All the generating functions (and statistical information) Enumerating compositions by number of occurrences, as containments of all possible pairs of offending compositions of, 3, 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, 2], nor the composition, [2, 1] Then infinity ----- 2 \ n x + 1 ) a(n) x = - ------ / -1 + x ----- n = 0 and in Maple format -(x^2+1)/(-1+x) n The asymptotic expression for a(n) is, 2.0000000000000000000 1. Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 2] and d occurrences (as containment) of the composition, [2, 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 3 2 3 3 3 2 X1 X2 x + X1 X2 x - X1 x - X2 x + x - x + x - 1 ----------------------------------------------------- 2 2 X1 X2 x - x + 2 x - 1 and in Maple format (X1*X2*x^3+X1*X2*x^2-X1*x^3-X2*x^3+x^3-x^2+x-1)/(X1*X2*x^2-x^2+2*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 2], are n - 1/2 + n/4, and , ----, respectively, while 16 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 1], are n - 1/2 + n/4, and , ----, respectively, while the asymptotic correlation betw\ 16 een these two random variables is 1 and in floating point 1. ------------------------------------------------- ------------------------ This ends this article, that took, 0.150, 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, 3, with , 3, parts By Shalosh B. Ekhad ------------------------ This ends this article, that took, 0., 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, 4, 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, 3], nor the composition, [2, 2] Then infinity ----- 2 \ n x - x + 1 ) a(n) x = --------------------- / 3 ----- (-1 + x) (x + x - 1) n = 0 and in Maple format (x^2-x+1)/(-1+x)/(x^3+x-1) The asymptotic expression for a(n) is, n 1.5076770638769435396 1.4655712318767680267 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 3] and d occurrences (as containment) of the composition, [2, 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 3 3 2 2 / 4 3 4 X1 X2 x - X2 x + X2 x - x + x - 1) / (X1 X2 x - X1 X2 x - X1 x / 4 3 4 2 3 2 - X2 x + 2 X2 x + x - X2 x - x + x - 2 x + 1) and in Maple format -(X1*X2*x^3-X2*x^3+X2*x^2-x^2+x-1)/(X1*X2*x^4-X1*X2*x^3-X1*x^4-X2*x^4+2*X2*x^3+ x^4-X2*x^2-x^3+x^2-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 3], are 3 n - 3/8 + n/8, and , - 3/64 + ---, respectively, while 64 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 2], are 23 7 n - 3/8 + n/8, and , - -- + ---, respectively, while the asymptotic correlatio\ 64 64 n between these two random variables is 1/2 1/2 3 7 --------- 21 and in floating point 0.2182178903 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 21 ate normal pair with correlation, ----- 21 1/2 21 23 i.e. , [[-----, 0], [0, --]] 21 21 ------------------------------------------------- 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, 3], nor the composition, [3, 1] Then infinity ----- 5 4 3 \ n x + x - x + x - 1 ) a(n) x = - --------------------- / 2 ----- (-1 + x) (x + x - 1) n = 0 and in Maple format -(x^5+x^4-x^3+x-1)/(-1+x)/(x^2+x-1) The asymptotic expression for a(n) is, n 0.72360679774997896964 1.6180339887498948482 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 3] and d occurrences (as containment) of the composition, [3, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 5 4 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x + X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 5 5 3 4 4 5 4 3 / - X1 x - X2 x + X1 X2 x - X1 x - X2 x + x + x - x + x - 1) / ( / 3 3 X1 X2 x - x + 2 x - 1) and in Maple format (X1*X2*x^5+X1*X2*x^4-X1*x^5-X2*x^5+X1*X2*x^3-X1*x^4-X2*x^4+x^5+x^4-x^3+x-1)/(X1 *X2*x^3-x^3+2*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 3], are 3 n - 3/8 + n/8, and , - 3/64 + ---, respectively, while 64 Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 1], are 3 n - 3/8 + n/8, and , - 3/64 + ---, respectively, while the asymptotic correlat\ 64 ion between these two random variables is 1 and in floating point 1. ------------------------------------------------- Theorem Number, 3, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 2], nor the composition, [3, 1] Then infinity ----- 2 \ n x - x + 1 ) a(n) x = --------------------- / 3 ----- (-1 + x) (x + x - 1) n = 0 and in Maple format (x^2-x+1)/(-1+x)/(x^3+x-1) The asymptotic expression for a(n) is, n 1.5076770638769435396 1.4655712318767680267 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 2] and d occurrences (as containment) of the composition, [3, 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 3 3 2 2 / 4 3 4 X1 X2 x - X1 x + X1 x - x + x - 1) / (X1 X2 x - X1 X2 x - X1 x / 4 3 4 2 3 2 - X2 x + 2 X1 x + x - X1 x - x + x - 2 x + 1) and in Maple format -(X1*X2*x^3-X1*x^3+X1*x^2-x^2+x-1)/(X1*X2*x^4-X1*X2*x^3-X1*x^4-X2*x^4+2*X1*x^3+ x^4-X1*x^2-x^3+x^2-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 2], are 23 7 n - 3/8 + n/8, and , - -- + ---, respectively, while 64 64 Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 1], are 3 n - 3/8 + n/8, and , - 3/64 + ---, respectively, while the asymptotic correlat\ 64 ion between these two random variables is 1/2 1/2 3 7 --------- 21 and in floating point 0.2182178903 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 21 ate normal pair with correlation, ----- 21 1/2 21 23 i.e. , [[-----, 0], [0, --]] 21 21 ------------------------------------------------- ------------------------ This ends this article, that took, 0.275, 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, 4, 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, 2], nor the composition, [1, 2, 1] Then infinity ----- 3 2 \ n x + x - x + 1 ) a(n) x = --------------- / 2 ----- (-1 + x) n = 0 and in Maple format (x^3+x^2-x+1)/(-1+x)^2 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 1, 2] and d occurrences (as containment) of the composition, [1, 2, 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 4 4 4 2 4 2 X1 X2 x - X1 x - X2 x + X1 X2 x + x - 2 x + 2 x - 1 - --------------------------------------------------------- 2 2 (-1 + x) (X1 X2 x - x + 2 x - 1) and in Maple format -(X1*X2*x^4-X1*x^4-X2*x^4+X1*X2*x^2+x^4-2*x^2+2*x-1)/(-1+x)/(X1*X2*x^2-x^2+2*x-\ 1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 1, 2], are n -1 + n/4, and , ----, respectively, while 16 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 2, 1], are n -1 + n/4, and , ----, respectively, while the asymptotic correlation between\ 16 these two random variables is 1 and in floating point 1. ------------------------------------------------- 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, 2], nor the composition, [2, 1, 1] Then infinity ----- 5 3 2 \ n x - x - x + x - 1 ) a(n) x = - -------------------- / 2 ----- (-1 + x) n = 0 and in Maple format -(x^5-x^3-x^2+x-1)/(-1+x)^2 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 1, 2] and d occurrences (as containment) of the composition, [2, 1, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 7 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 7 2 7 2 6 2 7 2 6 7 - 2 X1 X2 x - 2 X1 X2 x + X1 X2 x + X1 x + X1 X2 x + 4 X1 X2 x 2 7 2 6 6 7 2 6 7 6 + X2 x - X1 x - 4 X1 X2 x - 2 X1 x - X2 x - 2 X2 x + 3 X1 x 6 7 4 6 3 4 4 2 + 3 X2 x + x + X1 X2 x - 2 x - X1 X2 x - X1 x - X2 x + X1 X2 x 4 3 2 / 2 2 2 + x + 2 x - 4 x + 3 x - 1) / ((-1 + x) (X1 X2 x - x + 2 x - 1)) / and in Maple format (X1^2*X2^2*x^7-2*X1^2*X2*x^7-2*X1*X2^2*x^7+X1^2*X2*x^6+X1^2*x^7+X1*X2^2*x^6+4* X1*X2*x^7+X2^2*x^7-X1^2*x^6-4*X1*X2*x^6-2*X1*x^7-X2^2*x^6-2*X2*x^7+3*X1*x^6+3* X2*x^6+x^7+X1*X2*x^4-2*x^6-X1*X2*x^3-X1*x^4-X2*x^4+X1*X2*x^2+x^4+2*x^3-4*x^2+3* x-1)/(-1+x)^2/(X1*X2*x^2-x^2+2*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 1, 2], are n -1 + n/4, and , ----, respectively, while 16 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 1, 1], are n -1 + n/4, and , ----, respectively, while the asymptotic correlation between\ 16 these two random variables is 1 and in floating point 1. ------------------------------------------------- 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, 2, 1], nor the composition, [2, 1, 1] Then infinity ----- 3 2 \ n x + x - x + 1 ) a(n) x = --------------- / 2 ----- (-1 + x) n = 0 and in Maple format (x^3+x^2-x+1)/(-1+x)^2 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 2, 1] and d occurrences (as containment) of the composition, [2, 1, 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 4 4 4 2 4 2 X1 X2 x - X1 x - X2 x + X1 X2 x + x - 2 x + 2 x - 1 - --------------------------------------------------------- 2 2 (-1 + x) (X1 X2 x - x + 2 x - 1) and in Maple format -(X1*X2*x^4-X1*x^4-X2*x^4+X1*X2*x^2+x^4-2*x^2+2*x-1)/(-1+x)/(X1*X2*x^2-x^2+2*x-\ 1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 2, 1], are n -1 + n/4, and , ----, respectively, while 16 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 1, 1], are n -1 + n/4, and , ----, respectively, while the asymptotic correlation between\ 16 these two random variables is 1 and in floating point 1. ------------------------------------------------- ------------------------ This ends this article, that took, 0.370, 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, 4, with , 4, parts By Shalosh B. Ekhad ------------------------ This ends this article, that took, 0., 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, 5, 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, 4], nor the composition, [2, 3] Then infinity ----- 3 \ n x - x + 1 ) a(n) x = ------------------------------------ / 3 2 ----- (-1 + x) (x + 1) (x - x + 2 x - 1) n = 0 and in Maple format (x^3-x+1)/(-1+x)/(x+1)/(x^3-x^2+2*x-1) The asymptotic expression for a(n) is, n 0.87149424103766518898 1.7548776662466927601 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 4] and d occurrences (as containment) of the composition, [2, 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 4 4 3 3 / 5 4 5 X1 X2 x - X2 x + X2 x - x + x - 1) / (X1 X2 x - X1 X2 x - X1 x / 5 4 5 3 4 3 - X2 x + 2 X2 x + x - X2 x - x + x - 2 x + 1) and in Maple format -(X1*X2*x^4-X2*x^4+X2*x^3-x^3+x-1)/(X1*X2*x^5-X1*X2*x^4-X1*x^5-X2*x^5+2*X2*x^4+ x^5-X2*x^3-x^4+x^3-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 4], are n 9 n - 1/4 + ----, and , - 3/32 + ---, respectively, while 16 256 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 3], are n 13 13 n - 1/4 + ----, and , - -- + ----, respectively, while the asymptotic correlat\ 16 64 256 ion between these two random variables is 1/2 13 ----- 13 and in floating point 0.2773500981 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 13 ate normal pair with correlation, ----- 13 1/2 13 15 i.e. , [[-----, 0], [0, --]] 13 13 ------------------------------------------------- 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, 4], nor the composition, [3, 2] Then infinity ----- 4 2 \ n x + x - x + 1 ) a(n) x = - ----------------- / 3 2 ----- x - x + 2 x - 1 n = 0 and in Maple format -(x^4+x^2-x+1)/(x^3-x^2+2*x-1) The asymptotic expression for a(n) is, n 0.82299117732529152682 1.7548776662466927601 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 4] and d occurrences (as containment) of the composition, [3, 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 6 4 6 4 3 3 / X1 X2 x - X1 x - X2 x + X1 X2 x + x - X2 x + X2 x - x + x - 1) / / 5 4 5 5 4 5 3 4 3 (X1 X2 x - X1 X2 x - X1 x - X2 x + 2 X2 x + x - X2 x - x + x - 2 x + 1) and in Maple format -(X1*X2*x^6-X1*x^6-X2*x^6+X1*X2*x^4+x^6-X2*x^4+X2*x^3-x^3+x-1)/(X1*X2*x^5-X1*X2 *x^4-X1*x^5-X2*x^5+2*X2*x^4+x^5-X2*x^3-x^4+x^3-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 4], are n 9 n - 1/4 + ----, and , - 3/32 + ---, respectively, while 16 256 Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 2], are n 13 13 n - 1/4 + ----, and , - -- + ----, respectively, while the asymptotic correlat\ 16 64 256 ion between these two random variables is 1/2 13 ----- 13 and in floating point 0.2773500981 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 13 ate normal pair with correlation, ----- 13 1/2 13 15 i.e. , [[-----, 0], [0, --]] 13 13 ------------------------------------------------- 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, 4], nor the composition, [4, 1] Then infinity ----- 3 2 4 2 \ n (x + x - 1) (x + x - x + 1) ) a(n) x = - ------------------------------- / 3 2 ----- (-1 + x) (x + x + x - 1) n = 0 and in Maple format -(x^3+x^2-1)*(x^4+x^2-x+1)/(-1+x)/(x^3+x^2+x-1) The asymptotic expression for a(n) is, n 0.61841992231939255096 1.8392867552141611326 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 4] and d occurrences (as containment) of the composition, [4, 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 7 5 6 6 7 4 5 5 - X1 x - X2 x + X1 X2 x - X1 x - X2 x + x + X1 X2 x - X1 x - X2 x 6 5 4 / 4 4 + x + x - x + x - 1) / (X1 X2 x - x + 2 x - 1) / and in Maple format (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+X1*X2*x^4-X1*x^5 -X2*x^5+x^6+x^5-x^4+x-1)/(X1*X2*x^4-x^4+2*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 4], are n 9 n - 1/4 + ----, and , - 3/32 + ---, respectively, while 16 256 Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 1], are n 9 n - 1/4 + ----, and , - 3/32 + ---, respectively, while the asymptotic correla\ 16 256 tion between these two random variables is 1 and in floating point 1. ------------------------------------------------- Theorem Number, 4, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 3], nor the composition, [3, 2] Then infinity ----- 4 3 \ n x + x + 1 ) a(n) x = - -------------------- / 5 4 2 ----- x + x + x + x - 1 n = 0 and in Maple format -(x^4+x^3+1)/(x^5+x^4+x^2+x-1) The asymptotic expression for a(n) is, n 0.70545156891083810848 1.8124036192680426608 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 3] and d occurrences (as containment) of the composition, [3, 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 5 5 5 3 5 3 X1 X2 x - X1 x - X2 x + X1 X2 x + x - x + x - 1 - ----------------------------------------------------------------------- 6 6 6 4 6 3 4 3 X1 X2 x - X1 x - X2 x + X1 X2 x + x - X1 X2 x - x + x - 2 x + 1 and in Maple format -(X1*X2*x^5-X1*x^5-X2*x^5+X1*X2*x^3+x^5-x^3+x-1)/(X1*X2*x^6-X1*x^6-X2*x^6+X1*X2 *x^4+x^6-X1*X2*x^3-x^4+x^3-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 3], are n 13 13 n - 1/4 + ----, and , - -- + ----, respectively, while 16 64 256 Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 2], are n 13 13 n - 1/4 + ----, and , - -- + ----, respectively, while the asymptotic correlat\ 16 64 256 ion between these two random variables is 9/13 and in floating point 0.6923076923 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ ate normal pair with correlation, 9/13 331 i.e. , [[9/13, 0], [0, ---]] 169 ------------------------------------------------- Theorem Number, 5, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 3], nor the composition, [4, 1] Then infinity ----- 4 2 \ n x + x - x + 1 ) a(n) x = - ----------------- / 3 2 ----- x - x + 2 x - 1 n = 0 and in Maple format -(x^4+x^2-x+1)/(x^3-x^2+2*x-1) The asymptotic expression for a(n) is, n 0.82299117732529152682 1.7548776662466927601 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 3] and d occurrences (as containment) of the composition, [4, 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 6 6 6 4 6 4 3 3 / X1 X2 x - X1 x - X2 x + X1 X2 x + x - X1 x + X1 x - x + x - 1) / / 5 4 5 5 4 5 3 4 3 (X1 X2 x - X1 X2 x - X1 x - X2 x + 2 X1 x + x - X1 x - x + x - 2 x + 1) and in Maple format -(X1*X2*x^6-X1*x^6-X2*x^6+X1*X2*x^4+x^6-X1*x^4+X1*x^3-x^3+x-1)/(X1*X2*x^5-X1*X2 *x^4-X1*x^5-X2*x^5+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, [2, 3], are n 13 13 n - 1/4 + ----, and , - -- + ----, respectively, while 16 64 256 Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 1], are n 9 n - 1/4 + ----, and , - 3/32 + ---, respectively, while the asymptotic correla\ 16 256 tion between these two random variables is 1/2 13 ----- 13 and in floating point 0.2773500981 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 13 ate normal pair with correlation, ----- 13 1/2 13 15 i.e. , [[-----, 0], [0, --]] 13 13 ------------------------------------------------- Theorem Number, 6, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [3, 2], nor the composition, [4, 1] Then infinity ----- 3 \ n x - x + 1 ) a(n) x = ------------------------------------ / 3 2 ----- (-1 + x) (x + 1) (x - x + 2 x - 1) n = 0 and in Maple format (x^3-x+1)/(-1+x)/(x+1)/(x^3-x^2+2*x-1) The asymptotic expression for a(n) is, n 0.87149424103766518898 1.7548776662466927601 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [3, 2] and d occurrences (as containment) of the composition, [4, 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 4 4 3 3 / 5 4 5 X1 X2 x - X1 x + X1 x - x + x - 1) / (X1 X2 x - X1 X2 x - X1 x / 5 4 5 3 4 3 - X2 x + 2 X1 x + x - X1 x - x + x - 2 x + 1) and in Maple format -(X1*X2*x^4-X1*x^4+X1*x^3-x^3+x-1)/(X1*X2*x^5-X1*X2*x^4-X1*x^5-X2*x^5+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, 2], are n 13 13 n - 1/4 + ----, and , - -- + ----, respectively, while 16 64 256 Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 1], are n 9 n - 1/4 + ----, and , - 3/32 + ---, respectively, while the asymptotic correla\ 16 256 tion between these two random variables is 1/2 13 ----- 13 and in floating point 0.2773500981 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 13 ate normal pair with correlation, ----- 13 1/2 13 15 i.e. , [[-----, 0], [0, --]] 13 13 ------------------------------------------------- ------------------------ This ends this article, that took, 0.543, 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, 5, 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, 3], nor the composition, [1, 2, 2] Then infinity ----- 5 4 3 2 \ n x - 2 x + x - 2 x + 2 x - 1 ) a(n) x = ------------------------------- / 3 2 ----- (x + x - 1) (-1 + x) n = 0 and in Maple format (x^5-2*x^4+x^3-2*x^2+2*x-1)/(x^3+x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 3.2383381116555121159 1.4655712318767680267 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 1, 3] and d occurrences (as containment) of the composition, [1, 2, 2], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 5 4 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x - X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 5 5 3 4 4 5 3 4 2 - X1 x - X2 x + X1 X2 x + X1 x + 2 X2 x + x - 2 X2 x - 2 x + X2 x 3 2 / 4 3 4 4 + x - 2 x + 2 x - 1) / ((-1 + x) (X1 X2 x - X1 X2 x - X1 x - X2 x / 3 4 2 3 2 + 2 X2 x + x - X2 x - x + x - 2 x + 1)) and in Maple format (X1*X2*x^5-X1*X2*x^4-X1*x^5-X2*x^5+X1*X2*x^3+X1*x^4+2*X2*x^4+x^5-2*X2*x^3-2*x^4 +X2*x^2+x^3-2*x^2+2*x-1)/(-1+x)/(X1*X2*x^4-X1*X2*x^3-X1*x^4-X2*x^4+2*X2*x^3+x^4 -X2*x^2-x^3+x^2-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 1, 3], are 3 n - 5/8 + n/8, and , - 7/64 + ---, respectively, while 64 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 2, 2], are 35 7 n - 5/8 + n/8, and , - -- + ---, respectively, while the asymptotic correlatio\ 64 64 n between these two random variables is 1/2 1/2 3 7 --------- 21 and in floating point 0.2182178903 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 21 ate normal pair with correlation, ----- 21 1/2 21 23 i.e. , [[-----, 0], [0, --]] 21 21 ------------------------------------------------- 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, 3], nor the composition, [1, 3, 1] Then infinity ----- 6 5 3 2 \ n x + x - x - x + 2 x - 1 ) a(n) x = --------------------------- / 2 2 ----- (x + x - 1) (-1 + x) n = 0 and in Maple format (x^6+x^5-x^3-x^2+2*x-1)/(x^2+x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 1.1708203932499369089 1.6180339887498948482 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 1, 3] and d occurrences (as containment) of the composition, [1, 3, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 6 5 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x + X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 6 6 5 5 6 3 5 3 2 - X1 x - X2 x - X1 x - X2 x + x + X1 X2 x + x - x - x + 2 x - 1) / 3 3 / ((-1 + x) (X1 X2 x - x + 2 x - 1)) / and in Maple format -(X1*X2*x^6+X1*X2*x^5-X1*x^6-X2*x^6-X1*x^5-X2*x^5+x^6+X1*X2*x^3+x^5-x^3-x^2+2*x -1)/(-1+x)/(X1*X2*x^3-x^3+2*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 1, 3], are 3 n - 5/8 + n/8, and , - 7/64 + ---, respectively, while 64 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 3, 1], are 3 n - 5/8 + n/8, and , - 7/64 + ---, respectively, while the asymptotic correlat\ 64 ion between these two random variables is 1 and in floating point 1. ------------------------------------------------- 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, 3], nor the composition, [2, 1, 2] Then infinity ----- 6 5 2 \ n x - x - 2 x + 2 x - 1 ) a(n) x = ------------------------------------------- / 2 3 2 ----- (x + 1) (x - x + 1) (x + x - 1) (-1 + x) n = 0 and in Maple format (x^6-x^5-2*x^2+2*x-1)/(x+1)/(x^2-x+1)/(x^3+x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 2.8211001237292933382 1.4655712318767680267 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 1, 3] and d occurrences (as containment) of the composition, [2, 1, 2], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 7 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 7 2 7 2 6 7 2 7 6 - X1 X2 x - 2 X1 X2 x + 2 X1 X2 x + 2 X1 X2 x + X2 x - 3 X1 X2 x 2 6 7 6 2 5 6 5 6 3 - 2 X2 x - X2 x + X1 x + X2 x + 3 X2 x - 2 X2 x - x - X1 X2 x 5 3 2 2 / 4 3 4 + x + X2 x - X2 x + 2 x - 2 x + 1) / ((X1 X2 x - X1 X2 x - X1 x / 4 3 4 2 3 2 - X2 x + 2 X2 x + x - X2 x - x + x - 2 x + 1) 4 4 4 3 4 3 (X1 X2 x - X1 x - X2 x + X2 x + x - x + x - 1)) and in Maple format -(X1^2*X2^2*x^7-X1^2*X2*x^7-2*X1*X2^2*x^7+2*X1*X2^2*x^6+2*X1*X2*x^7+X2^2*x^7-3* X1*X2*x^6-2*X2^2*x^6-X2*x^7+X1*x^6+X2^2*x^5+3*X2*x^6-2*X2*x^5-x^6-X1*X2*x^3+x^5 +X2*x^3-X2*x^2+2*x^2-2*x+1)/(X1*X2*x^4-X1*X2*x^3-X1*x^4-X2*x^4+2*X2*x^3+x^4-X2* x^2-x^3+x^2-2*x+1)/(X1*X2*x^4-X1*x^4-X2*x^4+X2*x^3+x^4-x^3+x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 1, 3], are 3 n - 5/8 + n/8, and , - 7/64 + ---, respectively, while 64 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 1, 2], are 43 7 n - 5/8 + n/8, and , - -- + ---, respectively, while the asymptotic correlatio\ 64 64 n between these two random variables is 1/2 1/2 3 7 --------- 21 and in floating point 0.2182178903 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 21 ate normal pair with correlation, ----- 21 1/2 21 23 i.e. , [[-----, 0], [0, --]] 21 21 ------------------------------------------------- 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, 3], nor the composition, [2, 2, 1] Then infinity ----- 7 6 5 4 3 2 \ n x + x + x - 2 x + x - 2 x + 2 x - 1 ) a(n) x = ----------------------------------------- / 3 2 ----- (x + x - 1) (-1 + x) n = 0 and in Maple format (x^7+x^6+x^5-2*x^4+x^3-2*x^2+2*x-1)/(x^3+x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 2.2096081317784808200 1.4655712318767680267 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 1, 3] and d occurrences (as containment) of the composition, [2, 2, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 8 2 8 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x - X1 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 8 8 8 8 5 4 5 - 2 X1 X2 x + 2 X1 x + X2 x - x + 2 X1 X2 x - 2 X1 X2 x - 2 X1 x 5 3 4 4 5 3 4 2 - 3 X2 x + X1 X2 x + X1 x + 4 X2 x + 3 x - 3 X2 x - 3 x + X2 x 3 2 / 2 4 3 4 + 3 x - 4 x + 3 x - 1) / ((-1 + x) (X1 X2 x - X1 X2 x - X1 x / 4 3 4 2 3 2 - X2 x + 2 X2 x + x - X2 x - x + x - 2 x + 1)) and in Maple format -(X1^2*X2*x^8-X1^2*x^8-2*X1*X2*x^8+2*X1*x^8+X2*x^8-x^8+2*X1*X2*x^5-2*X1*X2*x^4-\ 2*X1*x^5-3*X2*x^5+X1*X2*x^3+X1*x^4+4*X2*x^4+3*x^5-3*X2*x^3-3*x^4+X2*x^2+3*x^3-4 *x^2+3*x-1)/(-1+x)^2/(X1*X2*x^4-X1*X2*x^3-X1*x^4-X2*x^4+2*X2*x^3+x^4-X2*x^2-x^3 +x^2-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 1, 3], are 3 n - 5/8 + n/8, and , - 7/64 + ---, respectively, while 64 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 2, 1], are 35 7 n - 5/8 + n/8, and , - -- + ---, respectively, while the asymptotic correlatio\ 64 64 n between these two random variables is 1/2 1/2 3 7 --------- 21 and in floating point 0.2182178903 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 21 ate normal pair with correlation, ----- 21 1/2 21 23 i.e. , [[-----, 0], [0, --]] 21 21 ------------------------------------------------- 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, 3], nor the composition, [3, 1, 1] Then infinity ----- 10 9 7 6 5 3 2 \ n x + 2 x - 2 x - 2 x - x + x + x - 2 x + 1 ) a(n) x = - ------------------------------------------------- / 2 2 ----- (x + x - 1) (-1 + x) n = 0 and in Maple format -(x^10+2*x^9-2*x^7-2*x^6-x^5+x^3+x^2-2*x+1)/(x^2+x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.72360679774997896964 1.6180339887498948482 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 1, 3] and d occurrences (as containment) of the composition, [3, 1, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 11 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 2 10 2 11 2 11 2 10 2 11 + X1 X2 x - 2 X1 X2 x - 2 X1 X2 x - 2 X1 X2 x + X1 x 2 10 11 2 11 2 9 2 10 2 9 - 2 X1 X2 x + 4 X1 X2 x + X2 x + X1 X2 x + X1 x + X1 X2 x 10 11 2 10 11 2 8 2 9 + 4 X1 X2 x - 2 X1 x + X2 x - 2 X2 x + X1 X2 x - X1 x 2 8 9 10 2 9 10 11 2 8 + X1 X2 x - 4 X1 X2 x - 2 X1 x - X2 x - 2 X2 x + x - X1 x 8 9 2 8 9 10 8 8 9 - 4 X1 X2 x + 3 X1 x - X2 x + 3 X2 x + x + 3 X1 x + 3 X2 x - 2 x 6 8 5 6 6 4 5 5 + X1 X2 x - 2 x + X1 X2 x - X1 x - X2 x - X1 X2 x - X1 x - X2 x 6 3 5 4 2 / 2 + x + X1 X2 x + x + x - 3 x + 3 x - 1) / ((-1 + x) / 3 3 (X1 X2 x - x + 2 x - 1)) and in Maple format (X1^2*X2^2*x^11+X1^2*X2^2*x^10-2*X1^2*X2*x^11-2*X1*X2^2*x^11-2*X1^2*X2*x^10+X1^ 2*x^11-2*X1*X2^2*x^10+4*X1*X2*x^11+X2^2*x^11+X1^2*X2*x^9+X1^2*x^10+X1*X2^2*x^9+ 4*X1*X2*x^10-2*X1*x^11+X2^2*x^10-2*X2*x^11+X1^2*X2*x^8-X1^2*x^9+X1*X2^2*x^8-4* X1*X2*x^9-2*X1*x^10-X2^2*x^9-2*X2*x^10+x^11-X1^2*x^8-4*X1*X2*x^8+3*X1*x^9-X2^2* x^8+3*X2*x^9+x^10+3*X1*x^8+3*X2*x^8-2*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-X2*x^5+x^6+X1*X2*x^3+x^5+x^4-3*x^2+3*x-1)/(-1+x)^2/(X1*X2*x ^3-x^3+2*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 1, 3], are 3 n - 5/8 + n/8, and , - 7/64 + ---, respectively, while 64 Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 1, 1], are 3 n - 5/8 + n/8, and , - 7/64 + ---, respectively, while the asymptotic correlat\ 64 ion between these two random variables is 1 and in floating point 1. ------------------------------------------------- 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, 2, 2], nor the composition, [1, 3, 1] Then infinity ----- 5 4 3 2 \ n x - 2 x + x - 2 x + 2 x - 1 ) a(n) x = ------------------------------- / 3 2 ----- (x + x - 1) (-1 + x) n = 0 and in Maple format (x^5-2*x^4+x^3-2*x^2+2*x-1)/(x^3+x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 3.2383381116555121159 1.4655712318767680267 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 2, 2] and d occurrences (as containment) of the composition, [1, 3, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 5 4 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x - X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 5 5 3 4 4 5 3 4 2 - X1 x - X2 x + X1 X2 x + 2 X1 x + X2 x + x - 2 X1 x - 2 x + X1 x 3 2 / 4 3 4 4 + x - 2 x + 2 x - 1) / ((-1 + x) (X1 X2 x - X1 X2 x - X1 x - X2 x / 3 4 2 3 2 + 2 X1 x + x - X1 x - x + x - 2 x + 1)) and in Maple format (X1*X2*x^5-X1*X2*x^4-X1*x^5-X2*x^5+X1*X2*x^3+2*X1*x^4+X2*x^4+x^5-2*X1*x^3-2*x^4 +X1*x^2+x^3-2*x^2+2*x-1)/(-1+x)/(X1*X2*x^4-X1*X2*x^3-X1*x^4-X2*x^4+2*X1*x^3+x^4 -X1*x^2-x^3+x^2-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 2, 2], are 35 7 n - 5/8 + n/8, and , - -- + ---, respectively, while 64 64 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 3, 1], are 3 n - 5/8 + n/8, and , - 7/64 + ---, respectively, while the asymptotic correlat\ 64 ion between these two random variables is 1/2 1/2 3 7 --------- 21 and in floating point 0.2182178903 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 21 ate normal pair with correlation, ----- 21 1/2 21 23 i.e. , [[-----, 0], [0, --]] 21 21 ------------------------------------------------- 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, 2, 2], nor the composition, [2, 1, 2] Then infinity ----- 2 \ n 2 x - 2 x + 1 ) a(n) x = ---------------------------------- / 2 2 ----- (-1 + x) (x - x + 1) (x + x - 1) n = 0 and in Maple format (2*x^2-2*x+1)/(-1+x)/(x^2-x+1)/(x^2+x-1) The asymptotic expression for a(n) is, n 1.3090169943749474241 1.6180339887498948482 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 2, 2] and d occurrences (as containment) of the composition, [2, 1, 2], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 5 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 5 4 5 3 4 2 3 - 2 X1 X2 x - X1 X2 x + X1 x + X1 X2 x + X2 x - X1 X2 x - X2 x 2 / 2 6 2 5 6 5 + 2 x - 2 x + 1) / (X1 X2 x - X1 X2 x - 2 X1 X2 x + X1 X2 x / 6 4 5 3 4 5 2 3 + X1 x + X1 X2 x + X2 x - 2 X1 X2 x - 2 X2 x - x + X1 X2 x + X2 x 4 3 2 + x + x - 3 x + 3 x - 1) and in Maple format -(X1*X2^2*x^5-2*X1*X2*x^5-X1*X2*x^4+X1*x^5+X1*X2*x^3+X2*x^4-X1*X2*x^2-X2*x^3+2* x^2-2*x+1)/(X1*X2^2*x^6-X1*X2^2*x^5-2*X1*X2*x^6+X1*X2*x^5+X1*x^6+X1*X2*x^4+X2*x ^5-2*X1*X2*x^3-2*X2*x^4-x^5+X1*X2*x^2+X2*x^3+x^4+x^3-3*x^2+3*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 2, 2], are 35 7 n - 5/8 + n/8, and , - -- + ---, respectively, while 64 64 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 1, 2], are 43 7 n - 5/8 + n/8, and , - -- + ---, respectively, while the asymptotic correlatio\ 64 64 n between these two random variables is 5/7 and in floating point 0.7142857143 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ ate normal pair with correlation, 5/7 99 i.e. , [[5/7, 0], [0, --]] 49 ------------------------------------------------- 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, 2, 2], nor the composition, [2, 2, 1] Then infinity ----- 7 6 5 4 3 2 \ n x - x + 2 x - 2 x + 3 x - 4 x + 3 x - 1 ) a(n) x = --------------------------------------------- / 3 2 2 ----- (x - x + 2 x - 1) (-1 + x) n = 0 and in Maple format (x^7-x^6+2*x^5-2*x^4+3*x^3-4*x^2+3*x-1)/(x^3-x^2+2*x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.72212441830311284116 1.7548776662466927601 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 2, 2] and d occurrences (as containment) of the composition, [2, 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 7 6 6 7 4 5 5 6 - X1 x - X2 x + X1 x + X2 x + x + 2 X1 X2 x - X1 x - X2 x - x 3 5 2 4 3 2 / - 2 X1 X2 x + 2 x + X1 X2 x - 2 x + 3 x - 4 x + 3 x - 1) / ( / 2 3 2 3 2 (-1 + x) (X1 X2 x - X1 X2 x - x + x - 2 x + 1)) and in Maple format -(X1*X2*x^7-X1*X2*x^6-X1*x^7-X2*x^7+X1*x^6+X2*x^6+x^7+2*X1*X2*x^4-X1*x^5-X2*x^5 -x^6-2*X1*X2*x^3+2*x^5+X1*X2*x^2-2*x^4+3*x^3-4*x^2+3*x-1)/(-1+x)^2/(X1*X2*x^3- X1*X2*x^2-x^3+x^2-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 2, 2], are 35 7 n - 5/8 + n/8, and , - -- + ---, respectively, while 64 64 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 2, 1], are 35 7 n - 5/8 + n/8, and , - -- + ---, respectively, while the asymptotic correlatio\ 64 64 n between these two random variables is 1 and in floating point 1. ------------------------------------------------- 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, 2, 2], nor the composition, [3, 1, 1] Then infinity ----- 7 6 5 4 3 2 \ n x + x + x - 2 x + x - 2 x + 2 x - 1 ) a(n) x = ----------------------------------------- / 3 2 ----- (x + x - 1) (-1 + x) n = 0 and in Maple format (x^7+x^6+x^5-2*x^4+x^3-2*x^2+2*x-1)/(x^3+x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 2.2096081317784808200 1.4655712318767680267 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 2, 2] and d occurrences (as containment) of the composition, [3, 1, 1], (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 2 8 8 8 8 5 4 - 2 X1 X2 x - X2 x + X1 x + 2 X2 x - x + 2 X1 X2 x - 2 X1 X2 x 5 5 3 4 4 5 3 4 - 3 X1 x - 2 X2 x + X1 X2 x + 4 X1 x + X2 x + 3 x - 3 X1 x - 3 x 2 3 2 / 2 4 3 + X1 x + 3 x - 4 x + 3 x - 1) / ((-1 + x) (X1 X2 x - X1 X2 x / 4 4 3 4 2 3 2 - X1 x - X2 x + 2 X1 x + x - X1 x - x + x - 2 x + 1)) and in Maple format -(X1*X2^2*x^8-2*X1*X2*x^8-X2^2*x^8+X1*x^8+2*X2*x^8-x^8+2*X1*X2*x^5-2*X1*X2*x^4-\ 3*X1*x^5-2*X2*x^5+X1*X2*x^3+4*X1*x^4+X2*x^4+3*x^5-3*X1*x^3-3*x^4+X1*x^2+3*x^3-4 *x^2+3*x-1)/(-1+x)^2/(X1*X2*x^4-X1*X2*x^3-X1*x^4-X2*x^4+2*X1*x^3+x^4-X1*x^2-x^3 +x^2-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 2, 2], are 35 7 n - 5/8 + n/8, and , - -- + ---, respectively, while 64 64 Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 1, 1], are 3 n - 5/8 + n/8, and , - 7/64 + ---, respectively, while the asymptotic correlat\ 64 ion between these two random variables is 1/2 1/2 3 7 --------- 21 and in floating point 0.2182178903 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 21 ate normal pair with correlation, ----- 21 1/2 21 23 i.e. , [[-----, 0], [0, --]] 21 21 ------------------------------------------------- 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, 3, 1], nor the composition, [2, 1, 2] Then infinity ----- 8 7 6 5 4 3 2 \ n x - x + 2 x - 4 x + 5 x - 5 x + 5 x - 3 x + 1 ) a(n) x = - ---------------------------------------------------- / 2 3 2 ----- (x - x + 1) (x + x - 1) (-1 + x) n = 0 and in Maple format -(x^8-x^7+2*x^6-4*x^5+5*x^4-5*x^3+5*x^2-3*x+1)/(x^2-x+1)/(x^3+x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 2.8211001237292933384 1.4655712318767680267 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 3, 1] and d occurrences (as containment) of the composition, [2, 1, 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 2 9 2 10 2 10 2 2 8 2 9 - X1 X2 x - 2 X1 X2 x - 2 X1 X2 x + X1 X2 x + X1 X2 x 2 10 2 9 10 2 10 2 2 7 2 8 + X1 x + 3 X1 X2 x + 4 X1 X2 x + X2 x - X1 X2 x - X1 X2 x 2 8 9 10 2 9 10 2 7 - 3 X1 X2 x - 4 X1 X2 x - 2 X1 x - 2 X2 x - 2 X2 x + X1 X2 x 2 7 8 9 2 8 9 10 + 4 X1 X2 x + 4 X1 X2 x + X1 x + 2 X2 x + 3 X2 x + x 2 6 7 8 2 7 8 9 6 - 2 X1 X2 x - 7 X1 X2 x - X1 x - 3 X2 x - 3 X2 x - x + 4 X1 X2 x 7 2 6 7 8 6 2 5 6 7 + 3 X1 x + 3 X2 x + 6 X2 x + x - 2 X1 x - X2 x - 6 X2 x - 3 x 4 5 6 3 4 5 3 2 - X1 X2 x + 2 X2 x + 3 x + X1 X2 x + X2 x - x - 2 X2 x + X2 x 3 2 / + 2 x - 4 x + 3 x - 1) / ((-1 + x) / 4 4 4 3 4 3 4 3 (X1 X2 x - X1 x - X2 x + X2 x + x - x + x - 1) (X1 X2 x - X1 X2 x 4 4 3 4 2 3 2 - X1 x - X2 x + 2 X2 x + x - X2 x - x + x - 2 x + 1)) and in Maple format -(X1^2*X2^2*x^10-X1^2*X2^2*x^9-2*X1^2*X2*x^10-2*X1*X2^2*x^10+X1^2*X2^2*x^8+X1^2 *X2*x^9+X1^2*x^10+3*X1*X2^2*x^9+4*X1*X2*x^10+X2^2*x^10-X1^2*X2^2*x^7-X1^2*X2*x^ 8-3*X1*X2^2*x^8-4*X1*X2*x^9-2*X1*x^10-2*X2^2*x^9-2*X2*x^10+X1^2*X2*x^7+4*X1*X2^ 2*x^7+4*X1*X2*x^8+X1*x^9+2*X2^2*x^8+3*X2*x^9+x^10-2*X1*X2^2*x^6-7*X1*X2*x^7-X1* x^8-3*X2^2*x^7-3*X2*x^8-x^9+4*X1*X2*x^6+3*X1*x^7+3*X2^2*x^6+6*X2*x^7+x^8-2*X1*x ^6-X2^2*x^5-6*X2*x^6-3*x^7-X1*X2*x^4+2*X2*x^5+3*x^6+X1*X2*x^3+X2*x^4-x^5-2*X2*x ^3+X2*x^2+2*x^3-4*x^2+3*x-1)/(-1+x)/(X1*X2*x^4-X1*x^4-X2*x^4+X2*x^3+x^4-x^3+x-1 )/(X1*X2*x^4-X1*X2*x^3-X1*x^4-X2*x^4+2*X2*x^3+x^4-X2*x^2-x^3+x^2-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 3, 1], are 3 n - 5/8 + n/8, and , - 7/64 + ---, respectively, while 64 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 1, 2], are 43 7 n - 5/8 + n/8, and , - -- + ---, respectively, while the asymptotic correlatio\ 64 64 n between these two random variables is 1/2 1/2 3 7 --------- 21 and in floating point 0.2182178903 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 21 ate normal pair with correlation, ----- 21 1/2 21 23 i.e. , [[-----, 0], [0, --]] 21 21 ------------------------------------------------- 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, 3, 1], nor the composition, [2, 2, 1] Then infinity ----- 5 4 3 2 \ n x - 2 x + x - 2 x + 2 x - 1 ) a(n) x = ------------------------------- / 3 2 ----- (x + x - 1) (-1 + x) n = 0 and in Maple format (x^5-2*x^4+x^3-2*x^2+2*x-1)/(x^3+x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 3.2383381116555121159 1.4655712318767680267 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 3, 1] and d occurrences (as containment) of the composition, [2, 2, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 5 4 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x - X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 5 5 3 4 4 5 3 4 2 - X1 x - X2 x + X1 X2 x + X1 x + 2 X2 x + x - 2 X2 x - 2 x + X2 x 3 2 / 4 3 4 4 + x - 2 x + 2 x - 1) / ((-1 + x) (X1 X2 x - X1 X2 x - X1 x - X2 x / 3 4 2 3 2 + 2 X2 x + x - X2 x - x + x - 2 x + 1)) and in Maple format (X1*X2*x^5-X1*X2*x^4-X1*x^5-X2*x^5+X1*X2*x^3+X1*x^4+2*X2*x^4+x^5-2*X2*x^3-2*x^4 +X2*x^2+x^3-2*x^2+2*x-1)/(-1+x)/(X1*X2*x^4-X1*X2*x^3-X1*x^4-X2*x^4+2*X2*x^3+x^4 -X2*x^2-x^3+x^2-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 3, 1], are 3 n - 5/8 + n/8, and , - 7/64 + ---, respectively, while 64 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 2, 1], are 35 7 n - 5/8 + n/8, and , - -- + ---, respectively, while the asymptotic correlatio\ 64 64 n between these two random variables is 1/2 1/2 3 7 --------- 21 and in floating point 0.2182178903 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 21 ate normal pair with correlation, ----- 