Generating functions and Growth rates for Weight-Enumerating Sequences Accor\ ding to the Number of occurrences of compositions of, 4 By Shalosh B. Ekhad The compositions of, 4, that yield polynomial growth (hence growth ratio 1, that is the lowest) are , [1, 1, 2], [1, 2, 1], [2, 1, 1], [1, 1, 1, 1] Theorem Number, 1, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 1, 2] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 2 \ n 2 x - 2 x + 1 ) a(n) x = - -------------- / 3 ----- (-1 + x) n = 0 and in Maple format -(2*x^2-2*x+1)/(-1+x)^3 The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 7, 11, 16, 22, 29, 37, 46, 56, 67, 79, 92, 106, 121, 137, 154, 172, 191, 211, 232, 254, 277, 301, 326, 352, 379, 407, 436] ---------------------------------------------------------------------------- The compositions of, 4, that yield the, 2, -th largest growth, that is, 1.6180339887498948482, are , [1, 3], [3, 1] Theorem Number, 2, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 3 \ n x - x + 1 ) a(n) x = --------------------- / 2 ----- (-1 + x) (x + x - 1) n = 0 and in Maple format (x^3-x+1)/(-1+x)/(x^2+x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 7, 12, 20, 33, 54, 88, 143, 232, 376, 609, 986, 1596, 2583, 4180, 6764, 10945, 17710, 28656, 46367, 75024, 121392, 196417, 317810, 514228, 832039, 1346268, 2178308] The limit of a(n+1)/a(n) as n goes to infinity is 1.61803398875 a(n) is asymptotic to 1.17082039325*1.61803398875^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [1, 3], denoted by the variable, X[1, 3], is 3 3 x X[1, 3] - x + x - 1 ------------------------- 3 3 x X[1, 3] - x + 2 x - 1 and in Maple format (x^3*X[1,3]-x^3+x-1)/(x^3*X[1,3]-x^3+2*x-1) Furthermore, the expectation of the random variable number of occurences (by\ containment) of , [1, 3], equals , - 3/8 + n/8 3 n The variance equals , - 3/64 + --- 64 The , 3, -th moment about the mean is , 3/64 39 27 2 21 The , 4, -th moment about the mean is , - ---- + ---- n - ---- n 4096 4096 1024 The compositions of, 4, that yield the, 3, -th largest growth, that is, 1.7548776662466927601, are , [2, 2] Theorem Number, 3, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 2] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 2 \ n x - x + 1 ) a(n) x = - ----------------- / 3 2 ----- x - x + 2 x - 1 n = 0 and in Maple format -(x^2-x+1)/(x^3-x^2+2*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 7, 12, 21, 37, 65, 114, 200, 351, 616, 1081, 1897, 3329, 5842, 10252, 17991, 31572, 55405, 97229, 170625, 299426, 525456, 922111, 1618192, 2839729, 4983377, 8745217, 15346786] The limit of a(n+1)/a(n) as n goes to infinity is 1.75487766625 a(n) is asymptotic to .722124418303*1.75487766625^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [2, 2], denoted by the variable, X[2, 2], is 2 2 x X[2, 2] - x + x - 1 - ------------------------------------------- 3 3 2 2 x X[2, 2] - x - x X[2, 2] + x - 2 x + 1 and in Maple format -(x^2*X[2,2]-x^2+x-1)/(x^3*X[2,2]-x^3-x^2*X[2,2]+x^2-2*x+1) Furthermore, the expectation of the random variable number of occurences (by\ containment) of , [2, 2], equals , - 3/8 + n/8 23 7 n The variance equals , - -- + --- 64 64 33 9 n The , 3, -th moment about the mean is , - --- + --- 128 128 1873 147 2 257 The , 4, -th moment about the mean is , ---- + ---- n - ---- n 4096 4096 1024 The compositions of, 4, that yield the, 4, -th largest growth, that is, 1.8392867552141611326, are , [4] Theorem Number, 4, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [4] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 1 ) a(n) x = - --------------- / 3 2 ----- x + x + x - 1 n = 0 and in Maple format -1/(x^3+x^2+x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 7, 13, 24, 44, 81, 149, 274, 504, 927, 1705, 3136, 5768, 10609, 19513, 35890, 66012, 121415, 223317, 410744, 755476, 1389537, 2555757, 4700770, 8646064, 15902591, 29249425, 53798080] The limit of a(n+1)/a(n) as n goes to infinity is 1.83928675521 a(n) is asymptotic to .618419922319*1.83928675521^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [4], denoted by the variable, X[4], is -1 + x ---------------------- 4 4 x X[4] - x + 2 x - 1 and in Maple format (-1+x)/(x^4*X[4]-x^4+2*x-1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [4], equals , - 1/8 + ---- 16 9 n The variance equals , - 1/32 + --- 256 33 15 n The , 3, -th moment about the mean is , ---- + ---- 1024 2048 53 243 2 525 The , 4, -th moment about the mean is , ---- + ----- n - ----- n 2048 65536 32768 This ends this article, that took, 0.198, seconds to generate. ---------------------------------------------------------- ---------------------------------------- Generating functions and Growth rates for Weight-Enumerating Sequences Accor\ ding to the Number of occurrences of compositions of, 5 By Shalosh B. Ekhad The compositions of, 5, that yield polynomial growth (hence growth ratio 1, that is the lowest) are , [1, 1, 1, 2], [1, 1, 2, 1], [1, 2, 1, 1], [2, 1, 1, 1], [1, 1, 1, 1, 1] Theorem Number, 1, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 1, 1, 2] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 4 3 2 \ n x - 2 x + 4 x - 3 x + 1 ) a(n) x = -------------------------- / 4 ----- (-1 + x) n = 0 and in Maple format (x^4-2*x^3+4*x^2-3*x+1)/(-1+x)^4 The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 15, 26, 42, 64, 93, 130, 176, 232, 299, 378, 470, 576, 697, 834, 988, 1160, 1351, 1562, 1794, 2048, 2325, 2626, 2952, 3304, 3683, 4090] ---------------------------------------------------------------------------- The compositions of, 5, that yield the, 2, -th largest growth, that is, 1.6180339887498948482, are , [1, 1, 3], [1, 3, 1], [3, 1, 1] Theorem Number, 2, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 1, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 3 2 \ n x + x - 2 x + 1 ) a(n) x = - ---------------------- / 2 2 ----- (x + x - 1) (-1 + x) n = 0 and in Maple format -(x^3+x^2-2*x+1)/(x^2+x-1)/(-1+x)^2 The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 15, 27, 47, 80, 134, 222, 365, 597, 973, 1582, 2568, 4164, 6747, 10927, 17691, 28636, 46346, 75002, 121369, 196393, 317785, 514202, 832012, 1346240, 2178279, 3524547] The limit of a(n+1)/a(n) as n goes to infinity is 1.61803398875 a(n) is asymptotic to 1.89442719100*1.61803398875^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [1, 1, 3], denoted by the variable, X[1, 1, 3], is 3 3 2 x X[1, 1, 3] - x - x + 2 x - 1 - --------------------------------------- 3 3 (-1 + x) (x X[1, 1, 3] - x + 2 x - 1) and in Maple format -(x^3*X[1,1,3]-x^3-x^2+2*x-1)/(-1+x)/(x^3*X[1,1,3]-x^3+2*x-1) Furthermore, the expectation of the random variable number of occurences (by\ containment) of , [1, 1, 3], equals , - 5/8 + n/8 3 n The variance equals , - 7/64 + --- 64 The , 3, -th moment about the mean is , 9/128 27 2 39 113 The , 4, -th moment about the mean is , ---- n - ---- n + ---- 4096 1024 4096 The compositions of, 5, that yield the, 3, -th largest growth, that is, 1.7548776662466927601, are , [2, 1, 2] Theorem Number, 3, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 2] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 5 2 \ n x + 2 x - 2 x + 1 ) a(n) x = - -------------------------------- / 3 3 2 ----- (x - x + 1) (x - x + 2 x - 1) n = 0 and in Maple format -(x^5+2*x^2-2*x+1)/(x^3-x+1)/(x^3-x^2+2*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 15, 27, 47, 81, 140, 244, 428, 753, 1325, 2329, 4089, 7174, 12584, 22076, 38735, 67975, 119295, 209361, 367416, 644776, 1131496, 1985617, 3484489, 6114833, 10730785, 18831242] The limit of a(n+1)/a(n) as n goes to infinity is 1.75487766625 a(n) is asymptotic to .886082866099*1.75487766625^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [2, 1, 2], denoted by the variable, X[2, 1, 2], is 5 2 5 5 2 2 / - (x X[2, 1, 2] - 2 x X[2, 1, 2] + x - x X[2, 1, 2] + 2 x - 2 x + 1) / / 3 3 ((x X[2, 1, 2] - x + x - 1) 3 3 2 2 (x X[2, 1, 2] - x - x X[2, 1, 2] + x - 2 x + 1)) and in Maple format -(x^5*X[2,1,2]^2-2*x^5*X[2,1,2]+x^5-x^2*X[2,1,2]+2*x^2-2*x+1)/(x^3*X[2,1,2]-x^3 +x-1)/(x^3*X[2,1,2]-x^3-x^2*X[2,1,2]+x^2-2*x+1) Furthermore, the expectation of the random variable number of occurences (by\ containment) of , [2, 1, 2], equals , - 5/8 + n/8 43 7 n The variance equals , - -- + --- 64 64 39 9 n The , 3, -th moment about the mean is , - -- + --- 64 128 147 2 5033 467 The , 4, -th moment about the mean is , ---- n + ---- - ---- n 4096 4096 1024 The compositions of, 5, that yield the, 4, -th largest growth, that is, 1.8392867552141611326, are , [1, 4], [4, 1] Theorem Number, 4, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 4] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 4 \ n x - x + 1 ) a(n) x = -------------------------- / 3 2 ----- (-1 + x) (x + x + x - 1) n = 0 and in Maple format (x^4-x+1)/(-1+x)/(x^3+x^2+x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 15, 28, 52, 96, 177, 326, 600, 1104, 2031, 3736, 6872, 12640, 23249, 42762, 78652, 144664, 266079, 489396, 900140, 1655616, 3045153, 5600910, 10301680, 18947744, 34850335, 64099760] The limit of a(n+1)/a(n) as n goes to infinity is 1.83928675521 a(n) is asymptotic to .736839844639*1.83928675521^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [1, 4], denoted by the variable, X[1, 4], is 4 4 x X[1, 4] - x + x - 1 ------------------------- 4 4 x X[1, 4] - x + 2 x - 1 and in Maple format (x^4*X[1,4]-x^4+x-1)/(x^4*X[1,4]-x^4+2*x-1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [1, 4], equals , - 1/4 + ---- 16 9 n The variance equals , - 3/32 + --- 256 15 15 n The , 3, -th moment about the mean is , --- + ---- 512 2048 153 957 243 2 The , 4, -th moment about the mean is , ---- - ----- n + ----- n 2048 32768 65536 The