - x + X1 X2 x - X2 x 6 7 8 5 6 7 4 6 + X1 X2 x - 2 X1 x + x - 2 X1 X2 x + X1 x + x + X1 X2 x - 2 x 5 4 3 2 / 2 + 2 x - x + x - 3 x + 3 x - 1) / ((-1 + x) / 6 5 6 4 5 4 (X1 X2 x + X1 X2 x - X1 x - X1 X2 x - x + x - 2 x + 1)) and in Maple format -(X1*X2*x^12-X1*x^12-X2*x^12-X1*X2*x^10+x^12+X1*x^10+X2*x^10-x^10+X1*X2*x^7-X2* x^8+X1*X2*x^6-2*X1*x^7+x^8-2*X1*X2*x^5+X1*x^6+x^7+X1*X2*x^4-2*x^6+2*x^5-x^4+x^3 -3*x^2+3*x-1)/(-1+x)^2/(X1*X2*x^6+X1*X2*x^5-X1*x^6-X1*X2*x^4-x^5+x^4-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 4, 1], are n 183 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 3, 4], are n 135 65 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 2 3 65 ------------ 39 and in floating point 0.7161148743 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 1/2 2 5 39 ate normal pair with correlation, ------------ 39 1/2 1/2 2 5 39 79 i.e. , [[------------, 0], [0, --]] 39 39 ------------------------------------------------- Theorem Number, 148, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 4, 1], nor the composition, [1, 4, 3] Then infinity ----- \ n ) a(n) x = / ----- n = 0 12 10 9 7 6 5 4 3 2 x - x + x + x - 2 x + 2 x - x + x - 3 x + 3 x - 1 - ------------------------------------------------------------ 7 6 5 3 2 3 (x + x + x + x + x + x - 1) (-1 + x) and in Maple format -(x^12-x^10+x^9+x^7-2*x^6+2*x^5-x^4+x^3-3*x^2+3*x-1)/(x^7+x^6+x^5+x^3+x^2+x-1)/ (-1+x)^3 The asymptotic expression for a(n) is, n 0.65557427343396184814 1.9221280043552247737 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 4, 1] and d occurrences (as containment) of the composition, [1, 4, 3], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 12 12 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x - X1 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 12 10 12 9 10 10 8 9 - X2 x - X1 X2 x + x + X1 X2 x + X1 x + X2 x - X1 X2 x - X1 x 9 10 7 8 9 6 7 5 - X2 x - x + X1 X2 x + X1 x + x + X1 X2 x - 2 X1 x - 2 X1 X2 x 6 7 4 6 5 4 3 2 / + X1 x + x + X1 X2 x - 2 x + 2 x - x + x - 3 x + 3 x - 1) / ( / 2 8 8 8 6 8 5 6 (-1 + x) (X1 X2 x - X1 x - X2 x + X1 X2 x + x + X1 X2 x - X1 x 4 5 4 - X1 X2 x - x + x - 2 x + 1)) and in Maple format -(X1*X2*x^12-X1*x^12-X2*x^12-X1*X2*x^10+x^12+X1*X2*x^9+X1*x^10+X2*x^10-X1*X2*x^ 8-X1*x^9-X2*x^9-x^10+X1*X2*x^7+X1*x^8+x^9+X1*X2*x^6-2*X1*x^7-2*X1*X2*x^5+X1*x^6 +x^7+X1*X2*x^4-2*x^6+2*x^5-x^4+x^3-3*x^2+3*x-1)/(-1+x)^2/(X1*X2*x^8-X1*x^8-X2*x ^8+X1*X2*x^6+x^8+X1*X2*x^5-X1*x^6-X1*X2*x^4-x^5+x^4-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 4, 1], are n 183 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 4, 3], are n 135 65 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 22 3 65 ------------- 585 and in floating point 0.5251509079 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 22 195 ate normal pair with correlation, --------- 585 1/2 22 195 2723 i.e. , [[---------, 0], [0, ----]] 585 1755 ------------------------------------------------- Theorem Number, 149, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 4, 1], nor the composition, [1, 5, 2] Then infinity ----- \ n ) a(n) x = / ----- n = 0 12 11 10 7 6 5 4 3 2 x - x + x + x - 2 x + 2 x - x + x - 3 x + 3 x - 1 - ------------------------------------------------------------- 9 8 5 4 2 (x + x - x + x - 2 x + 1) (-1 + x) and in Maple format -(x^12-x^11+x^10+x^7-2*x^6+2*x^5-x^4+x^3-3*x^2+3*x-1)/(x^9+x^8-x^5+x^4-2*x+1)/( -1+x)^2 The asymptotic expression for a(n) is, n 0.67714921393955080398 1.9159714162121856417 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 4, 1] and d occurrences (as containment) of the composition, [1, 5, 2], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 12 11 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x - X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 12 12 10 11 11 12 10 10 - X1 x - X2 x + X1 X2 x + X1 x + X2 x + x - X1 x - X2 x 11 8 10 7 8 6 7 - x - X1 X2 x + x + 2 X1 X2 x + X1 x - 2 X1 X2 x - 3 X1 x 5 6 7 5 6 4 5 4 3 2 + X1 X2 x + 4 X1 x + x - 3 X1 x - 2 x + X1 x + 2 x - x + x - 3 x / 2 9 8 9 9 8 + 3 x - 1) / ((-1 + x) (X1 X2 x + X1 X2 x - X1 x - X2 x - X1 x / 8 9 6 8 5 6 5 4 5 - X2 x + x + X1 X2 x + x - X1 X2 x - X1 x + 2 X1 x - X1 x - x 4 + x - 2 x + 1)) and in Maple format -(X1*X2*x^12-X1*X2*x^11-X1*x^12-X2*x^12+X1*X2*x^10+X1*x^11+X2*x^11+x^12-X1*x^10 -X2*x^10-x^11-X1*X2*x^8+x^10+2*X1*X2*x^7+X1*x^8-2*X1*X2*x^6-3*X1*x^7+X1*X2*x^5+ 4*X1*x^6+x^7-3*X1*x^5-2*x^6+X1*x^4+2*x^5-x^4+x^3-3*x^2+3*x-1)/(-1+x)^2/(X1*X2*x ^9+X1*X2*x^8-X1*x^9-X2*x^9-X1*x^8-X2*x^8+x^9+X1*X2*x^6+x^8-X1*X2*x^5-X1*x^6+2* X1*x^5-X1*x^4-x^5+x^4-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 4, 1], are n 183 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 5, 2], are n 111 57 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 14 19 -------- 171 and in floating point 0.3568689194 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 14 19 ate normal pair with correlation, -------- 171 1/2 14 19 1931 i.e. , [[--------, 0], [0, ----]] 171 1539 ------------------------------------------------- Theorem Number, 150, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 4, 1], nor the composition, [1, 6, 1] Then infinity ----- 8 7 6 5 4 2 \ n x - x - x + x - x - x + 2 x - 1 ) a(n) x = ------------------------------------- / 6 5 3 2 2 ----- (x + x + x + x + x - 1) (-1 + x) n = 0 and in Maple format (x^8-x^7-x^6+x^5-x^4-x^2+2*x-1)/(x^6+x^5+x^3+x^2+x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.69790391550484586467 1.9111834366854762350 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 4, 1] and d occurrences (as containment) of the composition, [1, 6, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 8 7 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x - X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 8 8 6 7 7 8 7 5 6 4 - X1 x - X2 x + X1 X2 x + X1 x + X2 x + x - x - X1 x - x + X1 x 5 4 2 / 7 6 7 + x - x - x + 2 x - 1) / ((-1 + x) (X1 X2 x - X1 X2 x - X1 x / 7 6 7 5 4 5 4 - X2 x + X1 x + x + X1 x - X1 x - x + x - 2 x + 1)) and in Maple format (X1*X2*x^8-X1*X2*x^7-X1*x^8-X2*x^8+X1*X2*x^6+X1*x^7+X2*x^7+x^8-x^7-X1*x^5-x^6+ X1*x^4+x^5-x^4-x^2+2*x-1)/(-1+x)/(X1*X2*x^7-X1*X2*x^6-X1*x^7-X2*x^7+X1*x^6+x^7+ X1*x^5-X1*x^4-x^5+x^4-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 4, 1], are n 183 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 6, 1], are n 49 53 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 8 3 53 ------------ 477 and in floating point 0.2114804230 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 8 159 ate normal pair with correlation, -------- 477 1/2 8 159 1559 i.e. , [[--------, 0], [0, ----]] 477 1431 ------------------------------------------------- Theorem Number, 151, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 4, 1], nor the composition, [2, 1, 5] Then infinity ----- 10 9 7 6 5 4 2 \ n x + x + x - 2 x + x - x - x + 2 x - 1 ) a(n) x = - --------------------------------------------- / 7 6 5 4 ----- (-1 + x) (x - x + x - x + 2 x - 1) n = 0 and in Maple format -(x^10+x^9+x^7-2*x^6+x^5-x^4-x^2+2*x-1)/(-1+x)/(x^7-x^6+x^5-x^4+2*x-1) The asymptotic expression for a(n) is, n 0.67750027158103274656 1.9132221246804735080 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 4, 1] and d occurrences (as containment) of the composition, [2, 1, 5], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 11 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 11 10 11 9 10 10 9 - 2 X1 X2 x - X1 X2 x + X1 x - X1 X2 x + X1 x + X2 x + X1 x 9 10 7 9 6 7 7 5 + X2 x - x - X1 X2 x - x + X1 X2 x + X1 x + X2 x - X1 X2 x 6 6 7 5 6 4 5 4 2 - 2 X1 x - X2 x - x + 2 X1 x + 2 x - X1 x - x + x + x - 2 x + 1) / 2 12 2 11 12 11 12 11 / (X1 X2 x - X1 X2 x - 2 X1 X2 x + 2 X1 X2 x + X1 x - X1 x / 8 7 8 8 6 7 7 8 - X1 X2 x + X1 X2 x + X1 x + X2 x - 2 X1 X2 x - X1 x - 2 X2 x - x 5 6 6 7 5 6 4 5 4 + X1 X2 x + 3 X1 x + X2 x + 2 x - 3 X1 x - 2 x + X1 x + 2 x - x 2 - 2 x + 3 x - 1) and in Maple format -(X1*X2^2*x^11-2*X1*X2*x^11-X1*X2*x^10+X1*x^11-X1*X2*x^9+X1*x^10+X2*x^10+X1*x^9 +X2*x^9-x^10-X1*X2*x^7-x^9+X1*X2*x^6+X1*x^7+X2*x^7-X1*X2*x^5-2*X1*x^6-X2*x^6-x^ 7+2*X1*x^5+2*x^6-X1*x^4-x^5+x^4+x^2-2*x+1)/(X1*X2^2*x^12-X1*X2^2*x^11-2*X1*X2*x ^12+2*X1*X2*x^11+X1*x^12-X1*x^11-X1*X2*x^8+X1*X2*x^7+X1*x^8+X2*x^8-2*X1*X2*x^6- X1*x^7-2*X2*x^7-x^8+X1*X2*x^5+3*X1*x^6+X2*x^6+2*x^7-3*X1*x^5-2*x^6+X1*x^4+2*x^5 -x^4-2*x^2+3*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 4, 1], are n 183 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 1, 5], are n 113 57 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 4 19 ------- 57 and in floating point 0.3058876452 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 4 19 ate normal pair with correlation, ------- 57 1/2 4 19 203 i.e. , [[-------, 0], [0, ---]] 57 171 ------------------------------------------------- Theorem Number, 152, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 4, 1], nor the composition, [2, 2, 4] Then infinity ----- 9 8 6 5 4 2 \ n x + x - x + x - x - x + 2 x - 1 ) a(n) x = - ------------------------------------- / 5 4 ----- (-1 + x) (x - x + 2 x - 1) n = 0 and in Maple format -(x^9+x^8-x^6+x^5-x^4-x^2+2*x-1)/(-1+x)/(x^5-x^4+2*x-1) The asymptotic expression for a(n) is, n 0.61915988369516142914 1.9331849818995204468 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 4, 1] and d occurrences (as containment) of the composition, [2, 2, 4], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 9 8 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x + X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 9 9 8 8 9 6 8 5 4 - X1 x - X2 x - X1 x - X2 x + x + X1 X2 x + x - X1 X2 x + X1 X2 x 6 5 4 2 / 7 6 7 - x + x - x - x + 2 x - 1) / (X1 X2 x - X1 X2 x - X1 x / 5 4 6 5 4 2 + 2 X1 X2 x - X1 X2 x + x - 2 x + x + 2 x - 3 x + 1) and in Maple format -(X1*X2*x^9+X1*X2*x^8-X1*x^9-X2*x^9-X1*x^8-X2*x^8+x^9+X1*X2*x^6+x^8-X1*X2*x^5+ X1*X2*x^4-x^6+x^5-x^4-x^2+2*x-1)/(X1*X2*x^7-X1*X2*x^6-X1*x^7+2*X1*X2*x^5-X1*X2* x^4+x^6-2*x^5+x^4+2*x^2-3*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 4, 1], are n 183 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 2, 4], are n 75 69 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 32 23 -------- 207 and in floating point 0.7413845834 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 32 23 ate normal pair with correlation, -------- 207 1/2 32 23 3911 i.e. , [[--------, 0], [0, ----]] 207 1863 ------------------------------------------------- Theorem Number, 153, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 4, 1], nor the composition, [2, 3, 3] Then infinity ----- 7 6 4 3 2 \ n x - 2 x + x - x - x + 2 x - 1 ) a(n) x = - ------------------------------------------- / 7 6 5 4 3 ----- (-1 + x) (x - x + x + x - x + 2 x - 1) n = 0 and in Maple format -(x^7-2*x^6+x^4-x^3-x^2+2*x-1)/(-1+x)/(x^7-x^6+x^5+x^4-x^3+2*x-1) The asymptotic expression for a(n) is, n 0.64732625001158940098 1.9249743097683141877 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 4, 1] and d occurrences (as containment) of the composition, [2, 3, 3], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 7 7 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x - X1 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 7 5 6 6 7 4 5 6 - X2 x - X1 X2 x + X1 x + X2 x + x + X1 X2 x + X2 x - 2 x 4 3 4 3 2 / 8 7 - 2 X2 x + X2 x + x - x - x + 2 x - 1) / (X1 X2 x - X1 X2 x / 8 8 7 7 8 5 6 6 7 - X1 x - X2 x + X1 x + 2 X2 x + x + 2 X1 X2 x - X1 x - X2 x - 2 x 4 5 6 4 3 4 3 2 - X1 X2 x - 2 X2 x + 2 x + 3 X2 x - X2 x - 2 x + x + 2 x - 3 x + 1 ) and in Maple format -(X1*X2*x^7-X1*x^7-X2*x^7-X1*X2*x^5+X1*x^6+X2*x^6+x^7+X1*X2*x^4+X2*x^5-2*x^6-2* X2*x^4+X2*x^3+x^4-x^3-x^2+2*x-1)/(X1*X2*x^8-X1*X2*x^7-X1*x^8-X2*x^8+X1*x^7+2*X2 *x^7+x^8+2*X1*X2*x^5-X1*x^6-X2*x^6-2*x^7-X1*X2*x^4-2*X2*x^5+2*x^6+3*X2*x^4-X2*x ^3-2*x^4+x^3+2*x^2-3*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 4, 1], are n 183 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 3, 3], are n 97 85 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 8 3 85 ------------ 255 and in floating point 0.5009794331 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 8 255 ate normal pair with correlation, -------- 255 1/2 8 255 383 i.e. , [[--------, 0], [0, ---]] 255 255 ------------------------------------------------- Theorem Number, 154, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 4, 1], nor the composition, [3, 1, 4] Then infinity ----- 13 12 11 10 9 4 2 \ n x + x + x - x - x + x + x - 2 x + 1 ) a(n) x = ---------------------------------------------- / 7 6 4 ----- (-1 + x) (x + x - x + 2 x - 1) n = 0 and in Maple format (x^13+x^12+x^11-x^10-x^9+x^4+x^2-2*x+1)/(-1+x)/(x^7+x^6-x^4+2*x-1) The asymptotic expression for a(n) is, n 0.66581546041682687250 1.9168163613635124211 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 4, 1] and d occurrences (as containment) of the composition, [3, 1, 4], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 16 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 2 15 2 16 2 16 2 15 2 16 + X1 X2 x - 2 X1 X2 x - X1 X2 x - 2 X1 X2 x + X1 x 2 15 16 2 2 13 2 14 2 15 - X1 X2 x + 2 X1 X2 x - X1 X2 x - X1 X2 x + X1 x 15 16 2 13 2 14 2 13 14 + 2 X1 X2 x - X1 x + 2 X1 X2 x + X1 x + X1 X2 x + 2 X1 X2 x 15 2 2 11 2 13 13 14 14 - X1 x + X1 X2 x - X1 x - 2 X1 X2 x - 2 X1 x - X2 x 2 2 10 2 11 13 14 2 10 2 10 - X1 X2 x - X1 X2 x + X1 x + x + X1 X2 x + X1 X2 x 11 2 9 10 11 11 9 - 3 X1 X2 x - X1 X2 x - X1 X2 x + 3 X1 x + 2 X2 x + 3 X1 X2 x 11 9 9 7 9 6 7 - 2 x - 2 X1 x - X2 x - X1 X2 x + x + 2 X1 X2 x + X2 x 5 6 4 5 5 4 3 2 - 2 X1 X2 x - 2 X2 x + X1 X2 x + X2 x + x - x + x - 3 x + 3 x - 1) / 2 2 12 2 2 11 2 12 2 2 10 / ((-1 + x) (X1 X2 x + X1 X2 x - 2 X1 X2 x - X1 X2 x / 2 11 2 12 2 11 2 10 2 11 2 10 - 2 X1 X2 x + X1 x + X1 X2 x + X1 X2 x + X1 x + X1 X2 x 11 2 9 10 11 9 8 - 2 X1 X2 x - X1 X2 x - X1 X2 x + X1 x + 2 X1 X2 x - X1 X2 x 9 7 8 8 6 7 8 5 - X1 x - X1 X2 x + X1 x + X2 x + X1 X2 x + X1 x - x - 2 X1 X2 x 6 4 5 6 5 4 2 - 2 X2 x + X1 X2 x + X2 x + x + x - x - 2 x + 3 x - 1)) and in Maple format -(X1^2*X2^2*x^16+X1^2*X2^2*x^15-2*X1^2*X2*x^16-X1*X2^2*x^16-2*X1^2*X2*x^15+X1^2 *x^16-X1*X2^2*x^15+2*X1*X2*x^16-X1^2*X2^2*x^13-X1^2*X2*x^14+X1^2*x^15+2*X1*X2*x ^15-X1*x^16+2*X1^2*X2*x^13+X1^2*x^14+X1*X2^2*x^13+2*X1*X2*x^14-X1*x^15+X1^2*X2^ 2*x^11-X1^2*x^13-2*X1*X2*x^13-2*X1*x^14-X2*x^14-X1^2*X2^2*x^10-X1^2*X2*x^11+X1* x^13+x^14+X1^2*X2*x^10+X1*X2^2*x^10-3*X1*X2*x^11-X1*X2^2*x^9-X1*X2*x^10+3*X1*x^ 11+2*X2*x^11+3*X1*X2*x^9-2*x^11-2*X1*x^9-X2*x^9-X1*X2*x^7+x^9+2*X1*X2*x^6+X2*x^ 7-2*X1*X2*x^5-2*X2*x^6+X1*X2*x^4+X2*x^5+x^5-x^4+x^3-3*x^2+3*x-1)/(-1+x)/(X1^2* X2^2*x^12+X1^2*X2^2*x^11-2*X1^2*X2*x^12-X1^2*X2^2*x^10-2*X1^2*X2*x^11+X1^2*x^12 +X1*X2^2*x^11+X1^2*X2*x^10+X1^2*x^11+X1*X2^2*x^10-2*X1*X2*x^11-X1*X2^2*x^9-X1* X2*x^10+X1*x^11+2*X1*X2*x^9-X1*X2*x^8-X1*x^9-X1*X2*x^7+X1*x^8+X2*x^8+X1*X2*x^6+ X1*x^7-x^8-2*X1*X2*x^5-2*X2*x^6+X1*X2*x^4+X2*x^5+x^6+x^5-x^4-2*x^2+3*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 4, 1], are n 183 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 1, 4], are n 141 65 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 16 3 65 ------------- 585 and in floating point 0.3819279330 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 16 195 ate normal pair with correlation, --------- 585 1/2 16 195 2267 i.e. , [[---------, 0], [0, ----]] 585 1755 ------------------------------------------------- Theorem Number, 155, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 4, 1], nor the composition, [3, 2, 3] Then infinity ----- 14 11 9 8 5 4 3 2 \ n x + x - 2 x + x + x - x + x + x - 2 x + 1 ) a(n) x = - --------------------------------------------------- / 8 4 3 2 3 ----- (-1 + x) (x - x - x - x - x + 1) (x - x + 1) n = 0 and in Maple format -(x^14+x^11-2*x^9+x^8+x^5-x^4+x^3+x^2-2*x+1)/(-1+x)/(x^8-x^4-x^3-x^2-x+1)/(x^3- x+1) The asymptotic expression for a(n) is, n 0.65153311498131888405 1.9217220658969757404 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 4, 1] and d occurrences (as containment) of the composition, [3, 2, 3], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 15 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 15 2 15 2 2 13 2 15 2 14 - 2 X1 X2 x - 2 X1 X2 x - X1 X2 x + X1 x + X1 X2 x 15 2 15 2 13 2 13 14 + 4 X1 X2 x + X2 x + X1 X2 x + 2 X1 X2 x - 2 X1 X2 x 15 2 14 15 2 2 11 2 12 13 - 2 X1 x - X2 x - 2 X2 x + X1 X2 x - X1 X2 x - 2 X1 X2 x 14 2 13 14 15 2 2 10 2 11 + X1 x - X2 x + 2 X2 x + x - X1 X2 x - X1 X2 x 2 11 12 2 12 13 14 2 10 - 2 X1 X2 x + 2 X1 X2 x + X2 x + X2 x - x + X1 X2 x 2 10 11 12 2 11 12 2 9 + 4 X1 X2 x + X1 X2 x - X1 x + X2 x - 2 X2 x - 2 X1 X2 x 10 11 2 10 12 9 10 - 6 X1 X2 x + X1 x - 3 X2 x + x + 4 X1 X2 x + 2 X1 x 2 9 10 11 9 2 8 9 10 + 3 X2 x + 5 X2 x - x - 2 X1 x - X2 x - 6 X2 x - 2 x 7 8 9 6 7 8 5 - X1 X2 x + 2 X2 x + 3 x + 2 X1 X2 x + X2 x - x - 2 X1 X2 x 6 4 5 6 4 5 3 4 2 - 3 X2 x + X1 X2 x + 4 X2 x + x - 3 X2 x - 2 x + X2 x + 2 x - 3 x / 2 2 12 2 2 11 2 12 + 3 x - 1) / ((-1 + x) (X1 X2 x + X1 X2 x - 2 X1 X2 x / 2 12 2 2 10 2 11 2 12 12 - 2 X1 X2 x - X1 X2 x - 2 X1 X2 x + X1 x + 4 X1 X2 x 2 12 2 10 2 11 2 10 12 2 11 + X2 x + X1 X2 x + X1 x + 4 X1 X2 x - 2 X1 x - X2 x 12 2 9 10 2 10 11 12 - 2 X2 x - 2 X1 X2 x - 6 X1 X2 x - 2 X2 x + 2 X2 x + x 9 10 2 9 10 11 8 9 + 3 X1 X2 x + 2 X1 x + 3 X2 x + 3 X2 x - x - X1 X2 x - X1 x 2 8 9 10 7 8 8 9 6 - X2 x - 5 X2 x - x - X1 X2 x + X1 x + 3 X2 x + 2 x + X1 X2 x 7 8 5 6 4 5 6 4 + X1 x - 2 x - 2 X1 X2 x - 2 X2 x + X1 X2 x + 3 X2 x + x - 3 X2 x 5 3 4 3 2 - x + X2 x + 2 x - x - 2 x + 3 x - 1)) and in Maple format -(X1^2*X2^2*x^15-2*X1^2*X2*x^15-2*X1*X2^2*x^15-X1^2*X2^2*x^13+X1^2*x^15+X1*X2^2 *x^14+4*X1*X2*x^15+X2^2*x^15+X1^2*X2*x^13+2*X1*X2^2*x^13-2*X1*X2*x^14-2*X1*x^15 -X2^2*x^14-2*X2*x^15+X1^2*X2^2*x^11-X1*X2^2*x^12-2*X1*X2*x^13+X1*x^14-X2^2*x^13 +2*X2*x^14+x^15-X1^2*X2^2*x^10-X1^2*X2*x^11-2*X1*X2^2*x^11+2*X1*X2*x^12+X2^2*x^ 12+X2*x^13-x^14+X1^2*X2*x^10+4*X1*X2^2*x^10+X1*X2*x^11-X1*x^12+X2^2*x^11-2*X2*x ^12-2*X1*X2^2*x^9-6*X1*X2*x^10+X1*x^11-3*X2^2*x^10+x^12+4*X1*X2*x^9+2*X1*x^10+3 *X2^2*x^9+5*X2*x^10-x^11-2*X1*x^9-X2^2*x^8-6*X2*x^9-2*x^10-X1*X2*x^7+2*X2*x^8+3 *x^9+2*X1*X2*x^6+X2*x^7-x^8-2*X1*X2*x^5-3*X2*x^6+X1*X2*x^4+4*X2*x^5+x^6-3*X2*x^ 4-2*x^5+X2*x^3+2*x^4-3*x^2+3*x-1)/(-1+x)/(X1^2*X2^2*x^12+X1^2*X2^2*x^11-2*X1^2* X2*x^12-2*X1*X2^2*x^12-X1^2*X2^2*x^10-2*X1^2*X2*x^11+X1^2*x^12+4*X1*X2*x^12+X2^ 2*x^12+X1^2*X2*x^10+X1^2*x^11+4*X1*X2^2*x^10-2*X1*x^12-X2^2*x^11-2*X2*x^12-2*X1 *X2^2*x^9-6*X1*X2*x^10-2*X2^2*x^10+2*X2*x^11+x^12+3*X1*X2*x^9+2*X1*x^10+3*X2^2* x^9+3*X2*x^10-x^11-X1*X2*x^8-X1*x^9-X2^2*x^8-5*X2*x^9-x^10-X1*X2*x^7+X1*x^8+3* X2*x^8+2*x^9+X1*X2*x^6+X1*x^7-2*x^8-2*X1*X2*x^5-2*X2*x^6+X1*X2*x^4+3*X2*x^5+x^6 -3*X2*x^4-x^5+X2*x^3+2*x^4-x^3-2*x^2+3*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 4, 1], are n 183 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 2, 3], are n 89 77 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 20 3 77 ------------- 693 and in floating point 0.4386344635 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 20 231 ate normal pair with correlation, --------- 693 1/2 20 231 2879 i.e. , [[---------, 0], [0, ----]] 693 2079 ------------------------------------------------- Theorem Number, 156, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 4, 1], nor the composition, [3, 3, 2] Then infinity ----- 7 6 4 3 2 \ n x - 2 x + x - x - x + 2 x - 1 ) a(n) x = ------------------------------------------------- / 10 7 6 5 4 3 ----- (-1 + x) (x - x + x - x - x + x - 2 x + 1) n = 0 and in Maple format (x^7-2*x^6+x^4-x^3-x^2+2*x-1)/(-1+x)/(x^10-x^7+x^6-x^5-x^4+x^3-2*x+1) The asymptotic expression for a(n) is, n 0.65532458744249045709 1.9223246520768555496 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 4, 1] and d occurrences (as containment) of the composition, [3, 3, 2], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 7 7 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x - X1 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 7 5 6 6 7 4 5 6 - X2 x - X1 X2 x + X1 x + X2 x + x + X1 X2 x + X2 x - 2 x 4 3 4 3 2 / 2 11 2 11 - 2 X2 x + X2 x + x - x - x + 2 x - 1) / (X1 X2 x - X1 x / 11 10 11 11 10 10 11 - 2 X1 X2 x + X1 X2 x + 2 X1 x + X2 x - X1 x - X2 x - x 8 10 7 8 8 7 7 8 + X1 X2 x + x - X1 X2 x - X1 x - X2 x + X1 x + 2 X2 x + x 5 6 6 7 4 5 6 4 + 2 X1 X2 x - X1 x - X2 x - 2 x - X1 X2 x - 2 X2 x + 2 x + 3 X2 x 3 4 3 2 - X2 x - 2 x + x + 2 x - 3 x + 1) and in Maple format -(X1*X2*x^7-X1*x^7-X2*x^7-X1*X2*x^5+X1*x^6+X2*x^6+x^7+X1*X2*x^4+X2*x^5-2*x^6-2* X2*x^4+X2*x^3+x^4-x^3-x^2+2*x-1)/(X1^2*X2*x^11-X1^2*x^11-2*X1*X2*x^11+X1*X2*x^ 10+2*X1*x^11+X2*x^11-X1*x^10-X2*x^10-x^11+X1*X2*x^8+x^10-X1*X2*x^7-X1*x^8-X2*x^ 8+X1*x^7+2*X2*x^7+x^8+2*X1*X2*x^5-X1*x^6-X2*x^6-2*x^7-X1*X2*x^4-2*X2*x^5+2*x^6+ 3*X2*x^4-X2*x^3-2*x^4+x^3+2*x^2-3*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 4, 1], are n 183 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 3, 2], are n 97 85 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 4 3 85 ------------ 153 and in floating point 0.4174828609 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 1/2 4 5 51 ate normal pair with correlation, ------------ 153 1/2 1/2 4 5 51 619 i.e. , [[------------, 0], [0, ---]] 153 459 ------------------------------------------------- Theorem Number, 157, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 4, 1], nor the composition, [4, 1, 3] Then infinity ----- \ n ) a(n) x = / ----- n = 0 13 12 11 10 8 4 2 x + x + x - x - x + x + x - 2 x + 1 - ----------------------------------------------------------- 10 9 8 7 6 3 2 2 (x + 2 x + 3 x + 2 x + x + x + x + x - 1) (-1 + x) and in Maple format -(x^13+x^12+x^11-x^10-x^8+x^4+x^2-2*x+1)/(x^10+2*x^9+3*x^8+2*x^7+x^6+x^3+x^2+x-\ 1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.66396839102705721566 1.9175371779461324874 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 4, 1] and d occurrences (as containment) of the composition, [4, 1, 3], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 16 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 2 15 2 16 2 16 2 15 2 16 + X1 X2 x - 2 X1 X2 x - X1 X2 x - 2 X1 X2 x + X1 x 2 15 16 2 14 2 15 15 16 - X1 X2 x + 2 X1 X2 x - X1 X2 x + X1 x + 2 X1 X2 x - X1 x 2 2 12 2 14 14 15 2 2 11 - X1 X2 x + X1 x + 2 X1 X2 x - X1 x + X1 X2 x 2 12 2 12 14 14 2 2 10 2 11 + 2 X1 X2 x + X1 X2 x - 2 X1 x - X2 x - X1 X2 x - X1 X2 x 2 12 12 14 2 10 2 10 11 - X1 x - 2 X1 X2 x + x + X1 X2 x + X1 X2 x - 3 X1 X2 x 12 2 9 11 11 9 10 10 + X1 x - X1 X2 x + 3 X1 x + 2 X2 x + X1 X2 x - X1 x - X2 x 11 8 9 10 7 8 8 9 - 2 x + X1 X2 x + X2 x + x - X1 X2 x - X1 x - X2 x - x 6 7 8 5 6 4 5 5 + 2 X1 X2 x + X2 x + x - 2 X1 X2 x - 2 X2 x + X1 X2 x + X2 x + x 4 3 2 / 2 2 14 2 2 13 - x + x - 3 x + 3 x - 1) / ((-1 + x) (X1 X2 x + X1 X2 x / 2 14 2 14 2 13 2 14 2 13 - 2 X1 X2 x - X1 X2 x - 2 X1 X2 x + X1 x - X1 X2 x 14 2 2 11 2 12 2 13 2 12 + 2 X1 X2 x + X1 X2 x - X1 X2 x + X1 x + X1 X2 x 13 14 2 2 10 2 11 2 12 2 11 + 2 X1 X2 x - X1 x - X1 X2 x - 2 X1 X2 x + X1 x + X1 X2 x 13 2 10 2 11 2 10 11 12 - X1 x + X1 X2 x + X1 x + X1 X2 x - 2 X1 X2 x - X1 x 12 2 9 10 11 12 9 9 - X2 x - X1 X2 x - X1 X2 x + X1 x + x + X1 x + 2 X2 x 7 9 6 7 5 6 4 - X1 X2 x - 2 x + X1 X2 x + X1 x - 2 X1 X2 x - 2 X2 x + X1 X2 x 5 6 5 4 2 + X2 x + x + x - x - 2 x + 3 x - 1)) and in Maple format -(X1^2*X2^2*x^16+X1^2*X2^2*x^15-2*X1^2*X2*x^16-X1*X2^2*x^16-2*X1^2*X2*x^15+X1^2 *x^16-X1*X2^2*x^15+2*X1*X2*x^16-X1^2*X2*x^14+X1^2*x^15+2*X1*X2*x^15-X1*x^16-X1^ 2*X2^2*x^12+X1^2*x^14+2*X1*X2*x^14-X1*x^15+X1^2*X2^2*x^11+2*X1^2*X2*x^12+X1*X2^ 2*x^12-2*X1*x^14-X2*x^14-X1^2*X2^2*x^10-X1^2*X2*x^11-X1^2*x^12-2*X1*X2*x^12+x^ 14+X1^2*X2*x^10+X1*X2^2*x^10-3*X1*X2*x^11+X1*x^12-X1*X2^2*x^9+3*X1*x^11+2*X2*x^ 11+X1*X2*x^9-X1*x^10-X2*x^10-2*x^11+X1*X2*x^8+X2*x^9+x^10-X1*X2*x^7-X1*x^8-X2*x ^8-x^9+2*X1*X2*x^6+X2*x^7+x^8-2*X1*X2*x^5-2*X2*x^6+X1*X2*x^4+X2*x^5+x^5-x^4+x^3 -3*x^2+3*x-1)/(-1+x)/(X1^2*X2^2*x^14+X1^2*X2^2*x^13-2*X1^2*X2*x^14-X1*X2^2*x^14 -2*X1^2*X2*x^13+X1^2*x^14-X1*X2^2*x^13+2*X1*X2*x^14+X1^2*X2^2*x^11-X1^2*X2*x^12 +X1^2*x^13+X1*X2^2*x^12+2*X1*X2*x^13-X1*x^14-X1^2*X2^2*x^10-2*X1^2*X2*x^11+X1^2 *x^12+X1*X2^2*x^11-X1*x^13+X1^2*X2*x^10+X1^2*x^11+X1*X2^2*x^10-2*X1*X2*x^11-X1* x^12-X2*x^12-X1*X2^2*x^9-X1*X2*x^10+X1*x^11+x^12+X1*x^9+2*X2*x^9-X1*X2*x^7-2*x^ 9+X1*X2*x^6+X1*x^7-2*X1*X2*x^5-2*X2*x^6+X1*X2*x^4+X2*x^5+x^6+x^5-x^4-2*x^2+3*x-\ 1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 4, 1], are n 183 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 1, 3], are n 141 65 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 16 3 65 ------------- 585 and in floating point 0.3819279330 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 16 195 ate normal pair with correlation, --------- 585 1/2 16 195 2267 i.e. , [[---------, 0], [0, ----]] 585 1755 ------------------------------------------------- Theorem Number, 158, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 4, 1], nor the composition, [4, 2, 2] Then infinity ----- 9 8 6 5 4 2 \ n x + x - x + x - x - x + 2 x - 1 ) a(n) x = ----------------------------------------------- / 11 10 9 5 4 ----- (-1 + x) (x + 2 x + x - x + x - 2 x + 1) n = 0 and in Maple format (x^9+x^8-x^6+x^5-x^4-x^2+2*x-1)/(-1+x)/(x^11+2*x^10+x^9-x^5+x^4-2*x+1) The asymptotic expression for a(n) is, n 0.65906522740498320917 1.9197449317290998989 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 4, 1] and d occurrences (as containment) of the composition, [4, 2, 2], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 9 8 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x + X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 9 9 8 8 9 6 8 5 4 - X1 x - X2 x - X1 x - X2 x + x + X1 X2 x + x - X1 X2 x + X1 X2 x 6 5 4 2 / 2 12 2 11 2 12 - x + x - x - x + 2 x - 1) / (X1 X2 x + X1 X2 x - X1 x / 12 2 11 11 12 12 10 - 2 X1 X2 x - X1 x - 2 X1 X2 x + 2 X1 x + X2 x + X1 X2 x 11 11 12 9 10 10 11 9 + 2 X1 x + X2 x - x + X1 X2 x - X1 x - X2 x - x - X1 x 9 10 7 9 6 7 5 4 - X2 x + x + X1 X2 x + x - X1 X2 x - X1 x + 2 X1 X2 x - X1 X2 x 6 5 4 2 + x - 2 x + x + 2 x - 3 x + 1) and in Maple format -(X1*X2*x^9+X1*X2*x^8-X1*x^9-X2*x^9-X1*x^8-X2*x^8+x^9+X1*X2*x^6+x^8-X1*X2*x^5+ X1*X2*x^4-x^6+x^5-x^4-x^2+2*x-1)/(X1^2*X2*x^12+X1^2*X2*x^11-X1^2*x^12-2*X1*X2*x ^12-X1^2*x^11-2*X1*X2*x^11+2*X1*x^12+X2*x^12+X1*X2*x^10+2*X1*x^11+X2*x^11-x^12+ X1*X2*x^9-X1*x^10-X2*x^10-x^11-X1*x^9-X2*x^9+x^10+X1*X2*x^7+x^9-X1*X2*x^6-X1*x^ 7+2*X1*X2*x^5-X1*X2*x^4+x^6-2*x^5+x^4+2*x^2-3*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 4, 1], are n 183 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 2, 2], are n 75 69 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 20 23 -------- 207 and in floating point 0.4633653646 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 20 23 ate normal pair with correlation, -------- 207 1/2 20 23 2663 i.e. , [[--------, 0], [0, ----]] 207 1863 ------------------------------------------------- Theorem Number, 159, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 4, 1], nor the composition, [4, 3, 1] Then infinity ----- 8 7 6 4 \ n x + x + x + x - x + 1 ) a(n) x = ----------------------------------------- / 7 6 5 3 2 ----- (-1 + x) (x + x + x + x + x + x - 1) n = 0 and in Maple format (x^8+x^7+x^6+x^4-x+1)/(-1+x)/(x^7+x^6+x^5+x^3+x^2+x-1) The asymptotic expression for a(n) is, n 0.65748289688935283811 1.9221280043552247737 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 4, 1] and d occurrences (as containment) of the composition, [4, 3, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 9 9 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x - X1 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 9 7 9 7 5 6 4 6 5 - X2 x + X1 X2 x + x - X1 x - X1 X2 x + X1 x + X1 X2 x - x + x 4 2 / 8 8 8 6 - x - x + 2 x - 1) / ((-1 + x) (X1 X2 x - X1 x - X2 x + X1 X2 x / 8 5 6 4 5 4 + x + X1 X2 x - X1 x - X1 X2 x - x + x - 2 x + 1)) and in Maple format (X1*X2*x^9-X1*x^9-X2*x^9+X1*X2*x^7+x^9-X1*x^7-X1*X2*x^5+X1*x^6+X1*X2*x^4-x^6+x^ 5-x^4-x^2+2*x-1)/(-1+x)/(X1*X2*x^8-X1*x^8-X2*x^8+X1*X2*x^6+x^8+X1*X2*x^5-X1*x^6 -X1*X2*x^4-x^5+x^4-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 4, 1], are n 183 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 3, 1], are n 135 65 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 22 3 65 ------------- 585 and in floating point 0.5251509079 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 22 195 ate normal pair with correlation, --------- 585 1/2 22 195 2723 i.e. , [[---------, 0], [0, ----]] 585 1755 ------------------------------------------------- Theorem Number, 160, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 4, 1], nor the composition, [5, 1, 2] Then infinity ----- \ n ) a(n) x = / ----- n = 0 9 8 7 6 5 4 2 x + x + x - 2 x + x - x - x + 2 x - 1 ----------------------------------------------------------------- 12 11 10 8 7 6 5 4 (-1 + x) (x + 2 x + 2 x - x - x + x - x + x - 2 x + 1) and in Maple format (x^9+x^8+x^7-2*x^6+x^5-x^4-x^2+2*x-1)/(-1+x)/(x^12+2*x^11+2*x^10-x^8-x^7+x^6-x^ 5+x^4-2*x+1) The asymptotic expression for a(n) is, n 0.67448318694762719220 1.9143463083540318048 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 4, 1] and d occurrences (as containment) of the composition, [5, 1, 2], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 14 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 2 13 2 14 2 14 2 13 2 14 + X1 X2 x - 2 X1 X2 x - X1 X2 x - 2 X1 X2 x + X1 x 2 13 14 2 13 13 14 2 11 - X1 X2 x + 2 X1 X2 x + X1 x + 2 X1 X2 x - X1 x + X1 X2 x 13 11 11 9 8 9 9 - X1 x - 2 X1 X2 x + X1 x - X1 X2 x - X1 X2 x + X1 x + X2 x 7 8 8 9 6 7 7 8 5 - X1 X2 x + X1 x + X2 x - x + X1 X2 x + X1 x + X2 x - x - X1 X2 x 6 6 7 5 6 4 5 4 2 - 2 X1 x - X2 x - x + 2 X1 x + 2 x - X1 x - x + x + x - 2 x + 1) / 2 2 15 2 15 2 15 2 2 13 2 15 / (X1 X2 x - 2 X1 X2 x - X1 X2 x - X1 X2 x + X1 x / 15 2 13 2 13 15 2 12 2 12 + 2 X1 X2 x + X1 X2 x + X1 X2 x - X1 x - X1 X2 x + X1 X2 x 2 12 2 11 13 13 11 12 12 + X1 x - X1 X2 x - X1 x - X2 x + 2 X1 X2 x - X1 x - X2 x 13 10 11 12 9 10 10 + x - 2 X1 X2 x - X1 x + x - X1 X2 x + 2 X1 x + 2 X2 x 9 9 10 7 9 6 7 7 + X1 x + X2 x - 2 x + X1 X2 x - x - 2 X1 X2 x - X1 x - 2 X2 x 5 6 6 7 5 6 4 5 4 + X1 X2 x + 3 X1 x + X2 x + 2 x - 3 X1 x - 2 x + X1 x + 2 x - x 2 - 2 x + 3 x - 1) and in Maple format -(X1^2*X2^2*x^14+X1^2*X2^2*x^13-2*X1^2*X2*x^14-X1*X2^2*x^14-2*X1^2*X2*x^13+X1^2 *x^14-X1*X2^2*x^13+2*X1*X2*x^14+X1^2*x^13+2*X1*X2*x^13-X1*x^14+X1*X2^2*x^11-X1* x^13-2*X1*X2*x^11+X1*x^11-X1*X2*x^9-X1*X2*x^8+X1*x^9+X2*x^9-X1*X2*x^7+X1*x^8+X2 *x^8-x^9+X1*X2*x^6+X1*x^7+X2*x^7-x^8-X1*X2*x^5-2*X1*x^6-X2*x^6-x^7+2*X1*x^5+2*x ^6-X1*x^4-x^5+x^4+x^2-2*x+1)/(X1^2*X2^2*x^15-2*X1^2*X2*x^15-X1*X2^2*x^15-X1^2* X2^2*x^13+X1^2*x^15+2*X1*X2*x^15+X1^2*X2*x^13+X1*X2^2*x^13-X1*x^15-X1^2*X2*x^12 +X1*X2^2*x^12+X1^2*x^12-X1*X2^2*x^11-X1*x^13-X2*x^13+2*X1*X2*x^11-X1*x^12-X2*x^ 12+x^13-2*X1*X2*x^10-X1*x^11+x^12-X1*X2*x^9+2*X1*x^10+2*X2*x^10+X1*x^9+X2*x^9-2 *x^10+X1*X2*x^7-x^9-2*X1*X2*x^6-X1*x^7-2*X2*x^7+X1*X2*x^5+3*X1*x^6+X2*x^6+2*x^7 -3*X1*x^5-2*x^6+X1*x^4+2*x^5-x^4-2*x^2+3*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 4, 1], are n 183 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [5, 1, 2], are n 113 57 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 4 19 ------- 57 and in floating point 0.3058876452 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 4 19 ate normal pair with correlation, ------- 57 1/2 4 19 203 i.e. , [[-------, 0], [0, ---]] 57 171 ------------------------------------------------- Theorem Number, 161, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 4, 1], nor the composition, [5, 2, 1] Then infinity ----- 10 9 6 5 4 2 \ n x + x - x + x - x - x + 2 x - 1 ) a(n) x = -------------------------------------- / 9 8 5 4 ----- (-1 + x) (x + x - x + x - 2 x + 1) n = 0 and in Maple format (x^10+x^9-x^6+x^5-x^4-x^2+2*x-1)/(-1+x)/(x^9+x^8-x^5+x^4-2*x+1) The asymptotic expression for a(n) is, n 0.67229840142428698322 1.9159714162121856417 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 4, 1] and d occurrences (as containment) of the composition, [5, 2, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 10 9 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x + X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 10 10 9 9 10 7 9 6 7 - X1 x - X2 x - X1 x - X2 x + x + X1 X2 x + x - X1 X2 x - X1 x 5 6 5 6 4 5 4 2 / + X1 X2 x + 2 X1 x - 2 X1 x - x + X1 x + x - x - x + 2 x - 1) / / 9 8 9 9 8 8 9 ((-1 + x) (X1 X2 x + X1 X2 x - X1 x - X2 x - X1 x - X2 x + x 6 8 5 6 5 4 5 4 + X1 X2 x + x - X1 X2 x - X1 x + 2 X1 x - X1 x - x + x - 2 x + 1)) and in Maple format (X1*X2*x^10+X1*X2*x^9-X1*x^10-X2*x^10-X1*x^9-X2*x^9+x^10+X1*X2*x^7+x^9-X1*X2*x^ 6-X1*x^7+X1*X2*x^5+2*X1*x^6-2*X1*x^5-x^6+X1*x^4+x^5-x^4-x^2+2*x-1)/(-1+x)/(X1* X2*x^9+X1*X2*x^8-X1*x^9-X2*x^9-X1*x^8-X2*x^8+x^9+X1*X2*x^6+x^8-X1*X2*x^5-X1*x^6 +2*X1*x^5-X1*x^4-x^5+x^4-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 4, 1], are n 183 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [5, 2, 1], are n 111 57 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 14 19 -------- 171 and in floating point 0.3568689194 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 14 19 ate normal pair with correlation, -------- 171 1/2 14 19 1931 i.e. , [[--------, 0], [0, ----]] 171 1539 ------------------------------------------------- Theorem Number, 162, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 4, 1], nor the composition, [6, 1, 1] Then infinity ----- 4 2 6 5 \ n (x + x - 2 x + 1) (x + x - 1) ) a(n) x = ------------------------------------- / 6 5 3 2 2 ----- (x + x + x + x + x - 1) (-1 + x) n = 0 and in Maple format (x^4+x^2-2*x+1)*(x^6+x^5-1)/(x^6+x^5+x^3+x^2+x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.68707893079649356212 1.9111834366854762350 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 4, 1] and d occurrences (as containment) of the composition, [6, 1, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 10 9 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x + X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 10 10 8 9 9 10 7 8 - X1 x - X2 x + X1 X2 x - X1 x - X2 x + x - X1 X2 x - X1 x 8 9 6 7 7 8 7 5 6 4 - X2 x + x + X1 X2 x + X1 x + X2 x + x - x - X1 x - x + X1 x 5 4 2 / 7 6 7 + x - x - x + 2 x - 1) / ((-1 + x) (X1 X2 x - X1 X2 x - X1 x / 7 6 7 5 4 5 4 - X2 x + X1 x + x + X1 x - X1 x - x + x - 2 x + 1)) and in Maple format (X1*X2*x^10+X1*X2*x^9-X1*x^10-X2*x^10+X1*X2*x^8-X1*x^9-X2*x^9+x^10-X1*X2*x^7-X1 *x^8-X2*x^8+x^9+X1*X2*x^6+X1*x^7+X2*x^7+x^8-x^7-X1*x^5-x^6+X1*x^4+x^5-x^4-x^2+2 *x-1)/(-1+x)/(X1*X2*x^7-X1*X2*x^6-X1*x^7-X2*x^7+X1*x^6+x^7+X1*x^5-X1*x^4-x^5+x^ 4-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 4, 1], are n 183 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [6, 1, 1], are n 49 53 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 8 3 53 ------------ 477 and in floating point 0.2114804230 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 8 159 ate normal pair with correlation, -------- 477 1/2 8 159 1559 i.e. , [[--------, 0], [0, ----]] 477 1431 ------------------------------------------------- Theorem Number, 163, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [3, 1, 3], nor the composition, [1, 1, 6] Then infinity ----- \ n 11 10 8 7 3 2 / ) a(n) x = (x + x - x - x - x - x + 2 x - 1) / ( / / ----- n = 0 7 6 5 4 2 7 6 5 4 2 (x + 2 x + 2 x + x + x + x - 1) (x + 2 x + 2 x + x + 1) (-1 + x) ) and in Maple format (x^11+x^10-x^8-x^7-x^3-x^2+2*x-1)/(x^7+2*x^6+2*x^5+x^4+x^2+x-1)/(x^7+2*x^6+2*x^ 5+x^4+1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.67485833261041293122 1.9131864632213745587 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [3, 1, 3] and d occurrences (as containment) of the composition, [1, 1, 6], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 14 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 2 13 2 14 2 14 2 13 2 14 + X1 X2 x - 2 X1 X2 x - X1 X2 x - 2 X1 X2 x + X1 x 2 13 14 2 13 13 14 2 11 - X1 X2 x + 2 X1 X2 x + X1 x + 2 X1 X2 x - X1 x + 2 X1 X2 x 13 2 10 2 11 11 2 10 - X1 x + 2 X1 X2 x - 2 X1 x - 3 X1 X2 x - 2 X1 x 10 11 11 10 10 11 2 8 10 - 3 X1 X2 x + 3 X1 x + X2 x + 3 X1 x + X2 x - x + X1 x - x 2 7 8 6 7 8 6 7 3 3 + X1 x - 2 X1 x - X1 X2 x - 2 X1 x + x + X1 x + x - X1 x + x 2 / 8 7 8 8 6 + x - 2 x + 1) / ((X1 X2 x + X1 X2 x - X1 x - X2 x - X1 X2 x / 7 7 8 6 7 5 4 5 3 4 3 - X1 x - X2 x + x + X1 x + x + X1 x + X1 x - x - X1 x - x + x 8 7 8 8 7 7 8 7 - 2 x + 1) (X1 X2 x + X1 X2 x - X1 x - X2 x - X1 x - X2 x + x + x 5 4 5 4 + X1 x + X1 x - x - x + x - 1)) and in Maple format -(X1^2*X2^2*x^14+X1^2*X2^2*x^13-2*X1^2*X2*x^14-X1*X2^2*x^14-2*X1^2*X2*x^13+X1^2 *x^14-X1*X2^2*x^13+2*X1*X2*x^14+X1^2*x^13+2*X1*X2*x^13-X1*x^14+2*X1^2*X2*x^11- X1*x^13+2*X1^2*X2*x^10-2*X1^2*x^11-3*X1*X2*x^11-2*X1^2*x^10-3*X1*X2*x^10+3*X1*x ^11+X2*x^11+3*X1*x^10+X2*x^10-x^11+X1^2*x^8-x^10+X1^2*x^7-2*X1*x^8-X1*X2*x^6-2* X1*x^7+x^8+X1*x^6+x^7-X1*x^3+x^3+x^2-2*x+1)/(X1*X2*x^8+X1*X2*x^7-X1*x^8-X2*x^8- X1*X2*x^6-X1*x^7-X2*x^7+x^8+X1*x^6+x^7+X1*x^5+X1*x^4-x^5-X1*x^3-x^4+x^3-2*x+1)/ (X1*X2*x^8+X1*X2*x^7-X1*x^8-X2*x^8-X1*x^7-X2*x^7+x^8+x^7+X1*x^5+X1*x^4-x^5-x^4+ x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 1, 3], are n 287 35 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 1, 6], are n 49 53 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 4 35 53 ------------- 1855 and in floating point 0.09287269204 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 4 1855 ate normal pair with correlation, --------- 1855 1/2 4 1855 1887 i.e. , [[---------, 0], [0, ----]] 1855 1855 ------------------------------------------------- Theorem Number, 164, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [3, 1, 3], nor the composition, [1, 2, 5] Then infinity ----- \ n ) a(n) x = / ----- n = 0 10 9 8 7 3 2 x + x - x - x - x - x + 2 x - 1 --------------------------------------------------------------------- 13 12 10 9 7 6 5 3 (-1 + x) (x + 2 x - 3 x - 2 x - x - 2 x - x + x - 2 x + 1) and in Maple format (x^10+x^9-x^8-x^7-x^3-x^2+2*x-1)/(-1+x)/(x^13+2*x^12-3*x^10-2*x^9-x^7-2*x^6-x^5 +x^3-2*x+1) The asymptotic expression for a(n) is, n 0.66081197302068638652 1.9169323599829195873 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [3, 1, 3] and d occurrences (as containment) of the composition, [1, 2, 5], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 12 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 12 2 12 2 12 12 2 10 - 2 X1 X2 x - X1 X2 x + X1 x + 2 X1 X2 x + 2 X1 X2 x 12 2 9 2 10 10 2 9 9 - X1 x + X1 X2 x - 2 X1 x - 3 X1 X2 x - X1 x - 2 X1 X2 x 10 10 2 8 9 9 10 2 7 7 + 3 X1 x + X2 x + X1 x + 2 X1 x + X2 x - x + X1 x - X1 X2 x 8 9 6 7 8 5 6 7 5 - 2 X1 x - x + X1 X2 x - X1 x + x - X1 X2 x - X1 x + x + X1 x 3 3 2 / 2 2 14 2 2 13 2 14 - X1 x + x + x - 2 x + 1) / (X1 X2 x + X1 X2 x - 2 X1 X2 x / 2 14 2 2 12 2 13 2 14 2 13 - 2 X1 X2 x - X1 X2 x - 2 X1 X2 x + X1 x - 2 X1 X2 x 14 2 14 2 12 2 13 2 12 + 4 X1 X2 x + X2 x + 4 X1 X2 x + X1 x + X1 X2 x 13 14 2 13 14 2 11 2 12 + 4 X1 X2 x - 2 X1 x + X2 x - 2 X2 x + 3 X1 X2 x - 3 X1 x 12 13 13 14 2 10 2 11 - 6 X1 X2 x - 2 X1 x - 2 X2 x + x - X1 X2 x - 3 X1 x 11 12 12 13 2 9 2 10 - 6 X1 X2 x + 5 X1 x + 2 X2 x + x - X1 X2 x + 2 X1 x 10 11 11 12 2 9 9 10 + X1 X2 x + 6 X1 x + 3 X2 x - 2 x + 3 X1 x + X1 X2 x - 3 X1 x 11 8 9 10 2 7 7 8 8 - 3 x - X1 X2 x - 5 X1 x + x - X1 x + X1 X2 x + X1 x + X2 x 9 6 7 8 5 6 7 5 6 + 2 x - 2 X1 X2 x + X1 x - x + X1 X2 x + X1 x - x - 2 X1 x + x 4 5 3 4 3 2 - X1 x + x + X1 x + x - x - 2 x + 3 x - 1) and in Maple format -(X1^2*X2^2*x^12-2*X1^2*X2*x^12-X1*X2^2*x^12+X1^2*x^12+2*X1*X2*x^12+2*X1^2*X2*x ^10-X1*x^12+X1^2*X2*x^9-2*X1^2*x^10-3*X1*X2*x^10-X1^2*x^9-2*X1*X2*x^9+3*X1*x^10 +X2*x^10+X1^2*x^8+2*X1*x^9+X2*x^9-x^10+X1^2*x^7-X1*X2*x^7-2*X1*x^8-x^9+X1*X2*x^ 6-X1*x^7+x^8-X1*X2*x^5-X1*x^6+x^7+X1*x^5-X1*x^3+x^3+x^2-2*x+1)/(X1^2*X2^2*x^14+ X1^2*X2^2*x^13-2*X1^2*X2*x^14-2*X1*X2^2*x^14-X1^2*X2^2*x^12-2*X1^2*X2*x^13+X1^2 *x^14-2*X1*X2^2*x^13+4*X1*X2*x^14+X2^2*x^14+4*X1^2*X2*x^12+X1^2*x^13+X1*X2^2*x^ 12+4*X1*X2*x^13-2*X1*x^14+X2^2*x^13-2*X2*x^14+3*X1^2*X2*x^11-3*X1^2*x^12-6*X1* X2*x^12-2*X1*x^13-2*X2*x^13+x^14-X1^2*X2*x^10-3*X1^2*x^11-6*X1*X2*x^11+5*X1*x^ 12+2*X2*x^12+x^13-X1^2*X2*x^9+2*X1^2*x^10+X1*X2*x^10+6*X1*x^11+3*X2*x^11-2*x^12 +3*X1^2*x^9+X1*X2*x^9-3*X1*x^10-3*x^11-X1*X2*x^8-5*X1*x^9+x^10-X1^2*x^7+X1*X2*x ^7+X1*x^8+X2*x^8+2*x^9-2*X1*X2*x^6+X1*x^7-x^8+X1*X2*x^5+X1*x^6-x^7-2*X1*x^5+x^6 -X1*x^4+x^5+X1*x^3+x^4-x^3-2*x^2+3*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 1, 3], are n 287 35 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 2, 5], are n 111 57 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 8 35 57 ------------- 1995 and in floating point 0.1791094652 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 8 1995 ate normal pair with correlation, --------- 1995 1/2 8 1995 2123 i.e. , [[---------, 0], [0, ----]] 1995 1995 ------------------------------------------------- Theorem Number, 165, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [3, 1, 3], nor the composition, [1, 3, 4] Then infinity ----- 2 7 6 5 4 3 2 \ n (x - x + 1) (x + x - x - 2 x - x + x + x - 1) ) a(n) x = - ---------------------------------------------------------- / 10 9 8 7 6 5 3 ----- (-1 + x) (x + 2 x + x + x + 2 x + x - x + 2 x - 1) n = 0 and in Maple format -(x^2-x+1)*(x^7+x^6-x^5-2*x^4-x^3+x^2+x-1)/(-1+x)/(x^10+2*x^9+x^8+x^7+2*x^6+x^5 -x^3+2*x-1) The asymptotic expression for a(n) is, n 0.64343319290120806019 1.9224267923147665937 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [3, 1, 3] and d occurrences (as containment) of the composition, [1, 3, 4], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 9 2 8 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x + X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 9 9 8 9 9 2 7 7 - X1 x - 2 X1 X2 x - 2 X1 X2 x + 2 X1 x + X2 x + X1 x - X1 X2 x 8 9 7 5 7 4 5 4 3 + X2 x - x - X1 x + X1 X2 x + x - X1 X2 x - X1 x + X1 x - X1 x 3 2 / 2 11 2 10 2 11 + x + x - 2 x + 1) / (X1 X2 x + 2 X1 X2 x - X1 x / 11 2 10 10 11 11 2 8 - 2 X1 X2 x - X1 x - 4 X1 X2 x + 2 X1 x + X2 x - X1 X2 x 2 9 9 10 10 11 2 8 8 + 2 X1 x - X1 X2 x + 2 X1 x + 2 X2 x - x + X1 x + X1 X2 x 9 9 10 2 7 7 8 9 7 - 3 X1 x + X2 x - x - X1 x + X1 X2 x - X1 x + x + X1 x 5 6 7 4 5 6 4 5 3 - 2 X1 X2 x - X1 x - x + X1 X2 x + X1 x + x - 2 X1 x + x + X1 x 4 3 2 + x - x - 2 x + 3 x - 1) and in Maple format -(X1^2*X2*x^9+X1^2*X2*x^8-X1^2*x^9-2*X1*X2*x^9-2*X1*X2*x^8+2*X1*x^9+X2*x^9+X1^2 *x^7-X1*X2*x^7+X2*x^8-x^9-X1*x^7+X1*X2*x^5+x^7-X1*X2*x^4-X1*x^5+X1*x^4-X1*x^3+x ^3+x^2-2*x+1)/(X1^2*X2*x^11+2*X1^2*X2*x^10-X1^2*x^11-2*X1*X2*x^11-X1^2*x^10-4* X1*X2*x^10+2*X1*x^11+X2*x^11-X1^2*X2*x^8+2*X1^2*x^9-X1*X2*x^9+2*X1*x^10+2*X2*x^ 10-x^11+X1^2*x^8+X1*X2*x^8-3*X1*x^9+X2*x^9-x^10-X1^2*x^7+X1*X2*x^7-X1*x^8+x^9+ X1*x^7-2*X1*X2*x^5-X1*x^6-x^7+X1*X2*x^4+X1*x^5+x^6-2*X1*x^4+x^5+X1*x^3+x^4-x^3-\ 2*x^2+3*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 1, 3], are n 287 35 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 3, 4], are n 135 65 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 16 7 13 ------------- 455 and in floating point 0.3354511477 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 16 91 ate normal pair with correlation, -------- 455 1/2 16 91 2787 i.e. , [[--------, 0], [0, ----]] 455 2275 ------------------------------------------------- Theorem Number, 166, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [3, 1, 3], nor the composition, [1, 4, 3] Then infinity ----- 2 7 6 5 4 3 2 \ n (x - x + 1) (x + x - x - 2 x - x + x + x - 1) ) a(n) x = - ---------------------------------------------------------- / 10 9 8 7 6 5 3 ----- (-1 + x) (x + 2 x + x + x + 2 x + x - x + 2 x - 1) n = 0 and in Maple format -(x^2-x+1)*(x^7+x^6-x^5-2*x^4-x^3+x^2+x-1)/(-1+x)/(x^10+2*x^9+x^8+x^7+2*x^6+x^5 -x^3+2*x-1) The asymptotic expression for a(n) is, n 0.64343319290120806019 1.9224267923147665937 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [3, 1, 3] and d occurrences (as containment) of the composition, [1, 4, 3], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 9 2 8 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x + X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 9 9 8 9 9 2 7 7 - X1 x - 2 X1 X2 x - 2 X1 X2 x + 2 X1 x + X2 x + X1 x - X1 X2 x 8 9 7 5 7 4 5 4 3 + X2 x - x - X1 x + X1 X2 x + x - X1 X2 x - X1 x + X1 x - X1 x 3 2 / 2 11 2 10 2 11 + x + x - 2 x + 1) / (X1 X2 x + 2 X1 X2 x - X1 x / 11 2 10 10 11 11 2 8 - 2 X1 X2 x - X1 x - 4 X1 X2 x + 2 X1 x + X2 x - X1 X2 x 2 9 9 10 10 11 2 8 8 + 2 X1 x - X1 X2 x + 2 X1 x + 2 X2 x - x + X1 x + X1 X2 x 9 9 10 2 7 7 8 9 7 - 3 X1 x + X2 x - x - X1 x + X1 X2 x - X1 x + x + X1 x 5 6 7 4 5 6 4 5 3 - 2 X1 X2 x - X1 x - x + X1 X2 x + X1 x + x - 2 X1 x + x + X1 x 4 3 2 + x - x - 2 x + 3 x - 1) and in Maple format -(X1^2*X2*x^9+X1^2*X2*x^8-X1^2*x^9-2*X1*X2*x^9-2*X1*X2*x^8+2*X1*x^9+X2*x^9+X1^2 *x^7-X1*X2*x^7+X2*x^8-x^9-X1*x^7+X1*X2*x^5+x^7-X1*X2*x^4-X1*x^5+X1*x^4-X1*x^3+x ^3+x^2-2*x+1)/(X1^2*X2*x^11+2*X1^2*X2*x^10-X1^2*x^11-2*X1*X2*x^11-X1^2*x^10-4* X1*X2*x^10+2*X1*x^11+X2*x^11-X1^2*X2*x^8+2*X1^2*x^9-X1*X2*x^9+2*X1*x^10+2*X2*x^ 10-x^11+X1^2*x^8+X1*X2*x^8-3*X1*x^9+X2*x^9-x^10-X1^2*x^7+X1*X2*x^7-X1*x^8+x^9+ X1*x^7-2*X1*X2*x^5-X1*x^6-x^7+X1*X2*x^4+X1*x^5+x^6-2*X1*x^4+x^5+X1*x^3+x^4-x^3-\ 2*x^2+3*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 1, 3], are n 287 35 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 4, 3], are n 135 65 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 16 7 13 ------------- 455 and in floating point 0.3354511477 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 16 91 ate normal pair with correlation, -------- 455 1/2 16 91 2787 i.e. , [[--------, 0], [0, ----]] 455 2275 ------------------------------------------------- Theorem Number, 167, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [3, 1, 3], nor the composition, [1, 5, 2] Then infinity ----- \ n ) a(n) x = / ----- n = 0 16 15 14 12 11 10 9 8 7 3 2 x + 2 x + x + x + x - 2 x - x + x + x + x + x - 2 x + 1 - ------------------------------------------------------------------------ 13 12 10 9 7 6 5 3 (-1 + x) (x + 2 x - 3 x - 2 x - x - 2 x - x + x - 2 x + 1) and in Maple format -(x^16+2*x^15+x^14+x^12+x^11-2*x^10-x^9+x^8+x^7+x^3+x^2-2*x+1)/(-1+x)/(x^13+2*x ^12-3*x^10-2*x^9-x^7-2*x^6-x^5+x^3-2*x+1) The asymptotic expression for a(n) is, n 0.66072243088480701000 1.9169323599829195873 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [3, 1, 3] and d occurrences (as containment) of the composition, [1, 5, 2], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 17 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 2 16 2 17 2 17 2 2 15 2 16 + X1 X2 x - 2 X1 X2 x - 2 X1 X2 x - X1 X2 x - 2 X1 X2 x 2 17 2 16 17 2 17 2 15 2 16 + X1 x - 2 X1 X2 x + 4 X1 X2 x + X2 x + 3 X1 X2 x + X1 x 2 15 16 17 2 16 17 2 2 13 + X1 X2 x + 4 X1 X2 x - 2 X1 x + X2 x - 2 X2 x + X1 X2 x 2 14 2 15 15 16 16 17 + X1 X2 x - 2 X1 x - 4 X1 X2 x - 2 X1 x - 2 X2 x + x 2 2 12 2 13 2 14 2 13 14 - X1 X2 x - 3 X1 X2 x - X1 x - X1 X2 x - 2 X1 X2 x 15 15 16 2 12 2 13 2 12 + 3 X1 x + X2 x + x + 2 X1 X2 x + 2 X1 x + X1 X2 x 13 14 14 15 2 11 2 12 + 4 X1 X2 x + 2 X1 x + X2 x - x + 2 X1 X2 x - X1 x 12 13 13 14 2 10 2 11 - 2 X1 X2 x - 3 X1 x - X2 x - x - X1 X2 x - 2 X1 x 11 12 13 2 9 2 10 10 - 5 X1 X2 x + X1 x + x - X1 X2 x + X1 x + 2 X1 X2 x 11 11 2 9 9 10 10 11 + 5 X1 x + 3 X2 x + 2 X1 x + 2 X1 X2 x - 2 X1 x - X2 x - 3 x 8 9 9 10 2 7 7 8 9 - X1 X2 x - 4 X1 x - X2 x + x - X1 x + 2 X1 X2 x + X1 x + 2 x 6 5 6 7 5 4 3 4 2 - 2 X1 X2 x + X1 X2 x + 2 X1 x - x - X1 x - X1 x + X1 x + x - 3 x / 2 2 14 2 2 13 2 14 + 3 x - 1) / ((-1 + x) (X1 X2 x + X1 X2 x - 2 X1 X2 x / 2 14 2 2 12 2 13 2 14 2 13 - 2 X1 X2 x - X1 X2 x - 2 X1 X2 x + X1 x - 2 X1 X2 x 14 2 14 2 12 2 13 2 12 + 4 X1 X2 x + X2 x + 4 X1 X2 x + X1 x + X1 X2 x 13 14 2 13 14 2 11 2 12 + 4 X1 X2 x - 2 X1 x + X2 x - 2 X2 x + 3 X1 X2 x - 3 X1 x 12 13 13 14 2 10 2 11 - 6 X1 X2 x - 2 X1 x - 2 X2 x + x - X1 X2 x - 3 X1 x 11 12 12 13 2 9 2 10 - 6 X1 X2 x + 5 X1 x + 2 X2 x + x - X1 X2 x + 2 X1 x 10 11 11 12 2 9 9 10 + X1 X2 x + 6 X1 x + 3 X2 x - 2 x + 3 X1 x + X1 X2 x - 3 X1 x 11 8 9 10 2 7 7 8 8 - 3 x - X1 X2 x - 5 X1 x + x - X1 x + X1 X2 x + X1 x + X2 x 9 6 7 8 5 6 7 5 6 + 2 x - 2 X1 X2 x + X1 x - x + X1 X2 x + X1 x - x - 2 X1 x + x 4 5 3 4 3 2 - X1 x + x + X1 x + x - x - 2 x + 3 x - 1)) and in Maple format -(X1^2*X2^2*x^17+X1^2*X2^2*x^16-2*X1^2*X2*x^17-2*X1*X2^2*x^17-X1^2*X2^2*x^15-2* X1^2*X2*x^16+X1^2*x^17-2*X1*X2^2*x^16+4*X1*X2*x^17+X2^2*x^17+3*X1^2*X2*x^15+X1^ 2*x^16+X1*X2^2*x^15+4*X1*X2*x^16-2*X1*x^17+X2^2*x^16-2*X2*x^17+X1^2*X2^2*x^13+ X1^2*X2*x^14-2*X1^2*x^15-4*X1*X2*x^15-2*X1*x^16-2*X2*x^16+x^17-X1^2*X2^2*x^12-3 *X1^2*X2*x^13-X1^2*x^14-X1*X2^2*x^13-2*X1*X2*x^14+3*X1*x^15+X2*x^15+x^16+2*X1^2 *X2*x^12+2*X1^2*x^13+X1*X2^2*x^12+4*X1*X2*x^13+2*X1*x^14+X2*x^14-x^15+2*X1^2*X2 *x^11-X1^2*x^12-2*X1*X2*x^12-3*X1*x^13-X2*x^13-x^14-X1^2*X2*x^10-2*X1^2*x^11-5* X1*X2*x^11+X1*x^12+x^13-X1^2*X2*x^9+X1^2*x^10+2*X1*X2*x^10+5*X1*x^11+3*X2*x^11+ 2*X1^2*x^9+2*X1*X2*x^9-2*X1*x^10-X2*x^10-3*x^11-X1*X2*x^8-4*X1*x^9-X2*x^9+x^10- X1^2*x^7+2*X1*X2*x^7+X1*x^8+2*x^9-2*X1*X2*x^6+X1*X2*x^5+2*X1*x^6-x^7-X1*x^5-X1* x^4+X1*x^3+x^4-3*x^2+3*x-1)/(-1+x)/(X1^2*X2^2*x^14+X1^2*X2^2*x^13-2*X1^2*X2*x^ 14-2*X1*X2^2*x^14-X1^2*X2^2*x^12-2*X1^2*X2*x^13+X1^2*x^14-2*X1*X2^2*x^13+4*X1* X2*x^14+X2^2*x^14+4*X1^2*X2*x^12+X1^2*x^13+X1*X2^2*x^12+4*X1*X2*x^13-2*X1*x^14+ X2^2*x^13-2*X2*x^14+3*X1^2*X2*x^11-3*X1^2*x^12-6*X1*X2*x^12-2*X1*x^13-2*X2*x^13 +x^14-X1^2*X2*x^10-3*X1^2*x^11-6*X1*X2*x^11+5*X1*x^12+2*X2*x^12+x^13-X1^2*X2*x^ 9+2*X1^2*x^10+X1*X2*x^10+6*X1*x^11+3*X2*x^11-2*x^12+3*X1^2*x^9+X1*X2*x^9-3*X1*x ^10-3*x^11-X1*X2*x^8-5*X1*x^9+x^10-X1^2*x^7+X1*X2*x^7+X1*x^8+X2*x^8+2*x^9-2*X1* X2*x^6+X1*x^7-x^8+X1*X2*x^5+X1*x^6-x^7-2*X1*x^5+x^6-X1*x^4+x^5+X1*x^3+x^4-x^3-2 *x^2+3*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 1, 3], are n 287 35 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 5, 2], are n 111 57 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 8 35 57 ------------- 1995 and in floating point 0.1791094652 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 8 1995 ate normal pair with correlation, --------- 1995 1/2 8 1995 2123 i.e. , [[---------, 0], [0, ----]] 1995 1995 ------------------------------------------------- Theorem Number, 168, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [3, 1, 3], nor the composition, [1, 6, 1] Then infinity ----- \ n 18 17 16 15 14 13 12 11 ) a(n) x = - (x + 3 x + 4 x + 3 x + x + x + 2 x - x / ----- n = 0 10 8 7 3 2 / - 2 x + x + x + x + x - 2 x + 1) / ( / 7 6 5 4 2 7 6 5 4 2 (x + 2 x + 2 x + x + x + x - 1) (x + 2 x + 2 x + x + 1) (-1 + x) ) and in Maple format -(x^18+3*x^17+4*x^16+3*x^15+x^14+x^13+2*x^12-x^11-2*x^10+x^8+x^7+x^3+x^2-2*x+1) /(x^7+2*x^6+2*x^5+x^4+x^2+x-1)/(x^7+2*x^6+2*x^5+x^4+1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.67485833261041293122 1.9131864632213745587 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [3, 1, 3] and d occurrences (as containment) of the composition, [1, 6, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 19 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 2 18 2 19 2 19 2 18 2 19 + 2 X1 X2 x - 2 X1 X2 x - 2 X1 X2 x - 4 X1 X2 x + X1 x 2 18 19 2 19 2 2 16 2 18 - 4 X1 X2 x + 4 X1 X2 x + X2 x - X1 X2 x + 2 X1 x 2 17 18 19 2 18 19 2 2 15 - X1 X2 x + 8 X1 X2 x - 2 X1 x + 2 X2 x - 2 X2 x + X1 X2 x 2 16 2 16 17 18 2 17 18 + 3 X1 X2 x + X1 X2 x + 2 X1 X2 x - 4 X1 x + X2 x - 4 X2 x 19 2 16 2 15 16 17 17 18 + x - 2 X1 x - X1 X2 x - 4 X1 X2 x - X1 x - 2 X2 x + 2 x 2 2 13 2 15 15 16 16 17 - X1 X2 x - X1 x - 2 X1 X2 x + 3 X1 x + X2 x + x 2 13 2 13 15 15 16 2 12 + X1 X2 x + X1 X2 x + 3 X1 x + 2 X2 x - x + 2 X1 X2 x 15 2 12 12 13 13 2 10 - 2 x - 2 X1 x - 5 X1 X2 x - X1 x - X2 x - 2 X1 X2 x 11 12 12 13 2 10 10 11 - X1 X2 x + 5 X1 x + 3 X2 x + x + 2 X1 x + 4 X1 X2 x + X1 x 11 12 2 9 10 10 11 9 10 + X2 x - 3 x + X1 x - 4 X1 x - 2 X2 x - x - 2 X1 x + 2 x 2 7 7 9 6 7 6 7 4 3 - X1 x - X1 X2 x + x + X1 X2 x + 3 X1 x - X1 x - x - X1 x + X1 x 4 2 / 8 7 8 8 + x - 3 x + 3 x - 1) / ((-1 + x) (X1 X2 x + X1 X2 x - X1 x - X2 x / 6 7 7 8 6 7 5 4 5 3 - X1 X2 x - X1 x - X2 x + x + X1 x + x + X1 x + X1 x - x - X1 x 4 3 8 7 8 8 7 7 - x + x - 2 x + 1) (X1 X2 x + X1 X2 x - X1 x - X2 x - X1 x - X2 x 8 7 5 4 5 4 + x + x + X1 x + X1 x - x - x + x - 1)) and in Maple format -(X1^2*X2^2*x^19+2*X1^2*X2^2*x^18-2*X1^2*X2*x^19-2*X1*X2^2*x^19-4*X1^2*X2*x^18+ X1^2*x^19-4*X1*X2^2*x^18+4*X1*X2*x^19+X2^2*x^19-X1^2*X2^2*x^16+2*X1^2*x^18-X1* X2^2*x^17+8*X1*X2*x^18-2*X1*x^19+2*X2^2*x^18-2*X2*x^19+X1^2*X2^2*x^15+3*X1^2*X2 *x^16+X1*X2^2*x^16+2*X1*X2*x^17-4*X1*x^18+X2^2*x^17-4*X2*x^18+x^19-2*X1^2*x^16- X1*X2^2*x^15-4*X1*X2*x^16-X1*x^17-2*X2*x^17+2*x^18-X1^2*X2^2*x^13-X1^2*x^15-2* X1*X2*x^15+3*X1*x^16+X2*x^16+x^17+X1^2*X2*x^13+X1*X2^2*x^13+3*X1*x^15+2*X2*x^15 -x^16+2*X1^2*X2*x^12-2*x^15-2*X1^2*x^12-5*X1*X2*x^12-X1*x^13-X2*x^13-2*X1^2*X2* x^10-X1*X2*x^11+5*X1*x^12+3*X2*x^12+x^13+2*X1^2*x^10+4*X1*X2*x^10+X1*x^11+X2*x^ 11-3*x^12+X1^2*x^9-4*X1*x^10-2*X2*x^10-x^11-2*X1*x^9+2*x^10-X1^2*x^7-X1*X2*x^7+ x^9+X1*X2*x^6+3*X1*x^7-X1*x^6-x^7-X1*x^4+X1*x^3+x^4-3*x^2+3*x-1)/(-1+x)/(X1*X2* x^8+X1*X2*x^7-X1*x^8-X2*x^8-X1*X2*x^6-X1*x^7-X2*x^7+x^8+X1*x^6+x^7+X1*x^5+X1*x^ 4-x^5-X1*x^3-x^4+x^3-2*x+1)/(X1*X2*x^8+X1*X2*x^7-X1*x^8-X2*x^8-X1*x^7-X2*x^7+x^ 8+x^7+X1*x^5+X1*x^4-x^5-x^4+x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 1, 3], are n 287 35 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 6, 1], are n 49 53 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 4 35 53 ------------- 1855 and in floating point 0.09287269204 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 4 1855 ate normal pair with correlation, --------- 1855 1/2 4 1855 1887 i.e. , [[---------, 0], [0, ----]] 1855 1855 ------------------------------------------------- Theorem Number, 169, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [3, 1, 3], nor the composition, [2, 1, 5] Then infinity ----- 10 8 7 3 2 \ n x - x - x - x - x + 2 x - 1 ) a(n) x = ---------------------------------------------------- / 7 5 4 3 7 5 4 ----- (x - x - x + x - 2 x + 1) (x - x - x + x - 1) n = 0 and in Maple format (x^10-x^8-x^7-x^3-x^2+2*x-1)/(x^7-x^5-x^4+x^3-2*x+1)/(x^7-x^5-x^4+x-1) The asymptotic expression for a(n) is, n 0.64476247294249366516 1.9235403491125894929 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [3, 1, 3] and d occurrences (as containment) of the composition, [2, 1, 5], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 12 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 2 11 2 12 2 12 2 11 2 12 + X1 X2 x - 2 X1 X2 x - X1 X2 x - 2 X1 X2 x + X1 x 12 2 10 2 11 12 2 9 2 10 + 2 X1 X2 x + 2 X1 X2 x + X1 x - X1 x + 2 X1 X2 x - 2 X1 x 10 2 9 9 10 10 2 8 - 3 X1 X2 x - 2 X1 x - 2 X1 X2 x + 3 X1 x + X2 x + X1 x 9 10 2 7 8 7 8 5 7 5 + 2 X1 x - x + X1 x - 2 X1 x - 2 X1 x + x - X1 X2 x + x + X1 x 3 3 2 / 7 6 7 7 - X1 x + x + x - 2 x + 1) / ((X1 X2 x + X1 X2 x - X1 x - X2 x / 5 6 7 5 4 5 3 4 3 - X1 X2 x - X1 x + x + 2 X1 x + X1 x - x - X1 x - x + x - 2 x + 1 7 6 7 7 6 7 5 4 5 ) (X1 X2 x + X1 X2 x - X1 x - X2 x - X1 x + x + X1 x + X1 x - x 4 - x + x - 1)) and in Maple format -(X1^2*X2^2*x^12+X1^2*X2^2*x^11-2*X1^2*X2*x^12-X1*X2^2*x^12-2*X1^2*X2*x^11+X1^2 *x^12+2*X1*X2*x^12+2*X1^2*X2*x^10+X1^2*x^11-X1*x^12+2*X1^2*X2*x^9-2*X1^2*x^10-3 *X1*X2*x^10-2*X1^2*x^9-2*X1*X2*x^9+3*X1*x^10+X2*x^10+X1^2*x^8+2*X1*x^9-x^10+X1^ 2*x^7-2*X1*x^8-2*X1*x^7+x^8-X1*X2*x^5+x^7+X1*x^5-X1*x^3+x^3+x^2-2*x+1)/(X1*X2*x ^7+X1*X2*x^6-X1*x^7-X2*x^7-X1*X2*x^5-X1*x^6+x^7+2*X1*x^5+X1*x^4-x^5-X1*x^3-x^4+ x^3-2*x+1)/(X1*X2*x^7+X1*X2*x^6-X1*x^7-X2*x^7-X1*x^6+x^7+X1*x^5+X1*x^4-x^5-x^4+ x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 1, 3], are n 287 35 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 1, 5], are n 113 57 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 2 35 57 ------------- 285 and in floating point 0.3134415640 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 1/2 2 7 285 ate normal pair with correlation, ------------- 285 1/2 1/2 2 7 285 341 i.e. , [[-------------, 0], [0, ---]] 285 285 ------------------------------------------------- Theorem Number, 170, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [3, 1, 3], nor the composition, [2, 2, 4] Then infinity ----- \ n ) a(n) x = / ----- n = 0 9 8 7 3 2 x - x - x - x - x + 2 x - 1 ----------------------------------------------------------------- 12 11 9 8 7 6 5 4 3 2 x - 2 x + 3 x - x - x + x + x + x - x - 2 x + 3 x - 1 and in Maple format (x^9-x^8-x^7-x^3-x^2+2*x-1)/(x^12-2*x^11+3*x^9-x^8-x^7+x^6+x^5+x^4-x^3-2*x^2+3* x-1) The asymptotic expression for a(n) is, n 0.63439518060791917574 1.9265719569048272535 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [3, 1, 3] and d occurrences (as containment) of the composition, [2, 2, 4], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 10 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 10 2 10 2 9 2 10 10 - 2 X1 X2 x - X1 X2 x + 2 X1 X2 x + X1 x + 2 X1 X2 x 2 8 2 9 9 10 8 9 9 + X1 X2 x - 2 X1 x - 3 X1 X2 x - X1 x - X1 X2 x + 3 X1 x + X2 x 2 7 8 9 6 7 8 5 6 7 + X1 x - X1 x - x - X1 X2 x - 2 X1 x + x + X1 X2 x + X1 x + x 4 5 4 3 3 2 / 2 2 12 - X1 X2 x - X1 x + X1 x - X1 x + x + x - 2 x + 1) / (X1 X2 x / 2 2 11 2 12 2 12 2 2 10 2 12 + X1 X2 x - 2 X1 X2 x - 2 X1 X2 x - X1 X2 x + X1 x 2 11 12 2 12 2 10 2 11 2 10 - X1 X2 x + 4 X1 X2 x + X2 x + 5 X1 X2 x - X1 x + X1 X2 x 11 12 12 2 9 2 10 10 - 2 X1 X2 x - 2 X1 x - 2 X2 x - X1 X2 x - 3 X1 x - 6 X1 X2 x 11 11 12 2 8 2 9 9 10 + 3 X1 x + 2 X2 x + x - X1 X2 x + 3 X1 x + 2 X1 X2 x + 3 X1 x 10 11 2 8 9 9 2 7 7 8 + X2 x - 2 x + X1 x - 6 X1 x - X2 x - X1 x - X1 X2 x + X2 x 9 6 7 8 5 6 7 4 + 3 x + X1 X2 x + 3 X1 x - x - 2 X1 X2 x - 2 X1 x - x + X1 X2 x 5 6 4 5 3 4 3 2 + X1 x + x - 2 X1 x + x + X1 x + x - x - 2 x + 3 x - 1) and in Maple format -(X1^2*X2^2*x^10-2*X1^2*X2*x^10-X1*X2^2*x^10+2*X1^2*X2*x^9+X1^2*x^10+2*X1*X2*x^ 10+X1^2*X2*x^8-2*X1^2*x^9-3*X1*X2*x^9-X1*x^10-X1*X2*x^8+3*X1*x^9+X2*x^9+X1^2*x^ 7-X1*x^8-x^9-X1*X2*x^6-2*X1*x^7+x^8+X1*X2*x^5+X1*x^6+x^7-X1*X2*x^4-X1*x^5+X1*x^ 4-X1*x^3+x^3+x^2-2*x+1)/(X1^2*X2^2*x^12+X1^2*X2^2*x^11-2*X1^2*X2*x^12-2*X1*X2^2 *x^12-X1^2*X2^2*x^10+X1^2*x^12-X1*X2^2*x^11+4*X1*X2*x^12+X2^2*x^12+5*X1^2*X2*x^ 10-X1^2*x^11+X1*X2^2*x^10-2*X1*X2*x^11-2*X1*x^12-2*X2*x^12-X1^2*X2*x^9-3*X1^2*x ^10-6*X1*X2*x^10+3*X1*x^11+2*X2*x^11+x^12-X1^2*X2*x^8+3*X1^2*x^9+2*X1*X2*x^9+3* X1*x^10+X2*x^10-2*x^11+X1^2*x^8-6*X1*x^9-X2*x^9-X1^2*x^7-X1*X2*x^7+X2*x^8+3*x^9 +X1*X2*x^6+3*X1*x^7-x^8-2*X1*X2*x^5-2*X1*x^6-x^7+X1*X2*x^4+X1*x^5+x^6-2*X1*x^4+ x^5+X1*x^3+x^4-x^3-2*x^2+3*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 1, 3], are n 287 35 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 2, 4], are n 75 69 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 6 35 69 ------------- 805 and in floating point 0.3662806174 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 1/2 6 3 805 ate normal pair with correlation, ------------- 805 1/2 1/2 6 3 805 1021 i.e. , [[-------------, 0], [0, ----]] 805 805 ------------------------------------------------- Theorem Number, 171, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [3, 1, 3], nor the composition, [2, 3, 3] Then infinity ----- 7 3 2 \ n x + x + x - 2 x + 1 ) a(n) x = - -------------------------------------------- / 9 7 6 5 4 3 2 ----- x - x + x + x + x - x - 2 x + 3 x - 1 n = 0 and in Maple format -(x^7+x^3+x^2-2*x+1)/(x^9-x^7+x^6+x^5+x^4-x^3-2*x^2+3*x-1) The asymptotic expression for a(n) is, n 0.62654038626736234360 1.9293902847278627598 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [3, 1, 3] and d occurrences (as containment) of the composition, [2, 3, 3], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 8 2 7 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x + X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 8 7 8 6 7 6 7 4 - 2 X1 X2 x - X1 X2 x + X2 x - X1 X2 x - X1 x + X1 x + x + X1 X2 x 3 4 3 2 / 2 10 2 9 - X1 X2 x - X1 x + x + x - 2 x + 1) / (X1 X2 x + 2 X1 X2 x / 10 9 10 2 7 9 9 9 - 2 X1 X2 x - 3 X1 X2 x + X2 x - X1 X2 x - X1 x + X2 x + x 6 7 6 7 4 5 6 3 + X1 X2 x + 2 X1 x - 2 X1 x - x - 2 X1 X2 x - X1 x + x + X1 X2 x 4 5 4 3 2 + X1 x + x + x - x - 2 x + 3 x - 1) and in Maple format -(X1^2*X2*x^8+X1^2*X2*x^7-2*X1*X2*x^8-X1*X2*x^7+X2*x^8-X1*X2*x^6-X1*x^7+X1*x^6+ x^7+X1*X2*x^4-X1*X2*x^3-X1*x^4+x^3+x^2-2*x+1)/(X1^2*X2*x^10+2*X1^2*X2*x^9-2*X1* X2*x^10-3*X1*X2*x^9+X2*x^10-X1^2*X2*x^7-X1*x^9+X2*x^9+x^9+X1*X2*x^6+2*X1*x^7-2* X1*x^6-x^7-2*X1*X2*x^4-X1*x^5+x^6+X1*X2*x^3+X1*x^4+x^5+x^4-x^3-2*x^2+3*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 1, 3], are n 287 35 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 3, 3], are n 97 85 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 26 7 17 ------------- 595 and in floating point 0.4766832187 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 26 119 ate normal pair with correlation, --------- 595 1/2 26 119 4327 i.e. , [[---------, 0], [0, ----]] 595 2975 ------------------------------------------------- Theorem Number, 172, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [3, 1, 3], nor the composition, [2, 4, 2] Then infinity ----- \ n ) a(n) x = / ----- n = 0 14 11 9 8 7 3 2 x + x - 2 x + x + x + x + x - 2 x + 1 - ----------------------------------------------------------------------- 12 11 10 9 8 7 6 5 4 3 2 x - 2 x - x + 3 x - x - x + x + x + x - x - 2 x + 3 x - 1 and in Maple format -(x^14+x^11-2*x^9+x^8+x^7+x^3+x^2-2*x+1)/(x^12-2*x^11-x^10+3*x^9-x^8-x^7+x^6+x^ 5+x^4-x^3-2*x^2+3*x-1) The asymptotic expression for a(n) is, n 0.64504985534481915960 1.9219536789860435545 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [3, 1, 3] and d occurrences (as containment) of the composition, [2, 4, 2], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 3 2 14 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 2 15 3 14 2 2 14 2 15 2 15 + X1 X2 x - 2 X1 X2 x - 2 X1 X2 x - 2 X1 X2 x - 2 X1 X2 x 3 13 3 14 2 2 13 2 14 2 15 2 14 + X1 X2 x + X1 x - X1 X2 x + 5 X1 X2 x + X1 x + X1 X2 x 15 2 15 3 13 2 14 2 13 14 + 4 X1 X2 x + X2 x - X1 x - 3 X1 x + X1 X2 x - 4 X1 X2 x 15 15 2 2 11 2 12 2 13 13 - 2 X1 x - 2 X2 x + X1 X2 x - X1 X2 x + X1 x - X1 X2 x 14 14 15 2 2 10 2 11 2 12 + 3 X1 x + X2 x + x - X1 X2 x - X1 X2 x + X1 x 2 11 12 14 2 10 2 10 12 - X1 X2 x + 2 X1 X2 x - x + 3 X1 X2 x + X1 X2 x - 2 X1 x 12 2 9 2 10 10 11 11 12 - X2 x - X1 X2 x - 2 X1 x - 5 X1 X2 x + X1 x + X2 x + x 2 8 2 9 9 10 10 11 2 8 - X1 X2 x + 2 X1 x + 3 X1 X2 x + 4 X1 x + 2 X2 x - x + X1 x 8 9 9 10 2 7 7 8 9 + X1 X2 x - 5 X1 x - 2 X2 x - 2 x - X1 x - X1 X2 x - X1 x + 3 x 6 7 5 6 7 4 5 + 2 X1 X2 x + 3 X1 x - 2 X1 X2 x - 2 X1 x - x + X1 X2 x + 2 X1 x 4 3 4 2 / 3 2 16 3 2 15 - 2 X1 x + X1 x + x - 3 x + 3 x - 1) / (X1 X2 x + X1 X2 x / 3 16 2 2 16 3 2 14 3 15 3 16 - 2 X1 X2 x - 2 X1 X2 x - X1 X2 x - X1 X2 x + X1 x 2 2 15 2 16 2 16 3 14 2 2 14 - 2 X1 X2 x + 4 X1 X2 x + X1 X2 x + 3 X1 X2 x + X1 X2 x 2 15 2 16 2 15 16 3 13 + 2 X1 X2 x - 2 X1 x + X1 X2 x - 2 X1 X2 x - X1 X2 x 3 14 2 2 13 2 14 15 16 3 13 - 2 X1 x + X1 X2 x - 4 X1 X2 x - X1 X2 x + X1 x + X1 x 2 14 2 13 14 2 2 11 2 12 + 3 X1 x - 2 X1 X2 x + X1 X2 x - 2 X1 X2 x + 2 X1 X2 x 2 13 2 12 13 14 2 13 2 2 10 - X1 x + X1 X2 x + 3 X1 X2 x - X1 x + X2 x + X1 X2 x 2 11 2 12 2 11 12 13 2 12 + 3 X1 X2 x - 2 X1 x + 2 X1 X2 x - 6 X1 X2 x - X1 x - X2 x 13 2 10 2 10 11 12 12 - 2 X2 x - 5 X1 X2 x - X1 X2 x - 3 X1 X2 x + 5 X1 x + 4 X2 x 13 2 10 10 11 12 2 8 2 9 + x + 5 X1 x + 8 X1 X2 x - X1 x - 3 x + X1 X2 x - 2 X1 x 9 10 10 11 2 8 8 9 - 2 X1 X2 x - 9 X1 x - 3 X2 x + x - 2 X1 x - X1 X2 x + 6 X1 x 9 10 2 7 7 8 8 9 + 2 X2 x + 4 x + X1 x + 2 X1 X2 x + 3 X1 x - X2 x - 4 x 6 7 5 6 7 4 5 - 3 X1 X2 x - 5 X1 x + 3 X1 X2 x + 3 X1 x + 2 x - X1 X2 x - 3 X1 x 4 3 4 3 2 + 3 X1 x - X1 x - 2 x - x + 5 x - 4 x + 1) and in Maple format -(X1^3*X2^2*x^14+X1^2*X2^2*x^15-2*X1^3*X2*x^14-2*X1^2*X2^2*x^14-2*X1^2*X2*x^15-\ 2*X1*X2^2*x^15+X1^3*X2*x^13+X1^3*x^14-X1^2*X2^2*x^13+5*X1^2*X2*x^14+X1^2*x^15+ X1*X2^2*x^14+4*X1*X2*x^15+X2^2*x^15-X1^3*x^13-3*X1^2*x^14+X1*X2^2*x^13-4*X1*X2* x^14-2*X1*x^15-2*X2*x^15+X1^2*X2^2*x^11-X1^2*X2*x^12+X1^2*x^13-X1*X2*x^13+3*X1* x^14+X2*x^14+x^15-X1^2*X2^2*x^10-X1^2*X2*x^11+X1^2*x^12-X1*X2^2*x^11+2*X1*X2*x^ 12-x^14+3*X1^2*X2*x^10+X1*X2^2*x^10-2*X1*x^12-X2*x^12-X1^2*X2*x^9-2*X1^2*x^10-5 *X1*X2*x^10+X1*x^11+X2*x^11+x^12-X1^2*X2*x^8+2*X1^2*x^9+3*X1*X2*x^9+4*X1*x^10+2 *X2*x^10-x^11+X1^2*x^8+X1*X2*x^8-5*X1*x^9-2*X2*x^9-2*x^10-X1^2*x^7-X1*X2*x^7-X1 *x^8+3*x^9+2*X1*X2*x^6+3*X1*x^7-2*X1*X2*x^5-2*X1*x^6-x^7+X1*X2*x^4+2*X1*x^5-2* X1*x^4+X1*x^3+x^4-3*x^2+3*x-1)/(X1^3*X2^2*x^16+X1^3*X2^2*x^15-2*X1^3*X2*x^16-2* X1^2*X2^2*x^16-X1^3*X2^2*x^14-X1^3*X2*x^15+X1^3*x^16-2*X1^2*X2^2*x^15+4*X1^2*X2 *x^16+X1*X2^2*x^16+3*X1^3*X2*x^14+X1^2*X2^2*x^14+2*X1^2*X2*x^15-2*X1^2*x^16+X1* X2^2*x^15-2*X1*X2*x^16-X1^3*X2*x^13-2*X1^3*x^14+X1^2*X2^2*x^13-4*X1^2*X2*x^14- X1*X2*x^15+X1*x^16+X1^3*x^13+3*X1^2*x^14-2*X1*X2^2*x^13+X1*X2*x^14-2*X1^2*X2^2* x^11+2*X1^2*X2*x^12-X1^2*x^13+X1*X2^2*x^12+3*X1*X2*x^13-X1*x^14+X2^2*x^13+X1^2* X2^2*x^10+3*X1^2*X2*x^11-2*X1^2*x^12+2*X1*X2^2*x^11-6*X1*X2*x^12-X1*x^13-X2^2*x ^12-2*X2*x^13-5*X1^2*X2*x^10-X1*X2^2*x^10-3*X1*X2*x^11+5*X1*x^12+4*X2*x^12+x^13 +5*X1^2*x^10+8*X1*X2*x^10-X1*x^11-3*x^12+X1^2*X2*x^8-2*X1^2*x^9-2*X1*X2*x^9-9* X1*x^10-3*X2*x^10+x^11-2*X1^2*x^8-X1*X2*x^8+6*X1*x^9+2*X2*x^9+4*x^10+X1^2*x^7+2 *X1*X2*x^7+3*X1*x^8-X2*x^8-4*x^9-3*X1*X2*x^6-5*X1*x^7+3*X1*X2*x^5+3*X1*x^6+2*x^ 7-X1*X2*x^4-3*X1*x^5+3*X1*x^4-X1*x^3-2*x^4-x^3+5*x^2-4*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 1, 3], are n 287 35 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 4, 2], are n 75 69 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 2 35 69 ------------- 345 and in floating point 0.2848849246 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 1/2 2 7 345 ate normal pair with correlation, ------------- 345 1/2 1/2 2 7 345 401 i.e. , [[-------------, 0], [0, ---]] 345 345 ------------------------------------------------- Theorem Number, 173, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [3, 1, 3], nor the composition, [2, 5, 1] Then infinity ----- \ n ) a(n) x = / ----- n = 0 16 15 14 12 11 10 9 8 7 3 2 x + 2 x + x + x + x - 2 x - x + x + x + x + x - 2 x + 1 - ------------------------------------------------------------------------ 13 12 10 9 7 6 5 3 (-1 + x) (x + 2 x - 3 x - 2 x - x - 2 x - x + x - 2 x + 1) and in Maple format -(x^16+2*x^15+x^14+x^12+x^11-2*x^10-x^9+x^8+x^7+x^3+x^2-2*x+1)/(-1+x)/(x^13+2*x ^12-3*x^10-2*x^9-x^7-2*x^6-x^5+x^3-2*x+1) The asymptotic expression for a(n) is, n 0.66072243088480701000 1.9169323599829195873 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [3, 1, 3] and d occurrences (as containment) of the composition, [2, 5, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 17 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 2 16 2 17 2 17 2 2 15 2 16 + X1 X2 x - 2 X1 X2 x - 2 X1 X2 x - X1 X2 x - 2 X1 X2 x 2 17 2 16 17 2 17 2 15 2 16 + X1 x - 2 X1 X2 x + 4 X1 X2 x + X2 x + 3 X1 X2 x + X1 x 2 15 16 17 2 16 17 2 2 13 + X1 X2 x + 4 X1 X2 x - 2 X1 x + X2 x - 2 X2 x + X1 X2 x 2 14 2 15 15 16 16 17 + X1 X2 x - 2 X1 x - 4 X1 X2 x - 2 X1 x - 2 X2 x + x 2 2 12 2 13 2 14 2 13 14 - X1 X2 x - 3 X1 X2 x - X1 x - X1 X2 x - 2 X1 X2 x 15 15 16 2 12 2 13 2 12 + 3 X1 x + X2 x + x + 2 X1 X2 x + 2 X1 x + X1 X2 x 13 14 14 15 2 11 2 12 + 4 X1 X2 x + 2 X1 x + X2 x - x + 2 X1 X2 x - X1 x 12 13 13 14 2 10 2 11 - 2 X1 X2 x - 3 X1 x - X2 x - x - X1 X2 x - 2 X1 x 11 12 13 2 9 2 10 10 - 5 X1 X2 x + X1 x + x - X1 X2 x + X1 x + 2 X1 X2 x 11 11 2 9 9 10 10 11 + 5 X1 x + 3 X2 x + 2 X1 x + 2 X1 X2 x - 2 X1 x - X2 x - 3 x 8 9 9 10 2 7 7 8 9 - X1 X2 x - 4 X1 x - X2 x + x - X1 x + 2 X1 X2 x + X1 x + 2 x 6 5 6 7 5 4 3 4 2 - 2 X1 X2 x + X1 X2 x + 2 X1 x - x - X1 x - X1 x + X1 x + x - 3 x / 2 2 14 2 2 13 2 14 + 3 x - 1) / ((-1 + x) (X1 X2 x + X1 X2 x - 2 X1 X2 x / 2 14 2 2 12 2 13 2 14 2 13 - 2 X1 X2 x - X1 X2 x - 2 X1 X2 x + X1 x - 2 X1 X2 x 14 2 14 2 12 2 13 2 12 + 4 X1 X2 x + X2 x + 4 X1 X2 x + X1 x + X1 X2 x 13 14 2 13 14 2 11 2 12 + 4 X1 X2 x - 2 X1 x + X2 x - 2 X2 x + 3 X1 X2 x - 3 X1 x 12 13 13 14 2 10 2 11 - 6 X1 X2 x - 2 X1 x - 2 X2 x + x - X1 X2 x - 3 X1 x 11 12 12 13 2 9 2 10 - 6 X1 X2 x + 5 X1 x + 2 X2 x + x - X1 X2 x + 2 X1 x 10 11 11 12 2 9 9 10 + X1 X2 x + 6 X1 x + 3 X2 x - 2 x + 3 X1 x + X1 X2 x - 3 X1 x 11 8 9 10 2 7 7 8 8 - 3 x - X1 X2 x - 5 X1 x + x - X1 x + X1 X2 x + X1 x + X2 x 9 6 7 8 5 6 7 5 6 + 2 x - 2 X1 X2 x + X1 x - x + X1 X2 x + X1 x - x - 2 X1 x + x 4 5 3 4 3 2 - X1 x + x + X1 x + x - x - 2 x + 3 x - 1)) and in Maple format -(X1^2*X2^2*x^17+X1^2*X2^2*x^16-2*X1^2*X2*x^17-2*X1*X2^2*x^17-X1^2*X2^2*x^15-2* X1^2*X2*x^16+X1^2*x^17-2*X1*X2^2*x^16+4*X1*X2*x^17+X2^2*x^17+3*X1^2*X2*x^15+X1^ 2*x^16+X1*X2^2*x^15+4*X1*X2*x^16-2*X1*x^17+X2^2*x^16-2*X2*x^17+X1^2*X2^2*x^13+ X1^2*X2*x^14-2*X1^2*x^15-4*X1*X2*x^15-2*X1*x^16-2*X2*x^16+x^17-X1^2*X2^2*x^12-3 *X1^2*X2*x^13-X1^2*x^14-X1*X2^2*x^13-2*X1*X2*x^14+3*X1*x^15+X2*x^15+x^16+2*X1^2 *X2*x^12+2*X1^2*x^13+X1*X2^2*x^12+4*X1*X2*x^13+2*X1*x^14+X2*x^14-x^15+2*X1^2*X2 *x^11-X1^2*x^12-2*X1*X2*x^12-3*X1*x^13-X2*x^13-x^14-X1^2*X2*x^10-2*X1^2*x^11-5* X1*X2*x^11+X1*x^12+x^13-X1^2*X2*x^9+X1^2*x^10+2*X1*X2*x^10+5*X1*x^11+3*X2*x^11+ 2*X1^2*x^9+2*X1*X2*x^9-2*X1*x^10-X2*x^10-3*x^11-X1*X2*x^8-4*X1*x^9-X2*x^9+x^10- X1^2*x^7+2*X1*X2*x^7+X1*x^8+2*x^9-2*X1*X2*x^6+X1*X2*x^5+2*X1*x^6-x^7-X1*x^5-X1* x^4+X1*x^3+x^4-3*x^2+3*x-1)/(-1+x)/(X1^2*X2^2*x^14+X1^2*X2^2*x^13-2*X1^2*X2*x^ 14-2*X1*X2^2*x^14-X1^2*X2^2*x^12-2*X1^2*X2*x^13+X1^2*x^14-2*X1*X2^2*x^13+4*X1* X2*x^14+X2^2*x^14+4*X1^2*X2*x^12+X1^2*x^13+X1*X2^2*x^12+4*X1*X2*x^13-2*X1*x^14+ X2^2*x^13-2*X2*x^14+3*X1^2*X2*x^11-3*X1^2*x^12-6*X1*X2*x^12-2*X1*x^13-2*X2*x^13 +x^14-X1^2*X2*x^10-3*X1^2*x^11-6*X1*X2*x^11+5*X1*x^12+2*X2*x^12+x^13-X1^2*X2*x^ 9+2*X1^2*x^10+X1*X2*x^10+6*X1*x^11+3*X2*x^11-2*x^12+3*X1^2*x^9+X1*X2*x^9-3*X1*x ^10-3*x^11-X1*X2*x^8-5*X1*x^9+x^10-X1^2*x^7+X1*X2*x^7+X1*x^8+X2*x^8+2*x^9-2*X1* X2*x^6+X1*x^7-x^8+X1*X2*x^5+X1*x^6-x^7-2*X1*x^5+x^6-X1*x^4+x^5+X1*x^3+x^4-x^3-2 *x^2+3*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 1, 3], are n 287 35 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 5, 1], are n 111 57 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 8 35 57 ------------- 1995 and in floating point 0.1791094652 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 8 1995 ate normal pair with correlation, --------- 1995 1/2 8 1995 2123 i.e. , [[---------, 0], [0, ----]] 1995 1995 ------------------------------------------------- Theorem Number, 174, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [3, 1, 3], nor the composition, [3, 3, 2] Then infinity ----- 7 3 2 \ n x + x + x - 2 x + 1 ) a(n) x = - -------------------------------------------- / 9 7 6 5 4 3 2 ----- x - x + x + x + x - x - 2 x + 3 x - 1 n = 0 and in Maple format -(x^7+x^3+x^2-2*x+1)/(x^9-x^7+x^6+x^5+x^4-x^3-2*x^2+3*x-1) The asymptotic expression for a(n) is, n 0.62654038626736234360 1.9293902847278627598 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [3, 1, 3] and d occurrences (as containment) of the composition, [3, 3, 2], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 8 2 7 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x + X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 8 7 8 6 7 6 7 4 - 2 X1 X2 x - X1 X2 x + X2 x - X1 X2 x - X1 x + X1 x + x + X1 X2 x 3 4 3 2 / 2 10 2 9 - X1 X2 x - X1 x + x + x - 2 x + 1) / (X1 X2 x + 2 X1 X2 x / 10 9 10 2 7 9 9 9 - 2 X1 X2 x - 3 X1 X2 x + X2 x - X1 X2 x - X1 x + X2 x + x 6 7 6 7 4 5 6 3 + X1 X2 x + 2 X1 x - 2 X1 x - x - 2 X1 X2 x - X1 x + x + X1 X2 x 4 5 4 3 2 + X1 x + x + x - x - 2 x + 3 x - 1) and in Maple format -(X1^2*X2*x^8+X1^2*X2*x^7-2*X1*X2*x^8-X1*X2*x^7+X2*x^8-X1*X2*x^6-X1*x^7+X1*x^6+ x^7+X1*X2*x^4-X1*X2*x^3-X1*x^4+x^3+x^2-2*x+1)/(X1^2*X2*x^10+2*X1^2*X2*x^9-2*X1* X2*x^10-3*X1*X2*x^9+X2*x^10-X1^2*X2*x^7-X1*x^9+X2*x^9+x^9+X1*X2*x^6+2*X1*x^7-2* X1*x^6-x^7-2*X1*X2*x^4-X1*x^5+x^6+X1*X2*x^3+X1*x^4+x^5+x^4-x^3-2*x^2+3*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 1, 3], are n 287 35 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 3, 2], are n 97 85 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 26 7 17 ------------- 595 and in floating point 0.4766832187 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 26 119 ate normal pair with correlation, --------- 595 1/2 26 119 4327 i.e. , [[---------, 0], [0, ----]] 595 2975 ------------------------------------------------- Theorem Number, 175, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [3, 1, 3], nor the composition, [3, 4, 1] Then infinity ----- 2 7 6 5 4 3 2 \ n (x - x + 1) (x + x - x - 2 x - x + x + x - 1) ) a(n) x = - ---------------------------------------------------------- / 10 9 8 7 6 5 3 ----- (-1 + x) (x + 2 x + x + x + 2 x + x - x + 2 x - 1) n = 0 and in Maple format -(x^2-x+1)*(x^7+x^6-x^5-2*x^4-x^3+x^2+x-1)/(-1+x)/(x^10+2*x^9+x^8+x^7+2*x^6+x^5 -x^3+2*x-1) The asymptotic expression for a(n) is, n 0.64343319290120806019 1.9224267923147665937 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [3, 1, 3] and d occurrences (as containment) of the composition, [3, 4, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 9 2 8 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x + X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 9 9 8 9 9 2 7 7 - X1 x - 2 X1 X2 x - 2 X1 X2 x + 2 X1 x + X2 x + X1 x - X1 X2 x 8 9 7 5 7 4 5 4 3 + X2 x - x - X1 x + X1 X2 x + x - X1 X2 x - X1 x + X1 x - X1 x 3 2 / 2 11 2 10 2 11 + x + x - 2 x + 1) / (X1 X2 x + 2 X1 X2 x - X1 x / 11 2 10 10 11 11 2 8 - 2 X1 X2 x - X1 x - 4 X1 X2 x + 2 X1 x + X2 x - X1 X2 x 2 9 9 10 10 11 2 8 8 + 2 X1 x - X1 X2 x + 2 X1 x + 2 X2 x - x + X1 x + X1 X2 x 9 9 10 2 7 7 8 9 7 - 3 X1 x + X2 x - x - X1 x + X1 X2 x - X1 x + x + X1 x 5 6 7 4 5 6 4 5 3 - 2 X1 X2 x - X1 x - x + X1 X2 x + X1 x + x - 2 X1 x + x + X1 x 4 3 2 + x - x - 2 x + 3 x - 1) and in Maple format -(X1^2*X2*x^9+X1^2*X2*x^8-X1^2*x^9-2*X1*X2*x^9-2*X1*X2*x^8+2*X1*x^9+X2*x^9+X1^2 *x^7-X1*X2*x^7+X2*x^8-x^9-X1*x^7+X1*X2*x^5+x^7-X1*X2*x^4-X1*x^5+X1*x^4-X1*x^3+x ^3+x^2-2*x+1)/(X1^2*X2*x^11+2*X1^2*X2*x^10-X1^2*x^11-2*X1*X2*x^11-X1^2*x^10-4* X1*X2*x^10+2*X1*x^11+X2*x^11-X1^2*X2*x^8+2*X1^2*x^9-X1*X2*x^9+2*X1*x^10+2*X2*x^ 10-x^11+X1^2*x^8+X1*X2*x^8-3*X1*x^9+X2*x^9-x^10-X1^2*x^7+X1*X2*x^7-X1*x^8+x^9+ X1*x^7-2*X1*X2*x^5-X1*x^6-x^7+X1*X2*x^4+X1*x^5+x^6-2*X1*x^4+x^5+X1*x^3+x^4-x^3-\ 2*x^2+3*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 1, 3], are n 287 35 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 4, 1], are n 135 65 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 16 7 13 ------------- 455 and in floating point 0.3354511477 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 16 91 ate normal pair with correlation, -------- 455 1/2 16 91 2787 i.e. , [[--------, 0], [0, ----]] 455 2275 ------------------------------------------------- Theorem Number, 176, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [3, 1, 3], nor the composition, [4, 2, 2] Then infinity ----- \ n ) a(n) x = / ----- n = 0 9 8 7 3 2 x - x - x - x - x + 2 x - 1 ----------------------------------------------------------------- 12 11 9 8 7 6 5 4 3 2 x - 2 x + 3 x - x - x + x + x + x - x - 2 x + 3 x - 1 and in Maple format (x^9-x^8-x^7-x^3-x^2+2*x-1)/(x^12-2*x^11+3*x^9-x^8-x^7+x^6+x^5+x^4-x^3-2*x^2+3* x-1) The asymptotic expression for a(n) is, n 0.63439518060791917574 1.9265719569048272535 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [3, 1, 3] and d occurrences (as containment) of the composition, [4, 2, 2], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 10 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 10 2 10 2 9 2 10 10 - 2 X1 X2 x - X1 X2 x + 2 X1 X2 x + X1 x + 2 X1 X2 x 2 8 2 9 9 10 8 9 9 + X1 X2 x - 2 X1 x - 3 X1 X2 x - X1 x - X1 X2 x + 3 X1 x + X2 x 2 7 8 9 6 7 8 5 6 7 + X1 x - X1 x - x - X1 X2 x - 2 X1 x + x + X1 X2 x + X1 x + x 4 5 4 3 3 2 / 2 2 12 - X1 X2 x - X1 x + X1 x - X1 x + x + x - 2 x + 1) / (X1 X2 x / 2 2 11 2 12 2 12 2 2 10 2 12 + X1 X2 x - 2 X1 X2 x - 2 X1 X2 x - X1 X2 x + X1 x 2 11 12 2 12 2 10 2 11 2 10 - X1 X2 x + 4 X1 X2 x + X2 x + 5 X1 X2 x - X1 x + X1 X2 x 11 12 12 2 9 2 10 10 - 2 X1 X2 x - 2 X1 x - 2 X2 x - X1 X2 x - 3 X1 x - 6 X1 X2 x 11 11 12 2 8 2 9 9 10 + 3 X1 x + 2 X2 x + x - X1 X2 x + 3 X1 x + 2 X1 X2 x + 3 X1 x 10 11 2 8 9 9 2 7 7 8 + X2 x - 2 x + X1 x - 6 X1 x - X2 x - X1 x - X1 X2 x + X2 x 9 6 7 8 5 6 7 4 + 3 x + X1 X2 x + 3 X1 x - x - 2 X1 X2 x - 2 X1 x - x + X1 X2 x 5 6 4 5 3 4 3 2 + X1 x + x - 2 X1 x + x + X1 x + x - x - 2 x + 3 x - 1) and in Maple format -(X1^2*X2^2*x^10-2*X1^2*X2*x^10-X1*X2^2*x^10+2*X1^2*X2*x^9+X1^2*x^10+2*X1*X2*x^ 10+X1^2*X2*x^8-2*X1^2*x^9-3*X1*X2*x^9-X1*x^10-X1*X2*x^8+3*X1*x^9+X2*x^9+X1^2*x^ 7-X1*x^8-x^9-X1*X2*x^6-2*X1*x^7+x^8+X1*X2*x^5+X1*x^6+x^7-X1*X2*x^4-X1*x^5+X1*x^ 4-X1*x^3+x^3+x^2-2*x+1)/(X1^2*X2^2*x^12+X1^2*X2^2*x^11-2*X1^2*X2*x^12-2*X1*X2^2 *x^12-X1^2*X2^2*x^10+X1^2*x^12-X1*X2^2*x^11+4*X1*X2*x^12+X2^2*x^12+5*X1^2*X2*x^ 10-X1^2*x^11+X1*X2^2*x^10-2*X1*X2*x^11-2*X1*x^12-2*X2*x^12-X1^2*X2*x^9-3*X1^2*x ^10-6*X1*X2*x^10+3*X1*x^11+2*X2*x^11+x^12-X1^2*X2*x^8+3*X1^2*x^9+2*X1*X2*x^9+3* X1*x^10+X2*x^10-2*x^11+X1^2*x^8-6*X1*x^9-X2*x^9-X1^2*x^7-X1*X2*x^7+X2*x^8+3*x^9 +X1*X2*x^6+3*X1*x^7-x^8-2*X1*X2*x^5-2*X1*x^6-x^7+X1*X2*x^4+X1*x^5+x^6-2*X1*x^4+ x^5+X1*x^3+x^4-x^3-2*x^2+3*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 1, 3], are n 287 35 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 2, 2], are n 75 69 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 6 35 69 ------------- 805 and in floating point 0.3662806174 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 1/2 6 3 805 ate normal pair with correlation, ------------- 805 1/2 1/2 6 3 805 1021 i.e. , [[-------------, 0], [0, ----]] 805 805 ------------------------------------------------- Theorem Number, 177, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [3, 1, 3], nor the composition, [4, 3, 1] Then infinity ----- 2 7 6 5 4 3 2 \ n (x - x + 1) (x + x - x - 2 x - x + x + x - 1) ) a(n) x = - ---------------------------------------------------------- / 10 9 8 7 6 5 3 ----- (-1 + x) (x + 2 x + x + x + 2 x + x - x + 2 x - 1) n = 0 and in Maple format -(x^2-x+1)*(x^7+x^6-x^5-2*x^4-x^3+x^2+x-1)/(-1+x)/(x^10+2*x^9+x^8+x^7+2*x^6+x^5 -x^3+2*x-1) The asymptotic expression for a(n) is, n 0.64343319290120806019 1.9224267923147665937 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [3, 1, 3] and d occurrences (as containment) of the composition, [4, 3, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 9 2 8 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x + X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 9 9 8 9 9 2 7 7 - X1 x - 2 X1 X2 x - 2 X1 X2 x + 2 X1 x + X2 x + X1 x - X1 X2 x 8 9 7 5 7 4 5 4 3 + X2 x - x - X1 x + X1 X2 x + x - X1 X2 x - X1 x + X1 x - X1 x 3 2 / 2 11 2 10 2 11 + x + x - 2 x + 1) / (X1 X2 x + 2 X1 X2 x - X1 x / 11 2 10 10 11 11 2 8 - 2 X1 X2 x - X1 x - 4 X1 X2 x + 2 X1 x + X2 x - X1 X2 x 2 9 9 10 10 11 2 8 8 + 2 X1 x - X1 X2 x + 2 X1 x + 2 X2 x - x + X1 x + X1 X2 x 9 9 10 2 7 7 8 9 7 - 3 X1 x + X2 x - x - X1 x + X1 X2 x - X1 x + x + X1 x 5 6 7 4 5 6 4 5 3 - 2 X1 X2 x - X1 x - x + X1 X2 x + X1 x + x - 2 X1 x + x + X1 x 4 3 2 + x - x - 2 x + 3 x - 1) and in Maple format -(X1^2*X2*x^9+X1^2*X2*x^8-X1^2*x^9-2*X1*X2*x^9-2*X1*X2*x^8+2*X1*x^9+X2*x^9+X1^2 *x^7-X1*X2*x^7+X2*x^8-x^9-X1*x^7+X1*X2*x^5+x^7-X1*X2*x^4-X1*x^5+X1*x^4-X1*x^3+x ^3+x^2-2*x+1)/(X1^2*X2*x^11+2*X1^2*X2*x^10-X1^2*x^11-2*X1*X2*x^11-X1^2*x^10-4* X1*X2*x^10+2*X1*x^11+X2*x^11-X1^2*X2*x^8+2*X1^2*x^9-X1*X2*x^9+2*X1*x^10+2*X2*x^ 10-x^11+X1^2*x^8+X1*X2*x^8-3*X1*x^9+X2*x^9-x^10-X1^2*x^7+X1*X2*x^7-X1*x^8+x^9+ X1*x^7-2*X1*X2*x^5-X1*x^6-x^7+X1*X2*x^4+X1*x^5+x^6-2*X1*x^4+x^5+X1*x^3+x^4-x^3-\ 2*x^2+3*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 1, 3], are n 287 35 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 3, 1], are n 135 65 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 16 7 13 ------------- 455 and in floating point 0.3354511477 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 16 91 ate normal pair with correlation, -------- 455 1/2 16 91 2787 i.e. , [[--------, 0], [0, ----]] 455 2275 ------------------------------------------------- Theorem Number, 178, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [3, 1, 3], nor the composition, [5, 1, 2] Then infinity ----- 10 8 7 3 2 \ n x - x - x - x - x + 2 x - 1 ) a(n) x = ---------------------------------------------------- / 7 5 4 3 7 5 4 ----- (x - x - x + x - 2 x + 1) (x - x - x + x - 1) n = 0 and in Maple format (x^10-x^8-x^7-x^3-x^2+2*x-1)/(x^7-x^5-x^4+x^3-2*x+1)/(x^7-x^5-x^4+x-1) The asymptotic expression for a(n) is, n 0.64476247294249366516 1.9235403491125894929 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [3, 1, 3] and d occurrences (as containment) of the composition, [5, 1, 2], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 12 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 2 11 2 12 2 12 2 11 2 12 + X1 X2 x - 2 X1 X2 x - X1 X2 x - 2 X1 X2 x + X1 x 12 2 10 2 11 12 2 9 2 10 + 2 X1 X2 x + 2 X1 X2 x + X1 x - X1 x + 2 X1 X2 x - 2 X1 x 10 2 9 9 10 10 2 8 - 3 X1 X2 x - 2 X1 x - 2 X1 X2 x + 3 X1 x + X2 x + X1 x 9 10 2 7 8 7 8 5 7 5 + 2 X1 x - x + X1 x - 2 X1 x - 2 X1 x + x - X1 X2 x + x + X1 x 3 3 2 / 7 6 7 7 - X1 x + x + x - 2 x + 1) / ((X1 X2 x + X1 X2 x - X1 x - X2 x / 6 7 5 4 5 4 7 6 - X1 x + x + X1 x + X1 x - x - x + x - 1) (X1 X2 x + X1 X2 x 7 7 5 6 7 5 4 5 3 - X1 x - X2 x - X1 X2 x - X1 x + x + 2 X1 x + X1 x - x - X1 x 4 3 - x + x - 2 x + 1)) and in Maple format -(X1^2*X2^2*x^12+X1^2*X2^2*x^11-2*X1^2*X2*x^12-X1*X2^2*x^12-2*X1^2*X2*x^11+X1^2 *x^12+2*X1*X2*x^12+2*X1^2*X2*x^10+X1^2*x^11-X1*x^12+2*X1^2*X2*x^9-2*X1^2*x^10-3 *X1*X2*x^10-2*X1^2*x^9-2*X1*X2*x^9+3*X1*x^10+X2*x^10+X1^2*x^8+2*X1*x^9-x^10+X1^ 2*x^7-2*X1*x^8-2*X1*x^7+x^8-X1*X2*x^5+x^7+X1*x^5-X1*x^3+x^3+x^2-2*x+1)/(X1*X2*x ^7+X1*X2*x^6-X1*x^7-X2*x^7-X1*x^6+x^7+X1*x^5+X1*x^4-x^5-x^4+x-1)/(X1*X2*x^7+X1* X2*x^6-X1*x^7-X2*x^7-X1*X2*x^5-X1*x^6+x^7+2*X1*x^5+X1*x^4-x^5-X1*x^3-x^4+x^3-2* x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 1, 3], are n 287 35 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [5, 1, 2], are n 113 57 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 2 35 57 ------------- 285 and in floating point 0.3134415640 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 1/2 2 7 285 ate normal pair with correlation, ------------- 285 1/2 1/2 2 7 285 341 i.e. , [[-------------, 0], [0, ---]] 285 285 ------------------------------------------------- Theorem Number, 179, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [3, 1, 3], nor the composition, [5, 2, 1] Then infinity ----- \ n ) a(n) x = / ----- n = 0 10 9 8 7 3 2 x + x - x - x - x - x + 2 x - 1 --------------------------------------------------------------------- 13 12 10 9 7 6 5 3 (-1 + x) (x + 2 x - 3 x - 2 x - x - 2 x - x + x - 2 x + 1) and in Maple format (x^10+x^9-x^8-x^7-x^3-x^2+2*x-1)/(-1+x)/(x^13+2*x^12-3*x^10-2*x^9-x^7-2*x^6-x^5 +x^3-2*x+1) The asymptotic expression for a(n) is, n 0.66081197302068638652 1.9169323599829195873 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [3, 1, 3] and d occurrences (as containment) of the composition, [5, 2, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 12 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 12 2 12 2 12 12 2 10 - 2 X1 X2 x - X1 X2 x + X1 x + 2 X1 X2 x + 2 X1 X2 x 12 2 9 2 10 10 2 9 9 - X1 x + X1 X2 x - 2 X1 x - 3 X1 X2 x - X1 x - 2 X1 X2 x 10 10 2 8 9 9 10 2 7 7 + 3 X1 x + X2 x + X1 x + 2 X1 x + X2 x - x + X1 x - X1 X2 x 8 9 6 7 8 5 6 7 5 - 2 X1 x - x + X1 X2 x - X1 x + x - X1 X2 x - X1 x + x + X1 x 3 3 2 / 2 2 14 2 2 13 2 14 - X1 x + x + x - 2 x + 1) / (X1 X2 x + X1 X2 x - 2 X1 X2 x / 2 14 2 2 12 2 13 2 14 2 13 - 2 X1 X2 x - X1 X2 x - 2 X1 X2 x + X1 x - 2 X1 X2 x 14 2 14 2 12 2 13 2 12 + 4 X1 X2 x + X2 x + 4 X1 X2 x + X1 x + X1 X2 x 13 14 2 13 14 2 11 2 12 + 4 X1 X2 x - 2 X1 x + X2 x - 2 X2 x + 3 X1 X2 x - 3 X1 x 12 13 13 14 2 10 2 11 - 6 X1 X2 x - 2 X1 x - 2 X2 x + x - X1 X2 x - 3 X1 x 11 12 12 13 2 9 2 10 - 6 X1 X2 x + 5 X1 x + 2 X2 x + x - X1 X2 x + 2 X1 x 10 11 11 12 2 9 9 10 + X1 X2 x + 6 X1 x + 3 X2 x - 2 x + 3 X1 x + X1 X2 x - 3 X1 x 11 8 9 10 2 7 7 8 8 - 3 x - X1 X2 x - 5 X1 x + x - X1 x + X1 X2 x + X1 x + X2 x 9 6 7 8 5 6 7 5 6 + 2 x - 2 X1 X2 x + X1 x - x + X1 X2 x + X1 x - x - 2 X1 x + x 4 5 3 4 3 2 - X1 x + x + X1 x + x - x - 2 x + 3 x - 1) and in Maple format -(X1^2*X2^2*x^12-2*X1^2*X2*x^12-X1*X2^2*x^12+X1^2*x^12+2*X1*X2*x^12+2*X1^2*X2*x ^10-X1*x^12+X1^2*X2*x^9-2*X1^2*x^10-3*X1*X2*x^10-X1^2*x^9-2*X1*X2*x^9+3*X1*x^10 +X2*x^10+X1^2*x^8+2*X1*x^9+X2*x^9-x^10+X1^2*x^7-X1*X2*x^7-2*X1*x^8-x^9+X1*X2*x^ 6-X1*x^7+x^8-X1*X2*x^5-X1*x^6+x^7+X1*x^5-X1*x^3+x^3+x^2-2*x+1)/(X1^2*X2^2*x^14+ X1^2*X2^2*x^13-2*X1^2*X2*x^14-2*X1*X2^2*x^14-X1^2*X2^2*x^12-2*X1^2*X2*x^13+X1^2 *x^14-2*X1*X2^2*x^13+4*X1*X2*x^14+X2^2*x^14+4*X1^2*X2*x^12+X1^2*x^13+X1*X2^2*x^ 12+4*X1*X2*x^13-2*X1*x^14+X2^2*x^13-2*X2*x^14+3*X1^2*X2*x^11-3*X1^2*x^12-6*X1* X2*x^12-2*X1*x^13-2*X2*x^13+x^14-X1^2*X2*x^10-3*X1^2*x^11-6*X1*X2*x^11+5*X1*x^ 12+2*X2*x^12+x^13-X1^2*X2*x^9+2*X1^2*x^10+X1*X2*x^10+6*X1*x^11+3*X2*x^11-2*x^12 +3*X1^2*x^9+X1*X2*x^9-3*X1*x^10-3*x^11-X1*X2*x^8-5*X1*x^9+x^10-X1^2*x^7+X1*X2*x ^7+X1*x^8+X2*x^8+2*x^9-2*X1*X2*x^6+X1*x^7-x^8+X1*X2*x^5+X1*x^6-x^7-2*X1*x^5+x^6 -X1*x^4+x^5+X1*x^3+x^4-x^3-2*x^2+3*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 1, 3], are n 287 35 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [5, 2, 1], are n 111 57 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 8 35 57 ------------- 1995 and in floating point 0.1791094652 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 8 1995 ate normal pair with correlation, --------- 1995 1/2 8 1995 2123 i.e. , [[---------, 0], [0, ----]] 1995 1995 ------------------------------------------------- Theorem Number, 180, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [3, 1, 3], nor the composition, [6, 1, 1] Then infinity ----- \ n 11 10 8 7 3 2 / ) a(n) x = (x + x - x - x - x - x + 2 x - 1) / ( / / ----- n = 0 7 6 5 4 2 7 6 5 4 2 (x + 2 x + 2 x + x + x + x - 1) (x + 2 x + 2 x + x + 1) (-1 + x) ) and in Maple format (x^11+x^10-x^8-x^7-x^3-x^2+2*x-1)/(x^7+2*x^6+2*x^5+x^4+x^2+x-1)/(x^7+2*x^6+2*x^ 5+x^4+1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.67485833261041293122 1.9131864632213745587 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [3, 1, 3] and d occurrences (as containment) of the composition, [6, 1, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 14 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 2 13 2 14 2 14 2 13 2 14 + X1 X2 x - 2 X1 X2 x - X1 X2 x - 2 X1 X2 x + X1 x 2 13 14 2 13 13 14 2 11 - X1 X2 x + 2 X1 X2 x + X1 x + 2 X1 X2 x - X1 x + 2 X1 X2 x 13 2 10 2 11 11 2 10 - X1 x + 2 X1 X2 x - 2 X1 x - 3 X1 X2 x - 2 X1 x 10 11 11 10 10 11 2 8 10 - 3 X1 X2 x + 3 X1 x + X2 x + 3 X1 x + X2 x - x + X1 x - x 2 7 8 6 7 8 6 7 3 3 + X1 x - 2 X1 x - X1 X2 x - 2 X1 x + x + X1 x + x - X1 x + x 2 / 8 7 8 8 7 7 + x - 2 x + 1) / ((X1 X2 x + X1 X2 x - X1 x - X2 x - X1 x - X2 x / 8 7 5 4 5 4 8 7 8 + x + x + X1 x + X1 x - x - x + x - 1) (X1 X2 x + X1 X2 x - X1 x 8 6 7 7 8 6 7 5 4 5 - X2 x - X1 X2 x - X1 x - X2 x + x + X1 x + x + X1 x + X1 x - x 3 4 3 - X1 x - x + x - 2 x + 1)) and in Maple format -(X1^2*X2^2*x^14+X1^2*X2^2*x^13-2*X1^2*X2*x^14-X1*X2^2*x^14-2*X1^2*X2*x^13+X1^2 *x^14-X1*X2^2*x^13+2*X1*X2*x^14+X1^2*x^13+2*X1*X2*x^13-X1*x^14+2*X1^2*X2*x^11- X1*x^13+2*X1^2*X2*x^10-2*X1^2*x^11-3*X1*X2*x^11-2*X1^2*x^10-3*X1*X2*x^10+3*X1*x ^11+X2*x^11+3*X1*x^10+X2*x^10-x^11+X1^2*x^8-x^10+X1^2*x^7-2*X1*x^8-X1*X2*x^6-2* X1*x^7+x^8+X1*x^6+x^7-X1*x^3+x^3+x^2-2*x+1)/(X1*X2*x^8+X1*X2*x^7-X1*x^8-X2*x^8- X1*x^7-X2*x^7+x^8+x^7+X1*x^5+X1*x^4-x^5-x^4+x-1)/(X1*X2*x^8+X1*X2*x^7-X1*x^8-X2 *x^8-X1*X2*x^6-X1*x^7-X2*x^7+x^8+X1*x^6+x^7+X1*x^5+X1*x^4-x^5-X1*x^3-x^4+x^3-2* x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 1, 3], are n 287 35 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [6, 1, 1], are n 49 53 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 4 35 53 ------------- 1855 and in floating point 0.09287269204 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 4 1855 ate normal pair with correlation, --------- 1855 1/2 4 1855 1887 i.e. , [[---------, 0], [0, ----]] 1855 1855 ------------------------------------------------- Theorem Number, 181, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [3, 2, 2], nor the composition, [1, 1, 6] Then infinity ----- \ n ) a(n) x = / ----- n = 0 14 13 12 9 7 6 4 3 2 x + x + x - x - x - x - x + x - 2 x + 2 x - 1 - ------------------------------------------------------------- 10 9 8 7 6 5 3 2 (-1 + x) (x + x + 2 x + x + x + x + x - x + 2 x - 1) and in Maple format -(x^14+x^13+x^12-x^9-x^7-x^6-x^4+x^3-2*x^2+2*x-1)/(-1+x)/(x^10+x^9+2*x^8+x^7+x^ 6+x^5+x^3-x^2+2*x-1) The asymptotic expression for a(n) is, n 0.67395690992279832777 1.9139965407423997306 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [3, 2, 2] and d occurrences (as containment) of the composition, [1, 1, 6], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 17 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 17 2 17 2 17 17 2 17 - 2 X1 X2 x - 2 X1 X2 x + X1 x + 4 X1 X2 x + X2 x 2 15 17 17 2 14 15 2 15 + X1 X2 x - 2 X1 x - 2 X2 x + X1 X2 x - 2 X1 X2 x - X2 x 17 2 13 2 13 14 15 2 14 + x + X1 X2 x - X1 X2 x - 2 X1 X2 x + X1 x - X2 x 15 2 13 14 2 13 14 15 13 13 + 2 X2 x - X1 x + X1 x + X2 x + 2 X2 x - x + X1 x - X2 x 14 11 11 11 9 11 8 9 - x + X1 X2 x - X1 x - X2 x - X1 X2 x + x + 2 X1 X2 x + X1 x 9 7 8 8 9 6 7 7 + X2 x - 2 X1 X2 x - 2 X1 x - 2 X2 x - x + X1 X2 x + 2 X1 x + X2 x 8 6 7 5 6 4 5 3 4 + 2 x - 2 X1 x - x + 2 X1 x + x - 2 X1 x - 2 x + X1 x + 2 x 2 / 2 13 13 2 13 - 3 x + 3 x - 1) / ((-1 + x) (X1 X2 x - 2 X1 X2 x - X2 x / 13 13 13 10 9 10 10 + X1 x + 2 X2 x - x + X1 X2 x + X1 X2 x - X1 x - X2 x 8 9 9 10 7 8 8 9 - X1 X2 x - X1 x - X2 x + x + 2 X1 X2 x + X1 x + X2 x + x 6 7 7 8 6 7 5 6 4 - X1 X2 x - 2 X1 x - X2 x - x + 2 X1 x + x - X1 x - x + 2 X1 x 5 3 4 3 2 + x - X1 x - 2 x + x + 2 x - 3 x + 1)) and in Maple format (X1^2*X2^2*x^17-2*X1^2*X2*x^17-2*X1*X2^2*x^17+X1^2*x^17+4*X1*X2*x^17+X2^2*x^17+ X1*X2^2*x^15-2*X1*x^17-2*X2*x^17+X1*X2^2*x^14-2*X1*X2*x^15-X2^2*x^15+x^17+X1^2* X2*x^13-X1*X2^2*x^13-2*X1*X2*x^14+X1*x^15-X2^2*x^14+2*X2*x^15-X1^2*x^13+X1*x^14 +X2^2*x^13+2*X2*x^14-x^15+X1*x^13-X2*x^13-x^14+X1*X2*x^11-X1*x^11-X2*x^11-X1*X2 *x^9+x^11+2*X1*X2*x^8+X1*x^9+X2*x^9-2*X1*X2*x^7-2*X1*x^8-2*X2*x^8-x^9+X1*X2*x^6 +2*X1*x^7+X2*x^7+2*x^8-2*X1*x^6-x^7+2*X1*x^5+x^6-2*X1*x^4-2*x^5+X1*x^3+2*x^4-3* x^2+3*x-1)/(-1+x)/(X1*X2^2*x^13-2*X1*X2*x^13-X2^2*x^13+X1*x^13+2*X2*x^13-x^13+ X1*X2*x^10+X1*X2*x^9-X1*x^10-X2*x^10-X1*X2*x^8-X1*x^9-X2*x^9+x^10+2*X1*X2*x^7+ X1*x^8+X2*x^8+x^9-X1*X2*x^6-2*X1*x^7-X2*x^7-x^8+2*X1*x^6+x^7-X1*x^5-x^6+2*X1*x^ 4+x^5-X1*x^3-2*x^4+x^3+2*x^2-3*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 2, 2], are n 315 39 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 1, 6], are n 49 53 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 2 39 53 ------------- 2067 and in floating point 0.04399058701 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 2 2067 ate normal pair with correlation, --------- 2067 1/2 2 2067 2075 i.e. , [[---------, 0], [0, ----]] 2067 2067 ------------------------------------------------- Theorem Number, 182, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [3, 2, 2], nor the composition, [1, 2, 5] Then infinity ----- \ n ) a(n) x = / ----- n = 0 13 12 11 8 7 6 5 3 x + x + x - x - x - x - x - x + x - 1 - ------------------------------------------------------------------------- 11 10 9 8 7 6 5 4 2 (-1 + x) (x + 2 x + 3 x + 3 x + 3 x + 2 x + x + x + x + x - 1) and in Maple format -(x^13+x^12+x^11-x^8-x^7-x^6-x^5-x^3+x-1)/(-1+x)/(x^11+2*x^10+3*x^9+3*x^8+3*x^7 +2*x^6+x^5+x^4+x^2+x-1) The asymptotic expression for a(n) is, n 0.64504238443304350890 1.9230256556702387387 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [3, 2, 2] and d occurrences (as containment) of the composition, [1, 2, 5], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 15 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 15 2 15 2 15 2 14 15 - 2 X1 X2 x - 2 X1 X2 x + X1 x + X1 X2 x + 4 X1 X2 x 2 15 14 15 2 14 15 2 12 + X2 x - 2 X1 X2 x - 2 X1 x - X2 x - 2 X2 x + X1 X2 x 14 14 15 2 12 12 14 11 + X1 x + 2 X2 x + x - X1 x - 2 X1 X2 x - x + X1 X2 x 12 12 10 11 11 12 9 + 2 X1 x + X2 x - X1 X2 x - X1 x - X2 x - x + X1 X2 x 10 10 11 8 9 9 10 7 + X1 x + X2 x + x - X1 X2 x - X1 x - X2 x - x + 2 X1 X2 x 8 9 6 7 5 6 5 6 + X1 x + x - 2 X1 X2 x - 2 X1 x + X1 X2 x + X1 x + X1 x + x 4 5 3 4 2 / - 2 X1 x - 2 x + X1 x + 2 x - 3 x + 3 x - 1) / ((-1 + x) ( / 2 13 13 2 13 13 13 13 10 X1 X2 x - 2 X1 X2 x - X2 x + X1 x + 2 X2 x - x + X1 X2 x 10 10 8 10 7 8 8 6 - X1 x - X2 x + X1 X2 x + x - X1 X2 x - X1 x - X2 x + 2 X1 X2 x 7 8 5 6 6 4 5 3 4 3 + X1 x + x - X1 X2 x - X1 x - x + 2 X1 x + x - X1 x - 2 x + x 2 + 2 x - 3 x + 1)) and in Maple format (X1^2*X2^2*x^15-2*X1^2*X2*x^15-2*X1*X2^2*x^15+X1^2*x^15+X1*X2^2*x^14+4*X1*X2*x^ 15+X2^2*x^15-2*X1*X2*x^14-2*X1*x^15-X2^2*x^14-2*X2*x^15+X1^2*X2*x^12+X1*x^14+2* X2*x^14+x^15-X1^2*x^12-2*X1*X2*x^12-x^14+X1*X2*x^11+2*X1*x^12+X2*x^12-X1*X2*x^ 10-X1*x^11-X2*x^11-x^12+X1*X2*x^9+X1*x^10+X2*x^10+x^11-X1*X2*x^8-X1*x^9-X2*x^9- x^10+2*X1*X2*x^7+X1*x^8+x^9-2*X1*X2*x^6-2*X1*x^7+X1*X2*x^5+X1*x^6+X1*x^5+x^6-2* X1*x^4-2*x^5+X1*x^3+2*x^4-3*x^2+3*x-1)/(-1+x)/(X1*X2^2*x^13-2*X1*X2*x^13-X2^2*x ^13+X1*x^13+2*X2*x^13-x^13+X1*X2*x^10-X1*x^10-X2*x^10+X1*X2*x^8+x^10-X1*X2*x^7- X1*x^8-X2*x^8+2*X1*X2*x^6+X1*x^7+x^8-X1*X2*x^5-X1*x^6-x^6+2*X1*x^4+x^5-X1*x^3-2 *x^4+x^3+2*x^2-3*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 2, 2], are n 315 39 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 2, 5], are n 111 57 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 4 13 19 ------------- 247 and in floating point 0.2545139052 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 4 247 ate normal pair with correlation, -------- 247 1/2 4 247 279 i.e. , [[--------, 0], [0, ---]] 247 247 ------------------------------------------------- Theorem Number, 183, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [3, 2, 2], nor the composition, [1, 3, 4] Then infinity ----- 8 7 6 5 3 \ n x + x + x + x + x - x + 1 ) a(n) x = - ----------------------------------- / 3 4 3 2 ----- (x - x + 1) (x + x + x + x - 1) n = 0 and in Maple format -(x^8+x^7+x^6+x^5+x^3-x+1)/(x^3-x+1)/(x^4+x^3+x^2+x-1) The asymptotic expression for a(n) is, n 0.63242246488998289333 1.9275619754829253043 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [3, 2, 2] and d occurrences (as containment) of the composition, [1, 3, 4], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 11 2 11 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x - X1 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 11 10 11 9 10 10 8 - X1 X2 x - X1 X2 x + X1 x + X1 X2 x + X1 x + X2 x - X1 X2 x 9 9 10 7 8 9 6 7 - X1 x - X2 x - x + X1 X2 x + X1 x + x + X1 X2 x - X1 x 5 6 4 5 6 4 5 3 - 2 X1 X2 x - 2 X1 x + X1 X2 x + 4 X1 x + x - 3 X1 x - 2 x + X1 x 4 2 / 8 7 8 + 2 x - 3 x + 3 x - 1) / ((-1 + x) (X1 X2 x - X1 X2 x - X1 x / 8 7 8 5 6 4 5 6 - X2 x + X1 x + x + 2 X1 X2 x + X1 x - X1 X2 x - 3 X1 x - x 4 5 3 4 3 2 + 3 X1 x + x - X1 x - 2 x + x + 2 x - 3 x + 1)) and in Maple format (X1^2*X2*x^11-X1^2*x^11-X1*X2*x^11-X1*X2*x^10+X1*x^11+X1*X2*x^9+X1*x^10+X2*x^10 -X1*X2*x^8-X1*x^9-X2*x^9-x^10+X1*X2*x^7+X1*x^8+x^9+X1*X2*x^6-X1*x^7-2*X1*X2*x^5 -2*X1*x^6+X1*X2*x^4+4*X1*x^5+x^6-3*X1*x^4-2*x^5+X1*x^3+2*x^4-3*x^2+3*x-1)/(-1+x )/(X1*X2*x^8-X1*X2*x^7-X1*x^8-X2*x^8+X1*x^7+x^8+2*X1*X2*x^5+X1*x^6-X1*X2*x^4-3* X1*x^5-x^6+3*X1*x^4+x^5-X1*x^3-2*x^4+x^3+2*x^2-3*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 2, 2], are n 315 39 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 3, 4], are n 135 65 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 4 3 5 ----------- 39 and in floating point 0.3972290613 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 1/2 4 3 5 ate normal pair with correlation, ----------- 39 1/2 1/2 4 3 5 667 i.e. , [[-----------, 0], [0, ---]] 39 507 ------------------------------------------------- Theorem Number, 184, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [3, 2, 2], nor the composition, [1, 4, 3] Then infinity ----- 8 7 6 5 3 \ n x + x + x + x + x - x + 1 ) a(n) x = - ----------------------------------- / 3 4 3 2 ----- (x - x + 1) (x + x + x + x - 1) n = 0 and in Maple format -(x^8+x^7+x^6+x^5+x^3-x+1)/(x^3-x+1)/(x^4+x^3+x^2+x-1) The asymptotic expression for a(n) is, n 0.63242246488998289333 1.9275619754829253043 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [3, 2, 2] and d occurrences (as containment) of the composition, [1, 4, 3], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 11 2 11 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x - X1 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 11 10 11 9 10 10 8 - X1 X2 x - X1 X2 x + X1 x + X1 X2 x + X1 x + X2 x - X1 X2 x 9 9 10 7 8 9 6 7 - X1 x - X2 x - x + X1 X2 x + X1 x + x + X1 X2 x - X1 x 5 6 4 5 6 4 5 3 - 2 X1 X2 x - 2 X1 x + X1 X2 x + 4 X1 x + x - 3 X1 x - 2 x + X1 x 4 2 / 8 7 8 + 2 x - 3 x + 3 x - 1) / ((-1 + x) (X1 X2 x - X1 X2 x - X1 x / 8 7 8 5 6 4 5 6 - X2 x + X1 x + x + 2 X1 X2 x + X1 x - X1 X2 x - 3 X1 x - x 4 5 3 4 3 2 + 3 X1 x + x - X1 x - 2 x + x + 2 x - 3 x + 1)) and in Maple format (X1^2*X2*x^11-X1^2*x^11-X1*X2*x^11-X1*X2*x^10+X1*x^11+X1*X2*x^9+X1*x^10+X2*x^10 -X1*X2*x^8-X1*x^9-X2*x^9-x^10+X1*X2*x^7+X1*x^8+x^9+X1*X2*x^6-X1*x^7-2*X1*X2*x^5 -2*X1*x^6+X1*X2*x^4+4*X1*x^5+x^6-3*X1*x^4-2*x^5+X1*x^3+2*x^4-3*x^2+3*x-1)/(-1+x )/(X1*X2*x^8-X1*X2*x^7-X1*x^8-X2*x^8+X1*x^7+x^8+2*X1*X2*x^5+X1*x^6-X1*X2*x^4-3* X1*x^5-x^6+3*X1*x^4+x^5-X1*x^3-2*x^4+x^3+2*x^2-3*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 2, 2], are n 315 39 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 4, 3], are n 135 65 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 4 3 5 ----------- 39 and in floating point 0.3972290613 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 1/2 4 3 5 ate normal pair with correlation, ----------- 39 1/2 1/2 4 3 5 667 i.e. , [[-----------, 0], [0, ---]] 39 507 ------------------------------------------------- Theorem Number, 185, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [3, 2, 2], nor the composition, [1, 5, 2] Then infinity ----- 8 7 6 5 3 \ n x + x + x + x + x - x + 1 ) a(n) x = - ----------------------------------- / 3 4 3 2 ----- (x - x + 1) (x + x + x + x - 1) n = 0 and in Maple format -(x^8+x^7+x^6+x^5+x^3-x+1)/(x^3-x+1)/(x^4+x^3+x^2+x-1) The asymptotic expression for a(n) is, n 0.63242246488998289333 1.9275619754829253043 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [3, 2, 2] and d occurrences (as containment) of the composition, [1, 5, 2], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 9 9 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x - X1 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 9 7 9 6 7 5 6 4 5 - X2 x + X1 X2 x + x - X1 X2 x - X1 x + X1 X2 x + X1 x - X1 x - x 3 4 3 2 / 8 7 8 8 + X1 x + x - x - x + 2 x - 1) / (X1 X2 x - X1 X2 x - X1 x - X2 x / 6 7 8 5 6 6 4 5 3 + 2 X1 X2 x + X1 x + x - X1 X2 x - X1 x - x + 2 X1 x + x - X1 x 4 3 2 - 2 x + x + 2 x - 3 x + 1) and in Maple format -(X1*X2*x^9-X1*x^9-X2*x^9+X1*X2*x^7+x^9-X1*X2*x^6-X1*x^7+X1*X2*x^5+X1*x^6-X1*x^ 4-x^5+X1*x^3+x^4-x^3-x^2+2*x-1)/(X1*X2*x^8-X1*X2*x^7-X1*x^8-X2*x^8+2*X1*X2*x^6+ X1*x^7+x^8-X1*X2*x^5-X1*x^6-x^6+2*X1*x^4+x^5-X1*x^3-2*x^4+x^3+2*x^2-3*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 2, 2], are n 315 39 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 5, 2], are n 111 57 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 16 13 19 -------------- 741 and in floating point 0.3393518736 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 16 247 ate normal pair with correlation, --------- 741 1/2 16 247 2735 i.e. , [[---------, 0], [0, ----]] 741 2223 ------------------------------------------------- Theorem Number, 186, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [3, 2, 2], nor the composition, [1, 6, 1] Then infinity ----- \ n ) a(n) x = / ----- n = 0 2 12 11 8 4 2 (x - x + 1) (x + x - x + x + x - 2 x + 1) - -------------------------------------------------------------- 10 9 8 7 6 5 3 2 2 (x + x + 2 x + x + x + x + x - x + 2 x - 1) (-1 + x) and in Maple format -(x^2-x+1)*(x^12+x^11-x^8+x^4+x^2-2*x+1)/(x^10+x^9+2*x^8+x^7+x^6+x^5+x^3-x^2+2* x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.67777121983368566754 1.9139965407423997306 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [3, 2, 2] and d occurrences (as containment) of the composition, [1, 6, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 16 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 16 2 16 2 14 16 16 2 13 - 2 X1 X2 x - X2 x - X1 X2 x + X1 x + 2 X2 x + X1 X2 x 14 2 14 16 13 14 2 13 14 + 2 X1 X2 x + X2 x - x - 2 X1 X2 x - X1 x - X2 x - 2 X2 x 12 13 13 14 12 12 13 12 + X1 X2 x + X1 x + 2 X2 x + x - X1 x - X2 x - x + x 9 8 9 9 7 8 8 + X1 X2 x - 2 X1 X2 x - X1 x - X2 x + 2 X1 X2 x + 2 X1 x + 2 X2 x 9 6 7 7 8 6 7 5 6 + x - X1 X2 x - 2 X1 x - X2 x - 2 x + 2 X1 x + x - 2 X1 x - x 4 5 3 4 2 / + 2 X1 x + 2 x - X1 x - 2 x + 3 x - 3 x + 1) / ((-1 + x) ( / 2 13 13 2 13 13 13 13 10 X1 X2 x - 2 X1 X2 x - X2 x + X1 x + 2 X2 x - x + X1 X2 x 9 10 10 8 9 9 10 7 + X1 X2 x - X1 x - X2 x - X1 X2 x - X1 x - X2 x + x + 2 X1 X2 x 8 8 9 6 7 7 8 6 7 + X1 x + X2 x + x - X1 X2 x - 2 X1 x - X2 x - x + 2 X1 x + x 5 6 4 5 3 4 3 2 - X1 x - x + 2 X1 x + x - X1 x - 2 x + x + 2 x - 3 x + 1)) and in Maple format -(X1*X2^2*x^16-2*X1*X2*x^16-X2^2*x^16-X1*X2^2*x^14+X1*x^16+2*X2*x^16+X1*X2^2*x^ 13+2*X1*X2*x^14+X2^2*x^14-x^16-2*X1*X2*x^13-X1*x^14-X2^2*x^13-2*X2*x^14+X1*X2*x ^12+X1*x^13+2*X2*x^13+x^14-X1*x^12-X2*x^12-x^13+x^12+X1*X2*x^9-2*X1*X2*x^8-X1*x ^9-X2*x^9+2*X1*X2*x^7+2*X1*x^8+2*X2*x^8+x^9-X1*X2*x^6-2*X1*x^7-X2*x^7-2*x^8+2* X1*x^6+x^7-2*X1*x^5-x^6+2*X1*x^4+2*x^5-X1*x^3-2*x^4+3*x^2-3*x+1)/(-1+x)/(X1*X2^ 2*x^13-2*X1*X2*x^13-X2^2*x^13+X1*x^13+2*X2*x^13-x^13+X1*X2*x^10+X1*X2*x^9-X1*x^ 10-X2*x^10-X1*X2*x^8-X1*x^9-X2*x^9+x^10+2*X1*X2*x^7+X1*x^8+X2*x^8+x^9-X1*X2*x^6 -2*X1*x^7-X2*x^7-x^8+2*X1*x^6+x^7-X1*x^5-x^6+2*X1*x^4+x^5-X1*x^3-2*x^4+x^3+2*x^ 2-3*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 2, 2], are n 315 39 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 6, 1], are n 49 53 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 2 39 53 ------------- 2067 and in floating point 0.04399058701 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 2 2067 ate normal pair with correlation, --------- 2067 1/2 2 2067 2075 i.e. , [[---------, 0], [0, ----]] 2067 2067 ------------------------------------------------- Theorem Number, 187, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [3, 2, 2], nor the composition, [2, 1, 5] Then infinity ----- \ n ) a(n) x = / ----- n = 0 13 12 11 9 8 7 6 5 3 x + x + x - x - x - x - 2 x - x - x + x - 1 - ---------------------------------------------------------------- 2 12 11 9 8 7 5 3 2 (x + 1) (x + x - x - x + x - 3 x + 3 x - x - 2 x + 1) and in Maple format -(x^13+x^12+x^11-x^9-x^8-x^7-2*x^6-x^5-x^3+x-1)/(x^2+1)/(x^12+x^11-x^9-x^8+x^7-\ 3*x^5+3*x^3-x^2-2*x+1) The asymptotic expression for a(n) is, n 0.64772484905004026630 1.9216163409913324968 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [3, 2, 2] and d occurrences (as containment) of the composition, [2, 1, 5], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 3 18 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 2 18 3 18 2 18 2 18 2 18 - 3 X1 X2 x - X1 X2 x + 3 X1 X2 x + 3 X1 X2 x - X1 x 18 18 2 15 2 14 15 2 15 - 3 X1 X2 x + X1 x + X1 X2 x - X1 X2 x - 2 X1 X2 x - X2 x 14 15 2 14 15 2 12 2 12 + 2 X1 X2 x + X1 x + X2 x + 2 X2 x - X1 X2 x - X1 X2 x 14 14 15 2 12 2 11 12 14 - X1 x - 2 X2 x - x + X1 x + X1 X2 x + 4 X1 X2 x + x 11 12 12 10 11 12 10 - 2 X1 X2 x - 3 X1 x - X2 x - X1 X2 x + X1 x + x + X1 x 10 8 10 7 8 8 6 + X2 x + X1 X2 x - x - 2 X1 X2 x - X1 x - X2 x + 2 X1 X2 x 7 7 8 5 6 6 7 5 + 2 X1 x + 2 X2 x + x - X1 X2 x - X1 x - X2 x - 2 x - X1 x 4 5 3 4 2 / 2 3 19 + 2 X1 x + 2 x - X1 x - 2 x + 3 x - 3 x + 1) / (X1 X2 x / 2 3 18 2 2 19 3 19 2 2 18 2 19 - X1 X2 x - 3 X1 X2 x - X1 X2 x + 3 X1 X2 x + 3 X1 X2 x 3 18 2 19 2 18 2 19 2 18 + X1 X2 x + 3 X1 X2 x - 3 X1 X2 x - X1 x - 3 X1 X2 x 19 2 18 18 19 2 2 15 2 16 - 3 X1 X2 x + X1 x + 3 X1 X2 x + X1 x - X1 X2 x + X1 X2 x 18 2 15 16 2 16 2 15 16 - X1 x + 2 X1 X2 x - 2 X1 X2 x - X2 x - X1 x + X1 x 2 15 16 2 13 2 13 15 16 2 13 + X2 x + 2 X2 x - X1 X2 x - X1 X2 x - 2 X2 x - x + X1 x 2 12 13 15 2 11 12 13 + 2 X1 X2 x + 4 X1 X2 x + x - X1 X2 x - 4 X1 X2 x - 3 X1 x 13 12 13 10 11 11 10 - X2 x + 2 X1 x + x + 2 X1 X2 x + X1 x + 2 X2 x - 2 X1 x 10 11 8 10 7 8 8 - 2 X2 x - 2 x - 2 X1 X2 x + 2 x + 3 X1 X2 x + 2 X1 x + 3 X2 x 6 7 7 8 5 6 6 7 - 3 X1 X2 x - 2 X1 x - 3 X2 x - 3 x + X1 X2 x + X1 x + X2 x + 2 x 5 6 4 5 3 4 3 2 + 2 X1 x + x - 3 X1 x - 3 x + X1 x + 3 x + x - 5 x + 4 x - 1) and in Maple format -(X1^2*X2^3*x^18-3*X1^2*X2^2*x^18-X1*X2^3*x^18+3*X1^2*X2*x^18+3*X1*X2^2*x^18-X1 ^2*x^18-3*X1*X2*x^18+X1*x^18+X1*X2^2*x^15-X1*X2^2*x^14-2*X1*X2*x^15-X2^2*x^15+2 *X1*X2*x^14+X1*x^15+X2^2*x^14+2*X2*x^15-X1^2*X2*x^12-X1*X2^2*x^12-X1*x^14-2*X2* x^14-x^15+X1^2*x^12+X1*X2^2*x^11+4*X1*X2*x^12+x^14-2*X1*X2*x^11-3*X1*x^12-X2*x^ 12-X1*X2*x^10+X1*x^11+x^12+X1*x^10+X2*x^10+X1*X2*x^8-x^10-2*X1*X2*x^7-X1*x^8-X2 *x^8+2*X1*X2*x^6+2*X1*x^7+2*X2*x^7+x^8-X1*X2*x^5-X1*x^6-X2*x^6-2*x^7-X1*x^5+2* X1*x^4+2*x^5-X1*x^3-2*x^4+3*x^2-3*x+1)/(X1^2*X2^3*x^19-X1^2*X2^3*x^18-3*X1^2*X2 ^2*x^19-X1*X2^3*x^19+3*X1^2*X2^2*x^18+3*X1^2*X2*x^19+X1*X2^3*x^18+3*X1*X2^2*x^ 19-3*X1^2*X2*x^18-X1^2*x^19-3*X1*X2^2*x^18-3*X1*X2*x^19+X1^2*x^18+3*X1*X2*x^18+ X1*x^19-X1^2*X2^2*x^15+X1*X2^2*x^16-X1*x^18+2*X1^2*X2*x^15-2*X1*X2*x^16-X2^2*x^ 16-X1^2*x^15+X1*x^16+X2^2*x^15+2*X2*x^16-X1^2*X2*x^13-X1*X2^2*x^13-2*X2*x^15-x^ 16+X1^2*x^13+2*X1*X2^2*x^12+4*X1*X2*x^13+x^15-X1*X2^2*x^11-4*X1*X2*x^12-3*X1*x^ 13-X2*x^13+2*X1*x^12+x^13+2*X1*X2*x^10+X1*x^11+2*X2*x^11-2*X1*x^10-2*X2*x^10-2* x^11-2*X1*X2*x^8+2*x^10+3*X1*X2*x^7+2*X1*x^8+3*X2*x^8-3*X1*X2*x^6-2*X1*x^7-3*X2 *x^7-3*x^8+X1*X2*x^5+X1*x^6+X2*x^6+2*x^7+2*X1*x^5+x^6-3*X1*x^4-3*x^5+X1*x^3+3*x ^4+x^3-5*x^2+4*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 2, 2], are n 315 39 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 1, 5], are n 113 57 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 10 13 19 -------------- 741 and in floating point 0.2120949210 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 10 247 ate normal pair with correlation, --------- 741 1/2 10 247 2423 i.e. , [[---------, 0], [0, ----]] 741 2223 ------------------------------------------------- Theorem Number, 188, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [3, 2, 2], nor the composition, [2, 2, 4] Then infinity ----- 10 6 4 3 2 \ n x - x - x + x - 2 x + 2 x - 1 ) a(n) x = - ------------------------------------------ / 11 7 5 4 3 2 ----- x - x - x + x - 2 x + 3 x - 3 x + 1 n = 0 and in Maple format -(x^10-x^6-x^4+x^3-2*x^2+2*x-1)/(x^11-x^7-x^5+x^4-2*x^3+3*x^2-3*x+1) The asymptotic expression for a(n) is, n 0.61318603545226866829 1.9346824023256656451 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [3, 2, 2] and d occurrences (as containment) of the composition, [2, 2, 4], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 13 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 13 2 13 2 13 2 12 13 - 2 X1 X2 x - 2 X1 X2 x + X1 x + X1 X2 x + 4 X1 X2 x 2 13 2 11 12 13 2 12 13 + X2 x + X1 X2 x - 2 X1 X2 x - 2 X1 x - X2 x - 2 X2 x 2 11 11 12 12 13 10 11 - X1 x - 2 X1 X2 x + X1 x + 2 X2 x + x + X1 X2 x + 2 X1 x 11 12 9 10 10 11 8 9 + X2 x - x - X1 X2 x - X1 x - X2 x - x + X1 X2 x + X1 x 9 10 7 8 8 9 6 7 8 + X2 x + x - X1 X2 x - X1 x - X2 x - x + 2 X1 X2 x + X1 x + x 5 6 4 5 6 4 5 3 - 2 X1 X2 x - 3 X1 x + X1 X2 x + 4 X1 x + x - 3 X1 x - 2 x + X1 x 4 2 / 2 2 14 2 14 2 14 + 2 x - 3 x + 3 x - 1) / (X1 X2 x - 2 X1 X2 x - 2 X1 X2 x / 2 14 2 13 14 2 14 2 12 13 + X1 x + X1 X2 x + 4 X1 X2 x + X2 x + X1 X2 x - 2 X1 X2 x 14 2 13 14 2 12 12 13 - 2 X1 x - X2 x - 2 X2 x - X1 x - 2 X1 X2 x + X1 x 13 14 11 12 12 13 10 + 2 X2 x + x + X1 X2 x + 2 X1 x + X2 x - x - X1 X2 x 11 11 12 9 10 10 11 8 - X1 x - X2 x - x + X1 X2 x + X1 x + X2 x + x - X1 X2 x 9 9 10 7 8 9 6 7 - X1 x - X2 x - x + 2 X1 X2 x + X1 x + x - 3 X1 X2 x - 3 X1 x 5 6 7 4 5 6 4 5 + 3 X1 X2 x + 5 X1 x + x - X1 X2 x - 6 X1 x - 2 x + 4 X1 x + 3 x 3 4 3 2 - X1 x - 3 x - x + 5 x - 4 x + 1) and in Maple format -(X1^2*X2^2*x^13-2*X1^2*X2*x^13-2*X1*X2^2*x^13+X1^2*x^13+X1*X2^2*x^12+4*X1*X2*x ^13+X2^2*x^13+X1^2*X2*x^11-2*X1*X2*x^12-2*X1*x^13-X2^2*x^12-2*X2*x^13-X1^2*x^11 -2*X1*X2*x^11+X1*x^12+2*X2*x^12+x^13+X1*X2*x^10+2*X1*x^11+X2*x^11-x^12-X1*X2*x^ 9-X1*x^10-X2*x^10-x^11+X1*X2*x^8+X1*x^9+X2*x^9+x^10-X1*X2*x^7-X1*x^8-X2*x^8-x^9 +2*X1*X2*x^6+X1*x^7+x^8-2*X1*X2*x^5-3*X1*x^6+X1*X2*x^4+4*X1*x^5+x^6-3*X1*x^4-2* x^5+X1*x^3+2*x^4-3*x^2+3*x-1)/(X1^2*X2^2*x^14-2*X1^2*X2*x^14-2*X1*X2^2*x^14+X1^ 2*x^14+X1*X2^2*x^13+4*X1*X2*x^14+X2^2*x^14+X1^2*X2*x^12-2*X1*X2*x^13-2*X1*x^14- X2^2*x^13-2*X2*x^14-X1^2*x^12-2*X1*X2*x^12+X1*x^13+2*X2*x^13+x^14+X1*X2*x^11+2* X1*x^12+X2*x^12-x^13-X1*X2*x^10-X1*x^11-X2*x^11-x^12+X1*X2*x^9+X1*x^10+X2*x^10+ x^11-X1*X2*x^8-X1*x^9-X2*x^9-x^10+2*X1*X2*x^7+X1*x^8+x^9-3*X1*X2*x^6-3*X1*x^7+3 *X1*X2*x^5+5*X1*x^6+x^7-X1*X2*x^4-6*X1*x^5-2*x^6+4*X1*x^4+3*x^5-X1*x^3-3*x^4-x^ 3+5*x^2-4*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 2, 2], are n 315 39 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 2, 4], are n 75 69 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 2 13 23 ------------- 69 and in floating point 0.5012062743 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 1/2 2 13 23 ate normal pair with correlation, ------------- 69 1/2 1/2 2 13 23 311 i.e. , [[-------------, 0], [0, ---]] 69 207 ------------------------------------------------- Theorem Number, 189, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [3, 2, 2], nor the composition, [2, 3, 3] Then infinity ----- 6 4 3 2 \ n x + x - x + 2 x - 2 x + 1 ) a(n) x = - ------------------------------------ / 7 5 4 3 2 ----- x + x - x + 2 x - 3 x + 3 x - 1 n = 0 and in Maple format -(x^6+x^4-x^3+2*x^2-2*x+1)/(x^7+x^5-x^4+2*x^3-3*x^2+3*x-1) The asymptotic expression for a(n) is, n 0.60945823930910114296 1.9365136360000287543 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [3, 2, 2] and d occurrences (as containment) of the composition, [2, 3, 3], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 10 2 10 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x - X1 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 10 9 10 8 9 9 7 - X1 X2 x - X1 X2 x + X1 x + X1 X2 x + X1 x + X2 x - X1 X2 x 8 8 9 6 7 8 5 6 - X1 x - X2 x - x + X1 X2 x + X1 x + x + X1 X2 x - 2 X1 x 4 5 6 3 5 4 2 / - 2 X1 X2 x + X1 x + x + X1 X2 x - 2 x + 2 x - 3 x + 3 x - 1) / ( / 2 11 2 11 11 10 11 9 10 X1 X2 x - X1 x - X1 X2 x - X1 X2 x + X1 x + X1 X2 x + X1 x 10 8 9 9 10 7 8 9 + X2 x - X1 X2 x - X1 x - X2 x - x + 2 X1 X2 x + X1 x + x 6 7 5 6 7 4 5 - X1 X2 x - 3 X1 x - 2 X1 X2 x + 3 X1 x + x + 3 X1 X2 x - X1 x 6 3 5 4 3 2 - 2 x - X1 X2 x + 3 x - 3 x - x + 5 x - 4 x + 1) and in Maple format -(X1^2*X2*x^10-X1^2*x^10-X1*X2*x^10-X1*X2*x^9+X1*x^10+X1*X2*x^8+X1*x^9+X2*x^9- X1*X2*x^7-X1*x^8-X2*x^8-x^9+X1*X2*x^6+X1*x^7+x^8+X1*X2*x^5-2*X1*x^6-2*X1*X2*x^4 +X1*x^5+x^6+X1*X2*x^3-2*x^5+2*x^4-3*x^2+3*x-1)/(X1^2*X2*x^11-X1^2*x^11-X1*X2*x^ 11-X1*X2*x^10+X1*x^11+X1*X2*x^9+X1*x^10+X2*x^10-X1*X2*x^8-X1*x^9-X2*x^9-x^10+2* X1*X2*x^7+X1*x^8+x^9-X1*X2*x^6-3*X1*x^7-2*X1*X2*x^5+3*X1*x^6+x^7+3*X1*X2*x^4-X1 *x^5-2*x^6-X1*X2*x^3+3*x^5-3*x^4-x^3+5*x^2-4*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 2, 2], are n 315 39 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 3, 3], are n 97 85 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 2 39 85 ------------- 195 and in floating point 0.5905234533 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 1/2 2 17 195 ate normal pair with correlation, -------------- 195 1/2 1/2 2 17 195 331 i.e. , [[--------------, 0], [0, ---]] 195 195 ------------------------------------------------- Theorem Number, 190, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [3, 2, 2], nor the composition, [2, 4, 2] Then infinity ----- 6 4 3 2 \ n x + x - x + 2 x - 2 x + 1 ) a(n) x = - ------------------------------------ / 7 5 4 3 2 ----- x + x - x + 2 x - 3 x + 3 x - 1 n = 0 and in Maple format -(x^6+x^4-x^3+2*x^2-2*x+1)/(x^7+x^5-x^4+2*x^3-3*x^2+3*x-1) The asymptotic expression for a(n) is, n 0.60945823930910114296 1.9365136360000287543 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [3, 2, 2] and d occurrences (as containment) of the composition, [2, 4, 2], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 8 8 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x - X1 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 8 6 8 5 6 4 5 4 - X2 x + X1 X2 x + x - X1 X2 x - X1 x + X1 X2 x + 2 X1 x - 2 X1 x 5 3 4 3 2 / 9 9 9 - x + X1 x + x - x - x + 2 x - 1) / (X1 X2 x - X1 x - X2 x / 7 9 6 7 5 6 4 + X1 X2 x + x - X1 X2 x - X1 x + 2 X1 X2 x + 2 X1 x - X1 X2 x 5 6 4 5 3 4 3 2 - 3 X1 x - x + 3 X1 x + x - X1 x - 2 x + x + 2 x - 3 x + 1) and in Maple format -(X1*X2*x^8-X1*x^8-X2*x^8+X1*X2*x^6+x^8-X1*X2*x^5-X1*x^6+X1*X2*x^4+2*X1*x^5-2* X1*x^4-x^5+X1*x^3+x^4-x^3-x^2+2*x-1)/(X1*X2*x^9-X1*x^9-X2*x^9+X1*X2*x^7+x^9-X1* X2*x^6-X1*x^7+2*X1*X2*x^5+2*X1*x^6-X1*X2*x^4-3*X1*x^5-x^6+3*X1*x^4+x^5-X1*x^3-2 *x^4+x^3+2*x^2-3*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 2, 2], are n 315 39 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 4, 2], are n 75 69 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 10 13 23 -------------- 299 and in floating point 0.5783149317 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 10 299 ate normal pair with correlation, --------- 299 1/2 10 299 499 i.e. , [[---------, 0], [0, ---]] 299 299 ------------------------------------------------- Theorem Number, 191, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [3, 2, 2], nor the composition, [2, 5, 1] Then infinity ----- \ n ) a(n) x = / ----- n = 0 8 7 6 5 3 x + x + x + x + x - x + 1 ------------------------------------------------------------------------- 11 10 9 8 7 6 5 4 2 (-1 + x) (x + 2 x + 3 x + 3 x + 3 x + 2 x + x + x + x + x - 1) and in Maple format (x^8+x^7+x^6+x^5+x^3-x+1)/(-1+x)/(x^11+2*x^10+3*x^9+3*x^8+3*x^7+2*x^6+x^5+x^4+x ^2+x-1) The asymptotic expression for a(n) is, n 0.64629602075846469709 1.9230256556702387387 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [3, 2, 2] and d occurrences (as containment) of the composition, [2, 5, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 9 9 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x - X1 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 9 7 9 6 7 5 6 4 5 - X2 x + X1 X2 x + x - X1 X2 x - X1 x + X1 X2 x + X1 x - X1 x - x 3 4 3 2 / 2 13 13 2 13 + X1 x + x - x - x + 2 x - 1) / (X1 X2 x - 2 X1 X2 x - X2 x / 13 13 13 10 10 10 8 10 + X1 x + 2 X2 x - x + X1 X2 x - X1 x - X2 x + X1 X2 x + x 7 8 8 6 7 8 5 6 - X1 X2 x - X1 x - X2 x + 2 X1 X2 x + X1 x + x - X1 X2 x - X1 x 6 4 5 3 4 3 2 - x + 2 X1 x + x - X1 x - 2 x + x + 2 x - 3 x + 1) and in Maple format -(X1*X2*x^9-X1*x^9-X2*x^9+X1*X2*x^7+x^9-X1*X2*x^6-X1*x^7+X1*X2*x^5+X1*x^6-X1*x^ 4-x^5+X1*x^3+x^4-x^3-x^2+2*x-1)/(X1*X2^2*x^13-2*X1*X2*x^13-X2^2*x^13+X1*x^13+2* X2*x^13-x^13+X1*X2*x^10-X1*x^10-X2*x^10+X1*X2*x^8+x^10-X1*X2*x^7-X1*x^8-X2*x^8+ 2*X1*X2*x^6+X1*x^7+x^8-X1*X2*x^5-X1*x^6-x^6+2*X1*x^4+x^5-X1*x^3-2*x^4+x^3+2*x^2 -3*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 2, 2], are n 315 39 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 5, 1], are n 111 57 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 4 13 19 ------------- 247 and in floating point 0.2545139052 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 4 247 ate normal pair with correlation, -------- 247 1/2 4 247 279 i.e. , [[--------, 0], [0, ---]] 247 247 ------------------------------------------------- Theorem Number, 192, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [3, 2, 2], nor the composition, [3, 1, 4] Then infinity ----- 7 6 4 3 2 \ n x - x + 2 x - 3 x + 4 x - 3 x + 1 ) a(n) x = - ---------------------------------------------------- / 9 7 6 5 4 3 2 ----- x - 2 x + 2 x + x - 3 x + 5 x - 6 x + 4 x - 1 n = 0 and in Maple format -(x^7-x^6+2*x^4-3*x^3+4*x^2-3*x+1)/(x^9-2*x^7+2*x^6+x^5-3*x^4+5*x^3-6*x^2+4*x-1 ) The asymptotic expression for a(n) is, n 0.63104184045431048184 1.9285748396761772314 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [3, 2, 2] and d occurrences (as containment) of the composition, [3, 1, 4], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 10 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 10 2 9 2 10 2 9 2 9 9 - 2 X1 X2 x + X1 X2 x + X1 x + X1 X2 x - X1 x - 3 X1 X2 x 8 9 8 8 6 8 5 6 + X1 X2 x + 2 X1 x - X1 x - X2 x - X1 X2 x + x + X1 X2 x + X1 x 6 4 5 5 6 4 5 3 4 + X2 x - X1 X2 x - 2 X1 x - X2 x - x + 2 X1 x + 2 x - X1 x - x 3 2 / 2 2 12 2 2 11 2 12 + x + x - 2 x + 1) / (X1 X2 x + X1 X2 x - 2 X1 X2 x / 2 2 10 2 11 2 12 2 11 2 10 - X1 X2 x - 2 X1 X2 x + X1 x + X1 X2 x + 3 X1 X2 x 2 11 2 10 11 2 9 2 10 2 9 + X1 x + X1 X2 x - 2 X1 X2 x - X1 X2 x - 2 X1 x - X1 X2 x 10 11 2 9 9 10 10 - 2 X1 X2 x + X1 x + X1 x + 4 X1 X2 x + X1 x - X2 x 8 9 9 10 7 8 8 9 - 2 X1 X2 x - 3 X1 x - X2 x + x - X1 X2 x + 2 X1 x + 2 X2 x + x 6 7 8 5 6 6 4 + X1 X2 x + X1 x - 2 x - 2 X1 X2 x - 2 X1 x - 2 X2 x + X1 X2 x 5 5 6 4 5 3 4 3 2 + 3 X1 x + X2 x + 3 x - 3 X1 x - 2 x + X1 x + 2 x - x - 2 x + 3 x - 1) and in Maple format -(X1^2*X2^2*x^10-2*X1^2*X2*x^10+X1^2*X2*x^9+X1^2*x^10+X1*X2^2*x^9-X1^2*x^9-3*X1 *X2*x^9+X1*X2*x^8+2*X1*x^9-X1*x^8-X2*x^8-X1*X2*x^6+x^8+X1*X2*x^5+X1*x^6+X2*x^6- X1*X2*x^4-2*X1*x^5-X2*x^5-x^6+2*X1*x^4+2*x^5-X1*x^3-x^4+x^3+x^2-2*x+1)/(X1^2*X2 ^2*x^12+X1^2*X2^2*x^11-2*X1^2*X2*x^12-X1^2*X2^2*x^10-2*X1^2*X2*x^11+X1^2*x^12+ X1*X2^2*x^11+3*X1^2*X2*x^10+X1^2*x^11+X1*X2^2*x^10-2*X1*X2*x^11-X1^2*X2*x^9-2* X1^2*x^10-X1*X2^2*x^9-2*X1*X2*x^10+X1*x^11+X1^2*x^9+4*X1*X2*x^9+X1*x^10-X2*x^10 -2*X1*X2*x^8-3*X1*x^9-X2*x^9+x^10-X1*X2*x^7+2*X1*x^8+2*X2*x^8+x^9+X1*X2*x^6+X1* x^7-2*x^8-2*X1*X2*x^5-2*X1*x^6-2*X2*x^6+X1*X2*x^4+3*X1*x^5+X2*x^5+3*x^6-3*X1*x^ 4-2*x^5+X1*x^3+2*x^4-x^3-2*x^2+3*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 2, 2], are n 315 39 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 1, 4], are n 141 65 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 6 3 5 ----------- 65 and in floating point 0.3575061550 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 1/2 6 3 5 ate normal pair with correlation, ----------- 65 1/2 1/2 6 3 5 1061 i.e. , [[-----------, 0], [0, ----]] 65 845 ------------------------------------------------- Theorem Number, 193, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [3, 2, 2], nor the composition, [3, 4, 1] Then infinity ----- 5 4 3 2 \ n x - x + x + x - 2 x + 1 ) a(n) x = -------------------------------------------- / 3 4 3 2 ----- (-1 + x) (x - x + 1) (x + x + x + x - 1) n = 0 and in Maple format (x^5-x^4+x^3+x^2-2*x+1)/(-1+x)/(x^3-x+1)/(x^4+x^3+x^2+x-1) The asymptotic expression for a(n) is, n 0.63758295460268560269 1.9275619754829253043 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [3, 2, 2] and d occurrences (as containment) of the composition, [3, 4, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 