21 1/2 21 23 i.e. , [[-----, 0], [0, --]] 21 21 ------------------------------------------------- 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, 3, 1], nor the composition, [3, 1, 1] Then infinity ----- 6 5 3 2 \ n x + x - x - x + 2 x - 1 ) a(n) x = --------------------------- / 2 2 ----- (x + x - 1) (-1 + x) n = 0 and in Maple format (x^6+x^5-x^3-x^2+2*x-1)/(x^2+x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 1.1708203932499369089 1.6180339887498948482 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 3, 1] and d occurrences (as containment) of the composition, [3, 1, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 6 5 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x + X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 6 6 5 5 6 3 5 3 2 - X1 x - X2 x - X1 x - X2 x + x + X1 X2 x + x - x - x + 2 x - 1) / 3 3 / ((-1 + x) (X1 X2 x - x + 2 x - 1)) / and in Maple format -(X1*X2*x^6+X1*X2*x^5-X1*x^6-X2*x^6-X1*x^5-X2*x^5+x^6+X1*X2*x^3+x^5-x^3-x^2+2*x -1)/(-1+x)/(X1*X2*x^3-x^3+2*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 3, 1], are 3 n - 5/8 + n/8, and , - 7/64 + ---, respectively, while 64 Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 1, 1], are 3 n - 5/8 + n/8, and , - 7/64 + ---, respectively, while the asymptotic correlat\ 64 ion between these two random variables is 1 and in floating point 1. ------------------------------------------------- Theorem Number, 13, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 1, 2], nor the composition, [2, 2, 1] Then infinity ----- 2 \ n 2 x - 2 x + 1 ) a(n) x = ---------------------------------- / 2 2 ----- (-1 + x) (x - x + 1) (x + x - 1) n = 0 and in Maple format (2*x^2-2*x+1)/(-1+x)/(x^2-x+1)/(x^2+x-1) The asymptotic expression for a(n) is, n 1.3090169943749474241 1.6180339887498948482 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 1, 2] and d occurrences (as containment) of the composition, [2, 2, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 5 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 5 4 5 3 4 2 3 - 2 X1 X2 x - X1 X2 x + X2 x + X1 X2 x + X1 x - X1 X2 x - X1 x 2 / 2 6 2 5 6 5 + 2 x - 2 x + 1) / (X1 X2 x - X1 X2 x - 2 X1 X2 x + X1 X2 x / 6 4 5 3 4 5 2 3 + X2 x + X1 X2 x + X1 x - 2 X1 X2 x - 2 X1 x - x + X1 X2 x + X1 x 4 3 2 + x + x - 3 x + 3 x - 1) and in Maple format -(X1^2*X2*x^5-2*X1*X2*x^5-X1*X2*x^4+X2*x^5+X1*X2*x^3+X1*x^4-X1*X2*x^2-X1*x^3+2* x^2-2*x+1)/(X1^2*X2*x^6-X1^2*X2*x^5-2*X1*X2*x^6+X1*X2*x^5+X2*x^6+X1*X2*x^4+X1*x ^5-2*X1*X2*x^3-2*X1*x^4-x^5+X1*X2*x^2+X1*x^3+x^4+x^3-3*x^2+3*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 1, 2], are 43 7 n - 5/8 + n/8, and , - -- + ---, respectively, while 64 64 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 2, 1], are 35 7 n - 5/8 + n/8, and , - -- + ---, respectively, while the asymptotic correlatio\ 64 64 n between these two random variables is 5/7 and in floating point 0.7142857143 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ ate normal pair with correlation, 5/7 99 i.e. , [[5/7, 0], [0, --]] 49 ------------------------------------------------- Theorem Number, 14, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 1, 2], nor the composition, [3, 1, 1] Then infinity ----- 6 5 2 \ n x - x - 2 x + 2 x - 1 ) a(n) x = ------------------------------------------- / 2 3 2 ----- (x + 1) (x - x + 1) (x + x - 1) (-1 + x) n = 0 and in Maple format (x^6-x^5-2*x^2+2*x-1)/(x+1)/(x^2-x+1)/(x^3+x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 2.8211001237292933382 1.4655712318767680267 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 1, 2] and d occurrences (as containment) of the composition, [3, 1, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 7 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 7 2 7 2 6 2 7 7 2 6 - 2 X1 X2 x - X1 X2 x + 2 X1 X2 x + X1 x + 2 X1 X2 x - 2 X1 x 6 7 2 5 6 6 5 6 3 - 3 X1 X2 x - X1 x + X1 x + 3 X1 x + X2 x - 2 X1 x - x - X1 X2 x 5 3 2 2 / + x + X1 x - X1 x + 2 x - 2 x + 1) / ( / 4 4 4 3 4 3 4 3 (X1 X2 x - X1 x - X2 x + X1 x + x - x + x - 1) (X1 X2 x - X1 X2 x 4 4 3 4 2 3 2 - X1 x - X2 x + 2 X1 x + x - X1 x - x + x - 2 x + 1)) and in Maple format -(X1^2*X2^2*x^7-2*X1^2*X2*x^7-X1*X2^2*x^7+2*X1^2*X2*x^6+X1^2*x^7+2*X1*X2*x^7-2* X1^2*x^6-3*X1*X2*x^6-X1*x^7+X1^2*x^5+3*X1*x^6+X2*x^6-2*X1*x^5-x^6-X1*X2*x^3+x^5 +X1*x^3-X1*x^2+2*x^2-2*x+1)/(X1*X2*x^4-X1*x^4-X2*x^4+X1*x^3+x^4-x^3+x-1)/(X1*X2 *x^4-X1*X2*x^3-X1*x^4-X2*x^4+2*X1*x^3+x^4-X1*x^2-x^3+x^2-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 1, 2], are 43 7 n - 5/8 + n/8, and , - -- + ---, respectively, while 64 64 Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 1, 1], are 3 n - 5/8 + n/8, and , - 7/64 + ---, respectively, while the asymptotic correlat\ 64 ion between these two random variables is 1/2 1/2 3 7 --------- 21 and in floating point 0.2182178903 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 21 ate normal pair with correlation, ----- 21 1/2 21 23 i.e. , [[-----, 0], [0, --]] 21 21 ------------------------------------------------- Theorem Number, 15, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 2, 1], nor the composition, [3, 1, 1] Then infinity ----- 5 4 3 2 \ n x - 2 x + x - 2 x + 2 x - 1 ) a(n) x = ------------------------------- / 3 2 ----- (x + x - 1) (-1 + x) n = 0 and in Maple format (x^5-2*x^4+x^3-2*x^2+2*x-1)/(x^3+x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 3.2383381116555121159 1.4655712318767680267 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 2, 1] and d occurrences (as containment) of the composition, [3, 1, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 5 4 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x - X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 5 5 3 4 4 5 3 4 2 - X1 x - X2 x + X1 X2 x + 2 X1 x + X2 x + x - 2 X1 x - 2 x + X1 x 3 2 / 4 3 4 4 + x - 2 x + 2 x - 1) / ((-1 + x) (X1 X2 x - X1 X2 x - X1 x - X2 x / 3 4 2 3 2 + 2 X1 x + x - X1 x - x + x - 2 x + 1)) and in Maple format (X1*X2*x^5-X1*X2*x^4-X1*x^5-X2*x^5+X1*X2*x^3+2*X1*x^4+X2*x^4+x^5-2*X1*x^3-2*x^4 +X1*x^2+x^3-2*x^2+2*x-1)/(-1+x)/(X1*X2*x^4-X1*X2*x^3-X1*x^4-X2*x^4+2*X1*x^3+x^4 -X1*x^2-x^3+x^2-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 2, 1], are 35 7 n - 5/8 + n/8, and , - -- + ---, respectively, while 64 64 Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 1, 1], are 3 n - 5/8 + n/8, and , - 7/64 + ---, respectively, while the asymptotic correlat\ 64 ion between these two random variables is 1/2 1/2 3 7 --------- 21 and in floating point 0.2182178903 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 21 ate normal pair with correlation, ----- 21 1/2 21 23 i.e. , [[-----, 0], [0, --]] 21 21 ------------------------------------------------- ------------------------ This ends this article, that took, 3.179, 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, 5, with , 4, 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, 1, 2], nor the composition, [1, 1, 2, 1] Then infinity ----- 4 2 \ n x + 2 x - 2 x + 1 ) a(n) x = - ------------------- / 3 ----- (-1 + x) n = 0 and in Maple format -(x^4+2*x^2-2*x+1)/(-1+x)^3 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 1, 1, 2] and d occurrences (as containment) of the composition, [1, 1, 2, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 5 4 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x + X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 5 5 3 5 2 4 3 2 - X1 x - X2 x - X1 X2 x + x + X1 X2 x - x + 2 x - 4 x + 3 x - 1) / 2 2 2 / ((-1 + x) (X1 X2 x - x + 2 x - 1)) / and in Maple format (X1*X2*x^5+X1*X2*x^4-X1*x^5-X2*x^5-X1*X2*x^3+x^5+X1*X2*x^2-x^4+2*x^3-4*x^2+3*x-\ 1)/(-1+x)^2/(X1*X2*x^2-x^2+2*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 1, 1, 2], are n - 3/2 + n/4, and , ----, respectively, while 16 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 1, 2, 1], are n - 3/2 + n/4, and , ----, respectively, while the asymptotic correlation betw\ 16 een these two random variables is 1 and in floating point 1. ------------------------------------------------- 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, 1, 2], nor the composition, [1, 2, 1, 1] Then infinity ----- 6 4 2 \ n x - x - 2 x + 2 x - 1 ) a(n) x = ------------------------ / 3 ----- (-1 + x) n = 0 and in Maple format (x^6-x^4-2*x^2+2*x-1)/(-1+x)^3 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 1, 1, 2] and d occurrences (as containment) of the composition, [1, 2, 1, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 8 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 8 2 8 2 7 2 8 2 7 8 - 2 X1 X2 x - 2 X1 X2 x + X1 X2 x + X1 x + X1 X2 x + 4 X1 X2 x 2 8 2 7 7 8 2 7 8 7 + X2 x - X1 x - 4 X1 X2 x - 2 X1 x - X2 x - 2 X2 x + 3 X1 x 7 8 7 4 5 5 3 5 + 3 X2 x + x - 2 x + 2 X1 X2 x - X1 x - X2 x - 2 X1 X2 x + 2 x 2 4 3 2 / 3 + X1 X2 x - 3 x + 6 x - 7 x + 4 x - 1) / ((-1 + x) / 2 2 (X1 X2 x - x + 2 x - 1)) and in Maple format -(X1^2*X2^2*x^8-2*X1^2*X2*x^8-2*X1*X2^2*x^8+X1^2*X2*x^7+X1^2*x^8+X1*X2^2*x^7+4* X1*X2*x^8+X2^2*x^8-X1^2*x^7-4*X1*X2*x^7-2*X1*x^8-X2^2*x^7-2*X2*x^8+3*X1*x^7+3* X2*x^7+x^8-2*x^7+2*X1*X2*x^4-X1*x^5-X2*x^5-2*X1*X2*x^3+2*x^5+X1*X2*x^2-3*x^4+6* x^3-7*x^2+4*x-1)/(-1+x)^3/(X1*X2*x^2-x^2+2*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 1, 1, 2], are n - 3/2 + n/4, and , ----, respectively, while 16 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 2, 1, 1], are n - 3/2 + n/4, and , ----, respectively, while the asymptotic correlation betw\ 16 een these two random variables is 1 and in floating point 1. ------------------------------------------------- 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, 1, 2], nor the composition, [2, 1, 1, 1] Then infinity ----- 8 7 6 4 2 \ n x - x - x + x + 2 x - 2 x + 1 ) a(n) x = - ---------------------------------- / 3 ----- (-1 + x) n = 0 and in Maple format -(x^8-x^7-x^6+x^4+2*x^2-2*x+1)/(-1+x)^3 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 1, 1, 2] and d occurrences (as containment) of the composition, [2, 1, 1, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 3 3 11 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 3 2 11 2 3 11 3 2 10 3 11 - 3 X1 X2 x - 3 X1 X2 x + 2 X1 X2 x + 3 X1 X2 x 2 3 10 2 2 11 3 11 3 10 3 11 + 2 X1 X2 x + 9 X1 X2 x + 3 X1 X2 x - 4 X1 X2 x - X1 x 2 2 10 2 11 3 10 2 11 3 11 - 12 X1 X2 x - 9 X1 X2 x - 4 X1 X2 x - 9 X1 X2 x - X2 x 3 9 3 10 2 2 9 2 10 2 11 + X1 X2 x + 2 X1 x + 3 X1 X2 x + 18 X1 X2 x + 3 X1 x 3 9 2 10 11 3 10 2 11 3 9 + X1 X2 x + 18 X1 X2 x + 9 X1 X2 x + 2 X2 x + 3 X2 x - X1 x 2 2 8 2 9 2 10 2 9 10 + X1 X2 x - 9 X1 X2 x - 8 X1 x - 9 X1 X2 x - 24 X1 X2 x 11 3 9 2 10 11 2 8 2 9 - 3 X1 x - X2 x - 8 X2 x - 3 X2 x - X1 X2 x + 6 X1 x 2 8 9 10 2 9 10 11 - X1 X2 x + 18 X1 X2 x + 10 X1 x + 6 X2 x + 10 X2 x + x 2 7 2 7 9 9 10 2 7 + X1 X2 x + X1 X2 x - 10 X1 x - 10 X2 x - 4 x - X1 x 7 8 2 7 8 9 7 7 8 - 3 X1 X2 x + X1 x - X2 x + X2 x + 5 x + 2 X1 x + 2 X2 x - x 5 6 6 7 4 5 5 6 - 2 X1 X2 x + X1 x + X2 x - x + 4 X1 X2 x - X1 x - X2 x - 2 x 3 5 2 4 3 2 / - 3 X1 X2 x + 5 x + X1 X2 x - 9 x + 13 x - 11 x + 5 x - 1) / ( / 4 2 2 (-1 + x) (X1 X2 x - x + 2 x - 1)) and in Maple format (X1^3*X2^3*x^11-3*X1^3*X2^2*x^11-3*X1^2*X2^3*x^11+2*X1^3*X2^2*x^10+3*X1^3*X2*x^ 11+2*X1^2*X2^3*x^10+9*X1^2*X2^2*x^11+3*X1*X2^3*x^11-4*X1^3*X2*x^10-X1^3*x^11-12 *X1^2*X2^2*x^10-9*X1^2*X2*x^11-4*X1*X2^3*x^10-9*X1*X2^2*x^11-X2^3*x^11+X1^3*X2* x^9+2*X1^3*x^10+3*X1^2*X2^2*x^9+18*X1^2*X2*x^10+3*X1^2*x^11+X1*X2^3*x^9+18*X1* X2^2*x^10+9*X1*X2*x^11+2*X2^3*x^10+3*X2^2*x^11-X1^3*x^9+X1^2*X2^2*x^8-9*X1^2*X2 *x^9-8*X1^2*x^10-9*X1*X2^2*x^9-24*X1*X2*x^10-3*X1*x^11-X2^3*x^9-8*X2^2*x^10-3* X2*x^11-X1^2*X2*x^8+6*X1^2*x^9-X1*X2^2*x^8+18*X1*X2*x^9+10*X1*x^10+6*X2^2*x^9+ 10*X2*x^10+x^11+X1^2*X2*x^7+X1*X2^2*x^7-10*X1*x^9-10*X2*x^9-4*x^10-X1^2*x^7-3* X1*X2*x^7+X1*x^8-X2^2*x^7+X2*x^8+5*x^9+2*X1*x^7+2*X2*x^7-x^8-2*X1*X2*x^5+X1*x^6 +X2*x^6-x^7+4*X1*X2*x^4-X1*x^5-X2*x^5-2*x^6-3*X1*X2*x^3+5*x^5+X1*X2*x^2-9*x^4+ 13*x^3-11*x^2+5*x-1)/(-1+x)^4/(X1*X2*x^2-x^2+2*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 1, 1, 2], are n - 3/2 + n/4, and , ----, respectively, while 16 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 1, 1, 1], are n - 3/2 + n/4, and , ----, respectively, while the asymptotic correlation betw\ 16 een these two random variables is 1 and in floating point 1. ------------------------------------------------- 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, 2, 1], nor the composition, [1, 2, 1, 1] Then infinity ----- 4 2 \ n x + 2 x - 2 x + 1 ) a(n) x = - ------------------- / 3 ----- (-1 + x) n = 0 and in Maple format -(x^4+2*x^2-2*x+1)/(-1+x)^3 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 1, 2, 1] and d occurrences (as containment) of the composition, [1, 2, 1, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 5 4 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x + X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 5 5 3 5 2 4 3 2 - X1 x - X2 x - X1 X2 x + x + X1 X2 x - x + 2 x - 4 x + 3 x - 1) / 2 2 2 / ((-1 + x) (X1 X2 x - x + 2 x - 1)) / and in Maple format (X1*X2*x^5+X1*X2*x^4-X1*x^5-X2*x^5-X1*X2*x^3+x^5+X1*X2*x^2-x^4+2*x^3-4*x^2+3*x-\ 1)/(-1+x)^2/(X1*X2*x^2-x^2+2*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 1, 2, 1], are n - 3/2 + n/4, and , ----, respectively, while 16 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 2, 1, 1], are n - 3/2 + n/4, and , ----, respectively, while the asymptotic correlation betw\ 16 een these two random variables is 1 and in floating point 1. ------------------------------------------------- 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, 2, 1], nor the composition, [2, 1, 1, 1] Then infinity ----- 6 4 2 \ n x - x - 2 x + 2 x - 1 ) a(n) x = ------------------------ / 3 ----- (-1 + x) n = 0 and in Maple format (x^6-x^4-2*x^2+2*x-1)/(-1+x)^3 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 1, 2, 1] and d occurrences (as containment) of the composition, [2, 1, 1, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 8 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 8 2 8 2 7 2 8 2 7 8 - 2 X1 X2 x - 2 X1 X2 x + X1 X2 x + X1 x + X1 X2 x + 4 X1 X2 x 2 8 2 7 7 8 2 7 8 7 + X2 x - X1 x - 4 X1 X2 x - 2 X1 x - X2 x - 2 X2 x + 3 X1 x 7 8 7 4 5 5 3 5 + 3 X2 x + x - 2 x + 2 X1 X2 x - X1 x - X2 x - 2 X1 X2 x + 2 x 2 4 3 2 / 3 + X1 X2 x - 3 x + 6 x - 7 x + 4 x - 1) / ((-1 + x) / 2 2 (X1 X2 x - x + 2 x - 1)) and in Maple format -(X1^2*X2^2*x^8-2*X1^2*X2*x^8-2*X1*X2^2*x^8+X1^2*X2*x^7+X1^2*x^8+X1*X2^2*x^7+4* X1*X2*x^8+X2^2*x^8-X1^2*x^7-4*X1*X2*x^7-2*X1*x^8-X2^2*x^7-2*X2*x^8+3*X1*x^7+3* X2*x^7+x^8-2*x^7+2*X1*X2*x^4-X1*x^5-X2*x^5-2*X1*X2*x^3+2*x^5+X1*X2*x^2-3*x^4+6* x^3-7*x^2+4*x-1)/(-1+x)^3/(X1*X2*x^2-x^2+2*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 1, 2, 1], are n - 3/2 + n/4, and , ----, respectively, while 16 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 1, 1, 1], are n - 3/2 + n/4, and , ----, respectively, while the asymptotic correlation betw\ 16 een these two random variables is 1 and in floating point 1. ------------------------------------------------- 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, 2, 1, 1], nor the composition, [2, 1, 1, 1] Then infinity ----- 4 2 \ n x + 2 x - 2 x + 1 ) a(n) x = - ------------------- / 3 ----- (-1 + x) n = 0 and in Maple format -(x^4+2*x^2-2*x+1)/(-1+x)^3 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 2, 1, 1] and d occurrences (as containment) of the composition, [2, 1, 1, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 5 4 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x + X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 5 5 3 5 2 4 3 2 - X1 x - X2 x - X1 X2 x + x + X1 X2 x - x + 2 x - 4 x + 3 x - 1) / 2 2 2 / ((-1 + x) (X1 X2 x - x + 2 x - 1)) / and in Maple format (X1*X2*x^5+X1*X2*x^4-X1*x^5-X2*x^5-X1*X2*x^3+x^5+X1*X2*x^2-x^4+2*x^3-4*x^2+3*x-\ 1)/(-1+x)^2/(X1*X2*x^2-x^2+2*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 2, 1, 1], are n - 3/2 + n/4, and , ----, respectively, while 16 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 1, 1, 1], are n - 3/2 + n/4, and , ----, respectively, while the asymptotic correlation betw\ 16 een these two random variables is 1 and in floating point 1. ------------------------------------------------- ------------------------ This ends this article, that took, 1.057, 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, 6, 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, 5], nor the composition, [2, 4] Then infinity ----- 4 \ n x - x + 1 ) a(n) x = ------------------------------- / 5 3 2 ----- (-1 + x) (x + x + x + x - 1) n = 0 and in Maple format (x^4-x+1)/(-1+x)/(x^5+x^3+x^2+x-1) The asymptotic expression for a(n) is, n 0.66926861995747078017 1.8885188454844147017 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 5] and d occurrences (as containment) of the composition, [2, 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 / 6 5 6 X1 X2 x - X2 x + X2 x - x + x - 1) / (X1 X2 x - X1 X2 x - X1 x / 6 5 6 4 5 4 - X2 x + 2 X2 x + x - X2 x - x + 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^6-X1*X2*x^5-X1*x^6-X2*x^6+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, [1, 5], are n 95 23 n - 5/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 4], are n 131 27 n - 5/32 + ----, and , - ---- + ----, respectively, while the asymptotic corre\ 32 1024 1024 lation between these two random variables is 1/2 1/2 23 3 ---------- 23 and in floating point 0.3611575593 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 1/2 23 3 ate normal pair with correlation, ---------- 23 1/2 1/2 23 3 29 i.e. , [[----------, 0], [0, --]] 23 23 ------------------------------------------------- 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, 5], nor the composition, [3, 3] Then infinity ----- 3 \ n x - x + 1 ) a(n) x = -------------------------------------- / 6 5 4 2 ----- (-1 + x) (x + 2 x + x + x + x - 1) n = 0 and in Maple format (x^3-x+1)/(-1+x)/(x^6+2*x^5+x^4+x^2+x-1) The asymptotic expression for a(n) is, n 0.66934247592948995161 1.8825804521685165608 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 5] and d occurrences (as containment) of the composition, [3, 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 5 5 3 3 / 7 6 7 X1 X2 x - X2 x + X2 x - x + x - 1) / (X1 X2 x + X1 X2 x - X1 x / 7 5 6 6 7 5 6 4 5 - X2 x - X1 X2 x - X1 x - X2 x + x + 2 X2 x + x + X2 x - x 3 4 3 - X2 x - x + x - 2 x + 1) and in Maple format -(X1*X2*x^5-X2*x^5+X2*x^3-x^3+x-1)/(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+2*X2*x^5+x^6+X2*x^4-x^5-X2*x^3-x^4+x^3-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 5], are n 95 23 n - 5/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 3], are n 195 35 n - 5/32 + ----, and , - ---- + ----, respectively, while the asymptotic corre\ 32 1024 1024 lation between these two random variables is 1/2 1/2 23 35 ----------- 161 and in floating point 0.1762268442 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 1/2 5 161 ate normal pair with correlation, ----------- 161 1/2 1/2 5 161 171 i.e. , [[-----------, 0], [0, ---]] 161 161 ------------------------------------------------- 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, 5], nor the composition, [4, 2] Then infinity ----- 8 7 4 \ n x + x - x + x - 1 ) a(n) x = - ------------------------------- / 5 3 2 ----- (-1 + x) (x + x + x + x - 1) n = 0 and in Maple format -(x^8+x^7-x^4+x-1)/(-1+x)/(x^5+x^3+x^2+x-1) The asymptotic expression for a(n) is, n 0.64750892987542444154 1.8885188454844147017 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 5] and d occurrences (as containment) of the composition, [4, 2], (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 7 7 8 5 7 5 4 4 - X1 x - X2 x - X1 x - X2 x + x + X1 X2 x + x - X2 x + X2 x - x / 6 5 6 6 5 6 4 + x - 1) / (X1 X2 x - X1 X2 x - X1 x - X2 x + 2 X2 x + x - X2 x / 5 4 - x + x - 2 x + 1) and in Maple format -(X1*X2*x^8+X1*X2*x^7-X1*x^8-X2*x^8-X1*x^7-X2*x^7+x^8+X1*X2*x^5+x^7-X2*x^5+X2*x ^4-x^4+x-1)/(X1*X2*x^6-X1*X2*x^5-X1*x^6-X2*x^6+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, [1, 5], are n 95 23 n - 5/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 2], are n 131 27 n - 5/32 + ----, and , - ---- + ----, respectively, while the asymptotic corre\ 32 1024 1024 lation between these two random variables is 1/2 1/2 23 3 ---------- 23 and in floating point 0.3611575593 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 1/2 23 3 ate normal pair with correlation, ---------- 23 1/2 1/2 23 3 29 i.e. , [[----------, 0], [0, --]] 23 23 ------------------------------------------------- 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, 5], nor the composition, [5, 1] Then infinity ----- 9 8 7 6 5 \ n x + x + x + x - x + x - 1 ) a(n) x = - ------------------------------- / 4 3 2 ----- (-1 + x) (x + x + x + x - 1) n = 0 and in Maple format -(x^9+x^8+x^7+x^6-x^5+x-1)/(-1+x)/(x^4+x^3+x^2+x-1) The asymptotic expression for a(n) is, n 0.56634288770265153485 1.9275619754829253043 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 5] and d occurrences (as containment) of the composition, [5, 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 7 8 8 9 6 7 7 - X1 x - X2 x + X1 X2 x - X1 x - X2 x + x + X1 X2 x - X1 x - X2 x 8 5 6 6 7 6 5 / + x + X1 X2 x - X1 x - X2 x + x + x - x + x - 1) / ( / 5 5 X1 X2 x - x + 2 x - 1) and in Maple format (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-X1*x^6-X2*x^6+x^7+x^6-x^5+x-1)/(X1*X2*x^5-x^5+2*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 5], are n 95 23 n - 5/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [5, 1], are n 95 23 n - 5/32 + ----, and , - ---- + ----, respectively, while the asymptotic corre\ 32 1024 1024 lation between these two random variables is 1 and in floating point 1. ------------------------------------------------- Theorem Number, 5, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 4], nor the composition, [3, 3] Then infinity ----- 3 \ n x - x + 1 ) a(n) x = -------------------------------- / 2 4 2 ----- (x - x + 1) (x - 2 x - x + 1) n = 0 and in Maple format (x^3-x+1)/(x^2-x+1)/(x^4-2*x^2-x+1) The asymptotic expression for a(n) is, n 0.62390494029615137372 1.9051661677540189096 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 4] and d occurrences (as containment) of the composition, [3, 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 4 4 3 3 / 6 5 6 X1 X2 x - X2 x + X2 x - x + x - 1) / (X1 X2 x + X1 X2 x - X1 x / 6 4 6 4 5 3 4 3 - X2 x - X1 X2 x + x + 2 X2 x - x - X2 x - x + x - 2 x + 1) and in Maple format -(X1*X2*x^4-X2*x^4+X2*x^3-x^3+x-1)/(X1*X2*x^6+X1*X2*x^5-X1*x^6-X2*x^6-X1*X2*x^4 +x^6+2*X2*x^4-x^5-X2*x^3-x^4+x^3-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 4], are n 131 27 n - 5/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 3], are n 195 35 n - 5/32 + ----, and , - ---- + ----, respectively, while the asymptotic corre\ 32 1024 1024 lation between these two random variables is 1/2 1/2 3 35 ---------- 21 and in floating point 0.4879500367 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 1/2 5 21 ate normal pair with correlation, ---------- 21 1/2 1/2 5 21 31 i.e. , [[----------, 0], [0, --]] 21 21 ------------------------------------------------- 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, 4], nor the composition, [4, 2] Then infinity ----- 7 6 4 \ n x + x - x + x - 1 ) a(n) x = - --------------------------- / 8 7 5 4 ----- x + x - x + x - 2 x + 1 n = 0 and in Maple format -(x^7+x^6-x^4+x-1)/(x^8+x^7-x^5+x^4-2*x+1) The asymptotic expression for a(n) is, n 0.62954739614193265823 1.8981723275022933773 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 4] and d occurrences (as containment) of the composition, [4, 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 7 6 6 7 4 6 4 / - X1 x - X2 x - X1 x - X2 x + x + X1 X2 x + x - x + x - 1) / ( / 8 7 8 8 7 7 8 5 7 X1 X2 x + X1 X2 x - X1 x - X2 x - X1 x - X2 x + x + X1 X2 x + x 4 5 4 - X1 X2 x - x + x - 2 x + 1) and in Maple format -(X1*X2*x^7+X1*X2*x^6-X1*x^7-X2*x^7-X1*x^6-X2*x^6+x^7+X1*X2*x^4+x^6-x^4+x-1)/( X1*X2*x^8+X1*X2*x^7-X1*x^8-X2*x^8-X1*x^7-X2*x^7+x^8+X1*X2*x^5+x^7-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], are n 131 27 n - 5/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 2], are n 131 27 n - 5/32 + ----, and , - ---- + ----, respectively, while the asymptotic corre\ 32 1024 1024 lation between these two random variables is 5/9 and in floating point 0.5555555556 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ ate normal pair with correlation, 5/9 131 i.e. , [[5/9, 0], [0, ---]] 81 ------------------------------------------------- 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, 4], nor the composition, [5, 1] Then infinity ----- 8 7 4 \ n x + x - x + x - 1 ) a(n) x = - ------------------------------- / 5 3 2 ----- (-1 + x) (x + x + x + x - 1) n = 0 and in Maple format -(x^8+x^7-x^4+x-1)/(-1+x)/(x^5+x^3+x^2+x-1) The asymptotic expression for a(n) is, n 0.64750892987542444154 1.8885188454844147017 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 4] and d occurrences (as containment) of the composition, [5, 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 7 7 8 5 7 5 4 4 - X1 x - X2 x - X1 x - X2 x + x + X1 X2 x + x - X1 x + X1 x - x / 6 5 6 6 5 6 4 + x - 1) / (X1 X2 x - X1 X2 x - X1 x - X2 x + 2 X1 x + x - X1 x / 5 4 - x + x - 2 x + 1) and in Maple format -(X1*X2*x^8+X1*X2*x^7-X1*x^8-X2*x^8-X1*x^7-X2*x^7+x^8+X1*X2*x^5+x^7-X1*x^5+X1*x ^4-x^4+x-1)/(X1*X2*x^6-X1*X2*x^5-X1*x^6-X2*x^6+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, [2, 4], are n 131 27 n - 5/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [5, 1], are n 95 23 n - 5/32 + ----, and , - ---- + ----, respectively, while the asymptotic corre\ 32 1024 1024 lation between these two random variables is 1/2 1/2 23 3 ---------- 23 and in floating point 0.3611575593 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 1/2 23 3 ate normal pair with correlation, ---------- 23 1/2 1/2 23 3 29 i.e. , [[----------, 0], [0, --]] 23 23 ------------------------------------------------- Theorem Number, 8, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [3, 3], nor the composition, [4, 2] Then infinity ----- 3 \ n x - x + 1 ) a(n) x = -------------------------------- / 2 4 2 ----- (x - x + 1) (x - 2 x - x + 1) n = 0 and in Maple format (x^3-x+1)/(x^2-x+1)/(x^4-2*x^2-x+1) The asymptotic expression for a(n) is, n 0.62390494029615137372 1.9051661677540189096 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [3, 3] and d occurrences (as containment) of the composition, [4, 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 4 4 3 3 / 6 5 6 X1 X2 x - X1 x + X1 x - x + x - 1) / (X1 X2 x + X1 X2 x - X1 x / 6 4 6 4 5 3 4 3 - X2 x - X1 X2 x + x + 2 X1 x - x - X1 x - x + x - 2 x + 1) and in Maple format -(X1*X2*x^4-X1*x^4+X1*x^3-x^3+x-1)/(X1*X2*x^6+X1*X2*x^5-X1*x^6-X2*x^6-X1*X2*x^4 +x^6+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], are n 195 35 n - 5/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 2], are n 131 27 n - 5/32 + ----, and , - ---- + ----, respectively, while the asymptotic corre\ 32 1024 1024 lation between these two random variables is 1/2 1/2 3 35 ---------- 21 and in floating point 0.4879500367 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 1/2 5 21 ate normal pair with correlation, ---------- 21 1/2 1/2 5 21 31 i.e. , [[----------, 0], [0, --]] 21 21 ------------------------------------------------- Theorem Number, 9, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [3, 3], nor the composition, [5, 1] Then infinity ----- 3 \ n x - x + 1 ) a(n) x = -------------------------------------- / 6 5 4 2 ----- (-1 + x) (x + 2 x + x + x + x - 1) n = 0 and in Maple format (x^3-x+1)/(-1+x)/(x^6+2*x^5+x^4+x^2+x-1) The asymptotic expression for a(n) is, n 0.66934247592948995161 1.8825804521685165608 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [3, 3] and d occurrences (as containment) of the composition, [5, 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 5 5 3 3 / 7 6 7 X1 X2 x - X1 x + X1 x - x + x - 1) / (X1 X2 x + X1 X2 x - X1 x / 7 5 6 6 7 5 6 4 5 - X2 x - X1 X2 x - X1 x - X2 x + x + 2 X1 x + x + X1 x - x 3 4 3 - X1 x - x + x - 2 x + 1) and in Maple format -(X1*X2*x^5-X1*x^5+X1*x^3-x^3+x-1)/(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+2*X1*x^5+x^6+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], are n 195 35 n - 5/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [5, 1], are n 95 23 n - 5/32 + ----, and , - ---- + ----, respectively, while the asymptotic corre\ 32 1024 1024 lation between these two random variables is 1/2 1/2 23 35 ----------- 161 and in floating point 0.1762268442 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 1/2 5 161 ate normal pair with correlation, ----------- 161 1/2 1/2 5 161 171 i.e. , [[-----------, 0], [0, ---]] 161 161 ------------------------------------------------- Theorem Number, 10, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [4, 2], nor the composition, [5, 1] Then infinity ----- 4 \ n x - x + 1 ) a(n) x = ------------------------------- / 5 3 2 ----- (-1 + x) (x + x + x + x - 1) n = 0 and in Maple format (x^4-x+1)/(-1+x)/(x^5+x^3+x^2+x-1) The asymptotic expression for a(n) is, n 0.66926861995747078017 1.8885188454844147017 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [4, 2] and d occurrences (as containment) of the composition, [5, 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 5 5 4 4 / 6 5 6 X1 X2 x - X1 x + X1 x - x + x - 1) / (X1 X2 x - X1 X2 x - X1 x / 6 5 6 4 5 4 - X2 x + 2 X1 x + x - X1 x - x + x - 2 x + 1) and in Maple format -(X1*X2*x^5-X1*x^5+X1*x^4-x^4+x-1)/(X1*X2*x^6-X1*X2*x^5-X1*x^6-X2*x^6+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, 2], are n 131 27 n - 5/32 + ----, and , - ---- + ----, respectively, while 32 1024 1024 Furthermore the average and variance of the random variable: Number of occur\ rences of, [5, 1], are n 95 23 n - 5/32 + ----, and , - ---- + ----, respectively, while the asymptotic corre\ 32 1024 1024 lation between these two random variables is 1/2 1/2 23 3 ---------- 23 and in floating point 0.3611575593 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 1/2 23 3 ate normal pair with correlation, ---------- 23 1/2 1/2 23 3 29 i.e. , [[----------, 0], [0, --]] 23 23 ------------------------------------------------- ------------------------ This ends this article, that took, 0.940, 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, 6, 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, 4], nor the composition, [1, 2, 3] Then infinity ----- 6 5 4 3 2 \ n x - 2 x + x - x - x + 2 x - 1 ) a(n) x = ------------------------------------- / 3 2 2 ----- (x + 1) (x - x + 2 x - 1) (-1 + x) n = 0 and in Maple format (x^6-2*x^5+x^4-x^3-x^2+2*x-1)/(x+1)/(x^3-x^2+2*x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 1.1544840707379761413 1.7548776662466927601 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 1, 4] and d occurrences (as containment) of the composition, [1, 2, 3], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 6 5 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x - X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 6 6 4 5 5 6 4 5 3 - X1 x - X2 x + X1 X2 x + X1 x + 2 X2 x + x - 2 X2 x - 2 x + X2 x 4 3 2 / 5 4 5 + x - x - x + 2 x - 1) / ((-1 + x) (X1 X2 x - X1 X2 x - X1 x / 5 4 5 3 4 3 - X2 x + 2 X2 x + x - X2 x - x + x - 2 x + 1)) and in Maple format (X1*X2*x^6-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)/(-1+x)/(X1*X2*x^5-X1*X2*x^4-X1*x^5-X2*x^5+2*X2*x^4+x ^5-X2*x^3-x^4+x^3-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 1, 4], are n 9 n - 3/8 + ----, and , - 5/32 + ---, respectively, while 16 256 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 2, 3], are n 19 13 n - 3/8 + ----, and , - -- + ----, respectively, while the asymptotic correlat\ 16 64 256 ion between these two random variables is 1/2 13 ----- 13 and in floating point 0.2773500981 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 13 ate normal pair with correlation, ----- 13 1/2 13 15 i.e. , [[-----, 0], [0, --]] 13 13 ------------------------------------------------- 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, 4], nor the composition, [1, 3, 2] Then infinity ----- 2 3 2 \ n (x + 1) (x + x - 2 x + 1) ) a(n) x = ---------------------------- / 3 2 ----- (-1 + x) (x - x + 2 x - 1) n = 0 and in Maple format (x^2+1)*(x^3+x^2-2*x+1)/(-1+x)/(x^3-x^2+2*x-1) The asymptotic expression for a(n) is, n 1.0902311912568087359 1.7548776662466927601 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 1, 4] and d occurrences (as containment) of the composition, [1, 3, 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 7 5 6 6 7 4 5 - X1 x - X2 x - X1 X2 x - X1 x - X2 x + x + X1 X2 x + X1 x 5 6 4 5 3 4 3 2 / + 2 X2 x + x - 2 X2 x - 2 x + X2 x + x - x - x + 2 x - 1) / ( / 5 4 5 5 4 5 3 4 (-1 + x) (X1 X2 x - X1 X2 x - X1 x - X2 x + 2 X2 x + x - X2 x - x 3 + x - 2 x + 1)) and in Maple format (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+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)/(-1+x)/(X1*X2*x^5-X1*X2* x^4-X1*x^5-X2*x^5+2*X2*x^4+x^5-X2*x^3-x^4+x^3-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 1, 4], are n 9 n - 3/8 + ----, and , - 5/32 + ---, respectively, while 16 256 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 3, 2], are n 19 13 n - 3/8 + ----, and , - -- + ----, respectively, while the asymptotic correlat\ 16 64 256 ion between these two random variables is 1/2 13 ----- 13 and in floating point 0.2773500981 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 13 ate normal pair with correlation, ----- 13 1/2 13 15 i.e. , [[-----, 0], [0, --]] 13 13 ------------------------------------------------- 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, 4], nor the composition, [1, 4, 1] Then infinity ----- 8 7 6 4 2 \ n x + x + x - x - x + 2 x - 1 ) a(n) x = -------------------------------- / 3 2 2 ----- (x + x + x - 1) (-1 + x) n = 0 and in Maple format (x^8+x^7+x^6-x^4-x^2+2*x-1)/(x^3+x^2+x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.73683984463878510196 1.8392867552141611326 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 1, 4] and d occurrences (as containment) of the composition, [1, 4, 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 6 6 7 - X1 x - X2 x + X1 X2 x - X1 x - X2 x + x - X1 x - X2 x + x 4 6 4 2 / + X1 X2 x + x - x - x + 2 x - 1) / ((-1 + x) / 4 4 (X1 X2 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-X1*x^6-X2*x^6+x ^7+X1*X2*x^4+x^6-x^4-x^2+2*x-1)/(-1+x)/(X1*X2*x^4-x^4+2*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 1, 4], are n 9 n - 3/8 + ----, and , - 5/32 + ---, respectively, while 16 256 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 4, 1], are n 9 n - 3/8 + ----, and , - 5/32 + ---, respectively, while the asymptotic correla\ 16 256 tion between these two random variables is 1 and in floating point 1. ------------------------------------------------- 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, 4], nor the composition, [2, 1, 3] Then infinity ----- 8 7 3 2 \ n x - x - x - x + 2 x - 1 ) a(n) x = ---------------------------------------------- / 3 2 4 2 ----- (x + 1) (x - x + 2 x - 1) (x + 1) (-1 + x) n = 0 and in Maple format (x^8-x^7-x^3-x^2+2*x-1)/(x+1)/(x^3-x^2+2*x-1)/(x^4+1)/(-1+x)^2 The asymptotic expression for a(n) is, n 1.1274912931215698825 1.7548776662466927601 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 1, 4] and d occurrences (as containment) of the composition, [2, 1, 3], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 9 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 9 2 9 2 8 9 2 9 8 - X1 X2 x - 2 X1 X2 x + 2 X1 X2 x + 2 X1 X2 x + X2 x - 3 X1 X2 x 2 8 9 8 2 7 8 7 8 7 - 2 X2 x - X2 x + X1 x + X2 x + 3 X2 x - 2 X2 x - x + x 4 4 3 3 2 / - X1 X2 x + X2 x - X2 x + x + x - 2 x + 1) / ( / 5 5 5 4 5 4 5 4 (X1 X2 x - X1 x - X2 x + X2 x + x - x + x - 1) (X1 X2 x - X1 X2 x 5 5 4 5 3 4 3 - X1 x - X2 x + 2 X2 x + x - X2 x - x + x - 2 x + 1)) and in Maple format -(X1^2*X2^2*x^9-X1^2*X2*x^9-2*X1*X2^2*x^9+2*X1*X2^2*x^8+2*X1*X2*x^9+X2^2*x^9-3* X1*X2*x^8-2*X2^2*x^8-X2*x^9+X1*x^8+X2^2*x^7+3*X2*x^8-2*X2*x^7-x^8+x^7-X1*X2*x^4 +X2*x^4-X2*x^3+x^3+x^2-2*x+1)/(X1*X2*x^5-X1*x^5-X2*x^5+X2*x^4+x^5-x^4+x-1)/(X1* X2*x^5-X1*X2*x^4-X1*x^5-X2*x^5+2*X2*x^4+x^5-X2*x^3-x^4+x^3-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 1, 4], are n 9 n - 3/8 + ----, and , - 5/32 + ---, respectively, while 16 256 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 1, 3], are n 21 13 n - 3/8 + ----, and , - -- + ----, respectively, while the asymptotic correlat\ 16 64 256 ion between these two random variables is 1/2 13 ----- 13 and in floating point 0.2773500981 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 13 ate normal pair with correlation, ----- 13 1/2 13 15 i.e. , [[-----, 0], [0, --]] 13 13 ------------------------------------------------- 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, 4], nor the composition, [2, 2, 2] Then infinity ----- 9 7 6 5 4 3 2 \ n x - x + x - x - x + x - 2 x + 2 x - 1 ) a(n) x = -------------------------------------------- / 7 6 5 4 3 2 ----- (x + 2 x + x + x + x + x - 1) (-1 + x) n = 0 and in Maple format (x^9-x^7+x^6-x^5-x^4+x^3-2*x^2+2*x-1)/(x^7+2*x^6+x^5+x^4+x^3+x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 1.0612450246603483778 1.7610793284505662547 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 1, 4] and d occurrences (as containment) of the composition, [2, 2, 2], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 9 2 9 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x - X1 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 9 9 9 7 9 6 7 7 - 2 X1 X2 x + 2 X1 x + X2 x + X1 X2 x - x - X1 X2 x - X1 x - X2 x 5 6 6 7 4 5 5 6 5 + X1 X2 x + X1 x + X2 x + x - X1 X2 x - X1 x - X2 x - x + x 3 4 2 3 2 / 2 9 2 9 + X2 x + x - X2 x - x + 2 x - 2 x + 1) / (X1 X2 x - X1 x / 9 9 9 7 9 6 7 - 2 X1 X2 x + 2 X1 x + X2 x + 2 X1 X2 x - x - X1 X2 x - 2 X1 x 7 5 6 6 7 4 5 5 - 2 X2 x + 2 X1 X2 x + X1 x + X2 x + 2 x - X1 X2 x - X1 x - X2 x 6 3 4 2 3 2 - x + 2 X2 x + x - X2 x - 2 x + 3 x - 3 x + 1) and in Maple format (X1^2*X2*x^9-X1^2*x^9-2*X1*X2*x^9+2*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-X1*X2*x^4-X1*x^5-X2*x^5-x^6+x^5+X2*x^3+x^4 -X2*x^2-x^3+2*x^2-2*x+1)/(X1^2*X2*x^9-X1^2*x^9-2*X1*X2*x^9+2*X1*x^9+X2*x^9+2*X1 *X2*x^7-x^9-X1*X2*x^6-2*X1*x^7-2*X2*x^7+2*X1*X2*x^5+X1*x^6+X2*x^6+2*x^7-X1*X2*x ^4-X1*x^5-X2*x^5-x^6+2*X2*x^3+x^4-X2*x^2-2*x^3+3*x^2-3*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 1, 4], are n 9 n - 3/8 + ----, and , - 5/32 + ---, respectively, while 16 256 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 2, 2], are n 23 25 n - 3/8 + ----, and , - -- + ----, respectively, while the asymptotic correlat\ 16 32 256 ion between these two random variables is 1/15 and in floating point 0.06666666667 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ ate normal pair with correlation, 1/15 227 i.e. , [[1/15, 0], [0, ---]] 225 ------------------------------------------------- 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, 4], nor the composition, [2, 3, 1] Then infinity ----- 8 6 4 3 2 \ n x + x - 2 x + 3 x - 4 x + 3 x - 1 ) a(n) x = -------------------------------------- / 3 2 2 ----- (x - x + 2 x - 1) (-1 + x) n = 0 and in Maple format (x^8+x^6-2*x^4+3*x^3-4*x^2+3*x-1)/(x^3-x^2+2*x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 1.0327532479435799189 1.7548776662466927601 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 1, 4] and d occurrences (as containment) of the composition, [2, 3, 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 10 10 10 6 5 - 2 X1 X2 x + 2 X1 x + X2 x - x + 2 X1 X2 x - 2 X1 X2 x 6 6 4 5 5 6 4 5 - 2 X1 x - 3 X2 x + X1 X2 x + X1 x + 4 X2 x + 3 x - 3 X2 x - 3 x 3 4 2 / 2 5 4 + X2 x + 2 x - 3 x + 3 x - 1) / ((-1 + x) (X1 X2 x - X1 X2 x / 5 5 4 5 3 4 3 - X1 x - X2 x + 2 X2 x + x - X2 x - x + x - 2 x + 1)) and in Maple format -(X1^2*X2*x^10-X1^2*x^10-2*X1*X2*x^10+2*X1*x^10+X2*x^10-x^10+2*X1*X2*x^6-2*X1* X2*x^5-2*X1*x^6-3*X2*x^6+X1*X2*x^4+X1*x^5+4*X2*x^5+3*x^6-3*X2*x^4-3*x^5+X2*x^3+ 2*x^4-3*x^2+3*x-1)/(-1+x)^2/(X1*X2*x^5-X1*X2*x^4-X1*x^5-X2*x^5+2*X2*x^4+x^5-X2* x^3-x^4+x^3-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 1, 4], are n 9 n - 3/8 + ----, and , - 5/32 + ---, respectively, while 16 256 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 3, 1], are n 19 13 n - 3/8 + ----, and , - -- + ----, respectively, while the asymptotic correlat\ 16 64 256 ion between these two random variables is 1/2 13 ----- 13 and in floating point 0.2773500981 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 13 ate normal pair with correlation, ----- 13 1/2 13 15 i.e. , [[-----, 0], [0, --]] 13 13 ------------------------------------------------- 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, 4], nor the composition, [3, 1, 2] Then infinity ----- \ n ) a(n) x = / ----- n = 0 11 9 8 7 6 5 4 3 2 x - 2 x + x - 3 x + 3 x - 3 x + 3 x - 3 x + 4 x - 3 x + 1 - ------------------------------------------------------------------- 3 2 4 2 (x - x + 2 x - 1) (x + 1) (-1 + x) and in Maple format -(x^11-2*x^9+x^8-3*x^7+3*x^6-3*x^5+3*x^4-3*x^3+4*x^2-3*x+1)/(x^3-x^2+2*x-1)/(x^ 4+1)/(-1+x)^2 The asymptotic expression for a(n) is, n 1.0054824915204388557 1.7548776662466927601 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 1, 4] and d occurrences (as containment) of the composition, [3, 1, 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 2 11 2 13 13 - 2 X1 X2 x - 2 X1 X2 x - X1 X2 x + X1 x + 4 X1 X2 x 2 13 2 2 10 2 11 2 11 13 + X2 x + X1 X2 x + 3 X1 X2 x + 3 X1 X2 x - 2 X1 x 13 2 2 9 2 10 2 11 2 10 - 2 X2 x - X1 X2 x - X1 X2 x - 2 X1 x - 3 X1 X2 x 11 2 11 13 2 9 2 9 10 - 8 X1 X2 x - 2 X2 x + x + X1 X2 x + 4 X1 X2 x + 4 X1 X2 x 11 2 10 11 2 8 9 10 + 5 X1 x + 2 X2 x + 5 X2 x - 2 X1 X2 x - 5 X1 X2 x - X1 x 2 9 10 11 8 9 2 8 9 - 3 X2 x - 3 X2 x - 3 x + 3 X1 X2 x + X1 x + 3 X2 x + 4 X2 x 10 7 8 2 7 8 9 7 7 8 + x + X1 X2 x - X1 x - X2 x - 5 X2 x - x - X1 x + X2 x + 2 x 5 4 5 4 3 4 2 - X1 X2 x + X1 X2 x + X2 x - 2 X2 x + X2 x + x - 3 x + 3 x - 1) / 5 4 5 5 4 5 3 / ((-1 + x) (X1 X2 x - X1 X2 x - X1 x - X2 x + 2 X2 x + x - X2 x / 4 3 5 5 5 4 5 4 - x + x - 2 x + 1) (X1 X2 x - X1 x - X2 x + X2 x + x - x + 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*X2^2*x^11+X1^2*x^13+4*X1*X2 *x^13+X2^2*x^13+X1^2*X2^2*x^10+3*X1^2*X2*x^11+3*X1*X2^2*x^11-2*X1*x^13-2*X2*x^ 13-X1^2*X2^2*x^9-X1^2*X2*x^10-2*X1^2*x^11-3*X1*X2^2*x^10-8*X1*X2*x^11-2*X2^2*x^ 11+x^13+X1^2*X2*x^9+4*X1*X2^2*x^9+4*X1*X2*x^10+5*X1*x^11+2*X2^2*x^10+5*X2*x^11-\ 2*X1*X2^2*x^8-5*X1*X2*x^9-X1*x^10-3*X2^2*x^9-3*X2*x^10-3*x^11+3*X1*X2*x^8+X1*x^ 9+3*X2^2*x^8+4*X2*x^9+x^10+X1*X2*x^7-X1*x^8-X2^2*x^7-5*X2*x^8-x^9-X1*x^7+X2*x^7 +2*x^8-X1*X2*x^5+X1*X2*x^4+X2*x^5-2*X2*x^4+X2*x^3+x^4-3*x^2+3*x-1)/(-1+x)/(X1* X2*x^5-X1*X2*x^4-X1*x^5-X2*x^5+2*X2*x^4+x^5-X2*x^3-x^4+x^3-2*x+1)/(X1*X2*x^5-X1 *x^5-X2*x^5+X2*x^4+x^5-x^4+x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 1, 4], are n 9 n - 3/8 + ----, and , - 5/32 + ---, respectively, while 16 256 Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 1, 2], are n 21 13 n - 3/8 + ----, and , - -- + ----, respectively, while the asymptotic correlat\ 16 64 256 ion between these two random variables is 1/2 13 ----- 13 and in floating point 0.2773500981 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 13 ate normal pair with correlation, ----- 13 1/2 13 15 i.e. , [[-----, 0], [0, --]] 13 13 ------------------------------------------------- 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, 4], nor the composition, [3, 2, 1] Then infinity ----- 11 8 6 4 3 2 \ n x - 2 x - x + 2 x - 3 x + 4 x - 3 x + 1 ) a(n) x = - ---------------------------------------------- / 3 2 2 ----- (x - x + 2 x - 1) (-1 + x) n = 0 and in Maple format -(x^11-2*x^8-x^6+2*x^4-3*x^3+4*x^2-3*x+1)/(x^3-x^2+2*x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.98591087239572105185 1.7548776662466927601 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 1, 4] and d occurrences (as containment) of the composition, [3, 2, 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 13 - 2 X1 X2 x - 2 X1 X2 x + X1 x + 4 X1 X2 x + X2 x 2 11 13 13 2 10 2 11 2 10 + X1 X2 x - 2 X1 x - 2 X2 x + X1 X2 x - X1 x + X1 X2 x 11 13 2 10 10 11 2 10 11 - 2 X1 X2 x + x - X1 x - 4 X1 X2 x + 2 X1 x - X2 x + X2 x 10 10 11 8 10 8 8 + 3 X1 x + 3 X2 x - x + X1 X2 x - 2 x - X1 x - X2 x 6 8 5 6 6 4 5 + 2 X1 X2 x + x - 2 X1 X2 x - 2 X1 x - 3 X2 x + X1 X2 x + X1 x 5 6 4 5 3 4 2 / + 4 X2 x + 3 x - 3 X2 x - 3 x + X2 x + 2 x - 3 x + 3 x - 1) / ( / 2 5 4 5 5 4 5 3 4 (-1 + x) (X1 X2 x - X1 X2 x - X1 x - X2 x + 2 X2 x + x - X2 x - x 3 + x - 2 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+4*X1*X2*x^13+X2^2*x^13 +X1^2*X2*x^11-2*X1*x^13-2*X2*x^13+X1^2*X2*x^10-X1^2*x^11+X1*X2^2*x^10-2*X1*X2*x ^11+x^13-X1^2*x^10-4*X1*X2*x^10+2*X1*x^11-X2^2*x^10+X2*x^11+3*X1*x^10+3*X2*x^10 -x^11+X1*X2*x^8-2*x^10-X1*x^8-X2*x^8+2*X1*X2*x^6+x^8-2*X1*X2*x^5-2*X1*x^6-3*X2* x^6+X1*X2*x^4+X1*x^5+4*X2*x^5+3*x^6-3*X2*x^4-3*x^5+X2*x^3+2*x^4-3*x^2+3*x-1)/(-\ 1+x)^2/(X1*X2*x^5-X1*X2*x^4-X1*x^5-X2*x^5+2*X2*x^4+x^5-X2*x^3-x^4+x^3-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 1, 4], are n 9 n - 3/8 + ----, and , - 5/32 + ---, respectively, while 16 256 Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 2, 1], are n 19 13 n - 3/8 + ----, and , - -- + ----, respectively, while the asymptotic correlat\ 16 64 256 ion between these two random variables is 1/2 13 ----- 13 and in floating point 0.2773500981 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 13 ate normal pair with correlation, ----- 13 1/2 13 15 i.e. , [[-----, 0], [0, --]] 13 13 ------------------------------------------------- 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, 4], nor the composition, [4, 1, 1] Then infinity ----- \ n 14 13 12 11 10 9 8 7 6 ) a(n) x = - (x + 2 x + 3 x + x - x - 3 x - 3 x - 2 x - x / ----- n = 0 4 2 / 3 2 2 + x + x - 2 x + 1) / ((x + x + x - 1) (-1 + x) ) / and in Maple format -(x^14+2*x^13+3*x^12+x^11-x^10-3*x^9-3*x^8-2*x^7-x^6+x^4+x^2-2*x+1)/(x^3+x^2+x-\ 1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.61841992231939255104 1.8392867552141611326 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 1, 4] and d occurrences (as containment) of the composition, [4, 1, 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 2 14 2 15 2 15 2 2 13 2 14 + X1 X2 x - 2 X1 X2 x - 2 X1 X2 x + X1 X2 x - 2 X1 X2 x 2 15 2 14 15 2 15 2 13 2 14 + X1 x - 2 X1 X2 x + 4 X1 X2 x + X2 x - 2 X1 X2 x + X1 x 2 13 14 15 2 14 15 2 12 - 2 X1 X2 x + 4 X1 X2 x - 2 X1 x + X2 x - 2 X2 x + X1 X2 x 2 13 2 12 13 14 2 13 14 15 + X1 x + X1 X2 x + 4 X1 X2 x - 2 X1 x + X2 x - 2 X2 x + x 2 11 2 12 2 11 12 13 2 12 + X1 X2 x - X1 x + X1 X2 x - 4 X1 X2 x - 2 X1 x - X2 x 13 14 2 10 2 11 2 10 11 - 2 X2 x + x + X1 X2 x - X1 x + X1 X2 x - 4 X1 X2 x 12 2 11 12 13 2 10 10 11 + 3 X1 x - X2 x + 3 X2 x + x - X1 x - 4 X1 X2 x + 3 X1 x 2 10 11 12 10 10 11 8 - X2 x + 3 X2 x - 2 x + 3 X1 x + 3 X2 x - 2 x + X1 X2 x 10 7 8 8 6 7 7 8 - 2 x + X1 X2 x - X1 x - X2 x + X1 X2 x - X1 x - X2 x + x 5 6 6 7 4 6 5 4 3 2 - X1 X2 x - X1 x - X2 x + x + X1 X2 x + x + x - x + x - 3 x / 2 4 4 + 3 x - 1) / ((-1 + x) (X1 X2 x - x + 2 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+X1^2*X2^2*x^13-2* X1^2*X2*x^14+X1^2*x^15-2*X1*X2^2*x^14+4*X1*X2*x^15+X2^2*x^15-2*X1^2*X2*x^13+X1^ 2*x^14-2*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*X2* x^11-X1^2*x^12+X1*X2^2*x^11-4*X1*X2*x^12-2*X1*x^13-X2^2*x^12-2*X2*x^13+x^14+X1^ 2*X2*x^10-X1^2*x^11+X1*X2^2*x^10-4*X1*X2*x^11+3*X1*x^12-X2^2*x^11+3*X2*x^12+x^ 13-X1^2*x^10-4*X1*X2*x^10+3*X1*x^11-X2^2*x^10+3*X2*x^11-2*x^12+3*X1*x^10+3*X2*x ^10-2*x^11+X1*X2*x^8-2*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*X2*x^5-X1*x^6-X2*x^6+x^7+X1*X2*x^4+x^6+x^5-x^4+x^3-3*x^2+3*x-1)/(-1+x)^2/( X1*X2*x^4-x^4+2*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 1, 4], are n 9 n - 3/8 + ----, and , - 5/32 + ---, respectively, while 16 256 Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 1, 1], are n 9 n - 3/8 + ----, and , - 5/32 + ---, respectively, while the asymptotic correla\ 16 256 tion between these two random variables is 1 and in floating point 1. ------------------------------------------------- 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, 2, 3], nor the composition, [1, 3, 2] Then infinity ----- 6 5 3 \ n x + x + x - x + 1 ) a(n) x = ------------------------------- / 5 4 2 ----- (-1 + x) (x + x + x + x - 1) n = 0 and in Maple format (x^6+x^5+x^3-x+1)/(-1+x)/(x^5+x^4+x^2+x-1) The asymptotic expression for a(n) is, n 0.86835109073791924063 1.8124036192680426608 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 2, 3] and d occurrences (as containment) of the composition, [1, 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 7 4 3 5 4 3 2 - X2 x + X1 X2 x + x - X1 X2 x + X1 X2 x - x + x - x - x + 2 x / - 1) / ((-1 + x) / 6 6 6 4 6 3 4 3 (X1 X2 x - X1 x - X2 x + X1 X2 x + x - X1 X2 x - x + x - 2 x + 1)) and in Maple format (X1*X2*x^7-X1*x^7-X2*x^7+X1*X2*x^5+x^7-X1*X2*x^4+X1*X2*x^3-x^5+x^4-x^3-x^2+2*x-\ 1)/(-1+x)/(X1*X2*x^6-X1*x^6-X2*x^6+X1*X2*x^4+x^6-X1*X2*x^3-x^4+x^3-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 2, 3], are n 19 13 n - 3/8 + ----, and , - -- + ----, respectively, while 16 64 256 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 3, 2], are n 19 13 n - 3/8 + ----, and , - -- + ----, respectively, while the asymptotic correlat\ 16 64 256 ion between these two random variables is 9/13 and in floating point 0.6923076923 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ ate normal pair with correlation, 9/13 331 i.e. , [[9/13, 0], [0, ---]] 169 ------------------------------------------------- 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, 2, 3], nor the composition, [1, 4, 1] Then infinity ----- 2 3 2 \ n (x + 1) (x + x - 2 x + 1) ) a(n) x = ---------------------------- / 3 2 ----- (-1 + x) (x - x + 2 x - 1) n = 0 and in Maple format (x^2+1)*(x^3+x^2-2*x+1)/(-1+x)/(x^3-x^2+2*x-1) The asymptotic expression for a(n) is, n 1.0902311912568087359 1.7548776662466927601 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 2, 3] and d occurrences (as containment) of the composition, [1, 4, 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 7 5 6 6 7 4 5 - X1 x - X2 x - X1 X2 x - X1 x - X2 x + x + X1 X2 x + 2 X1 x 5 6 4 5 3 4 3 2 / + X2 x + x - 2 X1 x - 2 x + X1 x + x - x - x + 2 x - 1) / ( / 5 4 5 5 4 5 3 4 (-1 + x) (X1 X2 x - X1 X2 x - X1 x - X2 x + 2 X1 x + x - X1 x - x 3 + x - 2 x + 1)) and in Maple format (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+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)/(-1+x)/(X1*X2*x^5-X1*X2* x^4-X1*x^5-X2*x^5+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, 2, 3], are n 19 13 n - 3/8 + ----, and , - -- + ----, respectively, while 16 64 256 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 4, 1], are n 9 n - 3/8 + ----, and , - 5/32 + ---, respectively, while the asymptotic correla\ 16 256 tion between these two random variables is 1/2 13 ----- 13 and in floating point 0.2773500981 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 13 ate normal pair with correlation, ----- 13 1/2 13 15 i.e. , [[-----, 0], [0, --]] 13 13 ------------------------------------------------- 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, 2, 3], nor the composition, [2, 1, 3] Then infinity ----- 3 2 \ n x + x - 2 x + 1 ) a(n) x = ---------------------------- / 5 3 ----- (-1 + x) (x - x + 2 x - 1) n = 0 and in Maple format (x^3+x^2-2*x+1)/(-1+x)/(x^5-x^3+2*x-1) The asymptotic expression for a(n) is, n 0.96634807298013408626 1.7845989333686468028 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 2, 3] and d occurrences (as containment) of the composition, [2, 1, 3], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 7 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 7 7 5 4 5 3 4 3 - 2 X1 X2 x + X1 x - X1 X2 x + X1 X2 x + X2 x - X1 X2 x - X2 x + x 2 / 2 8 2 7 8 7 + x - 2 x + 1) / (X1 X2 x - X1 X2 x - 2 X1 X2 x + 2 X1 X2 x / 8 6 7 5 6 6 4 + X1 x - X1 X2 x - X1 x + X1 X2 x + X1 x + X2 x - 2 X1 X2 x 5 6 3 4 5 4 3 2 - 2 X2 x - x + X1 X2 x + X2 x + x + x - x - 2 x + 3 x - 1) and in Maple format -(X1*X2^2*x^7-2*X1*X2*x^7+X1*x^7-X1*X2*x^5+X1*X2*x^4+X2*x^5-X1*X2*x^3-X2*x^4+x^ 3+x^2-2*x+1)/(X1*X2^2*x^8-X1*X2^2*x^7-2*X1*X2*x^8+2*X1*X2*x^7+X1*x^8-X1*X2*x^6- X1*x^7+X1*X2*x^5+X1*x^6+X2*x^6-2*X1*X2*x^4-2*X2*x^5-x^6+X1*X2*x^3+X2*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, [1, 2, 3], are n 19 13 n - 3/8 + ----, and , - -- + ----, respectively, while 16 64 256 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 1, 3], are n 21 13 n - 3/8 + ----, and , - -- + ----, respectively, while the asymptotic correlat\ 16 64 256 ion between these two random variables is 7/13 and in floating point 0.5384615385 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ ate normal pair with correlation, 7/13 267 i.e. , [[7/13, 0], [0, ---]] 169 ------------------------------------------------- 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, 2, 3], nor the composition, [2, 2, 2] Then infinity ----- 4 3 2 \ n x - x + 2 x - 2 x + 1 ) a(n) x = --------------------------------------- / 2 3 2 ----- (-1 + x) (x - x + 1) (x + x + x - 1) n = 0 and in Maple format (x^4-x^3+2*x^2-2*x+1)/(-1+x)/(x^2-x+1)/(x^3+x^2+x-1) The asymptotic expression for a(n) is, n 0.77591233811587484106 1.8392867552141611326 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 2, 3] and d occurrences (as containment) of the composition, [2, 2, 2], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 5 4 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x - X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 5 3 4 3 4 2 3 2 - X2 x + X1 X2 x + 2 X2 x - 2 X2 x - x + X2 x + x - 2 x + 2 x - 1) / 6 5 6 6 4 5 6 / (X1 X2 x - X1 X2 x - X1 x - X2 x + 2 X1 X2 x + 2 X2 x + x / 3 4 5 3 4 2 3 2 - X1 X2 x - 3 X2 x - x + 3 X2 x + x - X2 x - 2 x + 3 x - 3 x + 1) and in Maple format -(X1*X2*x^5-X1*X2*x^4-X2*x^5+X1*X2*x^3+2*X2*x^4-2*X2*x^3-x^4+X2*x^2+x^3-2*x^2+2 *x-1)/(X1*X2*x^6-X1*X2*x^5-X1*x^6-X2*x^6+2*X1*X2*x^4+2*X2*x^5+x^6-X1*X2*x^3-3* X2*x^4-x^5+3*X2*x^3+x^4-X2*x^2-2*x^3+3*x^2-3*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 2, 3], are n 19 13 n - 3/8 + ----, and , - -- + ----, respectively, while 16 64 256 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 2, 2], are n 23 25 n - 3/8 + ----, and , - -- + ----, respectively, while the asymptotic correlat\ 16 32 256 ion between these two random variables is 1/2 11 13 -------- 65 and in floating point 0.6101702157 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 11 13 ate normal pair with correlation, -------- 65 1/2 11 13 567 i.e. , [[--------, 0], [0, ---]] 65 325 ------------------------------------------------- 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, 2, 3], nor the composition, [2, 3, 1] Then infinity ----- 9 8 6 5 4 2 \ n x - x + 2 x - 2 x + 2 x - 3 x + 3 x - 1 ) a(n) x = --------------------------------------------- / 4 3 2 ----- (x - x + 2 x - 1) (-1 + x) n = 0 and in Maple format (x^9-x^8+2*x^6-2*x^5+2*x^4-3*x^2+3*x-1)/(x^4-x^3+2*x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.65776492763451364867 1.8667603991738620930 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 2, 3] and d occurrences (as containment) of the composition, [2, 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 8 5 6 6 - X1 x - X2 x + X1 x + X2 x + x - x + 2 X1 X2 x - X1 x - X2 x 4 6 3 5 4 2 / - 2 X1 X2 x + 2 x + X1 X2 x - 2 x + 2 x - 3 x + 3 x - 1) / ( / 2 4 3 4 3 (-1 + x) (X1 X2 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+2*X1*X2*x^5-X1*x^6-X2 *x^6-2*X1*X2*x^4+2*x^6+X1*X2*x^3-2*x^5+2*x^4-3*x^2+3*x-1)/(-1+x)^2/(X1*X2*x^4- X1*X2*x^3-x^4+x^3-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 2, 3], are n 19 13 n - 3/8 + ----, and , - -- + ----, respectively, while 16 64 256 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 3, 1], are n 19 13 n - 3/8 + ----, and , - -- + ----, respectively, while the asymptotic correlat\ 16 64 256 ion between these two random variables is 1 and in floating point 1. ------------------------------------------------- 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, 2, 3], nor the composition, [3, 1, 2] Then infinity ----- 13 12 10 9 8 6 3 2 \ n x + x - x + x + x + x - x - x + 2 x - 1 ) a(n) x = -------------------------------------------------- / 7 6 5 2 2 ----- (x + 2 x + x + x + x - 1) (-1 + x) n = 0 and in Maple format (x^13+x^12-x^10+x^9+x^8+x^6-x^3-x^2+2*x-1)/(x^7+2*x^6+x^5+x^2+x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.86386596006048776696 1.7927428383414338089 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 2, 3] and d occurrences (as containment) of the composition, [3, 1, 2], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 3 14 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 2 14 3 14 2 14 2 14 3 14 - 3 X1 X2 x - 2 X1 X2 x + 3 X1 X2 x + 6 X1 X2 x + X2 x 2 2 12 2 14 14 2 14 2 12 + X1 X2 x - X1 x - 6 X1 X2 x - 3 X2 x - 2 X1 X2 x 2 12 14 14 2 2 10 2 12 2 11 - 2 X1 X2 x + 2 X1 x + 3 X2 x + X1 X2 x + X1 x - X1 X2 x 12 2 12 14 2 2 9 2 10 11 + 4 X1 X2 x + X2 x - x - X1 X2 x - X1 X2 x + 2 X1 X2 x 12 2 11 12 2 9 2 9 10 - 2 X1 x + X2 x - 2 X2 x + 2 X1 X2 x + X1 X2 x - 2 X1 X2 x 11 2 10 11 12 2 9 2 8 9 - X1 x - X2 x - 2 X2 x + x - X1 x + X1 X2 x - 2 X1 X2 x 10 10 11 2 7 8 9 10 + 2 X1 x + 3 X2 x + x - X1 X2 x - X1 X2 x + X1 x - 2 x 7 8 7 8 5 6 7 4 + X1 X2 x - X2 x + X2 x + x + 2 X1 X2 x - X1 x - x - 2 X1 X2 x 5 6 3 4 4 2 / - 2 X2 x + x + X1 X2 x + X2 x + x - 3 x + 3 x - 1) / ((-1 + x) ( / 2 2 10 2 2 9 2 10 2 10 2 9 2 10 X1 X2 x - X1 X2 x - 2 X1 X2 x - X1 X2 x + X1 X2 x + X1 x 2 9 10 2 8 10 2 7 8 + X1 X2 x + 2 X1 X2 x + X1 X2 x - X1 x - X1 X2 x - 2 X1 X2 x 9 9 8 9 7 7 5 7 - X1 x - X2 x + X1 x + x + X1 x + 2 X2 x + X1 X2 x - 2 x 4 5 3 4 5 4 3 2 - 2 X1 X2 x - 2 X2 x + X1 X2 x + X2 x + x + x - x - 2 x + 3 x - 1) ) and in Maple format -(X1^2*X2^3*x^14-3*X1^2*X2^2*x^14-2*X1*X2^3*x^14+3*X1^2*X2*x^14+6*X1*X2^2*x^14+ X2^3*x^14+X1^2*X2^2*x^12-X1^2*x^14-6*X1*X2*x^14-3*X2^2*x^14-2*X1^2*X2*x^12-2*X1 *X2^2*x^12+2*X1*x^14+3*X2*x^14+X1^2*X2^2*x^10+X1^2*x^12-X1*X2^2*x^11+4*X1*X2*x^ 12+X2^2*x^12-x^14-X1^2*X2^2*x^9-X1^2*X2*x^10+2*X1*X2*x^11-2*X1*x^12+X2^2*x^11-2 *X2*x^12+2*X1^2*X2*x^9+X1*X2^2*x^9-2*X1*X2*x^10-X1*x^11-X2^2*x^10-2*X2*x^11+x^ 12-X1^2*x^9+X1*X2^2*x^8-2*X1*X2*x^9+2*X1*x^10+3*X2*x^10+x^11-X1*X2^2*x^7-X1*X2* x^8+X1*x^9-2*x^10+X1*X2*x^7-X2*x^8+X2*x^7+x^8+2*X1*X2*x^5-X1*x^6-x^7-2*X1*X2*x^ 4-2*X2*x^5+x^6+X1*X2*x^3+X2*x^4+x^4-3*x^2+3*x-1)/(-1+x)/(X1^2*X2^2*x^10-X1^2*X2 ^2*x^9-2*X1^2*X2*x^10-X1*X2^2*x^10+X1^2*X2*x^9+X1^2*x^10+X1*X2^2*x^9+2*X1*X2*x^ 10+X1*X2^2*x^8-X1*x^10-X1*X2^2*x^7-2*X1*X2*x^8-X1*x^9-X2*x^9+X1*x^8+x^9+X1*x^7+ 2*X2*x^7+X1*X2*x^5-2*x^7-2*X1*X2*x^4-2*X2*x^5+X1*X2*x^3+X2*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, [1, 2, 3], are n 19 13 n - 3/8 + ----, and , - -- + ----, respectively, while 16 64 256 Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 1, 2], are n 21 13 n - 3/8 + ----, and , - -- + ----, respectively, while the asymptotic correlat\ 16 64 256 ion between these two random variables is 7/13 and in floating point 0.5384615385 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ ate normal pair with correlation, 7/13 267 i.e. , [[7/13, 0], [0, ---]] 169 ------------------------------------------------- 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, 2, 3], nor the composition, [3, 2, 1] Then infinity ----- 11 8 7 5 4 3 2 \ n x + x + 2 x - x + x - x - x + 2 x - 1 ) a(n) x = --------------------------------------------- / 5 4 2 2 ----- (x + x + x + x - 1) (-1 + x) n = 0 and in Maple format (x^11+x^8+2*x^7-x^5+x^4-x^3-x^2+2*x-1)/(x^5+x^4+x^2+x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.79705424577713694206 1.8124036192680426608 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 2, 3] and d occurrences (as containment) of the composition, [3, 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 2 11 2 12 2 12 2 11 2 12 - X1 X2 x - 2 X1 X2 x - 2 X1 X2 x + 2 X1 X2 x + X1 x 2 11 12 2 12 2 11 11 12 + 2 X1 X2 x + 4 X1 X2 x + X2 x - X1 x - 4 X1 X2 x - 2 X1 x 2 11 12 2 9 2 9 11 11 12 - X2 x - 2 X2 x - X1 X2 x - X1 X2 x + 2 X1 x + 2 X2 x + x 2 9 9 2 9 11 8 9 9 + X1 x + 3 X1 X2 x + X2 x - x + X1 X2 x - 2 X1 x - 2 X2 x 7 8 8 9 6 7 7 8 - 2 X1 X2 x - X1 x - X2 x + x + X1 X2 x + 2 X1 x + 2 X2 x + x 5 7 4 6 3 5 4 2 - 2 X1 X2 x - 2 x + 2 X1 X2 x - x - X1 X2 x + 2 x - 2 x + 3 x / 2 - 3 x + 1) / ((-1 + x) / 6 6 6 4 6 3 4 3 (X1 X2 x - X1 x - X2 x + X1 X2 x + x - X1 X2 x - 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^12-2*X1*X2^2*x^12+2*X1^2*X2*x^11+X1^ 2*x^12+2*X1*X2^2*x^11+4*X1*X2*x^12+X2^2*x^12-X1^2*x^11-4*X1*X2*x^11-2*X1*x^12- X2^2*x^11-2*X2*x^12-X1^2*X2*x^9-X1*X2^2*x^9+2*X1*x^11+2*X2*x^11+x^12+X1^2*x^9+3 *X1*X2*x^9+X2^2*x^9-x^11+X1*X2*x^8-2*X1*x^9-2*X2*x^9-2*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^5-2*x^7+2*X1*X2*x^4-x^6-X1*X2*x^3 +2*x^5-2*x^4+3*x^2-3*x+1)/(-1+x)^2/(X1*X2*x^6-X1*x^6-X2*x^6+X1*X2*x^4+x^6-X1*X2 *x^3-x^4+x^3-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 2, 3], are n 19 13 n - 3/8 + ----, and , - -- + ----, respectively, while 16 64 256 Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 2, 1], are n 19 13 n - 3/8 + ----, and , - -- + ----, respectively, while the asymptotic correlat\ 16 64 256 ion between these two random variables is 9/13 and in floating point 0.6923076923 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ ate normal pair with correlation, 9/13 331 i.e. , [[9/13, 0], [0, ---]] 169 ------------------------------------------------- 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, 2, 3], nor the composition, [4, 1, 1] Then infinity ----- 11 8 6 4 3 2 \ n x - 2 x - x + 2 x - 3 x + 4 x - 3 x + 1 ) a(n) x = - ---------------------------------------------- / 3 2 2 ----- (x - x + 2 x - 1) (-1 + x) n = 0 and in Maple format -(x^11-2*x^8-x^6+2*x^4-3*x^3+4*x^2-3*x+1)/(x^3-x^2+2*x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.98591087239572105185 1.7548776662466927601 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 2, 3] and d occurrences (as containment) of the composition, [4, 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 13 - 2 X1 X2 x - 2 X1 X2 x + X1 x + 4 X1 X2 x + X2 x 2 11 13 13 2 10 2 10 11 + X1 X2 x - 2 X1 x - 2 X2 x + X1 X2 x + X1 X2 x - 2 X1 X2 x 2 11 13 2 10 10 11 2 10 11 - X2 x + x - X1 x - 4 X1 X2 x + X1 x - X2 x + 2 X2 x 10 10 11 8 10 8 8 + 3 X1 x + 3 X2 x - x + X1 X2 x - 2 x - X1 x - X2 x 6 8 5 6 6 4 5 + 2 X1 X2 x + x - 2 X1 X2 x - 3 X1 x - 2 X2 x + X1 X2 x + 4 X1 x 5 6 4 5 3 4 2 / + X2 x + 3 x - 3 X1 x - 3 x + X1 x + 2 x - 3 x + 3 x - 1) / ( / 2 5 4 5 5 4 5 3 4 (-1 + x) (X1 X2 x - X1 X2 x - X1 x - X2 x + 2 X1 x + x - X1 x - x 3 + x - 2 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+4*X1*X2*x^13+X2^2*x^13 +X1*X2^2*x^11-2*X1*x^13-2*X2*x^13+X1^2*X2*x^10+X1*X2^2*x^10-2*X1*X2*x^11-X2^2*x ^11+x^13-X1^2*x^10-4*X1*X2*x^10+X1*x^11-X2^2*x^10+2*X2*x^11+3*X1*x^10+3*X2*x^10 -x^11+X1*X2*x^8-2*x^10-X1*x^8-X2*x^8+2*X1*X2*x^6+x^8-2*X1*X2*x^5-3*X1*x^6-2*X2* x^6+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)/(-\ 1+x)^2/(X1*X2*x^5-X1*X2*x^4-X1*x^5-X2*x^5+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, 2, 3], are n 19 13 n - 3/8 + ----, and , - -- + ----, respectively, while 16 64 256 Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 1, 1], are n 9 n - 3/8 + ----, and , - 5/32 + ---, respectively, while the asymptotic correla\ 16 256 tion between these two random variables is 1/2 13 ----- 13 and in floating point 0.2773500981 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 13 ate normal pair with correlation, ----- 13 1/2 13 15 i.e. , [[-----, 0], [0, --]] 13 13 ------------------------------------------------- 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, 3, 2], nor the composition, [1, 4, 1] Then infinity ----- 6 5 4 3 2 \ n x - 2 x + x - x - x + 2 x - 1 ) a(n) x = ------------------------------------- / 3 2 2 ----- (x + 1) (x - x + 2 x - 1) (-1 + x) n = 0 and in Maple format (x^6-2*x^5+x^4-x^3-x^2+2*x-1)/(x+1)/(x^3-x^2+2*x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 1.1544840707379761413 1.7548776662466927601 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 3, 2] and d occurrences (as containment) of the composition, [1, 4, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 6 5 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x - X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 6 6 4 5 5 6 4 5 3 - X1 x - X2 x + X1 X2 x + 2 X1 x + X2 x + x - 2 X1 x - 2 x + X1 x 4 3 2 / 5 4 5 + x - x - x + 2 x - 1) / ((-1 + x) (X1 X2 x - X1 X2 x - X1 x / 5 4 5 3 4 3 - X2 x + 2 X1 x + x - X1 x - x + x - 2 x + 1)) and in Maple format (X1*X2*x^6-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)/(-1+x)/(X1*X2*x^5-X1*X2*x^4-X1*x^5-X2*x^5+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, 2], are n 19 13 n - 3/8 + ----, and , - -- + ----, respectively, while 16 64 256 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 4, 1], are n 9 n - 3/8 + ----, and , - 5/32 + ---, respectively, while the asymptotic correla\ 16 256 tion between these two random variables is 1/2 13 ----- 13 and in floating point 0.2773500981 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 13 ate normal pair with correlation, ----- 13 1/2 13 15 i.e. , [[-----, 0], [0, --]] 13 13 ------------------------------------------------- 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, 3, 2], nor the composition, [2, 1, 3] Then infinity ----- 5 4 3 \ n x + x + x - x + 1 ) a(n) x = -------------------------------------- / 7 6 5 2 ----- (-1 + x) (x + 2 x + x + x + x - 1) n = 0 and in Maple format (x^5+x^4+x^3-x+1)/(-1+x)/(x^7+2*x^6+x^5+x^2+x-1) The asymptotic expression for a(n) is, n 0.89858714643315996836 1.7927428383414338089 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 3, 2] and d occurrences (as containment) of the composition, [2, 1, 3], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 9 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 9 2 9 2 9 9 2 7 9 - 2 X1 X2 x - X1 X2 x + X1 x + 2 X1 X2 x + X1 X2 x - X1 x 7 6 7 5 6 6 4 - 2 X1 X2 x - X1 X2 x + X1 x - X1 X2 x + X1 x + X2 x + X1 X2 x 5 6 3 4 3 2 / 2 2 10 + X2 x - x - X1 X2 x - X2 x + x + x - 2 x + 1) / (X1 X2 x / 2 2 9 2 10 2 10 2 9 2 10 2 9 - X1 X2 x - 2 X1 X2 x - X1 X2 x + X1 X2 x + X1 x + X1 X2 x 10 2 8 10 2 7 8 9 + 2 X1 X2 x + X1 X2 x - X1 x - X1 X2 x - 2 X1 X2 x - X1 x 9 8 9 7 7 5 7 4 - X2 x + X1 x + x + X1 x + 2 X2 x + X1 X2 x - 2 x - 2 X1 X2 x 5 3 4 5 4 3 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^9-2*X1^2*X2*x^9-X1*X2^2*x^9+X1^2*x^9+2*X1*X2*x^9+X1*X2^2*x^7-X1*x ^9-2*X1*X2*x^7-X1*X2*x^6+X1*x^7-X1*X2*x^5+X1*x^6+X2*x^6+X1*X2*x^4+X2*x^5-x^6-X1 *X2*x^3-X2*x^4+x^3+x^2-2*x+1)/(X1^2*X2^2*x^10-X1^2*X2^2*x^9-2*X1^2*X2*x^10-X1* X2^2*x^10+X1^2*X2*x^9+X1^2*x^10+X1*X2^2*x^9+2*X1*X2*x^10+X1*X2^2*x^8-X1*x^10-X1 *X2^2*x^7-2*X1*X2*x^8-X1*x^9-X2*x^9+X1*x^8+x^9+X1*x^7+2*X2*x^7+X1*X2*x^5-2*x^7-\ 2*X1*X2*x^4-2*X2*x^5+X1*X2*x^3+X2*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, [1, 3, 2], are n 19 13 n - 3/8 + ----, and , - -- + ----, respectively, while 16 64 256 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 1, 3], are n 21 13 n - 3/8 + ----, and , - -- + ----, respectively, while the asymptotic correlat\ 16 64 256 ion between these two random variables is 7/13 and in floating point 0.5384615385 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ ate normal pair with correlation, 7/13 267 i.e. , [[7/13, 0], [0, ---]] 169 ------------------------------------------------- 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, 3, 2], nor the composition, [2, 2, 2] Then infinity ----- 4 3 2 \ n x - x + 2 x - 2 x + 1 ) a(n) x = --------------------------------------- / 2 3 2 ----- (-1 + x) (x - x + 1) (x + x + x - 1) n = 0 and in Maple format (x^4-x^3+2*x^2-2*x+1)/(-1+x)/(x^2-x+1)/(x^3+x^2+x-1) The asymptotic expression for a(n) is, n 0.77591233811587484106 1.8392867552141611326 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 3, 2] and d occurrences (as containment) of the composition, [2, 2, 2], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 5 4 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x - X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 5 3 4 3 4 2 3 2 - X2 x + X1 X2 x + 2 X2 x - 2 X2 x - x + X2 x + x - 2 x + 2 x - 1) / 6 5 6 6 4 5 6 / (X1 X2 x - X1 X2 x - X1 x - X2 x + 2 X1 X2 x + 2 X2 x + x / 3 4 5 3 4 2 3 2 - X1 X2 x - 3 X2 x - x + 3 X2 x + x - X2 x - 2 x + 3 x - 3 x + 1) and in Maple format -(X1*X2*x^5-X1*X2*x^4-X2*x^5+X1*X2*x^3+2*X2*x^4-2*X2*x^3-x^4+X2*x^2+x^3-2*x^2+2 *x-1)/(X1*X2*x^6-X1*X2*x^5-X1*x^6-X2*x^6+2*X1*X2*x^4+2*X2*x^5+x^6-X1*X2*x^3-3* X2*x^4-x^5+3*X2*x^3+x^4-X2*x^2-2*x^3+3*x^2-3*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 3, 2], are n 19 13 n - 3/8 + ----, and , - -- + ----, respectively, while 16 64 256 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 2, 2], are n 23 25 n - 3/8 + ----, and , - -- + ----, respectively, while the asymptotic correlat\ 16 32 256 ion between these two random variables is 1/2 11 13 -------- 65 and in floating point 0.6101702157 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 11 13 ate normal pair with correlation, -------- 65 1/2 11 13 567 i.e. , [[--------, 0], [0, ---]] 65 325 ------------------------------------------------- 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, 3, 2], nor the composition, [2, 3, 1] Then infinity ----- 9 8 7 6 5 4 2 \ n x - x + x + x - 2 x + 2 x - 3 x + 3 x - 1 ) a(n) x = - ------------------------------------------------ / 5 4 2 3 ----- (x + x + x + x - 1) (-1 + x) n = 0 and in Maple format -(x^9-x^8+x^7+x^6-2*x^5+2*x^4-3*x^2+3*x-1)/(x^5+x^4+x^2+x-1)/(-1+x)^3 The asymptotic expression for a(n) is, n 0.83890500626921008526 1.8124036192680426608 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 3, 2] and d occurrences (as containment) of the composition, [2, 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 7 8 8 9 6 7 7 - X1 x - X2 x + X1 X2 x + X1 x + X2 x + x - X1 X2 x - X1 x - X2 x 8 5 7 4 6 3 5 4 2 - x + 2 X1 X2 x + x - 2 X1 X2 x + x + X1 X2 x - 2 x + 2 x - 3 x / 2 + 3 x - 1) / ((-1 + x) / 6 6 6 4 6 3 4 3 (X1 X2 x - X1 x - X2 x + X1 X2 x + 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*X2*x^7+X1*x^8+X2*x^8+x^9-X1*X2*x^6-X1*x^ 7-X2*x^7-x^8+2*X1*X2*x^5+x^7-2*X1*X2*x^4+x^6+X1*X2*x^3-2*x^5+2*x^4-3*x^2+3*x-1) /(-1+x)^2/(X1*X2*x^6-X1*x^6-X2*x^6+X1*X2*x^4+x^6-X1*X2*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, 2], are n 19 13 n - 3/8 + ----, and , - -- + ----, respectively, while 16 64 256 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 3, 1], are n 19 13 n - 3/8 + ----, and , - -- + ----, respectively, while the asymptotic correlat\ 16 64 256 ion between these two random variables is 9/13 and in floating point 0.6923076923 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ ate normal pair with correlation, 9/13 331 i.e. , [[9/13, 0], [0, ---]] 169 ------------------------------------------------- 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, 3, 2], nor the composition, [3, 1, 2] Then infinity ----- 6 5 4 3 \ n x + x + x + x - x + 1 ) a(n) x = - ------------------------- / 5 3 ----- x - x + 2 x - 1 n = 0 and in Maple format -(x^6+x^5+x^4+x^3-x+1)/(x^5-x^3+2*x-1) The asymptotic expression for a(n) is, n 0.92094939015237528373 1.7845989333686468028 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 3, 2] and d occurrences (as containment) of the composition, [3, 1, 2], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 7 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 7 7 7 5 7 4 5 - 3 X1 X2 x + 2 X1 x + X2 x - X1 X2 x - x + X1 X2 x + X2 x 3 4 3 2 / 2 8 2 7 - X1 X2 x - X2 x + x + x - 2 x + 1) / (X1 X2 x - X1 X2 x / 8 7 8 6 7 5 6 - 2 X1 X2 x + 2 X1 X2 x + X1 x - X1 X2 x - X1 x + X1 X2 x + X1 x 6 4 5 6 3 4 5 4 3 + X2 x - 2 X1 X2 x - 2 X2 x - x + X1 X2 x + X2 x + x + x - x 2 - 2 x + 3 x - 1) and in Maple format -(X1*X2^2*x^7-3*X1*X2*x^7+2*X1*x^7+X2*x^7-X1*X2*x^5-x^7+X1*X2*x^4+X2*x^5-X1*X2* x^3-X2*x^4+x^3+x^2-2*x+1)/(X1*X2^2*x^8-X1*X2^2*x^7-2*X1*X2*x^8+2*X1*X2*x^7+X1*x ^8-X1*X2*x^6-X1*x^7+X1*X2*x^5+X1*x^6+X2*x^6-2*X1*X2*x^4-2*X2*x^5-x^6+X1*X2*x^3+ X2*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, [1, 3, 2], are n 19 13 n - 3/8 + ----, and , - -- + ----, respectively, while 16 64 256 Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 1, 2], are n 21 13 n - 3/8 + ----, and , - -- + ----, respectively, while the asymptotic correlat\ 16 64 256 ion between these two random variables is 7/13 and in floating point 0.5384615385 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ ate normal pair with correlation, 7/13 267 i.e. , [[7/13, 0], [0, ---]] 169 ------------------------------------------------- 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, 3, 2], nor the composition, [3, 2, 1] Then infinity ----- 9 8 6 5 4 2 \ n x - x + 2 x - 2 x + 2 x - 3 x + 3 x - 1 ) a(n) x = --------------------------------------------- / 4 3 2 ----- (x - x + 2 x - 1) (-1 + x) n = 0 and in Maple format (x^9-x^8+2*x^6-2*x^5+2*x^4-3*x^2+3*x-1)/(x^4-x^3+2*x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.65776492763451364867 1.8667603991738620930 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 3, 2] and d occurrences (as containment) of the composition, [3, 2, 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 8 5 6 6 - X1 x - X2 x + X1 x + X2 x + x - x + 2 X1 X2 x - X1 x - X2 x 4 6 3 5 4 2 / - 2 X1 X2 x + 2 x + X1 X2 x - 2 x + 2 x - 3 x + 3 x - 1) / ( / 2 4 3 4 3 (-1 + x) (X1 X2 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+2*X1*X2*x^5-X1*x^6-X2 *x^6-2*X1*X2*x^4+2*x^6+X1*X2*x^3-2*x^5+2*x^4-3*x^2+3*x-1)/(-1+x)^2/(X1*X2*x^4- X1*X2*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, 2], are n 19 13 n - 3/8 + ----, and , - -- + ----, respectively, while 16 64 256 Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 2, 1], are n 19 13 n - 3/8 + ----, and , - -- + ----, respectively, while the asymptotic correlat\ 16 64 256 ion between these two random variables is 1 and in floating point 1. ------------------------------------------------- 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, 3, 2], nor the composition, [4, 1, 1] Then infinity ----- 8 6 4 3 2 \ n x + x - 2 x + 3 x - 4 x + 3 x - 1 ) a(n) x = -------------------------------------- / 3 2 2 ----- (x - x + 2 x - 1) (-1 + x) n = 0 and in Maple format (x^8+x^6-2*x^4+3*x^3-4*x^2+3*x-1)/(x^3-x^2+2*x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 1.0327532479435799189 1.7548776662466927601 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 3, 2] and d occurrences (as containment) of the composition, [4, 1, 1], (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 10 2 10 10 10 10 6 - 2 X1 X2 x - X2 x + X1 x + 2 X2 x - x + 2 X1 X2 x 5 6 6 4 5 5 6 - 2 X1 X2 x - 3 X1 x - 2 X2 x + X1 X2 x + 4 X1 x + X2 x + 3 x 4 5 3 4 2 / 2 5 - 3 X1 x - 3 x + X1 x + 2 x - 3 x + 3 x - 1) / ((-1 + x) (X1 X2 x / 4 5 5 4 5 3 4 3 - X1 X2 x - X1 x - X2 x + 2 X1 x + x - X1 x - x + x - 2 x + 1)) and in Maple format -(X1*X2^2*x^10-2*X1*X2*x^10-X2^2*x^10+X1*x^10+2*X2*x^10-x^10+2*X1*X2*x^6-2*X1* X2*x^5-3*X1*x^6-2*X2*x^6+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)/(-1+x)^2/(X1*X2*x^5-X1*X2*x^4-X1*x^5-X2*x^5+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, 2], are n 19 13 n - 3/8 + ----, and , - -- + ----, respectively, while 16 64 256 Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 1, 1], are n 9 n - 3/8 + ----, and , - 5/32 + ---, respectively, while the asymptotic correla\ 16 256 tion between these two random variables is 1/2 13 ----- 13 and in floating point 0.2773500981 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 13 ate normal pair with correlation, ----- 13 1/2 13 15 i.e. , [[-----, 0], [0, --]] 13 13 ------------------------------------------------- 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, 4, 1], nor the composition, [2, 1, 3] Then infinity ----- \ n ) a(n) x = / ----- n = 0 11 8 7 6 5 4 3 2 x + x - 3 x + 3 x - 3 x + 3 x - 3 x + 4 x - 3 x + 1 - ------------------------------------------------------------ 3 2 4 2 (x - x + 2 x - 1) (x + 1) (-1 + x) and in Maple format -(x^11+x^8-3*x^7+3*x^6-3*x^5+3*x^4-3*x^3+4*x^2-3*x+1)/(x^3-x^2+2*x-1)/(x^4+1)/( -1+x)^2 The asymptotic expression for a(n) is, n 1.0647406982807962194 1.7548776662466927601 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 4, 1] and d occurrences (as containment) of the composition, [2, 1, 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 12 2 13 - 2 X1 X2 x - 2 X1 X2 x - X1 X2 x - X1 X2 x + X1 x 2 12 13 2 13 2 2 10 2 11 + X1 X2 x + 4 X1 X2 x + X2 x + X1 X2 x + 2 X1 X2 x 2 12 2 11 13 2 12 13 2 2 9 + X1 x + 2 X1 X2 x - 2 X1 x - X2 x - 2 X2 x - X1 X2 x 2 10 2 11 2 10 11 12 2 11 - X1 X2 x - X1 x - 3 X1 X2 x - 4 X1 X2 x - X1 x - X2 x 12 13 2 9 2 9 10 11 + X2 x + x + X1 X2 x + 4 X1 X2 x + 4 X1 X2 x + 2 X1 x 2 10 11 2 8 9 10 2 9 + 2 X2 x + 2 X2 x - 2 X1 X2 x - 7 X1 X2 x - X1 x - 3 X2 x 10 11 8 9 2 8 9 10 - 3 X2 x - x + 3 X1 X2 x + 3 X1 x + 3 X2 x + 6 X2 x + x 7 8 2 7 8 9 7 7 8 + X1 X2 x - X1 x - X2 x - 5 X2 x - 3 x - X1 x + X2 x + 2 x 5 4 5 4 3 4 2 - X1 X2 x + X1 X2 x + X2 x - 2 X2 x + X2 x + x - 3 x + 3 x - 1) / 5 4 5 5 4 5 3 / ((-1 + x) (X1 X2 x - X1 X2 x - X1 x - X2 x + 2 X2 x + x - X2 x / 4 3 5 5 5 4 5 4 - x + x - 2 x + 1) (X1 X2 x - X1 x - X2 x + X2 x + x - x + 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*X2^2*x^11-X1^2*X2*x^12+X1^2 *x^13+X1*X2^2*x^12+4*X1*X2*x^13+X2^2*x^13+X1^2*X2^2*x^10+2*X1^2*X2*x^11+X1^2*x^ 12+2*X1*X2^2*x^11-2*X1*x^13-X2^2*x^12-2*X2*x^13-X1^2*X2^2*x^9-X1^2*X2*x^10-X1^2 *x^11-3*X1*X2^2*x^10-4*X1*X2*x^11-X1*x^12-X2^2*x^11+X2*x^12+x^13+X1^2*X2*x^9+4* X1*X2^2*x^9+4*X1*X2*x^10+2*X1*x^11+2*X2^2*x^10+2*X2*x^11-2*X1*X2^2*x^8-7*X1*X2* x^9-X1*x^10-3*X2^2*x^9-3*X2*x^10-x^11+3*X1*X2*x^8+3*X1*x^9+3*X2^2*x^8+6*X2*x^9+ x^10+X1*X2*x^7-X1*x^8-X2^2*x^7-5*X2*x^8-3*x^9-X1*x^7+X2*x^7+2*x^8-X1*X2*x^5+X1* X2*x^4+X2*x^5-2*X2*x^4+X2*x^3+x^4-3*x^2+3*x-1)/(-1+x)/(X1*X2*x^5-X1*X2*x^4-X1*x ^5-X2*x^5+2*X2*x^4+x^5-X2*x^3-x^4+x^3-2*x+1)/(X1*X2*x^5-X1*x^5-X2*x^5+X2*x^4+x^ 5-x^4+x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 4, 1], are n 9 n - 3/8 + ----, and , - 5/32 + ---, respectively, while 16 256 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 1, 3], are n 21 13 n - 3/8 + ----, and , - -- + ----, respectively, while the asymptotic correlat\ 16 64 256 ion between these two random variables is 1/2 13 ----- 13 and in floating point 0.2773500981 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 13 ate normal pair with correlation, ----- 13 1/2 13 15 i.e. , [[-----, 0], [0, --]] 13 13 ------------------------------------------------- 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, 4, 1], nor the composition, [2, 2, 2] Then infinity ----- 4 3 2 6 5 3 \ n (x - x + x - x + 1) (x + x + x - x + 1) ) a(n) x = - --------------------------------------------- / 7 6 5 4 3 2 ----- (x + 2 x + x + x + x + x - 1) (-1 + x) n = 0 and in Maple format -(x^4-x^3+x^2-x+1)*(x^6+x^5+x^3-x+1)/(x^7+2*x^6+x^5+x^4+x^3+x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 1.0830425492496005336 1.7610793284505662547 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 4, 1] and d occurrences (as containment) of the composition, [2, 2, 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 2 10 2 11 11 2 9 2 10 10 - X1 X2 x - X1 x - 2 X1 X2 x + X1 X2 x + X1 x + 2 X1 X2 x 11 11 2 9 9 10 10 11 + 2 X1 x + X2 x - X1 x - X1 X2 x - 2 X1 x - X2 x - x 8 9 10 7 8 8 6 - X1 X2 x + X1 x + x + 2 X1 X2 x + X1 x + X2 x - 2 X1 X2 x 7 7 8 5 6 6 7 - 2 X1 x - 2 X2 x - x + 2 X1 X2 x + 2 X1 x + 2 X2 x + 2 x 4 5 5 6 4 3 4 2 3 - X1 X2 x - X1 x - X2 x - 2 x - X2 x + 2 X2 x + 2 x - X2 x - 3 x 2 / 2 9 2 9 9 9 + 4 x - 3 x + 1) / ((-1 + x) (X1 X2 x - X1 x - 2 X1 X2 x + 2 X1 x / 9 7 9 6 7 7 5 + X2 x + 2 X1 X2 x - x - X1 X2 x - 2 X1 x - 2 X2 x + 2 X1 X2 x 6 6 7 4 5 5 6 3 4 + X1 x + X2 x + 2 x - X1 X2 x - X1 x - X2 x - x + 2 X2 x + x 2 3 2 - X2 x - 2 x + 3 x - 3 x + 1)) and in Maple format -(X1^2*X2*x^11-X1^2*X2*x^10-X1^2*x^11-2*X1*X2*x^11+X1^2*X2*x^9+X1^2*x^10+2*X1* X2*x^10+2*X1*x^11+X2*x^11-X1^2*x^9-X1*X2*x^9-2*X1*x^10-X2*x^10-x^11-X1*X2*x^8+ X1*x^9+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+2*X1*X2 *x^5+2*X1*x^6+2*X2*x^6+2*x^7-X1*X2*x^4-X1*x^5-X2*x^5-2*x^6-X2*x^4+2*X2*x^3+2*x^ 4-X2*x^2-3*x^3+4*x^2-3*x+1)/(-1+x)/(X1^2*X2*x^9-X1^2*x^9-2*X1*X2*x^9+2*X1*x^9+ X2*x^9+2*X1*X2*x^7-x^9-X1*X2*x^6-2*X1*x^7-2*X2*x^7+2*X1*X2*x^5+X1*x^6+X2*x^6+2* x^7-X1*X2*x^4-X1*x^5-X2*x^5-x^6+2*X2*x^3+x^4-X2*x^2-2*x^3+3*x^2-3*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 4, 1], are n 9 n - 3/8 + ----, and , - 5/32 + ---, respectively, while 16 256 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 2, 2], are n 23 25 n - 3/8 + ----, and , - -- + ----, respectively, while the asymptotic correlat\ 16 32 256 ion between these two random variables is 1/15 and in floating point 0.06666666667 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ ate normal pair with correlation, 1/15 227 i.e. , [[1/15, 0], [0, ---]] 225 ------------------------------------------------- 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, 4, 1], nor the composition, [2, 3, 1] Then infinity ----- 6 5 4 3 2 \ n x - 2 x + x - x - x + 2 x - 1 ) a(n) x = ------------------------------------- / 3 2 2 ----- (x + 1) (x - x + 2 x - 1) (-1 + x) n = 0 and in Maple format (x^6-2*x^5+x^4-x^3-x^2+2*x-1)/(x+1)/(x^3-x^2+2*x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 1.1544840707379761413 1.7548776662466927601 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 4, 1] and d occurrences (as containment) of the composition, [2, 3, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 6 5 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x - X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 6 6 4 5 5 6 4 5 3 - X1 x - X2 x + X1 X2 x + X1 x + 2 X2 x + x - 2 X2 x - 2 x + X2 x 4 3 2 / 5 4 5 + x - x - x + 2 x - 1) / ((-1 + x) (X1 X2 x - X1 X2 x - X1 x / 5 4 5 3 4 3 - X2 x + 2 X2 x + x - X2 x - x + x - 2 x + 1)) and in Maple format (X1*X2*x^6-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)/(-1+x)/(X1*X2*x^5-X1*X2*x^4-X1*x^5-X2*x^5+2*X2*x^4+x ^5-X2*x^3-x^4+x^3-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 4, 1], are n 9 n - 3/8 + ----, and , - 5/32 + ---, respectively, while 16 256 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 3, 1], are n 19 13 n - 3/8 + ----, and , - -- + ----, respectively, while the asymptotic correlat\ 16 64 256 ion between these two random variables is 1/2 13 ----- 13 and in floating point 0.2773500981 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 13 ate normal pair with correlation, ----- 13 1/2 13 15 i.e. , [[-----, 0], [0, --]] 13 13 ------------------------------------------------- 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, 4, 1], nor the composition, [3, 1, 2] Then infinity ----- \ n ) a(n) x = / ----- n = 0 11 8 7 6 5 4 3 2 x + x - 3 x + 3 x - 3 x + 3 x - 3 x + 4 x - 3 x + 1 - ------------------------------------------------------------ 3 2 4 2 (x - x + 2 x - 1) (x + 1) (-1 + x) and in Maple format -(x^11+x^8-3*x^7+3*x^6-3*x^5+3*x^4-3*x^3+4*x^2-3*x+1)/(x^3-x^2+2*x-1)/(x^4+1)/( -1+x)^2 The asymptotic expression for a(n) is, n 1.0647406982807962194 1.7548776662466927601 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 4, 1] and d occurrences (as containment) of the composition, [3, 1, 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 2 11 2 12 2 13 - 2 X1 X2 x - 2 X1 X2 x - X1 X2 x - X1 X2 x + X1 x 2 12 13 2 13 2 2 10 2 11 + X1 X2 x + 4 X1 X2 x + X2 x + X1 X2 x + 2 X1 X2 x 2 12 2 11 13 2 12 13 2 2 9 + X1 x + 2 X1 X2 x - 2 X1 x - X2 x - 2 X2 x - X1 X2 x 2 10 2 11 2 10 11 12 2 11 - X1 X2 x - X1 x - 3 X1 X2 x - 4 X1 X2 x - X1 x - X2 x 12 13 2 9 2 9 10 11 + X2 x + x + X1 X2 x + 4 X1 X2 x + 4 X1 X2 x + 2 X1 x 2 10 11 2 8 9 10 2 9 + 2 X2 x + 2 X2 x - 2 X1 X2 x - 7 X1 X2 x - X1 x - 3 X2 x 10 11 8 9 2 8 9 10 - 3 X2 x - x + 3 X1 X2 x + 3 X1 x + 3 X2 x + 6 X2 x + x 7 8 2 7 8 9 7 7 8 + X1 X2 x - X1 x - X2 x - 5 X2 x - 3 x - X1 x + X2 x + 2 x 5 4 5 4 3 4 2 - X1 X2 x + X1 X2 x + X2 x - 2 X2 x + X2 x + x - 3 x + 3 x - 1) / 5 4 5 5 4 5 3 / ((-1 + x) (X1 X2 x - X1 X2 x - X1 x - X2 x + 2 X2 x + x - X2 x / 4 3 5 5 5 4 5 4 - x + x - 2 x + 1) (X1 X2 x - X1 x - X2 x + X2 x + x - x + 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*X2^2*x^11-X1^2*X2*x^12+X1^2 *x^13+X1*X2^2*x^12+4*X1*X2*x^13+X2^2*x^13+X1^2*X2^2*x^10+2*X1^2*X2*x^11+X1^2*x^ 12+2*X1*X2^2*x^11-2*X1*x^13-X2^2*x^12-2*X2*x^13-X1^2*X2^2*x^9-X1^2*X2*x^10-X1^2 *x^11-3*X1*X2^2*x^10-4*X1*X2*x^11-X1*x^12-X2^2*x^11+X2*x^12+x^13+X1^2*X2*x^9+4* X1*X2^2*x^9+4*X1*X2*x^10+2*X1*x^11+2*X2^2*x^10+2*X2*x^11-2*X1*X2^2*x^8-7*X1*X2* x^9-X1*x^10-3*X2^2*x^9-3*X2*x^10-x^11+3*X1*X2*x^8+3*X1*x^9+3*X2^2*x^8+6*X2*x^9+ x^10+X1*X2*x^7-X1*x^8-X2^2*x^7-5*X2*x^8-3*x^9-X1*x^7+X2*x^7+2*x^8-X1*X2*x^5+X1* X2*x^4+X2*x^5-2*X2*x^4+X2*x^3+x^4-3*x^2+3*x-1)/(-1+x)/(X1*X2*x^5-X1*X2*x^4-X1*x ^5-X2*x^5+2*X2*x^4+x^5-X2*x^3-x^4+x^3-2*x+1)/(X1*X2*x^5-X1*x^5-X2*x^5+X2*x^4+x^ 5-x^4+x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 4, 1], are n 9 n - 3/8 + ----, and , - 5/32 + ---, respectively, while 16 256 Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 1, 2], are n 21 13 n - 3/8 + ----, and , - -- + ----, respectively, while the asymptotic correlat\ 16 64 256 ion between these two random variables is 1/2 13 ----- 13 and in floating point 0.2773500981 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 13 ate normal pair with correlation, ----- 13 1/2 13 15 i.e. , [[-----, 0], [0, --]] 13 13 ------------------------------------------------- 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, 4, 1], nor the composition, [3, 2, 1] Then infinity ----- 2 3 2 \ n (x + 1) (x + x - 2 x + 1) ) a(n) x = ---------------------------- / 3 2 ----- (-1 + x) (x - x + 2 x - 1) n = 0 and in Maple format (x^2+1)*(x^3+x^2-2*x+1)/(-1+x)/(x^3-x^2+2*x-1) The asymptotic expression for a(n) is, n 1.0902311912568087359 1.7548776662466927601 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 4, 1] and d occurrences (as containment) of the composition, [3, 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 7 5 6 6 7 4 5 - X1 x - X2 x - X1 X2 x - X1 x - X2 x + x + X1 X2 x + X1 x 5 6 4 5 3 4 3 2 / + 2 X2 x + x - 2 X2 x - 2 x + X2 x + x - x - x + 2 x - 1) / ( / 5 4 5 5 4 5 3 4 (-1 + x) (X1 X2 x - X1 X2 x - X1 x - X2 x + 2 X2 x + x - X2 x - x 3 + x - 2 x + 1)) and in Maple format (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+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)/(-1+x)/(X1*X2*x^5-X1*X2* x^4-X1*x^5-X2*x^5+2*X2*x^4+x^5-X2*x^3-x^4+x^3-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 4, 1], are n 9 n - 3/8 + ----, and , - 5/32 + ---, respectively, while 16 256 Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 2, 1], are n 19 13 n - 3/8 + ----, and , - -- + ----, respectively, while the asymptotic correlat\ 16 64 256 ion between these two random variables is 1/2 13 ----- 13 and in floating point 0.2773500981 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 13 ate normal pair with correlation, ----- 13 1/2 13 15 i.e. , [[-----, 0], [0, --]] 13 13 ------------------------------------------------- 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, 4, 1], nor the composition, [4, 1, 1] Then infinity ----- 8 7 6 4 2 \ n x + x + x - x - x + 2 x - 1 ) a(n) x = -------------------------------- / 3 2 2 ----- (x + x + x - 1) (-1 + x) n = 0 and in Maple format (x^8+x^7+x^6-x^4-x^2+2*x-1)/(x^3+x^2+x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.73683984463878510196 1.8392867552141611326 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 4, 1] and d occurrences (as containment) of the composition, [4, 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 6 6 7 - X1 x - X2 x + X1 X2 x - X1 x - X2 x + x - X1 x - X2 x + x 4 6 4 2 / + X1 X2 x + x - x - x + 2 x - 1) / ((-1 + x) / 4 4 (X1 X2 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-X1*x^6-X2*x^6+x ^7+X1*X2*x^4+x^6-x^4-x^2+2*x-1)/(-1+x)/(X1*X2*x^4-x^4+2*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 4, 1], are n 9 n - 3/8 + ----, and , - 5/32 + ---, respectively, while 16 256 Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 1, 1], are n 9 n - 3/8 + ----, and , - 5/32 + ---, respectively, while the asymptotic correla\ 16 256 tion between these two random variables is 1 and in floating point 1. ------------------------------------------------- Theorem Number, 31, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 1, 3], nor the composition, [2, 2, 2] Then infinity ----- 6 5 4 3 2 \ n x - x + 2 x - x + 2 x - 2 x + 1 ) a(n) x = - ----------------------------------------------- / 7 6 5 4 3 2 ----- x - 2 x + 3 x - 2 x + 2 x - 3 x + 3 x - 1 n = 0 and in Maple format -(x^6-x^5+2*x^4-x^3+2*x^2-2*x+1)/(x^7-2*x^6+3*x^5-2*x^4+2*x^3-3*x^2+3*x-1) The asymptotic expression for a(n) is, n 0.79316020455689782171 1.8271232214769008001 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 1, 3] and d occurrences (as containment) of the composition, [2, 2, 2], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 7 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 7 6 7 5 6 6 4 - 2 X1 X2 x + X1 X2 x + X2 x - X1 X2 x - X1 x - X2 x + X1 X2 x 5 5 6 3 4 4 5 3 4 + X1 x + X2 x + x - X1 X2 x - X1 x - 2 X2 x - x + 2 X2 x + 2 x 2 3 2 / 2 8 2 7 8 - X2 x - x + 2 x - 2 x + 1) / (X1 X2 x - X1 X2 x - 2 X1 X2 x / 7 8 6 7 7 5 6 + 3 X1 X2 x + X2 x - 2 X1 X2 x - X1 x - 2 X2 x + X1 X2 x + 2 X1 x 6 7 4 5 5 6 3 4 + 2 X2 x + x - 2 X1 X2 x - 2 X1 x - 2 X2 x - 2 x + X1 X2 x + X1 x 4 5 3 4 2 3 2 + 3 X2 x + 3 x - 3 X2 x - 2 x + X2 x + 2 x - 3 x + 3 x - 1) and in Maple format -(X1^2*X2*x^7-2*X1*X2*x^7+X1*X2*x^6+X2*x^7-X1*X2*x^5-X1*x^6-X2*x^6+X1*X2*x^4+X1 *x^5+X2*x^5+x^6-X1*X2*x^3-X1*x^4-2*X2*x^4-x^5+2*X2*x^3+2*x^4-X2*x^2-x^3+2*x^2-2 *x+1)/(X1^2*X2*x^8-X1^2*X2*x^7-2*X1*X2*x^8+3*X1*X2*x^7+X2*x^8-2*X1*X2*x^6-X1*x^ 7-2*X2*x^7+X1*X2*x^5+2*X1*x^6+2*X2*x^6+x^7-2*X1*X2*x^4-2*X1*x^5-2*X2*x^5-2*x^6+ X1*X2*x^3+X1*x^4+3*X2*x^4+3*x^5-3*X2*x^3-2*x^4+X2*x^2+2*x^3-3*x^2+3*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 1, 3], are n 21 13 n - 3/8 + ----, and , - -- + ----, respectively, while 16 64 256 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 2, 2], are n 23 25 n - 3/8 + ----, and , - -- + ----, respectively, while the asymptotic correlat\ 16 32 256 ion between these two random variables is 1/2 9 13 ------- 65 and in floating point 0.4992301767 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 9 13 ate normal pair with correlation, ------- 65 1/2 9 13 487 i.e. , [[-------, 0], [0, ---]] 65 325 ------------------------------------------------- Theorem Number, 32, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 1, 3], nor the composition, [2, 3, 1] Then infinity ----- 6 5 4 3 \ n x + x + x + x - x + 1 ) a(n) x = - ------------------------- / 5 3 ----- x - x + 2 x - 1 n = 0 and in Maple format -(x^6+x^5+x^4+x^3-x+1)/(x^5-x^3+2*x-1) The asymptotic expression for a(n) is, n 0.92094939015237528373 1.7845989333686468028 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 1, 3] and d occurrences (as containment) of the composition, [2, 3, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 7 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 7 7 7 5 7 4 5 - 3 X1 X2 x + X1 x + 2 X2 x - X1 X2 x - x + X1 X2 x + X1 x 3 4 3 2 / 2 8 2 7 - X1 X2 x - X1 x + x + x - 2 x + 1) / (X1 X2 x - X1 X2 x / 8 7 8 6 7 5 6 - 2 X1 X2 x + 2 X1 X2 x + X2 x - X1 X2 x - X2 x + X1 X2 x + X1 x 6 4 5 6 3 4 5 4 3 + X2 x - 2 X1 X2 x - 2 X1 x - x + X1 X2 x + X1 x + x + x - x 2 - 2 x + 3 x - 1) and in Maple format -(X1^2*X2*x^7-3*X1*X2*x^7+X1*x^7+2*X2*x^7-X1*X2*x^5-x^7+X1*X2*x^4+X1*x^5-X1*X2* x^3-X1*x^4+x^3+x^2-2*x+1)/(X1^2*X2*x^8-X1^2*X2*x^7-2*X1*X2*x^8+2*X1*X2*x^7+X2*x ^8-X1*X2*x^6-X2*x^7+X1*X2*x^5+X1*x^6+X2*x^6-2*X1*X2*x^4-2*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, [2, 1, 3], are n 21 13 n - 3/8 + ----, and , - -- + ----, respectively, while 16 64 256 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 3, 1], are n 19 13 n - 3/8 + ----, and , - -- + ----, respectively, while the asymptotic correlat\ 16 64 256 ion between these two random variables is 7/13 and in floating point 0.5384615385 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ ate normal pair with correlation, 7/13 267 i.e. , [[7/13, 0], [0, ---]] 169 ------------------------------------------------- Theorem Number, 33, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 1, 3], nor the composition, [3, 1, 2] Then infinity ----- \ n ) a(n) x = / ----- n = 0 16 15 14 9 8 7 3 2 x + 2 x + x + x + x + x - x - x + 2 x - 1 - --------------------------------------------------------------- 5 4 2 12 9 7 5 4 2 (x + x + x + x - 1) (x - x + x + x - x - x + 2 x - 1) and in Maple format -(x^16+2*x^15+x^14+x^9+x^8+x^7-x^3-x^2+2*x-1)/(x^5+x^4+x^2+x-1)/(x^12-x^9+x^7+x ^5-x^4-x^2+2*x-1) The asymptotic expression for a(n) is, n 0.82278813730065777750 1.8124036192680426608 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 1, 3] and d occurrences (as containment) of the composition, [3, 1, 2], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 3 3 17 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 3 3 16 3 2 17 2 3 17 3 3 15 + X1 X2 x - 3 X1 X2 x - 3 X1 X2 x + X1 X2 x 3 2 16 3 17 2 3 16 2 2 17 - 3 X1 X2 x + 3 X1 X2 x - 3 X1 X2 x + 9 X1 X2 x 3 17 3 3 14 3 2 15 3 16 3 17 + 3 X1 X2 x + X1 X2 x - 2 X1 X2 x + 3 X1 X2 x - X1 x 2 3 15 2 2 16 2 17 3 16 - 2 X1 X2 x + 9 X1 X2 x - 9 X1 X2 x + 3 X1 X2 x 2 17 3 17 3 2 14 3 15 3 16 - 9 X1 X2 x - X2 x - 2 X1 X2 x + X1 X2 x - X1 x 2 3 14 2 2 15 2 16 2 17 3 15 - 2 X1 X2 x + 3 X1 X2 x - 9 X1 X2 x + 3 X1 x + X1 X2 x 2 16 17 3 16 2 17 3 3 12 - 9 X1 X2 x + 9 X1 X2 x - X2 x + 3 X2 x + X1 X2 x 3 14 2 2 14 2 16 3 14 16 + X1 X2 x + 3 X1 X2 x + 3 X1 x + X1 X2 x + 9 X1 X2 x 17 2 16 17 3 2 12 2 3 12 2 15 - 3 X1 x + 3 X2 x - 3 X2 x - X1 X2 x - X1 X2 x - X1 x 15 16 2 15 16 17 2 2 12 - 3 X1 X2 x - 3 X1 x - X2 x - 3 X2 x + x - 2 X1 X2 x 2 14 14 15 2 14 15 16 - X1 x - 3 X1 X2 x + 2 X1 x - X2 x + 2 X2 x + x 2 12 2 12 14 14 15 2 2 10 + 4 X1 X2 x + 4 X1 X2 x + 2 X1 x + 2 X2 x - x + X1 X2 x 2 12 12 2 12 14 2 10 2 10 - X1 x - 5 X1 X2 x - X2 x - x - 2 X1 X2 x - 2 X1 X2 x 12 12 2 2 8 2 10 10 2 10 + X1 x + X2 x - X1 X2 x + X1 x + 4 X1 X2 x + X2 x 2 2 7 10 10 8 10 7 8 + X1 X2 x - 2 X1 x - 2 X2 x + 3 X1 X2 x + x - 4 X1 X2 x - X1 x 8 7 7 7 4 3 4 2 - X2 x + 2 X1 x + 2 X2 x - x + X1 X2 x - X1 X2 x - x + 3 x - 3 x / + 1) / ( / 6 6 6 4 6 3 4 3 (X1 X2 x - X1 x - X2 x + X1 X2 x + x - X1 X2 x - x + x - 2 x + 1) ( 2 2 12 2 12 2 12 2 12 12 2 12 X1 X2 x - 2 X1 X2 x - 2 X1 X2 x + X1 x + 4 X1 X2 x + X2 x 2 2 9 12 12 2 9 2 9 12 9 + X1 X2 x - 2 X1 x - 2 X2 x - X1 X2 x - X1 X2 x + x + X1 x 9 7 9 7 7 5 7 4 5 + X2 x + X1 X2 x - x - X1 x - X2 x - X1 X2 x + x + X1 X2 x + x 4 2 - x - x + 2 x - 1)) and in Maple format -(X1^3*X2^3*x^17+X1^3*X2^3*x^16-3*X1^3*X2^2*x^17-3*X1^2*X2^3*x^17+X1^3*X2^3*x^ 15-3*X1^3*X2^2*x^16+3*X1^3*X2*x^17-3*X1^2*X2^3*x^16+9*X1^2*X2^2*x^17+3*X1*X2^3* x^17+X1^3*X2^3*x^14-2*X1^3*X2^2*x^15+3*X1^3*X2*x^16-X1^3*x^17-2*X1^2*X2^3*x^15+ 9*X1^2*X2^2*x^16-9*X1^2*X2*x^17+3*X1*X2^3*x^16-9*X1*X2^2*x^17-X2^3*x^17-2*X1^3* X2^2*x^14+X1^3*X2*x^15-X1^3*x^16-2*X1^2*X2^3*x^14+3*X1^2*X2^2*x^15-9*X1^2*X2*x^ 16+3*X1^2*x^17+X1*X2^3*x^15-9*X1*X2^2*x^16+9*X1*X2*x^17-X2^3*x^16+3*X2^2*x^17+ X1^3*X2^3*x^12+X1^3*X2*x^14+3*X1^2*X2^2*x^14+3*X1^2*x^16+X1*X2^3*x^14+9*X1*X2*x ^16-3*X1*x^17+3*X2^2*x^16-3*X2*x^17-X1^3*X2^2*x^12-X1^2*X2^3*x^12-X1^2*x^15-3* X1*X2*x^15-3*X1*x^16-X2^2*x^15-3*X2*x^16+x^17-2*X1^2*X2^2*x^12-X1^2*x^14-3*X1* X2*x^14+2*X1*x^15-X2^2*x^14+2*X2*x^15+x^16+4*X1^2*X2*x^12+4*X1*X2^2*x^12+2*X1*x ^14+2*X2*x^14-x^15+X1^2*X2^2*x^10-X1^2*x^12-5*X1*X2*x^12-X2^2*x^12-x^14-2*X1^2* X2*x^10-2*X1*X2^2*x^10+X1*x^12+X2*x^12-X1^2*X2^2*x^8+X1^2*x^10+4*X1*X2*x^10+X2^ 2*x^10+X1^2*X2^2*x^7-2*X1*x^10-2*X2*x^10+3*X1*X2*x^8+x^10-4*X1*X2*x^7-X1*x^8-X2 *x^8+2*X1*x^7+2*X2*x^7-x^7+X1*X2*x^4-X1*X2*x^3-x^4+3*x^2-3*x+1)/(X1*X2*x^6-X1*x ^6-X2*x^6+X1*X2*x^4+x^6-X1*X2*x^3-x^4+x^3-2*x+1)/(X1^2*X2^2*x^12-2*X1^2*X2*x^12 -2*X1*X2^2*x^12+X1^2*x^12+4*X1*X2*x^12+X2^2*x^12+X1^2*X2^2*x^9-2*X1*x^12-2*X2*x ^12-X1^2*X2*x^9-X1*X2^2*x^9+x^12+X1*x^9+X2*x^9+X1*X2*x^7-x^9-X1*x^7-X2*x^7-X1* X2*x^5+x^7+X1*X2*x^4+x^5-x^4-x^2+2*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 1, 3], are n 21 13 n - 3/8 + ----, and , - -- + ----, respectively, while 16 64 256 Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 1, 2], are n 21 13 n - 3/8 + ----, and , - -- + ----, respectively, while the asymptotic correlat\ 16 64 256 ion between these two random variables is 9/13 and in floating point 0.6923076923 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ ate normal pair with correlation, 9/13 331 i.e. , [[9/13, 0], [0, ---]] 169 ------------------------------------------------- Theorem Number, 34, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 1, 3], nor the composition, [3, 2, 1] Then infinity ----- 13 12 10 9 8 6 3 2 \ n x + x - x + x + x + x - x - x + 2 x - 1 ) a(n) x = -------------------------------------------------- / 7 6 5 2 2 ----- (x + 2 x + x + x + x - 1) (-1 + x) n = 0 and in Maple format (x^13+x^12-x^10+x^9+x^8+x^6-x^3-x^2+2*x-1)/(x^7+2*x^6+x^5+x^2+x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 0.86386596006048776696 1.7927428383414338089 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 1, 3] and d occurrences (as containment) of the composition, [3, 2, 1], (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 3 14 2 2 14 3 14 2 14 2 14 - 2 X1 X2 x - 3 X1 X2 x + X1 x + 6 X1 X2 x + 3 X1 X2 x 2 2 12 2 14 14 2 14 2 12 + X1 X2 x - 3 X1 x - 6 X1 X2 x - X2 x - 2 X1 X2 x 2 12 14 14 2 2 10 2 11 2 12 - 2 X1 X2 x + 3 X1 x + 2 X2 x + X1 X2 x - X1 X2 x + X1 x 12 2 12 14 2 2 9 2 11 2 10 + 4 X1 X2 x + X2 x - x - X1 X2 x + X1 x - X1 X2 x 11 12 12 2 9 2 10 2 9 + 2 X1 X2 x - 2 X1 x - 2 X2 x + X1 X2 x - X1 x + 2 X1 X2 x 10 11 11 12 2 8 9 - 2 X1 X2 x - 2 X1 x - X2 x + x + X1 X2 x - 2 X1 X2 x 10 2 9 10 11 2 7 8 9 + 3 X1 x - X2 x + 2 X2 x + x - X1 X2 x - X1 X2 x + X2 x 10 7 8 7 8 5 6 7 - 2 x + X1 X2 x - X1 x + X1 x + x + 2 X1 X2 x - X2 x - x 4 5 6 3 4 4 2 / - 2 X1 X2 x - 2 X1 x + x + X1 X2 x + X1 x + x - 3 x + 3 x - 1) / / 2 2 10 2 2 9 2 10 2 10 2 9 ((-1 + x) (X1 X2 x - X1 X2 x - X1 X2 x - 2 X1 X2 x + X1 X2 x 2 9 10 2 10 2 8 10 2 7 + X1 X2 x + 2 X1 X2 x + X2 x + X1 X2 x - X2 x - X1 X2 x 8 9 9 8 9 7 7 5 - 2 X1 X2 x - X1 x - X2 x + X2 x + x + 2 X1 x + X2 x + X1 X2 x 7 4 5 3 4 5 4 3 2 - 2 x - 2 X1 X2 x - 2 X1 x + X1 X2 x + X1 x + x + x - x - 2 x + 3 x - 1)) and in Maple format -(X1^3*X2^2*x^14-2*X1^3*X2*x^14-3*X1^2*X2^2*x^14+X1^3*x^14+6*X1^2*X2*x^14+3*X1* X2^2*x^14+X1^2*X2^2*x^12-3*X1^2*x^14-6*X1*X2*x^14-X2^2*x^14-2*X1^2*X2*x^12-2*X1 *X2^2*x^12+3*X1*x^14+2*X2*x^14+X1^2*X2^2*x^10-X1^2*X2*x^11+X1^2*x^12+4*X1*X2*x^ 12+X2^2*x^12-x^14-X1^2*X2^2*x^9+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-X1^2*x^10+2*X1*X2^2*x^9-2*X1*X2*x^10-2*X1*x^11-X2*x^11+x^ 12+X1^2*X2*x^8-2*X1*X2*x^9+3*X1*x^10-X2^2*x^9+2*X2*x^10+x^11-X1^2*X2*x^7-X1*X2* x^8+X2*x^9-2*x^10+X1*X2*x^7-X1*x^8+X1*x^7+x^8+2*X1*X2*x^5-X2*x^6-x^7-2*X1*X2*x^ 4-2*X1*x^5+x^6+X1*X2*x^3+X1*x^4+x^4-3*x^2+3*x-1)/(-1+x)/(X1^2*X2^2*x^10-X1^2*X2 ^2*x^9-X1^2*X2*x^10-2*X1*X2^2*x^10+X1^2*X2*x^9+X1*X2^2*x^9+2*X1*X2*x^10+X2^2*x^ 10+X1^2*X2*x^8-X2*x^10-X1^2*X2*x^7-2*X1*X2*x^8-X1*x^9-X2*x^9+X2*x^8+x^9+2*X1*x^ 7+X2*x^7+X1*X2*x^5-2*x^7-2*X1*X2*x^4-2*X1*x^5+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, [2, 1, 3], are n 21 13 n - 3/8 + ----, and , - -- + ----, respectively, while 16 64 256 Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 2, 1], are n 19 13 n - 3/8 + ----, and , - -- + ----, respectively, while the asymptotic correlat\ 16 64 256 ion between these two random variables is 7/13 and in floating point 0.5384615385 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ ate normal pair with correlation, 7/13 267 i.e. , [[7/13, 0], [0, ---]] 169 ------------------------------------------------- Theorem Number, 35, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 1, 3], nor the composition, [4, 1, 1] Then infinity ----- \ n ) a(n) x = / ----- n = 0 11 9 8 7 6 5 4 3 2 x - 2 x + x - 3 x + 3 x - 3 x + 3 x - 3 x + 4 x - 3 x + 1 - ------------------------------------------------------------------- 3 2 4 2 (x - x + 2 x - 1) (x + 1) (-1 + x) and in Maple format -(x^11-2*x^9+x^8-3*x^7+3*x^6-3*x^5+3*x^4-3*x^3+4*x^2-3*x+1)/(x^3-x^2+2*x-1)/(x^ 4+1)/(-1+x)^2 The asymptotic expression for a(n) is, n 1.0054824915204388557 1.7548776662466927601 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 1, 3] and d occurrences (as containment) of the composition, [4, 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 2 11 2 13 13 - 2 X1 X2 x - 2 X1 X2 x - X1 X2 x + X1 x + 4 X1 X2 x 2 13 2 2 10 2 11 2 11 13 + X2 x + X1 X2 x + 3 X1 X2 x + 3 X1 X2 x - 2 X1 x 13 2 2 9 2 10 2 11 2 10 - 2 X2 x - X1 X2 x - 3 X1 X2 x - 2 X1 x - X1 X2 x 11 2 11 13 2 9 2 10 2 9 - 8 X1 X2 x - 2 X2 x + x + 4 X1 X2 x + 2 X1 x + X1 X2 x 10 11 11 2 8 2 9 9 + 4 X1 X2 x + 5 X1 x + 5 X2 x - 2 X1 X2 x - 3 X1 x - 5 X1 X2 x 10 10 11 2 8 8 9 9 - 3 X1 x - X2 x - 3 x + 3 X1 x + 3 X1 X2 x + 4 X1 x + X2 x 10 2 7 7 8 8 9 7 7 8 + x - X1 x + X1 X2 x - 5 X1 x - X2 x - x + X1 x - X2 x + 2 x 5 4 5 4 3 4 2 - X1 X2 x + X1 X2 x + X1 x - 2 X1 x + X1 x + x - 3 x + 3 x - 1) / 5 5 5 4 5 4 / ((-1 + x) (X1 X2 x - X1 x - X2 x + X1 x + x - x + x - 1) ( / 5 4 5 5 4 5 3 4 3 X1 X2 x - X1 X2 x - X1 x - X2 x + 2 X1 x + x - X1 x - x + x - 2 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*X2^2*x^11+X1^2*x^13+4*X1*X2 *x^13+X2^2*x^13+X1^2*X2^2*x^10+3*X1^2*X2*x^11+3*X1*X2^2*x^11-2*X1*x^13-2*X2*x^ 13-X1^2*X2^2*x^9-3*X1^2*X2*x^10-2*X1^2*x^11-X1*X2^2*x^10-8*X1*X2*x^11-2*X2^2*x^ 11+x^13+4*X1^2*X2*x^9+2*X1^2*x^10+X1*X2^2*x^9+4*X1*X2*x^10+5*X1*x^11+5*X2*x^11-\ 2*X1^2*X2*x^8-3*X1^2*x^9-5*X1*X2*x^9-3*X1*x^10-X2*x^10-3*x^11+3*X1^2*x^8+3*X1* X2*x^8+4*X1*x^9+X2*x^9+x^10-X1^2*x^7+X1*X2*x^7-5*X1*x^8-X2*x^8-x^9+X1*x^7-X2*x^ 7+2*x^8-X1*X2*x^5+X1*X2*x^4+X1*x^5-2*X1*x^4+X1*x^3+x^4-3*x^2+3*x-1)/(-1+x)/(X1* X2*x^5-X1*x^5-X2*x^5+X1*x^4+x^5-x^4+x-1)/(X1*X2*x^5-X1*X2*x^4-X1*x^5-X2*x^5+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, [2, 1, 3], are n 21 13 n - 3/8 + ----, and , - -- + ----, respectively, while 16 64 256 Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 1, 1], are n 9 n - 3/8 + ----, and , - 5/32 + ---, respectively, while the asymptotic correla\ 16 256 tion between these two random variables is 1/2 13 ----- 13 and in floating point 0.2773500981 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 13 ate normal pair with correlation, ----- 13 1/2 13 15 i.e. , [[-----, 0], [0, --]] 13 13 ------------------------------------------------- Theorem Number, 36, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 2, 2], nor the composition, [2, 3, 1] Then infinity ----- 4 3 2 \ n x - x + 2 x - 2 x + 1 ) a(n) x = --------------------------------------- / 2 3 2 ----- (-1 + x) (x - x + 1) (x + x + x - 1) n = 0 and in Maple format (x^4-x^3+2*x^2-2*x+1)/(-1+x)/(x^2-x+1)/(x^3+x^2+x-1) The asymptotic expression for a(n) is, n 0.77591233811587484106 1.8392867552141611326 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 2, 2] and d occurrences (as containment) of the composition, [2, 3, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 5 4 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x - X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 5 3 4 3 4 2 3 2 - X1 x + X1 X2 x + 2 X1 x - 2 X1 x - x + X1 x + x - 2 x + 2 x - 1) / 6 5 6 6 4 5 6 / (X1 X2 x - X1 X2 x - X1 x - X2 x + 2 X1 X2 x + 2 X1 x + x / 3 4 5 3 4 2 3 2 - X1 X2 x - 3 X1 x - x + 3 X1 x + x - X1 x - 2 x + 3 x - 3 x + 1) and in Maple format -(X1*X2*x^5-X1*X2*x^4-X1*x^5+X1*X2*x^3+2*X1*x^4-2*X1*x^3-x^4+X1*x^2+x^3-2*x^2+2 *x-1)/(X1*X2*x^6-X1*X2*x^5-X1*x^6-X2*x^6+2*X1*X2*x^4+2*X1*x^5+x^6-X1*X2*x^3-3* X1*x^4-x^5+3*X1*x^3+x^4-X1*x^2-2*x^3+3*x^2-3*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 2, 2], are n 23 25 n - 3/8 + ----, and , - -- + ----, respectively, while 16 32 256 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 3, 1], are n 19 13 n - 3/8 + ----, and , - -- + ----, respectively, while the asymptotic correlat\ 16 64 256 ion between these two random variables is 1/2 11 13 -------- 65 and in floating point 0.6101702157 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 11 13 ate normal pair with correlation, -------- 65 1/2 11 13 567 i.e. , [[--------, 0], [0, ---]] 65 325 ------------------------------------------------- Theorem Number, 37, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 2, 2], nor the composition, [3, 1, 2] Then infinity ----- 6 5 4 3 2 \ n x - x + 2 x - x + 2 x - 2 x + 1 ) a(n) x = - ----------------------------------------------- / 7 6 5 4 3 2 ----- x - 2 x + 3 x - 2 x + 2 x - 3 x + 3 x - 1 n = 0 and in Maple format -(x^6-x^5+2*x^4-x^3+2*x^2-2*x+1)/(x^7-2*x^6+3*x^5-2*x^4+2*x^3-3*x^2+3*x-1) The asymptotic expression for a(n) is, n 0.79316020455689782171 1.8271232214769008001 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 2, 2] and d occurrences (as containment) of the composition, [3, 1, 2], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 7 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 7 6 7 5 6 6 4 - 2 X1 X2 x + X1 X2 x + X1 x - X1 X2 x - X1 x - X2 x + X1 X2 x 5 5 6 3 4 4 5 3 4 + X1 x + X2 x + x - X1 X2 x - 2 X1 x - X2 x - x + 2 X1 x + 2 x 2 3 2 / 2 8 2 7 8 - X1 x - x + 2 x - 2 x + 1) / (X1 X2 x - X1 X2 x - 2 X1 X2 x / 7 8 6 7 7 5 6 + 3 X1 X2 x + X1 x - 2 X1 X2 x - 2 X1 x - X2 x + X1 X2 x + 2 X1 x 6 7 4 5 5 6 3 + 2 X2 x + x - 2 X1 X2 x - 2 X1 x - 2 X2 x - 2 x + X1 X2 x 4 4 5 3 4 2 3 2 + 3 X1 x + X2 x + 3 x - 3 X1 x - 2 x + X1 x + 2 x - 3 x + 3 x - 1) and in Maple format -(X1*X2^2*x^7-2*X1*X2*x^7+X1*X2*x^6+X1*x^7-X1*X2*x^5-X1*x^6-X2*x^6+X1*X2*x^4+X1 *x^5+X2*x^5+x^6-X1*X2*x^3-2*X1*x^4-X2*x^4-x^5+2*X1*x^3+2*x^4-X1*x^2-x^3+2*x^2-2 *x+1)/(X1*X2^2*x^8-X1*X2^2*x^7-2*X1*X2*x^8+3*X1*X2*x^7+X1*x^8-2*X1*X2*x^6-2*X1* x^7-X2*x^7+X1*X2*x^5+2*X1*x^6+2*X2*x^6+x^7-2*X1*X2*x^4-2*X1*x^5-2*X2*x^5-2*x^6+ X1*X2*x^3+3*X1*x^4+X2*x^4+3*x^5-3*X1*x^3-2*x^4+X1*x^2+2*x^3-3*x^2+3*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 2, 2], are n 23 25 n - 3/8 + ----, and , - -- + ----, respectively, while 16 32 256 Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 1, 2], are n 21 13 n - 3/8 + ----, and , - -- + ----, respectively, while the asymptotic correlat\ 16 64 256 ion between these two random variables is 1/2 9 13 ------- 65 and in floating point 0.4992301767 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 9 13 ate normal pair with correlation, ------- 65 1/2 9 13 487 i.e. , [[-------, 0], [0, ---]] 65 325 ------------------------------------------------- Theorem Number, 38, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 2, 2], nor the composition, [3, 2, 1] Then infinity ----- 4 3 2 \ n x - x + 2 x - 2 x + 1 ) a(n) x = --------------------------------------- / 2 3 2 ----- (-1 + x) (x - x + 1) (x + x + x - 1) n = 0 and in Maple format (x^4-x^3+2*x^2-2*x+1)/(-1+x)/(x^2-x+1)/(x^3+x^2+x-1) The asymptotic expression for a(n) is, n 0.77591233811587484106 1.8392867552141611326 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 2, 2] and d occurrences (as containment) of the composition, [3, 2, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 5 4 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x - X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 5 3 4 3 4 2 3 2 - X1 x + X1 X2 x + 2 X1 x - 2 X1 x - x + X1 x + x - 2 x + 2 x - 1) / 6 5 6 6 4 5 6 / (X1 X2 x - X1 X2 x - X1 x - X2 x + 2 X1 X2 x + 2 X1 x + x / 3 4 5 3 4 2 3 2 - X1 X2 x - 3 X1 x - x + 3 X1 x + x - X1 x - 2 x + 3 x - 3 x + 1) and in Maple format -(X1*X2*x^5-X1*X2*x^4-X1*x^5+X1*X2*x^3+2*X1*x^4-2*X1*x^3-x^4+X1*x^2+x^3-2*x^2+2 *x-1)/(X1*X2*x^6-X1*X2*x^5-X1*x^6-X2*x^6+2*X1*X2*x^4+2*X1*x^5+x^6-X1*X2*x^3-3* X1*x^4-x^5+3*X1*x^3+x^4-X1*x^2-2*x^3+3*x^2-3*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 2, 2], are n 23 25 n - 3/8 + ----, and , - -- + ----, respectively, while 16 32 256 Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 2, 1], are n 19 13 n - 3/8 + ----, and , - -- + ----, respectively, while the asymptotic correlat\ 16 64 256 ion between these two random variables is 1/2 11 13 -------- 65 and in floating point 0.6101702157 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 11 13 ate normal pair with correlation, -------- 65 1/2 11 13 567 i.e. , [[--------, 0], [0, ---]] 65 325 ------------------------------------------------- Theorem Number, 39, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 2, 2], nor the composition, [4, 1, 1] Then infinity ----- 9 7 6 5 4 3 2 \ n x - x + x - x - x + x - 2 x + 2 x - 1 ) a(n) x = -------------------------------------------- / 7 6 5 4 3 2 ----- (x + 2 x + x + x + x + x - 1) (-1 + x) n = 0 and in Maple format (x^9-x^7+x^6-x^5-x^4+x^3-2*x^2+2*x-1)/(x^7+2*x^6+x^5+x^4+x^3+x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 1.0612450246603483778 1.7610793284505662547 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 2, 2] and d occurrences (as containment) of the composition, [4, 1, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 9 9 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x - 2 X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 9 9 9 7 9 6 7 7 - X2 x + X1 x + 2 X2 x + X1 X2 x - x - X1 X2 x - X1 x - X2 x 5 6 6 7 4 5 5 6 5 + X1 X2 x + X1 x + X2 x + x - X1 X2 x - X1 x - X2 x - x + x 3 4 2 3 2 / 2 9 9 + X1 x + x - X1 x - x + 2 x - 2 x + 1) / (X1 X2 x - 2 X1 X2 x / 2 9 9 9 7 9 6 7 - X2 x + X1 x + 2 X2 x + 2 X1 X2 x - x - X1 X2 x - 2 X1 x 7 5 6 6 7 4 5 5 - 2 X2 x + 2 X1 X2 x + X1 x + X2 x + 2 x - X1 X2 x - X1 x - X2 x 6 3 4 2 3 2 - x + 2 X1 x + x - X1 x - 2 x + 3 x - 3 x + 1) and in Maple format (X1*X2^2*x^9-2*X1*X2*x^9-X2^2*x^9+X1*x^9+2*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-X1*X2*x^4-X1*x^5-X2*x^5-x^6+x^5+X1*x^3+x^4 -X1*x^2-x^3+2*x^2-2*x+1)/(X1*X2^2*x^9-2*X1*X2*x^9-X2^2*x^9+X1*x^9+2*X2*x^9+2*X1 *X2*x^7-x^9-X1*X2*x^6-2*X1*x^7-2*X2*x^7+2*X1*X2*x^5+X1*x^6+X2*x^6+2*x^7-X1*X2*x ^4-X1*x^5-X2*x^5-x^6+2*X1*x^3+x^4-X1*x^2-2*x^3+3*x^2-3*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 2, 2], are n 23 25 n - 3/8 + ----, and , - -- + ----, respectively, while 16 32 256 Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 1, 1], are n 9 n - 3/8 + ----, and , - 5/32 + ---, respectively, while the asymptotic correla\ 16 256 tion between these two random variables is 1/15 and in floating point 0.06666666667 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ ate normal pair with correlation, 1/15 227 i.e. , [[1/15, 0], [0, ---]] 225 ------------------------------------------------- Theorem Number, 40, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 3, 1], nor the composition, [3, 1, 2] Then infinity ----- 5 4 3 \ n x + x + x - x + 1 ) a(n) x = -------------------------------------- / 7 6 5 2 ----- (-1 + x) (x + 2 x + x + x + x - 1) n = 0 and in Maple format (x^5+x^4+x^3-x+1)/(-1+x)/(x^7+2*x^6+x^5+x^2+x-1) The asymptotic expression for a(n) is, n 0.89858714643315996836 1.7927428383414338089 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 3, 1] and d occurrences (as containment) of the composition, [3, 1, 2], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 9 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 9 2 9 2 9 9 2 7 9 - 2 X1 X2 x - X1 X2 x + X1 x + 2 X1 X2 x + X1 X2 x - X1 x 7 6 7 5 6 6 4 - 2 X1 X2 x - X1 X2 x + X1 x - X1 X2 x + X1 x + X2 x + X1 X2 x 5 6 3 4 3 2 / 2 2 10 + X2 x - x - X1 X2 x - X2 x + x + x - 2 x + 1) / (X1 X2 x / 2 2 9 2 10 2 10 2 9 2 10 2 