compositions of, 5, that yield the, 5, -th largest growth, that is, 1.8667603991738620930, are , [2, 3], [3, 2] Theorem Number, 5, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 3 \ n x - x + 1 ) a(n) x = - ----------------- / 4 3 ----- x - x + 2 x - 1 n = 0 and in Maple format -(x^3-x+1)/(x^4-x^3+2*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 15, 28, 52, 97, 181, 338, 631, 1178, 2199, 4105, 7663, 14305, 26704, 49850, 93058, 173717, 324288, 605368, 1130077, 2109583, 3938086, 7351463, 13723420, 25618337, 47823297, 89274637] The limit of a(n+1)/a(n) as n goes to infinity is 1.86676039917 a(n) is asymptotic to .657764927635*1.86676039917^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [2, 3], denoted by the variable, X[2, 3], is 3 3 x X[2, 3] - x + x - 1 - ------------------------------------------- 4 4 3 3 x X[2, 3] - x - x X[2, 3] + x - 2 x + 1 and in Maple format -(x^3*X[2,3]-x^3+x-1)/(x^4*X[2,3]-x^4-x^3*X[2,3]+x^3-2*x+1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [2, 3], equals , - 1/4 + ---- 16 13 13 n The variance equals , - -- + ---- 64 256 15 63 n The , 3, -th moment about the mean is , - --- + ---- 128 2048 507 2 2017 605 The , 4, -th moment about the mean is , ----- n - ----- n + ---- 65536 32768 4096 The compositions of, 5, that yield the, 6, -th largest growth, that is, 1.9275619754829253043, are , [5] Theorem Number, 6, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [5] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 1 ) a(n) x = - -------------------- / 4 3 2 ----- x + x + x + x - 1 n = 0 and in Maple format -1/(x^4+x^3+x^2+x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 15, 29, 56, 108, 208, 401, 773, 1490, 2872, 5536, 10671, 20569, 39648, 76424, 147312, 283953, 547337, 1055026, 2033628, 3919944, 7555935, 14564533, 28074040, 54114452, 104308960, 201061985] The limit of a(n+1)/a(n) as n goes to infinity is 1.92756197548 a(n) is asymptotic to .566342887703*1.92756197548^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [5], denoted by the variable, X[5], is -1 + x ---------------------- 5 5 x X[5] - x + 2 x - 1 and in Maple format (-1+x)/(x^5*X[5]-x^5+2*x-1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [5], equals , - 3/32 + ---- 32 51 23 n The variance equals , - ---- + ---- 1024 1024 21 171 n The , 3, -th moment about the mean is , ----- + ----- 16384 16384 47217 1587 2 2413 The , 4, -th moment about the mean is , ------- + ------- n - ------ n 1048576 1048576 262144 This ends this article, that took, 0.231, seconds to generate. ---------------------------------------------------------- ---------------------------------------- Generating functions and Growth rates for Weight-Enumerating Sequences Accor\ ding to the Number of occurrences of compositions of, 6 By Shalosh B. Ekhad The compositions of, 6, that yield polynomial growth (hence growth ratio 1, that is the lowest) are , [1, 1, 1, 1, 2], [1, 1, 1, 2, 1], [1, 1, 2, 1, 1], [1, 2, 1, 1, 1], [2, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1] Theorem Number, 1, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 1, 1, 1, 2] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 2 2 \ n (x - x + 1) (3 x - 3 x + 1) ) a(n) x = - ----------------------------- / 5 ----- (-1 + x) n = 0 and in Maple format -(x^2-x+1)*(3*x^2-3*x+1)/(-1+x)^5 The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 31, 57, 99, 163, 256, 386, 562, 794, 1093, 1471, 1941, 2517, 3214, 4048, 5036, 6196, 7547, 9109, 10903, 12951, 15276, 17902, 20854, 24158, 27841] ---------------------------------------------------------------------------- The compositions of, 6, that yield the, 2, -th largest growth, that is, 1.6180339887498948482, are , [1, 1, 1, 3], [1, 1, 3, 1], [1, 3, 1, 1], [3, 1, 1, 1] Theorem Number, 2, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 1, 1, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 5 4 2 \ n x - x + 3 x - 3 x + 1 ) a(n) x = ------------------------ / 2 3 ----- (x + x - 1) (-1 + x) n = 0 and in Maple format (x^5-x^4+3*x^2-3*x+1)/(x^2+x-1)/(-1+x)^3 The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 31, 58, 105, 185, 319, 541, 906, 1503, 2476, 4058, 6626, 10790, 17537, 28464, 46155, 74791, 121137, 196139, 317508, 513901, 831686, 1345888, 2177900, 3524140, 5702419] The limit of a(n+1)/a(n) as n goes to infinity is 1.61803398875 a(n) is asymptotic to 3.06524758425*1.61803398875^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [1, 1, 1, 3], denoted by the variable, X[1, 1, 1, 3], is 5 5 4 4 3 2 (x X[1, 1, 1, 3] - x - x X[1, 1, 1, 3] + x + x X[1, 1, 1, 3] - 3 x + 3 x / 2 3 3 - 1) / ((-1 + x) (x X[1, 1, 1, 3] - x + 2 x - 1)) / and in Maple format (x^5*X[1,1,1,3]-x^5-x^4*X[1,1,1,3]+x^4+x^3*X[1,1,1,3]-3*x^2+3*x-1)/(-1+x)^2/(x^ 3*X[1,1,1,3]-x^3+2*x-1) Furthermore, the expectation of the random variable number of occurences (by\ containment) of , [1, 1, 1, 3], equals , - 7/8 + n/8 11 3 n The variance equals , - -- + --- 64 64 The , 3, -th moment about the mean is , 3/32 (9 n - 19) (3 n - 19) The , 4, -th moment about the mean is , --------------------- 4096 The compositions of, 6, that yield the, 3, -th largest growth, that is, 1.7548776662466927601, are , [2, 1, 1, 2] Theorem Number, 3, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 1, 2] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n ) a(n) x = / ----- n = 0 11 9 8 7 6 5 4 3 2 x + 2 x - x + 3 x - 4 x + 5 x - 6 x + 7 x - 7 x + 4 x - 1 - ------------------------------------------------------------------- 3 2 9 6 5 4 3 2 (x - x + 2 x - 1) (x - x + 2 x - x + x - 3 x + 3 x - 1) and in Maple format -(x^11+2*x^9-x^8+3*x^7-4*x^6+5*x^5-6*x^4+7*x^3-7*x^2+4*x-1)/(x^3-x^2+2*x-1)/(x^ 9-x^6+2*x^5-x^4+x^3-3*x^2+3*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 31, 58, 105, 185, 320, 549, 942, 1625, 2824, 4941, 8686, 15306, 26983, 47529, 83604, 146856, 257686, 451873, 792225, 1389052, 2436112, 4273686, 7499219, 13161377, 23100431] The limit of a(n+1)/a(n) as n goes to infinity is 1.75487766625 a(n) is asymptotic to 1.08726810186*1.75487766625^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [2, 1, 1, 2], denoted by the variable, X[2, 1, 1, 2], is 11 4 11 3 11 2 - (x X[2, 1, 1, 2] - 4 x X[2, 1, 1, 2] + 6 x X[2, 1, 1, 2] 11 9 3 11 9 2 - 4 x X[2, 1, 1, 2] - 2 x X[2, 1, 1, 2] + x + 6 x X[2, 1, 1, 2] 8 3 9 8 2 + x X[2, 1, 1, 2] - 6 x X[2, 1, 1, 2] - 3 x X[2, 1, 1, 2] 7 3 9 8 7 2 8 - x X[2, 1, 1, 2] + 2 x + 3 x X[2, 1, 1, 2] + 5 x X[2, 1, 1, 2] - x 7 6 2 7 6 - 7 x X[2, 1, 1, 2] - 3 x X[2, 1, 1, 2] + 3 x + 7 x X[2, 1, 1, 2] 5 2 6 5 5 + x X[2, 1, 1, 2] - 4 x - 6 x X[2, 1, 1, 2] + 5 x 4 4 3 3 2 + 5 x X[2, 1, 1, 2] - 6 x - 3 x X[2, 1, 1, 2] + 7 x + x X[2, 1, 1, 2] 2 / - 7 x + 4 x - 1) / ( / 3 3 2 2 (x X[2, 1, 1, 2] - x - x X[2, 1, 1, 2] + x - 2 x + 1) ( 9 3 9 2 9 9 x X[2, 1, 1, 2] - 3 x X[2, 1, 1, 2] + 3 x X[2, 1, 1, 2] - x 6 2 6 5 2 6 + x X[2, 1, 1, 2] - 2 x X[2, 1, 1, 2] - x X[2, 1, 1, 2] + x 5 5 4 4 3 2 + 3 x X[2, 1, 1, 2] - 2 x - x X[2, 1, 1, 2] + x - x + 3 x - 3 x + 1) ) and in Maple format -(x^11*X[2,1,1,2]^4-4*x^11*X[2,1,1,2]^3+6*x^11*X[2,1,1,2]^2-4*x^11*X[2,1,1,2]-2 *x^9*X[2,1,1,2]^3+x^11+6*x^9*X[2,1,1,2]^2+x^8*X[2,1,1,2]^3-6*x^9*X[2,1,1,2]-3*x ^8*X[2,1,1,2]^2-x^7*X[2,1,1,2]^3+2*x^9+3*x^8*X[2,1,1,2]+5*x^7*X[2,1,1,2]^2-x^8-\ 7*x^7*X[2,1,1,2]-3*x^6*X[2,1,1,2]^2+3*x^7+7*x^6*X[2,1,1,2]+x^5*X[2,1,1,2]^2-4*x ^6-6*x^5*X[2,1,1,2]+5*x^5+5*x^4*X[2,1,1,2]-6*x^4-3*x^3*X[2,1,1,2]+7*x^3+x^2*X[2 ,1,1,2]-7*x^2+4*x-1)/(x^3*X[2,1,1,2]-x^3-x^2*X[2,1,1,2]+x^2-2*x+1)/(x^9*X[2,1,1 ,2]^3-3*x^9*X[2,1,1,2]^2+3*x^9*X[2,1,1,2]-x^9+x^6*X[2,1,1,2]^2-2*x^6*X[2,1,1,2] -x^5*X[2,1,1,2]^2+x^6+3*x^5*X[2,1,1,2]-2*x^5-x^4*X[2,1,1,2]+x^4-x^3+3*x^2-3*x+1 ) Furthermore, the expectation of the random variable number of occurences (by\ containment) of , [2, 1, 1, 2], equals , - 7/8 + n/8 63 7 n The variance equals , - -- + --- 64 64 123 9 n The , 3, -th moment about the mean is , - --- + --- 128 128 147 2 677 10593 The , 4, -th moment about the mean is , ---- n - ---- n + ----- 4096 1024 4096 The compositions of, 6, that yield the, 4, -th largest growth, that is, 1.8392867552141611326, are , [1, 1, 4], [1, 4, 1], [4, 1, 1] Theorem Number, 4, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 1, 4] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 4 2 \ n x + x - 2 x + 1 ) a(n) x = - --------------------------- / 3 2 2 ----- (x + x + x - 1) (-1 + x) n = 0 and in Maple format -(x^4+x^2-2*x+1)/(x^3+x^2+x-1)/(-1+x)^2 The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 31, 59, 111, 207, 384, 710, 1310, 2414, 4445, 8181, 15053, 27693, 50942, 93704, 172356, 317020, 583099, 1072495, 1972635, 3628251, 6673404, 12274314, 22575994, 41523738, 76374073] The limit of a(n+1)/a(n) as n goes to infinity is 1.83928675521 a(n) is asymptotic to .877935747301*1.83928675521^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [1, 1, 4], denoted by the variable, X[1, 1, 4], is 4 4 2 x X[1, 1, 4] - x - x + 2 x - 1 - --------------------------------------- 4 4 (-1 + x) (x X[1, 1, 4] - x + 2 x - 1) and in Maple format -(x^4*X[1,1,4]-x^4-x^2+2*x-1)/(-1+x)/(x^4*X[1,1,4]-x^4+2*x-1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [1, 1, 4], equals , - 3/8 + ---- 16 9 n The variance equals , - 5/32 + --- 256 27 15 n The , 3, -th moment about the mean is , ---- + ---- 1024 2048 301 243 2 1389 The , 4, -th moment about the mean is , ---- + ----- n - ----- n 2048 65536 32768 The compositions of, 6, that yield the, 5, -th largest growth, that is, 1.8667603991738620930, are , [2, 1, 3], [3, 1, 2] Theorem Number, 5, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 7 3 2 \ n x + x + x - 2 x + 1 ) a(n) x = - -------------------------------- / 4 4 3 ----- (x - x + 1) (x - x + 2 x - 1) n = 0 and in Maple format -(x^7+x^3+x^2-2*x+1)/(x^4-x+1)/(x^4-x^3+2*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 31, 59, 111, 207, 385, 716, 1333, 2485, 4637, 8657, 16165, 30184, 56356, 105212, 196410, 366647, 684429, 1277643, 2385027, 4452251, 8311276, 15515173, 28963149, 54067321, 100930805] The limit of a(n+1)/a(n) as n goes to infinity is 1.86676039917 a(n) is asymptotic to .743646261889*1.86676039917^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [2, 1, 3], denoted by the variable, X[2, 1, 3], is 7 2 7 7 3 3 2 - (x X[2, 1, 3] - 2 x X[2, 1, 3] + x - x X[2, 1, 3] + x + x - 2 x + 1) / 4 4 / ((x X[2, 1, 3] - x + x - 1) / 4 4 3 3 (x X[2, 1, 3] - x - x X[2, 1, 3] + x - 2 x + 1)) and in Maple format -(x^7*X[2,1,3]^2-2*x^7*X[2,1,3]+x^7-x^3*X[2,1,3]+x^3+x^2-2*x+1)/(x^4*X[2,1,3]-x ^4+x-1)/(x^4*X[2,1,3]-x^4-x^3*X[2,1,3]+x^3-2*x+1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [2, 1, 3], equals , - 3/8 + ---- 16 21 13 n The variance equals , - -- + ---- 64 256 237 63 n The , 3, -th moment about the mean is , - ---- + ---- 1024 2048 1131 507 2 3265 The , 4, -th moment about the mean is , ---- + ----- n - ----- n 4096 65536 32768 The compositions of, 6, that yield the, 6, -th largest growth, that is, 1.9087907387871591034, are , [2, 2, 2] Theorem Number, 6, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 2, 2] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 4 3 2 \ n x - x + 2 x - 2 x + 1 ) a(n) x = - ------------------------------- / 5 4 3 2 ----- x - x + 2 x - 3 x + 3 x - 1 n = 0 and in Maple format -(x^4-x^3+2*x^2-2*x+1)/(x^5-x^4+2*x^3-3*x^2+3*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 31, 59, 112, 213, 406, 775, 1480, 2826, 5395, 10298, 19656, 37518, 71613, 136694, 260921, 498045, 950665, 1814621, 3463731, 6611536, 12620037, 24089009, 45980878, 87767876, 167530511] The limit of a(n+1)/a(n) as n goes to infinity is 1.90879073879 a(n) is asymptotic to .632913467972*1.90879073879^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [2, 2, 2], denoted by the variable, X[2, 2, 2], is 4 4 3 3 2 2 - (x X[2, 2, 2] - x - x X[2, 2, 2] + x + x X[2, 2, 2] - 2 x + 2 x - 1) / 5 5 4 4 3 3 / (x X[2, 2, 2] - x - x X[2, 2, 2] + x + 2 x X[2, 2, 2] - 2 x / 2 2 - x X[2, 2, 2] + 3 x - 3 x + 1) and in Maple format -(x^4*X[2,2,2]-x^4-x^3*X[2,2,2]+x^3+x^2*X[2,2,2]-2*x^2+2*x-1)/(x^5*X[2,2,2]-x^5 -x^4*X[2,2,2]+x^4+2*x^3*X[2,2,2]-2*x^3-x^2*X[2,2,2]+3*x^2-3*x+1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [2, 2, 2], equals , - 3/8 + ---- 16 23 25 n The variance equals , - -- + ---- 32 256 1581 351 n The , 3, -th moment about the mean is , - ---- + ----- 1024 2048 3299 1875 2 4381 The , 4, -th moment about the mean is , - ---- + ----- n - ----- n 2048 65536 32768 The compositions of, 6, that yield the, 7, -th largest growth, that is, 1.9275619754829253043, are , [1, 5], [5, 1] Theorem Number, 7, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 5] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 5 \ n x - x + 1 ) a(n) x = ------------------------------- / 4 3 2 ----- (-1 + x) (x + x + x + x - 1) n = 0 and in Maple format (x^5-x+1)/(-1+x)/(x^4+x^3+x^2+x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 31, 60, 116, 224, 432, 833, 1606, 3096, 5968, 11504, 22175, 42744, 82392, 158816, 306128, 590081, 1137418, 2192444, 4226072, 8146016, 15701951, 30266484, 58340524, 112454976, 216763936] The limit of a(n+1)/a(n) as n goes to infinity is 1.92756197548 a(n) is asymptotic to .610571479504*1.92756197548^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [1, 5], denoted by the variable, X[1, 5], is 5 5 x X[1, 5] - x + x - 1 ------------------------- 5 5 x X[1, 5] - x + 2 x - 1 and in Maple format (x^5*X[1,5]-x^5+x-1)/(x^5*X[1,5]-x^5+2*x-1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [1, 5], equals , - 5/32 + ---- 32 95 23 n The variance equals , - ---- + ---- 1024 1024 255 171 n The , 3, -th moment about the mean is , - ----- + ----- 16384 16384 76825 1587 2 3931 The , 4, -th moment about the mean is , ------- + ------- n - ------ n 1048576 1048576 262144 The compositions of, 6, that yield the, 8, -th largest growth, that is, 1.9331849818995204468, are , [2, 4], [4, 2] Theorem Number, 8, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 4] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 4 \ n x - x + 1 ) a(n) x = - ----------------- / 5 4 ----- x - x + 2 x - 1 n = 0 and in Maple format -(x^4-x+1)/(x^5-x^4+2*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 31, 60, 116, 224, 433, 837, 1618, 3128, 6047, 11690, 22599, 43688, 84457, 163271, 315633, 610177, 1179585, 2280356, 4408350, 8522156, 16474904, 31849037, 61570080, 119026354, 230099960] The limit of a(n+1)/a(n) as n goes to infinity is 1.93318498190 a(n) is asymptotic to .593901147371*1.93318498190^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [2, 4], denoted by the variable, X[2, 4], is 4 4 x X[2, 4] - x + x - 1 - ------------------------------------------- 5 5 4 4 x X[2, 4] - x - x X[2, 4] + x - 2 x + 1 and in Maple format -(x^4*X[2,4]-x^4+x-1)/(x^5*X[2,4]-x^5-x^4*X[2,4]+x^4-2*x+1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [2, 4], equals , - 5/32 + ---- 32 131 27 n The variance equals , - ---- + ---- 1024 1024 1317 297 n The , 3, -th moment about the mean is , - ----- + ----- 16384 16384 42385 2187 2 3855 The , 4, -th moment about the mean is , ------- + ------- n - ------ n 1048576 1048576 262144 The compositions of, 6, that yield the, 9, -th largest growth, that is, 1.9417130342786384772, are , [3, 3] Theorem Number, 9, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [3, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 3 \ n x - x + 1 ) a(n) x = - ---------------------- / 5 4 3 ----- x + x - x + 2 x - 1 n = 0 and in Maple format -(x^3-x+1)/(x^5+x^4-x^3+2*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 31, 60, 116, 225, 437, 849, 1649, 3202, 6217, 12071, 23438, 45510, 88368, 171586, 333171, 646922, 1256136, 2439055, 4735945, 9195847, 17855697, 34670640, 67320433, 130716961, 253814826] The limit of a(n+1)/a(n) as n goes to infinity is 1.94171303428 a(n) is asymptotic to .574071001256*1.94171303428^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [3, 3], denoted by the variable, X[3, 3], is 3 3 x X[3, 3] - x + x - 1 - ------------------------------------------------------------- 5 5 4 4 3 3 x X[3, 3] - x + x X[3, 3] - x - x X[3, 3] + x - 2 x + 1 and in Maple format -(x^3*X[3,3]-x^3+x-1)/(x^5*X[3,3]-x^5+x^4*X[3,3]-x^4-x^3*X[3,3]+x^3-2*x+1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [3, 3], equals , - 5/32 + ---- 32 195 35 n The variance equals , - ---- + ---- 1024 1024 4101 633 n The , 3, -th moment about the mean is , - ----- + ----- 16384 16384 237807 3675 2 989 The , 4, -th moment about the mean is , - ------- + ------- n + ------ n 1048576 1048576 262144 The compositions of, 6, that yield the, 10, -th largest growth, that is, 1.9659482366454853372, are , [6] Theorem Number, 10, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [6] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 1 ) a(n) x = - ------------------------- / 5 4 3 2 ----- x + x + x + x + x - 1 n = 0 and in Maple format -1/(x^5+x^4+x^3+x^2+x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 31, 61, 120, 236, 464, 912, 1793, 3525, 6930, 13624, 26784, 52656, 103519, 203513, 400096, 786568, 1546352, 3040048, 5976577, 11749641, 23099186, 45411804, 89277256, 175514464, 345052351] The limit of a(n+1)/a(n) as n goes to infinity is 1.96594823665 a(n) is asymptotic to .537926116819*1.96594823665^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [6], denoted by the variable, X[6], is -1 + x ---------------------- 6 6 x X[6] - x + 2 x - 1 and in Maple format (-1+x)/(x^6*X[6]-x^6+2*x-1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [6], equals , - 1/16 + ---- 64 23 53 n The variance equals , - --- + ---- 512 4096 153 141 n The , 3, -th moment about the mean is , - ---- + ----- 8192 16384 10321 8427 2 8969 The , 4, -th moment about the mean is , ------ + -------- n - ------- n 524288 16777216 8388608 This ends this article, that took, 0.477, seconds to generate. ---------------------------------------------------------- ---------------------------------------- Generating functions and Growth rates for Weight-Enumerating Sequences Accor\ ding to the Number of occurrences of compositions of, 7 By Shalosh B. Ekhad The compositions of, 7, that yield polynomial growth (hence growth ratio 1, that is the lowest) are , [1, 1, 1, 1, 1, 2], [1, 1, 1, 1, 2, 1], [1, 1, 1, 2, 1, 1], [1, 1, 2, 1, 1, 1], [1, 2, 1, 1, 1, 1], [2, 1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1] Theorem Number, 1, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 1, 1, 1, 1, 2] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 6 5 4 3 2 \ n x - 3 x + 9 x - 13 x + 11 x - 5 x + 1 ) a(n) x = ------------------------------------------ / 6 ----- (-1 + x) n = 0 and in Maple format (x^6-3*x^5+9*x^4-13*x^3+11*x^2-5*x+1)/(-1+x)^6 The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 63, 120, 219, 382, 638, 1024, 1586, 2380, 3473, 4944, 6885, 9402, 12616, 16664, 21700, 27896, 35443, 44552, 55455, 68406, 83682, 101584, 122438, 146596] ---------------------------------------------------------------------------- The compositions of, 7, that yield the, 2, -th largest growth, that is, 1.6180339887498948482, are , [1, 1, 1, 1, 3], [1, 1, 1, 3, 1], [1, 1, 3, 1, 1], [1, 3, 1, 1, 1], [3, 1, 1, 1, 1] Theorem Number, 2, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 1, 1, 1, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 5 4 3 2 \ n 2 x - x - 3 x + 6 x - 4 x + 1 ) a(n) x = - --------------------------------- / 2 4 ----- (x + x - 1) (-1 + x) n = 0 and in Maple format -(2*x^5-x^4-3*x^3+6*x^2-4*x+1)/(x^2+x-1)/(-1+x)^4 The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 63, 121, 226, 411, 730, 1271, 2177, 3680, 6156, 10214, 16840, 27630, 45167, 73631, 119786, 194577, 315714, 511853, 829361, 1343262, 2174948, 3520836, 5698736, 9222876] The limit of a(n+1)/a(n) as n goes to infinity is 1.61803398875 a(n) is asymptotic to 4.95967477525*1.61803398875^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [1, 1, 1, 1, 3], denoted by the variable, X[1, 1, 1, 1, 3], is 5 5 4 4 - (2 x X[1, 1, 1, 1, 3] - 2 x - 2 x X[1, 1, 1, 1, 3] + x 3 3 2 / 3 + x X[1, 1, 1, 1, 3] + 3 x - 6 x + 4 x - 1) / ((-1 + x) / 3 3 (x X[1, 1, 1, 1, 3] - x + 2 x - 1)) and in Maple format -(2*x^5*X[1,1,1,1,3]-2*x^5-2*x^4*X[1,1,1,1,3]+x^4+x^3*X[1,1,1,1,3]+3*x^3-6*x^2+ 4*x-1)/(-1+x)^3/(x^3*X[1,1,1,1,3]-x^3+2*x-1) Furthermore, the expectation of the random variable number of occurences (by\ containment) of , [1, 1, 1, 1, 3], equals , - 9/8 + n/8 15 3 n The variance equals , - -- + --- 64 64 15 The , 3, -th moment about the mean is , --- 128 705 27 2 75 The , 4, -th moment about the mean is , ---- + ---- n - ---- n 4096 4096 1024 The compositions of, 7, that yield the, 3, -th largest growth, that is, 1.7548776662466927601, are , [2, 1, 1, 1, 2] Theorem Number, 3, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 1, 1, 2] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 20 18 17 15 14 13 12 11 ) a(n) x = - (x + 2 x + x + 2 x - 3 x + 3 x - 3 x - x / ----- n = 0 10 9 8 7 6 5 4 3 2 + 6 x - 10 x + 9 x + 3 x - 24 x + 44 x - 51 x + 41 x - 22 x / 3 3 2 + 7 x - 1) / ((x - x + 1) (x - x + 2 x - 1) / 12 8 7 6 5 4 3 2 3 (x + x - x - x + x + x - 4 x + 6 x - 4 x + 1) (x + x - 1)) and in Maple format -(x^20+2*x^18+x^17+2*x^15-3*x^14+3*x^13-3*x^12-x^11+6*x^10-10*x^9+9*x^8+3*x^7-\ 24*x^6+44*x^5-51*x^4+41*x^3-22*x^2+7*x-1)/(x^3-x+1)/(x^3-x^2+2*x-1)/(x^12+x^8-x ^7-x^6+x^5+x^4-4*x^3+6*x^2-4*x+1)/(x^3+x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 63, 121, 226, 411, 730, 1272, 2187, 3734, 6368, 10897, 18764, 32549, 56855, 99863, 176050, 310927, 549289, 969586, 1708992, 3007219, 5283242, 9269885, 16250201, 28472951] The limit of a(n+1)/a(n) as n goes to infinity is 1.75487766625 a(n) is asymptotic to 1.33413247287*1.75487766625^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [2, 1, 1, 1, 2], denoted by the variable, X[2, 1, 1, 1, 2], is 20 7 20 20 20 18 - (x X[2, 1, 1, 1, 2] - 7 x %5 + 21 x %4 - 35 x %3 - 2 x %5 20 18 17 20 18 17 + 35 x %2 + 12 x %4 - x %5 - 21 x %1 - 30 x %3 + 6 x %4 16 20 18 17 16 + x %5 + 7 x X[2, 1, 1, 1, 2] + 40 x %2 - 15 x %3 - 5 x %4 15 20 18 17 16 15 - x %5 - x - 30 x %1 + 20 x %2 + 10 x %3 + 7 x %4 18 17 16 15 14 + 12 x X[2, 1, 1, 1, 2] - 15 x %1 - 10 x %2 - 20 x %3 - 3 x %4 18 17 16 15 14 - 2 x + 6 x X[2, 1, 1, 1, 2] + 5 x %1 + 30 x %2 + 15 x %3 13 17 16 15 14 13 + x %4 - x - x X[2, 1, 1, 1, 2] - 25 x %1 - 30 x %2 - 6 x %3 15 14 13 12 15 + 11 x X[2, 1, 1, 1, 2] + 30 x %1 + 15 x %2 - x %3 - 2 x 14 13 11 14 - 15 x X[2, 1, 1, 1, 2] - 19 x %1 + 4 x %3 + 3 x 13 12 11 10 13 + 12 x X[2, 1, 1, 1, 2] + 6 x %1 - 14 x %2 - 3 x %3 - 3 x 12 11 10 9 12 - 8 x X[2, 1, 1, 1, 2] + 17 x %1 + 17 x %2 + x %3 + 3 x 11 10 9 11 - 8 x X[2, 1, 1, 1, 2] - 31 x %1 - 12 x %2 + x 10 9 8 10 + 23 x X[2, 1, 1, 1, 2] + 31 x %1 + 5 x %2 - 6 x 9 8 7 9 - 30 x X[2, 1, 1, 1, 2] - 20 x %1 - x %2 + 10 x 8 7 8 7 6 + 24 x X[2, 1, 1, 1, 2] + 7 x %1 - 9 x - 4 x X[2, 1, 1, 1, 2] - x %1 7 6 6 5 5 - 3 x - 16 x X[2, 1, 1, 1, 2] + 24 x + 23 x X[2, 1, 1, 1, 2] - 44 x 4 4 3 3 - 16 x X[2, 1, 1, 1, 2] + 51 x + 6 x X[2, 1, 1, 1, 2] - 41 x 2 2 / - x X[2, 1, 1, 1, 2] + 22 x - 7 x + 1) / ( / 3 3 2 2 (x X[2, 1, 1, 1, 2] - x - x X[2, 1, 1, 1, 2] + x - 2 x + 1) 3 3 3 3 (x X[2, 1, 1, 1, 2] - x + x - 1) (x X[2, 1, 1, 1, 2] - x - x + 1) ( 12 12 12 12 12 8 8 x %3 - 4 x %2 + 6 x %1 - 4 x X[2, 1, 1, 1, 2] + x + x %2 - x %1 7 8 7 8 7 7 - x %2 - x X[2, 1, 1, 1, 2] + x %1 + x + x X[2, 1, 1, 1, 2] - x 6 6 5 5 4 3 2 + x X[2, 1, 1, 1, 2] - x - x X[2, 1, 1, 1, 2] + x + x - 4 x + 6 x - 4 x + 1)) 2 %1 := X[2, 1, 1, 1, 2] 3 %2 := X[2, 1, 1, 1, 2] 4 %3 := X[2, 1, 1, 1, 2] 5 %4 := X[2, 1, 1, 1, 2] 6 %5 := X[2, 1, 1, 1, 2] and in Maple format -(x^20*X[2,1,1,1,2]^7-7*x^20*X[2,1,1,1,2]^6+21*x^20*X[2,1,1,1,2]^5-35*x^20*X[2, 1,1,1,2]^4-2*x^18*X[2,1,1,1,2]^6+35*x^20*X[2,1,1,1,2]^3+12*x^18*X[2,1,1,1,2]^5- x^17*X[2,1,1,1,2]^6-21*x^20*X[2,1,1,1,2]^2-30*x^18*X[2,1,1,1,2]^4+6*x^17*X[2,1, 1,1,2]^5+x^16*X[2,1,1,1,2]^6+7*x^20*X[2,1,1,1,2]+40*x^18*X[2,1,1,1,2]^3-15*x^17 *X[2,1,1,1,2]^4-5*x^16*X[2,1,1,1,2]^5-x^15*X[2,1,1,1,2]^6-x^20-30*x^18*X[2,1,1, 1,2]^2+20*x^17*X[2,1,1,1,2]^3+10*x^16*X[2,1,1,1,2]^4+7*x^15*X[2,1,1,1,2]^5+12*x ^18*X[2,1,1,1,2]-15*x^17*X[2,1,1,1,2]^2-10*x^16*X[2,1,1,1,2]^3-20*x^15*X[2,1,1, 1,2]^4-3*x^14*X[2,1,1,1,2]^5-2*x^18+6*x^17*X[2,1,1,1,2]+5*x^16*X[2,1,1,1,2]^2+ 30*x^15*X[2,1,1,1,2]^3+15*x^14*X[2,1,1,1,2]^4+x^13*X[2,1,1,1,2]^5-x^17-x^16*X[2 ,1,1,1,2]-25*x^15*X[2,1,1,1,2]^2-30*x^14*X[2,1,1,1,2]^3-6*x^13*X[2,1,1,1,2]^4+ 11*x^15*X[2,1,1,1,2]+30*x^14*X[2,1,1,1,2]^2+15*x^13*X[2,1,1,1,2]^3-x^12*X[2,1,1 ,1,2]^4-2*x^15-15*x^14*X[2,1,1,1,2]-19*x^13*X[2,1,1,1,2]^2+4*x^11*X[2,1,1,1,2]^ 4+3*x^14+12*x^13*X[2,1,1,1,2]+6*x^12*X[2,1,1,1,2]^2-14*x^11*X[2,1,1,1,2]^3-3*x^ 10*X[2,1,1,1,2]^4-3*x^13-8*x^12*X[2,1,1,1,2]+17*x^11*X[2,1,1,1,2]^2+17*x^10*X[2 ,1,1,1,2]^3+x^9*X[2,1,1,1,2]^4+3*x^12-8*x^11*X[2,1,1,1,2]-31*x^10*X[2,1,1,1,2]^ 2-12*x^9*X[2,1,1,1,2]^3+x^11+23*x^10*X[2,1,1,1,2]+31*x^9*X[2,1,1,1,2]^2+5*x^8*X [2,1,1,1,2]^3-6*x^10-30*x^9*X[2,1,1,1,2]-20*x^8*X[2,1,1,1,2]^2-x^7*X[2,1,1,1,2] ^3+10*x^9+24*x^8*X[2,1,1,1,2]+7*x^7*X[2,1,1,1,2]^2-9*x^8-4*x^7*X[2,1,1,1,2]-x^6 *X[2,1,1,1,2]^2-3*x^7-16*x^6*X[2,1,1,1,2]+24*x^6+23*x^5*X[2,1,1,1,2]-44*x^5-16* x^4*X[2,1,1,1,2]+51*x^4+6*x^3*X[2,1,1,1,2]-41*x^3-x^2*X[2,1,1,1,2]+22*x^2-7*x+1 )/(x^3*X[2,1,1,1,2]-x^3-x^2*X[2,1,1,1,2]+x^2-2*x+1)/(x^3*X[2,1,1,1,2]-x^3+x-1)/ (x^3*X[2,1,1,1,2]-x^3-x+1)/(x^12*X[2,1,1,1,2]^4-4*x^12*X[2,1,1,1,2]^3+6*x^12*X[ 2,1,1,1,2]^2-4*x^12*X[2,1,1,1,2]+x^12+x^8*X[2,1,1,1,2]^3-x^8*X[2,1,1,1,2]^2-x^7 *X[2,1,1,1,2]^3-x^8*X[2,1,1,1,2]+x^7*X[2,1,1,1,2]^2+x^8+x^7*X[2,1,1,1,2]-x^7+x^ 6*X[2,1,1,1,2]-x^6-x^5*X[2,1,1,1,2]+x^5+x^4-4*x^3+6*x^2-4*x+1) Furthermore, the expectation of the random variable number of occurences (by\ containment) of , [2, 1, 1, 1, 2], equals , - 9/8 + n/8 83 7 n The variance equals , - -- + --- 64 64 21 9 n The , 3, -th moment about the mean is , - -- + --- 16 128 18553 147 2 887 The , 4, -th moment about the mean is , ----- + ---- n - ---- n 4096 4096 1024 The compositions of, 7, that yield the, 4, -th largest growth, that is, 1.8392867552141611326, are , [1, 1, 1, 4], [1, 1, 4, 1], [1, 4, 1, 1], [4, 1, 1, 1] Theorem Number, 4, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 1, 1, 4] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 6 5 4 3 2 \ n x - x + x - x + 3 x - 3 x + 1 ) a(n) x = ---------------------------------- / 3 2 3 ----- (x + x + x - 1) (-1 + x) n = 0 and in Maple format (x^6-x^5+x^4-x^3+3*x^2-3*x+1)/(x^3+x^2+x-1)/(-1+x)^3 The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 63, 122, 233, 440, 824, 1534, 2844, 5258, 9703, 17884, 32937, 60630, 111572, 205276, 377632, 694652, 1277751, 2350246, 4322881, 7951132, 14624536, 26898850, 49474844, 90998582] The limit of a(n+1)/a(n) as n goes to infinity is 1.83928675521 a(n) is asymptotic to 1.04604980580*1.83928675521^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [1, 1, 1, 4], denoted by the variable, X[1, 1, 1, 4], is 6 6 5 5 4 4 3 (x X[1, 1, 1, 4] - x - x X[1, 1, 1, 4] + x + x X[1, 1, 1, 4] - x + x 2 / 2 4 4 - 3 x + 3 x - 1) / ((-1 + x) (x X[1, 1, 1, 4] - x + 2 x - 1)) / and in Maple format (x^6*X[1,1,1,4]-x^6-x^5*X[1,1,1,4]+x^5+x^4*X[1,1,1,4]-x^4+x^3-3*x^2+3*x-1)/(-1+ x)^2/(x^4*X[1,1,1,4]-x^4+2*x-1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [1, 1, 1, 4], equals , - 1/2 + ---- 16 9 n The variance equals , - 7/32 + --- 256 15 n The , 3, -th moment about the mean is , 3/128 + ---- 2048 497 243 2 1821 The , 4, -th moment about the mean is , ---- + ----- n - ----- n 2048 65536 32768 The compositions of, 7, that yield the, 5, -th largest growth, that is, 1.8667603991738620930, are , [2, 1, 1, 3], [3, 1, 1, 2] Theorem Number, 5, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 1, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n ) a(n) x = / ----- n = 0 3 12 10 9 7 6 5 3 2 (x - x + 1) (x + x + x + x + x - 2 x + x - 3 x + 3 x - 1) - -------------------------------------------------------------------- 4 3 12 8 7 6 5 3 2 (x - x + 2 x - 1) (x - x + x + x - x + x - 3 x + 3 x - 1) and in Maple format -(x^3-x+1)*(x^12+x^10+x^9+x^7+x^6-2*x^5+x^3-3*x^2+3*x-1)/(x^4-x^3+2*x-1)/(x^12- x^8+x^7+x^6-x^5+x^3-3*x^2+3*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 63, 122, 233, 440, 824, 1535, 2852, 5296, 9841, 18309, 34108, 63610, 118722, 221683, 414015, 773226, 1443982, 2696289, 5034099, 9398052, 17543959, 32749370, 61132622, 114115265] The limit of a(n+1)/a(n) as n goes to infinity is 1.86676039917 a(n) is asymptotic to .840740726039*1.86676039917^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [2, 1, 1, 3], denoted by the variable, X[2, 1, 1, 3], is 15 4 15 3 15 2 - (x X[2, 1, 1, 3] - 4 x X[2, 1, 1, 3] + 6 x X[2, 1, 1, 3] 15 15 12 3 12 2 - 4 x X[2, 1, 1, 3] + x - 2 x X[2, 1, 1, 3] + 6 x X[2, 1, 1, 3] 11 3 12 11 2 + x X[2, 1, 1, 3] - 6 x X[2, 1, 1, 3] - 3 x X[2, 1, 1, 3] 10 3 12 11 10 2 - x X[2, 1, 1, 3] + 2 x + 3 x X[2, 1, 1, 3] + 3 x X[2, 1, 1, 3] 11 10 9 2 10 - x - 3 x X[2, 1, 1, 3] + 2 x X[2, 1, 1, 3] + x 9 8 2 9 8 - 4 x X[2, 1, 1, 3] - 3 x X[2, 1, 1, 3] + 2 x + 6 x X[2, 1, 1, 3] 7 2 8 7 6 6 + x X[2, 1, 1, 3] - 3 x - x X[2, 1, 1, 3] - 4 x X[2, 1, 1, 3] + 4 x 5 5 4 4 3 + 5 x X[2, 1, 1, 3] - 5 x - 3 x X[2, 1, 1, 3] + 2 x + x X[2, 1, 1, 3] 3 2 / + 3 x - 6 x + 4 