7 7 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x - X1 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 5 4 5 4 5 3 4 3 2 - X1 X2 x + X1 X2 x + 2 X1 x - 2 X1 x - x + X1 x + x - x - x / 8 7 8 8 7 8 + 2 x - 1) / (X1 X2 x - X1 X2 x - X1 x - X2 x + X1 x + x / 5 6 4 5 6 4 5 3 + 2 X1 X2 x + X1 x - X1 X2 x - 3 X1 x - x + 3 X1 x + x - X1 x 4 3 2 - 2 x + x + 2 x - 3 x + 1) and in Maple format -(X1*X2*x^7-X1*x^7-X1*X2*x^5+X1*X2*x^4+2*X1*x^5-2*X1*x^4-x^5+X1*x^3+x^4-x^3-x^2 +2*x-1)/(X1*X2*x^8-X1*X2*x^7-X1*x^8-X2*x^8+X1*x^7+x^8+2*X1*X2*x^5+X1*x^6-X1*X2* x^4-3*X1*x^5-x^6+3*X1*x^4+x^5-X1*x^3-2*x^4+x^3+2*x^2-3*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 2, 2], are n 315 39 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 4, 1], are n 135 65 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 4 3 5 ----------- 39 and in floating point 0.3972290613 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 1/2 4 3 5 ate normal pair with correlation, ----------- 39 1/2 1/2 4 3 5 667 i.e. , [[-----------, 0], [0, ---]] 39 507 ------------------------------------------------- Theorem Number, 194, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [3, 2, 2], nor the composition, [4, 1, 3] Then infinity ----- 7 6 4 3 2 \ n x - x + 2 x - 3 x + 4 x - 3 x + 1 ) a(n) x = - ---------------------------------------------------- / 9 7 6 5 4 3 2 ----- x - 2 x + 2 x + x - 3 x + 5 x - 6 x + 4 x - 1 n = 0 and in Maple format -(x^7-x^6+2*x^4-3*x^3+4*x^2-3*x+1)/(x^9-2*x^7+2*x^6+x^5-3*x^4+5*x^3-6*x^2+4*x-1 ) The asymptotic expression for a(n) is, n 0.63104184045431048184 1.9285748396761772314 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [3, 2, 2] and d occurrences (as containment) of the composition, [4, 1, 3], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 10 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 10 2 9 2 10 2 9 2 9 9 - 2 X1 X2 x + X1 X2 x + X1 x + X1 X2 x - X1 x - 3 X1 X2 x 8 9 8 8 6 8 5 6 + X1 X2 x + 2 X1 x - X1 x - X2 x - X1 X2 x + x + X1 X2 x + X1 x 6 4 5 5 6 4 5 3 4 + X2 x - X1 X2 x - 2 X1 x - X2 x - x + 2 X1 x + 2 x - X1 x - x 3 2 / 2 2 12 2 2 11 2 12 + x + x - 2 x + 1) / (X1 X2 x + X1 X2 x - 2 X1 X2 x / 2 2 10 2 11 2 12 2 11 2 10 - X1 X2 x - 2 X1 X2 x + X1 x + X1 X2 x + 3 X1 X2 x 2 11 2 10 11 2 9 2 10 2 9 + X1 x + X1 X2 x - 2 X1 X2 x - X1 X2 x - 2 X1 x - X1 X2 x 10 11 2 9 9 10 10 - 2 X1 X2 x + X1 x + X1 x + 4 X1 X2 x + X1 x - X2 x 8 9 9 10 7 8 8 9 - 2 X1 X2 x - 3 X1 x - X2 x + x - X1 X2 x + 2 X1 x + 2 X2 x + x 6 7 8 5 6 6 4 + X1 X2 x + X1 x - 2 x - 2 X1 X2 x - 2 X1 x - 2 X2 x + X1 X2 x 5 5 6 4 5 3 4 3 2 + 3 X1 x + X2 x + 3 x - 3 X1 x - 2 x + X1 x + 2 x - x - 2 x + 3 x - 1) and in Maple format -(X1^2*X2^2*x^10-2*X1^2*X2*x^10+X1^2*X2*x^9+X1^2*x^10+X1*X2^2*x^9-X1^2*x^9-3*X1 *X2*x^9+X1*X2*x^8+2*X1*x^9-X1*x^8-X2*x^8-X1*X2*x^6+x^8+X1*X2*x^5+X1*x^6+X2*x^6- X1*X2*x^4-2*X1*x^5-X2*x^5-x^6+2*X1*x^4+2*x^5-X1*x^3-x^4+x^3+x^2-2*x+1)/(X1^2*X2 ^2*x^12+X1^2*X2^2*x^11-2*X1^2*X2*x^12-X1^2*X2^2*x^10-2*X1^2*X2*x^11+X1^2*x^12+ X1*X2^2*x^11+3*X1^2*X2*x^10+X1^2*x^11+X1*X2^2*x^10-2*X1*X2*x^11-X1^2*X2*x^9-2* X1^2*x^10-X1*X2^2*x^9-2*X1*X2*x^10+X1*x^11+X1^2*x^9+4*X1*X2*x^9+X1*x^10-X2*x^10 -2*X1*X2*x^8-3*X1*x^9-X2*x^9+x^10-X1*X2*x^7+2*X1*x^8+2*X2*x^8+x^9+X1*X2*x^6+X1* x^7-2*x^8-2*X1*X2*x^5-2*X1*x^6-2*X2*x^6+X1*X2*x^4+3*X1*x^5+X2*x^5+3*x^6-3*X1*x^ 4-2*x^5+X1*x^3+2*x^4-x^3-2*x^2+3*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 2, 2], are n 315 39 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 1, 3], are n 141 65 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 6 3 5 ----------- 65 and in floating point 0.3575061550 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 1/2 6 3 5 ate normal pair with correlation, ----------- 65 1/2 1/2 6 3 5 1061 i.e. , [[-----------, 0], [0, ----]] 65 845 ------------------------------------------------- Theorem Number, 195, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [3, 2, 2], nor the composition, [4, 3, 1] Then infinity ----- 5 4 3 2 \ n x - x + x + x - 2 x + 1 ) a(n) x = -------------------------------------------- / 3 4 3 2 ----- (-1 + x) (x - x + 1) (x + x + x + x - 1) n = 0 and in Maple format (x^5-x^4+x^3+x^2-2*x+1)/(-1+x)/(x^3-x+1)/(x^4+x^3+x^2+x-1) The asymptotic expression for a(n) is, n 0.63758295460268560269 1.9275619754829253043 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [3, 2, 2] and d occurrences (as containment) of the composition, [4, 3, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 7 7 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x - X1 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 5 4 5 4 5 3 4 3 2 - X1 X2 x + X1 X2 x + 2 X1 x - 2 X1 x - x + X1 x + x - x - x / 8 7 8 8 7 8 + 2 x - 1) / (X1 X2 x - X1 X2 x - X1 x - X2 x + X1 x + x / 5 6 4 5 6 4 5 3 + 2 X1 X2 x + X1 x - X1 X2 x - 3 X1 x - x + 3 X1 x + x - X1 x 4 3 2 - 2 x + x + 2 x - 3 x + 1) and in Maple format -(X1*X2*x^7-X1*x^7-X1*X2*x^5+X1*X2*x^4+2*X1*x^5-2*X1*x^4-x^5+X1*x^3+x^4-x^3-x^2 +2*x-1)/(X1*X2*x^8-X1*X2*x^7-X1*x^8-X2*x^8+X1*x^7+x^8+2*X1*X2*x^5+X1*x^6-X1*X2* x^4-3*X1*x^5-x^6+3*X1*x^4+x^5-X1*x^3-2*x^4+x^3+2*x^2-3*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 2, 2], are n 315 39 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 3, 1], are n 135 65 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 4 3 5 ----------- 39 and in floating point 0.3972290613 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 1/2 4 3 5 ate normal pair with correlation, ----------- 39 1/2 1/2 4 3 5 667 i.e. , [[-----------, 0], [0, ---]] 39 507 ------------------------------------------------- Theorem Number, 196, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [3, 2, 2], nor the composition, [5, 1, 2] Then infinity ----- 9 7 6 5 4 3 2 \ n x - x + x + x - x + x + x - 2 x + 1 ) a(n) x = - ------------------------------------------------------ / 10 9 8 7 5 4 3 2 ----- x - x - x + 2 x - x + 2 x - x - 2 x + 3 x - 1 n = 0 and in Maple format -(x^9-x^7+x^6+x^5-x^4+x^3+x^2-2*x+1)/(x^10-x^9-x^8+2*x^7-x^5+2*x^4-x^3-2*x^2+3* x-1) The asymptotic expression for a(n) is, n 0.64281047729344093126 1.9255092871632283681 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [3, 2, 2] and d occurrences (as containment) of the composition, [5, 1, 2], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 11 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 11 11 9 9 9 7 9 - 2 X1 X2 x + X1 x + X1 X2 x - X1 x - X2 x - X1 X2 x + x 6 7 7 5 6 6 7 6 4 + X1 X2 x + X1 x + X2 x - X1 X2 x - X1 x - X2 x - x + x + X1 x 5 3 4 3 2 / 2 12 2 11 + x - X1 x - x + x + x - 2 x + 1) / (X1 X2 x - X1 X2 x / 12 11 12 10 11 9 - 2 X1 X2 x + 2 X1 X2 x + X1 x + X1 X2 x - X1 x - X1 X2 x 10 10 8 9 9 10 7 8 - X1 x - X2 x - X1 X2 x + X1 x + X2 x + x + X1 X2 x + X1 x 8 9 6 7 7 8 5 6 + X2 x - x - 2 X1 X2 x - X1 x - 2 X2 x - x + X1 X2 x + X1 x 6 7 4 5 3 4 3 2 + X2 x + 2 x - 2 X1 x - x + X1 x + 2 x - x - 2 x + 3 x - 1) and in Maple format -(X1*X2^2*x^11-2*X1*X2*x^11+X1*x^11+X1*X2*x^9-X1*x^9-X2*x^9-X1*X2*x^7+x^9+X1*X2 *x^6+X1*x^7+X2*x^7-X1*X2*x^5-X1*x^6-X2*x^6-x^7+x^6+X1*x^4+x^5-X1*x^3-x^4+x^3+x^ 2-2*x+1)/(X1*X2^2*x^12-X1*X2^2*x^11-2*X1*X2*x^12+2*X1*X2*x^11+X1*x^12+X1*X2*x^ 10-X1*x^11-X1*X2*x^9-X1*x^10-X2*x^10-X1*X2*x^8+X1*x^9+X2*x^9+x^10+X1*X2*x^7+X1* x^8+X2*x^8-x^9-2*X1*X2*x^6-X1*x^7-2*X2*x^7-x^8+X1*X2*x^5+X1*x^6+X2*x^6+2*x^7-2* X1*x^4-x^5+X1*x^3+2*x^4-x^3-2*x^2+3*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 2, 2], are n 315 39 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [5, 1, 2], are n 113 57 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 14 13 19 -------------- 741 and in floating point 0.2969328894 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 14 247 ate normal pair with correlation, --------- 741 1/2 14 247 2615 i.e. , [[---------, 0], [0, ----]] 741 2223 ------------------------------------------------- Theorem Number, 197, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [3, 2, 2], nor the composition, [5, 2, 1] Then infinity ----- 5 4 3 2 \ n x - x + x + x - 2 x + 1 ) a(n) x = -------------------------------------------- / 3 4 3 2 ----- (-1 + x) (x - x + 1) (x + x + x + x - 1) n = 0 and in Maple format (x^5-x^4+x^3+x^2-2*x+1)/(-1+x)/(x^3-x+1)/(x^4+x^3+x^2+x-1) The asymptotic expression for a(n) is, n 0.63758295460268560269 1.9275619754829253043 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [3, 2, 2] and d occurrences (as containment) of the composition, [5, 2, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 7 6 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x - X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 7 5 6 4 5 3 4 3 2 - X1 x + X1 X2 x + X1 x - X1 x - x + X1 x + x - x - x + 2 x - 1) / 8 7 8 8 6 7 8 / (X1 X2 x - X1 X2 x - X1 x - X2 x + 2 X1 X2 x + X1 x + x / 5 6 6 4 5 3 4 3 2 - X1 X2 x - X1 x - x + 2 X1 x + x - X1 x - 2 x + x + 2 x - 3 x + 1) and in Maple format -(X1*X2*x^7-X1*X2*x^6-X1*x^7+X1*X2*x^5+X1*x^6-X1*x^4-x^5+X1*x^3+x^4-x^3-x^2+2*x -1)/(X1*X2*x^8-X1*X2*x^7-X1*x^8-X2*x^8+2*X1*X2*x^6+X1*x^7+x^8-X1*X2*x^5-X1*x^6- x^6+2*X1*x^4+x^5-X1*x^3-2*x^4+x^3+2*x^2-3*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 2, 2], are n 315 39 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [5, 2, 1], are n 111 57 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 16 13 19 -------------- 741 and in floating point 0.3393518736 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 16 247 ate normal pair with correlation, --------- 741 1/2 16 247 2735 i.e. , [[---------, 0], [0, ----]] 741 2223 ------------------------------------------------- Theorem Number, 198, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [3, 2, 2], nor the composition, [6, 1, 1] Then infinity ----- \ n ) a(n) x = / ----- n = 0 13 10 8 7 5 4 3 2 x - x + x - x - x + x - x - x + 2 x - 1 ---------------------------------------------------------------------- 10 9 8 7 6 5 3 2 2 (x + 1) (x + x + 2 x + x + x + x + x - x + 2 x - 1) (-1 + x) and in Maple format (x^13-x^10+x^8-x^7-x^5+x^4-x^3-x^2+2*x-1)/(x+1)/(x^10+x^9+2*x^8+x^7+x^6+x^5+x^3 -x^2+2*x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.67919107669695186122 1.9139965407423997306 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [3, 2, 2] and d occurrences (as containment) of the composition, [6, 1, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 13 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 13 2 13 13 13 13 10 10 - 2 X1 X2 x - X2 x + X1 x + 2 X2 x - x + X1 X2 x - X1 x 10 8 10 7 8 8 6 7 - X2 x - X1 X2 x + x + X1 X2 x + X1 x + X2 x - X1 X2 x - X1 x 7 8 6 7 5 4 5 3 4 3 2 - X2 x - x + X1 x + x - X1 x + X1 x + x - X1 x - x + x + x / 2 13 13 2 13 13 13 - 2 x + 1) / (X1 X2 x - 2 X1 X2 x - X2 x + X1 x + 2 X2 x / 13 10 9 10 10 8 9 9 - x + X1 X2 x + X1 X2 x - X1 x - X2 x - X1 X2 x - X1 x - X2 x 10 7 8 8 9 6 7 7 8 + x + 2 X1 X2 x + X1 x + X2 x + x - X1 X2 x - 2 X1 x - X2 x - x 6 7 5 6 4 5 3 4 3 2 + 2 X1 x + x - X1 x - x + 2 X1 x + x - X1 x - 2 x + x + 2 x - 3 x + 1) and in Maple format (X1*X2^2*x^13-2*X1*X2*x^13-X2^2*x^13+X1*x^13+2*X2*x^13-x^13+X1*X2*x^10-X1*x^10- X2*x^10-X1*X2*x^8+x^10+X1*X2*x^7+X1*x^8+X2*x^8-X1*X2*x^6-X1*x^7-X2*x^7-x^8+X1*x ^6+x^7-X1*x^5+X1*x^4+x^5-X1*x^3-x^4+x^3+x^2-2*x+1)/(X1*X2^2*x^13-2*X1*X2*x^13- X2^2*x^13+X1*x^13+2*X2*x^13-x^13+X1*X2*x^10+X1*X2*x^9-X1*x^10-X2*x^10-X1*X2*x^8 -X1*x^9-X2*x^9+x^10+2*X1*X2*x^7+X1*x^8+X2*x^8+x^9-X1*X2*x^6-2*X1*x^7-X2*x^7-x^8 +2*X1*x^6+x^7-X1*x^5-x^6+2*X1*x^4+x^5-X1*x^3-2*x^4+x^3+2*x^2-3*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 2, 2], are n 315 39 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [6, 1, 1], are n 49 53 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 2 39 53 ------------- 2067 and in floating point 0.04399058701 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 2 2067 ate normal pair with correlation, --------- 2067 1/2 2 2067 2075 i.e. , [[---------, 0], [0, ----]] 2067 2067 ------------------------------------------------- Theorem Number, 199, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [3, 3, 1], nor the composition, [1, 1, 6] Then infinity ----- \ n ) a(n) x = / ----- n = 0 14 13 12 11 10 9 7 6 4 3 2 x + 2 x + 2 x + 2 x + x + x - x - x + x - x - x + 2 x - 1 ------------------------------------------------------------------------- 7 6 5 4 2 2 (x + 2 x + 2 x + x + x + x - 1) (-1 + x) and in Maple format (x^14+2*x^13+2*x^12+2*x^11+x^10+x^9-x^7-x^6+x^4-x^3-x^2+2*x-1)/(x^7+2*x^6+2*x^5 +x^4+x^2+x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.67544294785551085893 1.9131864632213745587 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [3, 3, 1] and d occurrences (as containment) of the composition, [1, 1, 6], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 15 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 14 15 2 15 14 15 2 14 + X1 X2 x - 2 X1 X2 x - X2 x - 2 X1 X2 x + X1 x - X2 x 15 14 14 15 14 11 11 11 + 2 X2 x + X1 x + 2 X2 x - x - x + X1 X2 x - X1 x - X2 x 9 11 8 9 9 7 8 8 + X1 X2 x + x + X1 X2 x - X1 x - X2 x - 2 X1 X2 x - X1 x - X2 x 9 6 7 7 8 5 6 4 5 3 + x + X1 X2 x + X1 x + X2 x + x + X1 x - x - 2 X1 x - x + X1 x 4 2 / 2 8 7 8 + 2 x - 3 x + 3 x - 1) / ((-1 + x) (X1 X2 x + X1 X2 x - X1 x / 8 6 7 7 8 6 7 5 4 5 - X2 x - X1 X2 x - X1 x - X2 x + x + X1 x + x + X1 x + X1 x - x 3 4 3 - X1 x - x + x - 2 x + 1)) and in Maple format -(X1*X2^2*x^15+X1*X2^2*x^14-2*X1*X2*x^15-X2^2*x^15-2*X1*X2*x^14+X1*x^15-X2^2*x^ 14+2*X2*x^15+X1*x^14+2*X2*x^14-x^15-x^14+X1*X2*x^11-X1*x^11-X2*x^11+X1*X2*x^9+x ^11+X1*X2*x^8-X1*x^9-X2*x^9-2*X1*X2*x^7-X1*x^8-X2*x^8+x^9+X1*X2*x^6+X1*x^7+X2*x ^7+x^8+X1*x^5-x^6-2*X1*x^4-x^5+X1*x^3+2*x^4-3*x^2+3*x-1)/(-1+x)^2/(X1*X2*x^8+X1 *X2*x^7-X1*x^8-X2*x^8-X1*X2*x^6-X1*x^7-X2*x^7+x^8+X1*x^6+x^7+X1*x^5+X1*x^4-x^5- X1*x^3-x^4+x^3-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 3, 1], are n 263 35 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 1, 6], are n 49 53 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 4 35 53 ------------- 1855 and in floating point 0.09287269204 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 4 1855 ate normal pair with correlation, --------- 1855 1/2 4 1855 1887 i.e. , [[---------, 0], [0, ----]] 1855 1855 ------------------------------------------------- Theorem Number, 200, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [3, 3, 1], nor the composition, [1, 2, 5] Then infinity ----- 13 12 8 6 5 4 2 \ n x - x - 2 x + x + x - 2 x + 3 x - 3 x + 1 ) a(n) x = -------------------------------------------------- / 7 5 4 3 2 ----- (x - x - x + x - 2 x + 1) (-1 + x) n = 0 and in Maple format (x^13-x^12-2*x^8+x^6+x^5-2*x^4+3*x^2-3*x+1)/(x^7-x^5-x^4+x^3-2*x+1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.64093877410291800826 1.9235403491125894929 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [3, 3, 1] and d occurrences (as containment) of the composition, [1, 2, 5], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 13 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 13 2 13 12 13 13 12 12 - 2 X1 X2 x - X2 x + X1 X2 x + X1 x + 2 X2 x - X1 x - X2 x 13 12 8 7 8 8 6 7 - x + x + X1 X2 x + X1 X2 x - X1 x - 2 X2 x - 2 X1 X2 x - 2 X1 x 7 8 5 6 6 4 5 3 4 + X2 x + 2 x + X1 X2 x + 3 X1 x - x - 2 X1 x - x + X1 x + 2 x 2 / 2 7 6 7 7 - 3 x + 3 x - 1) / ((-1 + x) (X1 X2 x + X1 X2 x - X1 x - X2 x / 5 6 7 5 4 5 3 4 3 - X1 X2 x - X1 x + x + 2 X1 x + X1 x - x - X1 x - x + x - 2 x + 1 )) and in Maple format -(X1*X2^2*x^13-2*X1*X2*x^13-X2^2*x^13+X1*X2*x^12+X1*x^13+2*X2*x^13-X1*x^12-X2*x ^12-x^13+x^12+X1*X2*x^8+X1*X2*x^7-X1*x^8-2*X2*x^8-2*X1*X2*x^6-2*X1*x^7+X2*x^7+2 *x^8+X1*X2*x^5+3*X1*x^6-x^6-2*X1*x^4-x^5+X1*x^3+2*x^4-3*x^2+3*x-1)/(-1+x)^2/(X1 *X2*x^7+X1*X2*x^6-X1*x^7-X2*x^7-X1*X2*x^5-X1*x^6+x^7+2*X1*x^5+X1*x^4-x^5-X1*x^3 -x^4+x^3-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 3, 1], are n 263 35 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 2, 5], are n 111 57 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 2 35 57 ------------- 285 and in floating point 0.3134415640 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 1/2 2 7 285 ate normal pair with correlation, ------------- 285 1/2 1/2 2 7 285 341 i.e. , [[-------------, 0], [0, ---]] 285 285 ------------------------------------------------- Theorem Number, 201, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [3, 3, 1], nor the composition, [1, 3, 4] Then infinity ----- 12 11 10 8 7 6 5 4 2 \ n x + x - x + x + x - x - x + 2 x - 3 x + 3 x - 1 ) a(n) x = ----------------------------------------------------------- / 5 4 3 2 ----- (x + x - x + 2 x - 1) (-1 + x) n = 0 and in Maple format (x^12+x^11-x^10+x^8+x^7-x^6-x^5+2*x^4-3*x^2+3*x-1)/(x^5+x^4-x^3+2*x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.59130364901708252896 1.9417130342786384772 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [3, 3, 1] and d occurrences (as containment) of the composition, [1, 3, 4], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 12 11 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x + X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 12 12 10 11 11 12 10 10 - X1 x - X2 x - X1 X2 x - X1 x - X2 x + x + X1 x + X2 x 11 10 7 8 6 7 8 5 7 + x - x + X1 X2 x - X2 x + X1 X2 x - 2 X1 x + x - 2 X1 X2 x + x 4 5 6 4 5 3 4 2 + X1 X2 x + 3 X1 x - x - 3 X1 x - x + X1 x + 2 x - 3 x + 3 x - 1) / 2 6 5 6 4 4 5 / ((-1 + x) (X1 X2 x + X1 X2 x - X1 x - X1 X2 x + 2 X1 x - x / 3 4 3 - X1 x - x + x - 2 x + 1)) and in Maple format -(X1*X2*x^12+X1*X2*x^11-X1*x^12-X2*x^12-X1*X2*x^10-X1*x^11-X2*x^11+x^12+X1*x^10 +X2*x^10+x^11-x^10+X1*X2*x^7-X2*x^8+X1*X2*x^6-2*X1*x^7+x^8-2*X1*X2*x^5+x^7+X1* X2*x^4+3*X1*x^5-x^6-3*X1*x^4-x^5+X1*x^3+2*x^4-3*x^2+3*x-1)/(-1+x)^2/(X1*X2*x^6+ X1*X2*x^5-X1*x^6-X1*X2*x^4+2*X1*x^4-x^5-X1*x^3-x^4+x^3-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 3, 1], are n 263 35 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 3, 4], are n 135 65 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 34 7 13 ------------- 455 and in floating point 0.7128336889 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 34 91 ate normal pair with correlation, -------- 455 1/2 34 91 4587 i.e. , [[--------, 0], [0, ----]] 455 2275 ------------------------------------------------- Theorem Number, 202, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [3, 3, 1], nor the composition, [1, 4, 3] Then infinity ----- 12 11 10 8 7 6 5 4 2 \ n x + x - x + x + x - x - x + 2 x - 3 x + 3 x - 1 ) a(n) x = ----------------------------------------------------------- / 5 4 3 2 ----- (x + x - x + 2 x - 1) (-1 + x) n = 0 and in Maple format (x^12+x^11-x^10+x^8+x^7-x^6-x^5+2*x^4-3*x^2+3*x-1)/(x^5+x^4-x^3+2*x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.59130364901708252896 1.9417130342786384772 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [3, 3, 1] and d occurrences (as containment) of the composition, [1, 4, 3], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 12 11 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x + X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 12 12 10 11 11 12 10 10 - X1 x - X2 x - X1 X2 x - X1 x - X2 x + x + X1 x + X2 x 11 10 7 8 6 7 8 5 7 + x - x + X1 X2 x - X2 x + X1 X2 x - 2 X1 x + x - 2 X1 X2 x + x 4 5 6 4 5 3 4 2 + X1 X2 x + 3 X1 x - x - 3 X1 x - x + X1 x + 2 x - 3 x + 3 x - 1) / 2 6 5 6 4 4 5 / ((-1 + x) (X1 X2 x + X1 X2 x - X1 x - X1 X2 x + 2 X1 x - x / 3 4 3 - X1 x - x + x - 2 x + 1)) and in Maple format -(X1*X2*x^12+X1*X2*x^11-X1*x^12-X2*x^12-X1*X2*x^10-X1*x^11-X2*x^11+x^12+X1*x^10 +X2*x^10+x^11-x^10+X1*X2*x^7-X2*x^8+X1*X2*x^6-2*X1*x^7+x^8-2*X1*X2*x^5+x^7+X1* X2*x^4+3*X1*x^5-x^6-3*X1*x^4-x^5+X1*x^3+2*x^4-3*x^2+3*x-1)/(-1+x)^2/(X1*X2*x^6+ X1*X2*x^5-X1*x^6-X1*X2*x^4+2*X1*x^4-x^5-X1*x^3-x^4+x^3-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 3, 1], are n 263 35 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 4, 3], are n 135 65 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 34 7 13 ------------- 455 and in floating point 0.7128336889 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 34 91 ate normal pair with correlation, -------- 455 1/2 34 91 4587 i.e. , [[--------, 0], [0, ----]] 455 2275 ------------------------------------------------- Theorem Number, 203, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [3, 3, 1], nor the composition, [1, 5, 2] Then infinity ----- 12 10 8 6 5 4 2 \ n x - x + 2 x - x - x + 2 x - 3 x + 3 x - 1 ) a(n) x = - -------------------------------------------------- / 7 5 4 3 2 ----- (x - x - x + x - 2 x + 1) (-1 + x) n = 0 and in Maple format -(x^12-x^10+2*x^8-x^6-x^5+2*x^4-3*x^2+3*x-1)/(x^7-x^5-x^4+x^3-2*x+1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.64616683293975167207 1.9235403491125894929 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [3, 3, 1] and d occurrences (as containment) of the composition, [1, 5, 2], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 12 12 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x - X1 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 12 10 12 10 10 8 10 7 - X2 x - X1 X2 x + x + X1 x + X2 x + X1 X2 x - x + X1 X2 x 8 8 6 7 7 8 5 - X1 x - 2 X2 x - 2 X1 X2 x - 2 X1 x + X2 x + 2 x + X1 X2 x 6 6 4 5 3 4 2 / + 3 X1 x - x - 2 X1 x - x + X1 x + 2 x - 3 x + 3 x - 1) / ( / 2 7 6 7 7 5 6 7 (-1 + x) (X1 X2 x + X1 X2 x - X1 x - X2 x - X1 X2 x - X1 x + x 5 4 5 3 4 3 + 2 X1 x + X1 x - x - X1 x - x + x - 2 x + 1)) and in Maple format -(X1*X2*x^12-X1*x^12-X2*x^12-X1*X2*x^10+x^12+X1*x^10+X2*x^10+X1*X2*x^8-x^10+X1* X2*x^7-X1*x^8-2*X2*x^8-2*X1*X2*x^6-2*X1*x^7+X2*x^7+2*x^8+X1*X2*x^5+3*X1*x^6-x^6 -2*X1*x^4-x^5+X1*x^3+2*x^4-3*x^2+3*x-1)/(-1+x)^2/(X1*X2*x^7+X1*X2*x^6-X1*x^7-X2 *x^7-X1*X2*x^5-X1*x^6+x^7+2*X1*x^5+X1*x^4-x^5-X1*x^3-x^4+x^3-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 3, 1], are n 263 35 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 5, 2], are n 111 57 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 2 35 57 ------------- 285 and in floating point 0.3134415640 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 1/2 2 7 285 ate normal pair with correlation, ------------- 285 1/2 1/2 2 7 285 341 i.e. , [[-------------, 0], [0, ---]] 285 285 ------------------------------------------------- Theorem Number, 204, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [3, 3, 1], nor the composition, [1, 6, 1] Then infinity ----- 9 7 6 4 3 2 \ n x - x - x + x - x - x + 2 x - 1 ) a(n) x = ---------------------------------------------- / 7 6 5 4 2 2 ----- (x + 2 x + 2 x + x + x + x - 1) (-1 + x) n = 0 and in Maple format (x^9-x^7-x^6+x^4-x^3-x^2+2*x-1)/(x^7+2*x^6+2*x^5+x^4+x^2+x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.68493773070922259842 1.9131864632213745587 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [3, 3, 1] and d occurrences (as containment) of the composition, [1, 6, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 9 9 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x - X1 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 9 7 9 6 7 7 7 6 4 - X2 x - X1 X2 x + x + X1 X2 x + X1 x + X2 x - x - x - X1 x 3 4 3 2 / 8 7 + X1 x + x - x - x + 2 x - 1) / ((-1 + x) (X1 X2 x + X1 X2 x / 8 8 6 7 7 8 6 7 5 - X1 x - X2 x - X1 X2 x - X1 x - X2 x + x + X1 x + x + X1 x 4 5 3 4 3 + X1 x - x - X1 x - x + x - 2 x + 1)) and in Maple format (X1*X2*x^9-X1*x^9-X2*x^9-X1*X2*x^7+x^9+X1*X2*x^6+X1*x^7+X2*x^7-x^7-x^6-X1*x^4+ X1*x^3+x^4-x^3-x^2+2*x-1)/(-1+x)/(X1*X2*x^8+X1*X2*x^7-X1*x^8-X2*x^8-X1*X2*x^6- X1*x^7-X2*x^7+x^8+X1*x^6+x^7+X1*x^5+X1*x^4-x^5-X1*x^3-x^4+x^3-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 3, 1], are n 263 35 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 6, 1], are n 49 53 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 4 35 53 ------------- 1855 and in floating point 0.09287269204 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 4 1855 ate normal pair with correlation, --------- 1855 1/2 4 1855 1887 i.e. , [[---------, 0], [0, ----]] 1855 1855 ------------------------------------------------- Theorem Number, 205, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [3, 3, 1], nor the composition, [2, 1, 5] Then infinity ----- \ n ) a(n) x = / ----- n = 0 16 14 10 9 8 7 6 5 4 2 x - x + x - x + x + 2 x - 2 x - x + 2 x - 3 x + 3 x - 1 -------------------------------------------------------------------- 8 6 5 4 3 2 (x - x + x + x - x + 2 x - 1) (-1 + x) and in Maple format (x^16-x^14+x^10-x^9+x^8+2*x^7-2*x^6-x^5+2*x^4-3*x^2+3*x-1)/(x^8-x^6+x^5+x^4-x^3 +2*x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.66664854364473939498 1.9158008597433206753 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [3, 3, 1] and d occurrences (as containment) of the composition, [2, 1, 5], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 16 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 16 2 16 2 14 16 16 14 - 2 X1 X2 x - X2 x - X1 X2 x + X1 x + 2 X2 x + 2 X1 X2 x 2 14 16 2 12 14 14 2 11 + X2 x - x - X1 X2 x - X1 x - 2 X2 x + X1 X2 x 12 14 11 12 10 11 9 + 2 X1 X2 x + x - 2 X1 X2 x - X1 x - X1 X2 x + X1 x + X1 X2 x 10 10 8 9 9 10 7 8 + X1 x + X2 x - X1 X2 x - X1 x - X2 x - x - X1 X2 x + X1 x 8 9 6 7 7 8 5 6 + X2 x + x + 2 X1 X2 x + 2 X1 x + X2 x - x - X1 X2 x - 3 X1 x 6 7 6 4 5 3 4 2 / - X2 x - 2 x + 2 x + 2 X1 x + x - X1 x - 2 x + 3 x - 3 x + 1) / / 2 12 2 11 12 11 12 ((-1 + x) (X1 X2 x - X1 X2 x - 2 X1 X2 x + 2 X1 X2 x + X1 x 11 9 8 9 9 8 8 9 - X1 x - X1 X2 x + X1 X2 x + X1 x + X2 x - X1 x - X2 x - x 6 7 8 5 6 6 7 5 6 - 2 X1 X2 x - X2 x + x + X1 X2 x + 3 X1 x + X2 x + x - X1 x - 2 x 4 3 4 3 2 - 2 X1 x + X1 x + 2 x - x - 2 x + 3 x - 1)) and in Maple format (X1*X2^2*x^16-2*X1*X2*x^16-X2^2*x^16-X1*X2^2*x^14+X1*x^16+2*X2*x^16+2*X1*X2*x^ 14+X2^2*x^14-x^16-X1*X2^2*x^12-X1*x^14-2*X2*x^14+X1*X2^2*x^11+2*X1*X2*x^12+x^14 -2*X1*X2*x^11-X1*x^12-X1*X2*x^10+X1*x^11+X1*X2*x^9+X1*x^10+X2*x^10-X1*X2*x^8-X1 *x^9-X2*x^9-x^10-X1*X2*x^7+X1*x^8+X2*x^8+x^9+2*X1*X2*x^6+2*X1*x^7+X2*x^7-x^8-X1 *X2*x^5-3*X1*x^6-X2*x^6-2*x^7+2*x^6+2*X1*x^4+x^5-X1*x^3-2*x^4+3*x^2-3*x+1)/(-1+ x)/(X1*X2^2*x^12-X1*X2^2*x^11-2*X1*X2*x^12+2*X1*X2*x^11+X1*x^12-X1*x^11-X1*X2*x ^9+X1*X2*x^8+X1*x^9+X2*x^9-X1*x^8-X2*x^8-x^9-2*X1*X2*x^6-X2*x^7+x^8+X1*X2*x^5+3 *X1*x^6+X2*x^6+x^7-X1*x^5-2*x^6-2*X1*x^4+X1*x^3+2*x^4-x^3-2*x^2+3*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 3, 1], are n 263 35 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 1, 5], are n 113 57 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 8 35 57 ------------- 1995 and in floating point 0.1791094652 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 8 1995 ate normal pair with correlation, --------- 1995 1/2 8 1995 2123 i.e. , [[---------, 0], [0, ----]] 1995 1995 ------------------------------------------------- Theorem Number, 206, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [3, 3, 1], nor the composition, [2, 2, 4] Then infinity ----- 11 9 7 6 4 3 2 \ n x + x + x - 2 x + x - x - x + 2 x - 1 ) a(n) x = - --------------------------------------------- / 7 6 5 4 3 ----- (-1 + x) (x - x + x + x - x + 2 x - 1) n = 0 and in Maple format -(x^11+x^9+x^7-2*x^6+x^4-x^3-x^2+2*x-1)/(-1+x)/(x^7-x^6+x^5+x^4-x^3+2*x-1) The asymptotic expression for a(n) is, n 0.64040732435378480241 1.9249743097683141877 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [3, 3, 1] and d occurrences (as containment) of the composition, [2, 2, 4], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 12 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 12 2 12 11 12 12 10 - 2 X1 X2 x - X2 x + X1 X2 x + X1 x + 2 X2 x - X1 X2 x 11 11 12 9 10 10 11 8 - X1 x - X2 x - x + X1 X2 x + X1 x + X2 x + x - X1 X2 x 9 9 10 7 8 8 9 6 7 - X1 x - X2 x - x + X1 X2 x + X1 x + X2 x + x + X1 X2 x - 2 X1 x 7 8 5 6 7 4 5 6 - 2 X2 x - x - 2 X1 X2 x + X2 x + 3 x + X1 X2 x + 3 X1 x - 2 x 4 5 3 4 2 / 8 - 3 X1 x - x + X1 x + 2 x - 3 x + 3 x - 1) / ((-1 + x) (X1 X2 x / 7 8 8 7 7 8 5 6 - X1 X2 x - X1 x - X2 x + X1 x + 2 X2 x + x + 2 X1 X2 x - X1 x 6 7 4 5 6 4 3 4 3 - X2 x - 2 x - X1 X2 x - 2 X1 x + 2 x + 3 X1 x - X1 x - 2 x + x 2 + 2 x - 3 x + 1)) and in Maple format (X1*X2^2*x^12-2*X1*X2*x^12-X2^2*x^12+X1*X2*x^11+X1*x^12+2*X2*x^12-X1*X2*x^10-X1 *x^11-X2*x^11-x^12+X1*X2*x^9+X1*x^10+X2*x^10+x^11-X1*X2*x^8-X1*x^9-X2*x^9-x^10+ X1*X2*x^7+X1*x^8+X2*x^8+x^9+X1*X2*x^6-2*X1*x^7-2*X2*x^7-x^8-2*X1*X2*x^5+X2*x^6+ 3*x^7+X1*X2*x^4+3*X1*x^5-2*x^6-3*X1*x^4-x^5+X1*x^3+2*x^4-3*x^2+3*x-1)/(-1+x)/( X1*X2*x^8-X1*X2*x^7-X1*x^8-X2*x^8+X1*x^7+2*X2*x^7+x^8+2*X1*X2*x^5-X1*x^6-X2*x^6 -2*x^7-X1*X2*x^4-2*X1*x^5+2*x^6+3*X1*x^4-X1*x^3-2*x^4+x^3+2*x^2-3*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 3, 1], are n 263 35 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 2, 4], are n 75 69 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 4 35 69 ------------- 483 and in floating point 0.4069784638 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 1/2 4 5 483 ate normal pair with correlation, ------------- 483 1/2 1/2 4 5 483 643 i.e. , [[-------------, 0], [0, ---]] 483 483 ------------------------------------------------- Theorem Number, 207, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [3, 3, 1], nor the composition, [2, 3, 3] Then infinity ----- 11 9 8 7 6 5 4 2 \ n x - x + x + x - x - x + 2 x - 3 x + 3 x - 1 ) a(n) x = ---------------------------------------------------- / 5 4 3 2 ----- (x + x - x + 2 x - 1) (-1 + x) n = 0 and in Maple format (x^11-x^9+x^8+x^7-x^6-x^5+2*x^4-3*x^2+3*x-1)/(x^5+x^4-x^3+2*x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.59752426117277830607 1.9417130342786384772 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [3, 3, 1] and d occurrences (as containment) of the composition, [2, 3, 3], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 11 11 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x - X1 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 11 9 11 8 9 9 7 8 - X2 x - X1 X2 x + x + X1 X2 x + X1 x + X2 x - X1 X2 x - X1 x 8 9 6 8 5 7 4 6 3 - X2 x - x + X1 X2 x + x + X1 X2 x + x - 2 X1 X2 x - x + X1 X2 x 5 4 2 / 7 6 7 - x + 2 x - 3 x + 3 x - 1) / ((-1 + x) (X1 X2 x - X1 X2 x - X1 x / 4 6 3 4 3 2 + 2 X1 X2 x + x - X1 X2 x - 2 x + x + 2 x - 3 x + 1)) and in Maple format (X1*X2*x^11-X1*x^11-X2*x^11-X1*X2*x^9+x^11+X1*X2*x^8+X1*x^9+X2*x^9-X1*X2*x^7-X1 *x^8-X2*x^8-x^9+X1*X2*x^6+x^8+X1*X2*x^5+x^7-2*X1*X2*x^4-x^6+X1*X2*x^3-x^5+2*x^4 -3*x^2+3*x-1)/(-1+x)/(X1*X2*x^7-X1*X2*x^6-X1*x^7+2*X1*X2*x^4+x^6-X1*X2*x^3-2*x^ 4+x^3+2*x^2-3*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 3, 1], are n 263 35 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 3, 3], are n 97 85 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 44 7 17 ------------- 595 and in floating point 0.8066946778 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 44 119 ate normal pair with correlation, --------- 595 1/2 44 119 6847 i.e. , [[---------, 0], [0, ----]] 595 2975 ------------------------------------------------- Theorem Number, 208, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [3, 3, 1], nor the composition, [2, 4, 2] Then infinity ----- 7 6 4 3 2 \ n x - 2 x + x - x - x + 2 x - 1 ) a(n) x = --------------------------------------------------- / 10 8 7 6 4 3 2 ----- x - x + 2 x - 2 x + 2 x - x - 2 x + 3 x - 1 n = 0 and in Maple format (x^7-2*x^6+x^4-x^3-x^2+2*x-1)/(x^10-x^8+2*x^7-2*x^6+2*x^4-x^3-2*x^2+3*x-1) The asymptotic expression for a(n) is, n 0.63012861965149722353 1.9302781753610965686 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [3, 3, 1] and d occurrences (as containment) of the composition, [2, 4, 2], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 7 7 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x - X1 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 7 5 6 6 7 4 5 6 - X2 x - X1 X2 x + X1 x + X2 x + x + X1 X2 x + X1 x - 2 x 4 3 4 3 2 / 10 10 - 2 X1 x + X1 x + x - x - x + 2 x - 1) / (X1 X2 x - X1 x / 10 8 10 7 8 8 7 7 - X2 x - X1 X2 x + x + X1 X2 x + X1 x + X2 x - X1 x - 2 X2 x 8 5 6 6 7 4 5 6 - x - 2 X1 X2 x + X1 x + X2 x + 2 x + X1 X2 x + 2 X1 x - 2 x 4 3 4 3 2 - 3 X1 x + X1 x + 2 x - x - 2 x + 3 x - 1) and in Maple format (X1*X2*x^7-X1*x^7-X2*x^7-X1*X2*x^5+X1*x^6+X2*x^6+x^7+X1*X2*x^4+X1*x^5-2*x^6-2* X1*x^4+X1*x^3+x^4-x^3-x^2+2*x-1)/(X1*X2*x^10-X1*x^10-X2*x^10-X1*X2*x^8+x^10+X1* X2*x^7+X1*x^8+X2*x^8-X1*x^7-2*X2*x^7-x^8-2*X1*X2*x^5+X1*x^6+X2*x^6+2*x^7+X1*X2* x^4+2*X1*x^5-2*x^6-3*X1*x^4+X1*x^3+2*x^4-x^3-2*x^2+3*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 3, 1], are n 263 35 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 4, 2], are n 75 69 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 8 35 69 ------------- 805 and in floating point 0.4883741565 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 1/2 8 3 805 ate normal pair with correlation, ------------- 805 1/2 1/2 8 3 805 1189 i.e. , [[-------------, 0], [0, ----]] 805 805 ------------------------------------------------- Theorem Number, 209, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [3, 3, 1], nor the composition, [2, 5, 1] Then infinity ----- 8 7 6 4 3 2 \ n x - x - x + x - x - x + 2 x - 1 ) a(n) x = -------------------------------------- / 7 5 4 3 ----- (-1 + x) (x - x - x + x - 2 x + 1) n = 0 and in Maple format (x^8-x^7-x^6+x^4-x^3-x^2+2*x-1)/(-1+x)/(x^7-x^5-x^4+x^3-2*x+1) The asymptotic expression for a(n) is, n 0.65342844522305416418 1.9235403491125894929 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [3, 3, 1] and d occurrences (as containment) of the composition, [2, 5, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 8 8 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x - X1 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 8 6 7 8 5 6 7 5 6 - X2 x - X1 X2 x + X2 x + x + X1 X2 x + 2 X1 x - x - X1 x - x 4 3 4 3 2 / 7 - X1 x + X1 x + x - x - x + 2 x - 1) / ((-1 + x) (X1 X2 x / 6 7 7 5 6 7 5 4 5 + X1 X2 x - X1 x - X2 x - X1 X2 x - X1 x + x + 2 X1 x + X1 x - x 3 4 3 - X1 x - x + x - 2 x + 1)) and in Maple format (X1*X2*x^8-X1*x^8-X2*x^8-X1*X2*x^6+X2*x^7+x^8+X1*X2*x^5+2*X1*x^6-x^7-X1*x^5-x^6 -X1*x^4+X1*x^3+x^4-x^3-x^2+2*x-1)/(-1+x)/(X1*X2*x^7+X1*X2*x^6-X1*x^7-X2*x^7-X1* X2*x^5-X1*x^6+x^7+2*X1*x^5+X1*x^4-x^5-X1*x^3-x^4+x^3-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 3, 1], are n 263 35 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 5, 1], are n 111 57 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 2 35 57 ------------- 285 and in floating point 0.3134415640 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 1/2 2 7 285 ate normal pair with correlation, ------------- 285 1/2 1/2 2 7 285 341 i.e. , [[-------------, 0], [0, ---]] 285 285 ------------------------------------------------- Theorem Number, 210, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [3, 3, 1], nor the composition, [3, 1, 4] Then infinity ----- 7 6 5 4 3 2 \ n x - x - x + x - x - x + 2 x - 1 ) a(n) x = - ---------------------------------------- / 8 7 4 3 ----- (-1 + x) (x + 2 x + x - x + 2 x - 1) n = 0 and in Maple format -(x^7-x^6-x^5+x^4-x^3-x^2+2*x-1)/(-1+x)/(x^8+2*x^7+x^4-x^3+2*x-1) The asymptotic expression for a(n) is, n 0.65444189970384155768 1.9203064027137528069 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [3, 3, 1] and d occurrences (as containment) of the composition, [3, 1, 4], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 10 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 9 10 9 10 9 7 7 + X1 X2 x - 2 X1 X2 x - 2 X1 X2 x + X1 x + X1 x - X1 X2 x + X1 x 7 5 6 7 4 5 5 6 4 + X2 x + X1 X2 x - X1 x - x - X1 X2 x - X1 x - X2 x + x + 2 X1 x 5 3 4 3 2 / 2 12 2 11 + x - X1 x - x + x + x - 2 x + 1) / (X1 X2 x + 2 X1 X2 x / 12 11 12 2 9 11 9 - 2 X1 X2 x - 4 X1 X2 x + X1 x - X1 X2 x + 2 X1 x + X1 X2 x 8 9 7 8 8 9 7 7 8 - X1 X2 x + X2 x + X1 X2 x + X1 x + X2 x - x - X1 x - 2 X2 x - x 5 6 6 7 4 5 5 4 - 2 X1 X2 x + X1 x - X2 x + 2 x + X1 X2 x + 2 X1 x + X2 x - 3 X1 x 5 3 4 3 2 - x + X1 x + 2 x - x - 2 x + 3 x - 1) and in Maple format -(X1*X2^2*x^10+X1*X2^2*x^9-2*X1*X2*x^10-2*X1*X2*x^9+X1*x^10+X1*x^9-X1*X2*x^7+X1 *x^7+X2*x^7+X1*X2*x^5-X1*x^6-x^7-X1*X2*x^4-X1*x^5-X2*x^5+x^6+2*X1*x^4+x^5-X1*x^ 3-x^4+x^3+x^2-2*x+1)/(X1*X2^2*x^12+2*X1*X2^2*x^11-2*X1*X2*x^12-4*X1*X2*x^11+X1* x^12-X1*X2^2*x^9+2*X1*x^11+X1*X2*x^9-X1*X2*x^8+X2*x^9+X1*X2*x^7+X1*x^8+X2*x^8-x ^9-X1*x^7-2*X2*x^7-x^8-2*X1*X2*x^5+X1*x^6-X2*x^6+2*x^7+X1*X2*x^4+2*X1*x^5+X2*x^ 5-3*X1*x^4-x^5+X1*x^3+2*x^4-x^3-2*x^2+3*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 3, 1], are n 263 35 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 1, 4], are n 141 65 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 16 7 13 ------------- 455 and in floating point 0.3354511477 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 16 91 ate normal pair with correlation, -------- 455 1/2 16 91 2787 i.e. , [[--------, 0], [0, ----]] 455 2275 ------------------------------------------------- Theorem Number, 211, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [3, 3, 1], nor the composition, [3, 2, 3] Then infinity ----- 5 4 3 2 \ n x - x + x + x - 2 x + 1 ) a(n) x = -------------------------------------------- / 3 4 3 2 ----- (-1 + x) (x - x + 1) (x + x + x + x - 1) n = 0 and in Maple format (x^5-x^4+x^3+x^2-2*x+1)/(-1+x)/(x^3-x+1)/(x^4+x^3+x^2+x-1) The asymptotic expression for a(n) is, n 0.63758295460268560269 1.9275619754829253043 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [3, 3, 1] and d occurrences (as containment) of the composition, [3, 2, 3], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 8 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 8 8 6 6 4 5 3 5 - 2 X1 X2 x + X1 x - X1 X2 x + X2 x + X1 X2 x - X2 x - X1 X2 x + x 4 3 2 / 2 10 2 9 10 - x + x + x - 2 x + 1) / (X1 X2 x + X1 X2 x - 2 X1 X2 x / 2 8 9 10 8 9 7 8 - X1 X2 x - 2 X1 X2 x + X1 x + X1 X2 x + X1 x - X1 X2 x + X2 x 6 7 8 6 4 5 6 3 + X1 X2 x + X1 x - x - 2 X2 x - 2 X1 X2 x + X2 x + x + X1 X2 x 5 4 3 2 - x + 2 x - x - 2 x + 3 x - 1) and in Maple format -(X1*X2^2*x^8-2*X1*X2*x^8+X1*x^8-X1*X2*x^6+X2*x^6+X1*X2*x^4-X2*x^5-X1*X2*x^3+x^ 5-x^4+x^3+x^2-2*x+1)/(X1*X2^2*x^10+X1*X2^2*x^9-2*X1*X2*x^10-X1*X2^2*x^8-2*X1*X2 *x^9+X1*x^10+X1*X2*x^8+X1*x^9-X1*X2*x^7+X2*x^8+X1*X2*x^6+X1*x^7-x^8-2*X2*x^6-2* X1*X2*x^4+X2*x^5+x^6+X1*X2*x^3-x^5+2*x^4-x^3-2*x^2+3*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 3, 1], are n 263 35 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 2, 3], are n 89 77 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 32 5 11 ------------- 385 and in floating point 0.6164113027 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 32 55 ate normal pair with correlation, -------- 385 1/2 32 55 4743 i.e. , [[--------, 0], [0, ----]] 385 2695 ------------------------------------------------- Theorem Number, 212, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [3, 3, 1], nor the composition, [4, 1, 3] Then infinity ----- 7 6 5 4 3 2 \ n x - x - x + x - x - x + 2 x - 1 ) a(n) x = - ---------------------------------------- / 8 7 4 3 ----- (-1 + x) (x + 2 x + x - x + 2 x - 1) n = 0 and in Maple format -(x^7-x^6-x^5+x^4-x^3-x^2+2*x-1)/(-1+x)/(x^8+2*x^7+x^4-x^3+2*x-1) The asymptotic expression for a(n) is, n 0.65444189970384155768 1.9203064027137528069 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [3, 3, 1] and d occurrences (as containment) of the composition, [4, 1, 3], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 10 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 9 10 9 10 9 7 7 + X1 X2 x - 2 X1 X2 x - 2 X1 X2 x + X1 x + X1 x - X1 X2 x + X1 x 7 5 6 7 4 5 5 6 4 + X2 x + X1 X2 x - X1 x - x - X1 X2 x - X1 x - X2 x + x + 2 X1 x 5 3 4 3 2 / 2 12 2 11 + x - X1 x - x + x + x - 2 x + 1) / (X1 X2 x + 2 X1 X2 x / 12 11 12 2 9 11 9 - 2 X1 X2 x - 4 X1 X2 x + X1 x - X1 X2 x + 2 X1 x + X1 X2 x 8 9 7 8 8 9 7 7 8 - X1 X2 x + X2 x + X1 X2 x + X1 x + X2 x - x - X1 x - 2 X2 x - x 5 6 6 7 4 5 5 4 - 2 X1 X2 x + X1 x - X2 x + 2 x + X1 X2 x + 2 X1 x + X2 x - 3 X1 x 5 3 4 3 2 - x + X1 x + 2 x - x - 2 x + 3 x - 1) and in Maple format -(X1*X2^2*x^10+X1*X2^2*x^9-2*X1*X2*x^10-2*X1*X2*x^9+X1*x^10+X1*x^9-X1*X2*x^7+X1 *x^7+X2*x^7+X1*X2*x^5-X1*x^6-x^7-X1*X2*x^4-X1*x^5-X2*x^5+x^6+2*X1*x^4+x^5-X1*x^ 3-x^4+x^3+x^2-2*x+1)/(X1*X2^2*x^12+2*X1*X2^2*x^11-2*X1*X2*x^12-4*X1*X2*x^11+X1* x^12-X1*X2^2*x^9+2*X1*x^11+X1*X2*x^9-X1*X2*x^8+X2*x^9+X1*X2*x^7+X1*x^8+X2*x^8-x ^9-X1*x^7-2*X2*x^7-x^8-2*X1*X2*x^5+X1*x^6-X2*x^6+2*x^7+X1*X2*x^4+2*X1*x^5+X2*x^ 5-3*X1*x^4-x^5+X1*x^3+2*x^4-x^3-2*x^2+3*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 3, 1], are n 263 35 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 1, 3], are n 141 65 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 16 7 13 ------------- 455 and in floating point 0.3354511477 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 16 91 ate normal pair with correlation, -------- 455 1/2 16 91 2787 i.e. , [[--------, 0], [0, ----]] 455 2275 ------------------------------------------------- Theorem Number, 213, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [3, 3, 1], nor the composition, [4, 2, 2] Then infinity ----- 7 6 4 3 2 \ n x - 2 x + x - x - x + 2 x - 1 ) a(n) x = - ------------------------------------------- / 7 6 5 4 3 ----- (-1 + x) (x - x + x + x - x + 2 x - 1) n = 0 and in Maple format -(x^7-2*x^6+x^4-x^3-x^2+2*x-1)/(-1+x)/(x^7-x^6+x^5+x^4-x^3+2*x-1) The asymptotic expression for a(n) is, n 0.64732625001158940098 1.9249743097683141877 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [3, 3, 1] and d occurrences (as containment) of the composition, [4, 2, 2], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 7 7 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x - X1 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 7 5 6 6 7 4 5 6 - X2 x - X1 X2 x + X1 x + X2 x + x + X1 X2 x + X1 x - 2 x 4 3 4 3 2 / 8 7 - 2 X1 x + X1 x + x - x - x + 2 x - 1) / (X1 X2 x - X1 X2 x / 8 8 7 7 8 5 6 6 7 - X1 x - X2 x + X1 x + 2 X2 x + x + 2 X1 X2 x - X1 x - X2 x - 2 x 4 5 6 4 3 4 3 2 - X1 X2 x - 2 X1 x + 2 x + 3 X1 x - X1 x - 2 x + x + 2 x - 3 x + 1 ) and in Maple format -(X1*X2*x^7-X1*x^7-X2*x^7-X1*X2*x^5+X1*x^6+X2*x^6+x^7+X1*X2*x^4+X1*x^5-2*x^6-2* X1*x^4+X1*x^3+x^4-x^3-x^2+2*x-1)/(X1*X2*x^8-X1*X2*x^7-X1*x^8-X2*x^8+X1*x^7+2*X2 *x^7+x^8+2*X1*X2*x^5-X1*x^6-X2*x^6-2*x^7-X1*X2*x^4-2*X1*x^5+2*x^6+3*X1*x^4-X1*x ^3-2*x^4+x^3+2*x^2-3*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 3, 1], are n 263 35 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 2, 2], are n 75 69 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 4 35 69 ------------- 483 and in floating point 0.4069784638 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 1/2 4 5 483 ate normal pair with correlation, ------------- 483 1/2 1/2 4 5 483 643 i.e. , [[-------------, 0], [0, ---]] 483 483 ------------------------------------------------- Theorem Number, 214, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [3, 3, 1], nor the composition, [5, 1, 2] Then infinity ----- 8 6 4 3 2 \ n x - 2 x + x - x - x + 2 x - 1 ) a(n) x = - ------------------------------------------- / 8 6 5 4 3 ----- (-1 + x) (x - x + x + x - x + 2 x - 1) n = 0 and in Maple format -(x^8-2*x^6+x^4-x^3-x^2+2*x-1)/(-1+x)/(x^8-x^6+x^5+x^4-x^3+2*x-1) The asymptotic expression for a(n) is, n 0.67267896620284337780 1.9158008597433206753 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [3, 3, 1] and d occurrences (as containment) of the composition, [5, 1, 2], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 11 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 11 11 8 8 8 6 8 - 2 X1 X2 x + X1 x - X1 X2 x + X1 x + X2 x + X1 X2 x - x 5 6 6 5 6 4 3 4 3 2 - X1 X2 x - 2 X1 x - X2 x + X1 x + 2 x + X1 x - X1 x - x + x + x / 2 12 2 11 12 11 - 2 x + 1) / (X1 X2 x - X1 X2 x - 2 X1 X2 x + 2 X1 X2 x / 12 11 9 8 9 9 8 8 + X1 x - X1 x - X1 X2 x + X1 X2 x + X1 x + X2 x - X1 x - X2 x 9 6 7 8 5 6 6 7 5 - x - 2 X1 X2 x - X2 x + x + X1 X2 x + 3 X1 x + X2 x + x - X1 x 6 4 3 4 3 2 - 2 x - 2 X1 x + X1 x + 2 x - x - 2 x + 3 x - 1) and in Maple format -(X1*X2^2*x^11-2*X1*X2*x^11+X1*x^11-X1*X2*x^8+X1*x^8+X2*x^8+X1*X2*x^6-x^8-X1*X2 *x^5-2*X1*x^6-X2*x^6+X1*x^5+2*x^6+X1*x^4-X1*x^3-x^4+x^3+x^2-2*x+1)/(X1*X2^2*x^ 12-X1*X2^2*x^11-2*X1*X2*x^12+2*X1*X2*x^11+X1*x^12-X1*x^11-X1*X2*x^9+X1*X2*x^8+ X1*x^9+X2*x^9-X1*x^8-X2*x^8-x^9-2*X1*X2*x^6-X2*x^7+x^8+X1*X2*x^5+3*X1*x^6+X2*x^ 6+x^7-X1*x^5-2*x^6-2*X1*x^4+X1*x^3+2*x^4-x^3-2*x^2+3*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 3, 1], are n 263 35 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [5, 1, 2], are n 113 57 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 8 35 57 ------------- 1995 and in floating point 0.1791094652 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 8 1995 ate normal pair with correlation, --------- 1995 1/2 8 1995 2123 i.e. , [[---------, 0], [0, ----]] 1995 1995 ------------------------------------------------- Theorem Number, 215, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [3, 3, 1], nor the composition, [5, 2, 1] Then infinity ----- 8 7 6 4 3 2 \ n x - x - x + x - x - x + 2 x - 1 ) a(n) x = -------------------------------------- / 7 5 4 3 ----- (-1 + x) (x - x - x + x - 2 x + 1) n = 0 and in Maple format (x^8-x^7-x^6+x^4-x^3-x^2+2*x-1)/(-1+x)/(x^7-x^5-x^4+x^3-2*x+1) The asymptotic expression for a(n) is, n 0.65342844522305416418 1.9235403491125894929 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [3, 3, 1] and d occurrences (as containment) of the composition, [5, 2, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 8 8 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x - X1 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 8 6 7 8 5 6 7 5 6 - X2 x - X1 X2 x + X2 x + x + X1 X2 x + 2 X1 x - x - X1 x - x 4 3 4 3 2 / 7 - X1 x + X1 x + x - x - x + 2 x - 1) / ((-1 + x) (X1 X2 x / 6 7 7 5 6 7 5 4 5 + X1 X2 x - X1 x - X2 x - X1 X2 x - X1 x + x + 2 X1 x + X1 x - x 3 4 3 - X1 x - x + x - 2 x + 1)) and in Maple format (X1*X2*x^8-X1*x^8-X2*x^8-X1*X2*x^6+X2*x^7+x^8+X1*X2*x^5+2*X1*x^6-x^7-X1*x^5-x^6 -X1*x^4+X1*x^3+x^4-x^3-x^2+2*x-1)/(-1+x)/(X1*X2*x^7+X1*X2*x^6-X1*x^7-X2*x^7-X1* X2*x^5-X1*x^6+x^7+2*X1*x^5+X1*x^4-x^5-X1*x^3-x^4+x^3-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 3, 1], are n 263 35 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [5, 2, 1], are n 111 57 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 2 35 57 ------------- 285 and in floating point 0.3134415640 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 1/2 2 7 285 ate normal pair with correlation, ------------- 285 1/2 1/2 2 7 285 341 i.e. , [[-------------, 0], [0, ---]] 285 285 ------------------------------------------------- Theorem Number, 216, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [3, 3, 1], nor the composition, [6, 1, 1] Then infinity ----- 9 7 6 4 3 2 \ n x - x - x + x - x - x + 2 x - 1 ) a(n) x = ---------------------------------------------- / 7 6 5 4 2 2 ----- (x + 2 x + 2 x + x + x + x - 1) (-1 + x) n = 0 and in Maple format (x^9-x^7-x^6+x^4-x^3-x^2+2*x-1)/(x^7+2*x^6+2*x^5+x^4+x^2+x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.68493773070922259842 1.9131864632213745587 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [3, 3, 1] and d occurrences (as containment) of the composition, [6, 1, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 9 9 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x - X1 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 9 7 9 6 7 7 7 6 4 - X2 x - X1 X2 x + x + X1 X2 x + X1 x + X2 x - x - x - X1 x 3 4 3 2 / 8 7 + X1 x + x - x - x + 2 x - 1) / ((-1 + x) (X1 X2 x + X1 X2 x / 8 8 6 7 7 8 6 7 5 - X1 x - X2 x - X1 X2 x - X1 x - X2 x + x + X1 x + x + X1 x 4 5 3 4 3 + X1 x - x - X1 x - x + x - 2 x + 1)) and in Maple format (X1*X2*x^9-X1*x^9-X2*x^9-X1*X2*x^7+x^9+X1*X2*x^6+X1*x^7+X2*x^7-x^7-x^6-X1*x^4+ X1*x^3+x^4-x^3-x^2+2*x-1)/(-1+x)/(X1*X2*x^8+X1*X2*x^7-X1*x^8-X2*x^8-X1*X2*x^6- X1*x^7-X2*x^7+x^8+X1*x^6+x^7+X1*x^5+X1*x^4-x^5-X1*x^3-x^4+x^3-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 3, 1], are n 263 35 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [6, 1, 1], are n 49 53 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 4 35 53 ------------- 1855 and in floating point 0.09287269204 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 4 1855 ate normal pair with correlation, --------- 1855 1/2 4 1855 1887 i.e. , [[---------, 0], [0, ----]] 1855 1855 ------------------------------------------------- Theorem Number, 217, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [4, 1, 2], nor the composition, [1, 1, 6] Then infinity ----- \ n 18 17 16 14 13 12 11 ) a(n) x = - (x + 2 x + 2 x - 3 x - 3 x - 2 x - 3 x / ----- n = 0 10 4 2 / 6 5 3 2 - 2 x + x + x - 2 x + 1) / ((x + x + x + x + x - 1) / 6 5 2 (x + x + 1) (-1 + x) ) and in Maple format -(x^18+2*x^17+2*x^16-3*x^14-3*x^13-2*x^12-3*x^11-2*x^10+x^4+x^2-2*x+1)/(x^6+x^5 +x^3+x^2+x-1)/(x^6+x^5+1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.67303521612186301918 1.9111834366854762350 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [4, 1, 2] and d occurrences (as containment) of the composition, [1, 1, 6], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 19 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 2 18 2 19 2 19 2 18 2 19 + X1 X2 x - 2 X1 X2 x - 2 X1 X2 x - 2 X1 X2 x + X1 x 2 18 19 2 19 2 2 16 2 18 - 2 X1 X2 x + 4 X1 X2 x + X2 x - X1 X2 x + X1 x 18 19 2 18 19 2 2 15 2 16 + 4 X1 X2 x - 2 X1 x + X2 x - 2 X2 x - X1 X2 x + 2 X1 X2 x 2 16 18 18 19 2 2 14 2 15 + 3 X1 X2 x - 2 X1 x - 2 X2 x + x + X1 X2 x + 3 X1 X2 x 2 16 2 15 16 2 16 18 2 2 13 - X1 x + 3 X1 X2 x - 6 X1 X2 x - 2 X2 x + x - X1 X2 x 2 14 2 15 2 14 15 16 - 2 X1 X2 x - 2 X1 x - X1 X2 x - 8 X1 X2 x + 3 X1 x 2 15 16 2 13 2 14 2 13 14 - 2 X2 x + 4 X2 x + X1 X2 x + X1 x + X1 X2 x + 2 X1 X2 x 15 15 16 2 12 14 15 + 5 X1 x + 5 X2 x - 2 x + 2 X1 X2 x - X1 x - 3 x 2 11 2 12 12 13 13 2 11 - 2 X1 X2 x - 2 X1 x - 3 X1 X2 x - X1 x - X2 x + 2 X1 x 11 12 12 13 2 10 10 11 + 3 X1 X2 x + 3 X1 x + X2 x + x + X1 x + X1 X2 x - 3 X1 x 11 12 2 9 9 10 10 11 9 - X2 x - x - X1 x + X1 X2 x - 3 X1 x - X2 x + x + X1 x 9 10 7 6 7 6 5 4 5 - X2 x + 2 x - X1 X2 x + X1 X2 x + X1 x - X1 x - X1 x + X1 x + x 4 3 2 / - x + x - 3 x + 3 x - 1) / ((-1 + x) / 7 7 7 7 5 5 7 6 (X1 X2 x - X1 x - X2 x + x + X1 x - x + x - 1) (X1 X2 x - X1 X2 x 7 7 6 7 5 4 5 4 - X1 x - X2 x + X1 x + x + X1 x - X1 x - x + x - 2 x + 1)) and in Maple format -(X1^2*X2^2*x^19+X1^2*X2^2*x^18-2*X1^2*X2*x^19-2*X1*X2^2*x^19-2*X1^2*X2*x^18+X1 ^2*x^19-2*X1*X2^2*x^18+4*X1*X2*x^19+X2^2*x^19-X1^2*X2^2*x^16+X1^2*x^18+4*X1*X2* x^18-2*X1*x^19+X2^2*x^18-2*X2*x^19-X1^2*X2^2*x^15+2*X1^2*X2*x^16+3*X1*X2^2*x^16 -2*X1*x^18-2*X2*x^18+x^19+X1^2*X2^2*x^14+3*X1^2*X2*x^15-X1^2*x^16+3*X1*X2^2*x^ 15-6*X1*X2*x^16-2*X2^2*x^16+x^18-X1^2*X2^2*x^13-2*X1^2*X2*x^14-2*X1^2*x^15-X1* X2^2*x^14-8*X1*X2*x^15+3*X1*x^16-2*X2^2*x^15+4*X2*x^16+X1^2*X2*x^13+X1^2*x^14+ X1*X2^2*x^13+2*X1*X2*x^14+5*X1*x^15+5*X2*x^15-2*x^16+2*X1^2*X2*x^12-X1*x^14-3*x ^15-2*X1^2*X2*x^11-2*X1^2*x^12-3*X1*X2*x^12-X1*x^13-X2*x^13+2*X1^2*x^11+3*X1*X2 *x^11+3*X1*x^12+X2*x^12+x^13+X1^2*x^10+X1*X2*x^10-3*X1*x^11-X2*x^11-x^12-X1^2*x ^9+X1*X2*x^9-3*X1*x^10-X2*x^10+x^11+X1*x^9-X2*x^9+2*x^10-X1*X2*x^7+X1*X2*x^6+X1 *x^7-X1*x^6-X1*x^5+X1*x^4+x^5-x^4+x^3-3*x^2+3*x-1)/(-1+x)/(X1*X2*x^7-X1*x^7-X2* x^7+x^7+X1*x^5-x^5+x-1)/(X1*X2*x^7-X1*X2*x^6-X1*x^7-X2*x^7+X1*x^6+x^7+X1*x^5-X1 *x^4-x^5+x^4-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 1, 2], are n 191 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 1, 6], are n 49 53 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 8 3 53 ------------ 477 and in floating point 0.2114804230 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 8 159 ate normal pair with correlation, -------- 477 1/2 8 159 1559 i.e. , [[--------, 0], [0, ----]] 477 1431 ------------------------------------------------- Theorem Number, 218, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [4, 1, 2], nor the composition, [1, 2, 5] Then infinity ----- \ n 21 20 19 18 17 14 13 12 ) a(n) x = (x + 2 x + x - x - 2 x + 2 x + 2 x - x / ----- n = 0 10 9 8 5 4 3 2 / - 2 x + x - x - x + x - x + 3 x - 3 x + 1) / ( / 14 13 12 11 10 9 8 7 6 4 (x + x + x + 2 x + x + x - x - x - x + x - 2 x + 1) 2 (-1 + x) ) and in Maple format (x^21+2*x^20+x^19-x^18-2*x^17+2*x^14+2*x^13-x^12-2*x^10+x^9-x^8-x^5+x^4-x^3+3*x ^2-3*x+1)/(x^14+x^13+x^12+2*x^11+x^10+x^9-x^8-x^7-x^6+x^4-2*x+1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.66293686307441587764 1.9144061473285935513 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [4, 1, 2] and d occurrences (as containment) of the composition, [1, 2, 5], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 3 2 21 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 3 2 20 3 21 2 2 21 3 2 19 + 2 X1 X2 x - 2 X1 X2 x - 3 X1 X2 x + X1 X2 x 3 20 3 21 2 2 20 2 21 2 21 - 4 X1 X2 x + X1 x - 6 X1 X2 x + 6 X1 X2 x + 3 X1 X2 x 3 19 3 20 2 2 19 2 20 2 21 - 2 X1 X2 x + 2 X1 x - 3 X1 X2 x + 12 X1 X2 x - 3 X1 x 2 20 21 2 21 3 19 2 2 18 + 6 X1 X2 x - 6 X1 X2 x - X2 x + X1 x + X1 X2 x 2 19 2 20 2 19 20 21 + 6 X1 X2 x - 6 X1 x + 3 X1 X2 x - 12 X1 X2 x + 3 X1 x 2 20 21 2 2 17 2 18 2 19 - 2 X2 x + 2 X2 x + 2 X1 X2 x - 2 X1 X2 x - 3 X1 x 2 18 19 20 2 19 20 21 - 2 X1 X2 x - 6 X1 X2 x + 6 X1 x - X2 x + 4 X2 x - x 2 17 2 18 2 17 18 19 2 18 - 4 X1 X2 x + X1 x - 4 X1 X2 x + 4 X1 X2 x + 3 X1 x + X2 x 19 20 2 17 17 18 2 17 + 2 X2 x - 2 x + 2 X1 x + 8 X1 X2 x - 2 X1 x + 2 X2 x 18 19 2 15 2 15 17 17 18 - 2 X2 x - x - X1 X2 x + X1 X2 x - 4 X1 x - 4 X2 x + x 2 2 13 2 14 2 15 2 14 2 15 17 - X1 X2 x + X1 X2 x + X1 x + X1 X2 x - X2 x + 2 x 2 13 2 14 2 13 14 15 2 14 + 3 X1 X2 x - X1 x + 2 X1 X2 x - 4 X1 X2 x - X1 x - X2 x 15 2 12 2 13 13 14 2 13 + X2 x - X1 X2 x - 2 X1 x - 6 X1 X2 x + 3 X1 x - X2 x 14 2 11 2 12 12 13 13 + 3 X2 x + X1 X2 x + X1 x + 2 X1 X2 x + 4 X1 x + 3 X2 x 14 2 10 2 11 11 12 12 13 - 2 x - X1 X2 x - X1 x - X1 X2 x - 2 X1 x - X2 x - 2 x 2 10 10 11 12 2 9 10 10 + 2 X1 x + 2 X1 X2 x + X1 x + x - X1 x - 4 X1 x - X2 x 9 10 7 8 9 6 7 8 + 2 X1 x + 2 x + 2 X1 X2 x - X2 x - x - 2 X1 X2 x - 2 X1 x + x 5 6 5 4 5 4 3 2 + X1 X2 x + 2 X1 x - 2 X1 x + X1 x + x - x + x - 3 x + 3 x - 1) / 2 2 15 2 15 2 15 2 2 13 / ((-1 + x) (X1 X2 x - 2 X1 X2 x - 2 X1 X2 x - X1 X2 x / 2 14 2 15 2 14 15 2 15 2 13 + X1 X2 x + X1 x - X1 X2 x + 4 X1 X2 x + X2 x + 2 X1 X2 x 2 14 2 13 15 2 14 15 2 12 - X1 x + X1 X2 x - 2 X1 x + X2 x - 2 X2 x - X1 X2 x 2 13 13 14 14 15 2 11 2 12 - X1 x - 2 X1 X2 x + X1 x - X2 x + x + X1 X2 x + X1 x 12 13 2 10 2 11 11 12 + 2 X1 X2 x + X1 x - X1 X2 x - X1 x - 2 X1 X2 x - 2 X1 x 12 2 10 11 11 12 2 9 9 - X2 x + 2 X1 x + 2 X1 x + X2 x + x - X1 x - X1 X2 x 10 10 11 9 9 7 9 6 - 2 X1 x + X2 x - x + 3 X1 x + X2 x + X1 X2 x - 2 x - 2 X1 X2 x 7 5 6 5 6 4 5 4 2 - X1 x + X1 X2 x + X1 x - 2 X1 x + x + X1 x + x - x - 2 x + 3 x - 1)) and in Maple format -(X1^3*X2^2*x^21+2*X1^3*X2^2*x^20-2*X1^3*X2*x^21-3*X1^2*X2^2*x^21+X1^3*X2^2*x^ 19-4*X1^3*X2*x^20+X1^3*x^21-6*X1^2*X2^2*x^20+6*X1^2*X2*x^21+3*X1*X2^2*x^21-2*X1 ^3*X2*x^19+2*X1^3*x^20-3*X1^2*X2^2*x^19+12*X1^2*X2*x^20-3*X1^2*x^21+6*X1*X2^2*x ^20-6*X1*X2*x^21-X2^2*x^21+X1^3*x^19+X1^2*X2^2*x^18+6*X1^2*X2*x^19-6*X1^2*x^20+ 3*X1*X2^2*x^19-12*X1*X2*x^20+3*X1*x^21-2*X2^2*x^20+2*X2*x^21+2*X1^2*X2^2*x^17-2 *X1^2*X2*x^18-3*X1^2*x^19-2*X1*X2^2*x^18-6*X1*X2*x^19+6*X1*x^20-X2^2*x^19+4*X2* x^20-x^21-4*X1^2*X2*x^17+X1^2*x^18-4*X1*X2^2*x^17+4*X1*X2*x^18+3*X1*x^19+X2^2*x ^18+2*X2*x^19-2*x^20+2*X1^2*x^17+8*X1*X2*x^17-2*X1*x^18+2*X2^2*x^17-2*X2*x^18-x ^19-X1^2*X2*x^15+X1*X2^2*x^15-4*X1*x^17-4*X2*x^17+x^18-X1^2*X2^2*x^13+X1^2*X2*x ^14+X1^2*x^15+X1*X2^2*x^14-X2^2*x^15+2*x^17+3*X1^2*X2*x^13-X1^2*x^14+2*X1*X2^2* x^13-4*X1*X2*x^14-X1*x^15-X2^2*x^14+X2*x^15-X1^2*X2*x^12-2*X1^2*x^13-6*X1*X2*x^ 13+3*X1*x^14-X2^2*x^13+3*X2*x^14+X1^2*X2*x^11+X1^2*x^12+2*X1*X2*x^12+4*X1*x^13+ 3*X2*x^13-2*x^14-X1^2*X2*x^10-X1^2*x^11-X1*X2*x^11-2*X1*x^12-X2*x^12-2*x^13+2* X1^2*x^10+2*X1*X2*x^10+X1*x^11+x^12-X1^2*x^9-4*X1*x^10-X2*x^10+2*X1*x^9+2*x^10+ 2*X1*X2*x^7-X2*x^8-x^9-2*X1*X2*x^6-2*X1*x^7+x^8+X1*X2*x^5+2*X1*x^6-2*X1*x^5+X1* x^4+x^5-x^4+x^3-3*x^2+3*x-1)/(-1+x)/(X1^2*X2^2*x^15-2*X1^2*X2*x^15-2*X1*X2^2*x^ 15-X1^2*X2^2*x^13+X1^2*X2*x^14+X1^2*x^15-X1*X2^2*x^14+4*X1*X2*x^15+X2^2*x^15+2* X1^2*X2*x^13-X1^2*x^14+X1*X2^2*x^13-2*X1*x^15+X2^2*x^14-2*X2*x^15-X1^2*X2*x^12- X1^2*x^13-2*X1*X2*x^13+X1*x^14-X2*x^14+x^15+X1^2*X2*x^11+X1^2*x^12+2*X1*X2*x^12 +X1*x^13-X1^2*X2*x^10-X1^2*x^11-2*X1*X2*x^11-2*X1*x^12-X2*x^12+2*X1^2*x^10+2*X1 *x^11+X2*x^11+x^12-X1^2*x^9-X1*X2*x^9-2*X1*x^10+X2*x^10-x^11+3*X1*x^9+X2*x^9+X1 *X2*x^7-2*x^9-2*X1*X2*x^6-X1*x^7+X1*X2*x^5+X1*x^6-2*X1*x^5+x^6+X1*x^4+x^5-x^4-2 *x^2+3*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 1, 2], are n 191 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 2, 5], are n 111 57 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 4 19 ------- 57 and in floating point 0.3058876452 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 4 19 ate normal pair with correlation, ------- 57 1/2 4 19 203 i.e. , [[-------, 0], [0, ---]] 57 171 ------------------------------------------------- Theorem Number, 219, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [4, 1, 2], nor the composition, [1, 3, 4] Then infinity ----- \ n 18 17 16 15 14 13 12 11 10 ) a(n) x = (x + 2 x + 2 x + x - x - x + x + 2 x + 2 x / ----- n = 0 8 4 2 / + x - x - x + 2 x - 1) / ( / 10 9 8 7 6 3 2 2 (x + 2 x + 3 x + 2 x + x + x + x + x - 1) (-1 + x) ) and in Maple format (x^18+2*x^17+2*x^16+x^15-x^14-x^13+x^12+2*x^11+2*x^10+x^8-x^4-x^2+2*x-1)/(x^10+ 2*x^9+3*x^8+2*x^7+x^6+x^3+x^2+x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.65353748308844606444 1.9175371779461324874 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [4, 1, 2] and d occurrences (as containment) of the composition, [1, 3, 4], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 3 2 19 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 3 2 18 3 19 2 2 19 3 18 3 19 + X1 X2 x - 2 X1 X2 x - 3 X1 X2 x - 2 X1 X2 x + X1 x 2 2 18 2 19 2 19 3 18 2 18 - 3 X1 X2 x + 6 X1 X2 x + 3 X1 X2 x + X1 x + 6 X1 X2 x 2 19 2 18 19 2 19 2 2 16 - 3 X1 x + 3 X1 X2 x - 6 X1 X2 x - X2 x + X1 X2 x 2 18 18 19 2 18 19 2 2 15 - 3 X1 x - 6 X1 X2 x + 3 X1 x - X2 x + 2 X2 x + X1 X2 x 2 16 2 16 18 18 19 2 2 14 - 2 X1 X2 x - 2 X1 X2 x + 3 X1 x + 2 X2 x - x - X1 X2 x 2 15 2 16 2 15 16 2 16 18 - 3 X1 X2 x + X1 x - 2 X1 X2 x + 4 X1 X2 x + X2 x - x 2 2 13 2 14 2 15 2 14 15 + X1 X2 x + X1 X2 x + 2 X1 x + 2 X1 X2 x + 6 X1 X2 x 16 2 15 16 2 2 12 2 13 14 - 2 X1 x + X2 x - 2 X2 x - X1 X2 x - X1 X2 x - 2 X1 X2 x 15 2 14 15 16 2 12 2 13 - 4 X1 x - X2 x - 3 X2 x + x + 2 X1 X2 x - X1 x 2 12 13 14 15 2 12 12 + 2 X1 X2 x - 2 X1 X2 x + X2 x + 2 x - X1 x - 4 X1 X2 x 13 2 12 13 2 10 12 12 13 + 3 X1 x - X2 x + 2 X2 x + X1 X2 x + 2 X1 x + 2 X2 x - 2 x 2 9 12 9 10 10 9 10 - X1 X2 x - x + X1 X2 x - 2 X1 x - X2 x + X1 x + 2 x 7 8 9 6 7 8 5 6 + X1 X2 x - X2 x - x + X1 X2 x - X1 x + x - 2 X1 X2 x - X1 x 4 5 5 4 3 2 / + X1 X2 x + X1 x + x - x + x - 3 x + 3 x - 1) / ((-1 + x) ( / 2 2 13 2 2 12 2 13 2 13 2 12 X1 X2 x - X1 X2 x - X1 X2 x - 2 X1 X2 x + X1 X2 x 2 12 13 2 13 13 2 10 12 + X1 X2 x + 2 X1 X2 x + X2 x - X2 x + X1 X2 x - X1 x 12 2 9 10 12 10 9 9 - X2 x - X1 X2 x - 2 X1 X2 x + x + X2 x + 2 X1 x + X2 x 7 9 7 5 6 4 5 6 + X1 X2 x - 2 x - X1 x - 2 X1 X2 x - X1 x + X1 X2 x + X1 x + x 5 4 2 + x - x - 2 x + 3 x - 1)) and in Maple format -(X1^3*X2^2*x^19+X1^3*X2^2*x^18-2*X1^3*X2*x^19-3*X1^2*X2^2*x^19-2*X1^3*X2*x^18+ X1^3*x^19-3*X1^2*X2^2*x^18+6*X1^2*X2*x^19+3*X1*X2^2*x^19+X1^3*x^18+6*X1^2*X2*x^ 18-3*X1^2*x^19+3*X1*X2^2*x^18-6*X1*X2*x^19-X2^2*x^19+X1^2*X2^2*x^16-3*X1^2*x^18 -6*X1*X2*x^18+3*X1*x^19-X2^2*x^18+2*X2*x^19+X1^2*X2^2*x^15-2*X1^2*X2*x^16-2*X1* X2^2*x^16+3*X1*x^18+2*X2*x^18-x^19-X1^2*X2^2*x^14-3*X1^2*X2*x^15+X1^2*x^16-2*X1 *X2^2*x^15+4*X1*X2*x^16+X2^2*x^16-x^18+X1^2*X2^2*x^13+X1^2*X2*x^14+2*X1^2*x^15+ 2*X1*X2^2*x^14+6*X1*X2*x^15-2*X1*x^16+X2^2*x^15-2*X2*x^16-X1^2*X2^2*x^12-X1*X2^ 2*x^13-2*X1*X2*x^14-4*X1*x^15-X2^2*x^14-3*X2*x^15+x^16+2*X1^2*X2*x^12-X1^2*x^13 +2*X1*X2^2*x^12-2*X1*X2*x^13+X2*x^14+2*x^15-X1^2*x^12-4*X1*X2*x^12+3*X1*x^13-X2 ^2*x^12+2*X2*x^13+X1^2*X2*x^10+2*X1*x^12+2*X2*x^12-2*x^13-X1^2*X2*x^9-x^12+X1* X2*x^9-2*X1*x^10-X2*x^10+X1*x^9+2*x^10+X1*X2*x^7-X2*x^8-x^9+X1*X2*x^6-X1*x^7+x^ 8-2*X1*X2*x^5-X1*x^6+X1*X2*x^4+X1*x^5+x^5-x^4+x^3-3*x^2+3*x-1)/(-1+x)/(X1^2*X2^ 2*x^13-X1^2*X2^2*x^12-X1^2*X2*x^13-2*X1*X2^2*x^13+X1^2*X2*x^12+X1*X2^2*x^12+2* X1*X2*x^13+X2^2*x^13-X2*x^13+X1^2*X2*x^10-X1*x^12-X2*x^12-X1^2*X2*x^9-2*X1*X2*x ^10+x^12+X2*x^10+2*X1*x^9+X2*x^9+X1*X2*x^7-2*x^9-X1*x^7-2*X1*X2*x^5-X1*x^6+X1* X2*x^4+X1*x^5+x^6+x^5-x^4-2*x^2+3*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 1, 2], are n 191 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 3, 4], are n 135 65 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 16 3 65 ------------- 585 and in floating point 0.3819279330 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 16 195 ate normal pair with correlation, --------- 585 1/2 16 195 2267 i.e. , [[---------, 0], [0, ----]] 585 1755 ------------------------------------------------- Theorem Number, 220, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [4, 1, 2], nor the composition, [1, 4, 3] Then infinity ----- 15 12 9 5 4 3 2 \ n x - x + x + x - x + x - 3 x + 3 x - 1 ) a(n) x = ---------------------------------------------- / 7 6 4 2 ----- (x + x - x + 2 x - 1) (-1 + x) n = 0 and in Maple format (x^15-x^12+x^9+x^5-x^4+x^3-3*x^2+3*x-1)/(x^7+x^6-x^4+2*x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.66071460672041514761 1.9168163613635124211 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [4, 1, 2] and d occurrences (as containment) of the composition, [1, 4, 3], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 15 2 15 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x - X1 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 15 15 15 2 12 15 2 12 - 2 X1 X2 x + 2 X1 x + X2 x - X1 X2 x - x + X1 x 12 2 10 12 12 2 9 10 + 2 X1 X2 x - X1 X2 x - 2 X1 x - X2 x + X1 X2 x + 2 X1 X2 x 12 9 10 8 9 9 7 + x - 3 X1 X2 x - X2 x + X1 X2 x + X1 x + 2 X2 x - X1 X2 x 8 9 6 7 5 6 4 5 - X1 x - x - X1 X2 x + X1 x + 2 X1 X2 x + X1 x - X1 X2 x - X1 x 5 4 3 2 / 2 10 2 9 - x + x - x + 3 x - 3 x + 1) / ((-1 + x) (X1 X2 x - X1 X2 x / 10 9 10 8 9 7 8 - 2 X1 X2 x + 2 X1 X2 x + X2 x - X1 X2 x - X2 x + X1 X2 x + X1 x 8 7 8 5 6 4 5 6 5 + X2 x - X1 x - x - 2 X1 X2 x - X1 x + X1 X2 x + X1 x + x + x 4 2 - x - 2 x + 3 x - 1)) and in Maple format (X1^2*X2*x^15-X1^2*x^15-2*X1*X2*x^15+2*X1*x^15+X2*x^15-X1^2*X2*x^12-x^15+X1^2*x ^12+2*X1*X2*x^12-X1^2*X2*x^10-2*X1*x^12-X2*x^12+X1^2*X2*x^9+2*X1*X2*x^10+x^12-3 *X1*X2*x^9-X2*x^10+X1*X2*x^8+X1*x^9+2*X2*x^9-X1*X2*x^7-X1*x^8-x^9-X1*X2*x^6+X1* x^7+2*X1*X2*x^5+X1*x^6-X1*X2*x^4-X1*x^5-x^5+x^4-x^3+3*x^2-3*x+1)/(-1+x)/(X1^2* X2*x^10-X1^2*X2*x^9-2*X1*X2*x^10+2*X1*X2*x^9+X2*x^10-X1*X2*x^8-X2*x^9+X1*X2*x^7 +X1*x^8+X2*x^8-X1*x^7-x^8-2*X1*X2*x^5-X1*x^6+X1*X2*x^4+X1*x^5+x^6+x^5-x^4-2*x^2 +3*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 1, 2], are n 191 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 4, 3], are n 135 65 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 16 3 65 ------------- 585 and in floating point 0.3819279330 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 16 195 ate normal pair with correlation, --------- 585 1/2 16 195 2267 i.e. , [[---------, 0], [0, ----]] 585 1755 ------------------------------------------------- Theorem Number, 221, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [4, 1, 2], nor the composition, [1, 5, 2] Then infinity ----- 2 8 7 5 4 2 \ n (x - x + 1) (2 x + 2 x - 2 x - 2 x + x + x - 1) ) a(n) x = - ----------------------------------------------------- / 10 9 7 6 4 ----- (-1 + x) (x - x + x + x - x + 2 x - 1) n = 0 and in Maple format -(x^2-x+1)*(2*x^8+2*x^7-2*x^5-2*x^4+x^2+x-1)/(-1+x)/(x^10-x^9+x^7+x^6-x^4+2*x-1 ) The asymptotic expression for a(n) is, n 0.67273009850846616931 1.9140011778740215244 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [4, 1, 2] and d occurrences (as containment) of the composition, [1, 5, 2], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 10 2 10 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x - X1 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 10 2 9 9 10 10 9 9 - 3 X1 X2 x + X1 x - X1 X2 x + 3 X1 x + 2 X2 x - X1 x + X2 x 10 7 6 7 5 6 5 4 - 2 x - X1 X2 x + X1 X2 x + X1 x - X1 X2 x - X1 x + X1 x - X1 x 4 2 / 2 11 2 10 2 11 11 + x + x - 2 x + 1) / (X1 X2 x - X1 X2 x - X1 x - 2 X1 X2 x / 2 10 10 11 11 2 9 10 10 + 2 X1 x + 2 X1 X2 x + 2 X1 x + X2 x - X1 x - 4 X1 x - X2 x 11 8 9 10 7 8 8 9 - x - X1 X2 x + 2 X1 x + 2 x + X1 X2 x + X1 x + X2 x - x 6 7 8 5 6 5 6 4 5 - 2 X1 X2 x - X1 x - x + X1 X2 x + X1 x - 2 X1 x + x + X1 x + x 4 2 - x - 2 x + 3 x - 1) and in Maple format -(X1^2*X2*x^10-X1^2*x^10-3*X1*X2*x^10+X1^2*x^9-X1*X2*x^9+3*X1*x^10+2*X2*x^10-X1 *x^9+X2*x^9-2*x^10-X1*X2*x^7+X1*X2*x^6+X1*x^7-X1*X2*x^5-X1*x^6+X1*x^5-X1*x^4+x^ 4+x^2-2*x+1)/(X1^2*X2*x^11-X1^2*X2*x^10-X1^2*x^11-2*X1*X2*x^11+2*X1^2*x^10+2*X1 *X2*x^10+2*X1*x^11+X2*x^11-X1^2*x^9-4*X1*x^10-X2*x^10-x^11-X1*X2*x^8+2*X1*x^9+2 *x^10+X1*X2*x^7+X1*x^8+X2*x^8-x^9-2*X1*X2*x^6-X1*x^7-x^8+X1*X2*x^5+X1*x^6-2*X1* x^5+x^6+X1*x^4+x^5-x^4-2*x^2+3*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 1, 2], are n 191 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 5, 2], are n 111 57 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 4 19 ------- 57 and in floating point 0.3058876452 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 4 19 ate normal pair with correlation, ------- 57 1/2 4 19 203 i.e. , [[-------, 0], [0, ---]] 57 171 ------------------------------------------------- Theorem Number, 222, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [4, 1, 2], nor the composition, [1, 6, 1] Then infinity ----- 17 16 15 12 11 10 4 2 \ n x + 2 x + x + x - 2 x - x + x + x - 2 x + 1 ) a(n) x = - --------------------------------------------------------- / 6 5 3 2 6 5 2 ----- (x + x + x + x + x - 1) (x + x + 1) (-1 + x) n = 0 and in Maple format -(x^17+2*x^16+x^15+x^12-2*x^11-x^10+x^4+x^2-2*x+1)/(x^6+x^5+x^3+x^2+x-1)/(x^6+x ^5+1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.68363894095775044155 1.9111834366854762350 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [4, 1, 2] and d occurrences (as containment) of the composition, [1, 6, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 18 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 18 2 18 2 18 2 17 18 - 2 X1 X2 x - 2 X1 X2 x + X1 x - X1 X2 x + 4 X1 X2 x 2 18 2 2 15 2 16 17 18 2 17 + X2 x - X1 X2 x + X1 X2 x + 2 X1 X2 x - 2 X1 x + X2 x 18 2 2 14 2 15 2 16 2 15 - 2 X2 x + X1 X2 x + 2 X1 X2 x - X1 x + 2 X1 X2 x 16 17 17 18 2 2 13 2 14 - 2 X1 X2 x - X1 x - 2 X2 x + x - X1 X2 x - 2 X1 X2 x 2 15 2 14 15 16 2 15 16 17 - X1 x - X1 X2 x - 4 X1 X2 x + 2 X1 x - X2 x + X2 x + x 2 13 2 14 2 13 14 15 15 + X1 X2 x + X1 x + X1 X2 x + 2 X1 X2 x + 2 X1 x + 2 X2 x 16 2 12 14 15 2 11 2 12 - x + 2 X1 X2 x - X1 x - x - 2 X1 X2 x - 2 X1 x 12 13 13 2 11 11 12 - 5 X1 X2 x - X1 x - X2 x + 2 X1 x + 3 X1 X2 x + 5 X1 x 12 13 2 10 11 11 12 2 9 9 + 3 X2 x + x + X1 x - 3 X1 x - X2 x - 3 x - X1 x + X1 X2 x 10 11 9 9 10 7 6 7 - 2 X1 x + x + X1 x - X2 x + x - X1 X2 x + X1 X2 x + X1 x 6 5 4 5 4 3 2 / - X1 x - X1 x + X1 x + x - x + x - 3 x + 3 x - 1) / ((-1 + x) ( / 7 6 7 7 6 7 5 4 5 4 X1 X2 x - X1 X2 x - X1 x - X2 x + X1 x + x + X1 x - X1 x - x + x 7 7 7 7 5 5 - 2 x + 1) (X1 X2 x - X1 x - X2 x + x + X1 x - x + x - 1)) and in Maple format -(X1^2*X2^2*x^18-2*X1^2*X2*x^18-2*X1*X2^2*x^18+X1^2*x^18-X1*X2^2*x^17+4*X1*X2*x ^18+X2^2*x^18-X1^2*X2^2*x^15+X1^2*X2*x^16+2*X1*X2*x^17-2*X1*x^18+X2^2*x^17-2*X2 *x^18+X1^2*X2^2*x^14+2*X1^2*X2*x^15-X1^2*x^16+2*X1*X2^2*x^15-2*X1*X2*x^16-X1*x^ 17-2*X2*x^17+x^18-X1^2*X2^2*x^13-2*X1^2*X2*x^14-X1^2*x^15-X1*X2^2*x^14-4*X1*X2* x^15+2*X1*x^16-X2^2*x^15+X2*x^16+x^17+X1^2*X2*x^13+X1^2*x^14+X1*X2^2*x^13+2*X1* X2*x^14+2*X1*x^15+2*X2*x^15-x^16+2*X1^2*X2*x^12-X1*x^14-x^15-2*X1^2*X2*x^11-2* X1^2*x^12-5*X1*X2*x^12-X1*x^13-X2*x^13+2*X1^2*x^11+3*X1*X2*x^11+5*X1*x^12+3*X2* x^12+x^13+X1^2*x^10-3*X1*x^11-X2*x^11-3*x^12-X1^2*x^9+X1*X2*x^9-2*X1*x^10+x^11+ X1*x^9-X2*x^9+x^10-X1*X2*x^7+X1*X2*x^6+X1*x^7-X1*x^6-X1*x^5+X1*x^4+x^5-x^4+x^3-\ 3*x^2+3*x-1)/(-1+x)/(X1*X2*x^7-X1*X2*x^6-X1*x^7-X2*x^7+X1*x^6+x^7+X1*x^5-X1*x^4 -x^5+x^4-2*x+1)/(X1*X2*x^7-X1*x^7-X2*x^7+x^7+X1*x^5-x^5+x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 1, 2], are n 191 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 6, 1], are n 49 53 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 8 3 53 ------------ 477 and in floating point 0.2114804230 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 8 159 ate normal pair with correlation, -------- 477 1/2 8 159 1559 i.e. , [[--------, 0], [0, ----]] 477 1431 ------------------------------------------------- Theorem Number, 223, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [4, 1, 2], nor the composition, [2, 1, 5] Then infinity ----- \ n 26 25 24 23 21 20 19 18 ) a(n) x = - (x + 4 x + 6 x + 4 x - 3 x - 3 x - x - x / ----- n = 0 17 16 15 14 11 10 9 5 4 3 2 - x + x + 2 x + x + x - 2 x - x - x + x - x + 3 x - 3 x / + 1) / ( / 18 17 16 13 12 10 9 6 5 2 (x + 2 x + x - x - x + x + x + x - x - x + 2 x - 1) 9 8 5 4 (x + x - x + x - 2 x + 1)) and in Maple format -(x^26+4*x^25+6*x^24+4*x^23-3*x^21-3*x^20-x^19-x^18-x^17+x^16+2*x^15+x^14+x^11-\ 2*x^10-x^9-x^5+x^4-x^3+3*x^2-3*x+1)/(x^18+2*x^17+x^16-x^13-x^12+x^10+x^9+x^6-x^ 5-x^2+2*x-1)/(x^9+x^8-x^5+x^4-2*x+1) The asymptotic expression for a(n) is, n 0.66036270785748552732 1.9159714162121856417 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [4, 1, 2] and d occurrences (as containment) of the composition, [2, 1, 5], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 3 3 26 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 3 3 25 3 2 26 2 3 26 3 3 24 + 4 X1 X2 x - 3 X1 X2 x - 3 X1 X2 x + 6 X1 X2 x 3 2 25 3 26 2 3 25 2 2 26 - 12 X1 X2 x + 3 X1 X2 x - 12 X1 X2 x + 9 X1 X2 x 3 26 3 3 23 3 2 24 3 25 3 26 + 3 X1 X2 x + 5 X1 X2 x - 18 X1 X2 x + 12 X1 X2 x - X1 x 2 3 24 2 2 25 2 26 3 25 - 18 X1 X2 x + 36 X1 X2 x - 9 X1 X2 x + 12 X1 X2 x 2 26 3 26 3 3 22 3 2 23 3 24 - 9 X1 X2 x - X2 x + 4 X1 X2 x - 15 X1 X2 x + 18 X1 X2 x 3 25 2 3 23 2 2 24 2 25 2 26 - 4 X1 x - 14 X1 X2 x + 54 X1 X2 x - 36 X1 X2 x + 3 X1 x 3 24 2 25 26 3 25 2 26 + 18 X1 X2 x - 36 X1 X2 x + 9 X1 X2 x - 4 X2 x + 3 X2 x 3 3 21 3 2 22 3 23 3 24 + 3 X1 X2 x - 11 X1 X2 x + 15 X1 X2 x - 6 X1 x 2 3 22 2 2 23 2 24 2 25 - 9 X1 X2 x + 42 X1 X2 x - 54 X1 X2 x + 12 X1 x 3 23 2 24 25 26 3 24 + 13 X1 X2 x - 54 X1 X2 x + 36 X1 X2 x - 3 X1 x - 6 X2 x 2 25 26 3 3 20 3 2 21 3 22 + 12 X2 x - 3 X2 x + X1 X2 x - 6 X1 X2 x + 10 X1 X2 x 3 23 2 3 21 2 2 22 2 23 2 24 - 5 X1 x - 6 X1 X2 x + 24 X1 X2 x - 42 X1 X2 x + 18 X1 x 3 22 2 23 24 25 3 23 + 6 X1 X2 x - 39 X1 X2 x + 54 X1 X2 x - 12 X1 x - 4 X2 x 2 24 25 26 3 3 19 3 21 3 22 + 18 X2 x - 12 X2 x + x + X1 X2 x + 3 X1 X2 x - 3 X1 x 2 3 20 2 2 21 2 22 2 23 - 2 X1 X2 x + 9 X1 X2 x - 21 X1 X2 x + 14 X1 x 3 21 2 22 23 24 3 22 + 3 X1 X2 x - 15 X1 X2 x + 39 X1 X2 x - 18 X1 x - X2 x 2 23 24 25 3 3 18 3 2 19 + 12 X2 x - 18 X2 x + 4 x + X1 X2 x - 2 X1 X2 x 3 20 2 3 19 2 2 20 2 22 3 20 - 3 X1 X2 x - X1 X2 x - 3 X1 X2 x + 6 X1 x + X1 X2 x 22 23 2 22 23 24 3 2 18 + 12 X1 X2 x - 13 X1 x + 2 X2 x - 12 X2 x + 6 x - X1 X2 x 3 19 3 20 2 3 18 2 20 2 21 + X1 X2 x + 2 X1 x - X1 X2 x + 12 X1 X2 x - 3 X1 x 2 20 21 22 2 21 22 23 + 6 X1 X2 x - 9 X1 X2 x - 3 X1 x - 3 X2 x - X2 x + 4 x 3 2 17 3 18 2 2 18 2 19 2 20 + 2 X1 X2 x - X1 X2 x - 2 X1 X2 x + 3 X1 X2 x - 7 X1 x 2 19 20 21 2 20 21 + 3 X1 X2 x - 15 X1 X2 x + 6 X1 x - 3 X2 x + 6 X2 x 3 17 3 18 2 2 17 2 18 2 19 - 3 X1 X2 x + X1 x - 6 X1 X2 x + 7 X1 X2 x - 2 X1 x 2 18 19 20 2 19 20 21 + 4 X1 X2 x - 6 X1 X2 x + 8 X1 x - X2 x + 6 X2 x - 3 x 3 16 3 17 2 17 2 18 2 17 + X1 X2 x + X1 x + 9 X1 X2 x - 4 X1 x + 6 X1 X2 x 18 19 2 18 19 20 3 16 - 8 X1 X2 x + 3 X1 x - X2 x + 2 X2 x - 3 x - X1 x 2 2 15 2 16 2 17 17 18 + 2 X1 X2 x - 3 X1 X2 x - 3 X1 x - 9 X1 X2 x + 4 X1 x 2 17 18 19 2 2 14 2 15 2 16 - 2 X2 x + 2 X2 x - x + X1 X2 x - 4 X1 X2 x + 3 X1 x 2 15 16 17 17 18 2 14 - 4 X1 X2 x + 3 X1 X2 x + 3 X1 x + 3 X2 x - x - 2 X1 X2 x 2 15 2 14 15 16 2 15 16 + 2 X1 x - 2 X1 X2 x + 8 X1 X2 x - 3 X1 x + 2 X2 x - X2 x 17 2 2 12 2 14 14 15 2 14 - x - X1 X2 x + X1 x + 4 X1 X2 x - 4 X1 x + X2 x 15 16 2 2 11 2 12 14 14 - 4 X2 x + x + X1 X2 x + 2 X1 X2 x - 2 X1 x - 2 X2 x 15 2 11 2 12 14 2 10 2 11 + 2 x - 4 X1 X2 x - X1 x + x + 2 X1 X2 x + 3 X1 x 11 2 10 10 11 11 2 9 + 3 X1 X2 x - 3 X1 x - 3 X1 X2 x - 3 X1 x - X2 x + X1 x 9 10 10 11 9 10 9 6 - 2 X1 X2 x + 5 X1 x + X2 x + x + 2 X2 x - 2 x - x + X1 X2 x 5 6 5 4 5 4 3 2 / - X1 X2 x - X1 x + 2 X1 x - X1 x - x + x - x + 3 x - 3 x + 1) / / 9 8 9 9 8 8 9 6 8 ((X1 X2 x + X1 X2 x - X1 x - X2 x - X1 x - X2 x + x + X1 X2 x + x 5 6 5 4 5 4 2 2 18 - X1 X2 x - X1 x + 2 X1 x - X1 x - x + x - 2 x + 1) (X1 X2 x 2 2 17 2 18 2 18 2 2 16 2 17 + 2 X1 X2 x - 2 X1 X2 x - 2 X1 X2 x + X1 X2 x - 4 X1 X2 x 2 18 2 17 18 2 18 2 16 + X1 x - 4 X1 X2 x + 4 X1 X2 x + X2 x - 2 X1 X2 x 2 17 2 16 17 18 2 17 18 + 2 X1 x - 2 X1 X2 x + 8 X1 X2 x - 2 X1 x + 2 X2 x - 2 X2 x 2 2 14 2 16 16 17 2 16 17 + X1 X2 x + X1 x + 4 X1 X2 x - 4 X1 x + X2 x - 4 X2 x 18 2 2 13 2 14 2 14 16 16 + x + X1 X2 x - 2 X1 X2 x - X1 X2 x - 2 X1 x - 2 X2 x 17 2 13 2 14 2 13 14 16 + 2 x - X1 X2 x + X1 x - X1 X2 x + 2 X1 X2 x + x 2 12 14 2 12 12 13 13 12 + X1 X2 x - X1 x - X1 x - 2 X1 X2 x + X1 x + X2 x + 2 X1 x 12 13 10 12 9 10 10 9 + X2 x - x + X1 X2 x - x + X1 X2 x - X1 x - X2 x - X1 x 9 10 7 9 6 7 6 5 6 - X2 x + x - X1 X2 x + x + X1 X2 x + X1 x - 2 X1 x + X1 x + x 5 2 - x - x + 2 x - 1)) and in Maple format -(X1^3*X2^3*x^26+4*X1^3*X2^3*x^25-3*X1^3*X2^2*x^26-3*X1^2*X2^3*x^26+6*X1^3*X2^3 *x^24-12*X1^3*X2^2*x^25+3*X1^3*X2*x^26-12*X1^2*X2^3*x^25+9*X1^2*X2^2*x^26+3*X1* X2^3*x^26+5*X1^3*X2^3*x^23-18*X1^3*X2^2*x^24+12*X1^3*X2*x^25-X1^3*x^26-18*X1^2* X2^3*x^24+36*X1^2*X2^2*x^25-9*X1^2*X2*x^26+12*X1*X2^3*x^25-9*X1*X2^2*x^26-X2^3* x^26+4*X1^3*X2^3*x^22-15*X1^3*X2^2*x^23+18*X1^3*X2*x^24-4*X1^3*x^25-14*X1^2*X2^ 3*x^23+54*X1^2*X2^2*x^24-36*X1^2*X2*x^25+3*X1^2*x^26+18*X1*X2^3*x^24-36*X1*X2^2 *x^25+9*X1*X2*x^26-4*X2^3*x^25+3*X2^2*x^26+3*X1^3*X2^3*x^21-11*X1^3*X2^2*x^22+ 15*X1^3*X2*x^23-6*X1^3*x^24-9*X1^2*X2^3*x^22+42*X1^2*X2^2*x^23-54*X1^2*X2*x^24+ 12*X1^2*x^25+13*X1*X2^3*x^23-54*X1*X2^2*x^24+36*X1*X2*x^25-3*X1*x^26-6*X2^3*x^ 24+12*X2^2*x^25-3*X2*x^26+X1^3*X2^3*x^20-6*X1^3*X2^2*x^21+10*X1^3*X2*x^22-5*X1^ 3*x^23-6*X1^2*X2^3*x^21+24*X1^2*X2^2*x^22-42*X1^2*X2*x^23+18*X1^2*x^24+6*X1*X2^ 3*x^22-39*X1*X2^2*x^23+54*X1*X2*x^24-12*X1*x^25-4*X2^3*x^23+18*X2^2*x^24-12*X2* x^25+x^26+X1^3*X2^3*x^19+3*X1^3*X2*x^21-3*X1^3*x^22-2*X1^2*X2^3*x^20+9*X1^2*X2^ 2*x^21-21*X1^2*X2*x^22+14*X1^2*x^23+3*X1*X2^3*x^21-15*X1*X2^2*x^22+39*X1*X2*x^ 23-18*X1*x^24-X2^3*x^22+12*X2^2*x^23-18*X2*x^24+4*x^25+X1^3*X2^3*x^18-2*X1^3*X2 ^2*x^19-3*X1^3*X2*x^20-X1^2*X2^3*x^19-3*X1^2*X2^2*x^20+6*X1^2*x^22+X1*X2^3*x^20 +12*X1*X2*x^22-13*X1*x^23+2*X2^2*x^22-12*X2*x^23+6*x^24-X1^3*X2^2*x^18+X1^3*X2* x^19+2*X1^3*x^20-X1^2*X2^3*x^18+12*X1^2*X2*x^20-3*X1^2*x^21+6*X1*X2^2*x^20-9*X1 *X2*x^21-3*X1*x^22-3*X2^2*x^21-X2*x^22+4*x^23+2*X1^3*X2^2*x^17-X1^3*X2*x^18-2* X1^2*X2^2*x^18+3*X1^2*X2*x^19-7*X1^2*x^20+3*X1*X2^2*x^19-15*X1*X2*x^20+6*X1*x^ 21-3*X2^2*x^20+6*X2*x^21-3*X1^3*X2*x^17+X1^3*x^18-6*X1^2*X2^2*x^17+7*X1^2*X2*x^ 18-2*X1^2*x^19+4*X1*X2^2*x^18-6*X1*X2*x^19+8*X1*x^20-X2^2*x^19+6*X2*x^20-3*x^21 +X1^3*X2*x^16+X1^3*x^17+9*X1^2*X2*x^17-4*X1^2*x^18+6*X1*X2^2*x^17-8*X1*X2*x^18+ 3*X1*x^19-X2^2*x^18+2*X2*x^19-3*x^20-X1^3*x^16+2*X1^2*X2^2*x^15-3*X1^2*X2*x^16-\ 3*X1^2*x^17-9*X1*X2*x^17+4*X1*x^18-2*X2^2*x^17+2*X2*x^18-x^19+X1^2*X2^2*x^14-4* X1^2*X2*x^15+3*X1^2*x^16-4*X1*X2^2*x^15+3*X1*X2*x^16+3*X1*x^17+3*X2*x^17-x^18-2 *X1^2*X2*x^14+2*X1^2*x^15-2*X1*X2^2*x^14+8*X1*X2*x^15-3*X1*x^16+2*X2^2*x^15-X2* x^16-x^17-X1^2*X2^2*x^12+X1^2*x^14+4*X1*X2*x^14-4*X1*x^15+X2^2*x^14-4*X2*x^15+x ^16+X1^2*X2^2*x^11+2*X1^2*X2*x^12-2*X1*x^14-2*X2*x^14+2*x^15-4*X1^2*X2*x^11-X1^ 2*x^12+x^14+2*X1^2*X2*x^10+3*X1^2*x^11+3*X1*X2*x^11-3*X1^2*x^10-3*X1*X2*x^10-3* X1*x^11-X2*x^11+X1^2*x^9-2*X1*X2*x^9+5*X1*x^10+X2*x^10+x^11+2*X2*x^9-2*x^10-x^9 +X1*X2*x^6-X1*X2*x^5-X1*x^6+2*X1*x^5-X1*x^4-x^5+x^4-x^3+3*x^2-3*x+1)/(X1*X2*x^9 +X1*X2*x^8-X1*x^9-X2*x^9-X1*x^8-X2*x^8+x^9+X1*X2*x^6+x^8-X1*X2*x^5-X1*x^6+2*X1* x^5-X1*x^4-x^5+x^4-2*x+1)/(X1^2*X2^2*x^18+2*X1^2*X2^2*x^17-2*X1^2*X2*x^18-2*X1* X2^2*x^18+X1^2*X2^2*x^16-4*X1^2*X2*x^17+X1^2*x^18-4*X1*X2^2*x^17+4*X1*X2*x^18+ X2^2*x^18-2*X1^2*X2*x^16+2*X1^2*x^17-2*X1*X2^2*x^16+8*X1*X2*x^17-2*X1*x^18+2*X2 ^2*x^17-2*X2*x^18+X1^2*X2^2*x^14+X1^2*x^16+4*X1*X2*x^16-4*X1*x^17+X2^2*x^16-4* X2*x^17+x^18+X1^2*X2^2*x^13-2*X1^2*X2*x^14-X1*X2^2*x^14-2*X1*x^16-2*X2*x^16+2*x ^17-X1^2*X2*x^13+X1^2*x^14-X1*X2^2*x^13+2*X1*X2*x^14+x^16+X1^2*X2*x^12-X1*x^14- X1^2*x^12-2*X1*X2*x^12+X1*x^13+X2*x^13+2*X1*x^12+X2*x^12-x^13+X1*X2*x^10-x^12+ X1*X2*x^9-X1*x^10-X2*x^10-X1*x^9-X2*x^9+x^10-X1*X2*x^7+x^9+X1*X2*x^6+X1*x^7-2* X1*x^6+X1*x^5+x^6-x^5-x^2+2*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 1, 2], are n 191 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 1, 5], are n 113 57 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 14 19 -------- 171 and in floating point 0.3568689194 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 14 19 ate normal pair with correlation, -------- 171 1/2 14 19 1931 i.e. , [[--------, 0], [0, ----]] 171 1539 ------------------------------------------------- Theorem Number, 224, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [4, 1, 2], nor the composition, [2, 2, 4] Then infinity ----- \ n ) a(n) x = / ----- n = 0 14 13 11 10 9 8 4 2 x + 2 x - 2 x - 2 x - x - x + x + x - 2 x + 1 - ---------------------------------------------------------------- 15 14 12 11 9 6 5 4 2 x + 2 x - 2 x - 2 x - x + x + x - x - 2 x + 3 x - 1 and in Maple format -(x^14+2*x^13-2*x^11-2*x^10-x^9-x^8+x^4+x^2-2*x+1)/(x^15+2*x^14-2*x^12-2*x^11-x ^9+x^6+x^5-x^4-2*x^2+3*x-1) The asymptotic expression for a(n) is, n 0.65170350411535370406 1.9191057201085461028 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [4, 1, 2] and d occurrences (as containment) of the composition, [2, 2, 4], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 3 2 16 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 3 2 15 3 16 2 2 16 3 15 2 2 15 + X1 X2 x - X1 X2 x - 3 X1 X2 x - X1 X2 x - 4 X1 X2 x 2 16 2 16 2 15 2 15 16 + 3 X1 X2 x + 3 X1 X2 x + 5 X1 X2 x + 5 X1 X2 x - 3 X1 X2 x 2 16 2 2 13 2 14 2 15 15 2 15 - X2 x + X1 X2 x + X1 X2 x - X1 x - 7 X1 X2 x - 2 X2 x 16 2 13 2 14 2 13 14 15 + X2 x - 2 X1 X2 x - X1 x - 3 X1 X2 x - 2 X1 X2 x + 2 X1 x 15 2 12 2 13 2 12 13 14 + 3 X2 x - X1 X2 x + X1 x - X1 X2 x + 6 X1 X2 x + 2 X1 x 2 13 14 15 2 12 12 13 2 12 + 2 X2 x + X2 x - x + X1 x + 4 X1 X2 x - 3 X1 x + X2 x 13 14 2 10 12 12 13 2 9 - 4 X2 x - x - X1 X2 x - 3 X1 x - 3 X2 x + 2 x + X1 X2 x 10 12 9 10 8 9 10 + X1 X2 x + 2 x - 2 X1 X2 x + X1 x - X1 X2 x + X2 x - x 7 8 8 6 7 8 5 + X1 X2 x + X1 x + X2 x - 2 X1 X2 x - X1 x - x + 2 X1 X2 x 6 4 5 5 4 3 2 / + 2 X1 x - X1 X2 x - X1 x - x + x - x + 3 x - 3 x + 1) / ( / 3 2 17 3 17 2 2 17 3 2 15 2 2 16 X1 X2 x - X1 X2 x - 3 X1 X2 x - X1 X2 x - X1 X2 x 2 17 2 17 3 15 2 2 15 2 16 + 3 X1 X2 x + 3 X1 X2 x + X1 X2 x + 3 X1 X2 x + 2 X1 X2 x 2 16 17 2 17 2 15 2 16 + 2 X1 X2 x - 3 X1 X2 x - X2 x - 2 X1 X2 x - X1 x 2 15 16 2 16 17 2 2 13 2 14 - 3 X1 X2 x - 4 X1 X2 x - X2 x + X2 x - X1 X2 x - X1 X2 x 2 15 2 14 15 16 2 15 16 - X1 x - X1 X2 x + X1 X2 x + 2 X1 x + X2 x + 2 X2 x 2 14 2 13 14 15 2 14 16 2 13 + X1 x + X1 X2 x + 4 X1 X2 x + 2 X1 x + X2 x - x + X1 x 13 14 14 15 2 11 13 + 2 X1 X2 x - 3 X1 x - 3 X2 x - x - X1 X2 x - 3 X1 x 13 14 2 10 13 2 9 10 - 2 X2 x + 2 x + 2 X1 X2 x + 2 x - X1 X2 x - 3 X1 X2 x 11 11 9 10 10 11 8 + 2 X1 x + X2 x + X1 X2 x - X1 x + X2 x - 2 x + X1 X2 x 9 10 7 8 9 6 7 + X1 x + x - 2 X1 X2 x - X1 x - x + 3 X1 X2 x + 3 X1 x 5 6 7 4 5 5 4 3 2 - 3 X1 X2 x - 3 X1 x - x + X1 X2 x + X1 x + 2 x - x + 2 x - 5 x + 4 x - 1) and in Maple format -(X1^3*X2^2*x^16+X1^3*X2^2*x^15-X1^3*X2*x^16-3*X1^2*X2^2*x^16-X1^3*X2*x^15-4*X1 ^2*X2^2*x^15+3*X1^2*X2*x^16+3*X1*X2^2*x^16+5*X1^2*X2*x^15+5*X1*X2^2*x^15-3*X1* X2*x^16-X2^2*x^16+X1^2*X2^2*x^13+X1^2*X2*x^14-X1^2*x^15-7*X1*X2*x^15-2*X2^2*x^ 15+X2*x^16-2*X1^2*X2*x^13-X1^2*x^14-3*X1*X2^2*x^13-2*X1*X2*x^14+2*X1*x^15+3*X2* x^15-X1^2*X2*x^12+X1^2*x^13-X1*X2^2*x^12+6*X1*X2*x^13+2*X1*x^14+2*X2^2*x^13+X2* x^14-x^15+X1^2*x^12+4*X1*X2*x^12-3*X1*x^13+X2^2*x^12-4*X2*x^13-x^14-X1^2*X2*x^ 10-3*X1*x^12-3*X2*x^12+2*x^13+X1^2*X2*x^9+X1*X2*x^10+2*x^12-2*X1*X2*x^9+X1*x^10 -X1*X2*x^8+X2*x^9-x^10+X1*X2*x^7+X1*x^8+X2*x^8-2*X1*X2*x^6-X1*x^7-x^8+2*X1*X2*x ^5+2*X1*x^6-X1*X2*x^4-X1*x^5-x^5+x^4-x^3+3*x^2-3*x+1)/(X1^3*X2^2*x^17-X1^3*X2*x ^17-3*X1^2*X2^2*x^17-X1^3*X2^2*x^15-X1^2*X2^2*x^16+3*X1^2*X2*x^17+3*X1*X2^2*x^ 17+X1^3*X2*x^15+3*X1^2*X2^2*x^15+2*X1^2*X2*x^16+2*X1*X2^2*x^16-3*X1*X2*x^17-X2^ 2*x^17-2*X1^2*X2*x^15-X1^2*x^16-3*X1*X2^2*x^15-4*X1*X2*x^16-X2^2*x^16+X2*x^17- X1^2*X2^2*x^13-X1^2*X2*x^14-X1^2*x^15-X1*X2^2*x^14+X1*X2*x^15+2*X1*x^16+X2^2*x^ 15+2*X2*x^16+X1^2*x^14+X1*X2^2*x^13+4*X1*X2*x^14+2*X1*x^15+X2^2*x^14-x^16+X1^2* x^13+2*X1*X2*x^13-3*X1*x^14-3*X2*x^14-x^15-X1^2*X2*x^11-3*X1*x^13-2*X2*x^13+2*x ^14+2*X1^2*X2*x^10+2*x^13-X1^2*X2*x^9-3*X1*X2*x^10+2*X1*x^11+X2*x^11+X1*X2*x^9- X1*x^10+X2*x^10-2*x^11+X1*X2*x^8+X1*x^9+x^10-2*X1*X2*x^7-X1*x^8-x^9+3*X1*X2*x^6 +3*X1*x^7-3*X1*X2*x^5-3*X1*x^6-x^7+X1*X2*x^4+X1*x^5+2*x^5-x^4+2*x^3-5*x^2+4*x-1 ) Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 1, 2], are n 191 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 2, 4], are n 75 69 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 2 23 ------- 23 and in floating point 0.4170288281 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 2 23 ate normal pair with correlation, ------- 23 1/2 2 23 31 i.e. , [[-------, 0], [0, --]] 23 23 ------------------------------------------------- Theorem Number, 225, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [4, 1, 2], nor the composition, [2, 3, 3] Then infinity ----- \ n ) a(n) x = - ( / ----- n = 0 14 11 10 9 8 7 6 5 4 3 2 / x + x + x + x - x + x - x - x + x - x - x + 2 x - 1) / ( / 15 14 12 11 9 8 7 5 4 3 2 x - x + x + x - x + 2 x - 2 x + x - 2 x + x + 2 x - 3 x + 1) and in Maple format -(x^14+x^11+x^10+x^9-x^8+x^7-x^6-x^5+x^4-x^3-x^2+2*x-1)/(x^15-x^14+x^12+x^11-x^ 9+2*x^8-2*x^7+x^5-2*x^4+x^3+2*x^2-3*x+1) The asymptotic expression for a(n) is, n 0.66099415014764611466 1.9170546374446963081 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [4, 1, 2] and d occurrences (as containment) of the composition, [2, 3, 3], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 3 15 3 15 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x - X1 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 15 2 14 2 15 15 2 14 - 3 X1 X2 x + X1 X2 x + 3 X1 x + 3 X1 X2 x - X1 x 14 15 15 2 12 14 14 15 - 2 X1 X2 x - 3 X1 x - X2 x - X1 X2 x + 2 X1 x + X2 x + x 2 12 12 14 2 10 12 12 2 9 + X1 x + 2 X1 X2 x - x - X1 X2 x - 2 X1 x - X2 x + X1 X2 x 10 12 9 10 8 9 9 + 2 X1 X2 x + x - 4 X1 X2 x - X2 x + 2 X1 X2 x + 2 X1 x + 3 X2 x 7 8 8 9 6 7 7 8 - X1 X2 x - 2 X1 x - 2 X2 x - 2 x - X1 X2 x + X1 x + 2 X2 x + 2 x 5 6 7 4 5 5 4 5 + 2 X1 X2 x + X1 x - 2 x - X1 X2 x - X1 x - 3 X2 x + 3 X2 x + 2 x 3 4 2 / 3 16 3 15 3 16 - X2 x - 2 x + 3 x - 3 x + 1) / (X1 X2 x - X1 X2 x - X1 x / 2 16 3 15 2 15 2 16 16 - 3 X1 X2 x + X1 x + 4 X1 X2 x + 3 X1 x + 3 X1 X2 x 2 14 2 15 15 16 16 2 13 - X1 X2 x - 4 X1 x - 5 X1 X2 x - 3 X1 x - X2 x - X1 X2 x 2 14 14 15 15 16 2 13 + X1 x + 2 X1 X2 x + 5 X1 x + 2 X2 x + x + X1 x 13 14 14 15 2 11 13 13 + 2 X1 X2 x - 2 X1 x - X2 x - 2 x - X1 X2 x - 2 X1 x - X2 x 14 2 10 11 13 2 9 10 11 + x + 2 X1 X2 x + X1 X2 x + x - X1 X2 x - 5 X1 X2 x + X1 x 9 10 10 11 8 9 9 + 5 X1 X2 x + X1 x + 3 X2 x - x - 3 X1 X2 x - 3 X1 x - 4 X2 x 10 7 8 8 9 6 7 8 - x + X1 X2 x + 3 X1 x + 4 X2 x + 3 x + 2 X1 X2 x - 3 X2 x - 4 x 5 6 6 7 4 5 5 6 - 3 X1 X2 x - 2 X1 x - X2 x + 2 x + X1 X2 x + X1 x + 5 X2 x + x 4 5 3 4 3 2 - 4 X2 x - 3 x + X2 x + 3 x + x - 5 x + 4 x - 1) and in Maple format -(X1^3*X2*x^15-X1^3*x^15-3*X1^2*X2*x^15+X1^2*X2*x^14+3*X1^2*x^15+3*X1*X2*x^15- X1^2*x^14-2*X1*X2*x^14-3*X1*x^15-X2*x^15-X1^2*X2*x^12+2*X1*x^14+X2*x^14+x^15+X1 ^2*x^12+2*X1*X2*x^12-x^14-X1^2*X2*x^10-2*X1*x^12-X2*x^12+X1^2*X2*x^9+2*X1*X2*x^ 10+x^12-4*X1*X2*x^9-X2*x^10+2*X1*X2*x^8+2*X1*x^9+3*X2*x^9-X1*X2*x^7-2*X1*x^8-2* X2*x^8-2*x^9-X1*X2*x^6+X1*x^7+2*X2*x^7+2*x^8+2*X1*X2*x^5+X1*x^6-2*x^7-X1*X2*x^4 -X1*x^5-3*X2*x^5+3*X2*x^4+2*x^5-X2*x^3-2*x^4+3*x^2-3*x+1)/(X1^3*X2*x^16-X1^3*X2 *x^15-X1^3*x^16-3*X1^2*X2*x^16+X1^3*x^15+4*X1^2*X2*x^15+3*X1^2*x^16+3*X1*X2*x^ 16-X1^2*X2*x^14-4*X1^2*x^15-5*X1*X2*x^15-3*X1*x^16-X2*x^16-X1^2*X2*x^13+X1^2*x^ 14+2*X1*X2*x^14+5*X1*x^15+2*X2*x^15+x^16+X1^2*x^13+2*X1*X2*x^13-2*X1*x^14-X2*x^ 14-2*x^15-X1^2*X2*x^11-2*X1*x^13-X2*x^13+x^14+2*X1^2*X2*x^10+X1*X2*x^11+x^13-X1 ^2*X2*x^9-5*X1*X2*x^10+X1*x^11+5*X1*X2*x^9+X1*x^10+3*X2*x^10-x^11-3*X1*X2*x^8-3 *X1*x^9-4*X2*x^9-x^10+X1*X2*x^7+3*X1*x^8+4*X2*x^8+3*x^9+2*X1*X2*x^6-3*X2*x^7-4* x^8-3*X1*X2*x^5-2*X1*x^6-X2*x^6+2*x^7+X1*X2*x^4+X1*x^5+5*X2*x^5+x^6-4*X2*x^4-3* x^5+X2*x^3+3*x^4+x^3-5*x^2+4*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 1, 2], are n 191 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 3, 3], are n 97 85 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 14 3 85 ------------- 765 and in floating point 0.2922380026 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 14 255 ate normal pair with correlation, --------- 765 1/2 14 255 2687 i.e. , [[---------, 0], [0, ----]] 765 2295 ------------------------------------------------- Theorem Number, 226, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [4, 1, 2], nor the composition, [2, 4, 2] Then infinity ----- 12 11 8 4 2 \ n x + x + x - x - x + 2 x - 1 ) a(n) x = - ------------------------------------------------ / 13 11 9 6 5 4 2 ----- x - x + 2 x - x - x + x + 2 x - 3 x + 1 n = 0 and in Maple format -(x^12+x^11+x^8-x^4-x^2+2*x-1)/(x^13-x^11+2*x^9-x^6-x^5+x^4+2*x^2-3*x+1) The asymptotic expression for a(n) is, n 0.66010691781921974090 1.9182218813446505063 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [4, 1, 2] and d occurrences (as containment) of the composition, [2, 4, 2], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 12 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 11 2 12 12 2 11 11 12 + X1 X2 x - X1 x - 2 X1 X2 x - X1 x - 2 X1 X2 x + 2 X1 x 12 2 9 11 11 12 9 11 + X2 x + X1 X2 x + 2 X1 x + X2 x - x - 2 X1 X2 x - x 8 9 8 8 6 8 5 6 - X1 X2 x + X2 x + X1 x + X2 x - X1 X2 x - x + X1 X2 x + X1 x 4 5 4 2 / 2 13 2 13 - X1 X2 x - X1 x + x + x - 2 x + 1) / (X1 X2 x - X1 x / 13 2 11 13 13 2 10 2 11 - 2 X1 X2 x - X1 X2 x + 2 X1 x + X2 x + X1 X2 x + X1 x 11 13 2 9 10 11 11 10 + 2 X1 X2 x - x - X1 X2 x - 2 X1 X2 x - 2 X1 x - X2 x + X2 x 11 9 9 7 9 6 7 5 + x + 2 X1 x + X2 x - X1 X2 x - 2 x + X1 X2 x + X1 x - 2 X1 X2 x 6 4 5 6 5 4 2 - 2 X1 x + X1 X2 x + X1 x + x + x - x - 2 x + 3 x - 1) and in Maple format -(X1^2*X2*x^12+X1^2*X2*x^11-X1^2*x^12-2*X1*X2*x^12-X1^2*x^11-2*X1*X2*x^11+2*X1* x^12+X2*x^12+X1^2*X2*x^9+2*X1*x^11+X2*x^11-x^12-2*X1*X2*x^9-x^11-X1*X2*x^8+X2*x ^9+X1*x^8+X2*x^8-X1*X2*x^6-x^8+X1*X2*x^5+X1*x^6-X1*X2*x^4-X1*x^5+x^4+x^2-2*x+1) /(X1^2*X2*x^13-X1^2*x^13-2*X1*X2*x^13-X1^2*X2*x^11+2*X1*x^13+X2*x^13+X1^2*X2*x^ 10+X1^2*x^11+2*X1*X2*x^11-x^13-X1^2*X2*x^9-2*X1*X2*x^10-2*X1*x^11-X2*x^11+X2*x^ 10+x^11+2*X1*x^9+X2*x^9-X1*X2*x^7-2*x^9+X1*X2*x^6+X1*x^7-2*X1*X2*x^5-2*X1*x^6+ X1*X2*x^4+X1*x^5+x^6+x^5-x^4-2*x^2+3*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 1, 2], are n 191 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 4, 2], are n 75 69 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 2 23 ------- 23 and in floating point 0.4170288281 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 2 23 ate normal pair with correlation, ------- 23 1/2 2 23 31 i.e. , [[-------, 0], [0, --]] 23 23 ------------------------------------------------- Theorem Number, 227, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [4, 1, 2], nor the composition, [2, 5, 1] Then infinity ----- \ n 14 12 10 8 4 2 / ) a(n) x = - (x - x - x - x + x + x - 2 x + 1) / ((-1 + x) / / ----- n = 0 14 13 12 11 10 9 8 7 6 4 (x + x + x + 2 x + x + x - x - x - x + x - 2 x + 1)) and in Maple format -(x^14-x^12-x^10-x^8+x^4+x^2-2*x+1)/(-1+x)/(x^14+x^13+x^12+2*x^11+x^10+x^9-x^8- x^7-x^6+x^4-2*x+1) The asymptotic expression for a(n) is, n 0.67221179878675203708 1.9144061473285935513 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [4, 1, 2] and d occurrences (as containment) of the composition, [2, 5, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 14 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 2 13 2 14 2 14 2 13 2 14 + X1 X2 x - 2 X1 X2 x - 2 X1 X2 x - X1 X2 x + X1 x 2 13 14 2 14 2 12 13 - 2 X1 X2 x + 4 X1 X2 x + X2 x + X1 X2 x + 2 X1 X2 x 14 2 13 14 2 12 12 13 14 - 2 X1 x + X2 x - 2 X2 x - X1 x - 2 X1 X2 x - X2 x + x 2 10 12 12 2 10 10 12 2 9 + X1 X2 x + 2 X1 x + X2 x - X1 x - 2 X1 X2 x - x + X1 x 9 10 10 8 9 9 10 7 - X1 X2 x + 2 X1 x + X2 x - X1 X2 x - X1 x + X2 x - x - X1 X2 x 8 8 6 7 8 5 6 5 4 + X1 x + X2 x + X1 X2 x + X1 x - x - X1 X2 x - X1 x + X1 x - X1 x 4 2 / 2 2 15 2 15 2 15 + x + x - 2 x + 1) / (X1 X2 x - 2 X1 X2 x - 2 X1 X2 x / 2 2 13 2 14 2 15 2 14 15 2 15 - X1 X2 x + X1 X2 x + X1 x - X1 X2 x + 4 X1 X2 x + X2 x 2 13 2 14 2 13 15 2 14 15 + 2 X1 X2 x - X1 x + X1 X2 x - 2 X1 x + X2 x - 2 X2 x 2 12 2 13 13 14 14 15 2 11 - X1 X2 x - X1 x - 2 X1 X2 x + X1 x - X2 x + x + X1 X2 x 2 12 12 13 2 10 2 11 11 + X1 x + 2 X1 X2 x + X1 x - X1 X2 x - X1 x - 2 X1 X2 x 12 12 2 10 11 11 12 2 9 - 2 X1 x - X2 x + 2 X1 x + 2 X1 x + X2 x + x - X1 x 9 10 10 11 9 9 7 9 - X1 X2 x - 2 X1 x + X2 x - x + 3 X1 x + X2 x + X1 X2 x - 2 x 6 7 5 6 5 6 4 5 4 - 2 X1 X2 x - X1 x + X1 X2 x + X1 x - 2 X1 x + x + X1 x + x - x 2 - 2 x + 3 x - 1) and in Maple format -(X1^2*X2^2*x^14+X1^2*X2^2*x^13-2*X1^2*X2*x^14-2*X1*X2^2*x^14-X1^2*X2*x^13+X1^2 *x^14-2*X1*X2^2*x^13+4*X1*X2*x^14+X2^2*x^14+X1^2*X2*x^12+2*X1*X2*x^13-2*X1*x^14 +X2^2*x^13-2*X2*x^14-X1^2*x^12-2*X1*X2*x^12-X2*x^13+x^14+X1^2*X2*x^10+2*X1*x^12 +X2*x^12-X1^2*x^10-2*X1*X2*x^10-x^12+X1^2*x^9-X1*X2*x^9+2*X1*x^10+X2*x^10-X1*X2 *x^8-X1*x^9+X2*x^9-x^10-X1*X2*x^7+X1*x^8+X2*x^8+X1*X2*x^6+X1*x^7-x^8-X1*X2*x^5- X1*x^6+X1*x^5-X1*x^4+x^4+x^2-2*x+1)/(X1^2*X2^2*x^15-2*X1^2*X2*x^15-2*X1*X2^2*x^ 15-X1^2*X2^2*x^13+X1^2*X2*x^14+X1^2*x^15-X1*X2^2*x^14+4*X1*X2*x^15+X2^2*x^15+2* X1^2*X2*x^13-X1^2*x^14+X1*X2^2*x^13-2*X1*x^15+X2^2*x^14-2*X2*x^15-X1^2*X2*x^12- X1^2*x^13-2*X1*X2*x^13+X1*x^14-X2*x^14+x^15+X1^2*X2*x^11+X1^2*x^12+2*X1*X2*x^12 +X1*x^13-X1^2*X2*x^10-X1^2*x^11-2*X1*X2*x^11-2*X1*x^12-X2*x^12+2*X1^2*x^10+2*X1 *x^11+X2*x^11+x^12-X1^2*x^9-X1*X2*x^9-2*X1*x^10+X2*x^10-x^11+3*X1*x^9+X2*x^9+X1 *X2*x^7-2*x^9-2*X1*X2*x^6-X1*x^7+X1*X2*x^5+X1*x^6-2*X1*x^5+x^6+X1*x^4+x^5-x^4-2 *x^2+3*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 1, 2], are n 191 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 5, 1], are n 111 57 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 4 19 ------- 57 and in floating point 0.3058876452 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 4 19 ate normal pair with correlation, ------- 57 1/2 4 19 203 i.e. , [[-------, 0], [0, ---]] 57 171 ------------------------------------------------- Theorem Number, 228, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [4, 1, 2], nor the composition, [3, 1, 4] Then infinity ----- \ n ) a(n) x = - ( / ----- n = 0 22 21 20 19 13 12 11 10 9 4 2 x + 2 x + 2 x + x + x + x + x + x + x - x - x + 2 x - 1) / 7 6 5 3 2 / ((x + x + x + x + x + x - 1) / 16 12 9 6 5 2 (x - x + x + x - x - x + 2 x - 1)) and in Maple format -(x^22+2*x^21+2*x^20+x^19+x^13+x^12+x^11+x^10+x^9-x^4-x^2+2*x-1)/(x^7+x^6+x^5+x ^3+x^2+x-1)/(x^16-x^12+x^9+x^6-x^5-x^2+2*x-1) The asymptotic expression for a(n) is, n 0.64880167659124523698 1.9221280043552247737 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [4, 1, 2] and d occurrences (as containment) of the composition, [3, 1, 4], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 3 3 23 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 3 3 22 3 2 23 2 3 23 3 3 21 + X1 X2 x - 3 X1 X2 x - 3 X1 X2 x + X1 X2 x 3 2 22 3 23 2 3 22 2 2 23 - 3 X1 X2 x + 3 X1 X2 x - 3 X1 X2 x + 9 X1 X2 x 3 23 3 3 20 3 2 21 3 22 3 23 + 3 X1 X2 x + X1 X2 x - 3 X1 X2 x + 3 X1 X2 x - X1 x 2 3 21 2 2 22 2 23 3 22 - 2 X1 X2 x + 9 X1 X2 x - 9 X1 X2 x + 3 X1 X2 x 2 23 3 23 3 3 19 3 2 20 3 21 - 9 X1 X2 x - X2 x + X1 X2 x - 2 X1 X2 x + 3 X1 X2 x 3 22 2 3 20 2 2 21 2 22 2 23 - X1 x - 2 X1 X2 x + 6 X1 X2 x - 9 X1 X2 x + 3 X1 x 3 21 2 22 23 3 22 2 23 + X1 X2 x - 9 X1 X2 x + 9 X1 X2 x - X2 x + 3 X2 x 3 2 19 3 20 3 21 2 3 19 2 2 20 - 2 X1 X2 x + X1 X2 x - X1 x - 2 X1 X2 x + 3 X1 X2 x 2 21 2 22 3 20 2 21 22 - 6 X1 X2 x + 3 X1 x + X1 X2 x - 3 X1 X2 x + 9 X1 X2 x 23 2 22 23 3 19 2 2 19 2 21 - 3 X1 x + 3 X2 x - 3 X2 x + X1 X2 x + 3 X1 X2 x + 2 X1 x 3 19 21 22 22 23 3 3 16 + X1 X2 x + 3 X1 X2 x - 3 X1 x - 3 X2 x + x + X1 X2 x 2 20 20 21 2 20 22 3 2 16 - X1 x - 3 X1 X2 x - X1 x - X2 x + x - X1 X2 x 2 3 16 2 19 19 20 2 19 20 - X1 X2 x - X1 x - 3 X1 X2 x + 2 X1 x - X2 x + 2 X2 x 2 2 16 19 19 20 2 16 2 16 - 2 X1 X2 x + 2 X1 x + 2 X2 x - x + 4 X1 X2 x + 4 X1 X2 x 19 2 16 16 2 16 2 2 13 2 14 - x - X1 x - 5 X1 X2 x - X2 x + 2 X1 X2 x - X1 X2 x 16 16 2 13 2 13 14 2 14 + X1 x + X2 x - 4 X1 X2 x - 2 X1 X2 x + 2 X1 X2 x + X2 x 2 2 11 2 13 13 14 14 2 11 - X1 X2 x + 2 X1 x + 4 X1 X2 x - X1 x - 2 X2 x + X1 X2 x 13 14 2 2 9 2 10 11 10 - 2 X1 x + x + X1 X2 x - X1 X2 x + X1 X2 x + 2 X1 X2 x 11 9 10 9 9 9 5 - X1 x - 4 X1 X2 x - X2 x + 2 X1 x + 2 X2 x - x + X1 X2 x 4 5 4 3 2 / 8 8 8 - X1 X2 x - x + x - x + 3 x - 3 x + 1) / ((X1 X2 x - X1 x - X2 x / 6 8 5 6 4 5 4 + X1 X2 x + x + X1 X2 x - X1 x - X1 X2 x - x + x - 2 x + 1) ( 2 2 16 2 16 2 16 2 16 16 2 16 X1 X2 x - 2 X1 X2 x - 2 X1 X2 x + X1 x + 4 X1 X2 x + X2 x 16 16 2 2 12 16 2 12 2 12 - 2 X1 x - 2 X2 x + X1 X2 x + x - X1 X2 x - X1 X2 x 12 12 12 9 9 9 7 9 7 + X1 x + X2 x - x + X1 X2 x - X1 x - X2 x - X1 X2 x + x + X1 x 5 6 6 5 2 + X1 X2 x - X1 x + x - x - x + 2 x - 1)) and in Maple format -(X1^3*X2^3*x^23+X1^3*X2^3*x^22-3*X1^3*X2^2*x^23-3*X1^2*X2^3*x^23+X1^3*X2^3*x^ 21-3*X1^3*X2^2*x^22+3*X1^3*X2*x^23-3*X1^2*X2^3*x^22+9*X1^2*X2^2*x^23+3*X1*X2^3* x^23+X1^3*X2^3*x^20-3*X1^3*X2^2*x^21+3*X1^3*X2*x^22-X1^3*x^23-2*X1^2*X2^3*x^21+ 9*X1^2*X2^2*x^22-9*X1^2*X2*x^23+3*X1*X2^3*x^22-9*X1*X2^2*x^23-X2^3*x^23+X1^3*X2 ^3*x^19-2*X1^3*X2^2*x^20+3*X1^3*X2*x^21-X1^3*x^22-2*X1^2*X2^3*x^20+6*X1^2*X2^2* x^21-9*X1^2*X2*x^22+3*X1^2*x^23+X1*X2^3*x^21-9*X1*X2^2*x^22+9*X1*X2*x^23-X2^3*x ^22+3*X2^2*x^23-2*X1^3*X2^2*x^19+X1^3*X2*x^20-X1^3*x^21-2*X1^2*X2^3*x^19+3*X1^2 *X2^2*x^20-6*X1^2*X2*x^21+3*X1^2*x^22+X1*X2^3*x^20-3*X1*X2^2*x^21+9*X1*X2*x^22-\ 3*X1*x^23+3*X2^2*x^22-3*X2*x^23+X1^3*X2*x^19+3*X1^2*X2^2*x^19+2*X1^2*x^21+X1*X2 ^3*x^19+3*X1*X2*x^21-3*X1*x^22-3*X2*x^22+x^23+X1^3*X2^3*x^16-X1^2*x^20-3*X1*X2* x^20-X1*x^21-X2^2*x^20+x^22-X1^3*X2^2*x^16-X1^2*X2^3*x^16-X1^2*x^19-3*X1*X2*x^ 19+2*X1*x^20-X2^2*x^19+2*X2*x^20-2*X1^2*X2^2*x^16+2*X1*x^19+2*X2*x^19-x^20+4*X1 ^2*X2*x^16+4*X1*X2^2*x^16-x^19-X1^2*x^16-5*X1*X2*x^16-X2^2*x^16+2*X1^2*X2^2*x^ 13-X1*X2^2*x^14+X1*x^16+X2*x^16-4*X1^2*X2*x^13-2*X1*X2^2*x^13+2*X1*X2*x^14+X2^2 *x^14-X1^2*X2^2*x^11+2*X1^2*x^13+4*X1*X2*x^13-X1*x^14-2*X2*x^14+X1^2*X2*x^11-2* X1*x^13+x^14+X1^2*X2^2*x^9-X1^2*X2*x^10+X1*X2*x^11+2*X1*X2*x^10-X1*x^11-4*X1*X2 *x^9-X2*x^10+2*X1*x^9+2*X2*x^9-x^9+X1*X2*x^5-X1*X2*x^4-x^5+x^4-x^3+3*x^2-3*x+1) /(X1*X2*x^8-X1*x^8-X2*x^8+X1*X2*x^6+x^8+X1*X2*x^5-X1*x^6-X1*X2*x^4-x^5+x^4-2*x+ 1)/(X1^2*X2^2*x^16-2*X1^2*X2*x^16-2*X1*X2^2*x^16+X1^2*x^16+4*X1*X2*x^16+X2^2*x^ 16-2*X1*x^16-2*X2*x^16+X1^2*X2^2*x^12+x^16-X1^2*X2*x^12-X1*X2^2*x^12+X1*x^12+X2 *x^12-x^12+X1*X2*x^9-X1*x^9-X2*x^9-X1*X2*x^7+x^9+X1*x^7+X1*X2*x^5-X1*x^6+x^6-x^ 5-x^2+2*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 1, 2], are n 191 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 1, 4], are n 141 65 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 22 3 65 ------------- 585 and in floating point 0.5251509079 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 22 195 ate normal pair with correlation, --------- 585 1/2 22 195 2723 i.e. , [[---------, 0], [0, ----]] 585 1755 ------------------------------------------------- Theorem Number, 229, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [4, 1, 2], nor the composition, [3, 2, 3] Then infinity ----- \ n ) a(n) x = / ----- n = 0 9 8 6 5 4 3 2 x - 2 x + x - 2 x + x - x - x + 2 x - 1 -------------------------------------------------------------------- 12 11 9 8 6 5 4 3 2 x - 2 x + 3 x - 3 x + 3 x - 2 x + 2 x - x - 2 x + 3 x - 1 and in Maple format (x^9-2*x^8+x^6-2*x^5+x^4-x^3-x^2+2*x-1)/(x^12-2*x^11+3*x^9-3*x^8+3*x^6-2*x^5+2* x^4-x^3-2*x^2+3*x-1) The asymptotic expression for a(n) is, n 0.65537287717026354174 1.9205873909649774992 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [4, 1, 2] and d occurrences (as containment) of the composition, [3, 2, 3], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 10 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 10 2 10 2 9 2 9 10 - X1 X2 x - 2 X1 X2 x + X1 X2 x + 2 X1 X2 x + 2 X1 X2 x 2 10 9 2 9 10 8 9 2 8 + X2 x - 5 X1 X2 x - 2 X2 x - X2 x + X1 X2 x + X1 x + X2 x 9 8 8 9 6 8 5 6 + 4 X2 x - X1 x - 3 X2 x - x - X1 X2 x + 2 x + X1 X2 x + X1 x 6 4 5 5 6 4 5 3 4 + X2 x - X1 X2 x - X1 x - 2 X2 x - x + 2 X2 x + 2 x - X2 x - x 3 2 / 2 2 12 2 2 11 2 12 + x + x - 2 x + 1) / (X1 X2 x + X1 X2 x - 2 X1 X2 x / 2 12 2 2 10 2 11 2 12 12 - 2 X1 X2 x - X1 X2 x - X1 X2 x + X1 x + 4 X1 X2 x 2 12 2 10 2 10 11 12 2 11 + X2 x + 2 X1 X2 x + 4 X1 X2 x - 2 X1 X2 x - 2 X1 x - X2 x 12 2 9 2 9 10 11 2 10 - 2 X2 x - X1 X2 x - 2 X1 X2 x - 7 X1 X2 x + 2 X1 x - 2 X2 x 11 12 9 10 2 9 10 11 + 3 X2 x + x + 6 X1 X2 x + X1 x + 3 X2 x + 3 X2 x - 2 x 8 9 2 8 9 7 8 8 - 2 X1 X2 x - 2 X1 x - X2 x - 7 X2 x - X1 X2 x + 2 X1 x + 4 X2 x 9 6 7 8 5 6 6 + 3 x + X1 X2 x + X1 x - 3 x - 2 X1 X2 x - 2 X1 x - 2 X2 x 4 5 5 6 4 5 3 4 3 + X1 X2 x + X1 x + 3 X2 x + 3 x - 3 X2 x - 2 x + X2 x + 2 x - x 2 - 2 x + 3 x - 1) and in Maple format -(X1^2*X2^2*x^10-X1^2*X2*x^10-2*X1*X2^2*x^10+X1^2*X2*x^9+2*X1*X2^2*x^9+2*X1*X2* x^10+X2^2*x^10-5*X1*X2*x^9-2*X2^2*x^9-X2*x^10+X1*X2*x^8+X1*x^9+X2^2*x^8+4*X2*x^ 9-X1*x^8-3*X2*x^8-x^9-X1*X2*x^6+2*x^8+X1*X2*x^5+X1*x^6+X2*x^6-X1*X2*x^4-X1*x^5-\ 2*X2*x^5-x^6+2*X2*x^4+2*x^5-X2*x^3-x^4+x^3+x^2-2*x+1)/(X1^2*X2^2*x^12+X1^2*X2^2 *x^11-2*X1^2*X2*x^12-2*X1*X2^2*x^12-X1^2*X2^2*x^10-X1^2*X2*x^11+X1^2*x^12+4*X1* X2*x^12+X2^2*x^12+2*X1^2*X2*x^10+4*X1*X2^2*x^10-2*X1*X2*x^11-2*X1*x^12-X2^2*x^ 11-2*X2*x^12-X1^2*X2*x^9-2*X1*X2^2*x^9-7*X1*X2*x^10+2*X1*x^11-2*X2^2*x^10+3*X2* x^11+x^12+6*X1*X2*x^9+X1*x^10+3*X2^2*x^9+3*X2*x^10-2*x^11-2*X1*X2*x^8-2*X1*x^9- X2^2*x^8-7*X2*x^9-X1*X2*x^7+2*X1*x^8+4*X2*x^8+3*x^9+X1*X2*x^6+X1*x^7-3*x^8-2*X1 *X2*x^5-2*X1*x^6-2*X2*x^6+X1*X2*x^4+X1*x^5+3*X2*x^5+3*x^6-3*X2*x^4-2*x^5+X2*x^3 +2*x^4-x^3-2*x^2+3*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 1, 2], are n 191 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 2, 3], are n 89 77 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 2 3 77 ------------ 77 and in floating point 0.3947710171 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 1/2 2 3 77 ate normal pair with correlation, ------------ 77 1/2 1/2 2 3 77 101 i.e. , [[------------, 0], [0, ---]] 77 77 ------------------------------------------------- Theorem Number, 230, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [4, 1, 2], nor the composition, [3, 3, 2] Then infinity ----- 8 7 6 5 4 3 2 \ n x - x + x + x - x + x + x - 2 x + 1 ) a(n) x = - -------------------------------------------------- / 9 8 7 5 4 3 2 ----- x - 2 x + 2 x - x + 2 x - x - 2 x + 3 x - 1 n = 0 and in Maple format -(x^8-x^7+x^6+x^5-x^4+x^3+x^2-2*x+1)/(x^9-2*x^8+2*x^7-x^5+2*x^4-x^3-2*x^2+3*x-1 ) The asymptotic expression for a(n) is, n 0.65556997991907913494 1.9211122916754298187 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [4, 1, 2] and d occurrences (as containment) of the composition, [3, 3, 2], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 9 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 9 8 9 7 8 8 7 7 - 2 X1 X2 x + X1 X2 x + X2 x - X1 X2 x - X1 x - X2 x + X1 x + X2 x 8 5 6 7 4 5 5 6 4 + x + X1 X2 x - X2 x - x - X1 X2 x - X1 x - X2 x + x + 2 X2 x 5 3 4 3 2 / 2 10 2 9 + x - X2 x - x + x + x - 2 x + 1) / (X1 X2 x - X1 X2 x / 10 9 10 8 9 9 - 2 X1 X2 x + 3 X1 X2 x + X2 x - 2 X1 X2 x - X1 x - 2 X2 x 7 8 8 9 7 7 8 5 + X1 X2 x + 2 X1 x + 2 X2 x + x - X1 x - 2 X2 x - 2 x - 2 X1 X2 x 6 6 7 4 5 5 4 5 3 - X1 x + X2 x + 2 x + X1 X2 x + X1 x + 2 X2 x - 3 X2 x - x + X2 x 4 3 2 + 2 x - x - 2 x + 3 x - 1) and in Maple format -(X1^2*X2*x^9-2*X1*X2*x^9+X1*X2*x^8+X2*x^9-X1*X2*x^7-X1*x^8-X2*x^8+X1*x^7+X2*x^ 7+x^8+X1*X2*x^5-X2*x^6-x^7-X1*X2*x^4-X1*x^5-X2*x^5+x^6+2*X2*x^4+x^5-X2*x^3-x^4+ x^3+x^2-2*x+1)/(X1^2*X2*x^10-X1^2*X2*x^9-2*X1*X2*x^10+3*X1*X2*x^9+X2*x^10-2*X1* X2*x^8-X1*x^9-2*X2*x^9+X1*X2*x^7+2*X1*x^8+2*X2*x^8+x^9-X1*x^7-2*X2*x^7-2*x^8-2* X1*X2*x^5-X1*x^6+X2*x^6+2*x^7+X1*X2*x^4+X1*x^5+2*X2*x^5-3*X2*x^4-x^5+X2*x^3+2*x ^4-x^3-2*x^2+3*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 1, 2], are n 191 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 3, 2], are n 97 85 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 2 3 85 ------------ 85 and in floating point 0.3757345747 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 1/2 2 3 85 ate normal pair with correlation, ------------ 85 1/2 1/2 2 3 85 109 i.e. , [[------------, 0], [0, ---]] 85 85 ------------------------------------------------- Theorem Number, 231, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [4, 1, 2], nor the composition, [3, 4, 1] Then infinity ----- 7 6 5 4 \ n x + x + x + x - x + 1 ) a(n) x = ---------------------------------------------------------- / 10 9 8 7 6 3 2 ----- (-1 + x) (x + 2 x + 3 x + 2 x + x + x + x + x - 1) n = 0 and in Maple format (x^7+x^6+x^5+x^4-x+1)/(-1+x)/(x^10+2*x^9+3*x^8+2*x^7+x^6+x^3+x^2+x-1) The asymptotic expression for a(n) is, n 0.66418334929247170970 1.9175371779461324874 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [4, 1, 2] and d occurrences (as containment) of the composition, [3, 4, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 12 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 12 2 12 12 2 12 12 2 9 - X1 X2 x - 2 X1 X2 x + 2 X1 X2 x + X2 x - X2 x + X1 X2 x 9 8 9 7 8 8 7 8 - 2 X1 X2 x - X1 X2 x + X2 x - X1 X2 x + X1 x + X2 x + X1 x - x 5 4 5 4 2 / 2 2 13 + X1 X2 x - X1 X2 x - X1 x + x + x - 2 x + 1) / (X1 X2 x / 2 2 12 2 13 2 13 2 12 2 12 - X1 X2 x - X1 X2 x - 2 X1 X2 x + X1 X2 x + X1 X2 x 13 2 13 13 2 10 12 12 + 2 X1 X2 x + X2 x - X2 x + X1 X2 x - X1 x - X2 x 2 9 10 12 10 9 9 7 - X1 X2 x - 2 X1 X2 x + x + X2 x + 2 X1 x + X2 x + X1 X2 x 9 7 5 6 4 5 6 5 4 - 2 x - X1 x - 2 X1 X2 x - X1 x + X1 X2 x + X1 x + x + x - x 2 - 2 x + 3 x - 1) and in Maple format -(X1^2*X2^2*x^12-X1^2*X2*x^12-2*X1*X2^2*x^12+2*X1*X2*x^12+X2^2*x^12-X2*x^12+X1^ 2*X2*x^9-2*X1*X2*x^9-X1*X2*x^8+X2*x^9-X1*X2*x^7+X1*x^8+X2*x^8+X1*x^7-x^8+X1*X2* x^5-X1*X2*x^4-X1*x^5+x^4+x^2-2*x+1)/(X1^2*X2^2*x^13-X1^2*X2^2*x^12-X1^2*X2*x^13 -2*X1*X2^2*x^13+X1^2*X2*x^12+X1*X2^2*x^12+2*X1*X2*x^13+X2^2*x^13-X2*x^13+X1^2* X2*x^10-X1*x^12-X2*x^12-X1^2*X2*x^9-2*X1*X2*x^10+x^12+X2*x^10+2*X1*x^9+X2*x^9+ X1*X2*x^7-2*x^9-X1*x^7-2*X1*X2*x^5-X1*x^6+X1*X2*x^4+X1*x^5+x^6+x^5-x^4-2*x^2+3* x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 1, 2], are n 191 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 4, 1], are n 135 65 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 16 3 65 ------------- 585 and in floating point 0.3819279330 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 16 195 ate normal pair with correlation, --------- 585 1/2 16 195 2267 i.e. , [[---------, 0], [0, ----]] 585 1755 ------------------------------------------------- Theorem Number, 232, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [4, 1, 2], nor the composition, [4, 3, 1] Then infinity ----- 4 2 \ n x + x - 2 x + 1 ) a(n) x = --------------------------------- / 7 6 4 ----- (-1 + x) (x + x - x + 2 x - 1) n = 0 and in Maple format (x^4+x^2-2*x+1)/(-1+x)/(x^7+x^6-x^4+2*x-1) The asymptotic expression for a(n) is, n 0.67238389941451345857 1.9168163613635124211 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [4, 1, 2] and d occurrences (as containment) of the composition, [4, 3, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 9 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 9 9 7 7 5 4 5 4 - 2 X1 X2 x + X2 x - X1 X2 x + X1 x + X1 X2 x - X1 X2 x - X1 x + x 2 / 2 10 2 9 10 9 + x - 2 x + 1) / (X1 X2 x - X1 X2 x - 2 X1 X2 x + 2 X1 X2 x / 10 8 9 7 8 8 7 8 + X2 x - X1 X2 x - X2 x + X1 X2 x + X1 x + X2 x - X1 x - x 5 6 4 5 6 5 4 2 - 2 X1 X2 x - X1 x + X1 X2 x + X1 x + x + x - x - 2 x + 3 x - 1) and in Maple format -(X1^2*X2*x^9-2*X1*X2*x^9+X2*x^9-X1*X2*x^7+X1*x^7+X1*X2*x^5-X1*X2*x^4-X1*x^5+x^ 4+x^2-2*x+1)/(X1^2*X2*x^10-X1^2*X2*x^9-2*X1*X2*x^10+2*X1*X2*x^9+X2*x^10-X1*X2*x ^8-X2*x^9+X1*X2*x^7+X1*x^8+X2*x^8-X1*x^7-x^8-2*X1*X2*x^5-X1*x^6+X1*X2*x^4+X1*x^ 5+x^6+x^5-x^4-2*x^2+3*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 1, 2], are n 191 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 3, 1], are n 135 65 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 16 3 65 ------------- 585 and in floating point 0.3819279330 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 16 195 ate normal pair with correlation, --------- 585 1/2 16 195 2267 i.e. , [[---------, 0], [0, ----]] 585 1755 ------------------------------------------------- Theorem Number, 233, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [4, 1, 2], nor the composition, [5, 2, 1] Then infinity ----- 10 9 4 2 \ n x - x - x - x + 2 x - 1 ) a(n) x = - -------------------------------------------- / 10 9 7 6 4 ----- (-1 + x) (x - x + x + x - x + 2 x - 1) n = 0 and in Maple format -(x^10-x^9-x^4-x^2+2*x-1)/(-1+x)/(x^10-x^9+x^7+x^6-x^4+2*x-1) The asymptotic expression for a(n) is, n 0.68264941498911845581 1.9140011778740215244 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [4, 1, 2] and d occurrences (as containment) of the composition, [5, 2, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 10 2 10 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x - X1 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 10 2 9 10 10 9 10 7 9 - 2 X1 X2 x + X1 x + 2 X1 x + X2 x - 2 X1 x - x - X1 X2 x + x 6 7 5 6 5 4 4 2 + X1 X2 x + X1 x - X1 X2 x - X1 x + X1 x - X1 x + x + x - 2 x + 1) / 2 11 2 10 2 11 11 2 10 / (X1 X2 x - X1 X2 x - X1 x - 2 X1 X2 x + 2 X1 x / 10 11 11 2 9 10 10 11 + 2 X1 X2 x + 2 X1 x + X2 x - X1 x - 4 X1 x - X2 x - x 8 9 10 7 8 8 9 6 - X1 X2 x + 2 X1 x + 2 x + X1 X2 x + X1 x + X2 x - x - 2 X1 X2 x 7 8 5 6 5 6 4 5 4 2 - X1 x - x + X1 X2 x + X1 x - 2 X1 x + x + X1 x + x - x - 2 x + 3 x - 1) and in Maple format -(X1^2*X2*x^10-X1^2*x^10-2*X1*X2*x^10+X1^2*x^9+2*X1*x^10+X2*x^10-2*X1*x^9-x^10- X1*X2*x^7+x^9+X1*X2*x^6+X1*x^7-X1*X2*x^5-X1*x^6+X1*x^5-X1*x^4+x^4+x^2-2*x+1)/( X1^2*X2*x^11-X1^2*X2*x^10-X1^2*x^11-2*X1*X2*x^11+2*X1^2*x^10+2*X1*X2*x^10+2*X1* x^11+X2*x^11-X1^2*x^9-4*X1*x^10-X2*x^10-x^11-X1*X2*x^8+2*X1*x^9+2*x^10+X1*X2*x^ 7+X1*x^8+X2*x^8-x^9-2*X1*X2*x^6-X1*x^7-x^8+X1*X2*x^5+X1*x^6-2*X1*x^5+x^6+X1*x^4 +x^5-x^4-2*x^2+3*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 1, 2], are n 191 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [5, 2, 1], are n 111 57 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 4 19 ------- 57 and in floating point 0.3058876452 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 4 19 ate normal pair with correlation, ------- 57 1/2 4 19 203 i.e. , [[-------, 0], [0, ---]] 57 171 ------------------------------------------------- Theorem Number, 234, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [4, 1, 2], nor the composition, [6, 1, 1] Then infinity ----- 11 9 4 2 \ n x - x - x - x + 2 x - 1 ) a(n) x = --------------------------------------------------- / 6 5 3 2 6 5 2 ----- (x + x + x + x + x - 1) (x + x + 1) (-1 + x) n = 0 and in Maple format (x^11-x^9-x^4-x^2+2*x-1)/(x^6+x^5+x^3+x^2+x-1)/(x^6+x^5+1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.69440972834504952286 1.9111834366854762350 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [4, 1, 2] and d occurrences (as containment) of the composition, [6, 1, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 13 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 13 2 13 2 13 13 2 11 - 2 X1 X2 x - X1 X2 x + X1 x + 2 X1 X2 x + 2 X1 X2 x 13 2 11 11 11 11 2 9 11 - X1 x - 2 X1 x - 3 X1 X2 x + 3 X1 x + X2 x + X1 x - x 9 9 6 6 4 4 2 / - 2 X1 x + x - X1 X2 x + X1 x - X1 x + x + x - 2 x + 1) / (( / 7 6 7 7 6 7 5 4 5 4 X1 X2 x - X1 X2 x - X1 x - X2 x + X1 x + x + X1 x - X1 x - x + x 7 7 7 7 5 5 - 2 x + 1) (X1 X2 x - X1 x - X2 x + x + X1 x - x + x - 1)) and in Maple format -(X1^2*X2^2*x^13-2*X1^2*X2*x^13-X1*X2^2*x^13+X1^2*x^13+2*X1*X2*x^13+2*X1^2*X2*x ^11-X1*x^13-2*X1^2*x^11-3*X1*X2*x^11+3*X1*x^11+X2*x^11+X1^2*x^9-x^11-2*X1*x^9+x ^9-X1*X2*x^6+X1*x^6-X1*x^4+x^4+x^2-2*x+1)/(X1*X2*x^7-X1*X2*x^6-X1*x^7-X2*x^7+X1 *x^6+x^7+X1*x^5-X1*x^4-x^5+x^4-2*x+1)/(X1*X2*x^7-X1*x^7-X2*x^7+x^7+X1*x^5-x^5+x -1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 1, 2], are n 191 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [6, 1, 1], are n 49 53 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 8 3 53 ------------ 477 and in floating point 0.2114804230 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 8 159 ate normal pair with correlation, -------- 477 1/2 8 159 1559 i.e. , [[--------, 0], [0, ----]] 477 1431 ------------------------------------------------- Theorem Number, 235, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [4, 2, 1], nor the composition, [1, 1, 6] Then infinity ----- \ n 18 17 16 15 13 12 11 10 ) a(n) x = - (x + 2 x + 2 x + x - 2 x - 3 x - 3 x - 2 x / ----- n = 0 9 8 7 6 5 4 2 / - x - x + x + x - x + x + x - 2 x + 1) / ( / 6 5 3 2 2 (x + x + x + x + x - 1) (-1 + x) ) and in Maple format -(x^18+2*x^17+2*x^16+x^15-2*x^13-3*x^12-3*x^11-2*x^10-x^9-x^8+x^7+x^6-x^5+x^4+x ^2-2*x+1)/(x^6+x^5+x^3+x^2+x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.67380351075430772877 1.9111834366854762350 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [4, 2, 1] and d occurrences (as containment) of the composition, [1, 1, 6], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 19 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 2 18 2 19 2 19 2 18 2 19 + X1 X2 x - 2 X1 X2 x - 2 X1 X2 x - 2 X1 X2 x + X1 x 2 18 19 2 19 2 18 18 19 - 2 X1 X2 x + 4 X1 X2 x + X2 x + X1 x + 4 X1 X2 x - 2 X1 x 2 18 19 2 16 18 18 19 2 15 + X2 x - 2 X2 x + X1 X2 x - 2 X1 x - 2 X2 x + x + X1 X2 x 16 2 16 18 2 14 2 14 15 - 2 X1 X2 x - X2 x + x + X1 X2 x + X1 X2 x - 2 X1 X2 x 16 2 15 16 2 13 2 14 14 + X1 x - X2 x + 2 X2 x + X1 X2 x - X1 x - 4 X1 X2 x 15 2 14 15 16 2 13 13 14 + X1 x - X2 x + 2 X2 x - x - X1 x - 2 X1 X2 x + 3 X1 x 14 15 13 13 14 11 13 10 + 3 X2 x - x + 2 X1 x + X2 x - 2 x + X1 X2 x - x + X1 X2 x 11 11 10 10 11 8 10 7 - X1 x - X2 x - X1 x - X2 x + x + 2 X1 X2 x + x - 2 X1 X2 x 8 8 6 7 7 8 6 5 - 2 X1 x - 2 X2 x + X1 X2 x + X1 x + X2 x + 2 x + X1 x - 2 X1 x 6 4 5 4 3 2 / 2 7 - 2 x + X1 x + 2 x - x + x - 3 x + 3 x - 1) / ((-1 + x) (X1 X2 x / 6 7 7 6 7 5 4 5 4 - X1 X2 x - X1 x - X2 x + X1 x + x + X1 x - X1 x - x + x - 2 x + 1)) and in Maple format -(X1^2*X2^2*x^19+X1^2*X2^2*x^18-2*X1^2*X2*x^19-2*X1*X2^2*x^19-2*X1^2*X2*x^18+X1 ^2*x^19-2*X1*X2^2*x^18+4*X1*X2*x^19+X2^2*x^19+X1^2*x^18+4*X1*X2*x^18-2*X1*x^19+ X2^2*x^18-2*X2*x^19+X1*X2^2*x^16-2*X1*x^18-2*X2*x^18+x^19+X1*X2^2*x^15-2*X1*X2* x^16-X2^2*x^16+x^18+X1^2*X2*x^14+X1*X2^2*x^14-2*X1*X2*x^15+X1*x^16-X2^2*x^15+2* X2*x^16+X1^2*X2*x^13-X1^2*x^14-4*X1*X2*x^14+X1*x^15-X2^2*x^14+2*X2*x^15-x^16-X1 ^2*x^13-2*X1*X2*x^13+3*X1*x^14+3*X2*x^14-x^15+2*X1*x^13+X2*x^13-2*x^14+X1*X2*x^ 11-x^13+X1*X2*x^10-X1*x^11-X2*x^11-X1*x^10-X2*x^10+x^11+2*X1*X2*x^8+x^10-2*X1* X2*x^7-2*X1*x^8-2*X2*x^8+X1*X2*x^6+X1*x^7+X2*x^7+2*x^8+X1*x^6-2*X1*x^5-2*x^6+X1 *x^4+2*x^5-x^4+x^3-3*x^2+3*x-1)/(-1+x)^2/(X1*X2*x^7-X1*X2*x^6-X1*x^7-X2*x^7+X1* x^6+x^7+X1*x^5-X1*x^4-x^5+x^4-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 2, 1], are n 183 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 1, 6], are n 49 53 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 8 3 53 ------------ 477 and in floating point 0.2114804230 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 8 159 ate normal pair with correlation, -------- 477 1/2 8 159 1559 i.e. , [[--------, 0], [0, ----]] 477 1431 ------------------------------------------------- Theorem Number, 236, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [4, 2, 1], nor the composition, [1, 2, 5] Then infinity ----- \ n 18 16 14 13 11 10 9 7 6 ) a(n) x = (x - x + x + 2 x + x - 2 x - x - x + 2 x / ----- n = 0 5 4 3 2 / 9 8 5 4 - 2 x + x - x + 3 x - 3 x + 1) / ((x + x - x + x - 2 x + 1) / 2 (-1 + x) ) and in Maple format (x^18-x^16+x^14+2*x^13+x^11-2*x^10-x^9-x^7+2*x^6-2*x^5+x^4-x^3+3*x^2-3*x+1)/(x^ 9+x^8-x^5+x^4-2*x+1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.65981530479575843941 1.9159714162121856417 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [4, 2, 1] and d occurrences (as containment) of the composition, [1, 2, 5], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 18 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 18 2 18 2 2 16 2 18 18 - 2 X1 X2 x - 2 X1 X2 x - X1 X2 x + X1 x + 4 X1 X2 x 2 18 2 16 2 16 18 18 2 16 + X2 x + 2 X1 X2 x + 2 X1 X2 x - 2 X1 x - 2 X2 x - X1 x 16 2 16 18 2 14 16 16 - 4 X1 X2 x - X2 x + x - X1 X2 x + 2 X1 x + 2 X2 x 2 13 2 13 14 2 14 16 2 12 - X1 X2 x - X1 X2 x + 2 X1 X2 x + X2 x - x - X1 X2 x 2 13 13 14 2 13 14 2 12 + X1 x + 4 X1 X2 x - X1 x + X2 x - 2 X2 x + X1 x 12 13 13 14 11 12 13 + X1 X2 x - 3 X1 x - 3 X2 x + x + X1 X2 x - X1 x + 2 x 10 11 11 9 10 10 11 - 2 X1 X2 x - X1 x - X2 x - X1 X2 x + 2 X1 x + 2 X2 x + x 8 9 9 10 7 8 9 6 + X1 X2 x + X1 x + X2 x - 2 x - 2 X1 X2 x - X1 x - x + 2 X1 X2 x 7 5 6 7 5 6 4 5 4 + 3 X1 x - X1 X2 x - 4 X1 x - x + 3 X1 x + 2 x - X1 x - 2 x + x 3 2 / 2 9 8 9 9 - x + 3 x - 3 x + 1) / ((-1 + x) (X1 X2 x + X1 X2 x - X1 x - X2 x / 8 8 9 6 8 5 6 5 4 - X1 x - X2 x + x + X1 X2 x + x - X1 X2 x - X1 x + 2 X1 x - X1 x 5 4 - x + x - 2 x + 1)) and in Maple format (X1^2*X2^2*x^18-2*X1^2*X2*x^18-2*X1*X2^2*x^18-X1^2*X2^2*x^16+X1^2*x^18+4*X1*X2* x^18+X2^2*x^18+2*X1^2*X2*x^16+2*X1*X2^2*x^16-2*X1*x^18-2*X2*x^18-X1^2*x^16-4*X1 *X2*x^16-X2^2*x^16+x^18-X1*X2^2*x^14+2*X1*x^16+2*X2*x^16-X1^2*X2*x^13-X1*X2^2*x ^13+2*X1*X2*x^14+X2^2*x^14-x^16-X1^2*X2*x^12+X1^2*x^13+4*X1*X2*x^13-X1*x^14+X2^ 2*x^13-2*X2*x^14+X1^2*x^12+X1*X2*x^12-3*X1*x^13-3*X2*x^13+x^14+X1*X2*x^11-X1*x^ 12+2*x^13-2*X1*X2*x^10-X1*x^11-X2*x^11-X1*X2*x^9+2*X1*x^10+2*X2*x^10+x^11+X1*X2 *x^8+X1*x^9+X2*x^9-2*x^10-2*X1*X2*x^7-X1*x^8-x^9+2*X1*X2*x^6+3*X1*x^7-X1*X2*x^5 -4*X1*x^6-x^7+3*X1*x^5+2*x^6-X1*x^4-2*x^5+x^4-x^3+3*x^2-3*x+1)/(-1+x)^2/(X1*X2* x^9+X1*X2*x^8-X1*x^9-X2*x^9-X1*x^8-X2*x^8+x^9+X1*X2*x^6+x^8-X1*X2*x^5-X1*x^6+2* X1*x^5-X1*x^4-x^5+x^4-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 2, 1], are n 183 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 2, 5], are n 111 57 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 14 19 -------- 171 and in floating point 0.3568689194 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 14 19 ate normal pair with correlation, -------- 171 1/2 14 19 1931 i.e. , [[--------, 0], [0, ----]] 171 1539 ------------------------------------------------- Theorem Number, 237, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [4, 2, 1], nor the composition, [1, 3, 4] Then infinity ----- 15 11 10 9 6 5 4 2 \ n x + x + 2 x + 2 x - x + x - x - x + 2 x - 1 ) a(n) x = ------------------------------------------------------ / 7 6 5 3 2 2 ----- (x + x + x + x + x + x - 1) (-1 + x) n = 0 and in Maple format (x^15+x^11+2*x^10+2*x^9-x^6+x^5-x^4-x^2+2*x-1)/(x^7+x^6+x^5+x^3+x^2+x-1)/(-1+x) ^2 The asymptotic expression for a(n) is, n 0.64232565556525224311 1.9221280043552247737 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [4, 2, 1] and d occurrences (as containment) of the composition, [1, 3, 4], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 16 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 2 15 2 16 2 16 2 15 2 16 - X1 X2 x - 2 X1 X2 x - 2 X1 X2 x + 2 X1 X2 x + X1 x 2 15 16 2 16 2 15 15 16 + 2 X1 X2 x + 4 X1 X2 x + X2 x - X1 x - 4 X1 X2 x - 2 X1 x 2 15 16 15 15 16 2 12 2 12 - X2 x - 2 X2 x + 2 X1 x + 2 X2 x + x - X1 X2 x - X1 X2 x 15 2 11 2 12 12 2 12 2 11 - x - X1 X2 x + X1 x + 3 X1 X2 x + X2 x + X1 x 11 12 12 11 11 12 9 + 2 X1 X2 x - 2 X1 x - 2 X2 x - 2 X1 x - X2 x + x - 2 X1 X2 x 11 8 9 9 7 8 9 6 + x + X1 X2 x + 2 X1 x + 2 X2 x - X1 X2 x - X1 x - 2 x - X1 X2 x 7 5 6 7 4 6 5 4 3 + 2 X1 x + 2 X1 X2 x - X1 x - x - X1 X2 x + 2 x - 2 x + x - x 2 / 2 8 8 8 6 8 + 3 x - 3 x + 1) / ((-1 + x) (X1 X2 x - X1 x - X2 x + X1 X2 x + x / 5 6 4 5 4 + X1 X2 x - X1 x - X1 X2 x - x + x - 2 x + 1)) and in Maple format (X1^2*X2^2*x^16-X1^2*X2^2*x^15-2*X1^2*X2*x^16-2*X1*X2^2*x^16+2*X1^2*X2*x^15+X1^ 2*x^16+2*X1*X2^2*x^15+4*X1*X2*x^16+X2^2*x^16-X1^2*x^15-4*X1*X2*x^15-2*X1*x^16- X2^2*x^15-2*X2*x^16+2*X1*x^15+2*X2*x^15+x^16-X1^2*X2*x^12-X1*X2^2*x^12-x^15-X1^ 2*X2*x^11+X1^2*x^12+3*X1*X2*x^12+X2^2*x^12+X1^2*x^11+2*X1*X2*x^11-2*X1*x^12-2* X2*x^12-2*X1*x^11-X2*x^11+x^12-2*X1*X2*x^9+x^11+X1*X2*x^8+2*X1*x^9+2*X2*x^9-X1* X2*x^7-X1*x^8-2*x^9-X1*X2*x^6+2*X1*x^7+2*X1*X2*x^5-X1*x^6-x^7-X1*X2*x^4+2*x^6-2 *x^5+x^4-x^3+3*x^2-3*x+1)/(-1+x)^2/(X1*X2*x^8-X1*x^8-X2*x^8+X1*X2*x^6+x^8+X1*X2 *x^5-X1*x^6-X1*X2*x^4-x^5+x^4-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 2, 1], are n 183 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 3, 4], are n 135 65 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 22 3 65 ------------- 585 and in floating point 0.5251509079 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 22 195 ate normal pair with correlation, --------- 585 1/2 22 195 2723 i.e. , [[---------, 0], [0, ----]] 585 1755 ------------------------------------------------- Theorem Number, 238, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [4, 2, 1], nor the composition, [1, 4, 3] Then infinity ----- 12 11 8 7 6 5 4 3 2 \ n x - x + x + x - 2 x + 2 x - x + x - 3 x + 3 x - 1 ) a(n) x = ------------------------------------------------------------ / 5 4 2 ----- (x - x + 2 x - 1) (-1 + x) n = 0 and in Maple format (x^12-x^11+x^8+x^7-2*x^6+2*x^5-x^4+x^3-3*x^2+3*x-1)/(x^5-x^4+2*x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.61442764446410597890 1.9331849818995204468 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [4, 2, 1] and d occurrences (as containment) of the composition, [1, 4, 3], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 11 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 12 2 11 11 12 12 11 11 + X1 X2 x - X1 x - 2 X1 X2 x - X1 x - X2 x + 2 X1 x + X2 x 12 11 7 8 6 7 8 5 + x - x + X1 X2 x - X2 x + X1 X2 x - 2 X1 x + x - 2 X1 X2 x 6 7 4 6 5 4 3 2 / + X1 x + x + X1 X2 x - 2 x + 2 x - x + x - 3 x + 3 x - 1) / ( / 2 6 5 6 4 5 4 (-1 + x) (X1 X2 x + X1 X2 x - X1 x - X1 X2 x - x + x - 2 x + 1)) and in Maple format -(X1^2*X2*x^11+X1*X2*x^12-X1^2*x^11-2*X1*X2*x^11-X1*x^12-X2*x^12+2*X1*x^11+X2*x ^11+x^12-x^11+X1*X2*x^7-X2*x^8+X1*X2*x^6-2*X1*x^7+x^8-2*X1*X2*x^5+X1*x^6+x^7+X1 *X2*x^4-2*x^6+2*x^5-x^4+x^3-3*x^2+3*x-1)/(-1+x)^2/(X1*X2*x^6+X1*X2*x^5-X1*x^6- X1*X2*x^4-x^5+x^4-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 2, 1], are n 183 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 4, 3], are n 135 65 