9 - X1 X2 x - 2 X1 X2 x - X1 X2 x + X1 X2 x + X1 x + X1 X2 x 10 2 8 10 2 7 8 9 + 2 X1 X2 x + X1 X2 x - X1 x - X1 X2 x - 2 X1 X2 x - X1 x 9 8 9 7 7 5 7 4 - X2 x + X1 x + x + X1 x + 2 X2 x + X1 X2 x - 2 x - 2 X1 X2 x 5 3 4 5 4 3 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^9-2*X1^2*X2*x^9-X1*X2^2*x^9+X1^2*x^9+2*X1*X2*x^9+X1*X2^2*x^7-X1*x ^9-2*X1*X2*x^7-X1*X2*x^6+X1*x^7-X1*X2*x^5+X1*x^6+X2*x^6+X1*X2*x^4+X2*x^5-x^6-X1 *X2*x^3-X2*x^4+x^3+x^2-2*x+1)/(X1^2*X2^2*x^10-X1^2*X2^2*x^9-2*X1^2*X2*x^10-X1* X2^2*x^10+X1^2*X2*x^9+X1^2*x^10+X1*X2^2*x^9+2*X1*X2*x^10+X1*X2^2*x^8-X1*x^10-X1 *X2^2*x^7-2*X1*X2*x^8-X1*x^9-X2*x^9+X1*x^8+x^9+X1*x^7+2*X2*x^7+X1*X2*x^5-2*x^7-\ 2*X1*X2*x^4-2*X2*x^5+X1*X2*x^3+X2*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, [2, 3, 1], are n 19 13 n - 3/8 + ----, and , - -- + ----, respectively, while 16 64 256 Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 1, 2], are n 21 13 n - 3/8 + ----, and , - -- + ----, respectively, while the asymptotic correlat\ 16 64 256 ion between these two random variables is 7/13 and in floating point 0.5384615385 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ ate normal pair with correlation, 7/13 267 i.e. , [[7/13, 0], [0, ---]] 169 ------------------------------------------------- Theorem Number, 41, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 3, 1], nor the composition, [3, 2, 1] Then infinity ----- 6 5 3 \ n x + x + x - x + 1 ) a(n) x = ------------------------------- / 5 4 2 ----- (-1 + x) (x + x + x + x - 1) n = 0 and in Maple format (x^6+x^5+x^3-x+1)/(-1+x)/(x^5+x^4+x^2+x-1) The asymptotic expression for a(n) is, n 0.86835109073791924063 1.8124036192680426608 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 3, 1] and d occurrences (as containment) of the composition, [3, 2, 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 7 5 7 4 3 5 4 3 2 - X2 x + X1 X2 x + x - X1 X2 x + X1 X2 x - x + x - x - x + 2 x / - 1) / ((-1 + x) / 6 6 6 4 6 3 4 3 (X1 X2 x - X1 x - X2 x + X1 X2 x + x - X1 X2 x - x + x - 2 x + 1)) and in Maple format (X1*X2*x^7-X1*x^7-X2*x^7+X1*X2*x^5+x^7-X1*X2*x^4+X1*X2*x^3-x^5+x^4-x^3-x^2+2*x-\ 1)/(-1+x)/(X1*X2*x^6-X1*x^6-X2*x^6+X1*X2*x^4+x^6-X1*X2*x^3-x^4+x^3-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 3, 1], are n 19 13 n - 3/8 + ----, and , - -- + ----, respectively, while 16 64 256 Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 2, 1], are n 19 13 n - 3/8 + ----, and , - -- + ----, respectively, while the asymptotic correlat\ 16 64 256 ion between these two random variables is 9/13 and in floating point 0.6923076923 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ ate normal pair with correlation, 9/13 331 i.e. , [[9/13, 0], [0, ---]] 169 ------------------------------------------------- Theorem Number, 42, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 3, 1], nor the composition, [4, 1, 1] Then infinity ----- 2 3 2 \ n (x + 1) (x + x - 2 x + 1) ) a(n) x = ---------------------------- / 3 2 ----- (-1 + x) (x - x + 2 x - 1) n = 0 and in Maple format (x^2+1)*(x^3+x^2-2*x+1)/(-1+x)/(x^3-x^2+2*x-1) The asymptotic expression for a(n) is, n 1.0902311912568087359 1.7548776662466927601 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 3, 1] and d occurrences (as containment) of the composition, [4, 1, 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 7 5 6 6 7 4 5 - X1 x - X2 x - X1 X2 x - X1 x - X2 x + x + X1 X2 x + 2 X1 x 5 6 4 5 3 4 3 2 / + X2 x + x - 2 X1 x - 2 x + X1 x + x - x - x + 2 x - 1) / ( / 5 4 5 5 4 5 3 4 (-1 + x) (X1 X2 x - X1 X2 x - X1 x - X2 x + 2 X1 x + x - X1 x - x 3 + x - 2 x + 1)) and in Maple format (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+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)/(-1+x)/(X1*X2*x^5-X1*X2* x^4-X1*x^5-X2*x^5+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, [2, 3, 1], are n 19 13 n - 3/8 + ----, and , - -- + ----, respectively, while 16 64 256 Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 1, 1], are n 9 n - 3/8 + ----, and , - 5/32 + ---, respectively, while the asymptotic correla\ 16 256 tion between these two random variables is 1/2 13 ----- 13 and in floating point 0.2773500981 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 13 ate normal pair with correlation, ----- 13 1/2 13 15 i.e. , [[-----, 0], [0, --]] 13 13 ------------------------------------------------- Theorem Number, 43, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [3, 1, 2], nor the composition, [3, 2, 1] Then infinity ----- 3 2 \ n x + x - 2 x + 1 ) a(n) x = ---------------------------- / 5 3 ----- (-1 + x) (x - x + 2 x - 1) n = 0 and in Maple format (x^3+x^2-2*x+1)/(-1+x)/(x^5-x^3+2*x-1) The asymptotic expression for a(n) is, n 0.96634807298013408626 1.7845989333686468028 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [3, 1, 2] and d occurrences (as containment) of the composition, [3, 2, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 7 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 7 7 5 4 5 3 4 3 - 2 X1 X2 x + X2 x - X1 X2 x + X1 X2 x + X1 x - X1 X2 x - X1 x + x 2 / 2 8 2 7 8 7 + x - 2 x + 1) / (X1 X2 x - X1 X2 x - 2 X1 X2 x + 2 X1 X2 x / 8 6 7 5 6 6 4 + X2 x - X1 X2 x - X2 x + X1 X2 x + X1 x + X2 x - 2 X1 X2 x 5 6 3 4 5 4 3 2 - 2 X1 x - x + X1 X2 x + X1 x + x + x - x - 2 x + 3 x - 1) and in Maple format -(X1^2*X2*x^7-2*X1*X2*x^7+X2*x^7-X1*X2*x^5+X1*X2*x^4+X1*x^5-X1*X2*x^3-X1*x^4+x^ 3+x^2-2*x+1)/(X1^2*X2*x^8-X1^2*X2*x^7-2*X1*X2*x^8+2*X1*X2*x^7+X2*x^8-X1*X2*x^6- X2*x^7+X1*X2*x^5+X1*x^6+X2*x^6-2*X1*X2*x^4-2*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, 2], are n 21 13 n - 3/8 + ----, and , - -- + ----, respectively, while 16 64 256 Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 2, 1], are n 19 13 n - 3/8 + ----, and , - -- + ----, respectively, while the asymptotic correlat\ 16 64 256 ion between these two random variables is 7/13 and in floating point 0.5384615385 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ ate normal pair with correlation, 7/13 267 i.e. , [[7/13, 0], [0, ---]] 169 ------------------------------------------------- Theorem Number, 44, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [3, 1, 2], nor the composition, [4, 1, 1] Then infinity ----- 8 7 3 2 \ n x - x - x - x + 2 x - 1 ) a(n) x = ---------------------------------------------- / 3 2 4 2 ----- (x + 1) (x - x + 2 x - 1) (x + 1) (-1 + x) n = 0 and in Maple format (x^8-x^7-x^3-x^2+2*x-1)/(x+1)/(x^3-x^2+2*x-1)/(x^4+1)/(-1+x)^2 The asymptotic expression for a(n) is, n 1.1274912931215698825 1.7548776662466927601 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [3, 1, 2] and d occurrences (as containment) of the composition, [4, 1, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 9 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 9 2 9 2 8 2 9 9 2 8 - 2 X1 X2 x - X1 X2 x + 2 X1 X2 x + X1 x + 2 X1 X2 x - 2 X1 x 8 9 2 7 8 8 7 8 7 - 3 X1 X2 x - X1 x + X1 x + 3 X1 x + X2 x - 2 X1 x - x + x 4 4 3 3 2 / - X1 X2 x + X1 x - X1 x + x + x - 2 x + 1) / ( / 5 5 5 4 5 4 5 4 (X1 X2 x - X1 x - X2 x + X1 x + x - x + x - 1) (X1 X2 x - X1 X2 x 5 5 4 5 3 4 3 - X1 x - X2 x + 2 X1 x + x - X1 x - x + x - 2 x + 1)) and in Maple format -(X1^2*X2^2*x^9-2*X1^2*X2*x^9-X1*X2^2*x^9+2*X1^2*X2*x^8+X1^2*x^9+2*X1*X2*x^9-2* X1^2*x^8-3*X1*X2*x^8-X1*x^9+X1^2*x^7+3*X1*x^8+X2*x^8-2*X1*x^7-x^8+x^7-X1*X2*x^4 +X1*x^4-X1*x^3+x^3+x^2-2*x+1)/(X1*X2*x^5-X1*x^5-X2*x^5+X1*x^4+x^5-x^4+x-1)/(X1* X2*x^5-X1*X2*x^4-X1*x^5-X2*x^5+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, 1, 2], are n 21 13 n - 3/8 + ----, and , - -- + ----, respectively, while 16 64 256 Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 1, 1], are n 9 n - 3/8 + ----, and , - 5/32 + ---, respectively, while the asymptotic correla\ 16 256 tion between these two random variables is 1/2 13 ----- 13 and in floating point 0.2773500981 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 13 ate normal pair with correlation, ----- 13 1/2 13 15 i.e. , [[-----, 0], [0, --]] 13 13 ------------------------------------------------- Theorem Number, 45, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [3, 2, 1], nor the composition, [4, 1, 1] Then infinity ----- 6 5 4 3 2 \ n x - 2 x + x - x - x + 2 x - 1 ) a(n) x = ------------------------------------- / 3 2 2 ----- (x + 1) (x - x + 2 x - 1) (-1 + x) n = 0 and in Maple format (x^6-2*x^5+x^4-x^3-x^2+2*x-1)/(x+1)/(x^3-x^2+2*x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 1.1544840707379761413 1.7548776662466927601 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [3, 2, 1] and d occurrences (as containment) of the composition, [4, 1, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 6 5 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x - X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 6 6 4 5 5 6 4 5 3 - X1 x - X2 x + X1 X2 x + 2 X1 x + X2 x + x - 2 X1 x - 2 x + X1 x 4 3 2 / 5 4 5 + x - x - x + 2 x - 1) / ((-1 + x) (X1 X2 x - X1 X2 x - X1 x / 5 4 5 3 4 3 - X2 x + 2 X1 x + x - X1 x - x + x - 2 x + 1)) and in Maple format (X1*X2*x^6-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)/(-1+x)/(X1*X2*x^5-X1*X2*x^4-X1*x^5-X2*x^5+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, 2, 1], are n 19 13 n - 3/8 + ----, and , - -- + ----, respectively, while 16 64 256 Furthermore the average and variance of the random variable: Number of occur\ rences of, [4, 1, 1], are n 9 n - 3/8 + ----, and , - 5/32 + ---, respectively, while the asymptotic correla\ 16 256 tion between these two random variables is 1/2 13 ----- 13 and in floating point 0.2773500981 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 13 ate normal pair with correlation, ----- 13 1/2 13 15 i.e. , [[-----, 0], [0, --]] 13 13 ------------------------------------------------- ------------------------ This ends this article, that took, 11.049, 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, 6, with , 4, 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, 1, 3], nor the composition, [1, 1, 2, 2] Then infinity ----- 5 4 3 2 \ n x - 3 x + 3 x - 4 x + 3 x - 1 ) a(n) x = - --------------------------------- / 3 3 ----- (x + x - 1) (-1 + x) n = 0 and in Maple format -(x^5-3*x^4+3*x^3-4*x^2+3*x-1)/(x^3+x-1)/(-1+x)^3 The asymptotic expression for a(n) is, n 6.9556233073109364759 1.4655712318767680267 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 1, 1, 3] and d occurrences (as containment) of the composition, [1, 1, 2, 2], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 5 ) | ) | ) A(n, c, d) x X1 X2 || = - (2 X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 4 5 5 3 4 4 5 3 - 2 X1 X2 x - X1 x - 2 X2 x + X1 X2 x + X1 x + 4 X2 x + x - 3 X2 x 4 2 3 2 / 2 4 - 3 x + X2 x + 3 x - 4 x + 3 x - 1) / ((-1 + x) (X1 X2 x / 3 4 4 3 4 2 3 2 - X1 X2 x - X1 x - X2 x + 2 X2 x + x - X2 x - x + x - 2 x + 1)) and in Maple format -(2*X1*X2*x^5-2*X1*X2*x^4-X1*x^5-2*X2*x^5+X1*X2*x^3+X1*x^4+4*X2*x^4+x^5-3*X2*x^ 3-3*x^4+X2*x^2+3*x^3-4*x^2+3*x-1)/(-1+x)^2/(X1*X2*x^4-X1*X2*x^3-X1*x^4-X2*x^4+2 *X2*x^3+x^4-X2*x^2-x^3+x^2-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 1, 1, 3], are 11 3 n - 7/8 + n/8, and , - -- + ---, respectively, while 64 64 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 1, 2, 2], are 47 7 n - 7/8 + n/8, and , - -- + ---, respectively, while the asymptotic correlatio\ 64 64 n between these two random variables is 1/2 1/2 3 7 --------- 21 and in floating point 0.2182178903 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 21 ate normal pair with correlation, ----- 21 1/2 21 23 i.e. , [[-----, 0], [0, --]] 21 21 ------------------------------------------------- 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, 1, 3], nor the composition, [1, 1, 3, 1] Then infinity ----- 7 6 5 4 2 \ n x + x - x + x - 3 x + 3 x - 1 ) a(n) x = - ---------------------------------- / 2 3 ----- (x + x - 1) (-1 + x) n = 0 and in Maple format -(x^7+x^6-x^5+x^4-3*x^2+3*x-1)/(x^2+x-1)/(-1+x)^3 The asymptotic expression for a(n) is, n 1.8944271909999158785 1.6180339887498948482 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 1, 1, 3] and d occurrences (as containment) of the composition, [1, 1, 3, 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 7 5 6 6 7 4 6 3 - X1 x - X2 x + X1 X2 x - X1 x - X2 x + x - X1 X2 x + x + X1 X2 x 5 4 2 / 2 3 3 - x + x - 3 x + 3 x - 1) / ((-1 + x) (X1 X2 x - x + 2 x - 1)) / and in Maple format (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-X1*X2*x^4+x^6+X1 *X2*x^3-x^5+x^4-3*x^2+3*x-1)/(-1+x)^2/(X1*X2*x^3-x^3+2*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 1, 1, 3], are 11 3 n - 7/8 + n/8, and , - -- + ---, respectively, while 64 64 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 1, 3, 1], are 11 3 n - 7/8 + n/8, and , - -- + ---, respectively, while the asymptotic correlatio\ 64 64 n between these two random variables is 1 and in floating point 1. ------------------------------------------------- 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, 1, 3], nor the composition, [1, 2, 1, 2] Then infinity ----- 9 8 7 6 5 3 2 \ n x - 3 x + 3 x - 3 x + 3 x - 2 x + 4 x - 3 x + 1 ) a(n) x = ------------------------------------------------------ / 2 3 3 ----- (x + 1) (x - x + 1) (x + x - 1) (-1 + x) n = 0 and in Maple format (x^9-3*x^8+3*x^7-3*x^6+3*x^5-2*x^3+4*x^2-3*x+1)/(x+1)/(x^2-x+1)/(x^3+x-1)/(-1+x )^3 The asymptotic expression for a(n) is, n 6.0594382353848054551 1.4655712318767680267 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 1, 1, 3] and d occurrences (as containment) of the composition, [1, 2, 1, 2], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 9 2 2 8 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x - X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 9 2 9 2 2 7 2 8 2 9 - 2 X1 X2 x - 2 X1 X2 x + X1 X2 x + 2 X1 X2 x + X1 x 2 8 9 2 9 2 7 2 8 2 7 + 4 X1 X2 x + 4 X1 X2 x + X2 x - X1 X2 x - X1 x - 4 X1 X2 x 8 9 2 8 9 2 6 7 - 8 X1 X2 x - 2 X1 x - 3 X2 x - 2 X2 x + 2 X1 X2 x + 6 X1 X2 x 8 2 7 8 9 6 7 2 6 + 4 X1 x + 4 X2 x + 6 X2 x + x - 4 X1 X2 x - 2 X1 x - 3 X2 x 7 8 6 2 5 6 7 4 5 - 7 X2 x - 3 x + 2 X1 x + X2 x + 6 X2 x + 3 x + X1 X2 x - X1 x 5 6 3 4 5 3 2 3 2 - 3 X2 x - 3 x - X1 X2 x - X2 x + 3 x + 2 X2 x - X2 x - 2 x + 4 x / - 3 x + 1) / ((-1 + x) / 4 4 4 3 4 3 4 3 (X1 X2 x - X1 x - X2 x + X2 x + x - x + x - 1) (X1 X2 x - X1 X2 x 4 4 3 4 2 3 2 - X1 x - X2 x + 2 X2 x + x - X2 x - x + x - 2 x + 1)) and in Maple format (X1^2*X2^2*x^9-X1^2*X2^2*x^8-2*X1^2*X2*x^9-2*X1*X2^2*x^9+X1^2*X2^2*x^7+2*X1^2* X2*x^8+X1^2*x^9+4*X1*X2^2*x^8+4*X1*X2*x^9+X2^2*x^9-X1^2*X2*x^7-X1^2*x^8-4*X1*X2 ^2*x^7-8*X1*X2*x^8-2*X1*x^9-3*X2^2*x^8-2*X2*x^9+2*X1*X2^2*x^6+6*X1*X2*x^7+4*X1* x^8+4*X2^2*x^7+6*X2*x^8+x^9-4*X1*X2*x^6-2*X1*x^7-3*X2^2*x^6-7*X2*x^7-3*x^8+2*X1 *x^6+X2^2*x^5+6*X2*x^6+3*x^7+X1*X2*x^4-X1*x^5-3*X2*x^5-3*x^6-X1*X2*x^3-X2*x^4+3 *x^5+2*X2*x^3-X2*x^2-2*x^3+4*x^2-3*x+1)/(-1+x)/(X1*X2*x^4-X1*x^4-X2*x^4+X2*x^3+ x^4-x^3+x-1)/(X1*X2*x^4-X1*X2*x^3-X1*x^4-X2*x^4+2*X2*x^3+x^4-X2*x^2-x^3+x^2-2*x +1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 1, 1, 3], are 11 3 n - 7/8 + n/8, and , - -- + ---, respectively, while 64 64 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 2, 1, 2], are 55 7 n - 7/8 + n/8, and , - -- + ---, respectively, while the asymptotic correlatio\ 64 64 n between these two random variables is 1/2 1/2 3 7 --------- 21 and in floating point 0.2182178903 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 21 ate normal pair with correlation, ----- 21 1/2 21 23 i.e. , [[-----, 0], [0, --]] 21 21 ------------------------------------------------- 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, 1, 3], nor the composition, [1, 2, 2, 1] Then infinity ----- 8 7 5 4 3 2 \ n x + x + x - 3 x + 3 x - 4 x + 3 x - 1 ) a(n) x = - ------------------------------------------- / 3 3 ----- (x + x - 1) (-1 + x) n = 0 and in Maple format -(x^8+x^7+x^5-3*x^4+3*x^3-4*x^2+3*x-1)/(x^3+x-1)/(-1+x)^3 The asymptotic expression for a(n) is, n 4.7460151755324556560 1.4655712318767680267 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 1, 1, 3] and d occurrences (as containment) of the composition, [1, 2, 2, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 9 2 9 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x - X1 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 9 9 9 7 9 6 7 - 2 X1 X2 x + 2 X1 x + X2 x + X1 X2 x - x - 2 X1 X2 x - X1 x 7 5 6 6 7 4 5 - X2 x + 4 X1 X2 x + X1 x + 2 X2 x + x - 3 X1 X2 x - 2 X1 x 5 6 3 4 4 5 3 4 - 6 X2 x - x + X1 X2 x + X1 x + 7 X2 x + 4 x - 4 X2 x - 6 x 2 3 2 / 3 4 3 + X2 x + 7 x - 7 x + 4 x - 1) / ((-1 + x) (X1 X2 x - X1 X2 x / 4 4 3 4 2 3 2 - X1 x - X2 x + 2 X2 x + x - X2 x - x + x - 2 x + 1)) and in Maple format (X1^2*X2*x^9-X1^2*x^9-2*X1*X2*x^9+2*X1*x^9+X2*x^9+X1*X2*x^7-x^9-2*X1*X2*x^6-X1* x^7-X2*x^7+4*X1*X2*x^5+X1*x^6+2*X2*x^6+x^7-3*X1*X2*x^4-2*X1*x^5-6*X2*x^5-x^6+X1 *X2*x^3+X1*x^4+7*X2*x^4+4*x^5-4*X2*x^3-6*x^4+X2*x^2+7*x^3-7*x^2+4*x-1)/(-1+x)^3 /(X1*X2*x^4-X1*X2*x^3-X1*x^4-X2*x^4+2*X2*x^3+x^4-X2*x^2-x^3+x^2-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 1, 1, 3], are 11 3 n - 7/8 + n/8, and , - -- + ---, respectively, while 64 64 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 2, 2, 1], are 47 7 n - 7/8 + n/8, and , - -- + ---, respectively, while the asymptotic correlatio\ 64 64 n between these two random variables is 1/2 1/2 3 7 --------- 21 and in floating point 0.2182178903 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 21 ate normal pair with correlation, ----- 21 1/2 21 23 i.e. , [[-----, 0], [0, --]] 21 21 ------------------------------------------------- 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, 1, 3], nor the composition, [1, 3, 1, 1] Then infinity ----- 11 10 8 7 6 5 4 2 \ n x + 2 x - 2 x - 2 x - x + x - x + 3 x - 3 x + 1 ) a(n) x = --------------------------------------------------------- / 2 3 ----- (x + x - 1) (-1 + x) n = 0 and in Maple format (x^11+2*x^10-2*x^8-2*x^7-x^6+x^5-x^4+3*x^2-3*x+1)/(x^2+x-1)/(-1+x)^3 The asymptotic expression for a(n) is, n 1.1708203932499369092 1.6180339887498948482 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 1, 1, 3] and d occurrences (as containment) of the composition, [1, 3, 1, 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 2 11 2 12 2 12 2 11 2 12 + X1 X2 x - 2 X1 X2 x - 2 X1 X2 x - 2 X1 X2 x + X1 x 2 11 12 2 12 2 10 2 11 2 10 - 2 X1 X2 x + 4 X1 X2 x + X2 x + X1 X2 x + X1 x + X1 X2 x 11 12 2 11 12 2 9 2 10 + 4 X1 X2 x - 2 X1 x + X2 x - 2 X2 x + X1 X2 x - X1 x 2 9 10 11 2 10 11 12 2 9 + X1 X2 x - 4 X1 X2 x - 2 X1 x - X2 x - 2 X2 x + x - X1 x 9 10 2 9 10 11 9 9 - 4 X1 X2 x + 3 X1 x - X2 x + 3 X2 x + x + 3 X1 x + 3 X2 x 10 7 9 7 7 5 6 6 - 2 x + X1 X2 x - 2 x - X1 x - X2 x + 2 X1 X2 x - X1 x - X2 x 7 4 6 3 5 4 3 2 + x - 2 X1 X2 x + 2 x + X1 X2 x - 2 x + x + 3 x - 6 x + 4 x - 1) / 3 3 3 / ((-1 + x) (X1 X2 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^12-2*X1*X2^2*x^12-2*X1^2*X2*x^11+X1 ^2*x^12-2*X1*X2^2*x^11+4*X1*X2*x^12+X2^2*x^12+X1^2*X2*x^10+X1^2*x^11+X1*X2^2*x^ 10+4*X1*X2*x^11-2*X1*x^12+X2^2*x^11-2*X2*x^12+X1^2*X2*x^9-X1^2*x^10+X1*X2^2*x^9 -4*X1*X2*x^10-2*X1*x^11-X2^2*x^10-2*X2*x^11+x^12-X1^2*x^9-4*X1*X2*x^9+3*X1*x^10 -X2^2*x^9+3*X2*x^10+x^11+3*X1*x^9+3*X2*x^9-2*x^10+X1*X2*x^7-2*x^9-X1*x^7-X2*x^7 +2*X1*X2*x^5-X1*x^6-X2*x^6+x^7-2*X1*X2*x^4+2*x^6+X1*X2*x^3-2*x^5+x^4+3*x^3-6*x^ 2+4*x-1)/(-1+x)^3/(X1*X2*x^3-x^3+2*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 1, 1, 3], are 11 3 n - 7/8 + n/8, and , - -- + ---, respectively, while 64 64 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 3, 1, 1], are 11 3 n - 7/8 + n/8, and , - -- + ---, respectively, while the asymptotic correlatio\ 64 64 n between these two random variables is 1 and in floating point 1. ------------------------------------------------- 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, 1, 3], nor the composition, [2, 1, 1, 2] Then infinity ----- \ n ) a(n) x = / ----- n = 0 12 11 9 8 6 5 4 3 2 x - x - 2 x + x + 3 x - x + 3 x - x + 2 x - 2 x + 1 -------------------------------------------------------------- 3 9 4 2 (x + x - 1) (x - x - 1) (-1 + x) and in Maple format (x^12-x^11-2*x^9+x^8+3*x^6-x^5+3*x^4-x^3+2*x^2-2*x+1)/(x^3+x-1)/(x^9-x^4-1)/(-1 +x)^2 The asymptotic expression for a(n) is, n 5.2787205554750117559 1.4655712318767680267 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 1, 1, 3] and d occurrences (as containment) of the composition, [2, 1, 1, 2], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 4 4 15 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 4 3 15 3 4 15 4 2 15 3 4 14 - 3 X1 X2 x - 4 X1 X2 x + 3 X1 X2 x + 4 X1 X2 x 3 3 15 2 4 15 4 15 3 3 14 + 12 X1 X2 x + 6 X1 X2 x - X1 X2 x - 13 X1 X2 x 3 2 15 2 4 14 2 3 15 4 15 - 12 X1 X2 x - 12 X1 X2 x - 18 X1 X2 x - 4 X1 X2 x 3 2 14 3 15 2 4 13 2 3 14 + 15 X1 X2 x + 4 X1 X2 x + 6 X1 X2 x + 39 X1 X2 x 2 2 15 4 14 3 15 4 15 3 3 12 + 18 X1 X2 x + 12 X1 X2 x + 12 X1 X2 x + X2 x - 2 X1 X2 x 3 14 2 3 13 2 2 14 2 15 - 7 X1 X2 x - 21 X1 X2 x - 45 X1 X2 x - 6 X1 X2 x 4 13 3 14 2 15 4 14 3 15 - 12 X1 X2 x - 39 X1 X2 x - 12 X1 X2 x - 4 X2 x - 3 X2 x 3 3 11 3 2 12 3 14 2 3 12 2 2 13 + X1 X2 x + 4 X1 X2 x + X1 x + 6 X1 X2 x + 27 X1 X2 x 2 14 4 12 3 13 2 14 + 21 X1 X2 x + 4 X1 X2 x + 42 X1 X2 x + 45 X1 X2 x 15 4 13 3 14 2 15 3 3 10 + 4 X1 X2 x + 6 X2 x + 13 X2 x + 3 X2 x - X1 X2 x 3 2 11 3 12 2 3 11 2 2 12 - 2 X1 X2 x - 2 X1 X2 x - 9 X1 X2 x - 12 X1 X2 x 2 13 2 14 3 12 2 13 14 - 15 X1 X2 x - 3 X1 x - 21 X1 X2 x - 54 X1 X2 x - 21 X1 X2 x 4 12 3 13 2 14 15 3 2 10 3 11 - 4 X2 x - 21 X2 x - 15 X2 x - X2 x + X1 X2 x + X1 X2 x 2 3 10 2 2 11 2 12 2 13 + 6 X1 X2 x + 20 X1 X2 x + 6 X1 X2 x + 3 X1 x 3 11 2 12 13 14 4 11 + 15 X1 X2 x + 33 X1 X2 x + 30 X1 X2 x + 3 X1 x + X2 x 3 12 2 13 14 2 3 9 2 2 10 + 17 X2 x + 27 X2 x + 7 X2 x - 3 X1 X2 x - 10 X1 X2 x 2 11 3 10 2 11 12 13 - 13 X1 X2 x - 15 X1 X2 x - 34 X1 X2 x - 19 X1 X2 x - 6 X1 x 3 11 2 12 13 14 2 2 9 2 10 - 11 X2 x - 25 X2 x - 15 X2 x - x + 7 X1 X2 x + 5 X1 X2 x 2 11 3 9 2 10 11 12 + 2 X1 x + 9 X1 X2 x + 33 X1 X2 x + 23 X1 X2 x + 3 X1 x 3 10 2 11 12 13 2 2 8 2 9 + 10 X2 x + 22 X2 x + 15 X2 x + 3 x - 3 X1 X2 x - 4 X1 X2 x 2 10 3 8 2 9 10 11 - X1 x - 3 X1 X2 x - 22 X1 X2 x - 24 X1 X2 x - 4 X1 x 3 9 2 10 11 12 2 2 7 2 8 - 8 X2 x - 24 X2 x - 15 X2 x - 3 x + X1 X2 x + 3 X1 X2 x 2 8 9 10 3 8 2 9 10 + 17 X1 X2 x + 15 X1 X2 x + 6 X1 x + 4 X2 x + 21 X2 x + 19 X2 x 11 2 7 2 7 8 9 3 7 + 3 x - X1 X2 x - 8 X1 X2 x - 17 X1 X2 x - 2 X1 x - X2 x 2 8 9 10 2 6 7 8 - 17 X2 x - 17 X2 x - 5 x + 2 X1 X2 x + 12 X1 X2 x + 3 X1 x 2 7 8 9 6 7 2 6 7 + 12 X2 x + 17 X2 x + 4 x - 7 X1 X2 x - 4 X1 x - 5 X2 x - 18 X2 x 8 5 6 2 5 6 7 4 - 4 x + 5 X1 X2 x + 4 X1 x + X2 x + 14 X2 x + 7 x - 3 X1 X2 x 5 5 6 3 4 4 5 3 - 3 X1 x - 11 X2 x - 8 x + X1 X2 x + X1 x + 8 X2 x + 8 x - 4 X2 x 4 2 3 2 / 4 3 4 - 7 x + X2 x + 7 x - 7 x + 4 x - 1) / ((X1 X2 x - X1 X2 x - X1 x / 4 3 4 2 3 2 3 3 12 - X2 x + 2 X2 x + x - X2 x - x + x - 2 x + 1) (X1 X2 x 3 2 12 2 3 12 3 12 2 3 11 - 3 X1 X2 x - 3 X1 X2 x + 3 X1 X2 x + 3 X1 X2 x 2 2 12 3 12 3 12 2 2 11 2 12 + 9 X1 X2 x + 3 X1 X2 x - X1 x - 9 X1 X2 x - 9 X1 X2 x 3 11 2 12 3 12 2 11 2 12 - 6 X1 X2 x - 9 X1 X2 x - X2 x + 9 X1 X2 x + 3 X1 x 3 10 2 11 12 3 11 2 12 + 3 X1 X2 x + 18 X1 X2 x + 9 X1 X2 x + 3 X2 x + 3 X2 x 2 11 2 10 11 12 3 10 - 3 X1 x - 9 X1 X2 x - 18 X1 X2 x - 3 X1 x - 3 X2 x 2 11 12 2 2 8 10 11 3 9 - 9 X2 x - 3 X2 x + X1 X2 x + 9 X1 X2 x + 6 X1 x + X2 x 2 10 11 12 2 2 7 2 8 2 8 + 9 X2 x + 9 X2 x + x - X1 X2 x - X1 X2 x - 2 X1 X2 x 10 2 9 10 11 2 7 2 7 - 3 X1 x - 3 X2 x - 9 X2 x - 3 x + X1 X2 x + 4 X1 X2 x 8 2 8 9 10 2 6 7 + 2 X1 X2 x + X2 x + 3 X2 x + 3 x - 2 X1 X2 x - 5 X1 X2 x 2 7 8 9 6 7 2 6 7 - 3 X2 x - X2 x - x + 4 X1 X2 x + X1 x + 3 X2 x + 4 X2 x 5 6 2 5 6 7 5 5 6 - X1 X2 x - 2 X1 x - X2 x - 6 X2 x - x + X1 x + 4 X2 x + 3 x 4 5 4 3 2 - X2 x - 3 x + x - x + 3 x - 3 x + 1)) and in Maple format -(X1^4*X2^4*x^15-3*X1^4*X2^3*x^15-4*X1^3*X2^4*x^15+3*X1^4*X2^2*x^15+4*X1^3*X2^4 *x^14+12*X1^3*X2^3*x^15+6*X1^2*X2^4*x^15-X1^4*X2*x^15-13*X1^3*X2^3*x^14-12*X1^3 *X2^2*x^15-12*X1^2*X2^4*x^14-18*X1^2*X2^3*x^15-4*X1*X2^4*x^15+15*X1^3*X2^2*x^14 +4*X1^3*X2*x^15+6*X1^2*X2^4*x^13+39*X1^2*X2^3*x^14+18*X1^2*X2^2*x^15+12*X1*X2^4 *x^14+12*X1*X2^3*x^15+X2^4*x^15-2*X1^3*X2^3*x^12-7*X1^3*X2*x^14-21*X1^2*X2^3*x^ 13-45*X1^2*X2^2*x^14-6*X1^2*X2*x^15-12*X1*X2^4*x^13-39*X1*X2^3*x^14-12*X1*X2^2* x^15-4*X2^4*x^14-3*X2^3*x^15+X1^3*X2^3*x^11+4*X1^3*X2^2*x^12+X1^3*x^14+6*X1^2* X2^3*x^12+27*X1^2*X2^2*x^13+21*X1^2*X2*x^14+4*X1*X2^4*x^12+42*X1*X2^3*x^13+45* X1*X2^2*x^14+4*X1*X2*x^15+6*X2^4*x^13+13*X2^3*x^14+3*X2^2*x^15-X1^3*X2^3*x^10-2 *X1^3*X2^2*x^11-2*X1^3*X2*x^12-9*X1^2*X2^3*x^11-12*X1^2*X2^2*x^12-15*X1^2*X2*x^ 13-3*X1^2*x^14-21*X1*X2^3*x^12-54*X1*X2^2*x^13-21*X1*X2*x^14-4*X2^4*x^12-21*X2^ 3*x^13-15*X2^2*x^14-X2*x^15+X1^3*X2^2*x^10+X1^3*X2*x^11+6*X1^2*X2^3*x^10+20*X1^ 2*X2^2*x^11+6*X1^2*X2*x^12+3*X1^2*x^13+15*X1*X2^3*x^11+33*X1*X2^2*x^12+30*X1*X2 *x^13+3*X1*x^14+X2^4*x^11+17*X2^3*x^12+27*X2^2*x^13+7*X2*x^14-3*X1^2*X2^3*x^9-\ 10*X1^2*X2^2*x^10-13*X1^2*X2*x^11-15*X1*X2^3*x^10-34*X1*X2^2*x^11-19*X1*X2*x^12 -6*X1*x^13-11*X2^3*x^11-25*X2^2*x^12-15*X2*x^13-x^14+7*X1^2*X2^2*x^9+5*X1^2*X2* x^10+2*X1^2*x^11+9*X1*X2^3*x^9+33*X1*X2^2*x^10+23*X1*X2*x^11+3*X1*x^12+10*X2^3* x^10+22*X2^2*x^11+15*X2*x^12+3*x^13-3*X1^2*X2^2*x^8-4*X1^2*X2*x^9-X1^2*x^10-3* X1*X2^3*x^8-22*X1*X2^2*x^9-24*X1*X2*x^10-4*X1*x^11-8*X2^3*x^9-24*X2^2*x^10-15* X2*x^11-3*x^12+X1^2*X2^2*x^7+3*X1^2*X2*x^8+17*X1*X2^2*x^8+15*X1*X2*x^9+6*X1*x^ 10+4*X2^3*x^8+21*X2^2*x^9+19*X2*x^10+3*x^11-X1^2*X2*x^7-8*X1*X2^2*x^7-17*X1*X2* x^8-2*X1*x^9-X2^3*x^7-17*X2^2*x^8-17*X2*x^9-5*x^10+2*X1*X2^2*x^6+12*X1*X2*x^7+3 *X1*x^8+12*X2^2*x^7+17*X2*x^8+4*x^9-7*X1*X2*x^6-4*X1*x^7-5*X2^2*x^6-18*X2*x^7-4 *x^8+5*X1*X2*x^5+4*X1*x^6+X2^2*x^5+14*X2*x^6+7*x^7-3*X1*X2*x^4-3*X1*x^5-11*X2*x ^5-8*x^6+X1*X2*x^3+X1*x^4+8*X2*x^4+8*x^5-4*X2*x^3-7*x^4+X2*x^2+7*x^3-7*x^2+4*x-\ 1)/(X1*X2*x^4-X1*X2*x^3-X1*x^4-X2*x^4+2*X2*x^3+x^4-X2*x^2-x^3+x^2-2*x+1)/(X1^3* X2^3*x^12-3*X1^3*X2^2*x^12-3*X1^2*X2^3*x^12+3*X1^3*X2*x^12+3*X1^2*X2^3*x^11+9* X1^2*X2^2*x^12+3*X1*X2^3*x^12-X1^3*x^12-9*X1^2*X2^2*x^11-9*X1^2*X2*x^12-6*X1*X2 ^3*x^11-9*X1*X2^2*x^12-X2^3*x^12+9*X1^2*X2*x^11+3*X1^2*x^12+3*X1*X2^3*x^10+18* X1*X2^2*x^11+9*X1*X2*x^12+3*X2^3*x^11+3*X2^2*x^12-3*X1^2*x^11-9*X1*X2^2*x^10-18 *X1*X2*x^11-3*X1*x^12-3*X2^3*x^10-9*X2^2*x^11-3*X2*x^12+X1^2*X2^2*x^8+9*X1*X2*x ^10+6*X1*x^11+X2^3*x^9+9*X2^2*x^10+9*X2*x^11+x^12-X1^2*X2^2*x^7-X1^2*X2*x^8-2* X1*X2^2*x^8-3*X1*x^10-3*X2^2*x^9-9*X2*x^10-3*x^11+X1^2*X2*x^7+4*X1*X2^2*x^7+2* X1*X2*x^8+X2^2*x^8+3*X2*x^9+3*x^10-2*X1*X2^2*x^6-5*X1*X2*x^7-3*X2^2*x^7-X2*x^8- x^9+4*X1*X2*x^6+X1*x^7+3*X2^2*x^6+4*X2*x^7-X1*X2*x^5-2*X1*x^6-X2^2*x^5-6*X2*x^6 -x^7+X1*x^5+4*X2*x^5+3*x^6-X2*x^4-3*x^5+x^4-x^3+3*x^2-3*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 1, 1, 3], are 11 3 n - 7/8 + n/8, and , - -- + ---, respectively, while 64 64 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 1, 1, 2], are 63 7 n - 7/8 + n/8, and , - -- + ---, respectively, while the asymptotic correlatio\ 64 64 n between these two random variables is 1/2 1/2 3 7 --------- 21 and in floating point 0.2182178903 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 21 ate normal pair with correlation, ----- 21 1/2 21 23 i.e. , [[-----, 0], [0, --]] 21 21 ------------------------------------------------- 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, 1, 3], nor the composition, [2, 1, 2, 1] Then infinity ----- 8 7 6 5 4 3 2 \ n x + 2 x - 4 x + 7 x - 10 x + 10 x - 8 x + 4 x - 1 ) a(n) x = - -------------------------------------------------------- / 2 3 3 ----- (x - x + 1) (x + x - 1) (-1 + x) n = 0 and in Maple format -(x^8+2*x^7-4*x^6+7*x^5-10*x^4+10*x^3-8*x^2+4*x-1)/(x^2-x+1)/(x^3+x-1)/(-1+x)^3 The asymptotic expression for a(n) is, n 4.1345231835816431372 1.4655712318767680267 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 1, 1, 3] and d occurrences (as containment) of the composition, [2, 1, 2, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 3 12 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 2 12 3 12 2 2 11 2 12 2 12 - X1 X2 x - X1 x + X1 X2 x - X1 X2 x + 2 X1 X2 x 2 2 10 2 11 2 12 2 11 12 + 2 X1 X2 x - X1 X2 x + 2 X1 x - 3 X1 X2 x - X1 X2 x 2 12 2 2 9 2 10 2 10 11 - X2 x - 4 X1 X2 x - 3 X1 X2 x - 4 X1 X2 x + 4 X1 X2 x 12 2 11 12 2 2 8 2 9 2 10 - X1 x + 2 X2 x + X2 x + 3 X1 X2 x + 7 X1 X2 x + X1 x 2 9 10 11 2 10 11 2 2 7 + 13 X1 X2 x + 6 X1 X2 x - X1 x + 2 X2 x - 3 X2 x - X1 X2 x 2 8 2 9 2 8 9 10 - 4 X1 X2 x - 3 X1 x - 14 X1 X2 x - 22 X1 X2 x - 2 X1 x 2 9 10 11 2 7 2 8 2 7 - 10 X2 x - 3 X2 x + x + X1 X2 x + X1 x + 8 X1 X2 x 8 9 2 8 9 10 2 6 + 22 X1 X2 x + 9 X1 x + 14 X2 x + 17 X2 x + x - 2 X1 X2 x 7 8 2 7 8 9 6 - 13 X1 X2 x - 8 X1 x - 11 X2 x - 24 X2 x - 7 x + 3 X1 X2 x 7 2 6 7 8 5 6 2 5 + 6 X1 x + 5 X2 x + 21 X2 x + 10 x + 3 X1 X2 x - 4 X1 x - X2 x 6 7 4 5 5 6 3 - 11 X2 x - 11 x - 3 X1 X2 x + X1 x - X2 x + 9 x + X1 X2 x 4 5 3 4 2 3 2 / + 6 X2 x - x - 4 X2 x - 8 x + X2 x + 13 x - 11 x + 5 x - 1) / ( / 3 4 4 4 3 4 3 4 (-1 + x) (X1 X2 x - X1 x - X2 x + X2 x + x - x + x - 1) (X1 X2 x 3 4 4 3 4 2 3 2 - X1 X2 x - X1 x - X2 x + 2 X2 x + x - X2 x - x + x - 2 x + 1)) and in Maple format -(X1^3*X2*x^12-X1^2*X2^2*x^12-X1^3*x^12+X1^2*X2^2*x^11-X1^2*X2*x^12+2*X1*X2^2*x ^12+2*X1^2*X2^2*x^10-X1^2*X2*x^11+2*X1^2*x^12-3*X1*X2^2*x^11-X1*X2*x^12-X2^2*x^ 12-4*X1^2*X2^2*x^9-3*X1^2*X2*x^10-4*X1*X2^2*x^10+4*X1*X2*x^11-X1*x^12+2*X2^2*x^ 11+X2*x^12+3*X1^2*X2^2*x^8+7*X1^2*X2*x^9+X1^2*x^10+13*X1*X2^2*x^9+6*X1*X2*x^10- X1*x^11+2*X2^2*x^10-3*X2*x^11-X1^2*X2^2*x^7-4*X1^2*X2*x^8-3*X1^2*x^9-14*X1*X2^2 *x^8-22*X1*X2*x^9-2*X1*x^10-10*X2^2*x^9-3*X2*x^10+x^11+X1^2*X2*x^7+X1^2*x^8+8* X1*X2^2*x^7+22*X1*X2*x^8+9*X1*x^9+14*X2^2*x^8+17*X2*x^9+x^10-2*X1*X2^2*x^6-13* X1*X2*x^7-8*X1*x^8-11*X2^2*x^7-24*X2*x^8-7*x^9+3*X1*X2*x^6+6*X1*x^7+5*X2^2*x^6+ 21*X2*x^7+10*x^8+3*X1*X2*x^5-4*X1*x^6-X2^2*x^5-11*X2*x^6-11*x^7-3*X1*X2*x^4+X1* x^5-X2*x^5+9*x^6+X1*X2*x^3+6*X2*x^4-x^5-4*X2*x^3-8*x^4+X2*x^2+13*x^3-11*x^2+5*x -1)/(-1+x)^3/(X1*X2*x^4-X1*x^4-X2*x^4+X2*x^3+x^4-x^3+x-1)/(X1*X2*x^4-X1*X2*x^3- X1*x^4-X2*x^4+2*X2*x^3+x^4-X2*x^2-x^3+x^2-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 1, 1, 3], are 11 3 n - 7/8 + n/8, and , - -- + ---, respectively, while 64 64 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 1, 2, 1], are 55 7 n - 7/8 + n/8, and , - -- + ---, respectively, while the asymptotic correlatio\ 64 64 n between these two random variables is 1/2 1/2 3 7 --------- 21 and in floating point 0.2182178903 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 21 ate normal pair with correlation, ----- 21 1/2 21 23 i.e. , [[-----, 0], [0, --]] 21 21 ------------------------------------------------- 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, 1, 3], nor the composition, [2, 2, 1, 1] Then infinity ----- \ n ) a(n) x = / ----- n = 0 5 3 2 7 6 5 4 3 2 (x + x - x + x - 1) (x + x - x - x - x - x + 2 x - 1) -------------------------------------------------------------- 3 3 (x + x - 1) (-1 + x) and in Maple format (x^5+x^3-x^2+x-1)*(x^7+x^6-x^5-x^4-x^3-x^2+2*x-1)/(x^3+x-1)/(-1+x)^3 The asymptotic expression for a(n) is, n 3.7172851956554243597 1.4655712318767680267 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 1, 1, 3] and d occurrences (as containment) of the composition, [2, 2, 1, 1], (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 3 14 2 2 14 3 14 2 2 13 2 14 - 2 X1 X2 x - 3 X1 X2 x + X1 x + X1 X2 x + 6 X1 X2 x 2 14 3 12 2 13 2 14 2 13 + 3 X1 X2 x + X1 X2 x - 2 X1 X2 x - 3 X1 x - 2 X1 X2 x 14 2 14 3 12 2 2 11 2 12 2 13 - 6 X1 X2 x - X2 x - X1 x + X1 X2 x - 3 X1 X2 x + X1 x 13 14 2 13 14 2 11 2 12 + 4 X1 X2 x + 3 X1 x + X2 x + 2 X2 x - 2 X1 X2 x + 3 X1 x 2 11 12 13 13 14 2 10 - X1 X2 x + 3 X1 X2 x - 2 X1 x - 2 X2 x - x + X1 X2 x 2 11 2 10 11 12 12 13 2 9 + X1 x - X1 X2 x + 2 X1 X2 x - 3 X1 x - X2 x + x + X1 X2 x 2 10 2 9 11 2 10 12 2 9 9 - X1 x + X1 X2 x - X1 x + X2 x + x - X1 x - 4 X1 X2 x 10 2 9 10 9 9 7 9 + X1 x - X2 x - X2 x + 3 X1 x + 3 X2 x + 3 X1 X2 x - 2 x 6 7 7 5 6 6 7 - 6 X1 X2 x - 2 X1 x - 3 X2 x + 7 X1 X2 x + 3 X1 x + 8 X2 x + 2 x 4 5 5 6 3 4 4 - 4 X1 X2 x - 3 X1 x - 13 X2 x - 5 x + X1 X2 x + X1 x + 11 X2 x 5 3 4 2 3 2 / + 10 x - 5 X2 x - 13 x + X2 x + 14 x - 11 x + 5 x - 1) / ( / 4 4 3 4 4 3 4 2 3 (-1 + x) (X1 X2 x - X1 X2 x - X1 x - X2 x + 2 X2 x + x - X2 x - x 2 + x - 2 x + 1)) and in Maple format -(X1^3*X2^2*x^14-2*X1^3*X2*x^14-3*X1^2*X2^2*x^14+X1^3*x^14+X1^2*X2^2*x^13+6*X1^ 2*X2*x^14+3*X1*X2^2*x^14+X1^3*X2*x^12-2*X1^2*X2*x^13-3*X1^2*x^14-2*X1*X2^2*x^13 -6*X1*X2*x^14-X2^2*x^14-X1^3*x^12+X1^2*X2^2*x^11-3*X1^2*X2*x^12+X1^2*x^13+4*X1* X2*x^13+3*X1*x^14+X2^2*x^13+2*X2*x^14-2*X1^2*X2*x^11+3*X1^2*x^12-X1*X2^2*x^11+3 *X1*X2*x^12-2*X1*x^13-2*X2*x^13-x^14+X1^2*X2*x^10+X1^2*x^11-X1*X2^2*x^10+2*X1* X2*x^11-3*X1*x^12-X2*x^12+x^13+X1^2*X2*x^9-X1^2*x^10+X1*X2^2*x^9-X1*x^11+X2^2*x ^10+x^12-X1^2*x^9-4*X1*X2*x^9+X1*x^10-X2^2*x^9-X2*x^10+3*X1*x^9+3*X2*x^9+3*X1* X2*x^7-2*x^9-6*X1*X2*x^6-2*X1*x^7-3*X2*x^7+7*X1*X2*x^5+3*X1*x^6+8*X2*x^6+2*x^7-\ 4*X1*X2*x^4-3*X1*x^5-13*X2*x^5-5*x^6+X1*X2*x^3+X1*x^4+11*X2*x^4+10*x^5-5*X2*x^3 -13*x^4+X2*x^2+14*x^3-11*x^2+5*x-1)/(-1+x)^4/(X1*X2*x^4-X1*X2*x^3-X1*x^4-X2*x^4 +2*X2*x^3+x^4-X2*x^2-x^3+x^2-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 1, 1, 3], are 11 3 n - 7/8 + n/8, and , - -- + ---, respectively, while 64 64 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 2, 1, 1], are 47 7 n - 7/8 + n/8, and , - -- + ---, respectively, while the asymptotic correlatio\ 64 64 n between these two random variables is 1/2 1/2 3 7 --------- 21 and in floating point 0.2182178903 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 21 ate normal pair with correlation, ----- 21 1/2 21 23 i.e. , [[-----, 0], [0, --]] 21 21 ------------------------------------------------- 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, 1, 3], nor the composition, [3, 1, 1, 1] Then infinity ----- \ n 15 14 13 12 11 10 9 8 ) a(n) x = - (x + 3 x + x - 5 x - 6 x - x + 3 x + 3 x / ----- n = 0 7 6 5 4 2 / 2 3 + 2 x + x - x + x - 3 x + 3 x - 1) / ((x + x - 1) (-1 + x) ) / and in Maple format -(x^15+3*x^14+x^13-5*x^12-6*x^11-x^10+3*x^9+3*x^8+2*x^7+x^6-x^5+x^4-3*x^2+3*x-1 )/(x^2+x-1)/(-1+x)^3 The asymptotic expression for a(n) is, n 0.72360679774997896961 1.6180339887498948482 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 1, 1, 3] and d occurrences (as containment) of the composition, [3, 1, 1, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 3 3 17 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 3 3 16 3 2 17 2 3 17 3 2 16 + X1 X2 x - 3 X1 X2 x - 3 X1 X2 x - 3 X1 X2 x 3 17 2 3 16 2 2 17 3 17 + 3 X1 X2 x - 3 X1 X2 x + 9 X1 X2 x + 3 X1 X2 x 3 2 15 3 16 3 17 2 3 15 2 2 16 + 2 X1 X2 x + 3 X1 X2 x - X1 x + 2 X1 X2 x + 9 X1 X2 x 2 17 3 16 2 17 3 17 3 2 14 - 9 X1 X2 x + 3 X1 X2 x - 9 X1 X2 x - X2 x + 2 X1 X2 x 3 15 3 16 2 3 14 2 2 15 2 16 - 4 X1 X2 x - X1 x + 2 X1 X2 x - 12 X1 X2 x - 9 X1 X2 x 2 17 3 15 2 16 17 3 16 + 3 X1 x - 4 X1 X2 x - 9 X1 X2 x + 9 X1 X2 x - X2 x 2 17 3 14 3 15 2 2 14 2 15 + 3 X2 x - 4 X1 X2 x + 2 X1 x - 12 X1 X2 x + 18 X1 X2 x 2 16 3 14 2 15 16 17 + 3 X1 x - 4 X1 X2 x + 18 X1 X2 x + 9 X1 X2 x - 3 X1 x 3 15 2 16 17 3 13 3 14 + 2 X2 x + 3 X2 x - 3 X2 x + X1 X2 x + 2 X1 x 2 2 13 2 14 2 15 3 13 2 14 + 3 X1 X2 x + 18 X1 X2 x - 8 X1 x + X1 X2 x + 18 X1 X2 x 15 16 3 14 2 15 16 17 - 24 X1 X2 x - 3 X1 x + 2 X2 x - 8 X2 x - 3 X2 x + x 3 12 3 13 2 2 12 2 13 2 14 + X1 X2 x - X1 x + 4 X1 X2 x - 9 X1 X2 x - 8 X1 x 3 12 2 13 14 15 3 13 + X1 X2 x - 9 X1 X2 x - 24 X1 X2 x + 10 X1 x - X2 x 2 14 15 16 3 12 2 2 11 2 12 - 8 X2 x + 10 X2 x + x - X1 x + X1 X2 x - 11 X1 X2 x 2 13 2 12 13 14 3 12 + 6 X1 x - 11 X1 X2 x + 18 X1 X2 x + 10 X1 x - X2 x 2 13 14 15 2 11 2 12 2 11 + 6 X2 x + 10 X2 x - 4 x - X1 X2 x + 7 X1 x - X1 X2 x 12 13 2 12 13 14 2 10 + 22 X1 X2 x - 10 X1 x + 7 X2 x - 10 X2 x - 4 x + 2 X1 X2 x 2 10 12 12 13 2 9 2 10 + 2 X1 X2 x - 12 X1 x - 12 X2 x + 5 x + X1 X2 x - 2 X1 x 2 9 10 11 2 10 11 12 2 9 + X1 X2 x - 8 X1 X2 x + X1 x - 2 X2 x + X2 x + 6 x - X1 x 9 10 2 9 10 11 9 9 - 3 X1 X2 x + 6 X1 x - X2 x + 6 X2 x - x + 2 X1 x + 2 X2 x 10 7 9 6 5 6 6 7 - 4 x + X1 X2 x - x - 2 X1 X2 x + 4 X1 X2 x - X1 x - X2 x - x 4 6 3 5 4 3 2 - 3 X1 X2 x + 4 x + X1 X2 x - 3 x - 2 x + 9 x - 10 x + 5 x - 1) / 4 3 3 / ((-1 + x) (X1 X2 x - x + 2 x - 1)) / and in Maple format (X1^3*X2^3*x^17+X1^3*X2^3*x^16-3*X1^3*X2^2*x^17-3*X1^2*X2^3*x^17-3*X1^3*X2^2*x^ 16+3*X1^3*X2*x^17-3*X1^2*X2^3*x^16+9*X1^2*X2^2*x^17+3*X1*X2^3*x^17+2*X1^3*X2^2* x^15+3*X1^3*X2*x^16-X1^3*x^17+2*X1^2*X2^3*x^15+9*X1^2*X2^2*x^16-9*X1^2*X2*x^17+ 3*X1*X2^3*x^16-9*X1*X2^2*x^17-X2^3*x^17+2*X1^3*X2^2*x^14-4*X1^3*X2*x^15-X1^3*x^ 16+2*X1^2*X2^3*x^14-12*X1^2*X2^2*x^15-9*X1^2*X2*x^16+3*X1^2*x^17-4*X1*X2^3*x^15 -9*X1*X2^2*x^16+9*X1*X2*x^17-X2^3*x^16+3*X2^2*x^17-4*X1^3*X2*x^14+2*X1^3*x^15-\ 12*X1^2*X2^2*x^14+18*X1^2*X2*x^15+3*X1^2*x^16-4*X1*X2^3*x^14+18*X1*X2^2*x^15+9* X1*X2*x^16-3*X1*x^17+2*X2^3*x^15+3*X2^2*x^16-3*X2*x^17+X1^3*X2*x^13+2*X1^3*x^14 +3*X1^2*X2^2*x^13+18*X1^2*X2*x^14-8*X1^2*x^15+X1*X2^3*x^13+18*X1*X2^2*x^14-24* X1*X2*x^15-3*X1*x^16+2*X2^3*x^14-8*X2^2*x^15-3*X2*x^16+x^17+X1^3*X2*x^12-X1^3*x ^13+4*X1^2*X2^2*x^12-9*X1^2*X2*x^13-8*X1^2*x^14+X1*X2^3*x^12-9*X1*X2^2*x^13-24* X1*X2*x^14+10*X1*x^15-X2^3*x^13-8*X2^2*x^14+10*X2*x^15+x^16-X1^3*x^12+X1^2*X2^2 *x^11-11*X1^2*X2*x^12+6*X1^2*x^13-11*X1*X2^2*x^12+18*X1*X2*x^13+10*X1*x^14-X2^3 *x^12+6*X2^2*x^13+10*X2*x^14-4*x^15-X1^2*X2*x^11+7*X1^2*x^12-X1*X2^2*x^11+22*X1 *X2*x^12-10*X1*x^13+7*X2^2*x^12-10*X2*x^13-4*x^14+2*X1^2*X2*x^10+2*X1*X2^2*x^10 -12*X1*x^12-12*X2*x^12+5*x^13+X1^2*X2*x^9-2*X1^2*x^10+X1*X2^2*x^9-8*X1*X2*x^10+ X1*x^11-2*X2^2*x^10+X2*x^11+6*x^12-X1^2*x^9-3*X1*X2*x^9+6*X1*x^10-X2^2*x^9+6*X2 *x^10-x^11+2*X1*x^9+2*X2*x^9-4*x^10+X1*X2*x^7-x^9-2*X1*X2*x^6+4*X1*X2*x^5-X1*x^ 6-X2*x^6-x^7-3*X1*X2*x^4+4*x^6+X1*X2*x^3-3*x^5-2*x^4+9*x^3-10*x^2+5*x-1)/(-1+x) ^4/(X1*X2*x^3-x^3+2*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 1, 1, 3], are 11 3 n - 7/8 + n/8, and , - -- + ---, respectively, while 64 64 Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 1, 1, 1], are 11 3 n - 7/8 + n/8, and , - -- + ---, respectively, while the asymptotic correlatio\ 64 64 n between these two random variables is 1 and in floating point 1. ------------------------------------------------- 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, 2, 2], nor the composition, [1, 1, 3, 1] Then infinity ----- 5 4 3 2 \ n x - 3 x + 3 x - 4 x + 3 x - 1 ) a(n) x = - --------------------------------- / 3 3 ----- (x + x - 1) (-1 + x) n = 0 and in Maple format -(x^5-3*x^4+3*x^3-4*x^2+3*x-1)/(x^3+x-1)/(-1+x)^3 The asymptotic expression for a(n) is, n 6.9556233073109364759 1.4655712318767680267 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 1, 2, 2] and d occurrences (as containment) of the composition, [1, 1, 3, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 5 ) | ) | ) A(n, c, d) x X1 X2 || = - (2 X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 4 5 5 3 4 4 5 3 - 2 X1 X2 x - 2 X1 x - X2 x + X1 X2 x + 4 X1 x + X2 x + x - 3 X1 x 4 2 3 2 / 2 4 - 3 x + X1 x + 3 x - 4 x + 3 x - 1) / ((-1 + x) (X1 X2 x / 3 4 4 3 4 2 3 2 - X1 X2 x - X1 x - X2 x + 2 X1 x + x - X1 x - x + x - 2 x + 1)) and in Maple format -(2*X1*X2*x^5-2*X1*X2*x^4-2*X1*x^5-X2*x^5+X1*X2*x^3+4*X1*x^4+X2*x^4+x^5-3*X1*x^ 3-3*x^4+X1*x^2+3*x^3-4*x^2+3*x-1)/(-1+x)^2/(X1*X2*x^4-X1*X2*x^3-X1*x^4-X2*x^4+2 *X1*x^3+x^4-X1*x^2-x^3+x^2-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 1, 2, 2], are 47 7 n - 7/8 + n/8, and , - -- + ---, respectively, while 64 64 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 1, 3, 1], are 11 3 n - 7/8 + n/8, and , - -- + ---, respectively, while the asymptotic correlatio\ 64 64 n between these two random variables is 1/2 1/2 3 7 --------- 21 and in floating point 0.2182178903 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 21 ate normal pair with correlation, ----- 21 1/2 21 23 i.e. , [[-----, 0], [0, --]] 21 21 ------------------------------------------------- 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, 2, 2], nor the composition, [1, 2, 1, 2] Then infinity ----- 6 5 3 2 \ n x - 2 x + 2 x - 4 x + 3 x - 1 ) a(n) x = ----------------------------------- / 2 2 2 ----- (x - x + 1) (x + x - 1) (-1 + x) n = 0 and in Maple format (x^6-2*x^5+2*x^3-4*x^2+3*x-1)/(x^2-x+1)/(x^2+x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 2.1180339887498948478 1.6180339887498948482 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 1, 2, 2] and d occurrences (as containment) of the composition, [1, 2, 1, 2], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 7 2 6 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x - X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 7 2 5 6 7 5 6 - 2 X1 X2 x + X1 X2 x + X1 X2 x + X1 x - X1 X2 x + X2 x 4 5 6 3 4 5 2 - 2 X1 X2 x - 2 X2 x - x + 2 X1 X2 x + 2 X2 x + 2 x - X1 X2 x 3 3 2 / 2 6 2 5 - X2 x - 2 x + 4 x - 3 x + 1) / ((-1 + x) (X1 X2 x - X1 X2 x / 6 5 6 4 5 3 4 - 2 X1 X2 x + X1 X2 x + X1 x + X1 X2 x + X2 x - 2 X1 X2 x - 2 X2 x 5 2 3 4 3 2 - x + X1 X2 x + X2 x + x + x - 3 x + 3 x - 1)) and in Maple format (X1*X2^2*x^7-X1*X2^2*x^6-2*X1*X2*x^7+X1*X2^2*x^5+X1*X2*x^6+X1*x^7-X1*X2*x^5+X2* x^6-2*X1*X2*x^4-2*X2*x^5-x^6+2*X1*X2*x^3+2*X2*x^4+2*x^5-X1*X2*x^2-X2*x^3-2*x^3+ 4*x^2-3*x+1)/(-1+x)/(X1*X2^2*x^6-X1*X2^2*x^5-2*X1*X2*x^6+X1*X2*x^5+X1*x^6+X1*X2 *x^4+X2*x^5-2*X1*X2*x^3-2*X2*x^4-x^5+X1*X2*x^2+X2*x^3+x^4+x^3-3*x^2+3*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 1, 2, 2], are 47 7 n - 7/8 + n/8, and , - -- + ---, respectively, while 64 64 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 2, 1, 2], are 55 7 n - 7/8 + n/8, and , - -- + ---, respectively, while the asymptotic correlatio\ 64 64 n between these two random variables is 5/7 and in floating point 0.7142857143 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ ate normal pair with correlation, 5/7 99 i.e. , [[5/7, 0], [0, --]] 49 ------------------------------------------------- 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, 2, 2], nor the composition, [1, 2, 2, 1] Then infinity ----- 8 7 6 5 4 3 2 \ n x - x + x + 2 x - 5 x + 7 x - 7 x + 4 x - 1 ) a(n) x = - -------------------------------------------------- / 3 2 3 ----- (x - x + 2 x - 1) (-1 + x) n = 0 and in Maple format -(x^8-x^7+x^6+2*x^5-5*x^4+7*x^3-7*x^2+4*x-1)/(x^3-x^2+2*x-1)/(-1+x)^3 The asymptotic expression for a(n) is, n 0.95661118429105013088 1.7548776662466927601 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 1, 2, 2] and d occurrences (as containment) of the composition, [1, 2, 2, 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 5 6 - X1 x - X2 x + X1 X2 x + X1 x + X2 x + x - 2 X1 X2 x - X1 x 6 7 4 6 3 5 2 4 - X2 x - x + 4 X1 X2 x + x - 3 X1 X2 x + 2 x + X1 X2 x - 5 x 3 2 / 3 + 7 x - 7 x + 4 x - 1) / ((-1 + x) / 3 2 3 2 (X1 X2 x - X1 X2 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^8-2*X1*X2*x^5-X1*x ^6-X2*x^6-x^7+4*X1*X2*x^4+x^6-3*X1*X2*x^3+2*x^5+X1*X2*x^2-5*x^4+7*x^3-7*x^2+4*x -1)/(-1+x)^3/(X1*X2*x^3-X1*X2*x^2-x^3+x^2-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 1, 2, 2], are 47 7 n - 7/8 + n/8, and , - -- + ---, respectively, while 64 64 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 2, 2, 1], are 47 7 n - 7/8 + n/8, and , - -- + ---, respectively, while the asymptotic correlatio\ 64 64 n between these two random variables is 1 and in floating point 1. ------------------------------------------------- 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, 2, 2], nor the composition, [1, 3, 1, 1] Then infinity ----- 8 7 5 4 3 2 \ n x + x + x - 3 x + 3 x - 4 x + 3 x - 1 ) a(n) x = - ------------------------------------------- / 3 3 ----- (x + x - 1) (-1 + x) n = 0 and in Maple format -(x^8+x^7+x^5-3*x^4+3*x^3-4*x^2+3*x-1)/(x^3+x-1)/(-1+x)^3 The asymptotic expression for a(n) is, n 4.7460151755324556560 1.4655712318767680267 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 1, 2, 2] and d occurrences (as containment) of the composition, [1, 3, 1, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 9 9 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x - 2 X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 9 9 9 7 9 6 7 7 - X2 x + X1 x + 2 X2 x + X1 X2 x - x - 2 X1 X2 x - X1 x - X2 x 5 6 6 7 4 5 5 6 + 4 X1 X2 x + 2 X1 x + X2 x + x - 3 X1 X2 x - 6 X1 x - 2 X2 x - x 3 4 4 5 3 4 2 3 2 + X1 X2 x + 7 X1 x + X2 x + 4 x - 4 X1 x - 6 x + X1 x + 7 x - 7 x / 3 4 3 4 4 3 + 4 x - 1) / ((-1 + x) (X1 X2 x - X1 X2 x - X1 x - X2 x + 2 X1 x / 4 2 3 2 + x - X1 x - x + x - 2 x + 1)) and in Maple format (X1*X2^2*x^9-2*X1*X2*x^9-X2^2*x^9+X1*x^9+2*X2*x^9+X1*X2*x^7-x^9-2*X1*X2*x^6-X1* x^7-X2*x^7+4*X1*X2*x^5+2*X1*x^6+X2*x^6+x^7-3*X1*X2*x^4-6*X1*x^5-2*X2*x^5-x^6+X1 *X2*x^3+7*X1*x^4+X2*x^4+4*x^5-4*X1*x^3-6*x^4+X1*x^2+7*x^3-7*x^2+4*x-1)/(-1+x)^3 /(X1*X2*x^4-X1*X2*x^3-X1*x^4-X2*x^4+2*X1*x^3+x^4-X1*x^2-x^3+x^2-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 1, 2, 2], are 47 7 n - 7/8 + n/8, and , - -- + ---, respectively, while 64 64 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 3, 1, 1], are 11 3 n - 7/8 + n/8, and , - -- + ---, respectively, while the asymptotic correlatio\ 64 64 n between these two random variables is 1/2 1/2 3 7 --------- 21 and in floating point 0.2182178903 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 21 ate normal pair with correlation, ----- 21 1/2 21 23 i.e. , [[-----, 0], [0, --]] 21 21 ------------------------------------------------- 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, 2, 2], nor the composition, [2, 1, 1, 2] Then infinity ----- 7 6 5 4 3 2 \ n 2 x - 3 x + 5 x - 6 x + 7 x - 7 x + 4 x - 1 ) a(n) x = - ------------------------------------------------- / 5 4 3 2 3 ----- (x - x + x - x + 2 x - 1) (-1 + x) n = 0 and in Maple format -(2*x^7-3*x^6+5*x^5-6*x^4+7*x^3-7*x^2+4*x-1)/(x^5-x^4+x^3-x^2+2*x-1)/(-1+x)^3 The asymptotic expression for a(n) is, n 1.2670431762094331495 1.6736485462998415616 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 1, 2, 2] and d occurrences (as containment) of the composition, [2, 1, 1, 2], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 4 11 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 3 11 2 3 10 2 2 11 2 3 9 2 2 10 - 4 X1 X2 x - X1 X2 x + 6 X1 X2 x - X1 X2 x + 2 X1 X2 x 2 11 3 10 2 3 8 2 2 9 2 10 - 4 X1 X2 x + X1 X2 x + X1 X2 x + 3 X1 X2 x - X1 X2 x 2 11 3 9 2 10 2 3 7 2 2 8 + X1 x - X1 X2 x - 2 X1 X2 x - X1 X2 x - 2 X1 X2 x 2 9 2 9 10 2 2 7 2 8 - 3 X1 X2 x + 3 X1 X2 x + X1 X2 x + X1 X2 x + 2 X1 X2 x 2 9 2 8 9 2 8 2 7 8 + X1 x - X1 X2 x - 3 X1 X2 x - X1 x + 4 X1 X2 x + X1 X2 x 9 2 6 7 2 5 6 7 + X1 x - 3 X1 X2 x - 5 X1 X2 x + X1 X2 x + 4 X1 X2 x + X1 x 7 5 6 6 7 4 5 - 2 X2 x - 3 X1 X2 x - X1 x + 3 X2 x + 2 x + 4 X1 X2 x - 3 X2 x 6 3 4 5 2 4 3 2 - 3 x - 3 X1 X2 x + X2 x + 5 x + X1 X2 x - 6 x + 7 x - 7 x + 4 x / 2 4 12 2 4 11 2 3 12 2 3 11 - 1) / (X1 X2 x - X1 X2 x - 4 X1 X2 x + 3 X1 X2 x / 2 2 12 2 2 11 2 12 3 11 2 3 9 + 6 X1 X2 x - 3 X1 X2 x - 4 X1 X2 x + X1 X2 x + X1 X2 x 2 2 10 2 11 2 12 3 10 2 11 + X1 X2 x + X1 X2 x + X1 x - 2 X1 X2 x - 3 X1 X2 x 2 3 8 2 2 9 2 10 3 9 2 10 - 2 X1 X2 x - 3 X1 X2 x - 2 X1 X2 x + X1 X2 x + 5 X1 X2 x 11 2 3 7 2 2 8 2 9 2 10 + 3 X1 X2 x + X1 X2 x + 3 X1 X2 x + 3 X1 X2 x + X1 x 2 9 10 11 2 2 7 2 8 2 9 - 3 X1 X2 x - 4 X1 X2 x - X1 x - X1 X2 x - X1 X2 x - X1 x 2 8 9 10 2 7 8 9 + 4 X1 X2 x + 3 X1 X2 x + X1 x - 6 X1 X2 x - 6 X1 X2 x - X1 x 2 6 7 8 8 2 5 6 + 4 X1 X2 x + 7 X1 X2 x + 2 X1 x - X2 x - X1 X2 x - 5 X1 X2 x 7 7 8 5 6 7 4 - X1 x + 4 X2 x + x + 5 X1 X2 x - 6 X2 x - 4 x - 6 X1 X2 x 5 6 3 4 5 2 4 3 + 4 X2 x + 7 x + 4 X1 X2 x - X2 x - 8 x - X1 X2 x + 9 x - 11 x 2 + 10 x - 5 x + 1) and in Maple format -(X1^2*X2^4*x^11-4*X1^2*X2^3*x^11-X1^2*X2^3*x^10+6*X1^2*X2^2*x^11-X1^2*X2^3*x^9 +2*X1^2*X2^2*x^10-4*X1^2*X2*x^11+X1*X2^3*x^10+X1^2*X2^3*x^8+3*X1^2*X2^2*x^9-X1^ 2*X2*x^10+X1^2*x^11-X1*X2^3*x^9-2*X1*X2^2*x^10-X1^2*X2^3*x^7-2*X1^2*X2^2*x^8-3* X1^2*X2*x^9+3*X1*X2^2*x^9+X1*X2*x^10+X1^2*X2^2*x^7+2*X1^2*X2*x^8+X1^2*x^9-X1*X2 ^2*x^8-3*X1*X2*x^9-X1^2*x^8+4*X1*X2^2*x^7+X1*X2*x^8+X1*x^9-3*X1*X2^2*x^6-5*X1* X2*x^7+X1*X2^2*x^5+4*X1*X2*x^6+X1*x^7-2*X2*x^7-3*X1*X2*x^5-X1*x^6+3*X2*x^6+2*x^ 7+4*X1*X2*x^4-3*X2*x^5-3*x^6-3*X1*X2*x^3+X2*x^4+5*x^5+X1*X2*x^2-6*x^4+7*x^3-7*x ^2+4*x-1)/(X1^2*X2^4*x^12-X1^2*X2^4*x^11-4*X1^2*X2^3*x^12+3*X1^2*X2^3*x^11+6*X1 ^2*X2^2*x^12-3*X1^2*X2^2*x^11-4*X1^2*X2*x^12+X1*X2^3*x^11+X1^2*X2^3*x^9+X1^2*X2 ^2*x^10+X1^2*X2*x^11+X1^2*x^12-2*X1*X2^3*x^10-3*X1*X2^2*x^11-2*X1^2*X2^3*x^8-3* X1^2*X2^2*x^9-2*X1^2*X2*x^10+X1*X2^3*x^9+5*X1*X2^2*x^10+3*X1*X2*x^11+X1^2*X2^3* x^7+3*X1^2*X2^2*x^8+3*X1^2*X2*x^9+X1^2*x^10-3*X1*X2^2*x^9-4*X1*X2*x^10-X1*x^11- X1^2*X2^2*x^7-X1^2*X2*x^8-X1^2*x^9+4*X1*X2^2*x^8+3*X1*X2*x^9+X1*x^10-6*X1*X2^2* x^7-6*X1*X2*x^8-X1*x^9+4*X1*X2^2*x^6+7*X1*X2*x^7+2*X1*x^8-X2*x^8-X1*X2^2*x^5-5* X1*X2*x^6-X1*x^7+4*X2*x^7+x^8+5*X1*X2*x^5-6*X2*x^6-4*x^7-6*X1*X2*x^4+4*X2*x^5+7 *x^6+4*X1*X2*x^3-X2*x^4-8*x^5-X1*X2*x^2+9*x^4-11*x^3+10*x^2-5*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 1, 2, 2], are 47 7 n - 7/8 + n/8, and , - -- + ---, respectively, while 64 64 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 1, 1, 2], are 63 7 n - 7/8 + n/8, and , - -- + ---, respectively, while the asymptotic correlatio\ 64 64 n between these two random variables is 5/7 and in floating point 0.7142857143 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ ate normal pair with correlation, 5/7 99 i.e. , [[5/7, 0], [0, --]] 49 ------------------------------------------------- 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, 2, 2], nor the composition, [2, 1, 2, 1] Then infinity ----- 7 6 5 4 2 \ n x + x - x + x - 3 x + 3 x - 1 ) a(n) x = - ---------------------------------- / 2 3 ----- (x + x - 1) (-1 + x) n = 0 and in Maple format -(x^7+x^6-x^5+x^4-3*x^2+3*x-1)/(x^2+x-1)/(-1+x)^3 The asymptotic expression for a(n) is, n 1.8944271909999158785 1.6180339887498948482 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 1, 2, 2] and d occurrences (as containment) of the composition, [2, 1, 2, 1], (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 2 9 2 10 2 10 2 9 2 10 - X1 X2 x - 3 X1 X2 x - X1 X2 x + 3 X1 X2 x + 2 X1 x 2 9 10 2 8 2 9 2 8 + 2 X1 X2 x + 4 X1 X2 x - X1 X2 x - 2 X1 x - 3 X1 X2 x 9 10 10 2 8 2 7 8 - 6 X1 X2 x - 3 X1 x - X2 x + X1 x + 4 X1 X2 x + 6 X1 X2 x 9 9 10 2 6 7 8 8 9 + 4 X1 x + X2 x + x - 3 X1 X2 x - 5 X1 X2 x - 3 X1 x + X2 x - x 2 5 6 7 7 8 5 6 + X1 X2 x + X1 X2 x + X1 x - 4 X2 x - x + 5 X1 X2 x + 7 X2 x 7 4 5 6 3 4 2 + 4 x - 7 X1 X2 x - 7 X2 x - 5 x + 4 X1 X2 x + 4 X2 x - X1 X2 x 3 4 3 2 / 3 2 6 - X2 x + 8 x - 13 x + 11 x - 5 x + 1) / ((-1 + x) (X1 X2 x / 2 5 6 5 6 4 5 - X1 X2 x - 2 X1 X2 x + X1 X2 x + X1 x + X1 X2 x + X2 x 3 4 5 2 3 4 3 2 - 2 X1 X2 x - 2 X2 x - x + X1 X2 x + X2 x + x + x - 3 x + 3 x - 1) ) and in Maple format (X1^2*X2^2*x^10-X1^2*X2^2*x^9-3*X1^2*X2*x^10-X1*X2^2*x^10+3*X1^2*X2*x^9+2*X1^2* x^10+2*X1*X2^2*x^9+4*X1*X2*x^10-X1^2*X2*x^8-2*X1^2*x^9-3*X1*X2^2*x^8-6*X1*X2*x^ 9-3*X1*x^10-X2*x^10+X1^2*x^8+4*X1*X2^2*x^7+6*X1*X2*x^8+4*X1*x^9+X2*x^9+x^10-3* X1*X2^2*x^6-5*X1*X2*x^7-3*X1*x^8+X2*x^8-x^9+X1*X2^2*x^5+X1*X2*x^6+X1*x^7-4*X2*x ^7-x^8+5*X1*X2*x^5+7*X2*x^6+4*x^7-7*X1*X2*x^4-7*X2*x^5-5*x^6+4*X1*X2*x^3+4*X2*x ^4-X1*X2*x^2-X2*x^3+8*x^4-13*x^3+11*x^2-5*x+1)/(-1+x)^3/(X1*X2^2*x^6-X1*X2^2*x^ 5-2*X1*X2*x^6+X1*X2*x^5+X1*x^6+X1*X2*x^4+X2*x^5-2*X1*X2*x^3-2*X2*x^4-x^5+X1*X2* x^2+X2*x^3+x^4+x^3-3*x^2+3*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 1, 2, 2], are 47 7 n - 7/8 + n/8, and , - -- + ---, respectively, while 64 64 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 1, 2, 1], are 55 7 n - 7/8 + n/8, and , - -- + ---, respectively, while the asymptotic correlatio\ 64 64 n between these two random variables is 5/7 and in floating point 0.7142857143 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ ate normal pair with correlation, 5/7 99 i.e. , [[5/7, 0], [0, --]] 49 ------------------------------------------------- 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, 2, 2], nor the composition, [2, 2, 1, 1] Then infinity ----- \ n ) a(n) x = / ----- n = 0 10 9 8 7 6 5 4 3 2 x - x + x - x - x - 2 x + 5 x - 7 x + 7 x - 4 x + 1 ------------------------------------------------------------- 3 2 3 (x - x + 2 x - 1) (-1 + x) and in Maple format (x^10-x^9+x^8-x^7-x^6-2*x^5+5*x^4-7*x^3+7*x^2-4*x+1)/(x^3-x^2+2*x-1)/(-1+x)^3 The asymptotic expression for a(n) is, n 0.72212441830311284143 1.7548776662466927601 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 1, 2, 2] and d occurrences (as containment) of the composition, [2, 2, 1, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 11 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 11 2 11 2 10 2 11 2 10 - 2 X1 X2 x - 2 X1 X2 x + X1 X2 x + X1 x + X1 X2 x 11 2 11 2 9 2 10 2 9 10 + 4 X1 X2 x + X2 x - X1 X2 x - X1 x - X1 X2 x - 4 X1 X2 x 11 2 10 11 2 8 2 9 2 8 - 2 X1 x - X2 x - 2 X2 x + X1 X2 x + X1 x + X1 X2 x 9 10 2 9 10 11 2 8 8 + 4 X1 X2 x + 3 X1 x + X2 x + 3 X2 x + x - X1 x - 4 X1 X2 x 9 2 8 9 10 8 8 9 - 3 X1 x - X2 x - 3 X2 x - 2 x + 3 X1 x + 3 X2 x + 2 x 6 8 5 6 6 4 6 + 3 X1 X2 x - 2 x - 6 X1 X2 x - X1 x - X2 x + 7 X1 X2 x - x 3 5 2 4 3 2 / - 4 X1 X2 x + 7 x + X1 X2 x - 12 x + 14 x - 11 x + 5 x - 1) / ( / 4 3 2 3 2 (-1 + x) (X1 X2 x - X1 X2 x - x + x - 2 x + 1)) and in Maple format -(X1^2*X2^2*x^11-2*X1^2*X2*x^11-2*X1*X2^2*x^11+X1^2*X2*x^10+X1^2*x^11+X1*X2^2*x ^10+4*X1*X2*x^11+X2^2*x^11-X1^2*X2*x^9-X1^2*x^10-X1*X2^2*x^9-4*X1*X2*x^10-2*X1* x^11-X2^2*x^10-2*X2*x^11+X1^2*X2*x^8+X1^2*x^9+X1*X2^2*x^8+4*X1*X2*x^9+3*X1*x^10 +X2^2*x^9+3*X2*x^10+x^11-X1^2*x^8-4*X1*X2*x^8-3*X1*x^9-X2^2*x^8-3*X2*x^9-2*x^10 +3*X1*x^8+3*X2*x^8+2*x^9+3*X1*X2*x^6-2*x^8-6*X1*X2*x^5-X1*x^6-X2*x^6+7*X1*X2*x^ 4-x^6-4*X1*X2*x^3+7*x^5+X1*X2*x^2-12*x^4+14*x^3-11*x^2+5*x-1)/(-1+x)^4/(X1*X2*x ^3-X1*X2*x^2-x^3+x^2-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 1, 2, 2], are 47 7 n - 7/8 + n/8, and , - -- + ---, respectively, while 64 64 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 2, 1, 1], are 47 7 n - 7/8 + n/8, and , - -- + ---, respectively, while the asymptotic correlatio\ 64 64 n between these two random variables is 1 and in floating point 1. ------------------------------------------------- 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, 2, 2], nor the composition, [3, 1, 1, 1] Then infinity ----- \ n ) a(n) x = / ----- n = 0 5 3 2 7 6 5 4 3 2 (x + x - x + x - 1) (x + x - x - x - x - x + 2 x - 1) -------------------------------------------------------------- 3 3 (x + x - 1) (-1 + x) and in Maple format (x^5+x^3-x^2+x-1)*(x^7+x^6-x^5-x^4-x^3-x^2+2*x-1)/(x^3+x-1)/(-1+x)^3 The asymptotic expression for a(n) is, n 3.7172851956554243597 1.4655712318767680267 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 1, 2, 2] and d occurrences (as containment) of the composition, [3, 1, 1, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 3 14 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 2 14 3 14 2 2 13 2 14 2 14 - 3 X1 X2 x - 2 X1 X2 x + X1 X2 x + 3 X1 X2 x + 6 X1 X2 x 3 14 2 13 2 14 3 12 2 13 + X2 x - 2 X1 X2 x - X1 x + X1 X2 x - 2 X1 X2 x 14 2 14 2 2 11 2 13 2 12 - 6 X1 X2 x - 3 X2 x + X1 X2 x + X1 x - 3 X1 X2 x 13 14 3 12 2 13 14 2 11 + 4 X1 X2 x + 2 X1 x - X2 x + X2 x + 3 X2 x - X1 X2 x 2 11 12 13 2 12 13 14 - 2 X1 X2 x + 3 X1 X2 x - 2 X1 x + 3 X2 x - 2 X2 x - x 2 10 2 10 11 12 2 11 12 - X1 X2 x + X1 X2 x + 2 X1 X2 x - X1 x + X2 x - 3 X2 x 13 2 9 2 10 2 9 2 10 11 12 2 9 + x + X1 X2 x + X1 x + X1 X2 x - X2 x - X2 x + x - X1 x 9 10 2 9 10 9 9 7 - 4 X1 X2 x - X1 x - X2 x + X2 x + 3 X1 x + 3 X2 x + 3 X1 X2 x 9 6 7 7 5 6 6 - 2 x - 6 X1 X2 x - 3 X1 x - 2 X2 x + 7 X1 X2 x + 8 X1 x + 3 X2 x 7 4 5 5 6 3 4 + 2 x - 4 X1 X2 x - 13 X1 x - 3 X2 x - 5 x + X1 X2 x + 11 X1 x 4 5 3 4 2 3 2 / + X2 x + 10 x - 5 X1 x - 13 x + X1 x + 14 x - 11 x + 5 x - 1) / ( / 4 4 3 4 4 3 4 2 3 (-1 + x) (X1 X2 x - X1 X2 x - X1 x - X2 x + 2 X1 x + x - X1 x - x 2 + x - 2 x + 1)) and in Maple format -(X1^2*X2^3*x^14-3*X1^2*X2^2*x^14-2*X1*X2^3*x^14+X1^2*X2^2*x^13+3*X1^2*X2*x^14+ 6*X1*X2^2*x^14+X2^3*x^14-2*X1^2*X2*x^13-X1^2*x^14+X1*X2^3*x^12-2*X1*X2^2*x^13-6 *X1*X2*x^14-3*X2^2*x^14+X1^2*X2^2*x^11+X1^2*x^13-3*X1*X2^2*x^12+4*X1*X2*x^13+2* X1*x^14-X2^3*x^12+X2^2*x^13+3*X2*x^14-X1^2*X2*x^11-2*X1*X2^2*x^11+3*X1*X2*x^12-\ 2*X1*x^13+3*X2^2*x^12-2*X2*x^13-x^14-X1^2*X2*x^10+X1*X2^2*x^10+2*X1*X2*x^11-X1* x^12+X2^2*x^11-3*X2*x^12+x^13+X1^2*X2*x^9+X1^2*x^10+X1*X2^2*x^9-X2^2*x^10-X2*x^ 11+x^12-X1^2*x^9-4*X1*X2*x^9-X1*x^10-X2^2*x^9+X2*x^10+3*X1*x^9+3*X2*x^9+3*X1*X2 *x^7-2*x^9-6*X1*X2*x^6-3*X1*x^7-2*X2*x^7+7*X1*X2*x^5+8*X1*x^6+3*X2*x^6+2*x^7-4* X1*X2*x^4-13*X1*x^5-3*X2*x^5-5*x^6+X1*X2*x^3+11*X1*x^4+X2*x^4+10*x^5-5*X1*x^3-\ 13*x^4+X1*x^2+14*x^3-11*x^2+5*x-1)/(-1+x)^4/(X1*X2*x^4-X1*X2*x^3-X1*x^4-X2*x^4+ 2*X1*x^3+x^4-X1*x^2-x^3+x^2-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 1, 2, 2], are 47 7 n - 7/8 + n/8, and , - -- + ---, respectively, while 64 64 Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 1, 1, 1], are 11 3 n - 7/8 + n/8, and , - -- + ---, respectively, while the asymptotic correlatio\ 64 64 n between these two random variables is 1/2 1/2 3 7 --------- 21 and in floating point 0.2182178903 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 21 ate normal pair with correlation, ----- 21 1/2 21 23 i.e. , [[-----, 0], [0, --]] 21 21 ------------------------------------------------- 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, 3, 1], nor the composition, [1, 2, 1, 2] Then infinity ----- 9 7 6 5 4 3 2 \ n x - 2 x + 4 x - 7 x + 10 x - 10 x + 8 x - 4 x + 1 ) a(n) x = -------------------------------------------------------- / 2 3 3 ----- (x - x + 1) (x + x - 1) (-1 + x) n = 0 and in Maple format (x^9-2*x^7+4*x^6-7*x^5+10*x^4-10*x^3+8*x^2-4*x+1)/(x^2-x+1)/(x^3+x-1)/(-1+x)^3 The asymptotic expression for a(n) is, n 6.0594382353848054548 1.4655712318767680267 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 1, 3, 1] and d occurrences (as containment) of the composition, [1, 2, 1, 2], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 11 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 11 2 11 2 2 9 2 10 2 11 - 2 X1 X2 x - 2 X1 X2 x - 2 X1 X2 x - X1 X2 x + X1 x 2 10 11 2 11 2 2 8 2 9 + X1 X2 x + 4 X1 X2 x + X2 x + 2 X1 X2 x + 4 X1 X2 x 2 10 2 9 11 2 10 11 2 2 7 + X1 x + 5 X1 X2 x - 2 X1 x - X2 x - 2 X2 x - X1 X2 x 2 8 2 9 2 8 9 10 2 9 - 3 X1 X2 x - 2 X1 x - 8 X1 X2 x - 10 X1 X2 x - X1 x - 3 X2 x 10 11 2 7 2 8 2 7 8 9 + X2 x + x + X1 X2 x + X1 x + 6 X1 X2 x + 12 X1 X2 x + 5 X1 x 2 8 9 2 6 7 8 2 7 + 7 X2 x + 6 X2 x - 2 X1 X2 x - 9 X1 X2 x - 4 X1 x - 7 X2 x 8 9 6 7 2 6 7 8 - 11 X2 x - 3 x + 4 X1 X2 x + 3 X1 x + 4 X2 x + 12 X2 x + 4 x 5 6 2 5 6 7 4 5 + X1 X2 x - 3 X1 x - X2 x - 9 X2 x - 5 x - 2 X1 X2 x + X1 x 5 6 3 4 5 3 4 2 + 2 X2 x + 6 x + X1 X2 x + 3 X2 x - 3 x - 3 X2 x - 2 x + X2 x 3 2 / 2 + 6 x - 7 x + 4 x - 1) / ((-1 + x) / 4 4 4 3 4 3 4 3 (X1 X2 x - X1 x - X2 x + X2 x + x - x + x - 1) (X1 X2 x - X1 X2 x 4 4 3 4 2 3 2 - X1 x - X2 x + 2 X2 x + x - X2 x - x + x - 2 x + 1)) and in Maple format (X1^2*X2^2*x^11-2*X1^2*X2*x^11-2*X1*X2^2*x^11-2*X1^2*X2^2*x^9-X1^2*X2*x^10+X1^2 *x^11+X1*X2^2*x^10+4*X1*X2*x^11+X2^2*x^11+2*X1^2*X2^2*x^8+4*X1^2*X2*x^9+X1^2*x^ 10+5*X1*X2^2*x^9-2*X1*x^11-X2^2*x^10-2*X2*x^11-X1^2*X2^2*x^7-3*X1^2*X2*x^8-2*X1 ^2*x^9-8*X1*X2^2*x^8-10*X1*X2*x^9-X1*x^10-3*X2^2*x^9+X2*x^10+x^11+X1^2*X2*x^7+ X1^2*x^8+6*X1*X2^2*x^7+12*X1*X2*x^8+5*X1*x^9+7*X2^2*x^8+6*X2*x^9-2*X1*X2^2*x^6-\ 9*X1*X2*x^7-4*X1*x^8-7*X2^2*x^7-11*X2*x^8-3*x^9+4*X1*X2*x^6+3*X1*x^7+4*X2^2*x^6 +12*X2*x^7+4*x^8+X1*X2*x^5-3*X1*x^6-X2^2*x^5-9*X2*x^6-5*x^7-2*X1*X2*x^4+X1*x^5+ 2*X2*x^5+6*x^6+X1*X2*x^3+3*X2*x^4-3*x^5-3*X2*x^3-2*x^4+X2*x^2+6*x^3-7*x^2+4*x-1 )/(-1+x)^2/(X1*X2*x^4-X1*x^4-X2*x^4+X2*x^3+x^4-x^3+x-1)/(X1*X2*x^4-X1*X2*x^3-X1 *x^4-X2*x^4+2*X2*x^3+x^4-X2*x^2-x^3+x^2-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 1, 3, 1], are 11 3 n - 7/8 + n/8, and , - -- + ---, respectively, while 64 64 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 2, 1, 2], are 55 7 n - 7/8 + n/8, and , - -- + ---, respectively, while the asymptotic correlatio\ 64 64 n between these two random variables is 1/2 1/2 3 7 --------- 21 and in floating point 0.2182178903 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 21 ate normal pair with correlation, ----- 21 1/2 21 23 i.e. , [[-----, 0], [0, --]] 21 21 ------------------------------------------------- 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, 1, 3, 1], nor the composition, [1, 2, 2, 1] Then infinity ----- 5 4 3 2 \ n x - 3 x + 3 x - 4 x + 3 x - 1 ) a(n) x = - --------------------------------- / 3 3 ----- (x + x - 1) (-1 + x) n = 0 and in Maple format -(x^5-3*x^4+3*x^3-4*x^2+3*x-1)/(x^3+x-1)/(-1+x)^3 The asymptotic expression for a(n) is, n 6.9556233073109364759 1.4655712318767680267 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 1, 3, 1] and d occurrences (as containment) of the composition, [1, 2, 2, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 5 ) | ) | ) A(n, c, d) x X1 X2 || = - (2 X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 4 5 5 3 4 4 5 3 - 2 X1 X2 x - X1 x - 2 X2 x + X1 X2 x + X1 x + 4 X2 x + x - 3 X2 x 4 2 3 2 / 2 4 - 3 x + X2 x + 3 x - 4 x + 3 x - 1) / ((-1 + x) (X1 X2 x / 3 4 4 3 4 2 3 2 - X1 X2 x - X1 x - X2 x + 2 X2 x + x - X2 x - x + x - 2 x + 1)) and in Maple format -(2*X1*X2*x^5-2*X1*X2*x^4-X1*x^5-2*X2*x^5+X1*X2*x^3+X1*x^4+4*X2*x^4+x^5-3*X2*x^ 3-3*x^4+X2*x^2+3*x^3-4*x^2+3*x-1)/(-1+x)^2/(X1*X2*x^4-X1*X2*x^3-X1*x^4-X2*x^4+2 *X2*x^3+x^4-X2*x^2-x^3+x^2-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 1, 3, 1], are 11 3 n - 7/8 + n/8, and , - -- + ---, respectively, while 64 64 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 2, 2, 1], are 47 7 n - 7/8 + n/8, and , - -- + ---, respectively, while the asymptotic correlatio\ 64 64 n between these two random variables is 1/2 1/2 3 7 --------- 21 and in floating point 0.2182178903 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 21 ate normal pair with correlation, ----- 21 1/2 21 23 i.e. , [[-----, 0], [0, --]] 21 21 ------------------------------------------------- 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, 1, 3, 1], nor the composition, [1, 3, 1, 1] Then infinity ----- 7 6 5 4 2 \ n x + x - x + x - 3 x + 3 x - 1 ) a(n) x = - ---------------------------------- / 2 3 ----- (x + x - 1) (-1 + x) n = 0 and in Maple format -(x^7+x^6-x^5+x^4-3*x^2+3*x-1)/(x^2+x-1)/(-1+x)^3 The asymptotic expression for a(n) is, n 1.8944271909999158785 1.6180339887498948482 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 1, 3, 1] and d occurrences (as containment) of the composition, [1, 3, 1, 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 7 5 6 6 7 4 6 3 - X1 x - X2 x + X1 X2 x - X1 x - X2 x + x - X1 X2 x + x + X1 X2 x 5 4 2 / 2 3 3 - x + x - 3 x + 3 x - 1) / ((-1 + x) (X1 X2 x - x + 2 x - 1)) / and in Maple format (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-X1*X2*x^4+x^6+X1 *X2*x^3-x^5+x^4-3*x^2+3*x-1)/(-1+x)^2/(X1*X2*x^3-x^3+2*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 1, 3, 1], are 11 3 n - 7/8 + n/8, and , - -- + ---, respectively, while 64 64 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 3, 1, 1], are 11 3 n - 7/8 + n/8, and , - -- + ---, respectively, while the asymptotic correlatio\ 64 64 n between these two random variables is 1 and in floating point 1. ------------------------------------------------- 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, 1, 3, 1], nor the composition, [2, 1, 1, 2] Then infinity ----- \ n 15 13 12 11 10 9 8 7 6 ) a(n) x = (x + 3 x - 4 x + x - 5 x + 3 x - x + 4 x - 4 x / ----- n = 0 5 4 3 2 / 3 9 4 + 4 x - 4 x + 3 x - 4 x + 3 x - 1) / ((x + x - 1) (x - x - 1) / 3 (-1 + x) ) and in Maple format (x^15+3*x^13-4*x^12+x^11-5*x^10+3*x^9-x^8+4*x^7-4*x^6+4*x^5-4*x^4+3*x^3-4*x^2+3 *x-1)/(x^3+x-1)/(x^9-x^4-1)/(-1+x)^3 The asymptotic expression for a(n) is, n 5.2787205554750117565 1.4655712318767680267 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 1, 3, 1] and d occurrences (as containment) of the composition, [2, 1, 1, 2], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 4 4 18 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 4 4 17 4 3 18 3 4 18 4 4 16 - 2 X1 X2 x - 4 X1 X2 x - 4 X1 X2 x + 2 X1 X2 x 4 3 17 4 2 18 3 4 17 3 3 18 + 6 X1 X2 x + 6 X1 X2 x + 11 X1 X2 x + 16 X1 X2 x 2 4 18 4 4 15 4 3 16 4 2 17 + 6 X1 X2 x - X1 X2 x - 6 X1 X2 x - 6 X1 X2 x 4 18 3 4 16 3 3 17 3 2 18 - 4 X1 X2 x - 15 X1 X2 x - 36 X1 X2 x - 24 X1 X2 x 2 4 17 2 3 18 4 18 4 3 15 - 21 X1 X2 x - 24 X1 X2 x - 4 X1 X2 x + 3 X1 X2 x 4 2 16 4 17 4 18 3 4 15 3 3 16 + 6 X1 X2 x + 2 X1 X2 x + X1 x + 12 X1 X2 x + 48 X1 X2 x 3 2 17 3 18 2 4 16 2 3 17 + 42 X1 X2 x + 16 X1 X2 x + 36 X1 X2 x + 72 X1 X2 x 2 2 18 4 17 3 18 4 18 4 2 15 + 36 X1 X2 x + 17 X1 X2 x + 16 X1 X2 x + X2 x - 3 X1 X2 x 4 16 3 4 14 3 3 15 3 2 16 - 2 X1 X2 x - 4 X1 X2 x - 42 X1 X2 x - 54 X1 X2 x 3 17 3 18 2 4 15 2 3 16 - 20 X1 X2 x - 4 X1 x - 39 X1 X2 x - 120 X1 X2 x 2 2 17 2 18 4 16 3 17 - 90 X1 X2 x - 24 X1 X2 x - 35 X1 X2 x - 60 X1 X2 x 2 18 4 17 3 18 4 15 3 3 14 - 24 X1 X2 x - 5 X2 x - 4 X2 x + X1 X2 x + 17 X1 X2 x 3 2 15 3 16 3 17 2 4 14 + 54 X1 X2 x + 24 X1 X2 x + 3 X1 x + 24 X1 X2 x 2 3 15 2 2 16 2 17 2 18 + 142 X1 X2 x + 144 X1 X2 x + 48 X1 X2 x + 6 X1 x 4 15 3 16 2 17 18 + 47 X1 X2 x + 120 X1 X2 x + 78 X1 X2 x + 16 X1 X2 x 4 16 3 17 2 18 3 3 13 3 2 14 + 12 X2 x + 18 X2 x + 6 X2 x - 5 X1 X2 x - 26 X1 X2 x 3 15 3 16 2 4 13 2 3 14 - 30 X1 X2 x - 3 X1 x - 6 X1 X2 x - 101 X1 X2 x 2 2 15 2 16 2 17 4 14 - 192 X1 X2 x - 72 X1 X2 x - 9 X1 x - 41 X1 X2 x 3 15 2 16 17 18 4 15 - 174 X1 X2 x - 150 X1 X2 x - 44 X1 X2 x - 4 X1 x - 19 X2 x 3 16 2 17 18 3 3 12 3 2 13 - 42 X2 x - 24 X2 x - 4 X2 x + 5 X1 X2 x + 11 X1 X2 x 3 14 3 15 2 3 13 2 2 14 + 16 X1 X2 x + 6 X1 x + 46 X1 X2 x + 156 X1 X2 x 2 15 2 16 4 13 3 14 + 114 X1 X2 x + 12 X1 x + 20 X1 X2 x + 171 X1 X2 x 2 15 16 17 4 14 3 15 + 240 X1 X2 x + 80 X1 X2 x + 9 X1 x + 21 X2 x + 71 X2 x 2 16 17 18 3 3 11 3 2 12 + 54 X2 x + 14 X2 x + x - 3 X1 X2 x - 9 X1 X2 x 3 13 3 14 2 3 12 2 2 13 - 6 X1 X2 x - 3 X1 x - 30 X1 X2 x - 89 X1 X2 x 2 14 2 15 4 12 3 13 - 102 X1 X2 x - 25 X1 x - 4 X1 X2 x - 111 X1 X2 x 2 14 15 16 4 13 3 14 - 264 X1 X2 x - 146 X1 X2 x - 15 X1 x - 15 X2 x - 87 X2 x 2 15 16 17 3 3 10 3 2 11 - 99 X2 x - 30 X2 x - 3 x + X1 X2 x + 4 X1 X2 x 3 12 2 3 11 2 2 12 2 13 + 4 X1 X2 x + 24 X1 X2 x + 67 X1 X2 x + 61 X1 X2 x 2 14 3 12 2 13 14 15 + 23 X1 x + 68 X1 X2 x + 199 X1 X2 x + 176 X1 X2 x + 33 X1 x 4 12 3 13 2 14 15 16 3 2 10 + 6 X2 x + 74 X2 x + 134 X2 x + 61 X2 x + 6 x - X1 X2 x 3 11 2 3 10 2 2 11 2 12 - X1 X2 x - 12 X1 X2 x - 50 X1 X2 x - 49 X1 X2 x 2 13 3 11 2 12 13 14 - 12 X1 x - 54 X1 X2 x - 152 X1 X2 x - 142 X1 X2 x - 42 X1 x 4 11 3 12 2 13 14 15 2 3 9 - X2 x - 51 X2 x - 127 X2 x - 90 X2 x - 14 x + 3 X1 X2 x 2 2 10 2 11 2 12 3 10 + 28 X1 X2 x + 35 X1 X2 x + 12 X1 x + 36 X1 X2 x 2 11 12 13 3 11 2 12 + 132 X1 X2 x + 123 X1 X2 x + 34 X1 x + 39 X2 x + 106 X2 x 13 14 2 2 9 2 10 2 11 + 91 X2 x + 22 x - 14 X1 X2 x - 18 X1 X2 x - 9 X1 x 3 9 2 10 11 12 3 10 - 15 X1 X2 x - 99 X1 X2 x - 110 X1 X2 x - 35 X1 x - 30 X2 x 2 11 12 13 2 2 8 2 9 - 98 X2 x - 86 X2 x - 23 x + 5 X1 X2 x + 11 X1 X2 x 2 10 3 8 2 9 10 11 + 2 X1 x + 3 X1 X2 x + 65 X1 X2 x + 81 X1 X2 x + 32 X1 x 3 9 2 10 11 12 2 2 7 2 8 + 17 X2 x + 87 X2 x + 86 X2 x + 25 x - X1 X2 x - 5 X1 X2 x 2 8 9 10 3 8 2 9 - 35 X1 X2 x - 65 X1 X2 x - 18 X1 x - 6 X2 x - 68 X2 x 10 11 2 7 2 7 8 9 - 78 X2 x - 26 x + X1 X2 x + 12 X1 X2 x + 51 X1 X2 x + 15 X1 x 3 7 2 8 9 10 2 6 7 + X2 x + 46 X2 x + 73 X2 x + 21 x - 2 X1 X2 x - 32 X1 X2 x 8 2 7 8 9 6 7 - 18 X1 x - 23 X2 x - 70 X2 x - 22 x + 20 X1 X2 x + 16 X1 x 2 6 7 8 5 6 2 5 6 + 7 X2 x + 58 X2 x + 29 x - 12 X1 X2 x - 11 X1 x - X2 x - 44 X2 x 7 4 5 5 6 3 4 - 32 x + 5 X1 X2 x + 5 X1 x + 31 X2 x + 31 x - X1 X2 x - X1 x 4 5 3 4 2 3 2 - 17 X2 x - 29 x + 6 X2 x + 28 x - X2 x - 25 x + 16 x - 6 x + 1) / 2 4 3 4 4 3 4 2 / ((-1 + x) (X1 X2 x - X1 X2 x - X1 x - X2 x + 2 X2 x + x - X2 x / 3 2 3 3 12 3 2 12 2 3 12 - x + x - 2 x + 1) (X1 X2 x - 3 X1 X2 x - 3 X1 X2 x 3 12 2 3 11 2 2 12 3 12 3 12 + 3 X1 X2 x + 3 X1 X2 x + 9 X1 X2 x + 3 X1 X2 x - X1 x 2 2 11 2 12 3 11 2 12 3 12 - 9 X1 X2 x - 9 X1 X2 x - 6 X1 X2 x - 9 X1 X2 x - X2 x 2 11 2 12 3 10 2 11 12 + 9 X1 X2 x + 3 X1 x + 3 X1 X2 x + 18 X1 X2 x + 9 X1 X2 x 3 11 2 12 2 11 2 10 11 + 3 X2 x + 3 X2 x - 3 X1 x - 9 X1 X2 x - 18 X1 X2 x 12 3 10 2 11 12 2 2 8 10 - 3 X1 x - 3 X2 x - 9 X2 x - 3 X2 x + X1 X2 x + 9 X1 X2 x 11 3 9 2 10 11 12 2 2 7 2 8 + 6 X1 x + X2 x + 9 X2 x + 9 X2 x + x - X1 X2 x - X1 X2 x 2 8 10 2 9 10 11 2 7 - 2 X1 X2 x - 3 X1 x - 3 X2 x - 9 X2 x - 3 x + X1 X2 x 2 7 8 2 8 9 10 2 6 + 4 X1 X2 x + 2 X1 X2 x + X2 x + 3 X2 x + 3 x - 2 X1 X2 x 7 2 7 8 9 6 7 2 6 - 5 X1 X2 x - 3 X2 x - X2 x - x + 4 X1 X2 x + X1 x + 3 X2 x 7 5 6 2 5 6 7 5 5 + 4 X2 x - X1 X2 x - 2 X1 x - X2 x - 6 X2 x - x + X1 x + 4 X2 x 6 4 5 4 3 2 + 3 x - X2 x - 3 x + x - x + 3 x - 3 x + 1)) and in Maple format (X1^4*X2^4*x^18-2*X1^4*X2^4*x^17-4*X1^4*X2^3*x^18-4*X1^3*X2^4*x^18+2*X1^4*X2^4* x^16+6*X1^4*X2^3*x^17+6*X1^4*X2^2*x^18+11*X1^3*X2^4*x^17+16*X1^3*X2^3*x^18+6*X1 ^2*X2^4*x^18-X1^4*X2^4*x^15-6*X1^4*X2^3*x^16-6*X1^4*X2^2*x^17-4*X1^4*X2*x^18-15 *X1^3*X2^4*x^16-36*X1^3*X2^3*x^17-24*X1^3*X2^2*x^18-21*X1^2*X2^4*x^17-24*X1^2* X2^3*x^18-4*X1*X2^4*x^18+3*X1^4*X2^3*x^15+6*X1^4*X2^2*x^16+2*X1^4*X2*x^17+X1^4* x^18+12*X1^3*X2^4*x^15+48*X1^3*X2^3*x^16+42*X1^3*X2^2*x^17+16*X1^3*X2*x^18+36* X1^2*X2^4*x^16+72*X1^2*X2^3*x^17+36*X1^2*X2^2*x^18+17*X1*X2^4*x^17+16*X1*X2^3*x ^18+X2^4*x^18-3*X1^4*X2^2*x^15-2*X1^4*X2*x^16-4*X1^3*X2^4*x^14-42*X1^3*X2^3*x^ 15-54*X1^3*X2^2*x^16-20*X1^3*X2*x^17-4*X1^3*x^18-39*X1^2*X2^4*x^15-120*X1^2*X2^ 3*x^16-90*X1^2*X2^2*x^17-24*X1^2*X2*x^18-35*X1*X2^4*x^16-60*X1*X2^3*x^17-24*X1* X2^2*x^18-5*X2^4*x^17-4*X2^3*x^18+X1^4*X2*x^15+17*X1^3*X2^3*x^14+54*X1^3*X2^2*x ^15+24*X1^3*X2*x^16+3*X1^3*x^17+24*X1^2*X2^4*x^14+142*X1^2*X2^3*x^15+144*X1^2* X2^2*x^16+48*X1^2*X2*x^17+6*X1^2*x^18+47*X1*X2^4*x^15+120*X1*X2^3*x^16+78*X1*X2 ^2*x^17+16*X1*X2*x^18+12*X2^4*x^16+18*X2^3*x^17+6*X2^2*x^18-5*X1^3*X2^3*x^13-26 *X1^3*X2^2*x^14-30*X1^3*X2*x^15-3*X1^3*x^16-6*X1^2*X2^4*x^13-101*X1^2*X2^3*x^14 -192*X1^2*X2^2*x^15-72*X1^2*X2*x^16-9*X1^2*x^17-41*X1*X2^4*x^14-174*X1*X2^3*x^ 15-150*X1*X2^2*x^16-44*X1*X2*x^17-4*X1*x^18-19*X2^4*x^15-42*X2^3*x^16-24*X2^2*x ^17-4*X2*x^18+5*X1^3*X2^3*x^12+11*X1^3*X2^2*x^13+16*X1^3*X2*x^14+6*X1^3*x^15+46 *X1^2*X2^3*x^13+156*X1^2*X2^2*x^14+114*X1^2*X2*x^15+12*X1^2*x^16+20*X1*X2^4*x^ 13+171*X1*X2^3*x^14+240*X1*X2^2*x^15+80*X1*X2*x^16+9*X1*x^17+21*X2^4*x^14+71*X2 ^3*x^15+54*X2^2*x^16+14*X2*x^17+x^18-3*X1^3*X2^3*x^11-9*X1^3*X2^2*x^12-6*X1^3* X2*x^13-3*X1^3*x^14-30*X1^2*X2^3*x^12-89*X1^2*X2^2*x^13-102*X1^2*X2*x^14-25*X1^ 2*x^15-4*X1*X2^4*x^12-111*X1*X2^3*x^13-264*X1*X2^2*x^14-146*X1*X2*x^15-15*X1*x^ 16-15*X2^4*x^13-87*X2^3*x^14-99*X2^2*x^15-30*X2*x^16-3*x^17+X1^3*X2^3*x^10+4*X1 ^3*X2^2*x^11+4*X1^3*X2*x^12+24*X1^2*X2^3*x^11+67*X1^2*X2^2*x^12+61*X1^2*X2*x^13 +23*X1^2*x^14+68*X1*X2^3*x^12+199*X1*X2^2*x^13+176*X1*X2*x^14+33*X1*x^15+6*X2^4 *x^12+74*X2^3*x^13+134*X2^2*x^14+61*X2*x^15+6*x^16-X1^3*X2^2*x^10-X1^3*X2*x^11-\ 12*X1^2*X2^3*x^10-50*X1^2*X2^2*x^11-49*X1^2*X2*x^12-12*X1^2*x^13-54*X1*X2^3*x^ 11-152*X1*X2^2*x^12-142*X1*X2*x^13-42*X1*x^14-X2^4*x^11-51*X2^3*x^12-127*X2^2*x ^13-90*X2*x^14-14*x^15+3*X1^2*X2^3*x^9+28*X1^2*X2^2*x^10+35*X1^2*X2*x^11+12*X1^ 2*x^12+36*X1*X2^3*x^10+132*X1*X2^2*x^11+123*X1*X2*x^12+34*X1*x^13+39*X2^3*x^11+ 106*X2^2*x^12+91*X2*x^13+22*x^14-14*X1^2*X2^2*x^9-18*X1^2*X2*x^10-9*X1^2*x^11-\ 15*X1*X2^3*x^9-99*X1*X2^2*x^10-110*X1*X2*x^11-35*X1*x^12-30*X2^3*x^10-98*X2^2*x ^11-86*X2*x^12-23*x^13+5*X1^2*X2^2*x^8+11*X1^2*X2*x^9+2*X1^2*x^10+3*X1*X2^3*x^8 +65*X1*X2^2*x^9+81*X1*X2*x^10+32*X1*x^11+17*X2^3*x^9+87*X2^2*x^10+86*X2*x^11+25 *x^12-X1^2*X2^2*x^7-5*X1^2*X2*x^8-35*X1*X2^2*x^8-65*X1*X2*x^9-18*X1*x^10-6*X2^3 *x^8-68*X2^2*x^9-78*X2*x^10-26*x^11+X1^2*X2*x^7+12*X1*X2^2*x^7+51*X1*X2*x^8+15* X1*x^9+X2^3*x^7+46*X2^2*x^8+73*X2*x^9+21*x^10-2*X1*X2^2*x^6-32*X1*X2*x^7-18*X1* x^8-23*X2^2*x^7-70*X2*x^8-22*x^9+20*X1*X2*x^6+16*X1*x^7+7*X2^2*x^6+58*X2*x^7+29 *x^8-12*X1*X2*x^5-11*X1*x^6-X2^2*x^5-44*X2*x^6-32*x^7+5*X1*X2*x^4+5*X1*x^5+31* X2*x^5+31*x^6-X1*X2*x^3-X1*x^4-17*X2*x^4-29*x^5+6*X2*x^3+28*x^4-X2*x^2-25*x^3+ 16*x^2-6*x+1)/(-1+x)^2/(X1*X2*x^4-X1*X2*x^3-X1*x^4-X2*x^4+2*X2*x^3+x^4-X2*x^2-x ^3+x^2-2*x+1)/(X1^3*X2^3*x^12-3*X1^3*X2^2*x^12-3*X1^2*X2^3*x^12+3*X1^3*X2*x^12+ 3*X1^2*X2^3*x^11+9*X1^2*X2^2*x^12+3*X1*X2^3*x^12-X1^3*x^12-9*X1^2*X2^2*x^11-9* X1^2*X2*x^12-6*X1*X2^3*x^11-9*X1*X2^2*x^12-X2^3*x^12+9*X1^2*X2*x^11+3*X1^2*x^12 +3*X1*X2^3*x^10+18*X1*X2^2*x^11+9*X1*X2*x^12+3*X2^3*x^11+3*X2^2*x^12-3*X1^2*x^ 11-9*X1*X2^2*x^10-18*X1*X2*x^11-3*X1*x^12-3*X2^3*x^10-9*X2^2*x^11-3*X2*x^12+X1^ 2*X2^2*x^8+9*X1*X2*x^10+6*X1*x^11+X2^3*x^9+9*X2^2*x^10+9*X2*x^11+x^12-X1^2*X2^2 *x^7-X1^2*X2*x^8-2*X1*X2^2*x^8-3*X1*x^10-3*X2^2*x^9-9*X2*x^10-3*x^11+X1^2*X2*x^ 7+4*X1*X2^2*x^7+2*X1*X2*x^8+X2^2*x^8+3*X2*x^9+3*x^10-2*X1*X2^2*x^6-5*X1*X2*x^7-\ 3*X2^2*x^7-X2*x^8-x^9+4*X1*X2*x^6+X1*x^7+3*X2^2*x^6+4*X2*x^7-X1*X2*x^5-2*X1*x^6 -X2^2*x^5-6*X2*x^6-x^7+X1*x^5+4*X2*x^5+3*x^6-X2*x^4-3*x^5+x^4-x^3+3*x^2-3*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 1, 3, 1], are 11 3 n - 7/8 + n/8, and , - -- + ---, respectively, while 64 64 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 1, 1, 2], are 63 7 n - 7/8 + n/8, and , - -- + ---, respectively, while the asymptotic correlatio\ 64 64 n between these two random variables is 1/2 1/2 3 7 --------- 21 and in floating point 0.2182178903 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 21 ate normal pair with correlation, ----- 21 1/2 21 23 i.e. , [[-----, 0], [0, --]] 21 21 ------------------------------------------------- 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, 1, 3, 1], nor the composition, [2, 1, 2, 1] Then infinity ----- 9 8 7 6 5 3 2 \ n x - 3 x + 3 x - 3 x + 3 x - 2 x + 4 x - 3 x + 1 ) a(n) x = ------------------------------------------------------ / 2 3 3 ----- (x + 1) (x - x + 1) (x + x - 1) (-1 + x) n = 0 and in Maple format (x^9-3*x^8+3*x^7-3*x^6+3*x^5-2*x^3+4*x^2-3*x+1)/(x+1)/(x^2-x+1)/(x^3+x-1)/(-1+x )^3 The asymptotic expression for a(n) is, n 6.0594382353848054551 1.4655712318767680267 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 1, 3, 1] and d occurrences (as containment) of the composition, [2, 1, 2, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 9 2 2 8 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x - X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 9 2 9 2 2 7 2 8 2 9 - 2 X1 X2 x - 2 X1 X2 x + X1 X2 x + 2 X1 X2 x + X1 x 2 8 9 2 9 2 7 2 8 2 7 + 4 X1 X2 x + 4 X1 X2 x + X2 x - X1 X2 x - X1 x - 4 X1 X2 x 8 9 2 8 9 2 6 7 - 8 X1 X2 x - 2 X1 x - 3 X2 x - 2 X2 x + 2 X1 X2 x + 6 X1 X2 x 8 2 7 8 9 6 7 2 6 + 4 X1 x + 4 X2 x + 6 X2 x + x - 4 X1 X2 x - 2 X1 x - 3 X2 x 7 8 6 2 5 6 7 