x - 1) / ( / 4 4 3 3 (x X[2, 1, 1, 3] - x - x X[2, 1, 1, 3] + x - 2 x + 1) ( 12 3 12 2 12 12 x X[2, 1, 1, 3] - 3 x X[2, 1, 1, 3] + 3 x X[2, 1, 1, 3] - x 8 2 8 7 2 8 + x X[2, 1, 1, 3] - 2 x X[2, 1, 1, 3] - x X[2, 1, 1, 3] + x 7 7 6 6 5 5 + 2 x X[2, 1, 1, 3] - x + x X[2, 1, 1, 3] - x - x X[2, 1, 1, 3] + x 3 2 - x + 3 x - 3 x + 1)) and in Maple format -(x^15*X[2,1,1,3]^4-4*x^15*X[2,1,1,3]^3+6*x^15*X[2,1,1,3]^2-4*x^15*X[2,1,1,3]+x ^15-2*x^12*X[2,1,1,3]^3+6*x^12*X[2,1,1,3]^2+x^11*X[2,1,1,3]^3-6*x^12*X[2,1,1,3] -3*x^11*X[2,1,1,3]^2-x^10*X[2,1,1,3]^3+2*x^12+3*x^11*X[2,1,1,3]+3*x^10*X[2,1,1, 3]^2-x^11-3*x^10*X[2,1,1,3]+2*x^9*X[2,1,1,3]^2+x^10-4*x^9*X[2,1,1,3]-3*x^8*X[2, 1,1,3]^2+2*x^9+6*x^8*X[2,1,1,3]+x^7*X[2,1,1,3]^2-3*x^8-x^7*X[2,1,1,3]-4*x^6*X[2 ,1,1,3]+4*x^6+5*x^5*X[2,1,1,3]-5*x^5-3*x^4*X[2,1,1,3]+2*x^4+x^3*X[2,1,1,3]+3*x^ 3-6*x^2+4*x-1)/(x^4*X[2,1,1,3]-x^4-x^3*X[2,1,1,3]+x^3-2*x+1)/(x^12*X[2,1,1,3]^3 -3*x^12*X[2,1,1,3]^2+3*x^12*X[2,1,1,3]-x^12+x^8*X[2,1,1,3]^2-2*x^8*X[2,1,1,3]-x ^7*X[2,1,1,3]^2+x^8+2*x^7*X[2,1,1,3]-x^7+x^6*X[2,1,1,3]-x^6-x^5*X[2,1,1,3]+x^5- x^3+3*x^2-3*x+1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [2, 1, 1, 3], equals , - 1/2 + ---- 16 29 13 n The variance equals , - -- + ---- 64 256 177 63 n The , 3, -th moment about the mean is , - --- + ---- 512 2048 2041 507 2 4513 The , 4, -th moment about the mean is , ---- + ----- n - ----- n 4096 65536 32768 The compositions of, 7, that yield the, 6, -th largest growth, that is, 1.8908049048980778372, are , [2, 1, 2, 2], [2, 2, 1, 2] Theorem Number, 6, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 2, 2] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 2 5 4 2 \ n (x - x + 1) (x + x + x - 2 x + 1) ) a(n) x = - ------------------------------------------------ / 8 7 6 5 4 3 2 ----- x - x + x + x - 3 x + 5 x - 6 x + 4 x - 1 n = 0 and in Maple format -(x^2-x+1)*(x^5+x^4+x^2-2*x+1)/(x^8-x^7+x^6+x^5-3*x^4+5*x^3-6*x^2+4*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 63, 122, 233, 441, 831, 1564, 2945, 5552, 10480, 19802, 37440, 70811, 133938, 253331, 479106, 906009, 1713170, 3239274, 6124701, 11580293, 21895548, 41399521, 78277699, 148007345] The limit of a(n+1)/a(n) as n goes to infinity is 1.89080490490 a(n) is asymptotic to .742822192761*1.89080490490^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [2, 1, 2, 2], denoted by the variable, X[2, 1, 2, 2], is 7 2 7 7 4 4 - (x X[2, 1, 2, 2] - 2 x X[2, 1, 2, 2] + x - 2 x X[2, 1, 2, 2] + 2 x 3 3 2 2 / + 2 x X[2, 1, 2, 2] - 3 x - x X[2, 1, 2, 2] + 4 x - 3 x + 1) / ( / 8 2 8 7 2 8 x X[2, 1, 2, 2] - 2 x X[2, 1, 2, 2] - x X[2, 1, 2, 2] + x 7 7 6 6 5 5 + 2 x X[2, 1, 2, 2] - x - x X[2, 1, 2, 2] + x - x X[2, 1, 2, 2] + x 4 4 3 3 2 + 3 x X[2, 1, 2, 2] - 3 x - 3 x X[2, 1, 2, 2] + 5 x + x X[2, 1, 2, 2] 2 - 6 x + 4 x - 1) and in Maple format -(x^7*X[2,1,2,2]^2-2*x^7*X[2,1,2,2]+x^7-2*x^4*X[2,1,2,2]+2*x^4+2*x^3*X[2,1,2,2] -3*x^3-x^2*X[2,1,2,2]+4*x^2-3*x+1)/(x^8*X[2,1,2,2]^2-2*x^8*X[2,1,2,2]-x^7*X[2,1 ,2,2]^2+x^8+2*x^7*X[2,1,2,2]-x^7-x^6*X[2,1,2,2]+x^6-x^5*X[2,1,2,2]+x^5+3*x^4*X[ 2,1,2,2]-3*x^4-3*x^3*X[2,1,2,2]+5*x^3+x^2*X[2,1,2,2]-6*x^2+4*x-1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [2, 1, 2, 2], equals , - 1/2 + ---- 16 53 21 n The variance equals , - -- + ---- 64 256 975 291 n The , 3, -th moment about the mean is , - --- + ----- 512 2048 13811 1323 2 2913 The , 4, -th moment about the mean is , - ----- + ----- n - ----- n 4096 65536 32768 The compositions of, 7, that yield the, 7, -th largest growth, that is, 1.9087907387871591034, are , [1, 2, 2, 2], [2, 2, 2, 1] Theorem Number, 7, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 2, 2, 2] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 2 4 2 \ n (x - x + 1) (x + x - 2 x + 1) ) a(n) x = ------------------------------------------ / 5 4 3 2 ----- (-1 + x) (x - x + 2 x - 3 x + 3 x - 1) n = 0 and in Maple format (x^2-x+1)*(x^4+x^2-2*x+1)/(-1+x)/(x^5-x^4+2*x^3-3*x^2+3*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 63, 122, 234, 447, 853, 1628, 3108, 5934, 11329, 21627, 41283, 78801, 150414, 287108, 548029, 1046074, 1996739, 3811360, 7275091, 13886627, 26506664, 50595673, 96576551, 184344427] The limit of a(n+1)/a(n) as n goes to infinity is 1.90879073879 a(n) is asymptotic to .696434768709*1.90879073879^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [1, 2, 2, 2], denoted by the variable, X[1, 2, 2, 2], is 6 6 5 5 4 4 (x X[1, 2, 2, 2] - x - x X[1, 2, 2, 2] + x + 2 x X[1, 2, 2, 2] - 2 x 3 3 2 2 / - 2 x X[1, 2, 2, 2] + 3 x + x X[1, 2, 2, 2] - 4 x + 3 x - 1) / ( / 5 5 4 4 (-1 + x) (x X[1, 2, 2, 2] - x - x X[1, 2, 2, 2] + x 3 3 2 2 + 2 x X[1, 2, 2, 2] - 2 x - x X[1, 2, 2, 2] + 3 x - 3 x + 1)) and in Maple format (x^6*X[1,2,2,2]-x^6-x^5*X[1,2,2,2]+x^5+2*x^4*X[1,2,2,2]-2*x^4-2*x^3*X[1,2,2,2]+ 3*x^3+x^2*X[1,2,2,2]-4*x^2+3*x-1)/(-1+x)/(x^5*X[1,2,2,2]-x^5-x^4*X[1,2,2,2]+x^4 +2*x^3*X[1,2,2,2]-2*x^3-x^2*X[1,2,2,2]+3*x^2-3*x+1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [1, 2, 2, 2], equals , - 1/2 + ---- 16 29 25 n The variance equals , - -- + ---- 32 256 237 351 n The , 3, -th moment about the mean is , - --- + ----- 128 2048 2339 1875 2 7981 The , 4, -th moment about the mean is , - ---- + ----- n - ----- n 2048 65536 32768 The compositions of, 7, that yield the, 8, -th largest growth, that is, 1.9275619754829253043, are , [1, 1, 5], [1, 5, 1], [5, 1, 1] Theorem Number, 8, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 1, 5] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 5 2 \ n x + x - 2 x + 1 ) a(n) x = - -------------------------------- / 4 3 2 2 ----- (x + x + x + x - 1) (-1 + x) n = 0 and in Maple format -(x^5+x^2-2*x+1)/(x^4+x^3+x^2+x-1)/(-1+x)^2 The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 63, 123, 239, 463, 895, 1728, 3334, 6430, 12398, 23902, 46077, 88821, 171213, 330029, 636157, 1226238, 2363656, 4556100, 8782172, 16928188, 32630139, 62896623, 121237147, 233692123] The limit of a(n+1)/a(n) as n goes to infinity is 1.92756197548 a(n) is asymptotic to .658254106618*1.92756197548^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [1, 1, 5], denoted by the variable, X[1, 1, 5], is 5 5 2 x X[1, 1, 5] - x - x + 2 x - 1 - --------------------------------------- 5 5 (-1 + x) (x X[1, 1, 5] - x + 2 x - 1) and in Maple format -(x^5*X[1,1,5]-x^5-x^2+2*x-1)/(-1+x)/(x^5*X[1,1,5]-x^5+2*x-1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [1, 1, 5], equals , - 7/32 + ---- 32 139 23 n The variance equals , - ---- + ---- 1024 1024 531 171 n The , 3, -th moment about the mean is , - ----- + ----- 16384 16384 118049 1587 2 5449 The , 4, -th moment about the mean is , ------- + ------- n - ------ n 1048576 1048576 262144 The compositions of, 7, that yield the, 9, -th largest growth, that is, 1.9331849818995204468, are , [2, 1, 4], [4, 1, 2] Theorem Number, 9, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 4] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 9 4 2 \ n x + x + x - 2 x + 1 ) a(n) x = - -------------------------------- / 5 5 4 ----- (x - x + 1) (x - x + 2 x - 1) n = 0 and in Maple format -(x^9+x^4+x^2-2*x+1)/(x^5-x+1)/(x^5-x^4+2*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 63, 123, 239, 463, 895, 1729, 3340, 6453, 12470, 24102, 46590, 90066, 174117, 336608, 650738, 1258013, 2431989, 4701499, 9088874, 17570469, 33966945, 65664349, 126941280, 245400916] The limit of a(n+1)/a(n) as n goes to infinity is 1.93318498190 a(n) is asymptotic to .633393735410*1.93318498190^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [2, 1, 4], denoted by the variable, X[2, 1, 4], is 9 2 9 9 4 4 2 - (x X[2, 1, 4] - 2 x X[2, 1, 4] + x - x X[2, 1, 4] + x + x - 2 x + 1) / 5 5 / ((x X[2, 1, 4] - x + x - 1) / 5 5 4 4 (x X[2, 1, 4] - x - x X[2, 1, 4] + x - 2 x + 1)) and in Maple format -(x^9*X[2,1,4]^2-2*x^9*X[2,1,4]+x^9-x^4*X[2,1,4]+x^4+x^2-2*x+1)/(x^5*X[2,1,4]-x ^5+x-1)/(x^5*X[2,1,4]-x^5-x^4*X[2,1,4]+x^4-2*x+1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [2, 1, 4], equals , - 7/32 + ---- 32 191 27 n The variance equals , - ---- + ---- 1024 1024 2145 297 n The , 3, -th moment about the mean is , - ----- + ----- 16384 16384 64345 2187 2 6285 The , 4, -th moment about the mean is , ------- + ------- n - ------ n 1048576 1048576 262144 The compositions of, 7, that yield the, 10, -th largest growth, that is, 1.9417130342786384772, are , [3, 1, 3] Theorem Number, 10, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [3, 1, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 8 7 3 2 \ n x + x + x + x - 2 x + 1 ) a(n) x = - ------------------------------------------ / 5 4 5 4 3 ----- (x + x - x + 1) (x + x - x + 2 x - 1) n = 0 and in Maple format -(x^8+x^7+x^3+x^2-2*x+1)/(x^5+x^4-x+1)/(x^5+x^4-x^3+2*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 63, 123, 239, 463, 896, 1735, 3363, 6525, 12669, 24607, 47798, 92839, 180301, 350120, 679834, 1319997, 2562947, 4976338, 9662422, 18761496, 36429440, 70735795, 137349260, 266693847] The limit of a(n+1)/a(n) as n goes to infinity is 1.94171303428 a(n) is asymptotic to .603201328844*1.94171303428^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [3, 1, 3], denoted by the variable, X[3, 1, 3], is 8 2 8 7 2 8 7 - (x X[3, 1, 3] - 2 x X[3, 1, 3] + x X[3, 1, 3] + x - 2 x X[3, 1, 3] 7 3 3 2 / + x - x X[3, 1, 3] + x + x - 2 x + 1) / ( / 5 5 4 4 (x X[3, 1, 3] - x + x X[3, 1, 3] - x + x - 1) 5 5 4 4 3 3 (x X[3, 1, 3] - x + x X[3, 1, 3] - x - x X[3, 1, 3] + x - 2 x + 1)) and in Maple format -(x^8*X[3,1,3]^2-2*x^8*X[3,1,3]+x^7*X[3,1,3]^2+x^8-2*x^7*X[3,1,3]+x^7-x^3*X[3,1 ,3]+x^3+x^2-2*x+1)/(x^5*X[3,1,3]-x^5+x^4*X[3,1,3]-x^4+x-1)/(x^5*X[3,1,3]-x^5+x^ 4*X[3,1,3]-x^4-x^3*X[3,1,3]+x^3-2*x+1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [3, 1, 3], equals , - 7/32 + ---- 32 