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 2 3 65 ------------ 39 and in floating point 0.7161148743 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 1/2 2 5 39 ate normal pair with correlation, ------------ 39 1/2 1/2 2 5 39 79 i.e. , [[------------, 0], [0, --]] 39 39 ------------------------------------------------- Theorem Number, 239, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [4, 2, 1], nor the composition, [1, 5, 2] Then infinity ----- 12 11 8 7 6 5 4 3 2 \ n x - x + x + x - 2 x + 2 x - x + x - 3 x + 3 x - 1 ) a(n) x = ------------------------------------------------------------ / 5 4 2 ----- (x - x + 2 x - 1) (-1 + x) n = 0 and in Maple format (x^12-x^11+x^8+x^7-2*x^6+2*x^5-x^4+x^3-3*x^2+3*x-1)/(x^5-x^4+2*x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.61442764446410597890 1.9331849818995204468 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [4, 2, 1] and d occurrences (as containment) of the composition, [1, 5, 2], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 12 11 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x - X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 12 12 11 11 12 11 7 8 - X1 x - X2 x + X1 x + X2 x + x - x + 2 X1 X2 x - X2 x 6 7 8 5 6 7 5 6 - 2 X1 X2 x - 3 X1 x + x + X1 X2 x + 4 X1 x + x - 3 X1 x - 2 x 4 5 4 3 2 / 2 + X1 x + 2 x - x + x - 3 x + 3 x - 1) / ((-1 + x) / 6 5 6 5 4 5 4 (X1 X2 x - X1 X2 x - X1 x + 2 X1 x - X1 x - x + x - 2 x + 1)) and in Maple format -(X1*X2*x^12-X1*X2*x^11-X1*x^12-X2*x^12+X1*x^11+X2*x^11+x^12-x^11+2*X1*X2*x^7- X2*x^8-2*X1*X2*x^6-3*X1*x^7+x^8+X1*X2*x^5+4*X1*x^6+x^7-3*X1*x^5-2*x^6+X1*x^4+2* x^5-x^4+x^3-3*x^2+3*x-1)/(-1+x)^2/(X1*X2*x^6-X1*X2*x^5-X1*x^6+2*X1*x^5-X1*x^4-x ^5+x^4-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 2, 1], are n 183 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 5, 2], are n 111 57 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 26 19 -------- 171 and in floating point 0.6627565645 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 26 19 ate normal pair with correlation, -------- 171 1/2 26 19 2891 i.e. , [[--------, 0], [0, ----]] 171 1539 ------------------------------------------------- Theorem Number, 240, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [4, 2, 1], nor the composition, [1, 6, 1] Then infinity ----- 4 2 6 5 \ n (x + x - 2 x + 1) (x + x - 1) ) a(n) x = ------------------------------------- / 6 5 3 2 2 ----- (x + x + x + x + x - 1) (-1 + x) n = 0 and in Maple format (x^4+x^2-2*x+1)*(x^6+x^5-1)/(x^6+x^5+x^3+x^2+x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.68707893079649356212 1.9111834366854762350 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [4, 2, 1] and d occurrences (as containment) of the composition, [1, 6, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 10 9 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x + X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 10 10 8 9 9 10 7 8 - X1 x - X2 x + X1 X2 x - X1 x - X2 x + x - X1 X2 x - X1 x 8 9 6 7 7 8 7 5 6 4 - X2 x + x + X1 X2 x + X1 x + X2 x + x - x - X1 x - x + X1 x 5 4 2 / 7 6 7 + x - x - x + 2 x - 1) / ((-1 + x) (X1 X2 x - X1 X2 x - X1 x / 7 6 7 5 4 5 4 - X2 x + X1 x + x + X1 x - X1 x - x + x - 2 x + 1)) and in Maple format (X1*X2*x^10+X1*X2*x^9-X1*x^10-X2*x^10+X1*X2*x^8-X1*x^9-X2*x^9+x^10-X1*X2*x^7-X1 *x^8-X2*x^8+x^9+X1*X2*x^6+X1*x^7+X2*x^7+x^8-x^7-X1*x^5-x^6+X1*x^4+x^5-x^4-x^2+2 *x-1)/(-1+x)/(X1*X2*x^7-X1*X2*x^6-X1*x^7-X2*x^7+X1*x^6+x^7+X1*x^5-X1*x^4-x^5+x^ 4-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 2, 1], are n 183 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 6, 1], are n 49 53 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 8 3 53 ------------ 477 and in floating point 0.2114804230 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 8 159 ate normal pair with correlation, -------- 477 1/2 8 159 1559 i.e. , [[--------, 0], [0, ----]] 477 1431 ------------------------------------------------- Theorem Number, 241, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [4, 2, 1], nor the composition, [2, 1, 5] Then infinity ----- \ n 22 21 20 18 17 16 15 14 13 ) a(n) x = (x + 2 x + x - x - 2 x - x + x + 2 x + x / ----- n = 0 11 7 6 5 4 3 2 / - x - 3 x + 3 x - 2 x + x - x + 3 x - 3 x + 1) / ( / 12 11 10 8 7 6 5 4 2 (x + 2 x + 2 x - x - x + x - x + x - 2 x + 1) (-1 + x) ) and in Maple format (x^22+2*x^21+x^20-x^18-2*x^17-x^16+x^15+2*x^14+x^13-x^11-3*x^7+3*x^6-2*x^5+x^4- x^3+3*x^2-3*x+1)/(x^12+2*x^11+2*x^10-x^8-x^7+x^6-x^5+x^4-2*x+1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.66523001633931812792 1.9143463083540318048 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [4, 2, 1] and d occurrences (as containment) of the composition, [2, 1, 5], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 3 22 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 3 21 2 2 22 3 22 2 3 20 + 2 X1 X2 x - 3 X1 X2 x - 2 X1 X2 x + X1 X2 x 2 2 21 2 22 3 21 2 22 3 22 - 6 X1 X2 x + 3 X1 X2 x - 4 X1 X2 x + 6 X1 X2 x + X2 x 2 2 20 2 21 2 22 3 20 2 21 - 3 X1 X2 x + 6 X1 X2 x - X1 x - 2 X1 X2 x + 12 X1 X2 x 22 3 21 2 22 2 20 2 21 - 6 X1 X2 x + 2 X2 x - 3 X2 x + 3 X1 X2 x - 2 X1 x 2 20 21 22 3 20 2 21 22 + 6 X1 X2 x - 12 X1 X2 x + 2 X1 x + X2 x - 6 X2 x + 3 X2 x 2 2 18 2 20 20 21 2 20 21 + X1 X2 x - X1 x - 6 X1 X2 x + 4 X1 x - 3 X2 x + 6 X2 x 22 2 2 17 2 18 2 18 20 20 - x + 2 X1 X2 x - 2 X1 X2 x - 2 X1 X2 x + 2 X1 x + 3 X2 x 21 2 17 2 18 2 17 18 2 18 - 2 x - 4 X1 X2 x + X1 x - 4 X1 X2 x + 4 X1 X2 x + X2 x 20 2 17 2 16 17 18 2 17 - x + 2 X1 x - X1 X2 x + 8 X1 X2 x - 2 X1 x + 2 X2 x 18 2 15 16 17 2 16 17 - 2 X2 x + X1 X2 x + 2 X1 X2 x - 4 X1 x + X2 x - 4 X2 x 18 2 2 13 2 14 2 14 15 16 + x - X1 X2 x + X1 X2 x + X1 X2 x - 2 X1 X2 x - X1 x 2 15 16 17 2 13 2 14 2 13 - X2 x - 2 X2 x + 2 x + 3 X1 X2 x - X1 x + X1 X2 x 14 15 2 14 15 16 2 13 2 12 - 4 X1 X2 x + X1 x - X2 x + 2 X2 x + x - 2 X1 x + X1 X2 x 13 14 14 15 2 11 12 - 4 X1 X2 x + 3 X1 x + 3 X2 x - x - X1 X2 x - 2 X1 X2 x 13 13 14 11 12 13 11 + 3 X1 x + X2 x - 2 x + 3 X1 X2 x + X1 x - x - 2 X1 x 11 11 7 6 7 7 5 - X2 x + x + 2 X1 X2 x - 2 X1 X2 x - 3 X1 x - 2 X2 x + X1 X2 x 6 6 7 5 6 4 5 4 3 2 + 4 X1 x + X2 x + 3 x - 3 X1 x - 3 x + X1 x + 2 x - x + x - 3 x / 2 2 15 2 15 2 15 + 3 x - 1) / ((-1 + x) (X1 X2 x - 2 X1 X2 x - X1 X2 x / 2 2 13 2 15 15 2 13 2 13 15 - X1 X2 x + X1 x + 2 X1 X2 x + X1 X2 x + X1 X2 x - X1 x 2 12 2 12 2 12 2 11 13 13 - X1 X2 x + X1 X2 x + X1 x - X1 X2 x - X1 x - X2 x 11 12 12 13 10 11 12 + 2 X1 X2 x - X1 x - X2 x + x - 2 X1 X2 x - X1 x + x 9 10 10 9 9 10 7 9 - X1 X2 x + 2 X1 x + 2 X2 x + X1 x + X2 x - 2 x + X1 X2 x - x 6 7 7 5 6 6 7 - 2 X1 X2 x - X1 x - 2 X2 x + X1 X2 x + 3 X1 x + X2 x + 2 x 5 6 4 5 4 2 - 3 X1 x - 2 x + X1 x + 2 x - x - 2 x + 3 x - 1)) and in Maple format -(X1^2*X2^3*x^22+2*X1^2*X2^3*x^21-3*X1^2*X2^2*x^22-2*X1*X2^3*x^22+X1^2*X2^3*x^ 20-6*X1^2*X2^2*x^21+3*X1^2*X2*x^22-4*X1*X2^3*x^21+6*X1*X2^2*x^22+X2^3*x^22-3*X1 ^2*X2^2*x^20+6*X1^2*X2*x^21-X1^2*x^22-2*X1*X2^3*x^20+12*X1*X2^2*x^21-6*X1*X2*x^ 22+2*X2^3*x^21-3*X2^2*x^22+3*X1^2*X2*x^20-2*X1^2*x^21+6*X1*X2^2*x^20-12*X1*X2*x ^21+2*X1*x^22+X2^3*x^20-6*X2^2*x^21+3*X2*x^22+X1^2*X2^2*x^18-X1^2*x^20-6*X1*X2* x^20+4*X1*x^21-3*X2^2*x^20+6*X2*x^21-x^22+2*X1^2*X2^2*x^17-2*X1^2*X2*x^18-2*X1* X2^2*x^18+2*X1*x^20+3*X2*x^20-2*x^21-4*X1^2*X2*x^17+X1^2*x^18-4*X1*X2^2*x^17+4* X1*X2*x^18+X2^2*x^18-x^20+2*X1^2*x^17-X1*X2^2*x^16+8*X1*X2*x^17-2*X1*x^18+2*X2^ 2*x^17-2*X2*x^18+X1*X2^2*x^15+2*X1*X2*x^16-4*X1*x^17+X2^2*x^16-4*X2*x^17+x^18- X1^2*X2^2*x^13+X1^2*X2*x^14+X1*X2^2*x^14-2*X1*X2*x^15-X1*x^16-X2^2*x^15-2*X2*x^ 16+2*x^17+3*X1^2*X2*x^13-X1^2*x^14+X1*X2^2*x^13-4*X1*X2*x^14+X1*x^15-X2^2*x^14+ 2*X2*x^15+x^16-2*X1^2*x^13+X1*X2^2*x^12-4*X1*X2*x^13+3*X1*x^14+3*X2*x^14-x^15- X1*X2^2*x^11-2*X1*X2*x^12+3*X1*x^13+X2*x^13-2*x^14+3*X1*X2*x^11+X1*x^12-x^13-2* X1*x^11-X2*x^11+x^11+2*X1*X2*x^7-2*X1*X2*x^6-3*X1*x^7-2*X2*x^7+X1*X2*x^5+4*X1*x ^6+X2*x^6+3*x^7-3*X1*x^5-3*x^6+X1*x^4+2*x^5-x^4+x^3-3*x^2+3*x-1)/(-1+x)/(X1^2* X2^2*x^15-2*X1^2*X2*x^15-X1*X2^2*x^15-X1^2*X2^2*x^13+X1^2*x^15+2*X1*X2*x^15+X1^ 2*X2*x^13+X1*X2^2*x^13-X1*x^15-X1^2*X2*x^12+X1*X2^2*x^12+X1^2*x^12-X1*X2^2*x^11 -X1*x^13-X2*x^13+2*X1*X2*x^11-X1*x^12-X2*x^12+x^13-2*X1*X2*x^10-X1*x^11+x^12-X1 *X2*x^9+2*X1*x^10+2*X2*x^10+X1*x^9+X2*x^9-2*x^10+X1*X2*x^7-x^9-2*X1*X2*x^6-X1*x ^7-2*X2*x^7+X1*X2*x^5+3*X1*x^6+X2*x^6+2*x^7-3*X1*x^5-2*x^6+X1*x^4+2*x^5-x^4-2*x ^2+3*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 2, 1], are n 183 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 1, 5], are n 113 57 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 4 19 ------- 57 and in floating point 0.3058876452 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 4 19 ate normal pair with correlation, ------- 57 1/2 4 19 203 i.e. , [[-------, 0], [0, ---]] 57 171 ------------------------------------------------- Theorem Number, 242, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [4, 2, 1], nor the composition, [2, 2, 4] Then infinity ----- \ n ) a(n) x = / ----- n = 0 14 13 11 10 9 8 6 5 4 2 x + 2 x - 2 x - x - x - x + x - x + x + x - 2 x + 1 - ----------------------------------------------------------------- 11 10 9 5 4 (-1 + x) (x + 2 x + x - x + x - 2 x + 1) and in Maple format -(x^14+2*x^13-2*x^11-x^10-x^9-x^8+x^6-x^5+x^4+x^2-2*x+1)/(-1+x)/(x^11+2*x^10+x^ 9-x^5+x^4-2*x+1) The asymptotic expression for a(n) is, n 0.65315694203017104538 1.9197449317290998989 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [4, 2, 1] and d occurrences (as containment) of the composition, [2, 2, 4], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 15 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 2 14 2 15 2 15 2 14 2 15 + X1 X2 x - 2 X1 X2 x - 2 X1 X2 x - 2 X1 X2 x + X1 x 2 14 15 2 15 2 13 2 14 2 13 - 2 X1 X2 x + 4 X1 X2 x + X2 x + X1 X2 x + X1 x + X1 X2 x 14 15 2 14 15 2 12 2 13 + 4 X1 X2 x - 2 X1 x + X2 x - 2 X2 x + X1 X2 x - X1 x 2 12 13 14 2 13 14 15 2 12 + X1 X2 x - 4 X1 X2 x - 2 X1 x - X2 x - 2 X2 x + x - X1 x 12 13 2 12 13 14 11 12 - 4 X1 X2 x + 3 X1 x - X2 x + 3 X2 x + x + X1 X2 x + 3 X1 x 12 13 11 11 12 11 8 7 + 3 X2 x - 2 x - X1 x - X2 x - 2 x + x + X1 X2 x - X1 X2 x 8 8 6 8 5 7 4 6 - X1 x - X2 x + 2 X1 X2 x + x - 2 X1 X2 x + x + X1 X2 x - 2 x 5 4 3 2 / 2 12 2 11 + 2 x - x + x - 3 x + 3 x - 1) / ((-1 + x) (X1 X2 x + X1 X2 x / 2 12 12 2 11 11 12 12 - X1 x - 2 X1 X2 x - X1 x - 2 X1 X2 x + 2 X1 x + X2 x 10 11 11 12 9 10 10 11 + X1 X2 x + 2 X1 x + X2 x - x + X1 X2 x - X1 x - X2 x - x 9 9 10 7 9 6 7 5 - X1 x - X2 x + x + X1 X2 x + x - X1 X2 x - X1 x + 2 X1 X2 x 4 6 5 4 2 - X1 X2 x + x - 2 x + x + 2 x - 3 x + 1)) and in Maple format (X1^2*X2^2*x^15+X1^2*X2^2*x^14-2*X1^2*X2*x^15-2*X1*X2^2*x^15-2*X1^2*X2*x^14+X1^ 2*x^15-2*X1*X2^2*x^14+4*X1*X2*x^15+X2^2*x^15+X1^2*X2*x^13+X1^2*x^14+X1*X2^2*x^ 13+4*X1*X2*x^14-2*X1*x^15+X2^2*x^14-2*X2*x^15+X1^2*X2*x^12-X1^2*x^13+X1*X2^2*x^ 12-4*X1*X2*x^13-2*X1*x^14-X2^2*x^13-2*X2*x^14+x^15-X1^2*x^12-4*X1*X2*x^12+3*X1* x^13-X2^2*x^12+3*X2*x^13+x^14+X1*X2*x^11+3*X1*x^12+3*X2*x^12-2*x^13-X1*x^11-X2* x^11-2*x^12+x^11+X1*X2*x^8-X1*X2*x^7-X1*x^8-X2*x^8+2*X1*X2*x^6+x^8-2*X1*X2*x^5+ x^7+X1*X2*x^4-2*x^6+2*x^5-x^4+x^3-3*x^2+3*x-1)/(-1+x)/(X1^2*X2*x^12+X1^2*X2*x^ 11-X1^2*x^12-2*X1*X2*x^12-X1^2*x^11-2*X1*X2*x^11+2*X1*x^12+X2*x^12+X1*X2*x^10+2 *X1*x^11+X2*x^11-x^12+X1*X2*x^9-X1*x^10-X2*x^10-x^11-X1*x^9-X2*x^9+x^10+X1*X2*x ^7+x^9-X1*X2*x^6-X1*x^7+2*X1*X2*x^5-X1*X2*x^4+x^6-2*x^5+x^4+2*x^2-3*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 2, 1], are n 183 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 2, 4], are n 75 69 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 20 23 -------- 207 and in floating point 0.4633653646 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 20 23 ate normal pair with correlation, -------- 207 1/2 20 23 2663 i.e. , [[--------, 0], [0, ----]] 207 1863 ------------------------------------------------- Theorem Number, 243, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [4, 2, 1], nor the composition, [2, 3, 3] Then infinity ----- 11 9 7 6 4 3 2 \ n x + x + x - 2 x + x - x - x + 2 x - 1 ) a(n) x = ------------------------------------------------- / 10 7 6 5 4 3 ----- (-1 + x) (x - x + x - x - x + x - 2 x + 1) n = 0 and in Maple format (x^11+x^9+x^7-2*x^6+x^4-x^3-x^2+2*x-1)/(-1+x)/(x^10-x^7+x^6-x^5-x^4+x^3-2*x+1) The asymptotic expression for a(n) is, n 0.64822251981642343554 1.9223246520768555496 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [4, 2, 1] and d occurrences (as containment) of the composition, [2, 3, 3], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 12 2 12 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x - X1 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 12 11 12 12 10 11 - 2 X1 X2 x + X1 X2 x + 2 X1 x + X2 x - X1 X2 x - X1 x 11 12 9 10 10 11 8 9 - X2 x - x + X1 X2 x + X1 x + X2 x + x - X1 X2 x - X1 x 9 10 7 8 8 9 6 7 - X2 x - x + X1 X2 x + X1 x + X2 x + x + X1 X2 x - 2 X1 x 7 8 5 6 7 4 5 6 - 2 X2 x - x - 2 X1 X2 x + X1 x + 3 x + X1 X2 x + 3 X2 x - 2 x 4 5 3 4 2 / 2 11 - 3 X2 x - x + X2 x + 2 x - 3 x + 3 x - 1) / ((-1 + x) (X1 X2 x / 2 11 11 10 11 11 10 10 - X1 x - 2 X1 X2 x + X1 X2 x + 2 X1 x + X2 x - X1 x - X2 x 11 8 10 7 8 8 7 7 8 - x + X1 X2 x + x - X1 X2 x - X1 x - X2 x + X1 x + 2 X2 x + x 5 6 6 7 4 5 6 4 + 2 X1 X2 x - X1 x - X2 x - 2 x - X1 X2 x - 2 X2 x + 2 x + 3 X2 x 3 4 3 2 - X2 x - 2 x + x + 2 x - 3 x + 1)) and in Maple format (X1^2*X2*x^12-X1^2*x^12-2*X1*X2*x^12+X1*X2*x^11+2*X1*x^12+X2*x^12-X1*X2*x^10-X1 *x^11-X2*x^11-x^12+X1*X2*x^9+X1*x^10+X2*x^10+x^11-X1*X2*x^8-X1*x^9-X2*x^9-x^10+ X1*X2*x^7+X1*x^8+X2*x^8+x^9+X1*X2*x^6-2*X1*x^7-2*X2*x^7-x^8-2*X1*X2*x^5+X1*x^6+ 3*x^7+X1*X2*x^4+3*X2*x^5-2*x^6-3*X2*x^4-x^5+X2*x^3+2*x^4-3*x^2+3*x-1)/(-1+x)/( X1^2*X2*x^11-X1^2*x^11-2*X1*X2*x^11+X1*X2*x^10+2*X1*x^11+X2*x^11-X1*x^10-X2*x^ 10-x^11+X1*X2*x^8+x^10-X1*X2*x^7-X1*x^8-X2*x^8+X1*x^7+2*X2*x^7+x^8+2*X1*X2*x^5- X1*x^6-X2*x^6-2*x^7-X1*X2*x^4-2*X2*x^5+2*x^6+3*X2*x^4-X2*x^3-2*x^4+x^3+2*x^2-3* x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 2, 1], are n 183 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 3, 3], are n 97 85 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 4 3 85 ------------ 153 and in floating point 0.4174828609 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 1/2 4 5 51 ate normal pair with correlation, ------------ 153 1/2 1/2 4 5 51 619 i.e. , [[------------, 0], [0, ---]] 153 459 ------------------------------------------------- Theorem Number, 244, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [4, 2, 1], nor the composition, [2, 4, 2] Then infinity ----- 9 8 6 5 4 2 \ n x + x - x + x - x - x + 2 x - 1 ) a(n) x = - ------------------------------------- / 5 4 ----- (-1 + x) (x - x + 2 x - 1) n = 0 and in Maple format -(x^9+x^8-x^6+x^5-x^4-x^2+2*x-1)/(-1+x)/(x^5-x^4+2*x-1) The asymptotic expression for a(n) is, n 0.61915988369516142914 1.9331849818995204468 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [4, 2, 1] and d occurrences (as containment) of the composition, [2, 4, 2], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 9 8 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x + X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 9 9 8 8 9 6 8 5 4 - X1 x - X2 x - X1 x - X2 x + x + X1 X2 x + x - X1 X2 x + X1 X2 x 6 5 4 2 / 7 6 7 - x + x - x - x + 2 x - 1) / (X1 X2 x - X1 X2 x - X1 x / 5 4 6 5 4 2 + 2 X1 X2 x - X1 X2 x + x - 2 x + x + 2 x - 3 x + 1) and in Maple format -(X1*X2*x^9+X1*X2*x^8-X1*x^9-X2*x^9-X1*x^8-X2*x^8+x^9+X1*X2*x^6+x^8-X1*X2*x^5+ X1*X2*x^4-x^6+x^5-x^4-x^2+2*x-1)/(X1*X2*x^7-X1*X2*x^6-X1*x^7+2*X1*X2*x^5-X1*X2* x^4+x^6-2*x^5+x^4+2*x^2-3*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 2, 1], are n 183 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 4, 2], are n 75 69 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 32 23 -------- 207 and in floating point 0.7413845834 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 32 23 ate normal pair with correlation, -------- 207 1/2 32 23 3911 i.e. , [[--------, 0], [0, ----]] 207 1863 ------------------------------------------------- Theorem Number, 245, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [4, 2, 1], nor the composition, [2, 5, 1] Then infinity ----- 10 9 6 5 4 2 \ n x + x - x + x - x - x + 2 x - 1 ) a(n) x = -------------------------------------- / 9 8 5 4 ----- (-1 + x) (x + x - x + x - 2 x + 1) n = 0 and in Maple format (x^10+x^9-x^6+x^5-x^4-x^2+2*x-1)/(-1+x)/(x^9+x^8-x^5+x^4-2*x+1) The asymptotic expression for a(n) is, n 0.67229840142428698322 1.9159714162121856417 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [4, 2, 1] and d occurrences (as containment) of the composition, [2, 5, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 10 9 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x + X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 10 10 9 9 10 7 9 6 7 - X1 x - X2 x - X1 x - X2 x + x + X1 X2 x + x - X1 X2 x - X1 x 5 6 5 6 4 5 4 2 / + X1 X2 x + 2 X1 x - 2 X1 x - x + X1 x + x - x - x + 2 x - 1) / / 9 8 9 9 8 8 9 ((-1 + x) (X1 X2 x + X1 X2 x - X1 x - X2 x - X1 x - X2 x + x 6 8 5 6 5 4 5 4 + X1 X2 x + x - X1 X2 x - X1 x + 2 X1 x - X1 x - x + x - 2 x + 1)) and in Maple format (X1*X2*x^10+X1*X2*x^9-X1*x^10-X2*x^10-X1*x^9-X2*x^9+x^10+X1*X2*x^7+x^9-X1*X2*x^ 6-X1*x^7+X1*X2*x^5+2*X1*x^6-2*X1*x^5-x^6+X1*x^4+x^5-x^4-x^2+2*x-1)/(-1+x)/(X1* X2*x^9+X1*X2*x^8-X1*x^9-X2*x^9-X1*x^8-X2*x^8+x^9+X1*X2*x^6+x^8-X1*X2*x^5-X1*x^6 +2*X1*x^5-X1*x^4-x^5+x^4-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 2, 1], are n 183 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 5, 1], are n 111 57 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 14 19 -------- 171 and in floating point 0.3568689194 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 14 19 ate normal pair with correlation, -------- 171 1/2 14 19 1931 i.e. , [[--------, 0], [0, ----]] 171 1539 ------------------------------------------------- Theorem Number, 246, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [4, 2, 1], nor the composition, [3, 1, 4] Then infinity ----- \ n ) a(n) x = / ----- n = 0 18 17 16 14 12 11 10 8 4 2 x + x + x - x + x + x + x + x - x - x + 2 x - 1 ---------------------------------------------------------------- 10 9 8 7 6 3 2 2 (x + 2 x + 3 x + 2 x + x + x + x + x - 1) (-1 + x) and in Maple format (x^18+x^17+x^16-x^14+x^12+x^11+x^10+x^8-x^4-x^2+2*x-1)/(x^10+2*x^9+3*x^8+2*x^7+ x^6+x^3+x^2+x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.65835090509051371772 1.9175371779461324874 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [4, 2, 1] and d occurrences (as containment) of the composition, [3, 1, 4], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 3 19 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 2 19 3 19 2 19 2 19 3 19 - 3 X1 X2 x - 2 X1 X2 x + 3 X1 X2 x + 6 X1 X2 x + X2 x 2 19 19 2 19 2 2 16 19 19 - X1 x - 6 X1 X2 x - 3 X2 x + X1 X2 x + 2 X1 x + 3 X2 x 2 16 2 16 19 2 16 2 15 16 - 2 X1 X2 x - 2 X1 X2 x - x + X1 x - X1 X2 x + 4 X1 X2 x 2 16 2 14 15 16 2 15 16 + X2 x + X1 X2 x + 2 X1 X2 x - 2 X1 x + X2 x - 2 X2 x 2 2 12 2 13 14 15 2 14 15 - X1 X2 x + X1 X2 x - 2 X1 X2 x - X1 x - X2 x - 2 X2 x 16 2 2 11 2 12 2 13 2 12 13 + x + X1 X2 x + 2 X1 X2 x - X1 x + X1 X2 x - 2 X1 X2 x 14 14 15 2 2 10 2 11 2 12 + X1 x + 2 X2 x + x - X1 X2 x - X1 X2 x - X1 x 12 13 13 14 2 10 2 10 - 2 X1 X2 x + 2 X1 x + X2 x - x + X1 X2 x + X1 X2 x 11 12 13 2 9 11 9 10 - X1 X2 x + X1 x - x - X1 X2 x + X1 x + X1 X2 x - X1 x 10 8 9 10 7 8 8 9 - X2 x + X1 X2 x + X2 x + x - X1 X2 x - X1 x - X2 x - x 6 7 8 5 6 4 5 5 + 2 X1 X2 x + X2 x + x - 2 X1 X2 x - 2 X2 x + X1 X2 x + X2 x + x 4 3 2 / 2 2 14 2 2 13 - x + x - 3 x + 3 x - 1) / ((-1 + x) (X1 X2 x + X1 X2 x / 2 14 2 14 2 13 2 14 2 13 - 2 X1 X2 x - X1 X2 x - 2 X1 X2 x + X1 x - X1 X2 x 14 2 2 11 2 12 2 13 2 12 + 2 X1 X2 x + X1 X2 x - X1 X2 x + X1 x + X1 X2 x 13 14 2 2 10 2 11 2 12 2 11 + 2 X1 X2 x - X1 x - X1 X2 x - 2 X1 X2 x + X1 x + X1 X2 x 13 2 10 2 11 2 10 11 12 - X1 x + X1 X2 x + X1 x + X1 X2 x - 2 X1 X2 x - X1 x 12 2 9 10 11 12 9 9 - X2 x - X1 X2 x - X1 X2 x + X1 x + x + X1 x + 2 X2 x 7 9 6 7 5 6 4 - X1 X2 x - 2 x + X1 X2 x + X1 x - 2 X1 X2 x - 2 X2 x + X1 X2 x 5 6 5 4 2 + X2 x + x + x - x - 2 x + 3 x - 1)) and in Maple format -(X1^2*X2^3*x^19-3*X1^2*X2^2*x^19-2*X1*X2^3*x^19+3*X1^2*X2*x^19+6*X1*X2^2*x^19+ X2^3*x^19-X1^2*x^19-6*X1*X2*x^19-3*X2^2*x^19+X1^2*X2^2*x^16+2*X1*x^19+3*X2*x^19 -2*X1^2*X2*x^16-2*X1*X2^2*x^16-x^19+X1^2*x^16-X1*X2^2*x^15+4*X1*X2*x^16+X2^2*x^ 16+X1*X2^2*x^14+2*X1*X2*x^15-2*X1*x^16+X2^2*x^15-2*X2*x^16-X1^2*X2^2*x^12+X1^2* X2*x^13-2*X1*X2*x^14-X1*x^15-X2^2*x^14-2*X2*x^15+x^16+X1^2*X2^2*x^11+2*X1^2*X2* x^12-X1^2*x^13+X1*X2^2*x^12-2*X1*X2*x^13+X1*x^14+2*X2*x^14+x^15-X1^2*X2^2*x^10- X1^2*X2*x^11-X1^2*x^12-2*X1*X2*x^12+2*X1*x^13+X2*x^13-x^14+X1^2*X2*x^10+X1*X2^2 *x^10-X1*X2*x^11+X1*x^12-x^13-X1*X2^2*x^9+X1*x^11+X1*X2*x^9-X1*x^10-X2*x^10+X1* X2*x^8+X2*x^9+x^10-X1*X2*x^7-X1*x^8-X2*x^8-x^9+2*X1*X2*x^6+X2*x^7+x^8-2*X1*X2*x ^5-2*X2*x^6+X1*X2*x^4+X2*x^5+x^5-x^4+x^3-3*x^2+3*x-1)/(-1+x)/(X1^2*X2^2*x^14+X1 ^2*X2^2*x^13-2*X1^2*X2*x^14-X1*X2^2*x^14-2*X1^2*X2*x^13+X1^2*x^14-X1*X2^2*x^13+ 2*X1*X2*x^14+X1^2*X2^2*x^11-X1^2*X2*x^12+X1^2*x^13+X1*X2^2*x^12+2*X1*X2*x^13-X1 *x^14-X1^2*X2^2*x^10-2*X1^2*X2*x^11+X1^2*x^12+X1*X2^2*x^11-X1*x^13+X1^2*X2*x^10 +X1^2*x^11+X1*X2^2*x^10-2*X1*X2*x^11-X1*x^12-X2*x^12-X1*X2^2*x^9-X1*X2*x^10+X1* x^11+x^12+X1*x^9+2*X2*x^9-X1*X2*x^7-2*x^9+X1*X2*x^6+X1*x^7-2*X1*X2*x^5-2*X2*x^6 +X1*X2*x^4+X2*x^5+x^6+x^5-x^4-2*x^2+3*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 2, 1], are n 183 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 1, 4], are n 141 65 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 16 3 65 ------------- 585 and in floating point 0.3819279330 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 16 195 ate normal pair with correlation, --------- 585 1/2 16 195 2267 i.e. , [[---------, 0], [0, ----]] 585 1755 ------------------------------------------------- Theorem Number, 247, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [4, 2, 1], nor the composition, [3, 2, 3] Then infinity ----- 2 7 5 4 3 2 \ n (x - x + 1) (x - x - x - x + x + x - 1) ) a(n) x = ------------------------------------------------- / 8 4 3 2 3 ----- (-1 + x) (x - x - x - x - x + 1) (x - x + 1) n = 0 and in Maple format (x^2-x+1)*(x^7-x^5-x^4-x^3+x^2+x-1)/(-1+x)/(x^8-x^4-x^3-x^2-x+1)/(x^3-x+1) The asymptotic expression for a(n) is, n 0.65527718001580393906 1.9217220658969757404 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [4, 2, 1] and d occurrences (as containment) of the composition, [3, 2, 3], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 10 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 10 2 10 2 9 10 2 10 - X1 X2 x - 2 X1 X2 x + 2 X1 X2 x + 2 X1 X2 x + X2 x 9 2 9 10 9 2 8 9 8 9 - 3 X1 X2 x - 2 X2 x - X2 x + X1 x + X2 x + 3 X2 x - 2 X2 x - x 6 8 5 6 4 5 4 5 - X1 X2 x + x + X1 X2 x + X2 x - X1 X2 x - 2 X2 x + 2 X2 x + x 3 4 3 2 / 2 2 12 2 2 11 - X2 x - x + x + x - 2 x + 1) / (X1 X2 x + X1 X2 x / 2 12 2 12 2 2 10 2 11 2 12 - 2 X1 X2 x - 2 X1 X2 x - X1 X2 x - 2 X1 X2 x + X1 x 12 2 12 2 10 2 11 2 10 12 + 4 X1 X2 x + X2 x + X1 X2 x + X1 x + 4 X1 X2 x - 2 X1 x 2 11 12 2 9 10 2 10 11 - X2 x - 2 X2 x - 2 X1 X2 x - 6 X1 X2 x - 2 X2 x + 2 X2 x 12 9 10 2 9 10 11 8 + x + 3 X1 X2 x + 2 X1 x + 3 X2 x + 3 X2 x - x - X1 X2 x 9 2 8 9 10 7 8 8 9 - X1 x - X2 x - 5 X2 x - x - X1 X2 x + X1 x + 3 X2 x + 2 x 6 7 8 5 6 4 5 6 + X1 X2 x + X1 x - 2 x - 2 X1 X2 x - 2 X2 x + X1 X2 x + 3 X2 x + x 4 5 3 4 3 2 - 3 X2 x - x + X2 x + 2 x - x - 2 x + 3 x - 1) and in Maple format -(X1^2*X2^2*x^10-X1^2*X2*x^10-2*X1*X2^2*x^10+2*X1*X2^2*x^9+2*X1*X2*x^10+X2^2*x^ 10-3*X1*X2*x^9-2*X2^2*x^9-X2*x^10+X1*x^9+X2^2*x^8+3*X2*x^9-2*X2*x^8-x^9-X1*X2*x ^6+x^8+X1*X2*x^5+X2*x^6-X1*X2*x^4-2*X2*x^5+2*X2*x^4+x^5-X2*x^3-x^4+x^3+x^2-2*x+ 1)/(X1^2*X2^2*x^12+X1^2*X2^2*x^11-2*X1^2*X2*x^12-2*X1*X2^2*x^12-X1^2*X2^2*x^10-\ 2*X1^2*X2*x^11+X1^2*x^12+4*X1*X2*x^12+X2^2*x^12+X1^2*X2*x^10+X1^2*x^11+4*X1*X2^ 2*x^10-2*X1*x^12-X2^2*x^11-2*X2*x^12-2*X1*X2^2*x^9-6*X1*X2*x^10-2*X2^2*x^10+2* X2*x^11+x^12+3*X1*X2*x^9+2*X1*x^10+3*X2^2*x^9+3*X2*x^10-x^11-X1*X2*x^8-X1*x^9- X2^2*x^8-5*X2*x^9-x^10-X1*X2*x^7+X1*x^8+3*X2*x^8+2*x^9+X1*X2*x^6+X1*x^7-2*x^8-2 *X1*X2*x^5-2*X2*x^6+X1*X2*x^4+3*X2*x^5+x^6-3*X2*x^4-x^5+X2*x^3+2*x^4-x^3-2*x^2+ 3*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 2, 1], are n 183 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 2, 3], are n 89 77 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 20 3 77 ------------- 693 and in floating point 0.4386344635 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 20 231 ate normal pair with correlation, --------- 693 1/2 20 231 2879 i.e. , [[---------, 0], [0, ----]] 693 2079 ------------------------------------------------- Theorem Number, 248, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [4, 2, 1], nor the composition, [3, 3, 2] Then infinity ----- 7 6 4 3 2 \ n x - 2 x + x - x - x + 2 x - 1 ) a(n) x = - ------------------------------------------- / 7 6 5 4 3 ----- (-1 + x) (x - x + x + x - x + 2 x - 1) n = 0 and in Maple format -(x^7-2*x^6+x^4-x^3-x^2+2*x-1)/(-1+x)/(x^7-x^6+x^5+x^4-x^3+2*x-1) The asymptotic expression for a(n) is, n 0.64732625001158940098 1.9249743097683141877 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [4, 2, 1] and d occurrences (as containment) of the composition, [3, 3, 2], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 7 7 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x - X1 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 7 5 6 6 7 4 5 6 - X2 x - X1 X2 x + X1 x + X2 x + x + X1 X2 x + X2 x - 2 x 4 3 4 3 2 / 8 7 - 2 X2 x + X2 x + x - x - x + 2 x - 1) / (X1 X2 x - X1 X2 x / 8 8 7 7 8 5 6 6 7 - X1 x - X2 x + X1 x + 2 X2 x + x + 2 X1 X2 x - X1 x - X2 x - 2 x 4 5 6 4 3 4 3 2 - X1 X2 x - 2 X2 x + 2 x + 3 X2 x - X2 x - 2 x + x + 2 x - 3 x + 1 ) and in Maple format -(X1*X2*x^7-X1*x^7-X2*x^7-X1*X2*x^5+X1*x^6+X2*x^6+x^7+X1*X2*x^4+X2*x^5-2*x^6-2* X2*x^4+X2*x^3+x^4-x^3-x^2+2*x-1)/(X1*X2*x^8-X1*X2*x^7-X1*x^8-X2*x^8+X1*x^7+2*X2 *x^7+x^8+2*X1*X2*x^5-X1*x^6-X2*x^6-2*x^7-X1*X2*x^4-2*X2*x^5+2*x^6+3*X2*x^4-X2*x ^3-2*x^4+x^3+2*x^2-3*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 2, 1], are n 183 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 3, 2], are n 97 85 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 8 3 85 ------------ 255 and in floating point 0.5009794331 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 8 255 ate normal pair with correlation, -------- 255 1/2 8 255 383 i.e. , [[--------, 0], [0, ---]] 255 255 ------------------------------------------------- Theorem Number, 249, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [4, 2, 1], nor the composition, [3, 4, 1] Then infinity ----- 8 7 6 4 \ n x + x + x + x - x + 1 ) a(n) x = ----------------------------------------- / 7 6 5 3 2 ----- (-1 + x) (x + x + x + x + x + x - 1) n = 0 and in Maple format (x^8+x^7+x^6+x^4-x+1)/(-1+x)/(x^7+x^6+x^5+x^3+x^2+x-1) The asymptotic expression for a(n) is, n 0.65748289688935283811 1.9221280043552247737 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [4, 2, 1] and d occurrences (as containment) of the composition, [3, 4, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 9 9 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x - X1 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 9 7 9 7 5 6 4 6 5 - X2 x + X1 X2 x + x - X1 x - X1 X2 x + X1 x + X1 X2 x - x + x 4 2 / 8 8 8 6 - x - x + 2 x - 1) / ((-1 + x) (X1 X2 x - X1 x - X2 x + X1 X2 x / 8 5 6 4 5 4 + x + X1 X2 x - X1 x - X1 X2 x - x + x - 2 x + 1)) and in Maple format (X1*X2*x^9-X1*x^9-X2*x^9+X1*X2*x^7+x^9-X1*x^7-X1*X2*x^5+X1*x^6+X1*X2*x^4-x^6+x^ 5-x^4-x^2+2*x-1)/(-1+x)/(X1*X2*x^8-X1*x^8-X2*x^8+X1*X2*x^6+x^8+X1*X2*x^5-X1*x^6 -X1*X2*x^4-x^5+x^4-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 2, 1], are n 183 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 4, 1], are n 135 65 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 22 3 65 ------------- 585 and in floating point 0.5251509079 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 22 195 ate normal pair with correlation, --------- 585 1/2 22 195 2723 i.e. , [[---------, 0], [0, ----]] 585 1755 ------------------------------------------------- Theorem Number, 250, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [4, 2, 1], nor the composition, [4, 1, 3] Then infinity ----- 4 2 \ n x + x - 2 x + 1 ) a(n) x = --------------------------------- / 7 6 4 ----- (-1 + x) (x + x - x + 2 x - 1) n = 0 and in Maple format (x^4+x^2-2*x+1)/(-1+x)/(x^7+x^6-x^4+2*x-1) The asymptotic expression for a(n) is, n 0.67238389941451345857 1.9168163613635124211 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [4, 2, 1] and d occurrences (as containment) of the composition, [4, 1, 3], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 10 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 10 2 9 10 9 10 9 - X1 X2 x + X1 X2 x - X1 X2 x - 2 X1 X2 x + X1 x + X1 x 6 5 6 4 5 4 2 / - X1 X2 x + X1 X2 x + X2 x - X1 X2 x - X2 x + x + x - 2 x + 1) / / 2 2 12 2 2 11 2 12 2 2 10 2 11 (X1 X2 x + X1 X2 x - 2 X1 X2 x - X1 X2 x - 2 X1 X2 x 2 12 2 11 2 10 2 11 2 10 11 + X1 x + X1 X2 x + X1 X2 x + X1 x + X1 X2 x - 2 X1 X2 x 2 9 10 11 9 8 9 - X1 X2 x - X1 X2 x + X1 x + 2 X1 X2 x - X1 X2 x - X1 x 7 8 8 6 7 8 5 6 - X1 X2 x + X1 x + X2 x + X1 X2 x + X1 x - x - 2 X1 X2 x - 2 X2 x 4 5 6 5 4 2 + X1 X2 x + X2 x + x + x - x - 2 x + 3 x - 1) and in Maple format -(X1^2*X2^2*x^10-X1^2*X2*x^10+X1*X2^2*x^9-X1*X2*x^10-2*X1*X2*x^9+X1*x^10+X1*x^9 -X1*X2*x^6+X1*X2*x^5+X2*x^6-X1*X2*x^4-X2*x^5+x^4+x^2-2*x+1)/(X1^2*X2^2*x^12+X1^ 2*X2^2*x^11-2*X1^2*X2*x^12-X1^2*X2^2*x^10-2*X1^2*X2*x^11+X1^2*x^12+X1*X2^2*x^11 +X1^2*X2*x^10+X1^2*x^11+X1*X2^2*x^10-2*X1*X2*x^11-X1*X2^2*x^9-X1*X2*x^10+X1*x^ 11+2*X1*X2*x^9-X1*X2*x^8-X1*x^9-X1*X2*x^7+X1*x^8+X2*x^8+X1*X2*x^6+X1*x^7-x^8-2* X1*X2*x^5-2*X2*x^6+X1*X2*x^4+X2*x^5+x^6+x^5-x^4-2*x^2+3*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 2, 1], are n 183 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 1, 3], are n 141 65 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 16 3 65 ------------- 585 and in floating point 0.3819279330 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 16 195 ate normal pair with correlation, --------- 585 1/2 16 195 2267 i.e. , [[---------, 0], [0, ----]] 585 1755 ------------------------------------------------- Theorem Number, 251, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [4, 2, 1], nor the composition, [5, 1, 2] Then infinity ----- 7 6 5 4 2 \ n x - 2 x + x - x - x + 2 x - 1 ) a(n) x = - -------------------------------------- / 7 6 5 4 ----- (-1 + x) (x - x + x - x + 2 x - 1) n = 0 and in Maple format -(x^7-2*x^6+x^5-x^4-x^2+2*x-1)/(-1+x)/(x^7-x^6+x^5-x^4+2*x-1) The asymptotic expression for a(n) is, n 0.68788768175297946158 1.9132221246804735080 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [4, 2, 1] and d occurrences (as containment) of the composition, [5, 1, 2], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 11 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 11 11 7 6 7 7 5 - 2 X1 X2 x + X1 x - X1 X2 x + X1 X2 x + X1 x + X2 x - X1 X2 x 6 6 7 5 6 4 5 4 2 - 2 X1 x - X2 x - x + 2 X1 x + 2 x - X1 x - x + x + x - 2 x + 1) / 2 12 2 11 12 11 12 11 / (X1 X2 x - X1 X2 x - 2 X1 X2 x + 2 X1 X2 x + X1 x - X1 x / 8 7 8 8 6 7 7 8 - X1 X2 x + X1 X2 x + X1 x + X2 x - 2 X1 X2 x - X1 x - 2 X2 x - x 5 6 6 7 5 6 4 5 4 + X1 X2 x + 3 X1 x + X2 x + 2 x - 3 X1 x - 2 x + X1 x + 2 x - x 2 - 2 x + 3 x - 1) and in Maple format -(X1*X2^2*x^11-2*X1*X2*x^11+X1*x^11-X1*X2*x^7+X1*X2*x^6+X1*x^7+X2*x^7-X1*X2*x^5 -2*X1*x^6-X2*x^6-x^7+2*X1*x^5+2*x^6-X1*x^4-x^5+x^4+x^2-2*x+1)/(X1*X2^2*x^12-X1* X2^2*x^11-2*X1*X2*x^12+2*X1*X2*x^11+X1*x^12-X1*x^11-X1*X2*x^8+X1*X2*x^7+X1*x^8+ X2*x^8-2*X1*X2*x^6-X1*x^7-2*X2*x^7-x^8+X1*X2*x^5+3*X1*x^6+X2*x^6+2*x^7-3*X1*x^5 -2*x^6+X1*x^4+2*x^5-x^4-2*x^2+3*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 2, 1], are n 183 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [5, 1, 2], are n 113 57 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 4 19 ------- 57 and in floating point 0.3058876452 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 4 19 ate normal pair with correlation, ------- 57 1/2 4 19 203 i.e. , [[-------, 0], [0, ---]] 57 171 ------------------------------------------------- Theorem Number, 252, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [4, 2, 1], nor the composition, [6, 1, 1] Then infinity ----- 8 7 6 5 4 2 \ n x - x - x + x - x - x + 2 x - 1 ) a(n) x = ------------------------------------- / 6 5 3 2 2 ----- (x + x + x + x + x - 1) (-1 + x) n = 0 and in Maple format (x^8-x^7-x^6+x^5-x^4-x^2+2*x-1)/(x^6+x^5+x^3+x^2+x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.69790391550484586467 1.9111834366854762350 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [4, 2, 1] and d occurrences (as containment) of the composition, [6, 1, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 8 7 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x - X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 8 8 6 7 7 8 7 5 6 4 - X1 x - X2 x + X1 X2 x + X1 x + X2 x + x - x - X1 x - x + X1 x 5 4 2 / 7 6 7 + x - x - x + 2 x - 1) / ((-1 + x) (X1 X2 x - X1 X2 x - X1 x / 7 6 7 5 4 5 4 - X2 x + X1 x + x + X1 x - X1 x - x + x - 2 x + 1)) and in Maple format (X1*X2*x^8-X1*X2*x^7-X1*x^8-X2*x^8+X1*X2*x^6+X1*x^7+X2*x^7+x^8-x^7-X1*x^5-x^6+ X1*x^4+x^5-x^4-x^2+2*x-1)/(-1+x)/(X1*X2*x^7-X1*X2*x^6-X1*x^7-X2*x^7+X1*x^6+x^7+ X1*x^5-X1*x^4-x^5+x^4-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 2, 1], are n 183 27 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [6, 1, 1], are n 49 53 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 8 3 53 ------------ 477 and in floating point 0.2114804230 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 8 159 ate normal pair with correlation, -------- 477 1/2 8 159 1559 i.e. , [[--------, 0], [0, ----]] 477 1431 ------------------------------------------------- Theorem Number, 253, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [5, 1, 1], nor the composition, [1, 1, 6] Then infinity ----- \ n 20 19 18 17 16 15 14 13 ) a(n) x = - (x + 2 x + 3 x + 4 x + 3 x + x - x - 3 x / ----- n = 0 12 11 10 9 8 5 2 / - 4 x - 4 x - 3 x - 2 x - x + x + x - 2 x + 1) / ( / 4 3 2 2 (x + x + x + x - 1) (-1 + x) ) and in Maple format -(x^20+2*x^19+3*x^18+4*x^17+3*x^16+x^15-x^14-3*x^13-4*x^12-4*x^11-3*x^10-2*x^9- x^8+x^5+x^2-2*x+1)/(x^4+x^3+x^2+x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.60970919018057873430 1.9275619754829253043 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [5, 1, 1] and d occurrences (as containment) of the composition, [1, 1, 6], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 21 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 2 20 2 21 2 21 2 2 19 2 20 + X1 X2 x - 2 X1 X2 x - 2 X1 X2 x + X1 X2 x - 2 X1 X2 x 2 21 2 20 21 2 21 2 2 18 + X1 x - 2 X1 X2 x + 4 X1 X2 x + X2 x + X1 X2 x 2 19 2 20 2 19 20 21 2 20 - 2 X1 X2 x + X1 x - 2 X1 X2 x + 4 X1 X2 x - 2 X1 x + X2 x 21 2 18 2 19 2 18 19 - 2 X2 x - 2 X1 X2 x + X1 x - 2 X1 X2 x + 4 X1 X2 x 20 2 19 20 21 2 18 2 17 18 - 2 X1 x + X2 x - 2 X2 x + x + X1 x + X1 X2 x + 4 X1 X2 x 19 2 18 19 20 2 16 2 16 - 2 X1 x + X2 x - 2 X2 x + x + X1 X2 x + X1 X2 x 17 18 2 17 18 19 2 15 2 16 - 2 X1 X2 x - 2 X1 x - X2 x - 2 X2 x + x + X1 X2 x - X1 x 2 15 16 17 2 16 17 18 + X1 X2 x - 4 X1 X2 x + X1 x - X2 x + 2 X2 x + x 2 14 2 15 2 14 15 16 2 15 + X1 X2 x - X1 x + X1 X2 x - 4 X1 X2 x + 3 X1 x - X2 x 16 17 2 13 2 14 14 15 2 14 + 3 X2 x - x + X1 X2 x - X1 x - 4 X1 X2 x + 3 X1 x - X2 x 15 16 2 13 13 14 14 15 + 3 X2 x - 2 x - X1 x - 2 X1 X2 x + 3 X1 x + 3 X2 x - 2 x 13 13 14 11 13 10 11 + 2 X1 x + X2 x - 2 x + X1 X2 x - x + X1 X2 x - X1 x 11 9 10 10 11 8 9 9 - X2 x + X1 X2 x - X1 x - X2 x + x + X1 X2 x - X1 x - X2 x 10 7 8 8 9 6 7 8 6 + x - X1 X2 x - X1 x - X2 x + x + X1 X2 x + X1 x + x - 2 X1 x 5 6 5 3 2 / 2 + X1 x + x - x + x - 3 x + 3 x - 1) / ((-1 + x) / 6 6 5 5 (X1 X2 x - X1 x + X1 x - x + 2 x - 1)) and in Maple format (X1^2*X2^2*x^21+X1^2*X2^2*x^20-2*X1^2*X2*x^21-2*X1*X2^2*x^21+X1^2*X2^2*x^19-2* X1^2*X2*x^20+X1^2*x^21-2*X1*X2^2*x^20+4*X1*X2*x^21+X2^2*x^21+X1^2*X2^2*x^18-2* X1^2*X2*x^19+X1^2*x^20-2*X1*X2^2*x^19+4*X1*X2*x^20-2*X1*x^21+X2^2*x^20-2*X2*x^ 21-2*X1^2*X2*x^18+X1^2*x^19-2*X1*X2^2*x^18+4*X1*X2*x^19-2*X1*x^20+X2^2*x^19-2* X2*x^20+x^21+X1^2*x^18+X1*X2^2*x^17+4*X1*X2*x^18-2*X1*x^19+X2^2*x^18-2*X2*x^19+ x^20+X1^2*X2*x^16+X1*X2^2*x^16-2*X1*X2*x^17-2*X1*x^18-X2^2*x^17-2*X2*x^18+x^19+ X1^2*X2*x^15-X1^2*x^16+X1*X2^2*x^15-4*X1*X2*x^16+X1*x^17-X2^2*x^16+2*X2*x^17+x^ 18+X1^2*X2*x^14-X1^2*x^15+X1*X2^2*x^14-4*X1*X2*x^15+3*X1*x^16-X2^2*x^15+3*X2*x^ 16-x^17+X1^2*X2*x^13-X1^2*x^14-4*X1*X2*x^14+3*X1*x^15-X2^2*x^14+3*X2*x^15-2*x^ 16-X1^2*x^13-2*X1*X2*x^13+3*X1*x^14+3*X2*x^14-2*x^15+2*X1*x^13+X2*x^13-2*x^14+ X1*X2*x^11-x^13+X1*X2*x^10-X1*x^11-X2*x^11+X1*X2*x^9-X1*x^10-X2*x^10+x^11+X1*X2 *x^8-X1*x^9-X2*x^9+x^10-X1*X2*x^7-X1*x^8-X2*x^8+x^9+X1*X2*x^6+X1*x^7+x^8-2*X1*x ^6+X1*x^5+x^6-x^5+x^3-3*x^2+3*x-1)/(-1+x)^2/(X1*X2*x^6-X1*x^6+X1*x^5-x^5+2*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [5, 1, 1], are n 139 23 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 1, 6], are n 49 53 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 22 23 53 -------------- 1219 and in floating point 0.6301164650 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 22 1219 ate normal pair with correlation, ---------- 1219 1/2 22 1219 2187 i.e. , [[----------, 0], [0, ----]] 1219 1219 ------------------------------------------------- Theorem Number, 254, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [5, 1, 1], nor the composition, [1, 2, 5] Then infinity ----- \ n 3 15 14 13 12 11 10 ) a(n) x = - (x + x - 1) (x + 2 x + 2 x + 2 x + x - x / ----- n = 0 9 8 7 6 5 3 / 4 3 2 - 2 x - 2 x - 2 x - 2 x - x - x + x - 1) / ((x + x + x + x - 1) / 2 (-1 + x) ) and in Maple format -(x^3+x-1)*(x^15+2*x^14+2*x^13+2*x^12+x^11-x^10-2*x^9-2*x^8-2*x^7-2*x^6-x^5-x^3 +x-1)/(x^4+x^3+x^2+x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.61223359561691903626 1.9275619754829253043 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [5, 1, 1] and d occurrences (as containment) of the composition, [1, 2, 5], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 19 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 2 18 2 19 2 19 2 2 17 2 18 + X1 X2 x - 2 X1 X2 x - 2 X1 X2 x + X1 X2 x - 2 X1 X2 x 2 19 2 18 19 2 19 2 17 2 18 + X1 x - 2 X1 X2 x + 4 X1 X2 x + X2 x - 2 X1 X2 x + X1 x 2 17 18 19 2 18 19 2 17 - 2 X1 X2 x + 4 X1 X2 x - 2 X1 x + X2 x - 2 X2 x + X1 x 17 18 2 17 18 19 2 15 + 4 X1 X2 x - 2 X1 x + X2 x - 2 X2 x + x + X1 X2 x 2 15 17 17 18 2 14 2 15 + X1 X2 x - 2 X1 x - 2 X2 x + x + X1 X2 x - X1 x 2 14 15 2 15 17 2 13 2 14 + X1 X2 x - 4 X1 X2 x - X2 x + x + X1 X2 x - X1 x 2 13 14 15 2 14 15 2 12 + X1 X2 x - 4 X1 X2 x + 3 X1 x - X2 x + 3 X2 x + X1 X2 x 2 13 13 14 2 13 14 15 2 12 - X1 x - 4 X1 X2 x + 3 X1 x - X2 x + 3 X2 x - 2 x - X1 x 12 13 13 14 11 12 - 2 X1 X2 x + 3 X1 x + 3 X2 x - 2 x + X1 X2 x + 2 X1 x 12 13 10 11 11 12 9 10 + X2 x - 2 x + X1 X2 x - X1 x - X2 x - x + X1 X2 x - X1 x 10 11 9 9 10 7 8 9 - X2 x + x - X1 x - X2 x + x + 2 X1 X2 x - X2 x + x 6 7 8 5 6 6 5 3 2 - 2 X1 X2 x - 2 X1 x + x + X1 X2 x + X1 x + x - x + x - 3 x + 3 x / 2 6 5 6 5 - 1) / ((-1 + x) (X1 X2 x - X1 X2 x - X1 x + x - 2 x + 1)) / and in Maple format -(X1^2*X2^2*x^19+X1^2*X2^2*x^18-2*X1^2*X2*x^19-2*X1*X2^2*x^19+X1^2*X2^2*x^17-2* X1^2*X2*x^18+X1^2*x^19-2*X1*X2^2*x^18+4*X1*X2*x^19+X2^2*x^19-2*X1^2*X2*x^17+X1^ 2*x^18-2*X1*X2^2*x^17+4*X1*X2*x^18-2*X1*x^19+X2^2*x^18-2*X2*x^19+X1^2*x^17+4*X1 *X2*x^17-2*X1*x^18+X2^2*x^17-2*X2*x^18+x^19+X1^2*X2*x^15+X1*X2^2*x^15-2*X1*x^17 -2*X2*x^17+x^18+X1^2*X2*x^14-X1^2*x^15+X1*X2^2*x^14-4*X1*X2*x^15-X2^2*x^15+x^17 +X1^2*X2*x^13-X1^2*x^14+X1*X2^2*x^13-4*X1*X2*x^14+3*X1*x^15-X2^2*x^14+3*X2*x^15 +X1^2*X2*x^12-X1^2*x^13-4*X1*X2*x^13+3*X1*x^14-X2^2*x^13+3*X2*x^14-2*x^15-X1^2* x^12-2*X1*X2*x^12+3*X1*x^13+3*X2*x^13-2*x^14+X1*X2*x^11+2*X1*x^12+X2*x^12-2*x^ 13+X1*X2*x^10-X1*x^11-X2*x^11-x^12+X1*X2*x^9-X1*x^10-X2*x^10+x^11-X1*x^9-X2*x^9 +x^10+2*X1*X2*x^7-X2*x^8+x^9-2*X1*X2*x^6-2*X1*x^7+x^8+X1*X2*x^5+X1*x^6+x^6-x^5+ x^3-3*x^2+3*x-1)/(-1+x)^2/(X1*X2*x^6-X1*X2*x^5-X1*x^6+x^5-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [5, 1, 1], are n 139 23 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 2, 5], are n 111 57 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 8 23 57 ------------- 437 and in floating point 0.6628418121 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 1/2 8 3 437 ate normal pair with correlation, ------------- 437 1/2 1/2 8 3 437 821 i.e. , [[-------------, 0], [0, ---]] 437 437 ------------------------------------------------- Theorem Number, 255, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [5, 1, 1], nor the composition, [1, 3, 4] Then infinity ----- \ n ) a(n) x = / ----- n = 0 3 2 13 11 9 7 6 4 3 2 (x + x - 1) (x + x - 2 x - 2 x - x - x + x - 2 x + 2 x - 1) - ----------------------------------------------------------------------- 6 5 3 2 2 (x + x + x + x + x - 1) (-1 + x) and in Maple format -(x^3+x^2-1)*(x^13+x^11-2*x^9-2*x^7-x^6-x^4+x^3-2*x^2+2*x-1)/(x^6+x^5+x^3+x^2+x -1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.67598850878507360158 1.9111834366854762350 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [5, 1, 1] and d occurrences (as containment) of the composition, [1, 3, 4], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 17 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 17 2 17 2 17 17 2 17 17 - 2 X1 X2 x - 2 X1 X2 x + X1 x + 4 X1 X2 x + X2 x - 2 X1 x 17 2 14 17 2 13 2 14 2 13 - 2 X2 x + X1 X2 x + x + X1 X2 x - X1 x + X1 X2 x 14 2 12 2 13 13 14 2 13 - 2 X1 X2 x + X1 X2 x - X1 x - 4 X1 X2 x + 2 X1 x - X2 x 14 2 12 12 13 13 14 11 + X2 x - X1 x - 2 X1 X2 x + 3 X1 x + 3 X2 x - x + X1 X2 x 12 12 13 10 11 11 12 10 + 2 X1 x + X2 x - 2 x + X1 X2 x - X1 x - X2 x - x - X1 x 10 11 8 10 7 8 8 6 - X2 x + x + X1 X2 x + x + X1 X2 x - X1 x - 2 X2 x - 2 X1 X2 x 7 8 5 6 6 5 6 4 - X1 x + 2 x + X1 X2 x + X1 x + 3 X2 x - 3 X2 x - 2 x + X2 x 5 4 3 2 / 2 7 6 + 2 x - x + x - 3 x + 3 x - 1) / ((-1 + x) (X1 X2 x + X1 X2 x / 7 7 5 6 7 5 4 5 4 - X1 x - X2 x - X1 X2 x - X1 x + x + 2 X2 x - X2 x - x + x - 2 x + 1)) and in Maple format -(X1^2*X2^2*x^17-2*X1^2*X2*x^17-2*X1*X2^2*x^17+X1^2*x^17+4*X1*X2*x^17+X2^2*x^17 -2*X1*x^17-2*X2*x^17+X1^2*X2*x^14+x^17+X1^2*X2*x^13-X1^2*x^14+X1*X2^2*x^13-2*X1 *X2*x^14+X1^2*X2*x^12-X1^2*x^13-4*X1*X2*x^13+2*X1*x^14-X2^2*x^13+X2*x^14-X1^2*x ^12-2*X1*X2*x^12+3*X1*x^13+3*X2*x^13-x^14+X1*X2*x^11+2*X1*x^12+X2*x^12-2*x^13+ X1*X2*x^10-X1*x^11-X2*x^11-x^12-X1*x^10-X2*x^10+x^11+X1*X2*x^8+x^10+X1*X2*x^7- X1*x^8-2*X2*x^8-2*X1*X2*x^6-X1*x^7+2*x^8+X1*X2*x^5+X1*x^6+3*X2*x^6-3*X2*x^5-2*x ^6+X2*x^4+2*x^5-x^4+x^3-3*x^2+3*x-1)/(-1+x)^2/(X1*X2*x^7+X1*X2*x^6-X1*x^7-X2*x^ 7-X1*X2*x^5-X1*x^6+x^7+2*X2*x^5-X2*x^4-x^5+x^4-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [5, 1, 1], are n 139 23 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 3, 4], are n 135 65 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 12 23 65 -------------- 1495 and in floating point 0.3103563600 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 12 1495 ate normal pair with correlation, ---------- 1495 1/2 12 1495 1783 i.e. , [[----------, 0], [0, ----]] 1495 1495 ------------------------------------------------- Theorem Number, 256, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [5, 1, 1], nor the composition, [1, 4, 3] Then infinity ----- \ n ) a(n) x = / ----- n = 0 12 11 10 9 8 7 6 5 4 2 x + 2 x + x + x + x - x - x + x - x - x + 2 x - 1 -------------------------------------------------------------- 6 5 3 2 2 (x + x + x + x + x - 1) (-1 + x) and in Maple format (x^12+2*x^11+x^10+x^9+x^8-x^7-x^6+x^5-x^4-x^2+2*x-1)/(x^6+x^5+x^3+x^2+x-1)/(-1+ x)^2 The asymptotic expression for a(n) is, n 0.68216970258446089521 1.9111834366854762350 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [5, 1, 1] and d occurrences (as containment) of the composition, [1, 4, 3], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 13 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 12 2 13 13 2 12 12 13 + X1 X2 x - X1 x - 2 X1 X2 x - X1 x - 2 X1 X2 x + 2 X1 x 13 11 12 12 13 11 11 12 + X2 x + X1 X2 x + 2 X1 x + X2 x - x - X1 x - X2 x - x 11 8 7 8 8 6 7 8 + x + X1 X2 x + X1 X2 x - X1 x - 2 X2 x - 2 X1 X2 x - X1 x + 2 x 5 6 6 5 6 4 5 4 3 + X1 X2 x + X1 x + 3 X2 x - 3 X2 x - 2 x + X2 x + 2 x - x + x 2 / 2 7 6 7 7 - 3 x + 3 x - 1) / ((-1 + x) (X1 X2 x + X1 X2 x - X1 x - X2 x / 5 6 7 5 4 5 4 - X1 X2 x - X1 x + x + 2 X2 x - X2 x - x + x - 2 x + 1)) and in Maple format -(X1^2*X2*x^13+X1^2*X2*x^12-X1^2*x^13-2*X1*X2*x^13-X1^2*x^12-2*X1*X2*x^12+2*X1* x^13+X2*x^13+X1*X2*x^11+2*X1*x^12+X2*x^12-x^13-X1*x^11-X2*x^11-x^12+x^11+X1*X2* x^8+X1*X2*x^7-X1*x^8-2*X2*x^8-2*X1*X2*x^6-X1*x^7+2*x^8+X1*X2*x^5+X1*x^6+3*X2*x^ 6-3*X2*x^5-2*x^6+X2*x^4+2*x^5-x^4+x^3-3*x^2+3*x-1)/(-1+x)^2/(X1*X2*x^7+X1*X2*x^ 6-X1*x^7-X2*x^7-X1*X2*x^5-X1*x^6+x^7+2*X2*x^5-X2*x^4-x^5+x^4-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [5, 1, 1], are n 139 23 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 4, 3], are n 135 65 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 12 23 65 -------------- 1495 and in floating point 0.3103563600 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 12 1495 ate normal pair with correlation, ---------- 1495 1/2 12 1495 1783 i.e. , [[----------, 0], [0, ----]] 1495 1495 ------------------------------------------------- Theorem Number, 257, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [5, 1, 1], nor the composition, [1, 5, 2] Then infinity ----- 11 10 9 8 5 2 \ n x + x + x + x - x - x + 2 x - 1 ) a(n) x = --------------------------------------- / 4 3 2 2 ----- (x + x + x + x - 1) (-1 + x) n = 0 and in Maple format (x^11+x^10+x^9+x^8-x^5-x^2+2*x-1)/(x^4+x^3+x^2+x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.63351683346155297547 1.9275619754829253043 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [5, 1, 1] and d occurrences (as containment) of the composition, [1, 5, 2], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 12 2 12 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x - X1 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 12 12 12 12 7 8 6 - 2 X1 X2 x + 2 X1 x + X2 x - x + 2 X1 X2 x - X2 x - 2 X1 X2 x 7 8 5 6 6 5 3 2 / - 2 X1 x + x + X1 X2 x + X1 x + x - x + x - 3 x + 3 x - 1) / ( / 2 6 5 6 5 (-1 + x) (X1 X2 x - X1 X2 x - X1 x + x - 2 x + 1)) and in Maple format -(X1^2*X2*x^12-X1^2*x^12-2*X1*X2*x^12+2*X1*x^12+X2*x^12-x^12+2*X1*X2*x^7-X2*x^8 -2*X1*X2*x^6-2*X1*x^7+x^8+X1*X2*x^5+X1*x^6+x^6-x^5+x^3-3*x^2+3*x-1)/(-1+x)^2/( X1*X2*x^6-X1*X2*x^5-X1*x^6+x^5-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [5, 1, 1], are n 139 23 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 5, 2], are n 111 57 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 8 23 57 ------------- 437 and in floating point 0.6628418121 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 1/2 8 3 437 ate normal pair with correlation, ------------- 437 1/2 1/2 8 3 437 821 i.e. , [[-------------, 0], [0, ---]] 437 437 ------------------------------------------------- Theorem Number, 258, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [5, 1, 1], nor the composition, [1, 6, 1] Then infinity ----- 11 10 9 8 5 2 \ n x + x + x + x - x - x + 2 x - 1 ) a(n) x = --------------------------------------- / 4 3 2 2 ----- (x + x + x + x - 1) (-1 + x) n = 0 and in Maple format (x^11+x^10+x^9+x^8-x^5-x^2+2*x-1)/(x^4+x^3+x^2+x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.63351683346155297547 1.9275619754829253043 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [5, 1, 1] and d occurrences (as containment) of the composition, [1, 6, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 11 10 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x + X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 11 11 9 10 10 11 8 9 - X1 x - X2 x + X1 X2 x - X1 x - X2 x + x + X1 X2 x - X1 x 9 10 8 8 9 6 8 6 5 5 - X2 x + x - X1 x - X2 x + x + X1 X2 x + x - X1 x + X1 x - x 2 / 6 6 5 5 - x + 2 x - 1) / ((-1 + x) (X1 X2 x - X1 x + X1 x - x + 2 x - 1)) / and in Maple format -(X1*X2*x^11+X1*X2*x^10-X1*x^11-X2*x^11+X1*X2*x^9-X1*x^10-X2*x^10+x^11+X1*X2*x^ 8-X1*x^9-X2*x^9+x^10-X1*x^8-X2*x^8+x^9+X1*X2*x^6+x^8-X1*x^6+X1*x^5-x^5-x^2+2*x-\ 1)/(-1+x)/(X1*X2*x^6-X1*x^6+X1*x^5-x^5+2*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [5, 1, 1], are n 139 23 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 6, 1], are n 49 53 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 22 23 53 -------------- 1219 and in floating point 0.6301164650 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 22 1219 ate normal pair with correlation, ---------- 1219 1/2 22 1219 2187 i.e. , [[----------, 0], [0, ----]] 1219 1219 ------------------------------------------------- Theorem Number, 259, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [5, 1, 1], nor the composition, [2, 1, 5] Then infinity ----- \ n 3 15 14 13 12 10 9 ) a(n) x = - (x + x - 1) (x + 2 x + 2 x + x - x - 2 x / ----- n = 0 8 7 6 5 3 / 4 3 2 - 2 x - 2 x - 2 x - x - x + x - 1) / ((x + x + x + x - 1) / 2 (-1 + x) ) and in Maple format -(x^3+x-1)*(x^15+2*x^14+2*x^13+x^12-x^10-2*x^9-2*x^8-2*x^7-2*x^6-x^5-x^3+x-1)/( x^4+x^3+x^2+x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.61316322549247699946 1.9275619754829253043 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [5, 1, 1] and d occurrences (as containment) of the composition, [2, 1, 5], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 19 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 2 18 2 19 2 19 2 2 17 2 18 + X1 X2 x - 2 X1 X2 x - 2 X1 X2 x + X1 X2 x - 2 X1 X2 x 2 19 2 18 19 2 19 2 17 2 18 + X1 x - 2 X1 X2 x + 4 X1 X2 x + X2 x - 2 X1 X2 x + X1 x 2 17 18 19 2 18 19 2 2 15 - 2 X1 X2 x + 4 X1 X2 x - 2 X1 x + X2 x - 2 X2 x - X1 X2 x 2 17 2 16 17 18 2 17 18 19 + X1 x + X1 X2 x + 4 X1 X2 x - 2 X1 x + X2 x - 2 X2 x + x 2 2 14 2 15 2 15 16 17 - X1 X2 x + 3 X1 X2 x + 2 X1 X2 x - 2 X1 X2 x - 2 X1 x 2 16 17 18 2 2 13 2 14 2 15 - X2 x - 2 X2 x + x - X1 X2 x + 3 X1 X2 x - 2 X1 x 2 14 15 16 2 15 16 17 + 2 X1 X2 x - 6 X1 X2 x + X1 x - X2 x + 2 X2 x + x 2 2 12 2 13 2 14 2 13 14 + X1 X2 x + 3 X1 X2 x - 2 X1 x + X1 X2 x - 6 X1 X2 x 15 2 14 15 16 2 2 11 2 12 + 4 X1 x - X2 x + 3 X2 x - x - X1 X2 x - X1 X2 x 2 13 13 14 14 15 2 11 - 2 X1 x - 4 X1 X2 x + 4 X1 x + 3 X2 x - 2 x + X1 X2 x 12 13 13 14 11 12 13 - X1 X2 x + 3 X1 x + X2 x - 2 x + X1 X2 x + X1 x - x 10 11 9 10 10 8 9 + X1 X2 x - X1 x + X1 X2 x - X1 x - X2 x + X1 X2 x - X1 x 9 10 8 8 9 6 8 5 6 5 - X2 x + x - X1 x - X2 x + x - X1 X2 x + x + X1 X2 x + x - x 3 2 / + x - 3 x + 3 x - 1) / ((-1 + x) / 6 5 6 5 6 6 (X1 X2 x - X1 X2 x - X1 x + x - 2 x + 1) (X1 X2 x - X1 x + x - 1)) and in Maple format -(X1^2*X2^2*x^19+X1^2*X2^2*x^18-2*X1^2*X2*x^19-2*X1*X2^2*x^19+X1^2*X2^2*x^17-2* X1^2*X2*x^18+X1^2*x^19-2*X1*X2^2*x^18+4*X1*X2*x^19+X2^2*x^19-2*X1^2*X2*x^17+X1^ 2*x^18-2*X1*X2^2*x^17+4*X1*X2*x^18-2*X1*x^19+X2^2*x^18-2*X2*x^19-X1^2*X2^2*x^15 +X1^2*x^17+X1*X2^2*x^16+4*X1*X2*x^17-2*X1*x^18+X2^2*x^17-2*X2*x^18+x^19-X1^2*X2 ^2*x^14+3*X1^2*X2*x^15+2*X1*X2^2*x^15-2*X1*X2*x^16-2*X1*x^17-X2^2*x^16-2*X2*x^ 17+x^18-X1^2*X2^2*x^13+3*X1^2*X2*x^14-2*X1^2*x^15+2*X1*X2^2*x^14-6*X1*X2*x^15+ X1*x^16-X2^2*x^15+2*X2*x^16+x^17+X1^2*X2^2*x^12+3*X1^2*X2*x^13-2*X1^2*x^14+X1* X2^2*x^13-6*X1*X2*x^14+4*X1*x^15-X2^2*x^14+3*X2*x^15-x^16-X1^2*X2^2*x^11-X1^2* X2*x^12-2*X1^2*x^13-4*X1*X2*x^13+4*X1*x^14+3*X2*x^14-2*x^15+X1^2*X2*x^11-X1*X2* x^12+3*X1*x^13+X2*x^13-2*x^14+X1*X2*x^11+X1*x^12-x^13+X1*X2*x^10-X1*x^11+X1*X2* x^9-X1*x^10-X2*x^10+X1*X2*x^8-X1*x^9-X2*x^9+x^10-X1*x^8-X2*x^8+x^9-X1*X2*x^6+x^ 8+X1*X2*x^5+x^6-x^5+x^3-3*x^2+3*x-1)/(-1+x)/(X1*X2*x^6-X1*X2*x^5-X1*x^6+x^5-2*x +1)/(X1*X2*x^6-X1*x^6+x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [5, 1, 1], are n 139 23 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 1, 5], are n 113 57 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 8 23 57 ------------- 437 and in floating point 0.6628418121 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 1/2 8 3 437 ate normal pair with correlation, ------------- 437 1/2 1/2 8 3 437 821 i.e. , [[-------------, 0], [0, ---]] 437 437 ------------------------------------------------- Theorem Number, 260, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [5, 1, 1], nor the composition, [2, 2, 4] Then infinity ----- \ n 16 15 14 12 11 10 9 7 ) a(n) x = - (x + 2 x + 2 x - 2 x - 3 x - x - x - x / ----- n = 0 6 5 4 2 / + 2 x - x + x + x - 2 x + 1) / ( / 9 8 7 5 3 2 2 (x + x + x + x + x + x + x - 1) (-1 + x) ) and in Maple format -(x^16+2*x^15+2*x^14-2*x^12-3*x^11-x^10-x^9-x^7+2*x^6-x^5+x^4+x^2-2*x+1)/(x^9+x ^8+x^7+x^5+x^3+x^2+x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.68190830405220103442 1.9099139227361088013 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [5, 1, 1] and d occurrences (as containment) of the composition, [2, 2, 4], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 17 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 2 16 2 17 2 17 2 16 2 17 + X1 X2 x - 2 X1 X2 x - 2 X1 X2 x - 2 X1 X2 x + X1 x 2 16 17 2 17 2 16 16 17 - 2 X1 X2 x + 4 X1 X2 x + X2 x + X1 x + 4 X1 X2 x - 2 X1 x 2 16 17 2 14 2 14 16 16 17 + X2 x - 2 X2 x + X1 X2 x + X1 X2 x - 2 X1 x - 2 X2 x + x 2 13 2 14 2 13 14 2 14 16 + X1 X2 x - X1 x + X1 X2 x - 4 X1 X2 x - X2 x + x 2 12 2 13 13 14 2 13 14 + X1 X2 x - X1 x - 4 X1 X2 x + 3 X1 x - X2 x + 3 X2 x 2 11 2 12 12 13 13 14 - X1 X2 x - X1 x - 2 X1 X2 x + 3 X1 x + 3 X2 x - 2 x 2 11 11 12 12 13 11 11 + X1 x + 3 X1 X2 x + 2 X1 x + X2 x - 2 x - 3 X1 x - 2 X2 x 12 9 11 8 9 9 7 8 - x + X1 X2 x + 2 x - X1 X2 x - X1 x - X2 x + 2 X1 X2 x + X1 x 8 9 6 7 7 8 5 6 + X2 x + x - 2 X1 X2 x - 2 X1 x - 3 X2 x - x + X1 X2 x + X1 x 6 7 5 6 4 5 4 3 2 + 4 X2 x + 3 x - 3 X2 x - 3 x + X2 x + 2 x - x + x - 3 x + 3 x / 2 11 2 11 11 10 - 1) / ((-1 + x) (X1 X2 x - X1 x - 2 X1 X2 x + X1 X2 x / 11 11 10 10 11 8 10 7 + 2 X1 x + X2 x - X1 x - X2 x - x + X1 X2 x + x - X1 X2 x 8 8 6 7 7 8 5 6 - X1 x - X2 x + 2 X1 X2 x + X1 x + 2 X2 x + x - X1 X2 x - X1 x 6 7 5 6 4 5 4 2 - 3 X2 x - 2 x + 3 X2 x + 2 x - X2 x - 2 x + x + 2 x - 3 x + 1)) and in Maple format (X1^2*X2^2*x^17+X1^2*X2^2*x^16-2*X1^2*X2*x^17-2*X1*X2^2*x^17-2*X1^2*X2*x^16+X1^ 2*x^17-2*X1*X2^2*x^16+4*X1*X2*x^17+X2^2*x^17+X1^2*x^16+4*X1*X2*x^16-2*X1*x^17+ X2^2*x^16-2*X2*x^17+X1^2*X2*x^14+X1*X2^2*x^14-2*X1*x^16-2*X2*x^16+x^17+X1^2*X2* x^13-X1^2*x^14+X1*X2^2*x^13-4*X1*X2*x^14-X2^2*x^14+x^16+X1^2*X2*x^12-X1^2*x^13-\ 4*X1*X2*x^13+3*X1*x^14-X2^2*x^13+3*X2*x^14-X1^2*X2*x^11-X1^2*x^12-2*X1*X2*x^12+ 3*X1*x^13+3*X2*x^13-2*x^14+X1^2*x^11+3*X1*X2*x^11+2*X1*x^12+X2*x^12-2*x^13-3*X1 *x^11-2*X2*x^11-x^12+X1*X2*x^9+2*x^11-X1*X2*x^8-X1*x^9-X2*x^9+2*X1*X2*x^7+X1*x^ 8+X2*x^8+x^9-2*X1*X2*x^6-2*X1*x^7-3*X2*x^7-x^8+X1*X2*x^5+X1*x^6+4*X2*x^6+3*x^7-\ 3*X2*x^5-3*x^6+X2*x^4+2*x^5-x^4+x^3-3*x^2+3*x-1)/(-1+x)/(X1^2*X2*x^11-X1^2*x^11 -2*X1*X2*x^11+X1*X2*x^10+2*X1*x^11+X2*x^11-X1*x^10-X2*x^10-x^11+X1*X2*x^8+x^10- X1*X2*x^7-X1*x^8-X2*x^8+2*X1*X2*x^6+X1*x^7+2*X2*x^7+x^8-X1*X2*x^5-X1*x^6-3*X2*x ^6-2*x^7+3*X2*x^5+2*x^6-X2*x^4-2*x^5+x^4+2*x^2-3*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [5, 1, 1], are n 139 23 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 2, 4], are n 75 69 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 10 3 ------- 69 and in floating point 0.2510218562 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 10 3 ate normal pair with correlation, ------- 69 1/2 10 3 1787 i.e. , [[-------, 0], [0, ----]] 69 1587 ------------------------------------------------- Theorem Number, 261, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [5, 1, 1], nor the composition, [2, 3, 3] Then infinity ----- 12 11 10 9 8 6 4 3 2 \ n x + 2 x + x - x + x - 2 x + x - x - x + 2 x - 1 ) a(n) x = ----------------------------------------------------------- / 9 8 7 6 5 4 2 2 ----- (x + 2 x + x + x + 2 x + x + x + x - 1) (-1 + x) n = 0 and in Maple format (x^12+2*x^11+x^10-x^9+x^8-2*x^6+x^4-x^3-x^2+2*x-1)/(x^9+2*x^8+x^7+x^6+2*x^5+x^4 +x^2+x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.69729201967249542857 1.9073680513412163549 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [5, 1, 1] and d occurrences (as containment) of the composition, [2, 3, 3], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 13 2 12 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x + X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 13 13 2 11 2 12 12 13 - X1 x - 2 X1 X2 x - X1 X2 x - X1 x - 2 X1 X2 x + 2 X1 x 13 2 11 11 12 12 13 10 + X2 x + X1 x + 2 X1 X2 x + 2 X1 x + X2 x - x + 2 X1 X2 x 11 11 12 9 10 10 11 - 2 X1 x - X2 x - x - 2 X1 X2 x - 2 X1 x - 2 X2 x + x 8 9 9 10 7 8 8 9 + X1 X2 x + 2 X1 x + 2 X2 x + 2 x + X1 X2 x - X1 x - X2 x - 2 x 6 7 7 8 5 6 6 7 - 2 X1 X2 x - X1 x - 2 X2 x + x + X1 X2 x + X1 x + 3 X2 x + 2 x 6 4 5 3 4 2 / - 2 x - 2 X2 x - x + X2 x + 2 x - 3 x + 3 x - 1) / ((-1 + x) ( / 2 11 2 11 11 11 11 9 11 X1 X2 x - X1 x - 2 X1 X2 x + 2 X1 x + X2 x + 2 X1 X2 x - x 8 9 9 8 8 9 6 7 - X1 X2 x - 2 X1 x - 2 X2 x + X1 x + X2 x + 2 x + 2 X1 X2 x + X2 x 8 5 6 6 7 5 6 4 3 - x - X1 X2 x - X1 x - 3 X2 x - x + X2 x + 2 x + 2 X2 x - X2 x 4 3 2 - 2 x + x + 2 x - 3 x + 1)) and in Maple format (X1^2*X2*x^13+X1^2*X2*x^12-X1^2*x^13-2*X1*X2*x^13-X1^2*X2*x^11-X1^2*x^12-2*X1* X2*x^12+2*X1*x^13+X2*x^13+X1^2*x^11+2*X1*X2*x^11+2*X1*x^12+X2*x^12-x^13+2*X1*X2 *x^10-2*X1*x^11-X2*x^11-x^12-2*X1*X2*x^9-2*X1*x^10-2*X2*x^10+x^11+X1*X2*x^8+2* X1*x^9+2*X2*x^9+2*x^10+X1*X2*x^7-X1*x^8-X2*x^8-2*x^9-2*X1*X2*x^6-X1*x^7-2*X2*x^ 7+x^8+X1*X2*x^5+X1*x^6+3*X2*x^6+2*x^7-2*x^6-2*X2*x^4-x^5+X2*x^3+2*x^4-3*x^2+3*x -1)/(-1+x)/(X1^2*X2*x^11-X1^2*x^11-2*X1*X2*x^11+2*X1*x^11+X2*x^11+2*X1*X2*x^9-x ^11-X1*X2*x^8-2*X1*x^9-2*X2*x^9+X1*x^8+X2*x^8+2*x^9+2*X1*X2*x^6+X2*x^7-x^8-X1* X2*x^5-X1*x^6-3*X2*x^6-x^7+X2*x^5+2*x^6+2*X2*x^4-X2*x^3-2*x^4+x^3+2*x^2-3*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [5, 1, 1], are n 139 23 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 3, 3], are n 97 85 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 6 23 85 ------------- 1955 and in floating point 0.1356993819 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 6 1955 ate normal pair with correlation, --------- 1955 1/2 6 1955 2027 i.e. , [[---------, 0], [0, ----]] 1955 1955 ------------------------------------------------- Theorem Number, 262, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [5, 1, 1], nor the composition, [2, 4, 2] Then infinity ----- 11 9 7 6 5 4 2 \ n x + x + x - 2 x + x - x - x + 2 x - 1 ) a(n) x = ----------------------------------------------- / 9 8 7 5 3 2 2 ----- (x + x + x + x + x + x + x - 1) (-1 + x) n = 0 and in Maple format (x^11+x^9+x^7-2*x^6+x^5-x^4-x^2+2*x-1)/(x^9+x^8+x^7+x^5+x^3+x^2+x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.69057460186873174687 1.9099139227361088013 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [5, 1, 1] and d occurrences (as containment) of the composition, [2, 4, 2], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 11 2 11 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x - X1 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 11 11 11 9 11 9 9 - 2 X1 X2 x + 2 X1 x + X2 x - X1 X2 x - x + X1 x + X2 x 7 9 6 7 7 5 6 6 - X1 X2 x - x + X1 X2 x + X1 x + X2 x - X1 X2 x - X1 x - 2 X2 x 7 5 6 4 5 4 2 / 2 11 - x + 2 X2 x + 2 x - X2 x - x + x + x - 2 x + 1) / (X1 X2 x / 2 11 11 10 11 11 10 10 - X1 x - 2 X1 X2 x + X1 X2 x + 2 X1 x + X2 x - X1 x - X2 x 11 8 10 7 8 8 6 7 - x + X1 X2 x + x - X1 X2 x - X1 x - X2 x + 2 X1 X2 x + X1 x 7 8 5 6 6 7 5 6 + 2 X2 x + x - X1 X2 x - X1 x - 3 X2 x - 2 x + 3 X2 x + 2 x 4 5 4 2 - X2 x - 2 x + x + 2 x - 3 x + 1) and in Maple format (X1^2*X2*x^11-X1^2*x^11-2*X1*X2*x^11+2*X1*x^11+X2*x^11-X1*X2*x^9-x^11+X1*x^9+X2 *x^9-X1*X2*x^7-x^9+X1*X2*x^6+X1*x^7+X2*x^7-X1*X2*x^5-X1*x^6-2*X2*x^6-x^7+2*X2*x ^5+2*x^6-X2*x^4-x^5+x^4+x^2-2*x+1)/(X1^2*X2*x^11-X1^2*x^11-2*X1*X2*x^11+X1*X2*x ^10+2*X1*x^11+X2*x^11-X1*x^10-X2*x^10-x^11+X1*X2*x^8+x^10-X1*X2*x^7-X1*x^8-X2*x ^8+2*X1*X2*x^6+X1*x^7+2*X2*x^7+x^8-X1*X2*x^5-X1*x^6-3*X2*x^6-2*x^7+3*X2*x^5+2*x ^6-X2*x^4-2*x^5+x^4+2*x^2-3*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [5, 1, 1], are n 139 23 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 4, 2], are n 75 69 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 10 3 ------- 69 and in floating point 0.2510218562 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 10 3 ate normal pair with correlation, ------- 69 1/2 10 3 1787 i.e. , [[-------, 0], [0, ----]] 69 1587 ------------------------------------------------- Theorem Number, 263, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [5, 1, 1], nor the composition, [2, 5, 1] Then infinity ----- 10 9 8 5 2 \ n x + x + x - x - x + 2 x - 1 ) a(n) x = --------------------------------- / 4 3 2 2 ----- (x + x + x + x - 1) (-1 + x) n = 0 and in Maple format (x^10+x^9+x^8-x^5-x^2+2*x-1)/(x^4+x^3+x^2+x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.63530875266095142903 1.9275619754829253043 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [5, 1, 1] and d occurrences (as containment) of the composition, [2, 5, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 10 9 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x + X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 10 10 8 9 9 10 7 8 - X1 x - X2 x + X1 X2 x - X1 x - X2 x + x + X1 X2 x - X1 x 8 9 6 7 8 5 6 5 2 - X2 x + x - X1 X2 x - X1 x + x + X1 X2 x + X1 x - x - x + 2 x / 6 5 6 5 - 1) / ((-1 + x) (X1 X2 x - X1 X2 x - X1 x + x - 2 x + 1)) / and in Maple format (X1*X2*x^10+X1*X2*x^9-X1*x^10-X2*x^10+X1*X2*x^8-X1*x^9-X2*x^9+x^10+X1*X2*x^7-X1 *x^8-X2*x^8+x^9-X1*X2*x^6-X1*x^7+x^8+X1*X2*x^5+X1*x^6-x^5-x^2+2*x-1)/(-1+x)/(X1 *X2*x^6-X1*X2*x^5-X1*x^6+x^5-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [5, 1, 1], are n 139 23 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 5, 1], are n 111 57 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 8 23 57 ------------- 437 and in floating point 0.6628418121 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 1/2 8 3 437 ate normal pair with correlation, ------------- 437 1/2 1/2 8 3 437 821 i.e. , [[-------------, 0], [0, ---]] 437 437 ------------------------------------------------- Theorem Number, 264, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [5, 1, 1], nor the composition, [3, 1, 4] Then infinity ----- \ n ) a(n) x = / ----- n = 0 16 15 14 13 12 11 10 4 2 x + x - x - 2 x - x - 2 x - x + x + x - 2 x + 1 - --------------------------------------------------------------- 6 5 3 2 6 5 2 (x + x + x + x + x - 1) (x + x + 1) (-1 + x) and in Maple format -(x^16+x^15-x^14-2*x^13-x^12-2*x^11-x^10+x^4+x^2-2*x+1)/(x^6+x^5+x^3+x^2+x-1)/( x^6+x^5+1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.68033974934962079394 1.9111834366854762350 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [5, 1, 1] and d occurrences (as containment) of the composition, [3, 1, 4], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 17 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 17 2 17 2 2 15 2 17 17 - 2 X1 X2 x - 2 X1 X2 x - X1 X2 x + X1 x + 4 X1 X2 x 2 17 2 2 14 2 15 2 15 17 + X2 x - X1 X2 x + 2 X1 X2 x + 3 X1 X2 x - 2 X1 x 17 2 2 13 2 14 2 15 2 14 - 2 X2 x + X1 X2 x + 3 X1 X2 x - X1 x + X1 X2 x 15 2 15 17 2 13 2 14 2 13 - 6 X1 X2 x - 2 X2 x + x - X1 X2 x - 2 X1 x - 3 X1 X2 x 14 15 15 2 2 11 2 12 - 4 X1 X2 x + 3 X1 x + 4 X2 x - X1 X2 x + 2 X1 X2 x 13 14 2 13 14 15 2 11 + 4 X1 X2 x + 3 X1 x + 2 X2 x + X2 x - 2 x + X1 X2 x 2 11 12 13 2 12 13 14 + 2 X1 X2 x - 3 X1 X2 x - X1 x - 2 X2 x - 3 X2 x - x 2 10 11 12 12 13 10 - 2 X1 X2 x - 2 X1 X2 x + X1 x + 3 X2 x + x + 3 X1 X2 x 2 10 11 12 9 10 2 9 10 11 + 2 X2 x - X2 x - x + X1 X2 x - X1 x - X2 x - 3 X2 x + x 9 9 10 6 5 6 5 4 5 - X1 x + X2 x + x - X1 X2 x + X1 X2 x + X2 x - 2 X2 x + X2 x + x 4 3 2 / 7 6 7 - x + x - 3 x + 3 x - 1) / ((-1 + x) (X1 X2 x + X1 X2 x - X1 x / 7 5 6 7 5 4 5 4 - X2 x - X1 X2 x - X1 x + x + 2 X2 x - X2 x - x + x - 2 x + 1) 7 6 7 7 6 7 5 5 (X1 X2 x + X1 X2 x - X1 x - X2 x - X1 x + x + X2 x - x + x - 1)) and in Maple format -(X1^2*X2^2*x^17-2*X1^2*X2*x^17-2*X1*X2^2*x^17-X1^2*X2^2*x^15+X1^2*x^17+4*X1*X2 *x^17+X2^2*x^17-X1^2*X2^2*x^14+2*X1^2*X2*x^15+3*X1*X2^2*x^15-2*X1*x^17-2*X2*x^ 17+X1^2*X2^2*x^13+3*X1^2*X2*x^14-X1^2*x^15+X1*X2^2*x^14-6*X1*X2*x^15-2*X2^2*x^ 15+x^17-X1^2*X2*x^13-2*X1^2*x^14-3*X1*X2^2*x^13-4*X1*X2*x^14+3*X1*x^15+4*X2*x^ 15-X1^2*X2^2*x^11+2*X1*X2^2*x^12+4*X1*X2*x^13+3*X1*x^14+2*X2^2*x^13+X2*x^14-2*x ^15+X1^2*X2*x^11+2*X1*X2^2*x^11-3*X1*X2*x^12-X1*x^13-2*X2^2*x^12-3*X2*x^13-x^14 -2*X1*X2^2*x^10-2*X1*X2*x^11+X1*x^12+3*X2*x^12+x^13+3*X1*X2*x^10+2*X2^2*x^10-X2 *x^11-x^12+X1*X2*x^9-X1*x^10-X2^2*x^9-3*X2*x^10+x^11-X1*x^9+X2*x^9+x^10-X1*X2*x ^6+X1*X2*x^5+X2*x^6-2*X2*x^5+X2*x^4+x^5-x^4+x^3-3*x^2+3*x-1)/(-1+x)/(X1*X2*x^7+ X1*X2*x^6-X1*x^7-X2*x^7-X1*X2*x^5-X1*x^6+x^7+2*X2*x^5-X2*x^4-x^5+x^4-2*x+1)/(X1 *X2*x^7+X1*X2*x^6-X1*x^7-X2*x^7-X1*x^6+x^7+X2*x^5-x^5+x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [5, 1, 1], are n 139 23 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 1, 4], are n 141 65 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 12 23 65 -------------- 1495 and in floating point 0.3103563600 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 12 1495 ate normal pair with correlation, ---------- 1495 1/2 12 1495 1783 i.e. , [[----------, 0], [0, ----]] 1495 1495 ------------------------------------------------- Theorem Number, 265, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [5, 1, 1], nor the composition, [3, 2, 3] Then infinity ----- \ n 11 10 9 8 7 6 5 4 3 2 ) a(n) x = (x + x - x - x + x - x - x + x - x - x + 2 x - 1 / ----- n = 0 / ) / ( / 12 11 10 9 8 7 6 5 4 2 (x + 3 x + 3 x + 2 x + 3 x + 3 x + x + x + x + x + x - 1) 2 (-1 + x) ) and in Maple format (x^11+x^10-x^9-x^8+x^7-x^6-x^5+x^4-x^3-x^2+2*x-1)/(x^12+3*x^11+3*x^10+2*x^9+3*x ^8+3*x^7+x^6+x^5+x^4+x^2+x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.70407803205746399354 1.9060446822436648581 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [5, 1, 1] and d occurrences (as containment) of the composition, [3, 2, 3], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 12 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 12 2 12 2 11 12 2 12 2 11 - X1 X2 x - 2 X1 X2 x + X1 X2 x + 2 X1 X2 x + X2 x - X1 x 2 10 11 12 10 11 2 10 + 2 X1 X2 x - 2 X1 X2 x - X2 x - 3 X1 X2 x + 2 X1 x - 2 X2 x 11 9 10 10 11 9 2 8 9 + X2 x + X1 X2 x + X1 x + 3 X2 x - x - X1 x + X2 x - X2 x 10 7 8 9 6 7 7 8 5 - x - X1 X2 x - 2 X2 x + x + X1 X2 x + X1 x + X2 x + x - X1 X2 x 6 6 7 6 4 5 3 4 3 2 - X1 x - X2 x - x + x + X2 x + x - X2 x - x + x + x - 2 x + 1) / 2 2 14 2 2 13 2 14 2 14 2 2 12 / (X1 X2 x + X1 X2 x - 2 X1 X2 x - 2 X1 X2 x - X1 X2 x / 2 13 2 14 2 13 14 2 14 - 2 X1 X2 x + X1 x - 2 X1 X2 x + 4 X1 X2 x + X2 x 2 12 2 13 2 12 13 14 2 13 + X1 X2 x + X1 x + 4 X1 X2 x + 4 X1 X2 x - 2 X1 x + X2 x 14 2 11 2 11 12 13 - 2 X2 x - X1 X2 x + 2 X1 X2 x - 6 X1 X2 x - 2 X1 x 2 12 13 14 2 11 2 10 11 - 3 X2 x - 2 X2 x + x + X1 x - 2 X1 X2 x - 2 X1 X2 x 12 2 11 12 13 10 2 10 + 2 X1 x - 2 X2 x + 5 X2 x + x + 3 X1 X2 x + 3 X2 x 11 12 9 10 2 9 10 11 + 3 X2 x - 2 x - 2 X1 X2 x - X1 x + X2 x - 5 X2 x - x 8 9 2 8 10 7 8 8 9 - X1 X2 x + 2 X1 x - X2 x + 2 x + X1 X2 x + X1 x + 3 X2 x - x 6 7 7 8 5 6 6 7 - 2 X1 X2 x - X1 x - 2 X2 x - 2 x + X1 X2 x + X1 x + X2 x + 2 x 4 5 3 4 3 2 - 2 X2 x - x + X2 x + 2 x - x - 2 x + 3 x - 1) and in Maple format -(X1^2*X2^2*x^12-X1^2*X2*x^12-2*X1*X2^2*x^12+X1^2*X2*x^11+2*X1*X2*x^12+X2^2*x^ 12-X1^2*x^11+2*X1*X2^2*x^10-2*X1*X2*x^11-X2*x^12-3*X1*X2*x^10+2*X1*x^11-2*X2^2* x^10+X2*x^11+X1*X2*x^9+X1*x^10+3*X2*x^10-x^11-X1*x^9+X2^2*x^8-X2*x^9-x^10-X1*X2 *x^7-2*X2*x^8+x^9+X1*X2*x^6+X1*x^7+X2*x^7+x^8-X1*X2*x^5-X1*x^6-X2*x^6-x^7+x^6+ X2*x^4+x^5-X2*x^3-x^4+x^3+x^2-2*x+1)/(X1^2*X2^2*x^14+X1^2*X2^2*x^13-2*X1^2*X2*x ^14-2*X1*X2^2*x^14-X1^2*X2^2*x^12-2*X1^2*X2*x^13+X1^2*x^14-2*X1*X2^2*x^13+4*X1* X2*x^14+X2^2*x^14+X1^2*X2*x^12+X1^2*x^13+4*X1*X2^2*x^12+4*X1*X2*x^13-2*X1*x^14+ X2^2*x^13-2*X2*x^14-X1^2*X2*x^11+2*X1*X2^2*x^11-6*X1*X2*x^12-2*X1*x^13-3*X2^2*x ^12-2*X2*x^13+x^14+X1^2*x^11-2*X1*X2^2*x^10-2*X1*X2*x^11+2*X1*x^12-2*X2^2*x^11+ 5*X2*x^12+x^13+3*X1*X2*x^10+3*X2^2*x^10+3*X2*x^11-2*x^12-2*X1*X2*x^9-X1*x^10+X2 ^2*x^9-5*X2*x^10-x^11-X1*X2*x^8+2*X1*x^9-X2^2*x^8+2*x^10+X1*X2*x^7+X1*x^8+3*X2* x^8-x^9-2*X1*X2*x^6-X1*x^7-2*X2*x^7-2*x^8+X1*X2*x^5+X1*x^6+X2*x^6+2*x^7-2*X2*x^ 4-x^5+X2*x^3+2*x^4-x^3-2*x^2+3*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [5, 1, 1], are n 139 23 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 2, 3], are n 89 77 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 6 23 77 ------------- 1771 and in floating point 0.1425745369 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 6 1771 ate normal pair with correlation, --------- 1771 1/2 6 1771 1843 i.e. , [[---------, 0], [0, ----]] 1771 1771 ------------------------------------------------- Theorem Number, 266, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [5, 1, 1], nor the composition, [3, 3, 2] Then infinity ----- 2 9 8 7 6 4 3 2 \ n (x - x + 1) (x + x - x - x - x - x + x + x - 1) ) a(n) x = -------------------------------------------------------- / 9 8 7 6 5 4 2 2 ----- (x + 2 x + x + x + 2 x + x + x + x - 1) (-1 + x) n = 0 and in Maple format (x^2-x+1)*(x^9+x^8-x^7-x^6-x^4-x^3+x^2+x-1)/(x^9+2*x^8+x^7+x^6+2*x^5+x^4+x^2+x-\ 1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.70325653317007746417 1.9073680513412163549 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [5, 1, 1] and d occurrences (as containment) of the composition, [3, 3, 2], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 11 2 11 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x - X1 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 11 11 11 9 11 8 9 - 2 X1 X2 x + 2 X1 x + X2 x + X1 X2 x - x - X1 X2 x - X1 x 9 8 8 9 6 8 5 6 6 - X2 x + X1 x + X2 x + x + X1 X2 x - x - X1 X2 x - X1 x - 2 X2 x 5 6 4 3 4 3 2 / 2 11 + X2 x + 2 x + X2 x - X2 x - x + x + x - 2 x + 1) / (X1 X2 x / 2 11 11 11 11 9 11 8 - X1 x - 2 X1 X2 x + 2 X1 x + X2 x + 2 X1 X2 x - x - X1 X2 x 9 9 8 8 9 6 7 8 - 2 X1 x - 2 X2 x + X1 x + X2 x + 2 x + 2 X1 X2 x + X2 x - x 5 6 6 7 5 6 4 3 4 - X1 X2 x - X1 x - 3 X2 x - x + X2 x + 2 x + 2 X2 x - X2 x - 2 x 3 2 + x + 2 x - 3 x + 1) and in Maple format (X1^2*X2*x^11-X1^2*x^11-2*X1*X2*x^11+2*X1*x^11+X2*x^11+X1*X2*x^9-x^11-X1*X2*x^8 -X1*x^9-X2*x^9+X1*x^8+X2*x^8+x^9+X1*X2*x^6-x^8-X1*X2*x^5-X1*x^6-2*X2*x^6+X2*x^5 +2*x^6+X2*x^4-X2*x^3-x^4+x^3+x^2-2*x+1)/(X1^2*X2*x^11-X1^2*x^11-2*X1*X2*x^11+2* X1*x^11+X2*x^11+2*X1*X2*x^9-x^11-X1*X2*x^8-2*X1*x^9-2*X2*x^9+X1*x^8+X2*x^8+2*x^ 9+2*X1*X2*x^6+X2*x^7-x^8-X1*X2*x^5-X1*x^6-3*X2*x^6-x^7+X2*x^5+2*x^6+2*X2*x^4-X2 *x^3-2*x^4+x^3+2*x^2-3*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [5, 1, 1], are n 139 23 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 3, 2], are n 97 85 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 1/2 6 23 85 ------------- 1955 and in floating point 0.1356993819 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 6 1955 ate normal pair with correlation, --------- 1955 1/2 6 1955 2027 i.e. , [[---------, 0], [0, ----]] 1955 1955 ------------------------------------------------- Theorem Number, 267, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [5, 1, 1], nor the composition, [3, 4, 1] Then infinity ----- 8 7 6 4 \ n x + 2 x + x + x - x + 1 ) a(n) x = ------------------------------------ / 6 5 3 2 ----- (-1 + x) (x + x + x + x + x - 1) n = 0 and in Maple format (x^8+2*x^7+x^6+x^4-x+1)/(-1+x)/(x^6+x^5+x^3+x^2+x-1) The asymptotic expression for a(n) is, n 0.69079734461128122568 1.9111834366854762350 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [5, 1, 1] and d occurrences (as containment) of the composition, [3, 4, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 9 8 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x + X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 9 9 8 8 9 6 7 8 5 - X1 x - X2 x - X1 x - X2 x + x - X1 X2 x + X2 x + x + X1 X2 x 6 6 7 5 6 4 5 4 2 / + X1 x + X2 x - x - 2 X2 x - x + X2 x + x - x - x + 2 x - 1) / / 7 6 7 7 5 6 7 ((-1 + x) (X1 X2 x + X1 X2 x - X1 x - X2 x - X1 X2 x - X1 x + x 5 4 5 4 + 2 X2 x - X2 x - x + x - 2 x + 1)) and in Maple format (X1*X2*x^9+X1*X2*x^8-X1*x^9-X2*x^9-X1*x^8-X2*x^8+x^9-X1*X2*x^6+X2*x^7+x^8+X1*X2 *x^5+X1*x^6+X2*x^6-x^7-2*X2*x^5-x^6+X2*x^4+x^5-x^4-x^2+2*x-1)/(-1+x)/(X1*X2*x^7 +X1*X2*x^6-X1*x^7-X2*x^7-X1*X2*x^5-X1*x^6+x^7+2*X2*x^5-X2*x^4-x^5+x^4-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [5, 1, 1], are n 139 23 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 4, 1], are n 135 65 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 12 23 65 -------------- 1495 and in floating point 0.3103563600 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 12 1495 ate normal pair with correlation, ---------- 1495 1/2 12 1495 1783 i.e. , [[----------, 0], [0, ----]] 1495 1495 ------------------------------------------------- Theorem Number, 268, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [5, 1, 1], nor the composition, [4, 1, 3] Then infinity ----- 11 9 4 2 \ n x - x - x - x + 2 x - 1 ) a(n) x = --------------------------------------------------- / 6 5 3 2 6 5 2 ----- (x + x + x + x + x - 1) (x + x + 1) (-1 + x) n = 0 and in Maple format (x^11-x^9-x^4-x^2+2*x-1)/(x^6+x^5+x^3+x^2+x-1)/(x^6+x^5+1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.69440972834504952286 1.9111834366854762350 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [5, 1, 1] and d occurrences (as containment) of the composition, [4, 1, 3], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 12 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 2 11 2 12 2 12 2 11 12 + X1 X2 x - X1 X2 x - 2 X1 X2 x - X1 X2 x + 2 X1 X2 x 2 12 2 10 11 2 11 12 10 + X2 x + 2 X1 X2 x - X1 X2 x - X2 x - X2 x - 3 X1 X2 x 11 2 10 11 10 2 9 10 11 9 + X1 x - X2 x + 2 X2 x + X1 x + X2 x + X2 x - x - 2 X2 x 9 5 5 4 4 2 / + x - X1 X2 x + X2 x - X2 x + x + x - 2 x + 1) / ( / 7 6 7 7 6 7 5 5 (X1 X2 x + X1 X2 x - X1 x - X2 x - X1 x + x + X2 x - x + x - 1) ( 7 6 7 7 5 6 7 5 X1 X2 x + X1 X2 x - X1 x - X2 x - X1 X2 x - X1 x + x + 2 X2 x 4 5 4 - X2 x - x + x - 2 x + 1)) and in Maple format -(X1^2*X2^2*x^12+X1^2*X2^2*x^11-X1^2*X2*x^12-2*X1*X2^2*x^12-X1^2*X2*x^11+2*X1* X2*x^12+X2^2*x^12+2*X1*X2^2*x^10-X1*X2*x^11-X2^2*x^11-X2*x^12-3*X1*X2*x^10+X1*x ^11-X2^2*x^10+2*X2*x^11+X1*x^10+X2^2*x^9+X2*x^10-x^11-2*X2*x^9+x^9-X1*X2*x^5+X2 *x^5-X2*x^4+x^4+x^2-2*x+1)/(X1*X2*x^7+X1*X2*x^6-X1*x^7-X2*x^7-X1*x^6+x^7+X2*x^5 -x^5+x-1)/(X1*X2*x^7+X1*X2*x^6-X1*x^7-X2*x^7-X1*X2*x^5-X1*x^6+x^7+2*X2*x^5-X2*x ^4-x^5+x^4-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [5, 1, 1], are n 139 23 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 1, 3], are n 141 65 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 12 23 65 -------------- 1495 and in floating point 0.3103563600 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 12 1495 ate normal pair with correlation, ---------- 1495 1/2 12 1495 1783 i.e. , [[----------, 0], [0, ----]] 1495 1495 ------------------------------------------------- Theorem Number, 269, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [5, 1, 1], nor the composition, [4, 2, 2] Then infinity ----- 11 10 7 6 5 4 2 \ n x - x + x - 2 x + x - x - x + 2 x - 1 ) a(n) x = ----------------------------------------------- / 9 8 7 5 3 2 2 ----- (x + x + x + x + x + x + x - 1) (-1 + x) n = 0 and in Maple format (x^11-x^10+x^7-2*x^6+x^5-x^4-x^2+2*x-1)/(x^9+x^8+x^7+x^5+x^3+x^2+x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.70132476238170317564 1.9099139227361088013 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [5, 1, 1] and d occurrences (as containment) of the composition, [4, 2, 2], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 11 2 11 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x - X1 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 11 10 11 11 10 10 11 - 2 X1 X2 x + X1 X2 x + 2 X1 x + X2 x - X1 x - X2 x - x 10 7 6 7 7 5 6 6 + x - X1 X2 x + X1 X2 x + X1 x + X2 x - X1 X2 x - X1 x - 2 X2 x 7 5 6 4 5 4 2 / 2 11 - x + 2 X2 x + 2 x - X2 x - x + x + x - 2 x + 1) / (X1 X2 x / 2 11 11 10 11 11 10 10 - X1 x - 2 X1 X2 x + X1 X2 x + 2 X1 x + X2 x - X1 x - X2 x 11 8 10 7 8 8 6 7 - x + X1 X2 x + x - X1 X2 x - X1 x - X2 x + 2 X1 X2 x + X1 x 7 8 5 6 6 7 5 6 + 2 X2 x + x - X1 X2 x - X1 x - 3 X2 x - 2 x + 3 X2 x + 2 x 4 5 4 2 - X2 x - 2 x + x + 2 x - 3 x + 1) and in Maple format (X1^2*X2*x^11-X1^2*x^11-2*X1*X2*x^11+X1*X2*x^10+2*X1*x^11+X2*x^11-X1*x^10-X2*x^ 10-x^11+x^10-X1*X2*x^7+X1*X2*x^6+X1*x^7+X2*x^7-X1*X2*x^5-X1*x^6-2*X2*x^6-x^7+2* X2*x^5+2*x^6-X2*x^4-x^5+x^4+x^2-2*x+1)/(X1^2*X2*x^11-X1^2*x^11-2*X1*X2*x^11+X1* X2*x^10+2*X1*x^11+X2*x^11-X1*x^10-X2*x^10-x^11+X1*X2*x^8+x^10-X1*X2*x^7-X1*x^8- X2*x^8+2*X1*X2*x^6+X1*x^7+2*X2*x^7+x^8-X1*X2*x^5-X1*x^6-3*X2*x^6-2*x^7+3*X2*x^5 +2*x^6-X2*x^4-2*x^5+x^4+2*x^2-3*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [5, 1, 1], are n 139 23 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 2, 2], are n 75 69 n - 1/8 + ----, and , - --- + ----, respectively, while the asymptotic correla\ 64 512 4096 tion between these two random variables is 1/2 10 3 ------- 69 and in floating point 0.2510218562 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 10 3 ate normal pair with correlation, ------- 69 1/2 10 3 1787 i.e. , [[-------, 0], [0, ----]] 69 1587 ------------------------------------------------- Theorem Number, 270, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [5, 1, 1], nor the composition, [4, 3, 1] Then infinity ----- 8 7 6 5 4 2 \ n x - x - x + x - x - x + 2 x - 1 ) a(n) x = ------------------------------------- / 6 5 3 2 2 ----- (x + x + x + x + x - 1) (-1 + x) n = 0 and in Maple format (x^8-x^7-x^6+x^5-x^4-x^2+2*x-1)/(x^6+x^5+x^3+x^2+x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.69790391550484586467 1.9111834366854762350 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [5, 1, 1] and d occurrences (as containment) of the composition, [4, 3, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 8 8 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x - X1 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 8 6 7 8 5 6 6 7 5 - X2 x - X1 X2 x + X2 x + x + X1 X2 x + X1 x + X2 x - x - 2 X2 x 6 4 5 4 2 / 7 6 - x + X2 x + x - x - x + 2 x - 1) / ((-1 + x) (X1 X2 x + X1 X2 x / 7 7 5 6 7 5 4 5 4 - X1 x - X2 x - X1 X2 x - X1 x + x + 2 X2 x - X2 x - x + x - 2 x + 1)) and in Maple format (X1*X2*x^8-X1*x^8-X2*x^8-X1*X2*x^6+X2*x^7+x^8+X1*X2*x^5+X1*x^6+X2*x^6-x^7-2*X2* x^5-x^6+X2*x^4+x^5-x^4-x^2+2*x-1)/(-1+x)/(X1*X2*x^7+X1*X2*x^6-X1*x^7-X2*x^7-X1* X2*x^5-X1*x^6+x^7+2*X2*x^5-X2*x^4-x^5+x^4-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [5, 1, 1], are n 139 23 n - 7/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 3, 1], are n 135 65 n - 1/8 + ----, and , - ---- + ----, respectively, while the asymptotic correl\ 64 1024 4096 ation between these two random variables is 1/2 1/2 12 23 65 -------------- 1495 and in floating point 0.3103563600 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 12 1495 ate normal pair with correlation, ---------- 1495 1/2 12 1495 1783 i.e. , [[----------, 0], [0, ----]] 1495 1495 ------------------------------------------------- ------------------------ This ends this article, that took, 72.597, to generate. This took, 75.926, seconds