4 5 - 7 X2 x - 3 x + 2 X1 x + X2 x + 6 X2 x + 3 x + X1 X2 x - X1 x 5 6 3 4 5 3 2 3 2 - 3 X2 x - 3 x - X1 X2 x - X2 x + 3 x + 2 X2 x - X2 x - 2 x + 4 x / - 3 x + 1) / ((-1 + x) / 4 4 4 3 4 3 4 3 (X1 X2 x - X1 x - X2 x + X2 x + x - x + x - 1) (X1 X2 x - X1 X2 x 4 4 3 4 2 3 2 - X1 x - X2 x + 2 X2 x + x - X2 x - x + x - 2 x + 1)) and in Maple format (X1^2*X2^2*x^9-X1^2*X2^2*x^8-2*X1^2*X2*x^9-2*X1*X2^2*x^9+X1^2*X2^2*x^7+2*X1^2* X2*x^8+X1^2*x^9+4*X1*X2^2*x^8+4*X1*X2*x^9+X2^2*x^9-X1^2*X2*x^7-X1^2*x^8-4*X1*X2 ^2*x^7-8*X1*X2*x^8-2*X1*x^9-3*X2^2*x^8-2*X2*x^9+2*X1*X2^2*x^6+6*X1*X2*x^7+4*X1* x^8+4*X2^2*x^7+6*X2*x^8+x^9-4*X1*X2*x^6-2*X1*x^7-3*X2^2*x^6-7*X2*x^7-3*x^8+2*X1 *x^6+X2^2*x^5+6*X2*x^6+3*x^7+X1*X2*x^4-X1*x^5-3*X2*x^5-3*x^6-X1*X2*x^3-X2*x^4+3 *x^5+2*X2*x^3-X2*x^2-2*x^3+4*x^2-3*x+1)/(-1+x)/(X1*X2*x^4-X1*x^4-X2*x^4+X2*x^3+ x^4-x^3+x-1)/(X1*X2*x^4-X1*X2*x^3-X1*x^4-X2*x^4+2*X2*x^3+x^4-X2*x^2-x^3+x^2-2*x +1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 1, 3, 1], are 11 3 n - 7/8 + n/8, and , - -- + ---, respectively, while 64 64 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 1, 2, 1], are 55 7 n - 7/8 + n/8, and , - -- + ---, respectively, while the asymptotic correlatio\ 64 64 n between these two random variables is 1/2 1/2 3 7 --------- 21 and in floating point 0.2182178903 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 21 ate normal pair with correlation, ----- 21 1/2 21 23 i.e. , [[-----, 0], [0, --]] 21 21 ------------------------------------------------- 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, 1, 3, 1], nor the composition, [2, 2, 1, 1] Then infinity ----- 8 7 5 4 3 2 \ n x + x + x - 3 x + 3 x - 4 x + 3 x - 1 ) a(n) x = - ------------------------------------------- / 3 3 ----- (x + x - 1) (-1 + x) n = 0 and in Maple format -(x^8+x^7+x^5-3*x^4+3*x^3-4*x^2+3*x-1)/(x^3+x-1)/(-1+x)^3 The asymptotic expression for a(n) is, n 4.7460151755324556560 1.4655712318767680267 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 1, 3, 1] and d occurrences (as containment) of the composition, [2, 2, 1, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 9 2 9 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x - X1 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 9 9 9 7 9 6 7 - 2 X1 X2 x + 2 X1 x + X2 x + X1 X2 x - x - 2 X1 X2 x - X1 x 7 5 6 6 7 4 5 - X2 x + 4 X1 X2 x + X1 x + 2 X2 x + x - 3 X1 X2 x - 2 X1 x 5 6 3 4 4 5 3 4 - 6 X2 x - x + X1 X2 x + X1 x + 7 X2 x + 4 x - 4 X2 x - 6 x 2 3 2 / 3 4 3 + X2 x + 7 x - 7 x + 4 x - 1) / ((-1 + x) (X1 X2 x - X1 X2 x / 4 4 3 4 2 3 2 - X1 x - X2 x + 2 X2 x + x - X2 x - x + x - 2 x + 1)) and in Maple format (X1^2*X2*x^9-X1^2*x^9-2*X1*X2*x^9+2*X1*x^9+X2*x^9+X1*X2*x^7-x^9-2*X1*X2*x^6-X1* x^7-X2*x^7+4*X1*X2*x^5+X1*x^6+2*X2*x^6+x^7-3*X1*X2*x^4-2*X1*x^5-6*X2*x^5-x^6+X1 *X2*x^3+X1*x^4+7*X2*x^4+4*x^5-4*X2*x^3-6*x^4+X2*x^2+7*x^3-7*x^2+4*x-1)/(-1+x)^3 /(X1*X2*x^4-X1*X2*x^3-X1*x^4-X2*x^4+2*X2*x^3+x^4-X2*x^2-x^3+x^2-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 1, 3, 1], are 11 3 n - 7/8 + n/8, and , - -- + ---, respectively, while 64 64 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 2, 1, 1], are 47 7 n - 7/8 + n/8, and , - -- + ---, respectively, while the asymptotic correlatio\ 64 64 n between these two random variables is 1/2 1/2 3 7 --------- 21 and in floating point 0.2182178903 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 21 ate normal pair with correlation, ----- 21 1/2 21 23 i.e. , [[-----, 0], [0, --]] 21 21 ------------------------------------------------- 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, 1, 3, 1], nor the composition, [3, 1, 1, 1] Then infinity ----- 11 10 8 7 6 5 4 2 \ n x + 2 x - 2 x - 2 x - x + x - x + 3 x - 3 x + 1 ) a(n) x = --------------------------------------------------------- / 2 3 ----- (x + x - 1) (-1 + x) n = 0 and in Maple format (x^11+2*x^10-2*x^8-2*x^7-x^6+x^5-x^4+3*x^2-3*x+1)/(x^2+x-1)/(-1+x)^3 The asymptotic expression for a(n) is, n 1.1708203932499369092 1.6180339887498948482 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 1, 3, 1] and d occurrences (as containment) of the composition, [3, 1, 1, 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 2 11 2 12 2 12 2 11 2 12 + X1 X2 x - 2 X1 X2 x - 2 X1 X2 x - 2 X1 X2 x + X1 x 2 11 12 2 12 2 10 2 11 2 10 - 2 X1 X2 x + 4 X1 X2 x + X2 x + X1 X2 x + X1 x + X1 X2 x 11 12 2 11 12 2 9 2 10 + 4 X1 X2 x - 2 X1 x + X2 x - 2 X2 x + X1 X2 x - X1 x 2 9 10 11 2 10 11 12 2 9 + X1 X2 x - 4 X1 X2 x - 2 X1 x - X2 x - 2 X2 x + x - X1 x 9 10 2 9 10 11 9 9 - 4 X1 X2 x + 3 X1 x - X2 x + 3 X2 x + x + 3 X1 x + 3 X2 x 10 7 9 7 7 5 6 6 - 2 x + X1 X2 x - 2 x - X1 x - X2 x + 2 X1 X2 x - X1 x - X2 x 7 4 6 3 5 4 3 2 + x - 2 X1 X2 x + 2 x + X1 X2 x - 2 x + x + 3 x - 6 x + 4 x - 1) / 3 3 3 / ((-1 + x) (X1 X2 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^12-2*X1*X2^2*x^12-2*X1^2*X2*x^11+X1 ^2*x^12-2*X1*X2^2*x^11+4*X1*X2*x^12+X2^2*x^12+X1^2*X2*x^10+X1^2*x^11+X1*X2^2*x^ 10+4*X1*X2*x^11-2*X1*x^12+X2^2*x^11-2*X2*x^12+X1^2*X2*x^9-X1^2*x^10+X1*X2^2*x^9 -4*X1*X2*x^10-2*X1*x^11-X2^2*x^10-2*X2*x^11+x^12-X1^2*x^9-4*X1*X2*x^9+3*X1*x^10 -X2^2*x^9+3*X2*x^10+x^11+3*X1*x^9+3*X2*x^9-2*x^10+X1*X2*x^7-2*x^9-X1*x^7-X2*x^7 +2*X1*X2*x^5-X1*x^6-X2*x^6+x^7-2*X1*X2*x^4+2*x^6+X1*X2*x^3-2*x^5+x^4+3*x^3-6*x^ 2+4*x-1)/(-1+x)^3/(X1*X2*x^3-x^3+2*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 1, 3, 1], are 11 3 n - 7/8 + n/8, and , - -- + ---, respectively, while 64 64 Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 1, 1, 1], are 11 3 n - 7/8 + n/8, and , - -- + ---, respectively, while the asymptotic correlatio\ 64 64 n between these two random variables is 1 and in floating point 1. ------------------------------------------------- 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, 1, 2], nor the composition, [1, 2, 2, 1] Then infinity ----- 6 5 3 2 \ n x - 2 x + 2 x - 4 x + 3 x - 1 ) a(n) x = ----------------------------------- / 2 2 2 ----- (x - x + 1) (x + x - 1) (-1 + x) n = 0 and in Maple format (x^6-2*x^5+2*x^3-4*x^2+3*x-1)/(x^2-x+1)/(x^2+x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 2.1180339887498948478 1.6180339887498948482 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 2, 1, 2] and d occurrences (as containment) of the composition, [1, 2, 2, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 7 2 6 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x - X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 7 2 5 6 7 5 6 - 2 X1 X2 x + X1 X2 x + X1 X2 x + X2 x - X1 X2 x + X1 x 4 5 6 3 4 5 2 - 2 X1 X2 x - 2 X1 x - x + 2 X1 X2 x + 2 X1 x + 2 x - X1 X2 x 3 3 2 / 2 6 2 5 - X1 x - 2 x + 4 x - 3 x + 1) / ((-1 + x) (X1 X2 x - X1 X2 x / 6 5 6 4 5 3 4 - 2 X1 X2 x + X1 X2 x + X2 x + X1 X2 x + X1 x - 2 X1 X2 x - 2 X1 x 5 2 3 4 3 2 - x + X1 X2 x + X1 x + x + x - 3 x + 3 x - 1)) and in Maple format (X1^2*X2*x^7-X1^2*X2*x^6-2*X1*X2*x^7+X1^2*X2*x^5+X1*X2*x^6+X2*x^7-X1*X2*x^5+X1* x^6-2*X1*X2*x^4-2*X1*x^5-x^6+2*X1*X2*x^3+2*X1*x^4+2*x^5-X1*X2*x^2-X1*x^3-2*x^3+ 4*x^2-3*x+1)/(-1+x)/(X1^2*X2*x^6-X1^2*X2*x^5-2*X1*X2*x^6+X1*X2*x^5+X2*x^6+X1*X2 *x^4+X1*x^5-2*X1*X2*x^3-2*X1*x^4-x^5+X1*X2*x^2+X1*x^3+x^4+x^3-3*x^2+3*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 2, 1, 2], are 55 7 n - 7/8 + n/8, and , - -- + ---, respectively, while 64 64 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 2, 2, 1], are 47 7 n - 7/8 + n/8, and , - -- + ---, respectively, while the asymptotic correlatio\ 64 64 n between these two random variables is 5/7 and in floating point 0.7142857143 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ ate normal pair with correlation, 5/7 99 i.e. , [[5/7, 0], [0, --]] 49 ------------------------------------------------- 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, 1, 2], nor the composition, [1, 3, 1, 1] Then infinity ----- 9 8 7 6 5 3 2 \ n x - 3 x + 3 x - 3 x + 3 x - 2 x + 4 x - 3 x + 1 ) a(n) x = ------------------------------------------------------ / 2 3 3 ----- (x + 1) (x - x + 1) (x + x - 1) (-1 + x) n = 0 and in Maple format (x^9-3*x^8+3*x^7-3*x^6+3*x^5-2*x^3+4*x^2-3*x+1)/(x+1)/(x^2-x+1)/(x^3+x-1)/(-1+x )^3 The asymptotic expression for a(n) is, n 6.0594382353848054551 1.4655712318767680267 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 2, 1, 2] and d occurrences (as containment) of the composition, [1, 3, 1, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 9 2 2 8 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x - X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 9 2 9 2 2 7 2 8 2 9 - 2 X1 X2 x - 2 X1 X2 x + X1 X2 x + 4 X1 X2 x + X1 x 2 8 9 2 9 2 7 2 8 2 7 + 2 X1 X2 x + 4 X1 X2 x + X2 x - 4 X1 X2 x - 3 X1 x - X1 X2 x 8 9 2 8 9 2 6 2 7 - 8 X1 X2 x - 2 X1 x - X2 x - 2 X2 x + 2 X1 X2 x + 4 X1 x 7 8 8 9 2 6 6 7 + 6 X1 X2 x + 6 X1 x + 4 X2 x + x - 3 X1 x - 4 X1 X2 x - 7 X1 x 7 8 2 5 6 6 7 4 5 - 2 X2 x - 3 x + X1 x + 6 X1 x + 2 X2 x + 3 x + X1 X2 x - 3 X1 x 5 6 3 4 5 3 2 3 2 - X2 x - 3 x - X1 X2 x - X1 x + 3 x + 2 X1 x - X1 x - 2 x + 4 x / 4 3 4 4 3 - 3 x + 1) / ((-1 + x) (X1 X2 x - X1 X2 x - X1 x - X2 x + 2 X1 x / 4 2 3 2 + x - X1 x - x + x - 2 x + 1) 4 4 4 3 4 3 (X1 X2 x - X1 x - X2 x + X1 x + x - x + x - 1)) and in Maple format (X1^2*X2^2*x^9-X1^2*X2^2*x^8-2*X1^2*X2*x^9-2*X1*X2^2*x^9+X1^2*X2^2*x^7+4*X1^2* X2*x^8+X1^2*x^9+2*X1*X2^2*x^8+4*X1*X2*x^9+X2^2*x^9-4*X1^2*X2*x^7-3*X1^2*x^8-X1* X2^2*x^7-8*X1*X2*x^8-2*X1*x^9-X2^2*x^8-2*X2*x^9+2*X1^2*X2*x^6+4*X1^2*x^7+6*X1* X2*x^7+6*X1*x^8+4*X2*x^8+x^9-3*X1^2*x^6-4*X1*X2*x^6-7*X1*x^7-2*X2*x^7-3*x^8+X1^ 2*x^5+6*X1*x^6+2*X2*x^6+3*x^7+X1*X2*x^4-3*X1*x^5-X2*x^5-3*x^6-X1*X2*x^3-X1*x^4+ 3*x^5+2*X1*x^3-X1*x^2-2*x^3+4*x^2-3*x+1)/(-1+x)/(X1*X2*x^4-X1*X2*x^3-X1*x^4-X2* x^4+2*X1*x^3+x^4-X1*x^2-x^3+x^2-2*x+1)/(X1*X2*x^4-X1*x^4-X2*x^4+X1*x^3+x^4-x^3+ x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 2, 1, 2], are 55 7 n - 7/8 + n/8, and , - -- + ---, respectively, while 64 64 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 3, 1, 1], are 11 3 n - 7/8 + n/8, and , - -- + ---, respectively, while the asymptotic correlatio\ 64 64 n between these two random variables is 1/2 1/2 3 7 --------- 21 and in floating point 0.2182178903 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 21 ate normal pair with correlation, ----- 21 1/2 21 23 i.e. , [[-----, 0], [0, --]] 21 21 ------------------------------------------------- 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, 1, 2], nor the composition, [2, 1, 1, 2] Then infinity ----- 7 6 4 3 2 \ n x - x - x + 2 x - 4 x + 3 x - 1 ) a(n) x = - --------------------------------------------- / 7 6 5 3 2 ----- (-1 + x) (x - x + x + x - 3 x + 3 x - 1) n = 0 and in Maple format -(x^7-x^6-x^4+2*x^3-4*x^2+3*x-1)/(-1+x)/(x^7-x^6+x^5+x^3-3*x^2+3*x-1) The asymptotic expression for a(n) is, n 1.5121946674937959604 1.6530424890094669421 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 2, 1, 2] and d occurrences (as containment) of the composition, [2, 1, 1, 2], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 4 11 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 3 11 2 3 10 2 2 11 2 3 9 2 2 10 - 4 X1 X2 x - X1 X2 x + 6 X1 X2 x - X1 X2 x + 3 X1 X2 x 2 11 3 10 2 3 8 2 2 9 2 10 - 4 X1 X2 x + X1 X2 x + X1 X2 x + 2 X1 X2 x - 3 X1 X2 x 2 11 3 9 2 10 2 3 7 2 2 8 2 9 + X1 x - X1 X2 x - 3 X1 X2 x - X1 X2 x - X1 X2 x - X1 X2 x 2 10 2 9 10 2 2 7 2 8 + X1 x + 4 X1 X2 x + 3 X1 X2 x + X1 X2 x + X1 X2 x 2 8 9 10 2 7 2 7 9 - 2 X1 X2 x - 5 X1 X2 x - X1 x - 2 X1 X2 x + 4 X1 X2 x + 2 X1 x 2 6 2 6 7 8 2 5 2 5 + 2 X1 X2 x - 3 X1 X2 x - X1 X2 x + 2 X2 x - X1 X2 x + X1 X2 x 6 7 7 8 5 6 6 7 + X1 X2 x + X1 x - 4 X2 x - x - 2 X1 X2 x - 3 X1 x + 4 X2 x + 2 x 4 5 5 6 3 4 4 5 + 4 X1 X2 x + 4 X1 x - 3 X2 x - x - 3 X1 X2 x - 3 X1 x + X2 x + x 2 3 4 3 2 / 2 4 12 + X1 X2 x + X1 x - 3 x + 6 x - 7 x + 4 x - 1) / (X1 X2 x / 2 4 11 2 3 12 2 3 11 2 2 12 - X1 X2 x - 4 X1 X2 x + 3 X1 X2 x + 6 X1 X2 x 2 2 11 2 12 3 11 2 3 9 2 11 - 3 X1 X2 x - 4 X1 X2 x + X1 X2 x + X1 X2 x + X1 X2 x 2 12 3 10 2 11 2 3 8 2 2 9 + X1 x - 2 X1 X2 x - 3 X1 X2 x - 2 X1 X2 x - X1 X2 x 3 9 2 10 11 2 3 7 2 2 8 + X1 X2 x + 6 X1 X2 x + 3 X1 X2 x + X1 X2 x + 2 X1 X2 x 2 9 10 11 2 2 7 2 8 - 5 X1 X2 x - 6 X1 X2 x - X1 x - X1 X2 x - X1 X2 x 2 8 9 10 2 7 2 7 + 5 X1 X2 x + 5 X1 X2 x + 2 X1 x + 3 X1 X2 x - 6 X1 X2 x 8 9 9 2 6 2 6 7 - 3 X1 X2 x - X1 x + X2 x - 3 X1 X2 x + 4 X1 X2 x + X1 X2 x 8 9 2 5 2 5 6 7 7 8 - 4 X2 x - x + X1 X2 x - X1 X2 x - X1 X2 x - X1 x + 7 X2 x + 3 x 5 6 6 7 4 5 5 + 4 X1 X2 x + 4 X1 x - 7 X2 x - 4 x - 6 X1 X2 x - 6 X1 x + 4 X2 x 6 3 4 4 5 2 3 4 + 3 x + 4 X1 X2 x + 4 X1 x - X2 x - 2 x - X1 X2 x - X1 x + 5 x 3 2 - 10 x + 10 x - 5 x + 1) and in Maple format -(X1^2*X2^4*x^11-4*X1^2*X2^3*x^11-X1^2*X2^3*x^10+6*X1^2*X2^2*x^11-X1^2*X2^3*x^9 +3*X1^2*X2^2*x^10-4*X1^2*X2*x^11+X1*X2^3*x^10+X1^2*X2^3*x^8+2*X1^2*X2^2*x^9-3* X1^2*X2*x^10+X1^2*x^11-X1*X2^3*x^9-3*X1*X2^2*x^10-X1^2*X2^3*x^7-X1^2*X2^2*x^8- X1^2*X2*x^9+X1^2*x^10+4*X1*X2^2*x^9+3*X1*X2*x^10+X1^2*X2^2*x^7+X1^2*X2*x^8-2*X1 *X2^2*x^8-5*X1*X2*x^9-X1*x^10-2*X1^2*X2*x^7+4*X1*X2^2*x^7+2*X1*x^9+2*X1^2*X2*x^ 6-3*X1*X2^2*x^6-X1*X2*x^7+2*X2*x^8-X1^2*X2*x^5+X1*X2^2*x^5+X1*X2*x^6+X1*x^7-4* X2*x^7-x^8-2*X1*X2*x^5-3*X1*x^6+4*X2*x^6+2*x^7+4*X1*X2*x^4+4*X1*x^5-3*X2*x^5-x^ 6-3*X1*X2*x^3-3*X1*x^4+X2*x^4+x^5+X1*X2*x^2+X1*x^3-3*x^4+6*x^3-7*x^2+4*x-1)/(X1 ^2*X2^4*x^12-X1^2*X2^4*x^11-4*X1^2*X2^3*x^12+3*X1^2*X2^3*x^11+6*X1^2*X2^2*x^12-\ 3*X1^2*X2^2*x^11-4*X1^2*X2*x^12+X1*X2^3*x^11+X1^2*X2^3*x^9+X1^2*X2*x^11+X1^2*x^ 12-2*X1*X2^3*x^10-3*X1*X2^2*x^11-2*X1^2*X2^3*x^8-X1^2*X2^2*x^9+X1*X2^3*x^9+6*X1 *X2^2*x^10+3*X1*X2*x^11+X1^2*X2^3*x^7+2*X1^2*X2^2*x^8-5*X1*X2^2*x^9-6*X1*X2*x^ 10-X1*x^11-X1^2*X2^2*x^7-X1^2*X2*x^8+5*X1*X2^2*x^8+5*X1*X2*x^9+2*X1*x^10+3*X1^2 *X2*x^7-6*X1*X2^2*x^7-3*X1*X2*x^8-X1*x^9+X2*x^9-3*X1^2*X2*x^6+4*X1*X2^2*x^6+X1* X2*x^7-4*X2*x^8-x^9+X1^2*X2*x^5-X1*X2^2*x^5-X1*X2*x^6-X1*x^7+7*X2*x^7+3*x^8+4* X1*X2*x^5+4*X1*x^6-7*X2*x^6-4*x^7-6*X1*X2*x^4-6*X1*x^5+4*X2*x^5+3*x^6+4*X1*X2*x ^3+4*X1*x^4-X2*x^4-2*x^5-X1*X2*x^2-X1*x^3+5*x^4-10*x^3+10*x^2-5*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 2, 1, 2], are 55 7 n - 7/8 + n/8, and , - -- + ---, respectively, while 64 64 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 1, 1, 2], are 63 7 n - 7/8 + n/8, and , - -- + ---, respectively, while the asymptotic correlatio\ 64 64 n between these two random variables is 5/7 and in floating point 0.7142857143 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ ate normal pair with correlation, 5/7 99 i.e. , [[5/7, 0], [0, --]] 49 ------------------------------------------------- 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, 1, 2], nor the composition, [2, 1, 2, 1] Then infinity ----- \ n ) a(n) x = / ----- n = 0 11 10 9 8 7 6 4 3 2 x - x + x + 2 x - 6 x + 6 x - 8 x + 13 x - 11 x + 5 x - 1 - -------------------------------------------------------------------- 3 3 2 3 (x - x + 1) (x - x + 2 x - 1) (-1 + x) and in Maple format -(x^11-x^10+x^9+2*x^8-6*x^7+6*x^6-8*x^4+13*x^3-11*x^2+5*x-1)/(x^3-x+1)/(x^3-x^2 +2*x-1)/(-1+x)^3 The asymptotic expression for a(n) is, n 0.88608286609883249205 1.7548776662466927601 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 2, 1, 2] and d occurrences (as containment) of the composition, [2, 1, 2, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 11 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 2 10 2 11 2 11 2 10 2 10 - X1 X2 x - X1 X2 x - X1 X2 x + X1 X2 x + X1 X2 x 2 2 8 2 9 2 9 11 11 2 2 7 - 2 X1 X2 x + X1 X2 x + X1 X2 x + X1 x + X2 x + 4 X1 X2 x 2 8 2 8 9 10 10 11 - X1 X2 x - X1 X2 x - 5 X1 X2 x - X1 x - X2 x - x 2 2 6 8 9 9 10 2 2 5 - 3 X1 X2 x + 8 X1 X2 x + 2 X1 x + 2 X2 x + x + X1 X2 x 7 8 8 9 6 7 7 8 - 8 X1 X2 x - X1 x - X2 x - x + 7 X1 X2 x - X1 x - X2 x - 2 x 5 6 6 7 4 6 3 - 2 X1 X2 x + X1 x + X2 x + 6 x - 3 X1 X2 x - 6 x + 3 X1 X2 x 2 4 3 2 / 3 - X1 X2 x + 8 x - 13 x + 11 x - 5 x + 1) / ((-1 + x) / 3 3 3 2 3 2 (X1 X2 x - x + x - 1) (X1 X2 x - X1 X2 x - x + x - 2 x + 1)) and in Maple format (X1^2*X2^2*x^11-X1^2*X2^2*x^10-X1^2*X2*x^11-X1*X2^2*x^11+X1^2*X2*x^10+X1*X2^2*x ^10-2*X1^2*X2^2*x^8+X1^2*X2*x^9+X1*X2^2*x^9+X1*x^11+X2*x^11+4*X1^2*X2^2*x^7-X1^ 2*X2*x^8-X1*X2^2*x^8-5*X1*X2*x^9-X1*x^10-X2*x^10-x^11-3*X1^2*X2^2*x^6+8*X1*X2*x ^8+2*X1*x^9+2*X2*x^9+x^10+X1^2*X2^2*x^5-8*X1*X2*x^7-X1*x^8-X2*x^8-x^9+7*X1*X2*x ^6-X1*x^7-X2*x^7-2*x^8-2*X1*X2*x^5+X1*x^6+X2*x^6+6*x^7-3*X1*X2*x^4-6*x^6+3*X1* X2*x^3-X1*X2*x^2+8*x^4-13*x^3+11*x^2-5*x+1)/(-1+x)^3/(X1*X2*x^3-x^3+x-1)/(X1*X2 *x^3-X1*X2*x^2-x^3+x^2-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 2, 1, 2], are 55 7 n - 7/8 + n/8, and , - -- + ---, respectively, while 64 64 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 1, 2, 1], are 55 7 n - 7/8 + n/8, and , - -- + ---, respectively, while the asymptotic correlatio\ 64 64 n between these two random variables is 1 and in floating point 1. ------------------------------------------------- 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, 1, 2], nor the composition, [2, 2, 1, 1] Then infinity ----- 7 6 5 4 2 \ n x + x - x + x - 3 x + 3 x - 1 ) a(n) x = - ---------------------------------- / 2 3 ----- (x + x - 1) (-1 + x) n = 0 and in Maple format -(x^7+x^6-x^5+x^4-3*x^2+3*x-1)/(x^2+x-1)/(-1+x)^3 The asymptotic expression for a(n) is, n 1.8944271909999158785 1.6180339887498948482 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 2, 1, 2] and d occurrences (as containment) of the composition, [2, 2, 1, 1], (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 2 9 2 10 2 10 2 9 2 9 - X1 X2 x - X1 X2 x - 3 X1 X2 x + 2 X1 X2 x + 3 X1 X2 x 10 2 10 2 8 2 8 9 10 + 4 X1 X2 x + 2 X2 x - 3 X1 X2 x - X1 X2 x - 6 X1 X2 x - X1 x 2 9 10 2 7 8 9 2 8 - 2 X2 x - 3 X2 x + 4 X1 X2 x + 6 X1 X2 x + X1 x + X2 x 9 10 2 6 7 8 8 9 + 4 X2 x + x - 3 X1 X2 x - 5 X1 X2 x + X1 x - 3 X2 x - x 2 5 6 7 7 8 5 6 + X1 X2 x + X1 X2 x - 4 X1 x + X2 x - x + 5 X1 X2 x + 7 X1 x 7 4 5 6 3 4 2 + 4 x - 7 X1 X2 x - 7 X1 x - 5 x + 4 X1 X2 x + 4 X1 x - X1 X2 x 3 4 3 2 / 3 2 6 - X1 x + 8 x - 13 x + 11 x - 5 x + 1) / ((-1 + x) (X1 X2 x / 2 5 6 5 6 4 5 - X1 X2 x - 2 X1 X2 x + X1 X2 x + X2 x + X1 X2 x + X1 x 3 4 5 2 3 4 3 2 - 2 X1 X2 x - 2 X1 x - x + X1 X2 x + X1 x + x + x - 3 x + 3 x - 1) ) and in Maple format (X1^2*X2^2*x^10-X1^2*X2^2*x^9-X1^2*X2*x^10-3*X1*X2^2*x^10+2*X1^2*X2*x^9+3*X1*X2 ^2*x^9+4*X1*X2*x^10+2*X2^2*x^10-3*X1^2*X2*x^8-X1*X2^2*x^8-6*X1*X2*x^9-X1*x^10-2 *X2^2*x^9-3*X2*x^10+4*X1^2*X2*x^7+6*X1*X2*x^8+X1*x^9+X2^2*x^8+4*X2*x^9+x^10-3* X1^2*X2*x^6-5*X1*X2*x^7+X1*x^8-3*X2*x^8-x^9+X1^2*X2*x^5+X1*X2*x^6-4*X1*x^7+X2*x ^7-x^8+5*X1*X2*x^5+7*X1*x^6+4*x^7-7*X1*X2*x^4-7*X1*x^5-5*x^6+4*X1*X2*x^3+4*X1*x ^4-X1*X2*x^2-X1*x^3+8*x^4-13*x^3+11*x^2-5*x+1)/(-1+x)^3/(X1^2*X2*x^6-X1^2*X2*x^ 5-2*X1*X2*x^6+X1*X2*x^5+X2*x^6+X1*X2*x^4+X1*x^5-2*X1*X2*x^3-2*X1*x^4-x^5+X1*X2* x^2+X1*x^3+x^4+x^3-3*x^2+3*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 2, 1, 2], are 55 7 n - 7/8 + n/8, and , - -- + ---, respectively, while 64 64 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 2, 1, 1], are 47 7 n - 7/8 + n/8, and , - -- + ---, respectively, while the asymptotic correlatio\ 64 64 n between these two random variables is 5/7 and in floating point 0.7142857143 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ ate normal pair with correlation, 5/7 99 i.e. , [[5/7, 0], [0, --]] 49 ------------------------------------------------- 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, 1, 2], nor the composition, [3, 1, 1, 1] Then infinity ----- 8 7 6 5 4 3 2 \ n x + 2 x - 4 x + 7 x - 10 x + 10 x - 8 x + 4 x - 1 ) a(n) x = - -------------------------------------------------------- / 2 3 3 ----- (x - x + 1) (x + x - 1) (-1 + x) n = 0 and in Maple format -(x^8+2*x^7-4*x^6+7*x^5-10*x^4+10*x^3-8*x^2+4*x-1)/(x^2-x+1)/(x^3+x-1)/(-1+x)^3 The asymptotic expression for a(n) is, n 4.1345231835816431372 1.4655712318767680267 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 2, 1, 2] and d occurrences (as containment) of the composition, [3, 1, 1, 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 3 12 2 2 11 2 12 2 12 3 12 - X1 X2 x - X1 X2 x - 2 X1 X2 x + X1 X2 x + X2 x 2 2 10 2 11 2 12 2 11 12 - 2 X1 X2 x + 3 X1 X2 x + X1 x + X1 X2 x + X1 X2 x 2 12 2 2 9 2 10 2 11 2 10 - 2 X2 x + 4 X1 X2 x + 4 X1 X2 x - 2 X1 x + 3 X1 X2 x 11 12 12 2 2 8 2 9 2 10 - 4 X1 X2 x - X1 x + X2 x - 3 X1 X2 x - 13 X1 X2 x - 2 X1 x 2 9 10 11 2 10 11 2 2 7 - 7 X1 X2 x - 6 X1 X2 x + 3 X1 x - X2 x + X2 x + X1 X2 x 2 8 2 9 2 8 9 10 + 14 X1 X2 x + 10 X1 x + 4 X1 X2 x + 22 X1 X2 x + 3 X1 x 2 9 10 11 2 7 2 8 2 7 + 3 X2 x + 2 X2 x - x - 8 X1 X2 x - 14 X1 x - X1 X2 x 8 9 2 8 9 10 2 6 - 22 X1 X2 x - 17 X1 x - X2 x - 9 X2 x - x + 2 X1 X2 x 2 7 7 8 8 9 2 6 + 11 X1 x + 13 X1 X2 x + 24 X1 x + 8 X2 x + 7 x - 5 X1 x 6 7 7 8 2 5 5 6 - 3 X1 X2 x - 21 X1 x - 6 X2 x - 10 x + X1 x - 3 X1 X2 x + 11 X1 x 6 7 4 5 5 6 3 4 + 4 X2 x + 11 x + 3 X1 X2 x + X1 x - X2 x - 9 x - X1 X2 x - 6 X1 x 5 3 4 2 3 2 / 3 + x + 4 X1 x + 8 x - X1 x - 13 x + 11 x - 5 x + 1) / ((-1 + x) / 4 4 4 3 4 3 4 3 (X1 X2 x - X1 x - X2 x + X1 x + x - x + x - 1) (X1 X2 x - X1 X2 x 4 4 3 4 2 3 2 - X1 x - X2 x + 2 X1 x + x - X1 x - x + x - 2 x + 1)) and in Maple format (X1^2*X2^2*x^12-X1*X2^3*x^12-X1^2*X2^2*x^11-2*X1^2*X2*x^12+X1*X2^2*x^12+X2^3*x^ 12-2*X1^2*X2^2*x^10+3*X1^2*X2*x^11+X1^2*x^12+X1*X2^2*x^11+X1*X2*x^12-2*X2^2*x^ 12+4*X1^2*X2^2*x^9+4*X1^2*X2*x^10-2*X1^2*x^11+3*X1*X2^2*x^10-4*X1*X2*x^11-X1*x^ 12+X2*x^12-3*X1^2*X2^2*x^8-13*X1^2*X2*x^9-2*X1^2*x^10-7*X1*X2^2*x^9-6*X1*X2*x^ 10+3*X1*x^11-X2^2*x^10+X2*x^11+X1^2*X2^2*x^7+14*X1^2*X2*x^8+10*X1^2*x^9+4*X1*X2 ^2*x^8+22*X1*X2*x^9+3*X1*x^10+3*X2^2*x^9+2*X2*x^10-x^11-8*X1^2*X2*x^7-14*X1^2*x ^8-X1*X2^2*x^7-22*X1*X2*x^8-17*X1*x^9-X2^2*x^8-9*X2*x^9-x^10+2*X1^2*X2*x^6+11* X1^2*x^7+13*X1*X2*x^7+24*X1*x^8+8*X2*x^8+7*x^9-5*X1^2*x^6-3*X1*X2*x^6-21*X1*x^7 -6*X2*x^7-10*x^8+X1^2*x^5-3*X1*X2*x^5+11*X1*x^6+4*X2*x^6+11*x^7+3*X1*X2*x^4+X1* x^5-X2*x^5-9*x^6-X1*X2*x^3-6*X1*x^4+x^5+4*X1*x^3+8*x^4-X1*x^2-13*x^3+11*x^2-5*x +1)/(-1+x)^3/(X1*X2*x^4-X1*x^4-X2*x^4+X1*x^3+x^4-x^3+x-1)/(X1*X2*x^4-X1*X2*x^3- X1*x^4-X2*x^4+2*X1*x^3+x^4-X1*x^2-x^3+x^2-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 2, 1, 2], are 55 7 n - 7/8 + n/8, and , - -- + ---, respectively, while 64 64 Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 1, 1, 1], are 11 3 n - 7/8 + n/8, and , - -- + ---, respectively, while the asymptotic correlatio\ 64 64 n between these two random variables is 1/2 1/2 3 7 --------- 21 and in floating point 0.2182178903 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 21 ate normal pair with correlation, ----- 21 1/2 21 23 i.e. , [[-----, 0], [0, --]] 21 21 ------------------------------------------------- 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, 2, 1], nor the composition, [1, 3, 1, 1] Then infinity ----- 5 4 3 2 \ n x - 3 x + 3 x - 4 x + 3 x - 1 ) a(n) x = - --------------------------------- / 3 3 ----- (x + x - 1) (-1 + x) n = 0 and in Maple format -(x^5-3*x^4+3*x^3-4*x^2+3*x-1)/(x^3+x-1)/(-1+x)^3 The asymptotic expression for a(n) is, n 6.9556233073109364759 1.4655712318767680267 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 2, 2, 1] and d occurrences (as containment) of the composition, [1, 3, 1, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 5 ) | ) | ) A(n, c, d) x X1 X2 || = - (2 X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 4 5 5 3 4 4 5 3 - 2 X1 X2 x - 2 X1 x - X2 x + X1 X2 x + 4 X1 x + X2 x + x - 3 X1 x 4 2 3 2 / 2 4 - 3 x + X1 x + 3 x - 4 x + 3 x - 1) / ((-1 + x) (X1 X2 x / 3 4 4 3 4 2 3 2 - X1 X2 x - X1 x - X2 x + 2 X1 x + x - X1 x - x + x - 2 x + 1)) and in Maple format -(2*X1*X2*x^5-2*X1*X2*x^4-2*X1*x^5-X2*x^5+X1*X2*x^3+4*X1*x^4+X2*x^4+x^5-3*X1*x^ 3-3*x^4+X1*x^2+3*x^3-4*x^2+3*x-1)/(-1+x)^2/(X1*X2*x^4-X1*X2*x^3-X1*x^4-X2*x^4+2 *X1*x^3+x^4-X1*x^2-x^3+x^2-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 2, 2, 1], are 47 7 n - 7/8 + n/8, and , - -- + ---, respectively, while 64 64 Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 3, 1, 1], are 11 3 n - 7/8 + n/8, and , - -- + ---, respectively, while the asymptotic correlatio\ 64 64 n between these two random variables is 1/2 1/2 3 7 --------- 21 and in floating point 0.2182178903 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 21 ate normal pair with correlation, ----- 21 1/2 21 23 i.e. , [[-----, 0], [0, --]] 21 21 ------------------------------------------------- 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, 2, 1], nor the composition, [2, 1, 1, 2] Then infinity ----- \ n ) a(n) x = / ----- n = 0 14 10 8 7 6 5 4 3 2 x + x - 4 x + 6 x - 8 x + 11 x - 13 x + 14 x - 11 x + 5 x - 1 ------------------------------------------------------------------------ 5 4 3 2 4 (x - x + x - x + 2 x - 1) (-1 + x) and in Maple format (x^14+x^10-4*x^8+6*x^7-8*x^6+11*x^5-13*x^4+14*x^3-11*x^2+5*x-1)/(x^5-x^4+x^3-x^ 2+2*x-1)/(-1+x)^4 The asymptotic expression for a(n) is, n 1.2195277363024998027 1.6736485462998415616 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 2, 2, 1] and d occurrences (as containment) of the composition, [2, 1, 1, 2], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 4 16 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 4 15 2 3 16 4 16 2 3 15 2 2 16 - X1 X2 x - 4 X1 X2 x - X1 X2 x + 3 X1 X2 x + 6 X1 X2 x 4 15 3 16 2 4 13 2 2 15 2 16 + X1 X2 x + 4 X1 X2 x - X1 X2 x - 3 X1 X2 x - 4 X1 X2 x 3 15 2 16 2 4 12 2 3 13 - 2 X1 X2 x - 6 X1 X2 x + 2 X1 X2 x + 5 X1 X2 x 2 2 14 2 15 2 16 3 14 16 + X1 X2 x + X1 X2 x + X1 x - 2 X1 X2 x + 4 X1 X2 x 3 15 2 4 11 2 3 12 2 2 13 2 14 - X2 x - X1 X2 x - 9 X1 X2 x - 10 X1 X2 x - 2 X1 X2 x 2 14 15 16 3 14 2 15 + 4 X1 X2 x + 2 X1 X2 x - X1 x + 2 X2 x + 3 X2 x 2 3 11 2 2 12 2 13 2 14 3 12 + 4 X1 X2 x + 15 X1 X2 x + 9 X1 X2 x + X1 x + X1 X2 x 2 13 14 15 3 13 2 14 15 + 2 X1 X2 x - 2 X1 X2 x - X1 x - X2 x - 5 X2 x - 3 X2 x 2 3 10 2 2 11 2 12 2 13 3 11 - 2 X1 X2 x - 6 X1 X2 x - 12 X1 X2 x - 3 X1 x + 2 X1 X2 x 2 12 13 2 13 14 15 2 3 9 - 2 X1 X2 x - 4 X1 X2 x + 2 X2 x + 4 X2 x + x + 4 X1 X2 x 2 2 10 2 11 2 12 3 10 2 11 + 4 X1 X2 x + 6 X1 X2 x + 4 X1 x - 3 X1 X2 x - 9 X1 X2 x 12 13 2 12 13 14 2 3 8 + 3 X1 X2 x + 2 X1 x - X2 x - X2 x - x - 3 X1 X2 x 2 2 9 2 10 2 11 3 9 2 10 - 7 X1 X2 x - 5 X1 X2 x - 3 X1 x + X1 X2 x + 14 X1 X2 x 11 12 2 11 12 2 3 7 2 2 8 + 8 X1 X2 x - 2 X1 x + 3 X2 x + X2 x + X1 X2 x + 4 X1 X2 x 2 9 2 10 2 9 10 11 + 6 X1 X2 x + 3 X1 x - 11 X1 X2 x - 14 X1 X2 x - X1 x 2 10 11 2 2 7 2 8 2 9 - 3 X2 x - 4 X2 x - X1 X2 x - 2 X1 X2 x - 3 X1 x 2 8 9 10 2 9 10 11 2 8 + 12 X1 X2 x + 10 X1 X2 x + 3 X1 x + X2 x + 4 X2 x + x + X1 x 2 7 8 9 10 2 6 7 - 11 X1 X2 x - 12 X1 X2 x + 3 X2 x - x + 5 X1 X2 x + 15 X1 X2 x 8 9 2 5 6 7 7 8 - 10 X2 x - 4 x - X1 X2 x - 14 X1 X2 x - 2 X1 x + 12 X2 x + 10 x 5 6 6 7 4 5 6 + 14 X1 X2 x + X1 x - 10 X2 x - 14 x - 11 X1 X2 x + 5 X2 x + 19 x 3 4 5 2 4 3 2 + 5 X1 X2 x - X2 x - 24 x - X1 X2 x + 27 x - 25 x + 16 x - 6 x + 1) / 2 2 4 12 2 4 11 2 3 12 2 3 11 / ((-1 + x) (X1 X2 x - X1 X2 x - 4 X1 X2 x + 3 X1 X2 x / 2 2 12 2 2 11 2 12 3 11 2 3 9 + 6 X1 X2 x - 3 X1 X2 x - 4 X1 X2 x + X1 X2 x + X1 X2 x 2 2 10 2 11 2 12 3 10 2 11 + X1 X2 x + X1 X2 x + X1 x - 2 X1 X2 x - 3 X1 X2 x 2 3 8 2 2 9 2 10 3 9 2 10 - 2 X1 X2 x - 3 X1 X2 x - 2 X1 X2 x + X1 X2 x + 5 X1 X2 x 11 2 3 7 2 2 8 2 9 2 10 + 3 X1 X2 x + X1 X2 x + 3 X1 X2 x + 3 X1 X2 x + X1 x 2 9 10 11 2 2 7 2 8 2 9 - 3 X1 X2 x - 4 X1 X2 x - X1 x - X1 X2 x - X1 X2 x - X1 x 2 8 9 10 2 7 8 9 + 4 X1 X2 x + 3 X1 X2 x + X1 x - 6 X1 X2 x - 6 X1 X2 x - X1 x 2 6 7 8 8 2 5 6 + 4 X1 X2 x + 7 X1 X2 x + 2 X1 x - X2 x - X1 X2 x - 5 X1 X2 x 7 7 8 5 6 7 4 - X1 x + 4 X2 x + x + 5 X1 X2 x - 6 X2 x - 4 x - 6 X1 X2 x 5 6 3 4 5 2 4 3 + 4 X2 x + 7 x + 4 X1 X2 x - X2 x - 8 x - X1 X2 x + 9 x - 11 x 2 + 10 x - 5 x + 1)) and in Maple format (X1^2*X2^4*x^16-X1^2*X2^4*x^15-4*X1^2*X2^3*x^16-X1*X2^4*x^16+3*X1^2*X2^3*x^15+6 *X1^2*X2^2*x^16+X1*X2^4*x^15+4*X1*X2^3*x^16-X1^2*X2^4*x^13-3*X1^2*X2^2*x^15-4* X1^2*X2*x^16-2*X1*X2^3*x^15-6*X1*X2^2*x^16+2*X1^2*X2^4*x^12+5*X1^2*X2^3*x^13+X1 ^2*X2^2*x^14+X1^2*X2*x^15+X1^2*x^16-2*X1*X2^3*x^14+4*X1*X2*x^16-X2^3*x^15-X1^2* X2^4*x^11-9*X1^2*X2^3*x^12-10*X1^2*X2^2*x^13-2*X1^2*X2*x^14+4*X1*X2^2*x^14+2*X1 *X2*x^15-X1*x^16+2*X2^3*x^14+3*X2^2*x^15+4*X1^2*X2^3*x^11+15*X1^2*X2^2*x^12+9* X1^2*X2*x^13+X1^2*x^14+X1*X2^3*x^12+2*X1*X2^2*x^13-2*X1*X2*x^14-X1*x^15-X2^3*x^ 13-5*X2^2*x^14-3*X2*x^15-2*X1^2*X2^3*x^10-6*X1^2*X2^2*x^11-12*X1^2*X2*x^12-3*X1 ^2*x^13+2*X1*X2^3*x^11-2*X1*X2^2*x^12-4*X1*X2*x^13+2*X2^2*x^13+4*X2*x^14+x^15+4 *X1^2*X2^3*x^9+4*X1^2*X2^2*x^10+6*X1^2*X2*x^11+4*X1^2*x^12-3*X1*X2^3*x^10-9*X1* X2^2*x^11+3*X1*X2*x^12+2*X1*x^13-X2^2*x^12-X2*x^13-x^14-3*X1^2*X2^3*x^8-7*X1^2* X2^2*x^9-5*X1^2*X2*x^10-3*X1^2*x^11+X1*X2^3*x^9+14*X1*X2^2*x^10+8*X1*X2*x^11-2* X1*x^12+3*X2^2*x^11+X2*x^12+X1^2*X2^3*x^7+4*X1^2*X2^2*x^8+6*X1^2*X2*x^9+3*X1^2* x^10-11*X1*X2^2*x^9-14*X1*X2*x^10-X1*x^11-3*X2^2*x^10-4*X2*x^11-X1^2*X2^2*x^7-2 *X1^2*X2*x^8-3*X1^2*x^9+12*X1*X2^2*x^8+10*X1*X2*x^9+3*X1*x^10+X2^2*x^9+4*X2*x^ 10+x^11+X1^2*x^8-11*X1*X2^2*x^7-12*X1*X2*x^8+3*X2*x^9-x^10+5*X1*X2^2*x^6+15*X1* X2*x^7-10*X2*x^8-4*x^9-X1*X2^2*x^5-14*X1*X2*x^6-2*X1*x^7+12*X2*x^7+10*x^8+14*X1 *X2*x^5+X1*x^6-10*X2*x^6-14*x^7-11*X1*X2*x^4+5*X2*x^5+19*x^6+5*X1*X2*x^3-X2*x^4 -24*x^5-X1*X2*x^2+27*x^4-25*x^3+16*x^2-6*x+1)/(-1+x)^2/(X1^2*X2^4*x^12-X1^2*X2^ 4*x^11-4*X1^2*X2^3*x^12+3*X1^2*X2^3*x^11+6*X1^2*X2^2*x^12-3*X1^2*X2^2*x^11-4*X1 ^2*X2*x^12+X1*X2^3*x^11+X1^2*X2^3*x^9+X1^2*X2^2*x^10+X1^2*X2*x^11+X1^2*x^12-2* X1*X2^3*x^10-3*X1*X2^2*x^11-2*X1^2*X2^3*x^8-3*X1^2*X2^2*x^9-2*X1^2*X2*x^10+X1* X2^3*x^9+5*X1*X2^2*x^10+3*X1*X2*x^11+X1^2*X2^3*x^7+3*X1^2*X2^2*x^8+3*X1^2*X2*x^ 9+X1^2*x^10-3*X1*X2^2*x^9-4*X1*X2*x^10-X1*x^11-X1^2*X2^2*x^7-X1^2*X2*x^8-X1^2*x ^9+4*X1*X2^2*x^8+3*X1*X2*x^9+X1*x^10-6*X1*X2^2*x^7-6*X1*X2*x^8-X1*x^9+4*X1*X2^2 *x^6+7*X1*X2*x^7+2*X1*x^8-X2*x^8-X1*X2^2*x^5-5*X1*X2*x^6-X1*x^7+4*X2*x^7+x^8+5* X1*X2*x^5-6*X2*x^6-4*x^7-6*X1*X2*x^4+4*X2*x^5+7*x^6+4*X1*X2*x^3-X2*x^4-8*x^5-X1 *X2*x^2+9*x^4-11*x^3+10*x^2-5*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 2, 2, 1], are 47 7 n - 7/8 + n/8, and , - -- + ---, respectively, while 64 64 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 1, 1, 2], are 63 7 n - 7/8 + n/8, and , - -- + ---, respectively, while the asymptotic correlatio\ 64 64 n between these two random variables is 5/7 and in floating point 0.7142857143 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ ate normal pair with correlation, 5/7 99 i.e. , [[5/7, 0], [0, --]] 49 ------------------------------------------------- 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, 2, 1], nor the composition, [2, 1, 2, 1] Then infinity ----- 6 5 3 2 \ n x - 2 x + 2 x - 4 x + 3 x - 1 ) a(n) x = ----------------------------------- / 2 2 2 ----- (x - x + 1) (x + x - 1) (-1 + x) n = 0 and in Maple format (x^6-2*x^5+2*x^3-4*x^2+3*x-1)/(x^2-x+1)/(x^2+x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 2.1180339887498948478 1.6180339887498948482 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 2, 2, 1] and d occurrences (as containment) of the composition, [2, 1, 2, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 7 2 6 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x - X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 7 2 5 6 7 5 6 - 2 X1 X2 x + X1 X2 x + X1 X2 x + X1 x - X1 X2 x + X2 x 4 5 6 3 4 5 2 - 2 X1 X2 x - 2 X2 x - x + 2 X1 X2 x + 2 X2 x + 2 x - X1 X2 x 3 3 2 / 2 6 2 5 - X2 x - 2 x + 4 x - 3 x + 1) / ((-1 + x) (X1 X2 x - X1 X2 x / 6 5 6 4 5 3 4 - 2 X1 X2 x + X1 X2 x + X1 x + X1 X2 x + X2 x - 2 X1 X2 x - 2 X2 x 5 2 3 4 3 2 - x + X1 X2 x + X2 x + x + x - 3 x + 3 x - 1)) and in Maple format (X1*X2^2*x^7-X1*X2^2*x^6-2*X1*X2*x^7+X1*X2^2*x^5+X1*X2*x^6+X1*x^7-X1*X2*x^5+X2* x^6-2*X1*X2*x^4-2*X2*x^5-x^6+2*X1*X2*x^3+2*X2*x^4+2*x^5-X1*X2*x^2-X2*x^3-2*x^3+ 4*x^2-3*x+1)/(-1+x)/(X1*X2^2*x^6-X1*X2^2*x^5-2*X1*X2*x^6+X1*X2*x^5+X1*x^6+X1*X2 *x^4+X2*x^5-2*X1*X2*x^3-2*X2*x^4-x^5+X1*X2*x^2+X2*x^3+x^4+x^3-3*x^2+3*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 2, 2, 1], are 47 7 n - 7/8 + n/8, and , - -- + ---, respectively, while 64 64 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 1, 2, 1], are 55 7 n - 7/8 + n/8, and , - -- + ---, respectively, while the asymptotic correlatio\ 64 64 n between these two random variables is 5/7 and in floating point 0.7142857143 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ ate normal pair with correlation, 5/7 99 i.e. , [[5/7, 0], [0, --]] 49 ------------------------------------------------- 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, 2, 1], nor the composition, [2, 2, 1, 1] Then infinity ----- 8 7 6 5 4 3 2 \ n x - x + x + 2 x - 5 x + 7 x - 7 x + 4 x - 1 ) a(n) x = - -------------------------------------------------- / 3 2 3 ----- (x - x + 2 x - 1) (-1 + x) n = 0 and in Maple format -(x^8-x^7+x^6+2*x^5-5*x^4+7*x^3-7*x^2+4*x-1)/(x^3-x^2+2*x-1)/(-1+x)^3 The asymptotic expression for a(n) is, n 0.95661118429105013088 1.7548776662466927601 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 2, 2, 1] and d occurrences (as containment) of the composition, [2, 2, 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 5 6 - X1 x - X2 x + X1 X2 x + X1 x + X2 x + x - 2 X1 X2 x - X1 x 6 7 4 6 3 5 2 4 - X2 x - x + 4 X1 X2 x + x - 3 X1 X2 x + 2 x + X1 X2 x - 5 x 3 2 / 3 + 7 x - 7 x + 4 x - 1) / ((-1 + x) / 3 2 3 2 (X1 X2 x - X1 X2 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^8-2*X1*X2*x^5-X1*x ^6-X2*x^6-x^7+4*X1*X2*x^4+x^6-3*X1*X2*x^3+2*x^5+X1*X2*x^2-5*x^4+7*x^3-7*x^2+4*x -1)/(-1+x)^3/(X1*X2*x^3-X1*X2*x^2-x^3+x^2-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 2, 2, 1], are 47 7 n - 7/8 + n/8, and , - -- + ---, respectively, while 64 64 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 2, 1, 1], are 47 7 n - 7/8 + n/8, and , - -- + ---, respectively, while the asymptotic correlatio\ 64 64 n between these two random variables is 1 and in floating point 1. ------------------------------------------------- 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, 2, 1], nor the composition, [3, 1, 1, 1] Then infinity ----- 8 7 5 4 3 2 \ n x + x + x - 3 x + 3 x - 4 x + 3 x - 1 ) a(n) x = - ------------------------------------------- / 3 3 ----- (x + x - 1) (-1 + x) n = 0 and in Maple format -(x^8+x^7+x^5-3*x^4+3*x^3-4*x^2+3*x-1)/(x^3+x-1)/(-1+x)^3 The asymptotic expression for a(n) is, n 4.7460151755324556560 1.4655712318767680267 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 2, 2, 1] and d occurrences (as containment) of the composition, [3, 1, 1, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 9 9 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x - 2 X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 9 9 9 7 9 6 7 7 - X2 x + X1 x + 2 X2 x + X1 X2 x - x - 2 X1 X2 x - X1 x - X2 x 5 6 6 7 4 5 5 6 + 4 X1 X2 x + 2 X1 x + X2 x + x - 3 X1 X2 x - 6 X1 x - 2 X2 x - x 3 4 4 5 3 4 2 3 2 + X1 X2 x + 7 X1 x + X2 x + 4 x - 4 X1 x - 6 x + X1 x + 7 x - 7 x / 3 4 3 4 4 3 + 4 x - 1) / ((-1 + x) (X1 X2 x - X1 X2 x - X1 x - X2 x + 2 X1 x / 4 2 3 2 + x - X1 x - x + x - 2 x + 1)) and in Maple format (X1*X2^2*x^9-2*X1*X2*x^9-X2^2*x^9+X1*x^9+2*X2*x^9+X1*X2*x^7-x^9-2*X1*X2*x^6-X1* x^7-X2*x^7+4*X1*X2*x^5+2*X1*x^6+X2*x^6+x^7-3*X1*X2*x^4-6*X1*x^5-2*X2*x^5-x^6+X1 *X2*x^3+7*X1*x^4+X2*x^4+4*x^5-4*X1*x^3-6*x^4+X1*x^2+7*x^3-7*x^2+4*x-1)/(-1+x)^3 /(X1*X2*x^4-X1*X2*x^3-X1*x^4-X2*x^4+2*X1*x^3+x^4-X1*x^2-x^3+x^2-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 2, 2, 1], are 47 7 n - 7/8 + n/8, and , - -- + ---, respectively, while 64 64 Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 1, 1, 1], are 11 3 n - 7/8 + n/8, and , - -- + ---, respectively, while the asymptotic correlatio\ 64 64 n between these two random variables is 1/2 1/2 3 7 --------- 21 and in floating point 0.2182178903 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 21 ate normal pair with correlation, ----- 21 1/2 21 23 i.e. , [[-----, 0], [0, --]] 21 21 ------------------------------------------------- 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, 3, 1, 1], nor the composition, [2, 1, 1, 2] Then infinity ----- \ n 15 13 12 11 10 9 8 7 6 ) a(n) x = (x + 3 x - 4 x + x - 5 x + 3 x - x + 4 x - 4 x / ----- n = 0 5 4 3 2 / 3 9 4 + 4 x - 4 x + 3 x - 4 x + 3 x - 1) / ((x + x - 1) (x - x - 1) / 3 (-1 + x) ) and in Maple format (x^15+3*x^13-4*x^12+x^11-5*x^10+3*x^9-x^8+4*x^7-4*x^6+4*x^5-4*x^4+3*x^3-4*x^2+3 *x-1)/(x^3+x-1)/(x^9-x^4-1)/(-1+x)^3 The asymptotic expression for a(n) is, n 5.2787205554750117565 1.4655712318767680267 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 3, 1, 1] and d occurrences (as containment) of the composition, [2, 1, 1, 2], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 4 4 18 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 4 4 17 4 3 18 3 4 18 4 4 16 - 2 X1 X2 x - 4 X1 X2 x - 4 X1 X2 x + 2 X1 X2 x 4 3 17 4 2 18 3 4 17 3 3 18 + 6 X1 X2 x + 6 X1 X2 x + 11 X1 X2 x + 16 X1 X2 x 2 4 18 4 4 15 4 3 16 4 2 17 + 6 X1 X2 x - X1 X2 x - 6 X1 X2 x - 6 X1 X2 x 4 18 3 4 16 3 3 17 3 2 18 - 4 X1 X2 x - 15 X1 X2 x - 36 X1 X2 x - 24 X1 X2 x 2 4 17 2 3 18 4 18 4 3 15 - 21 X1 X2 x - 24 X1 X2 x - 4 X1 X2 x + 3 X1 X2 x 4 2 16 4 17 4 18 3 4 15 3 3 16 + 6 X1 X2 x + 2 X1 X2 x + X1 x + 12 X1 X2 x + 48 X1 X2 x 3 2 17 3 18 2 4 16 2 3 17 + 42 X1 X2 x + 16 X1 X2 x + 36 X1 X2 x + 72 X1 X2 x 2 2 18 4 17 3 18 4 18 4 2 15 + 36 X1 X2 x + 17 X1 X2 x + 16 X1 X2 x + X2 x - 3 X1 X2 x 4 16 3 4 14 3 3 15 3 2 16 - 2 X1 X2 x - 4 X1 X2 x - 42 X1 X2 x - 54 X1 X2 x 3 17 3 18 2 4 15 2 3 16 - 20 X1 X2 x - 4 X1 x - 39 X1 X2 x - 120 X1 X2 x 2 2 17 