287 35 n The variance equals , - ---- + ---- 1024 1024 6513 633 n The , 3, -th moment about the mean is , - ----- + ----- 16384 16384 3675 2 381671 3841 The , 4, -th moment about the mean is , ------- n - ------- - ------ n 1048576 1048576 262144 The compositions of, 7, that yield the, 11, -th largest growth, that is, 1.9454365275632690792, are , [2, 2, 3], [2, 3, 2], [3, 2, 2] Theorem Number, 11, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 2, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 5 4 3 2 \ n x - x + x + x - 2 x + 1 ) a(n) x = - ------------------------------------ / 6 5 4 3 2 ----- x - x + 2 x - x - 2 x + 3 x - 1 n = 0 and in Maple format -(x^5-x^4+x^3+x^2-2*x+1)/(x^6-x^5+2*x^4-x^3-2*x^2+3*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 63, 123, 239, 464, 901, 1751, 3405, 6624, 12888, 25076, 48788, 94918, 184659, 359241, 698875, 1359608, 2645021, 5145713, 10010657, 19475106, 37887600, 73707944, 143394148, 278964224] The limit of a(n+1)/a(n) as n goes to infinity is 1.94543652756 a(n) is asymptotic to .595712202984*1.94543652756^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [2, 2, 3], denoted by the variable, X[2, 2, 3], is 5 5 4 4 3 3 2 - (x X[2, 2, 3] - x - x X[2, 2, 3] + x + x X[2, 2, 3] - x - x + 2 x - 1) / 6 6 5 5 4 4 / (x X[2, 2, 3] - x - x X[2, 2, 3] + x + 2 x X[2, 2, 3] - 2 x / 3 3 2 - x X[2, 2, 3] + x + 2 x - 3 x + 1) and in Maple format -(x^5*X[2,2,3]-x^5-x^4*X[2,2,3]+x^4+x^3*X[2,2,3]-x^3-x^2+2*x-1)/(x^6*X[2,2,3]-x ^6-x^5*X[2,2,3]+x^5+2*x^4*X[2,2,3]-2*x^4-x^3*X[2,2,3]+x^3+2*x^2-3*x+1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [2, 2, 3], equals , - 7/32 + ---- 32 315 39 n The variance equals , - ---- + ---- 1024 1024 7851 819 n The , 3, -th moment about the mean is , - ----- + ----- 16384 16384 493311 4563 2 1257 The , 4, -th moment about the mean is , - ------- + ------- n - ------ n 1048576 1048576 262144 The compositions of, 7, that yield the, 12, -th largest growth, that is, 1.9659482366454853372, are , [1, 6], [6, 1] Theorem Number, 12, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 6] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 6 \ n x - x + 1 ) a(n) x = ------------------------------------ / 5 4 3 2 ----- (-1 + x) (x + x + x + x + x - 1) n = 0 and in Maple format (x^6-x+1)/(-1+x)/(x^5+x^4+x^3+x^2+x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 63, 124, 244, 480, 944, 1856, 3649, 7174, 14104, 27728, 54512, 107168, 210687, 414200, 814296, 1600864, 3147216, 6187264, 12163841, 23913482, 47012668, 92424472, 181701728, 357216192] The limit of a(n+1)/a(n) as n goes to infinity is 1.96594823665 a(n) is asymptotic to .556889175229*1.96594823665^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [1, 6], denoted by the variable, X[1, 6], is 6 6 x X[1, 6] - x + x - 1 ------------------------- 6 6 x X[1, 6] - x + 2 x - 1 and in Maple format (x^6*X[1,6]-x^6+x-1)/(x^6*X[1,6]-x^6+2*x-1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [1, 6], equals , - 3/32 + ---- 64 53 n The variance equals , - 9/128 + ---- 4096 1137 141 n The , 3, -th moment about the mean is , - ----- + ----- 32768 16384 13419 8427 2 25505 The , 4, -th moment about the mean is , ------ + -------- n - ------- n 524288 16777216 8388608 The compositions of, 7, that yield the, 13, -th largest growth, that is, 1.9671682128139660358, are , [2, 5], [5, 2] Theorem Number, 13, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 5] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 5 \ n x - x + 1 ) a(n) x = - ----------------- / 6 5 ----- x - x + 2 x - 1 n = 0 and in Maple format -(x^5-x+1)/(x^6-x^5+2*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 63, 124, 244, 480, 944, 1857, 3653, 7186, 14136, 27808, 54703, 107610, 211687, 416424, 819176, 1611457, 3170007, 6235937, 12267137, 24131522, 47470763, 93382976, 183700022, 361368844] The limit of a(n+1)/a(n) as n goes to infinity is 1.96716821281 a(n) is asymptotic to .552975339403*1.96716821281^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [2, 5], denoted by the variable, X[2, 5], is 5 5 x X[2, 5] - x + x - 1 - ------------------------------------------- 6 6 5 5 x X[2, 5] - x - x X[2, 5] + x - 2 x + 1 and in Maple format -(x^5*X[2,5]-x^5+x-1)/(x^6*X[2,5]-x^6-x^5*X[2,5]+x^5-2*x+1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [2, 5], equals , - 3/32 + ---- 64 83 57 n The variance equals , - ---- + ---- 1024 4096 483 357 n The , 3, -th moment about the mean is , - ---- + ----- 8192 32768 9747 2 3869 7269 The , 4, -th moment about the mean is , -------- n - ------- - ------- n 16777216 1048576 8388608 The compositions of, 7, that yield the, 14, -th largest growth, that is, 1.9693144732632464526, are , [3, 4], [4, 3] Theorem Number, 14, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [3, 4] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 4 \ n x - x + 1 ) a(n) x = - ---------------------- / 6 5 4 ----- x + x - x + 2 x - 1 n = 0 and in Maple format -(x^4-x+1)/(x^6+x^5-x^4+2*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 63, 124, 244, 480, 945, 1861, 3665, 7218, 14215, 27994, 55129, 108566, 213800, 421039, 829158, 1632873, 3215641, 6332609, 12470899, 24559122, 48364634, 95245173, 187567697, 369379780] The limit of a(n+1)/a(n) as n goes to infinity is 1.96931447326 a(n) is asymptotic to .547042335160*1.96931447326^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [3, 4], denoted by the variable, X[3, 4], is 4 4 x X[3, 4] - x + x - 1 - ------------------------------------------------------------- 6 6 5 5 4 4 x X[3, 4] - x + x X[3, 4] - x - x X[3, 4] + x - 2 x + 1 and in Maple format -(x^4*X[3,4]-x^4+x-1)/(x^6*X[3,4]-x^6+x^5*X[3,4]-x^5-x^4*X[3,4]+x^4-2*x+1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [3, 4], equals , - 3/32 + ---- 64 103 65 n The variance equals , - ---- + ---- 1024 4096 915 33 n The , 3, -th moment about the mean is , - ---- + ---- 8192 2048 100285 12675 2 52939 The , 4, -th moment about the mean is , - ------- + -------- n + ------- n 1048576 16777216 8388608 The compositions of, 7, that yield the, 15, -th largest growth, that is, 1.9835828434243263304, are , [7] Theorem Number, 15, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [7] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 1 ) a(n) x = - ------------------------------ / 6 5 4 3 2 ----- x + x + x + x + x + x - 1 n = 0 and in Maple format -1/(x^6+x^5+x^4+x^3+x^2+x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 63, 125, 248, 492, 976, 1936, 3840, 7617, 15109, 29970, 59448, 117920, 233904, 463968, 920319, 1825529, 3621088, 7182728, 14247536, 28261168, 56058368, 111196417, 220567305, 437513522] The limit of a(n+1)/a(n) as n goes to infinity is 1.98358284342 a(n) is asymptotic to .521772494287*1.98358284342^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [7], denoted by the variable, X[7], is -1 + x ---------------------- 7 7 x X[7] - x + 2 x - 1 and in Maple format (-1+x)/(x^7*X[7]-x^7+2*x-1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [7], equals , - 5/128 + --- 128 535 115 n The variance equals , - ----- + ----- 16384 16384 1425 2943 n The , 3, -th moment about the mean is , - ----- + ------ 65536 524288 39675 2 127837 435295 The , 4, -th moment about the mean is , --------- n + -------- n - --------- 268435456 67108864 268435456 This ends this article, that took, 1.112, seconds to generate. ---------------------------------------------------------- ---------------------------------------- Generating functions and Growth rates for Weight-Enumerating Sequences Accor\ ding to the Number of occurrences of compositions of, 8 By Shalosh B. Ekhad The compositions of, 8, that yield polynomial growth (hence growth ratio 1, that is the lowest) are , [2, 1, 1, 1, 1, 2] Theorem Number, 1, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 1, 1, 1, 2] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 32 30 29 28 27 26 25 24 23 ) a(n) x = - (x + 2 x + x + x + x - x + x - x - 2 x / ----- n = 0 22 21 20 19 18 17 16 15 14 + 2 x + 2 x - 6 x + 8 x - 6 x + 5 x - 12 x + 32 x - 57 x 13 12 11 10 9 8 7 6 + 60 x - 22 x - 54 x + 161 x - 306 x + 488 x - 652 x + 702 x 5 4 3 2 / 3 2 - 590 x + 376 x - 175 x + 56 x - 11 x + 1) / ((x - x + 2 x - 1) / 15 10 9 8 7 5 4 3 2 (x - x + 3 x - 3 x + x + x - 5 x + 10 x - 10 x + 5 x - 1) 15 10 9 8 7 6 5 4 3 2 (x + 2 x - 5 x + 3 x + x - x + x - 5 x + 10 x - 10 x + 5 x - 1) ) and in Maple format -(x^32+2*x^30+x^29+x^28+x^27-x^26+x^25-x^24-2*x^23+2*x^22+2*x^21-6*x^20+8*x^19-\ 6*x^18+5*x^17-12*x^16+32*x^15-57*x^14+60*x^13-22*x^12-54*x^11+161*x^10-306*x^9+ 488*x^8-652*x^7+702*x^6-590*x^5+376*x^4-175*x^3+56*x^2-11*x+1)/(x^3-x^2+2*x-1)/ (x^15-x^10+3*x^9-3*x^8+x^7+x^5-5*x^4+10*x^3-10*x^2+5*x-1)/(x^15+2*x^10-5*x^9+3* x^8+x^7-x^6+x^5-5*x^4+10*x^3-10*x^2+5*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 127, 248, 474, 885, 1615, 2886, 5064, 8755, 14975, 25455, 43190, 73428, 125451, 215775, 373875, 652374, 1145083, 2018792, 3569071, 6317906, 11184155, 19780765, 34932849] ---------------------------------------------------------------------------- The compositions of, 8, that yield the, 2, -th largest growth, that is, 1.6180339887498948482, are , [1, 1, 1, 1, 1, 3], [1, 1, 1, 1, 3, 1], [1, 1, 1, 3, 1, 1], [1, 1, 3, 1, 1, 1], [1, 3, 1, 1, 1, 1], [3, 1, 1, 1, 1, 1] Theorem Number, 2, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 1, 1, 1, 1, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 7 6 5 4 3 2 \ n x - 2 x + 3 x + 2 x - 9 x + 10 x - 5 x + 1 ) a(n) x = ------------------------------------------------ / 2 5 ----- (x + x - 1) (-1 + x) n = 0 and in Maple format (x^7-2*x^6+3*x^5+2*x^4-9*x^3+10*x^2-5*x+1)/(x^2+x-1)/(-1+x)^5 The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 127, 248, 474, 885, 