2 18 4 16 3 17 - 90 X1 X2 x - 24 X1 X2 x - 35 X1 X2 x - 60 X1 X2 x 2 18 4 17 3 18 4 15 3 3 14 - 24 X1 X2 x - 5 X2 x - 4 X2 x + X1 X2 x + 17 X1 X2 x 3 2 15 3 16 3 17 2 4 14 + 54 X1 X2 x + 24 X1 X2 x + 3 X1 x + 24 X1 X2 x 2 3 15 2 2 16 2 17 2 18 + 142 X1 X2 x + 144 X1 X2 x + 48 X1 X2 x + 6 X1 x 4 15 3 16 2 17 18 + 47 X1 X2 x + 120 X1 X2 x + 78 X1 X2 x + 16 X1 X2 x 4 16 3 17 2 18 3 3 13 3 2 14 + 12 X2 x + 18 X2 x + 6 X2 x - 5 X1 X2 x - 26 X1 X2 x 3 15 3 16 2 4 13 2 3 14 - 30 X1 X2 x - 3 X1 x - 6 X1 X2 x - 101 X1 X2 x 2 2 15 2 16 2 17 4 14 - 192 X1 X2 x - 72 X1 X2 x - 9 X1 x - 41 X1 X2 x 3 15 2 16 17 18 4 15 - 174 X1 X2 x - 150 X1 X2 x - 44 X1 X2 x - 4 X1 x - 19 X2 x 3 16 2 17 18 3 3 12 3 2 13 - 42 X2 x - 24 X2 x - 4 X2 x + 5 X1 X2 x + 11 X1 X2 x 3 14 3 15 2 3 13 2 2 14 + 16 X1 X2 x + 6 X1 x + 46 X1 X2 x + 156 X1 X2 x 2 15 2 16 4 13 3 14 + 114 X1 X2 x + 12 X1 x + 20 X1 X2 x + 171 X1 X2 x 2 15 16 17 4 14 3 15 + 240 X1 X2 x + 80 X1 X2 x + 9 X1 x + 21 X2 x + 71 X2 x 2 16 17 18 3 3 11 3 2 12 + 54 X2 x + 14 X2 x + x - 3 X1 X2 x - 9 X1 X2 x 3 13 3 14 2 3 12 2 2 13 - 6 X1 X2 x - 3 X1 x - 30 X1 X2 x - 89 X1 X2 x 2 14 2 15 4 12 3 13 - 102 X1 X2 x - 25 X1 x - 4 X1 X2 x - 111 X1 X2 x 2 14 15 16 4 13 3 14 - 264 X1 X2 x - 146 X1 X2 x - 15 X1 x - 15 X2 x - 87 X2 x 2 15 16 17 3 3 10 3 2 11 - 99 X2 x - 30 X2 x - 3 x + X1 X2 x + 4 X1 X2 x 3 12 2 3 11 2 2 12 2 13 + 4 X1 X2 x + 24 X1 X2 x + 67 X1 X2 x + 61 X1 X2 x 2 14 3 12 2 13 14 15 + 23 X1 x + 68 X1 X2 x + 199 X1 X2 x + 176 X1 X2 x + 33 X1 x 4 12 3 13 2 14 15 16 3 2 10 + 6 X2 x + 74 X2 x + 134 X2 x + 61 X2 x + 6 x - X1 X2 x 3 11 2 3 10 2 2 11 2 12 - X1 X2 x - 12 X1 X2 x - 50 X1 X2 x - 49 X1 X2 x 2 13 3 11 2 12 13 14 - 12 X1 x - 54 X1 X2 x - 152 X1 X2 x - 142 X1 X2 x - 42 X1 x 4 11 3 12 2 13 14 15 2 3 9 - X2 x - 51 X2 x - 127 X2 x - 90 X2 x - 14 x + 3 X1 X2 x 2 2 10 2 11 2 12 3 10 + 28 X1 X2 x + 35 X1 X2 x + 12 X1 x + 36 X1 X2 x 2 11 12 13 3 11 2 12 + 132 X1 X2 x + 123 X1 X2 x + 34 X1 x + 39 X2 x + 106 X2 x 13 14 2 2 9 2 10 2 11 + 91 X2 x + 22 x - 14 X1 X2 x - 18 X1 X2 x - 9 X1 x 3 9 2 10 11 12 3 10 - 15 X1 X2 x - 99 X1 X2 x - 110 X1 X2 x - 35 X1 x - 30 X2 x 2 11 12 13 2 2 8 2 9 - 98 X2 x - 86 X2 x - 23 x + 5 X1 X2 x + 11 X1 X2 x 2 10 3 8 2 9 10 11 + 2 X1 x + 3 X1 X2 x + 65 X1 X2 x + 81 X1 X2 x + 32 X1 x 3 9 2 10 11 12 2 2 7 2 8 + 17 X2 x + 87 X2 x + 86 X2 x + 25 x - X1 X2 x - 5 X1 X2 x 2 8 9 10 3 8 2 9 - 35 X1 X2 x - 65 X1 X2 x - 18 X1 x - 6 X2 x - 68 X2 x 10 11 2 7 2 7 8 9 - 78 X2 x - 26 x + X1 X2 x + 12 X1 X2 x + 51 X1 X2 x + 15 X1 x 3 7 2 8 9 10 2 6 7 + X2 x + 46 X2 x + 73 X2 x + 21 x - 2 X1 X2 x - 32 X1 X2 x 8 2 7 8 9 6 7 - 18 X1 x - 23 X2 x - 70 X2 x - 22 x + 20 X1 X2 x + 16 X1 x 2 6 7 8 5 6 2 5 6 + 7 X2 x + 58 X2 x + 29 x - 12 X1 X2 x - 11 X1 x - X2 x - 44 X2 x 7 4 5 5 6 3 4 - 32 x + 5 X1 X2 x + 5 X1 x + 31 X2 x + 31 x - X1 X2 x - X1 x 4 5 3 4 2 3 2 - 17 X2 x - 29 x + 6 X2 x + 28 x - X2 x - 25 x + 16 x - 6 x + 1) / 2 4 3 4 4 3 4 2 / ((-1 + x) (X1 X2 x - X1 X2 x - X1 x - X2 x + 2 X2 x + x - X2 x / 3 2 3 3 12 3 2 12 2 3 12 - x + x - 2 x + 1) (X1 X2 x - 3 X1 X2 x - 3 X1 X2 x 3 12 2 3 11 2 2 12 3 12 3 12 + 3 X1 X2 x + 3 X1 X2 x + 9 X1 X2 x + 3 X1 X2 x - X1 x 2 2 11 2 12 3 11 2 12 3 12 - 9 X1 X2 x - 9 X1 X2 x - 6 X1 X2 x - 9 X1 X2 x - X2 x 2 11 2 12 3 10 2 11 12 + 9 X1 X2 x + 3 X1 x + 3 X1 X2 x + 18 X1 X2 x + 9 X1 X2 x 3 11 2 12 2 11 2 10 11 + 3 X2 x + 3 X2 x - 3 X1 x - 9 X1 X2 x - 18 X1 X2 x 12 3 10 2 11 12 2 2 8 10 - 3 X1 x - 3 X2 x - 9 X2 x - 3 X2 x + X1 X2 x + 9 X1 X2 x 11 3 9 2 10 11 12 2 2 7 2 8 + 6 X1 x + X2 x + 9 X2 x + 9 X2 x + x - X1 X2 x - X1 X2 x 2 8 10 2 9 10 11 2 7 - 2 X1 X2 x - 3 X1 x - 3 X2 x - 9 X2 x - 3 x + X1 X2 x 2 7 8 2 8 9 10 2 6 + 4 X1 X2 x + 2 X1 X2 x + X2 x + 3 X2 x + 3 x - 2 X1 X2 x 7 2 7 8 9 6 7 2 6 - 5 X1 X2 x - 3 X2 x - X2 x - x + 4 X1 X2 x + X1 x + 3 X2 x 7 5 6 2 5 6 7 5 5 + 4 X2 x - X1 X2 x - 2 X1 x - X2 x - 6 X2 x - x + X1 x + 4 X2 x 6 4 5 4 3 2 + 3 x - X2 x - 3 x + x - x + 3 x - 3 x + 1)) and in Maple format (X1^4*X2^4*x^18-2*X1^4*X2^4*x^17-4*X1^4*X2^3*x^18-4*X1^3*X2^4*x^18+2*X1^4*X2^4* x^16+6*X1^4*X2^3*x^17+6*X1^4*X2^2*x^18+11*X1^3*X2^4*x^17+16*X1^3*X2^3*x^18+6*X1 ^2*X2^4*x^18-X1^4*X2^4*x^15-6*X1^4*X2^3*x^16-6*X1^4*X2^2*x^17-4*X1^4*X2*x^18-15 *X1^3*X2^4*x^16-36*X1^3*X2^3*x^17-24*X1^3*X2^2*x^18-21*X1^2*X2^4*x^17-24*X1^2* X2^3*x^18-4*X1*X2^4*x^18+3*X1^4*X2^3*x^15+6*X1^4*X2^2*x^16+2*X1^4*X2*x^17+X1^4* x^18+12*X1^3*X2^4*x^15+48*X1^3*X2^3*x^16+42*X1^3*X2^2*x^17+16*X1^3*X2*x^18+36* X1^2*X2^4*x^16+72*X1^2*X2^3*x^17+36*X1^2*X2^2*x^18+17*X1*X2^4*x^17+16*X1*X2^3*x ^18+X2^4*x^18-3*X1^4*X2^2*x^15-2*X1^4*X2*x^16-4*X1^3*X2^4*x^14-42*X1^3*X2^3*x^ 15-54*X1^3*X2^2*x^16-20*X1^3*X2*x^17-4*X1^3*x^18-39*X1^2*X2^4*x^15-120*X1^2*X2^ 3*x^16-90*X1^2*X2^2*x^17-24*X1^2*X2*x^18-35*X1*X2^4*x^16-60*X1*X2^3*x^17-24*X1* X2^2*x^18-5*X2^4*x^17-4*X2^3*x^18+X1^4*X2*x^15+17*X1^3*X2^3*x^14+54*X1^3*X2^2*x ^15+24*X1^3*X2*x^16+3*X1^3*x^17+24*X1^2*X2^4*x^14+142*X1^2*X2^3*x^15+144*X1^2* X2^2*x^16+48*X1^2*X2*x^17+6*X1^2*x^18+47*X1*X2^4*x^15+120*X1*X2^3*x^16+78*X1*X2 ^2*x^17+16*X1*X2*x^18+12*X2^4*x^16+18*X2^3*x^17+6*X2^2*x^18-5*X1^3*X2^3*x^13-26 *X1^3*X2^2*x^14-30*X1^3*X2*x^15-3*X1^3*x^16-6*X1^2*X2^4*x^13-101*X1^2*X2^3*x^14 -192*X1^2*X2^2*x^15-72*X1^2*X2*x^16-9*X1^2*x^17-41*X1*X2^4*x^14-174*X1*X2^3*x^ 15-150*X1*X2^2*x^16-44*X1*X2*x^17-4*X1*x^18-19*X2^4*x^15-42*X2^3*x^16-24*X2^2*x ^17-4*X2*x^18+5*X1^3*X2^3*x^12+11*X1^3*X2^2*x^13+16*X1^3*X2*x^14+6*X1^3*x^15+46 *X1^2*X2^3*x^13+156*X1^2*X2^2*x^14+114*X1^2*X2*x^15+12*X1^2*x^16+20*X1*X2^4*x^ 13+171*X1*X2^3*x^14+240*X1*X2^2*x^15+80*X1*X2*x^16+9*X1*x^17+21*X2^4*x^14+71*X2 ^3*x^15+54*X2^2*x^16+14*X2*x^17+x^18-3*X1^3*X2^3*x^11-9*X1^3*X2^2*x^12-6*X1^3* X2*x^13-3*X1^3*x^14-30*X1^2*X2^3*x^12-89*X1^2*X2^2*x^13-102*X1^2*X2*x^14-25*X1^ 2*x^15-4*X1*X2^4*x^12-111*X1*X2^3*x^13-264*X1*X2^2*x^14-146*X1*X2*x^15-15*X1*x^ 16-15*X2^4*x^13-87*X2^3*x^14-99*X2^2*x^15-30*X2*x^16-3*x^17+X1^3*X2^3*x^10+4*X1 ^3*X2^2*x^11+4*X1^3*X2*x^12+24*X1^2*X2^3*x^11+67*X1^2*X2^2*x^12+61*X1^2*X2*x^13 +23*X1^2*x^14+68*X1*X2^3*x^12+199*X1*X2^2*x^13+176*X1*X2*x^14+33*X1*x^15+6*X2^4 *x^12+74*X2^3*x^13+134*X2^2*x^14+61*X2*x^15+6*x^16-X1^3*X2^2*x^10-X1^3*X2*x^11-\ 12*X1^2*X2^3*x^10-50*X1^2*X2^2*x^11-49*X1^2*X2*x^12-12*X1^2*x^13-54*X1*X2^3*x^ 11-152*X1*X2^2*x^12-142*X1*X2*x^13-42*X1*x^14-X2^4*x^11-51*X2^3*x^12-127*X2^2*x ^13-90*X2*x^14-14*x^15+3*X1^2*X2^3*x^9+28*X1^2*X2^2*x^10+35*X1^2*X2*x^11+12*X1^ 2*x^12+36*X1*X2^3*x^10+132*X1*X2^2*x^11+123*X1*X2*x^12+34*X1*x^13+39*X2^3*x^11+ 106*X2^2*x^12+91*X2*x^13+22*x^14-14*X1^2*X2^2*x^9-18*X1^2*X2*x^10-9*X1^2*x^11-\ 15*X1*X2^3*x^9-99*X1*X2^2*x^10-110*X1*X2*x^11-35*X1*x^12-30*X2^3*x^10-98*X2^2*x ^11-86*X2*x^12-23*x^13+5*X1^2*X2^2*x^8+11*X1^2*X2*x^9+2*X1^2*x^10+3*X1*X2^3*x^8 +65*X1*X2^2*x^9+81*X1*X2*x^10+32*X1*x^11+17*X2^3*x^9+87*X2^2*x^10+86*X2*x^11+25 *x^12-X1^2*X2^2*x^7-5*X1^2*X2*x^8-35*X1*X2^2*x^8-65*X1*X2*x^9-18*X1*x^10-6*X2^3 *x^8-68*X2^2*x^9-78*X2*x^10-26*x^11+X1^2*X2*x^7+12*X1*X2^2*x^7+51*X1*X2*x^8+15* X1*x^9+X2^3*x^7+46*X2^2*x^8+73*X2*x^9+21*x^10-2*X1*X2^2*x^6-32*X1*X2*x^7-18*X1* x^8-23*X2^2*x^7-70*X2*x^8-22*x^9+20*X1*X2*x^6+16*X1*x^7+7*X2^2*x^6+58*X2*x^7+29 *x^8-12*X1*X2*x^5-11*X1*x^6-X2^2*x^5-44*X2*x^6-32*x^7+5*X1*X2*x^4+5*X1*x^5+31* X2*x^5+31*x^6-X1*X2*x^3-X1*x^4-17*X2*x^4-29*x^5+6*X2*x^3+28*x^4-X2*x^2-25*x^3+ 16*x^2-6*x+1)/(-1+x)^2/(X1*X2*x^4-X1*X2*x^3-X1*x^4-X2*x^4+2*X2*x^3+x^4-X2*x^2-x ^3+x^2-2*x+1)/(X1^3*X2^3*x^12-3*X1^3*X2^2*x^12-3*X1^2*X2^3*x^12+3*X1^3*X2*x^12+ 3*X1^2*X2^3*x^11+9*X1^2*X2^2*x^12+3*X1*X2^3*x^12-X1^3*x^12-9*X1^2*X2^2*x^11-9* X1^2*X2*x^12-6*X1*X2^3*x^11-9*X1*X2^2*x^12-X2^3*x^12+9*X1^2*X2*x^11+3*X1^2*x^12 +3*X1*X2^3*x^10+18*X1*X2^2*x^11+9*X1*X2*x^12+3*X2^3*x^11+3*X2^2*x^12-3*X1^2*x^ 11-9*X1*X2^2*x^10-18*X1*X2*x^11-3*X1*x^12-3*X2^3*x^10-9*X2^2*x^11-3*X2*x^12+X1^ 2*X2^2*x^8+9*X1*X2*x^10+6*X1*x^11+X2^3*x^9+9*X2^2*x^10+9*X2*x^11+x^12-X1^2*X2^2 *x^7-X1^2*X2*x^8-2*X1*X2^2*x^8-3*X1*x^10-3*X2^2*x^9-9*X2*x^10-3*x^11+X1^2*X2*x^ 7+4*X1*X2^2*x^7+2*X1*X2*x^8+X2^2*x^8+3*X2*x^9+3*x^10-2*X1*X2^2*x^6-5*X1*X2*x^7-\ 3*X2^2*x^7-X2*x^8-x^9+4*X1*X2*x^6+X1*x^7+3*X2^2*x^6+4*X2*x^7-X1*X2*x^5-2*X1*x^6 -X2^2*x^5-6*X2*x^6-x^7+X1*x^5+4*X2*x^5+3*x^6-X2*x^4-3*x^5+x^4-x^3+3*x^2-3*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 3, 1, 1], are 11 3 n - 7/8 + n/8, and , - -- + ---, respectively, while 64 64 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 1, 1, 2], are 63 7 n - 7/8 + n/8, and , - -- + ---, respectively, while the asymptotic correlatio\ 64 64 n between these two random variables is 1/2 1/2 3 7 --------- 21 and in floating point 0.2182178903 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 21 ate normal pair with correlation, ----- 21 1/2 21 23 i.e. , [[-----, 0], [0, --]] 21 21 ------------------------------------------------- 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, 1, 1], nor the composition, [2, 1, 2, 1] Then infinity ----- 9 7 6 5 4 3 2 \ n x - 2 x + 4 x - 7 x + 10 x - 10 x + 8 x - 4 x + 1 ) a(n) x = -------------------------------------------------------- / 2 3 3 ----- (x - x + 1) (x + x - 1) (-1 + x) n = 0 and in Maple format (x^9-2*x^7+4*x^6-7*x^5+10*x^4-10*x^3+8*x^2-4*x+1)/(x^2-x+1)/(x^3+x-1)/(-1+x)^3 The asymptotic expression for a(n) is, n 6.0594382353848054548 1.4655712318767680267 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 3, 1, 1] and d occurrences (as containment) of the composition, [2, 1, 2, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 11 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 11 2 11 2 2 9 2 10 2 11 - 2 X1 X2 x - 2 X1 X2 x - 2 X1 X2 x - X1 X2 x + X1 x 2 10 11 2 11 2 2 8 2 9 + X1 X2 x + 4 X1 X2 x + X2 x + 2 X1 X2 x + 4 X1 X2 x 2 10 2 9 11 2 10 11 2 2 7 + X1 x + 5 X1 X2 x - 2 X1 x - X2 x - 2 X2 x - X1 X2 x 2 8 2 9 2 8 9 10 2 9 - 3 X1 X2 x - 2 X1 x - 8 X1 X2 x - 10 X1 X2 x - X1 x - 3 X2 x 10 11 2 7 2 8 2 7 8 9 + X2 x + x + X1 X2 x + X1 x + 6 X1 X2 x + 12 X1 X2 x + 5 X1 x 2 8 9 2 6 7 8 2 7 + 7 X2 x + 6 X2 x - 2 X1 X2 x - 9 X1 X2 x - 4 X1 x - 7 X2 x 8 9 6 7 2 6 7 8 - 11 X2 x - 3 x + 4 X1 X2 x + 3 X1 x + 4 X2 x + 12 X2 x + 4 x 5 6 2 5 6 7 4 5 + X1 X2 x - 3 X1 x - X2 x - 9 X2 x - 5 x - 2 X1 X2 x + X1 x 5 6 3 4 5 3 4 2 + 2 X2 x + 6 x + X1 X2 x + 3 X2 x - 3 x - 3 X2 x - 2 x + X2 x 3 2 / 2 + 6 x - 7 x + 4 x - 1) / ((-1 + x) / 4 4 4 3 4 3 4 3 (X1 X2 x - X1 x - X2 x + X2 x + x - x + x - 1) (X1 X2 x - X1 X2 x 4 4 3 4 2 3 2 - X1 x - X2 x + 2 X2 x + x - X2 x - x + x - 2 x + 1)) and in Maple format (X1^2*X2^2*x^11-2*X1^2*X2*x^11-2*X1*X2^2*x^11-2*X1^2*X2^2*x^9-X1^2*X2*x^10+X1^2 *x^11+X1*X2^2*x^10+4*X1*X2*x^11+X2^2*x^11+2*X1^2*X2^2*x^8+4*X1^2*X2*x^9+X1^2*x^ 10+5*X1*X2^2*x^9-2*X1*x^11-X2^2*x^10-2*X2*x^11-X1^2*X2^2*x^7-3*X1^2*X2*x^8-2*X1 ^2*x^9-8*X1*X2^2*x^8-10*X1*X2*x^9-X1*x^10-3*X2^2*x^9+X2*x^10+x^11+X1^2*X2*x^7+ X1^2*x^8+6*X1*X2^2*x^7+12*X1*X2*x^8+5*X1*x^9+7*X2^2*x^8+6*X2*x^9-2*X1*X2^2*x^6-\ 9*X1*X2*x^7-4*X1*x^8-7*X2^2*x^7-11*X2*x^8-3*x^9+4*X1*X2*x^6+3*X1*x^7+4*X2^2*x^6 +12*X2*x^7+4*x^8+X1*X2*x^5-3*X1*x^6-X2^2*x^5-9*X2*x^6-5*x^7-2*X1*X2*x^4+X1*x^5+ 2*X2*x^5+6*x^6+X1*X2*x^3+3*X2*x^4-3*x^5-3*X2*x^3-2*x^4+X2*x^2+6*x^3-7*x^2+4*x-1 )/(-1+x)^2/(X1*X2*x^4-X1*x^4-X2*x^4+X2*x^3+x^4-x^3+x-1)/(X1*X2*x^4-X1*X2*x^3-X1 *x^4-X2*x^4+2*X2*x^3+x^4-X2*x^2-x^3+x^2-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 3, 1, 1], are 11 3 n - 7/8 + n/8, and , - -- + ---, respectively, while 64 64 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 1, 2, 1], are 55 7 n - 7/8 + n/8, and , - -- + ---, respectively, while the asymptotic correlatio\ 64 64 n between these two random variables is 1/2 1/2 3 7 --------- 21 and in floating point 0.2182178903 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 21 ate normal pair with correlation, ----- 21 1/2 21 23 i.e. , [[-----, 0], [0, --]] 21 21 ------------------------------------------------- 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, 1, 1], nor the composition, [2, 2, 1, 1] Then infinity ----- 5 4 3 2 \ n x - 3 x + 3 x - 4 x + 3 x - 1 ) a(n) x = - --------------------------------- / 3 3 ----- (x + x - 1) (-1 + x) n = 0 and in Maple format -(x^5-3*x^4+3*x^3-4*x^2+3*x-1)/(x^3+x-1)/(-1+x)^3 The asymptotic expression for a(n) is, n 6.9556233073109364759 1.4655712318767680267 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 3, 1, 1] and d occurrences (as containment) of the composition, [2, 2, 1, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 5 ) | ) | ) A(n, c, d) x X1 X2 || = - (2 X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 4 5 5 3 4 4 5 3 - 2 X1 X2 x - X1 x - 2 X2 x + X1 X2 x + X1 x + 4 X2 x + x - 3 X2 x 4 2 3 2 / 2 4 - 3 x + X2 x + 3 x - 4 x + 3 x - 1) / ((-1 + x) (X1 X2 x / 3 4 4 3 4 2 3 2 - X1 X2 x - X1 x - X2 x + 2 X2 x + x - X2 x - x + x - 2 x + 1)) and in Maple format -(2*X1*X2*x^5-2*X1*X2*x^4-X1*x^5-2*X2*x^5+X1*X2*x^3+X1*x^4+4*X2*x^4+x^5-3*X2*x^ 3-3*x^4+X2*x^2+3*x^3-4*x^2+3*x-1)/(-1+x)^2/(X1*X2*x^4-X1*X2*x^3-X1*x^4-X2*x^4+2 *X2*x^3+x^4-X2*x^2-x^3+x^2-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 3, 1, 1], are 11 3 n - 7/8 + n/8, and , - -- + ---, respectively, while 64 64 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 2, 1, 1], are 47 7 n - 7/8 + n/8, and , - -- + ---, respectively, while the asymptotic correlatio\ 64 64 n between these two random variables is 1/2 1/2 3 7 --------- 21 and in floating point 0.2182178903 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 21 ate normal pair with correlation, ----- 21 1/2 21 23 i.e. , [[-----, 0], [0, --]] 21 21 ------------------------------------------------- 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, 1, 1], nor the composition, [3, 1, 1, 1] Then infinity ----- 7 6 5 4 2 \ n x + x - x + x - 3 x + 3 x - 1 ) a(n) x = - ---------------------------------- / 2 3 ----- (x + x - 1) (-1 + x) n = 0 and in Maple format -(x^7+x^6-x^5+x^4-3*x^2+3*x-1)/(x^2+x-1)/(-1+x)^3 The asymptotic expression for a(n) is, n 1.8944271909999158785 1.6180339887498948482 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [1, 3, 1, 1] and d occurrences (as containment) of the composition, [3, 1, 1, 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 7 5 6 6 7 4 6 3 - X1 x - X2 x + X1 X2 x - X1 x - X2 x + x - X1 X2 x + x + X1 X2 x 5 4 2 / 2 3 3 - x + x - 3 x + 3 x - 1) / ((-1 + x) (X1 X2 x - x + 2 x - 1)) / and in Maple format (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-X1*X2*x^4+x^6+X1 *X2*x^3-x^5+x^4-3*x^2+3*x-1)/(-1+x)^2/(X1*X2*x^3-x^3+2*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [1, 3, 1, 1], are 11 3 n - 7/8 + n/8, and , - -- + ---, respectively, while 64 64 Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 1, 1, 1], are 11 3 n - 7/8 + n/8, and , - -- + ---, respectively, while the asymptotic correlatio\ 64 64 n between these two random variables is 1 and in floating point 1. ------------------------------------------------- Theorem Number, 40, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 1, 1, 2], nor the composition, [2, 1, 2, 1] Then infinity ----- 7 6 4 3 2 \ n x - x - x + 2 x - 4 x + 3 x - 1 ) a(n) x = - --------------------------------------------- / 7 6 5 3 2 ----- (-1 + x) (x - x + x + x - 3 x + 3 x - 1) n = 0 and in Maple format -(x^7-x^6-x^4+2*x^3-4*x^2+3*x-1)/(-1+x)/(x^7-x^6+x^5+x^3-3*x^2+3*x-1) The asymptotic expression for a(n) is, n 1.5121946674937959604 1.6530424890094669421 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 1, 1, 2] and d occurrences (as containment) of the composition, [2, 1, 2, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 4 2 11 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 3 2 11 3 2 10 2 2 11 3 2 9 3 10 - 4 X1 X2 x - X1 X2 x + 6 X1 X2 x - X1 X2 x + X1 X2 x 2 2 10 2 11 3 2 8 3 9 2 2 9 + 3 X1 X2 x - 4 X1 X2 x + X1 X2 x - X1 X2 x + 2 X1 X2 x 2 10 2 10 2 11 3 2 7 2 2 8 - 3 X1 X2 x - 3 X1 X2 x + X2 x - X1 X2 x - X1 X2 x 2 9 2 9 10 2 10 2 2 7 + 4 X1 X2 x - X1 X2 x + 3 X1 X2 x + X2 x + X1 X2 x 2 8 2 8 9 10 2 7 - 2 X1 X2 x + X1 X2 x - 5 X1 X2 x - X2 x + 4 X1 X2 x 2 7 9 2 6 2 6 7 8 - 2 X1 X2 x + 2 X2 x - 3 X1 X2 x + 2 X1 X2 x - X1 X2 x + 2 X1 x 2 5 2 5 6 7 7 8 5 + X1 X2 x - X1 X2 x + X1 X2 x - 4 X1 x + X2 x - x - 2 X1 X2 x 6 6 7 4 5 5 6 + 4 X1 x - 3 X2 x + 2 x + 4 X1 X2 x - 3 X1 x + 4 X2 x - x 3 4 4 5 2 3 4 3 - 3 X1 X2 x + X1 x - 3 X2 x + x + X1 X2 x + X2 x - 3 x + 6 x 2 / 4 2 12 4 2 11 3 2 12 - 7 x + 4 x - 1) / (X1 X2 x - X1 X2 x - 4 X1 X2 x / 3 2 11 2 2 12 3 11 2 2 11 + 3 X1 X2 x + 6 X1 X2 x + X1 X2 x - 3 X1 X2 x 2 12 3 2 9 3 10 2 11 2 11 - 4 X1 X2 x + X1 X2 x - 2 X1 X2 x - 3 X1 X2 x + X1 X2 x 2 12 3 2 8 3 9 2 2 9 2 10 + X2 x - 2 X1 X2 x + X1 X2 x - X1 X2 x + 6 X1 X2 x 11 3 2 7 2 2 8 2 9 10 + 3 X1 X2 x + X1 X2 x + 2 X1 X2 x - 5 X1 X2 x - 6 X1 X2 x 11 2 2 7 2 8 2 8 9 10 - X2 x - X1 X2 x + 5 X1 X2 x - X1 X2 x + 5 X1 X2 x + 2 X2 x 2 7 2 7 8 9 9 2 6 - 6 X1 X2 x + 3 X1 X2 x - 3 X1 X2 x + X1 x - X2 x + 4 X1 X2 x 2 6 7 8 9 2 5 2 5 6 - 3 X1 X2 x + X1 X2 x - 4 X1 x - x - X1 X2 x + X1 X2 x - X1 X2 x 7 7 8 5 6 6 7 + 7 X1 x - X2 x + 3 x + 4 X1 X2 x - 7 X1 x + 4 X2 x - 4 x 4 5 5 6 3 4 4 - 6 X1 X2 x + 4 X1 x - 6 X2 x + 3 x + 4 X1 X2 x - X1 x + 4 X2 x 5 2 3 4 3 2 - 2 x - X1 X2 x - X2 x + 5 x - 10 x + 10 x - 5 x + 1) and in Maple format -(X1^4*X2^2*x^11-4*X1^3*X2^2*x^11-X1^3*X2^2*x^10+6*X1^2*X2^2*x^11-X1^3*X2^2*x^9 +X1^3*X2*x^10+3*X1^2*X2^2*x^10-4*X1*X2^2*x^11+X1^3*X2^2*x^8-X1^3*X2*x^9+2*X1^2* X2^2*x^9-3*X1^2*X2*x^10-3*X1*X2^2*x^10+X2^2*x^11-X1^3*X2^2*x^7-X1^2*X2^2*x^8+4* X1^2*X2*x^9-X1*X2^2*x^9+3*X1*X2*x^10+X2^2*x^10+X1^2*X2^2*x^7-2*X1^2*X2*x^8+X1* X2^2*x^8-5*X1*X2*x^9-X2*x^10+4*X1^2*X2*x^7-2*X1*X2^2*x^7+2*X2*x^9-3*X1^2*X2*x^6 +2*X1*X2^2*x^6-X1*X2*x^7+2*X1*x^8+X1^2*X2*x^5-X1*X2^2*x^5+X1*X2*x^6-4*X1*x^7+X2 *x^7-x^8-2*X1*X2*x^5+4*X1*x^6-3*X2*x^6+2*x^7+4*X1*X2*x^4-3*X1*x^5+4*X2*x^5-x^6-\ 3*X1*X2*x^3+X1*x^4-3*X2*x^4+x^5+X1*X2*x^2+X2*x^3-3*x^4+6*x^3-7*x^2+4*x-1)/(X1^4 *X2^2*x^12-X1^4*X2^2*x^11-4*X1^3*X2^2*x^12+3*X1^3*X2^2*x^11+6*X1^2*X2^2*x^12+X1 ^3*X2*x^11-3*X1^2*X2^2*x^11-4*X1*X2^2*x^12+X1^3*X2^2*x^9-2*X1^3*X2*x^10-3*X1^2* X2*x^11+X1*X2^2*x^11+X2^2*x^12-2*X1^3*X2^2*x^8+X1^3*X2*x^9-X1^2*X2^2*x^9+6*X1^2 *X2*x^10+3*X1*X2*x^11+X1^3*X2^2*x^7+2*X1^2*X2^2*x^8-5*X1^2*X2*x^9-6*X1*X2*x^10- X2*x^11-X1^2*X2^2*x^7+5*X1^2*X2*x^8-X1*X2^2*x^8+5*X1*X2*x^9+2*X2*x^10-6*X1^2*X2 *x^7+3*X1*X2^2*x^7-3*X1*X2*x^8+X1*x^9-X2*x^9+4*X1^2*X2*x^6-3*X1*X2^2*x^6+X1*X2* x^7-4*X1*x^8-x^9-X1^2*X2*x^5+X1*X2^2*x^5-X1*X2*x^6+7*X1*x^7-X2*x^7+3*x^8+4*X1* X2*x^5-7*X1*x^6+4*X2*x^6-4*x^7-6*X1*X2*x^4+4*X1*x^5-6*X2*x^5+3*x^6+4*X1*X2*x^3- X1*x^4+4*X2*x^4-2*x^5-X1*X2*x^2-X2*x^3+5*x^4-10*x^3+10*x^2-5*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 1, 1, 2], are 63 7 n - 7/8 + n/8, and , - -- + ---, respectively, while 64 64 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 1, 2, 1], are 55 7 n - 7/8 + n/8, and , - -- + ---, respectively, while the asymptotic correlatio\ 64 64 n between these two random variables is 5/7 and in floating point 0.7142857143 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ ate normal pair with correlation, 5/7 99 i.e. , [[5/7, 0], [0, --]] 49 ------------------------------------------------- Theorem Number, 41, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 1, 1, 2], nor the composition, [2, 2, 1, 1] Then infinity ----- 7 6 5 4 3 2 \ n 2 x - 3 x + 5 x - 6 x + 7 x - 7 x + 4 x - 1 ) a(n) x = - ------------------------------------------------- / 5 4 3 2 3 ----- (x - x + x - x + 2 x - 1) (-1 + x) n = 0 and in Maple format -(2*x^7-3*x^6+5*x^5-6*x^4+7*x^3-7*x^2+4*x-1)/(x^5-x^4+x^3-x^2+2*x-1)/(-1+x)^3 The asymptotic expression for a(n) is, n 1.2670431762094331495 1.6736485462998415616 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 1, 1, 2] and d occurrences (as containment) of the composition, [2, 2, 1, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 4 2 11 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 3 2 11 3 2 10 2 2 11 3 2 9 3 10 - 4 X1 X2 x - X1 X2 x + 6 X1 X2 x - X1 X2 x + X1 X2 x 2 2 10 2 11 3 2 8 3 9 2 2 9 + 2 X1 X2 x - 4 X1 X2 x + X1 X2 x - X1 X2 x + 3 X1 X2 x 2 10 2 10 2 11 3 2 7 2 2 8 - 2 X1 X2 x - X1 X2 x + X2 x - X1 X2 x - 2 X1 X2 x 2 9 2 9 10 2 2 7 2 8 + 3 X1 X2 x - 3 X1 X2 x + X1 X2 x + X1 X2 x - X1 X2 x 2 8 9 2 9 2 7 8 2 8 + 2 X1 X2 x - 3 X1 X2 x + X2 x + 4 X1 X2 x + X1 X2 x - X2 x 9 2 6 7 2 5 6 7 + X2 x - 3 X1 X2 x - 5 X1 X2 x + X1 X2 x + 4 X1 X2 x - 2 X1 x 7 5 6 6 7 4 5 + X2 x - 3 X1 X2 x + 3 X1 x - X2 x + 2 x + 4 X1 X2 x - 3 X1 x 6 3 4 5 2 4 3 2 - 3 x - 3 X1 X2 x + X1 x + 5 x + X1 X2 x - 6 x + 7 x - 7 x + 4 x / 4 2 12 4 2 11 3 2 12 3 2 11 - 1) / (X1 X2 x - X1 X2 x - 4 X1 X2 x + 3 X1 X2 x / 2 2 12 3 11 2 2 11 2 12 3 2 9 + 6 X1 X2 x + X1 X2 x - 3 X1 X2 x - 4 X1 X2 x + X1 X2 x 3 10 2 2 10 2 11 2 11 2 12 - 2 X1 X2 x + X1 X2 x - 3 X1 X2 x + X1 X2 x + X2 x 3 2 8 3 9 2 2 9 2 10 2 10 - 2 X1 X2 x + X1 X2 x - 3 X1 X2 x + 5 X1 X2 x - 2 X1 X2 x 11 3 2 7 2 2 8 2 9 2 9 + 3 X1 X2 x + X1 X2 x + 3 X1 X2 x - 3 X1 X2 x + 3 X1 X2 x 10 2 10 11 2 2 7 2 8 2 8 - 4 X1 X2 x + X2 x - X2 x - X1 X2 x + 4 X1 X2 x - X1 X2 x 9 2 9 10 2 7 8 9 + 3 X1 X2 x - X2 x + X2 x - 6 X1 X2 x - 6 X1 X2 x - X2 x 2 6 7 8 8 2 5 6 + 4 X1 X2 x + 7 X1 X2 x - X1 x + 2 X2 x - X1 X2 x - 5 X1 X2 x 7 7 8 5 6 7 4 + 4 X1 x - X2 x + x + 5 X1 X2 x - 6 X1 x - 4 x - 6 X1 X2 x 5 6 3 4 5 2 4 3 + 4 X1 x + 7 x + 4 X1 X2 x - X1 x - 8 x - X1 X2 x + 9 x - 11 x 2 + 10 x - 5 x + 1) and in Maple format -(X1^4*X2^2*x^11-4*X1^3*X2^2*x^11-X1^3*X2^2*x^10+6*X1^2*X2^2*x^11-X1^3*X2^2*x^9 +X1^3*X2*x^10+2*X1^2*X2^2*x^10-4*X1*X2^2*x^11+X1^3*X2^2*x^8-X1^3*X2*x^9+3*X1^2* X2^2*x^9-2*X1^2*X2*x^10-X1*X2^2*x^10+X2^2*x^11-X1^3*X2^2*x^7-2*X1^2*X2^2*x^8+3* X1^2*X2*x^9-3*X1*X2^2*x^9+X1*X2*x^10+X1^2*X2^2*x^7-X1^2*X2*x^8+2*X1*X2^2*x^8-3* X1*X2*x^9+X2^2*x^9+4*X1^2*X2*x^7+X1*X2*x^8-X2^2*x^8+X2*x^9-3*X1^2*X2*x^6-5*X1* X2*x^7+X1^2*X2*x^5+4*X1*X2*x^6-2*X1*x^7+X2*x^7-3*X1*X2*x^5+3*X1*x^6-X2*x^6+2*x^ 7+4*X1*X2*x^4-3*X1*x^5-3*x^6-3*X1*X2*x^3+X1*x^4+5*x^5+X1*X2*x^2-6*x^4+7*x^3-7*x ^2+4*x-1)/(X1^4*X2^2*x^12-X1^4*X2^2*x^11-4*X1^3*X2^2*x^12+3*X1^3*X2^2*x^11+6*X1 ^2*X2^2*x^12+X1^3*X2*x^11-3*X1^2*X2^2*x^11-4*X1*X2^2*x^12+X1^3*X2^2*x^9-2*X1^3* X2*x^10+X1^2*X2^2*x^10-3*X1^2*X2*x^11+X1*X2^2*x^11+X2^2*x^12-2*X1^3*X2^2*x^8+X1 ^3*X2*x^9-3*X1^2*X2^2*x^9+5*X1^2*X2*x^10-2*X1*X2^2*x^10+3*X1*X2*x^11+X1^3*X2^2* x^7+3*X1^2*X2^2*x^8-3*X1^2*X2*x^9+3*X1*X2^2*x^9-4*X1*X2*x^10+X2^2*x^10-X2*x^11- X1^2*X2^2*x^7+4*X1^2*X2*x^8-X1*X2^2*x^8+3*X1*X2*x^9-X2^2*x^9+X2*x^10-6*X1^2*X2* x^7-6*X1*X2*x^8-X2*x^9+4*X1^2*X2*x^6+7*X1*X2*x^7-X1*x^8+2*X2*x^8-X1^2*X2*x^5-5* X1*X2*x^6+4*X1*x^7-X2*x^7+x^8+5*X1*X2*x^5-6*X1*x^6-4*x^7-6*X1*X2*x^4+4*X1*x^5+7 *x^6+4*X1*X2*x^3-X1*x^4-8*x^5-X1*X2*x^2+9*x^4-11*x^3+10*x^2-5*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 1, 1, 2], are 63 7 n - 7/8 + n/8, and , - -- + ---, respectively, while 64 64 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 2, 1, 1], are 47 7 n - 7/8 + n/8, and , - -- + ---, respectively, while the asymptotic correlatio\ 64 64 n between these two random variables is 5/7 and in floating point 0.7142857143 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ ate normal pair with correlation, 5/7 99 i.e. , [[5/7, 0], [0, --]] 49 ------------------------------------------------- Theorem Number, 42, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 1, 1, 2], nor the composition, [3, 1, 1, 1] Then infinity ----- \ n ) a(n) x = / ----- n = 0 12 11 9 8 6 5 4 3 2 x - x - 2 x + x + 3 x - x + 3 x - x + 2 x - 2 x + 1 -------------------------------------------------------------- 3 9 4 2 (x + x - 1) (x - x - 1) (-1 + x) and in Maple format (x^12-x^11-2*x^9+x^8+3*x^6-x^5+3*x^4-x^3+2*x^2-2*x+1)/(x^3+x-1)/(x^9-x^4-1)/(-1 +x)^2 The asymptotic expression for a(n) is, n 5.2787205554750117559 1.4655712318767680267 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 1, 1, 2] and d occurrences (as containment) of the composition, [3, 1, 1, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 4 4 15 ) | ) | ) A(n, c, d) x X1 X2 || = - (X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 4 3 15 3 4 15 4 3 14 4 2 15 - 4 X1 X2 x - 3 X1 X2 x + 4 X1 X2 x + 6 X1 X2 x 3 3 15 2 4 15 4 2 14 4 15 + 12 X1 X2 x + 3 X1 X2 x - 12 X1 X2 x - 4 X1 X2 x 3 3 14 3 2 15 2 3 15 4 15 - 13 X1 X2 x - 18 X1 X2 x - 12 X1 X2 x - X1 X2 x 4 2 13 4 14 4 15 3 2 14 3 15 + 6 X1 X2 x + 12 X1 X2 x + X1 x + 39 X1 X2 x + 12 X1 X2 x 2 3 14 2 2 15 3 15 4 13 + 15 X1 X2 x + 18 X1 X2 x + 4 X1 X2 x - 12 X1 X2 x 4 14 3 3 12 3 2 13 3 14 3 15 - 4 X1 x - 2 X1 X2 x - 21 X1 X2 x - 39 X1 X2 x - 3 X1 x 2 2 14 2 15 3 14 2 15 - 45 X1 X2 x - 12 X1 X2 x - 7 X1 X2 x - 6 X1 X2 x 4 12 4 13 3 3 11 3 2 12 3 13 + 4 X1 X2 x + 6 X1 x + X1 X2 x + 6 X1 X2 x + 42 X1 X2 x 3 14 2 3 12 2 2 13 2 14 2 15 + 13 X1 x + 4 X1 X2 x + 27 X1 X2 x + 45 X1 X2 x + 3 X1 x 2 14 15 3 14 4 12 3 3 10 + 21 X1 X2 x + 4 X1 X2 x + X2 x - 4 X1 x - X1 X2 x 3 2 11 3 12 3 13 2 3 11 - 9 X1 X2 x - 21 X1 X2 x - 21 X1 x - 2 X1 X2 x 2 2 12 2 13 2 14 3 12 - 12 X1 X2 x - 54 X1 X2 x - 15 X1 x - 2 X1 X2 x 2 13 14 15 2 14 4 11 - 15 X1 X2 x - 21 X1 X2 x - X1 x - 3 X2 x + X1 x 3 2 10 3 11 3 12 2 3 10 + 6 X1 X2 x + 15 X1 X2 x + 17 X1 x + X1 X2 x 2 2 11 2 12 2 13 3 11 2 12 + 20 X1 X2 x + 33 X1 X2 x + 27 X1 x + X1 X2 x + 6 X1 X2 x 13 14 2 13 14 3 2 9 + 30 X1 X2 x + 7 X1 x + 3 X2 x + 3 X2 x - 3 X1 X2 x 3 10 3 11 2 2 10 2 11 2 12 - 15 X1 X2 x - 11 X1 x - 10 X1 X2 x - 34 X1 X2 x - 25 X1 x 2 11 12 13 13 14 3 9 - 13 X1 X2 x - 19 X1 X2 x - 15 X1 x - 6 X2 x - x + 9 X1 X2 x 3 10 2 2 9 2 10 2 11 2 10 + 10 X1 x + 7 X1 X2 x + 33 X1 X2 x + 22 X1 x + 5 X1 X2 x 11 12 2 11 12 13 3 8 + 23 X1 X2 x + 15 X1 x + 2 X2 x + 3 X2 x + 3 x - 3 X1 X2 x 3 9 2 2 8 2 9 2 10 2 9 - 8 X1 x - 3 X1 X2 x - 22 X1 X2 x - 24 X1 x - 4 X1 X2 x 10 11 2 10 11 12 3 8 - 24 X1 X2 x - 15 X1 x - X2 x - 4 X2 x - 3 x + 4 X1 x 2 2 7 2 8 2 9 2 8 9 + X1 X2 x + 17 X1 X2 x + 21 X1 x + 3 X1 X2 x + 15 X1 X2 x 10 10 11 3 7 2 7 2 8 + 19 X1 x + 6 X2 x + 3 x - X1 x - 8 X1 X2 x - 17 X1 x 2 7 8 9 9 10 2 6 - X1 X2 x - 17 X1 X2 x - 17 X1 x - 2 X2 x - 5 x + 2 X1 X2 x 2 7 7 8 8 9 2 6 + 12 X1 x + 12 X1 X2 x + 17 X1 x + 3 X2 x + 4 x - 5 X1 x 6 7 7 8 2 5 5 6 - 7 X1 X2 x - 18 X1 x - 4 X2 x - 4 x + X1 x + 5 X1 X2 x + 14 X1 x 6 7 4 5 5 6 3 + 4 X2 x + 7 x - 3 X1 X2 x - 11 X1 x - 3 X2 x - 8 x + X1 X2 x 4 4 5 3 4 2 3 2 + 8 X1 x + X2 x + 8 x - 4 X1 x - 7 x + X1 x + 7 x - 7 x + 4 x - 1) / 4 3 4 4 3 4 2 3 2 / ((X1 X2 x - X1 X2 x - X1 x - X2 x + 2 X1 x + x - X1 x - x + x / 3 3 12 3 2 12 2 3 12 3 2 11 - 2 x + 1) (X1 X2 x - 3 X1 X2 x - 3 X1 X2 x + 3 X1 X2 x 3 12 2 2 12 3 12 3 11 3 12 + 3 X1 X2 x + 9 X1 X2 x + 3 X1 X2 x - 6 X1 X2 x - X1 x 2 2 11 2 12 2 12 3 12 3 10 - 9 X1 X2 x - 9 X1 X2 x - 9 X1 X2 x - X2 x + 3 X1 X2 x 3 11 2 11 2 12 2 11 12 + 3 X1 x + 18 X1 X2 x + 3 X1 x + 9 X1 X2 x + 9 X1 X2 x 2 12 3 10 2 10 2 11 11 + 3 X2 x - 3 X1 x - 9 X1 X2 x - 9 X1 x - 18 X1 X2 x 12 2 11 12 3 9 2 2 8 2 10 - 3 X1 x - 3 X2 x - 3 X2 x + X1 x + X1 X2 x + 9 X1 x 10 11 11 12 2 2 7 2 8 + 9 X1 X2 x + 9 X1 x + 6 X2 x + x - X1 X2 x - 2 X1 X2 x 2 9 2 8 10 10 11 2 7 - 3 X1 x - X1 X2 x - 9 X1 x - 3 X2 x - 3 x + 4 X1 X2 x 2 8 2 7 8 9 10 2 6 + X1 x + X1 X2 x + 2 X1 X2 x + 3 X1 x + 3 x - 2 X1 X2 x 2 7 7 8 9 2 6 6 7 - 3 X1 x - 5 X1 X2 x - X1 x - x + 3 X1 x + 4 X1 X2 x + 4 X1 x 7 2 5 5 6 6 7 5 5 + X2 x - X1 x - X1 X2 x - 6 X1 x - 2 X2 x - x + 4 X1 x + X2 x 6 4 5 4 3 2 + 3 x - X1 x - 3 x + x - x + 3 x - 3 x + 1)) and in Maple format -(X1^4*X2^4*x^15-4*X1^4*X2^3*x^15-3*X1^3*X2^4*x^15+4*X1^4*X2^3*x^14+6*X1^4*X2^2 *x^15+12*X1^3*X2^3*x^15+3*X1^2*X2^4*x^15-12*X1^4*X2^2*x^14-4*X1^4*X2*x^15-13*X1 ^3*X2^3*x^14-18*X1^3*X2^2*x^15-12*X1^2*X2^3*x^15-X1*X2^4*x^15+6*X1^4*X2^2*x^13+ 12*X1^4*X2*x^14+X1^4*x^15+39*X1^3*X2^2*x^14+12*X1^3*X2*x^15+15*X1^2*X2^3*x^14+ 18*X1^2*X2^2*x^15+4*X1*X2^3*x^15-12*X1^4*X2*x^13-4*X1^4*x^14-2*X1^3*X2^3*x^12-\ 21*X1^3*X2^2*x^13-39*X1^3*X2*x^14-3*X1^3*x^15-45*X1^2*X2^2*x^14-12*X1^2*X2*x^15 -7*X1*X2^3*x^14-6*X1*X2^2*x^15+4*X1^4*X2*x^12+6*X1^4*x^13+X1^3*X2^3*x^11+6*X1^3 *X2^2*x^12+42*X1^3*X2*x^13+13*X1^3*x^14+4*X1^2*X2^3*x^12+27*X1^2*X2^2*x^13+45* X1^2*X2*x^14+3*X1^2*x^15+21*X1*X2^2*x^14+4*X1*X2*x^15+X2^3*x^14-4*X1^4*x^12-X1^ 3*X2^3*x^10-9*X1^3*X2^2*x^11-21*X1^3*X2*x^12-21*X1^3*x^13-2*X1^2*X2^3*x^11-12* X1^2*X2^2*x^12-54*X1^2*X2*x^13-15*X1^2*x^14-2*X1*X2^3*x^12-15*X1*X2^2*x^13-21* X1*X2*x^14-X1*x^15-3*X2^2*x^14+X1^4*x^11+6*X1^3*X2^2*x^10+15*X1^3*X2*x^11+17*X1 ^3*x^12+X1^2*X2^3*x^10+20*X1^2*X2^2*x^11+33*X1^2*X2*x^12+27*X1^2*x^13+X1*X2^3*x ^11+6*X1*X2^2*x^12+30*X1*X2*x^13+7*X1*x^14+3*X2^2*x^13+3*X2*x^14-3*X1^3*X2^2*x^ 9-15*X1^3*X2*x^10-11*X1^3*x^11-10*X1^2*X2^2*x^10-34*X1^2*X2*x^11-25*X1^2*x^12-\ 13*X1*X2^2*x^11-19*X1*X2*x^12-15*X1*x^13-6*X2*x^13-x^14+9*X1^3*X2*x^9+10*X1^3*x ^10+7*X1^2*X2^2*x^9+33*X1^2*X2*x^10+22*X1^2*x^11+5*X1*X2^2*x^10+23*X1*X2*x^11+ 15*X1*x^12+2*X2^2*x^11+3*X2*x^12+3*x^13-3*X1^3*X2*x^8-8*X1^3*x^9-3*X1^2*X2^2*x^ 8-22*X1^2*X2*x^9-24*X1^2*x^10-4*X1*X2^2*x^9-24*X1*X2*x^10-15*X1*x^11-X2^2*x^10-\ 4*X2*x^11-3*x^12+4*X1^3*x^8+X1^2*X2^2*x^7+17*X1^2*X2*x^8+21*X1^2*x^9+3*X1*X2^2* x^8+15*X1*X2*x^9+19*X1*x^10+6*X2*x^10+3*x^11-X1^3*x^7-8*X1^2*X2*x^7-17*X1^2*x^8 -X1*X2^2*x^7-17*X1*X2*x^8-17*X1*x^9-2*X2*x^9-5*x^10+2*X1^2*X2*x^6+12*X1^2*x^7+ 12*X1*X2*x^7+17*X1*x^8+3*X2*x^8+4*x^9-5*X1^2*x^6-7*X1*X2*x^6-18*X1*x^7-4*X2*x^7 -4*x^8+X1^2*x^5+5*X1*X2*x^5+14*X1*x^6+4*X2*x^6+7*x^7-3*X1*X2*x^4-11*X1*x^5-3*X2 *x^5-8*x^6+X1*X2*x^3+8*X1*x^4+X2*x^4+8*x^5-4*X1*x^3-7*x^4+X1*x^2+7*x^3-7*x^2+4* x-1)/(X1*X2*x^4-X1*X2*x^3-X1*x^4-X2*x^4+2*X1*x^3+x^4-X1*x^2-x^3+x^2-2*x+1)/(X1^ 3*X2^3*x^12-3*X1^3*X2^2*x^12-3*X1^2*X2^3*x^12+3*X1^3*X2^2*x^11+3*X1^3*X2*x^12+9 *X1^2*X2^2*x^12+3*X1*X2^3*x^12-6*X1^3*X2*x^11-X1^3*x^12-9*X1^2*X2^2*x^11-9*X1^2 *X2*x^12-9*X1*X2^2*x^12-X2^3*x^12+3*X1^3*X2*x^10+3*X1^3*x^11+18*X1^2*X2*x^11+3* X1^2*x^12+9*X1*X2^2*x^11+9*X1*X2*x^12+3*X2^2*x^12-3*X1^3*x^10-9*X1^2*X2*x^10-9* X1^2*x^11-18*X1*X2*x^11-3*X1*x^12-3*X2^2*x^11-3*X2*x^12+X1^3*x^9+X1^2*X2^2*x^8+ 9*X1^2*x^10+9*X1*X2*x^10+9*X1*x^11+6*X2*x^11+x^12-X1^2*X2^2*x^7-2*X1^2*X2*x^8-3 *X1^2*x^9-X1*X2^2*x^8-9*X1*x^10-3*X2*x^10-3*x^11+4*X1^2*X2*x^7+X1^2*x^8+X1*X2^2 *x^7+2*X1*X2*x^8+3*X1*x^9+3*x^10-2*X1^2*X2*x^6-3*X1^2*x^7-5*X1*X2*x^7-X1*x^8-x^ 9+3*X1^2*x^6+4*X1*X2*x^6+4*X1*x^7+X2*x^7-X1^2*x^5-X1*X2*x^5-6*X1*x^6-2*X2*x^6-x ^7+4*X1*x^5+X2*x^5+3*x^6-X1*x^4-3*x^5+x^4-x^3+3*x^2-3*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 1, 1, 2], are 63 7 n - 7/8 + n/8, and , - -- + ---, respectively, while 64 64 Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 1, 1, 1], are 11 3 n - 7/8 + n/8, and , - -- + ---, respectively, while the asymptotic correlatio\ 64 64 n between these two random variables is 1/2 1/2 3 7 --------- 21 and in floating point 0.2182178903 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 21 ate normal pair with correlation, ----- 21 1/2 21 23 i.e. , [[-----, 0], [0, --]] 21 21 ------------------------------------------------- Theorem Number, 43, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 1, 2, 1], nor the composition, [2, 2, 1, 1] Then infinity ----- 6 5 3 2 \ n x - 2 x + 2 x - 4 x + 3 x - 1 ) a(n) x = ----------------------------------- / 2 2 2 ----- (x - x + 1) (x + x - 1) (-1 + x) n = 0 and in Maple format (x^6-2*x^5+2*x^3-4*x^2+3*x-1)/(x^2-x+1)/(x^2+x-1)/(-1+x)^2 The asymptotic expression for a(n) is, n 2.1180339887498948478 1.6180339887498948482 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 1, 2, 1] and d occurrences (as containment) of the composition, [2, 2, 1, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 7 2 6 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x - X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 7 2 5 6 7 5 6 - 2 X1 X2 x + X1 X2 x + X1 X2 x + X2 x - X1 X2 x + X1 x 4 5 6 3 4 5 2 - 2 X1 X2 x - 2 X1 x - x + 2 X1 X2 x + 2 X1 x + 2 x - X1 X2 x 3 3 2 / 2 6 2 5 - X1 x - 2 x + 4 x - 3 x + 1) / ((-1 + x) (X1 X2 x - X1 X2 x / 6 5 6 4 5 3 4 - 2 X1 X2 x + X1 X2 x + X2 x + X1 X2 x + X1 x - 2 X1 X2 x - 2 X1 x 5 2 3 4 3 2 - x + X1 X2 x + X1 x + x + x - 3 x + 3 x - 1)) and in Maple format (X1^2*X2*x^7-X1^2*X2*x^6-2*X1*X2*x^7+X1^2*X2*x^5+X1*X2*x^6+X2*x^7-X1*X2*x^5+X1* x^6-2*X1*X2*x^4-2*X1*x^5-x^6+2*X1*X2*x^3+2*X1*x^4+2*x^5-X1*X2*x^2-X1*x^3-2*x^3+ 4*x^2-3*x+1)/(-1+x)/(X1^2*X2*x^6-X1^2*X2*x^5-2*X1*X2*x^6+X1*X2*x^5+X2*x^6+X1*X2 *x^4+X1*x^5-2*X1*X2*x^3-2*X1*x^4-x^5+X1*X2*x^2+X1*x^3+x^4+x^3-3*x^2+3*x-1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 1, 2, 1], are 55 7 n - 7/8 + n/8, and , - -- + ---, respectively, while 64 64 Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 2, 1, 1], are 47 7 n - 7/8 + n/8, and , - -- + ---, respectively, while the asymptotic correlatio\ 64 64 n between these two random variables is 5/7 and in floating point 0.7142857143 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ ate normal pair with correlation, 5/7 99 i.e. , [[5/7, 0], [0, --]] 49 ------------------------------------------------- Theorem Number, 44, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 1, 2, 1], nor the composition, [3, 1, 1, 1] Then infinity ----- 9 8 7 6 5 3 2 \ n x - 3 x + 3 x - 3 x + 3 x - 2 x + 4 x - 3 x + 1 ) a(n) x = ------------------------------------------------------ / 2 3 3 ----- (x + 1) (x - x + 1) (x + x - 1) (-1 + x) n = 0 and in Maple format (x^9-3*x^8+3*x^7-3*x^6+3*x^5-2*x^3+4*x^2-3*x+1)/(x+1)/(x^2-x+1)/(x^3+x-1)/(-1+x )^3 The asymptotic expression for a(n) is, n 6.0594382353848054551 1.4655712318767680267 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 1, 2, 1] and d occurrences (as containment) of the composition, [3, 1, 1, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 2 2 9 2 2 8 ) | ) | ) A(n, c, d) x X1 X2 || = (X1 X2 x - X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 2 9 2 9 2 2 7 2 8 2 9 - 2 X1 X2 x - 2 X1 X2 x + X1 X2 x + 4 X1 X2 x + X1 x 2 8 9 2 9 2 7 2 8 2 7 + 2 X1 X2 x + 4 X1 X2 x + X2 x - 4 X1 X2 x - 3 X1 x - X1 X2 x 8 9 2 8 9 2 6 2 7 - 8 X1 X2 x - 2 X1 x - X2 x - 2 X2 x + 2 X1 X2 x + 4 X1 x 7 8 8 9 2 6 6 7 + 6 X1 X2 x + 6 X1 x + 4 X2 x + x - 3 X1 x - 4 X1 X2 x - 7 X1 x 7 8 2 5 6 6 7 4 5 - 2 X2 x - 3 x + X1 x + 6 X1 x + 2 X2 x + 3 x + X1 X2 x - 3 X1 x 5 6 3 4 5 3 2 3 2 - X2 x - 3 x - X1 X2 x - X1 x + 3 x + 2 X1 x - X1 x - 2 x + 4 x / - 3 x + 1) / ((-1 + x) / 4 4 4 3 4 3 4 3 (X1 X2 x - X1 x - X2 x + X1 x + x - x + x - 1) (X1 X2 x - X1 X2 x 4 4 3 4 2 3 2 - X1 x - X2 x + 2 X1 x + x - X1 x - x + x - 2 x + 1)) and in Maple format (X1^2*X2^2*x^9-X1^2*X2^2*x^8-2*X1^2*X2*x^9-2*X1*X2^2*x^9+X1^2*X2^2*x^7+4*X1^2* X2*x^8+X1^2*x^9+2*X1*X2^2*x^8+4*X1*X2*x^9+X2^2*x^9-4*X1^2*X2*x^7-3*X1^2*x^8-X1* X2^2*x^7-8*X1*X2*x^8-2*X1*x^9-X2^2*x^8-2*X2*x^9+2*X1^2*X2*x^6+4*X1^2*x^7+6*X1* X2*x^7+6*X1*x^8+4*X2*x^8+x^9-3*X1^2*x^6-4*X1*X2*x^6-7*X1*x^7-2*X2*x^7-3*x^8+X1^ 2*x^5+6*X1*x^6+2*X2*x^6+3*x^7+X1*X2*x^4-3*X1*x^5-X2*x^5-3*x^6-X1*X2*x^3-X1*x^4+ 3*x^5+2*X1*x^3-X1*x^2-2*x^3+4*x^2-3*x+1)/(-1+x)/(X1*X2*x^4-X1*x^4-X2*x^4+X1*x^3 +x^4-x^3+x-1)/(X1*X2*x^4-X1*X2*x^3-X1*x^4-X2*x^4+2*X1*x^3+x^4-X1*x^2-x^3+x^2-2* x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 1, 2, 1], are 55 7 n - 7/8 + n/8, and , - -- + ---, respectively, while 64 64 Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 1, 1, 1], are 11 3 n - 7/8 + n/8, and , - -- + ---, respectively, while the asymptotic correlatio\ 64 64 n between these two random variables is 1/2 1/2 3 7 --------- 21 and in floating point 0.2182178903 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 21 ate normal pair with correlation, ----- 21 1/2 21 23 i.e. , [[-----, 0], [0, --]] 21 21 ------------------------------------------------- Theorem Number, 45, : The following statements are true Let a(n) be the number of compositions of n that contain neither the composi\ tion, [2, 2, 1, 1], nor the composition, [3, 1, 1, 1] Then infinity ----- 5 4 3 2 \ n x - 3 x + 3 x - 4 x + 3 x - 1 ) a(n) x = - --------------------------------- / 3 3 ----- (x + x - 1) (-1 + x) n = 0 and in Maple format -(x^5-3*x^4+3*x^3-4*x^2+3*x-1)/(x^3+x-1)/(-1+x)^3 The asymptotic expression for a(n) is, n 6.9556233073109364759 1.4655712318767680267 Let , A(n, c, d), be the number of compositions of n with c occurrences (as\ containment) of the composition , [2, 2, 1, 1] and d occurrences (as containment) of the composition, [3, 1, 1, 1], (Note that A(n,0,0)=a(n)), Then infinity ------- /infinity /infinity \\ \ | ----- | ----- || \ | \ | \ n c d|| 5 ) | ) | ) A(n, c, d) x X1 X2 || = - (2 X1 X2 x / | / | / || / | ----- | ----- || ------- \ d = 0 \ c = 0 // n = 0 4 5 5 3 4 4 5 3 - 2 X1 X2 x - 2 X1 x - X2 x + X1 X2 x + 4 X1 x + X2 x + x - 3 X1 x 4 2 3 2 / 2 4 - 3 x + X1 x + 3 x - 4 x + 3 x - 1) / ((-1 + x) (X1 X2 x / 3 4 4 3 4 2 3 2 - X1 X2 x - X1 x - X2 x + 2 X1 x + x - X1 x - x + x - 2 x + 1)) and in Maple format -(2*X1*X2*x^5-2*X1*X2*x^4-2*X1*x^5-X2*x^5+X1*X2*x^3+4*X1*x^4+X2*x^4+x^5-3*X1*x^ 3-3*x^4+X1*x^2+3*x^3-4*x^2+3*x-1)/(-1+x)^2/(X1*X2*x^4-X1*X2*x^3-X1*x^4-X2*x^4+2 *X1*x^3+x^4-X1*x^2-x^3+x^2-2*x+1) Furthermore the average and variance of the random variable: Number of occur\ rences of, [2, 2, 1, 1], are 47 7 n - 7/8 + n/8, and , - -- + ---, respectively, while 64 64 Furthermore the average and variance of the random variable: Number of occur\ rences of, [3, 1, 1, 1], are 11 3 n - 7/8 + n/8, and , - -- + ---, respectively, while the asymptotic correlatio\ 64 64 n between these two random variables is 1/2 1/2 3 7 --------- 21 and in floating point 0.2182178903 The asymptotic standardized mixed moments are indeed as they are for bi-vari\ 1/2 21 ate normal pair with correlation, ----- 21 1/2 21 23 i.e. , [[-----, 0], [0, --]] 21 21 ------------------------------------------------- ------------------------ This ends this article, that took, 22.866, to generate. This took, 40.515, seconds