1615, 2886, 5063, 8743, 14899, 25113, 41953, 69583, 114750, 188381, 308167, 502744, 818458, 1330311, 2159672, 3502934, 5677882, 9198718, 14897454] The limit of a(n+1)/a(n) as n goes to infinity is 1.61803398875 a(n) is asymptotic to 8.02492235950*1.61803398875^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [1, 1, 1, 1, 1, 3], denoted by the variable, X[1, 1, 1, 1, 1, 3], is 7 7 6 6 5 5 4 4 3 3 (x %1 - x - 2 x %1 + 2 x + 4 x %1 - 3 x - 3 x %1 - 2 x + x %1 + 9 x 2 / 4 3 3 - 10 x + 5 x - 1) / ((-1 + x) (x %1 - x + 2 x - 1)) / %1 := X[1, 1, 1, 1, 1, 3] and in Maple format (x^7*X[1,1,1,1,1,3]-x^7-2*x^6*X[1,1,1,1,1,3]+2*x^6+4*x^5*X[1,1,1,1,1,3]-3*x^5-3 *x^4*X[1,1,1,1,1,3]-2*x^4+x^3*X[1,1,1,1,1,3]+9*x^3-10*x^2+5*x-1)/(-1+x)^4/(x^3* X[1,1,1,1,1,3]-x^3+2*x-1) Furthermore, the expectation of the random variable number of occurences (by\ containment) of , [1, 1, 1, 1, 1, 3], equals , - 11/8 + n/8 19 3 n The variance equals , - -- + --- 64 64 The , 3, -th moment about the mean is , 9/64 1145 27 2 93 The , 4, -th moment about the mean is , ---- + ---- n - ---- n 4096 4096 1024 The compositions of, 8, that yield the, 3, -th largest growth, that is, 1.7548776662466927601, are , [1, 1, 2, 1, 1, 2], [1, 2, 1, 1, 2, 1], [2, 1, 1, 2, 1, 1] Theorem Number, 3, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 1, 2, 1, 1, 2] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 2 ) a(n) x = - (x - x + 1) / ----- n = 0 11 9 8 7 6 5 4 3 2 (2 x + 3 x - 2 x + 3 x - 7 x + 7 x - 7 x + 10 x - 10 x + 5 x - 1) / 3 2 9 6 5 4 3 2 / ((x - x + 2 x - 1) (x - x + 2 x - x + x - 3 x + 3 x - 1) / 2 (-1 + x) ) and in Maple format -(x^2-x+1)*(2*x^11+3*x^9-2*x^8+3*x^7-7*x^6+7*x^5-7*x^4+10*x^3-10*x^2+5*x-1)/(x^ 3-x^2+2*x-1)/(x^9-x^6+2*x^5-x^4+x^3-3*x^2+3*x-1)/(-1+x)^2 The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 127, 248, 474, 885, 1616, 2896, 5118, 8965, 15636, 27248, 47546, 83150, 145737, 255853, 449573, 790149, 1388411, 2438546, 4280906, 7512318, 13179842, 23121052, 40561481] The limit of a(n+1)/a(n) as n goes to infinity is 1.75487766625 a(n) is asymptotic to 1.90802250917*1.75487766625^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [1, 1, 2, 1, 1, 2], denoted by the variable, X[1, 1, 2, 1, 1, 2], is 13 4 13 3 12 4 13 2 12 3 11 4 - (2 x %1 - 8 x %1 - 2 x %1 + 12 x %1 + 8 x %1 + x %1 13 12 2 11 3 13 12 11 2 - 8 x %1 - 12 x %1 - 8 x %1 + 2 x + 8 x %1 + 18 x %1 10 3 12 11 10 2 9 3 11 + 5 x %1 - 2 x - 16 x %1 - 15 x %1 - 5 x %1 + 5 x 10 9 2 8 3 10 9 8 2 7 3 + 15 x %1 + 18 x %1 + 3 x %1 - 5 x - 21 x %1 - 17 x %1 - x %1 9 8 7 2 8 7 6 2 7 + 8 x + 26 x %1 + 12 x %1 - 12 x - 28 x %1 - 5 x %1 + 17 x 6 5 2 6 5 5 4 4 + 25 x %1 + x %1 - 21 x - 19 x %1 + 24 x + 12 x %1 - 27 x 3 3 2 2 / 2 - 5 x %1 + 25 x + x %1 - 16 x + 6 x - 1) / ((-1 + x) / 3 3 2 2 9 3 9 2 9 9 (x %1 - x - x %1 + x - 2 x + 1) (x %1 - 3 x %1 + 3 x %1 - x 6 2 6 5 2 6 5 5 4 4 3 2 + x %1 - 2 x %1 - x %1 + x + 3 x %1 - 2 x - x %1 + x - x + 3 x - 3 x + 1)) %1 := X[1, 1, 2, 1, 1, 2] and in Maple format -(2*x^13*X[1,1,2,1,1,2]^4-8*x^13*X[1,1,2,1,1,2]^3-2*x^12*X[1,1,2,1,1,2]^4+12*x^ 13*X[1,1,2,1,1,2]^2+8*x^12*X[1,1,2,1,1,2]^3+x^11*X[1,1,2,1,1,2]^4-8*x^13*X[1,1, 2,1,1,2]-12*x^12*X[1,1,2,1,1,2]^2-8*x^11*X[1,1,2,1,1,2]^3+2*x^13+8*x^12*X[1,1,2 ,1,1,2]+18*x^11*X[1,1,2,1,1,2]^2+5*x^10*X[1,1,2,1,1,2]^3-2*x^12-16*x^11*X[1,1,2 ,1,1,2]-15*x^10*X[1,1,2,1,1,2]^2-5*x^9*X[1,1,2,1,1,2]^3+5*x^11+15*x^10*X[1,1,2, 1,1,2]+18*x^9*X[1,1,2,1,1,2]^2+3*x^8*X[1,1,2,1,1,2]^3-5*x^10-21*x^9*X[1,1,2,1,1 ,2]-17*x^8*X[1,1,2,1,1,2]^2-x^7*X[1,1,2,1,1,2]^3+8*x^9+26*x^8*X[1,1,2,1,1,2]+12 *x^7*X[1,1,2,1,1,2]^2-12*x^8-28*x^7*X[1,1,2,1,1,2]-5*x^6*X[1,1,2,1,1,2]^2+17*x^ 7+25*x^6*X[1,1,2,1,1,2]+x^5*X[1,1,2,1,1,2]^2-21*x^6-19*x^5*X[1,1,2,1,1,2]+24*x^ 5+12*x^4*X[1,1,2,1,1,2]-27*x^4-5*x^3*X[1,1,2,1,1,2]+25*x^3+x^2*X[1,1,2,1,1,2]-\ 16*x^2+6*x-1)/(-1+x)^2/(x^3*X[1,1,2,1,1,2]-x^3-x^2*X[1,1,2,1,1,2]+x^2-2*x+1)/(x ^9*X[1,1,2,1,1,2]^3-3*x^9*X[1,1,2,1,1,2]^2+3*x^9*X[1,1,2,1,1,2]-x^9+x^6*X[1,1,2 ,1,1,2]^2-2*x^6*X[1,1,2,1,1,2]-x^5*X[1,1,2,1,1,2]^2+x^6+3*x^5*X[1,1,2,1,1,2]-2* x^5-x^4*X[1,1,2,1,1,2]+x^4-x^3+3*x^2-3*x+1) Furthermore, the expectation of the random variable number of occurences (by\ containment) of , [1, 1, 2, 1, 1, 2], equals , - 11/8 + n/8 87 7 n The variance equals , - -- + --- 64 64 141 9 n The , 3, -th moment about the mean is , - --- + --- 128 128 22353 147 2 929 The , 4, -th moment about the mean is , ----- + ---- n - ---- n 4096 4096 1024 The compositions of, 8, that yield the, 4, -th largest growth, that is, 1.8392867552141611326, are , [1, 1, 1, 1, 4], [1, 1, 1, 4, 1], [1, 1, 4, 1, 1], [1, 4, 1, 1, 1], [4, 1, 1, 1, 1] Theorem Number, 4, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 1, 1, 1, 4] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 6 5 4 3 2 \ n 2 x - 2 x + 2 x - 4 x + 6 x - 4 x + 1 ) a(n) x = - ------------------------------------------ / 3 2 4 ----- (x + x + x - 1) (-1 + x) n = 0 and in Maple format -(2*x^6-2*x^5+2*x^4-4*x^3+6*x^2-4*x+1)/(x^3+x^2+x-1)/(-1+x)^4 The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 127, 249, 482, 922, 1746, 3280, 6124, 11382, 21085, 38969, 71906, 132536, 244108, 449384, 827016, 1521668, 2799419, 5149665, 9472546, 17423678, 32048214, 58947064, 108421908] The limit of a(n+1)/a(n) as n goes to infinity is 1.83928675521 a(n) is asymptotic to 1.24635566962*1.83928675521^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [1, 1, 1, 1, 4], denoted by the variable, X[1, 1, 1, 1, 4], is 6 6 5 5 - (2 x X[1, 1, 1, 1, 4] - 2 x - 2 x X[1, 1, 1, 1, 4] + 2 x 4 4 3 2 / 3 + x X[1, 1, 1, 1, 4] - 2 x + 4 x - 6 x + 4 x - 1) / ((-1 + x) / 4 4 (x X[1, 1, 1, 1, 4] - x + 2 x - 1)) and in Maple format -(2*x^6*X[1,1,1,1,4]-2*x^6-2*x^5*X[1,1,1,1,4]+2*x^5+x^4*X[1,1,1,1,4]-2*x^4+4*x^ 3-6*x^2+4*x-1)/(-1+x)^3/(x^4*X[1,1,1,1,4]-x^4+2*x-1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [1, 1, 1, 1, 4], equals , - 5/8 + ---- 16 9 n The variance equals , - 9/32 + --- 256 21 15 n The , 3, -th moment about the mean is , ---- + ---- 1024 2048 741 243 2 2253 The , 4, -th moment about the mean is , ---- + ----- n - ----- n 2048 65536 32768 The compositions of, 8, that yield the, 5, -th largest growth, that is, 1.8667603991738620930, are , [2, 1, 1, 1, 3], [3, 1, 1, 1, 2] Theorem Number, 5, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 1, 1, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 27 24 23 22 21 20 19 18 17 ) a(n) x = - (x + 2 x + x - x + 2 x + x - 3 x + x + x / ----- n = 0 16 15 14 13 12 11 10 9 8 + 2 x - 8 x + 5 x + 5 x - 10 x + 4 x + 4 x - 5 x + 8 x 7 6 5 4 3 2 / - 17 x + 16 x + 5 x - 29 x + 34 x - 21 x + 7 x - 1) / ( / 4 4 3 4 (x - x + 1) (x - x + 2 x - 1) (x + x - 1) 16 11 10 9 7 6 4 3 2 (x - x + 3 x - 2 x - x + x + x - 4 x + 6 x - 4 x + 1)) and in Maple format -(x^27+2*x^24+x^23-x^22+2*x^21+x^20-3*x^19+x^18+x^17+2*x^16-8*x^15+5*x^14+5*x^ 13-10*x^12+4*x^11+4*x^10-5*x^9+8*x^8-17*x^7+16*x^6+5*x^5-29*x^4+34*x^3-21*x^2+7 *x-1)/(x^4-x+1)/(x^4-x^3+2*x-1)/(x^4+x-1)/(x^16-x^11+3*x^10-2*x^9-x^7+x^6+x^4-4 *x^3+6*x^2-4*x+1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 127, 249, 482, 922, 1746, 3280, 6125, 11392, 21142, 39210, 72748, 135113, 251271, 467904, 872301, 1627646, 3038906, 5675820, 10602440, 19805504, 36993860, 69090383, 129016404] The limit of a(n+1)/a(n) as n goes to infinity is 1.86676039917 a(n) is asymptotic to .950512366759*1.86676039917^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [2, 1, 1, 1, 3], denoted by the variable, X[2, 1, 1, 1, 3], is 27 7 27 27 27 27 - (x X[2, 1, 1, 1, 3] - 7 x %5 + 21 x %4 - 35 x %3 + 35 x %2 24 27 24 23 27 - 2 x %5 - 21 x %1 + 12 x %4 - x %5 + 7 x X[2, 1, 1, 1, 3] 24 23 22 27 24 23 22 - 30 x %3 + 6 x %4 + x %5 - x + 40 x %2 - 15 x %3 - 6 x %4 21 24 23 22 21 - x %5 - 30 x %1 + 20 x %2 + 15 x %3 + 7 x %4 24 23 22 21 20 + 12 x X[2, 1, 1, 1, 3] - 15 x %1 - 20 x %2 - 20 x %3 + x %4 24 23 22 21 20 - 2 x + 6 x X[2, 1, 1, 1, 3] + 15 x %1 + 30 x %2 - 5 x %3 19 23 22 21 20 - 3 x %4 - x - 6 x X[2, 1, 1, 1, 3] - 25 x %1 + 10 x %2 19 18 22 21 20 + 15 x %3 + x %4 + x + 11 x X[2, 1, 1, 1, 3] - 10 x %1 19 18 21 20 19 - 30 x %2 - 5 x %3 - 2 x + 5 x X[2, 1, 1, 1, 3] + 30 x %1 18 17 20 19 18 + 10 x %2 - x %3 - x - 15 x X[2, 1, 1, 1, 3] - 10 x %1 17 16 19 18 17 16 + 4 x %2 - x %3 + 3 x + 5 x X[2, 1, 1, 1, 3] - 6 x %1 + 5 x %2 15 18 17 16 15 + 4 x %3 - x + 4 x X[2, 1, 1, 1, 3] - 9 x %1 - 20 x %2 14 17 16 15 14 13 - 3 x %3 - x + 7 x X[2, 1, 1, 1, 3] + 36 x %1 + 14 x %2 + x %3 16 15 14 13 15 - 2 x - 28 x X[2, 1, 1, 1, 3] - 24 x %1 + x %2 + 8 x 14 13 12 14 + 18 x X[2, 1, 1, 1, 3] - 10 x %1 - 8 x %2 - 5 x 13 12 11 13 + 13 x X[2, 1, 1, 1, 3] + 26 x %1 + 5 x %2 - 5 x 12 11 10 12 - 28 x X[2, 1, 1, 1, 3] - 14 x %1 - x %2 + 10 x 11 10 11 10 + 13 x X[2, 1, 1, 1, 3] - 2 x %1 - 4 x + 7 x X[2, 1, 1, 1, 3] 9 10 9 8 9 + 4 x %1 - 4 x - 9 x X[2, 1, 1, 1, 3] - x %1 + 5 x 8 8 7 7 + 9 x X[2, 1, 1, 1, 3] - 8 x - 18 x X[2, 1, 1, 1, 3] + 17 x 6 6 5 5 + 23 x X[2, 1, 1, 1, 3] - 16 x - 16 x X[2, 1, 1, 1, 3] - 5 x 4 4 3 3 2 + 6 x X[2, 1, 1, 1, 3] + 29 x - x X[2, 1, 1, 1, 3] - 34 x + 21 x / - 7 x + 1) / ( / 4 4 3 3 (x X[2, 1, 1, 1, 3] - x - x X[2, 1, 1, 1, 3] + x - 2 x + 1) 4 4 4 4 (x X[2, 1, 1, 1, 3] - x + x - 1) (x X[2, 1, 1, 1, 3] - x - x + 1) ( 16 16 16 16 16 11 x %3 - 4 x %2 + 6 x %1 - 4 x X[2, 1, 1, 1, 3] + x + x %2 11 10 11 10 11 - 3 x %1 - x %2 + 3 x X[2, 1, 1, 1, 3] + 5 x %1 - x 10 9 10 9 9 - 7 x X[2, 1, 1, 1, 3] - 2 x %1 + 3 x + 4 x X[2, 1, 1, 1, 3] - 2 x 7 7 6 6 4 3 2 + x X[2, 1, 1, 1, 3] - x - x X[2, 1, 1, 1, 3] + x + x - 4 x + 6 x - 4 x + 1)) 2 %1 := X[2, 1, 1, 1, 3] 3 %2 := X[2, 1, 1, 1, 3] 4 %3 := X[2, 1, 1, 1, 3] 5 %4 := X[2, 1, 1, 1, 3] 6 %5 := X[2, 1, 1, 1, 3] and in Maple format -(x^27*X[2,1,1,1,3]^7-7*x^27*X[2,1,1,1,3]^6+21*x^27*X[2,1,1,1,3]^5-35*x^27*X[2, 1,1,1,3]^4+35*x^27*X[2,1,1,1,3]^3-2*x^24*X[2,1,1,1,3]^6-21*x^27*X[2,1,1,1,3]^2+ 12*x^24*X[2,1,1,1,3]^5-x^23*X[2,1,1,1,3]^6+7*x^27*X[2,1,1,1,3]-30*x^24*X[2,1,1, 1,3]^4+6*x^23*X[2,1,1,1,3]^5+x^22*X[2,1,1,1,3]^6-x^27+40*x^24*X[2,1,1,1,3]^3-15 *x^23*X[2,1,1,1,3]^4-6*x^22*X[2,1,1,1,3]^5-x^21*X[2,1,1,1,3]^6-30*x^24*X[2,1,1, 1,3]^2+20*x^23*X[2,1,1,1,3]^3+15*x^22*X[2,1,1,1,3]^4+7*x^21*X[2,1,1,1,3]^5+12*x ^24*X[2,1,1,1,3]-15*x^23*X[2,1,1,1,3]^2-20*x^22*X[2,1,1,1,3]^3-20*x^21*X[2,1,1, 1,3]^4+x^20*X[2,1,1,1,3]^5-2*x^24+6*x^23*X[2,1,1,1,3]+15*x^22*X[2,1,1,1,3]^2+30 *x^21*X[2,1,1,1,3]^3-5*x^20*X[2,1,1,1,3]^4-3*x^19*X[2,1,1,1,3]^5-x^23-6*x^22*X[ 2,1,1,1,3]-25*x^21*X[2,1,1,1,3]^2+10*x^20*X[2,1,1,1,3]^3+15*x^19*X[2,1,1,1,3]^4 +x^18*X[2,1,1,1,3]^5+x^22+11*x^21*X[2,1,1,1,3]-10*x^20*X[2,1,1,1,3]^2-30*x^19*X [2,1,1,1,3]^3-5*x^18*X[2,1,1,1,3]^4-2*x^21+5*x^20*X[2,1,1,1,3]+30*x^19*X[2,1,1, 1,3]^2+10*x^18*X[2,1,1,1,3]^3-x^17*X[2,1,1,1,3]^4-x^20-15*x^19*X[2,1,1,1,3]-10* x^18*X[2,1,1,1,3]^2+4*x^17*X[2,1,1,1,3]^3-x^16*X[2,1,1,1,3]^4+3*x^19+5*x^18*X[2 ,1,1,1,3]-6*x^17*X[2,1,1,1,3]^2+5*x^16*X[2,1,1,1,3]^3+4*x^15*X[2,1,1,1,3]^4-x^ 18+4*x^17*X[2,1,1,1,3]-9*x^16*X[2,1,1,1,3]^2-20*x^15*X[2,1,1,1,3]^3-3*x^14*X[2, 1,1,1,3]^4-x^17+7*x^16*X[2,1,1,1,3]+36*x^15*X[2,1,1,1,3]^2+14*x^14*X[2,1,1,1,3] ^3+x^13*X[2,1,1,1,3]^4-2*x^16-28*x^15*X[2,1,1,1,3]-24*x^14*X[2,1,1,1,3]^2+x^13* X[2,1,1,1,3]^3+8*x^15+18*x^14*X[2,1,1,1,3]-10*x^13*X[2,1,1,1,3]^2-8*x^12*X[2,1, 1,1,3]^3-5*x^14+13*x^13*X[2,1,1,1,3]+26*x^12*X[2,1,1,1,3]^2+5*x^11*X[2,1,1,1,3] ^3-5*x^13-28*x^12*X[2,1,1,1,3]-14*x^11*X[2,1,1,1,3]^2-x^10*X[2,1,1,1,3]^3+10*x^ 12+13*x^11*X[2,1,1,1,3]-2*x^10*X[2,1,1,1,3]^2-4*x^11+7*x^10*X[2,1,1,1,3]+4*x^9* X[2,1,1,1,3]^2-4*x^10-9*x^9*X[2,1,1,1,3]-x^8*X[2,1,1,1,3]^2+5*x^9+9*x^8*X[2,1,1 ,1,3]-8*x^8-18*x^7*X[2,1,1,1,3]+17*x^7+23*x^6*X[2,1,1,1,3]-16*x^6-16*x^5*X[2,1, 1,1,3]-5*x^5+6*x^4*X[2,1,1,1,3]+29*x^4-x^3*X[2,1,1,1,3]-34*x^3+21*x^2-7*x+1)/(x ^4*X[2,1,1,1,3]-x^4-x^3*X[2,1,1,1,3]+x^3-2*x+1)/(x^4*X[2,1,1,1,3]-x^4+x-1)/(x^4 *X[2,1,1,1,3]-x^4-x+1)/(x^16*X[2,1,1,1,3]^4-4*x^16*X[2,1,1,1,3]^3+6*x^16*X[2,1, 1,1,3]^2-4*x^16*X[2,1,1,1,3]+x^16+x^11*X[2,1,1,1,3]^3-3*x^11*X[2,1,1,1,3]^2-x^ 10*X[2,1,1,1,3]^3+3*x^11*X[2,1,1,1,3]+5*x^10*X[2,1,1,1,3]^2-x^11-7*x^10*X[2,1,1 ,1,3]-2*x^9*X[2,1,1,1,3]^2+3*x^10+4*x^9*X[2,1,1,1,3]-2*x^9+x^7*X[2,1,1,1,3]-x^7 -x^6*X[2,1,1,1,3]+x^6+x^4-4*x^3+6*x^2-4*x+1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [2, 1, 1, 1, 3], equals , - 5/8 + ---- 16 37 13 n The variance equals , - -- + ---- 64 256 471 63 n The , 3, -th moment about the mean is , - ---- + ---- 1024 2048 3335 507 2 5761 The , 4, -th moment about the mean is , ---- + ----- n - ----- n 4096 65536 32768 The compositions of, 8, that yield the, 6, -th largest growth, that is, 1.8908049048980778372, are , [1, 2, 1, 2, 2], [1, 2, 2, 1, 2], [2, 1, 2, 2, 1], [2, 2, 1, 2, 1] Theorem Number, 6, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 2, 1, 2, 2] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 9 8 7 5 4 3 2 \ n x - x + 2 x - 2 x + 5 x - 7 x + 7 x - 4 x + 1 ) a(n) x = ----------------------------------------------------------- / 8 7 6 5 4 3 2 ----- (-1 + x) (x - x + x + x - 3 x + 5 x - 6 x + 4 x - 1) n = 0 and in Maple format (x^9-x^8+2*x^7-2*x^5+5*x^4-7*x^3+7*x^2-4*x+1)/(-1+x)/(x^8-x^7+x^6+x^5-3*x^4+5*x ^3-6*x^2+4*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 127, 249, 482, 923, 1754, 3318, 6263, 11815, 22295, 42097, 79537, 150348, 284286, 537617, 1016723, 1922732, 3635902, 6875176, 12999877, 24580170, 46475718, 87875239, 166152938] The limit of a(n+1)/a(n) as n goes to infinity is 1.89080490490 a(n) is asymptotic to .833877528824*1.89080490490^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [1, 2, 1, 2, 2], denoted by the variable, X[1, 2, 1, 2, 2], is 9 9 8 9 8 7 (x %1 - 2 x X[1, 2, 1, 2, 2] - x %1 + x + 2 x X[1, 2, 1, 2, 2] + x %1 8 7 7 5 5 - x - 3 x X[1, 2, 1, 2, 2] + 2 x + 2 x X[1, 2, 1, 2, 2] - 2 x 4 4 3 3 - 4 x X[1, 2, 1, 2, 2] + 5 x + 3 x X[1, 2, 1, 2, 2] - 7 x 2 2 / 8 - x X[1, 2, 1, 2, 2] + 7 x - 4 x + 1) / ((-1 + x) (x %1 / 8 7 8 7 7 - 2 x X[1, 2, 1, 2, 2] - x %1 + x + 2 x X[1, 2, 1, 2, 2] - x 6 6 5 5 - x X[1, 2, 1, 2, 2] + x - x X[1, 2, 1, 2, 2] + x 4 4 3 3 + 3 x X[1, 2, 1, 2, 2] - 3 x - 3 x X[1, 2, 1, 2, 2] + 5 x 2 2 + x X[1, 2, 1, 2, 2] - 6 x + 4 x - 1)) 2 %1 := X[1, 2, 1, 2, 2] and in Maple format (x^9*X[1,2,1,2,2]^2-2*x^9*X[1,2,1,2,2]-x^8*X[1,2,1,2,2]^2+x^9+2*x^8*X[1,2,1,2,2 ]+x^7*X[1,2,1,2,2]^2-x^8-3*x^7*X[1,2,1,2,2]+2*x^7+2*x^5*X[1,2,1,2,2]-2*x^5-4*x^ 4*X[1,2,1,2,2]+5*x^4+3*x^3*X[1,2,1,2,2]-7*x^3-x^2*X[1,2,1,2,2]+7*x^2-4*x+1)/(-1 +x)/(x^8*X[1,2,1,2,2]^2-2*x^8*X[1,2,1,2,2]-x^7*X[1,2,1,2,2]^2+x^8+2*x^7*X[1,2,1 ,2,2]-x^7-x^6*X[1,2,1,2,2]+x^6-x^5*X[1,2,1,2,2]+x^5+3*x^4*X[1,2,1,2,2]-3*x^4-3* x^3*X[1,2,1,2,2]+5*x^3+x^2*X[1,2,1,2,2]-6*x^2+4*x-1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [1, 2, 1, 2, 2], equals , - 5/8 + ---- 16 63 21 n The variance equals , - -- + ---- 64 256 2211 291 n The , 3, -th moment about the mean is , - ---- + ----- 1024 2048 12531 1323 2 5433 The , 4, -th moment about the mean is , - ----- + ----- n - ----- n 4096 65536 32768 The compositions of, 8, that yield the, 7, -th largest growth, that is, 1.8923110706522823122, are , [2, 1, 1, 2, 2], [2, 2, 1, 1, 2] Theorem Number, 7, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 1, 2, 2] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n ) a(n) x = / ----- n = 0 10 8 6 5 4 3 2 x + x + x - 2 x + 5 x - 7 x + 7 x - 4 x + 1 - --------------------------------------------------------------------- 11 10 9 8 7 6 5 4 3 2 x - x + x - x + x - x + 4 x - 8 x + 11 x - 10 x + 5 x - 1 and in Maple format -(x^10+x^8+x^6-2*x^5+5*x^4-7*x^3+7*x^2-4*x+1)/(x^11-x^10+x^9-x^8+x^7-x^6+4*x^5-\ 8*x^4+11*x^3-10*x^2+5*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 127, 249, 482, 923, 1753, 3311, 6235, 11730, 22077, 41599, 78493, 148301, 280481, 530840, 1005052, 1903140, 3603611, 6822606, 12914983, 24443975, 46259097, 87535889, 165635897] The limit of a(n+1)/a(n) as n goes to infinity is 1.89231107065 a(n) is asymptotic to .811466235213*1.89231107065^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [2, 1, 1, 2, 2], denoted by the variable, X[2, 1, 1, 2, 2], is 11 11 10 11 10 9 11 - (x %2 - 3 x %1 - x %2 + 3 x X[2, 1, 1, 2, 2] + 3 x %1 + x %2 - x 10 9 10 9 8 - 3 x X[2, 1, 1, 2, 2] - 3 x %1 + x + 3 x X[2, 1, 1, 2, 2] + x %1 9 8 8 7 7 - x - 2 x X[2, 1, 1, 2, 2] + x + x X[2, 1, 1, 2, 2] - x 6 6 5 5 - 3 x X[2, 1, 1, 2, 2] + 3 x + 6 x X[2, 1, 1, 2, 2] - 7 x 4 4 3 3 - 7 x X[2, 1, 1, 2, 2] + 12 x + 4 x X[2, 1, 1, 2, 2] - 14 x 2 2 / 12 12 11 - x X[2, 1, 1, 2, 2] + 11 x - 5 x + 1) / (x %2 - 3 x %1 - x %2 / 12 11 10 12 + 3 x X[2, 1, 1, 2, 2] + 4 x %1 + 2 x %2 - x 11 10 9 11 - 5 x X[2, 1, 1, 2, 2] - 6 x %1 - x %2 + 2 x 10 9 10 9 8 + 6 x X[2, 1, 1, 2, 2] + 4 x %1 - 2 x - 5 x X[2, 1, 1, 2, 2] - x %1 9 8 8 7 7 + 2 x + 3 x X[2, 1, 1, 2, 2] - 2 x - 2 x X[2, 1, 1, 2, 2] + 2 x 6 6 5 5 + 5 x X[2, 1, 1, 2, 2] - 5 x - 10 x X[2, 1, 1, 2, 2] + 12 x 4 4 3 3 + 10 x X[2, 1, 1, 2, 2] - 19 x - 5 x X[2, 1, 1, 2, 2] + 21 x 2 2 + x X[2, 1, 1, 2, 2] - 15 x + 6 x - 1) 2 %1 := X[2, 1, 1, 2, 2] 3 %2 := X[2, 1, 1, 2, 2] and in Maple format -(x^11*X[2,1,1,2,2]^3-3*x^11*X[2,1,1,2,2]^2-x^10*X[2,1,1,2,2]^3+3*x^11*X[2,1,1, 2,2]+3*x^10*X[2,1,1,2,2]^2+x^9*X[2,1,1,2,2]^3-x^11-3*x^10*X[2,1,1,2,2]-3*x^9*X[ 2,1,1,2,2]^2+x^10+3*x^9*X[2,1,1,2,2]+x^8*X[2,1,1,2,2]^2-x^9-2*x^8*X[2,1,1,2,2]+ x^8+x^7*X[2,1,1,2,2]-x^7-3*x^6*X[2,1,1,2,2]+3*x^6+6*x^5*X[2,1,1,2,2]-7*x^5-7*x^ 4*X[2,1,1,2,2]+12*x^4+4*x^3*X[2,1,1,2,2]-14*x^3-x^2*X[2,1,1,2,2]+11*x^2-5*x+1)/ (x^12*X[2,1,1,2,2]^3-3*x^12*X[2,1,1,2,2]^2-x^11*X[2,1,1,2,2]^3+3*x^12*X[2,1,1,2 ,2]+4*x^11*X[2,1,1,2,2]^2+2*x^10*X[2,1,1,2,2]^3-x^12-5*x^11*X[2,1,1,2,2]-6*x^10 *X[2,1,1,2,2]^2-x^9*X[2,1,1,2,2]^3+2*x^11+6*x^10*X[2,1,1,2,2]+4*x^9*X[2,1,1,2,2 ]^2-2*x^10-5*x^9*X[2,1,1,2,2]-x^8*X[2,1,1,2,2]^2+2*x^9+3*x^8*X[2,1,1,2,2]-2*x^8 -2*x^7*X[2,1,1,2,2]+2*x^7+5*x^6*X[2,1,1,2,2]-5*x^6-10*x^5*X[2,1,1,2,2]+12*x^5+ 10*x^4*X[2,1,1,2,2]-19*x^4-5*x^3*X[2,1,1,2,2]+21*x^3+x^2*X[2,1,1,2,2]-15*x^2+6* x-1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [2, 1, 1, 2, 2], equals , - 5/8 + ---- 16 67 21 n The variance equals , - -- + ---- 64 256 2415 279 n The , 3, -th moment about the mean is , - ---- + ----- 1024 2048 13603 1323 2 7449 The , 4, -th moment about the mean is , - ----- + ----- n - ----- n 4096 65536 32768 The compositions of, 8, that yield the, 8, -th largest growth, that is, 1.9087907387871591034, are , [2, 1, 2, 1, 2] Theorem Number, 8, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 2, 1, 2] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n ) a(n) x = - ( / ----- n = 0 14 11 10 9 8 7 5 4 3 2 x + x - x + 2 x - x - x + 9 x - 21 x + 24 x - 16 x + 6 x - 1) / 5 4 3 2 / ((x - x + 2 x - 3 x + 3 x - 1) / 10 8 7 5 4 3 2 (x - x + 2 x - 3 x + 2 x + 3 x - 6 x + 4 x - 1)) and in Maple format -(x^14+x^11-x^10+2*x^9-x^8-x^7+9*x^5-21*x^4+24*x^3-16*x^2+6*x-1)/(x^5-x^4+2*x^3 -3*x^2+3*x-1)/(x^10-x^8+2*x^7-3*x^5+2*x^4+3*x^3-6*x^2+4*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 127, 249, 482, 923, 1755, 3326, 6301, 11953, 22720, 43270, 82535, 157592, 301064, 575232, 1098948, 2098976, 4007962, 7651424, 14604699, 27874367, 53199129, 101533210, 193787063] The limit of a(n+1)/a(n) as n goes to infinity is 1.90879073879 a(n) is asymptotic to .732089369699*1.90879073879^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [2, 1, 2, 1, 2], denoted by the variable, X[2, 1, 2, 1, 2], is 14 14 14 14 11 11 - (x %2 - 3 x %1 + 3 x X[2, 1, 2, 1, 2] - x + 2 x %2 - 5 x %1 10 11 10 9 11 - 2 x %2 + 4 x X[2, 1, 2, 1, 2] + 5 x %1 + x %2 - x 10 9 10 9 8 - 4 x X[2, 1, 2, 1, 2] - 3 x %1 + x + 4 x X[2, 1, 2, 1, 2] - 3 x %1 9 8 7 8 7 - 2 x + 2 x X[2, 1, 2, 1, 2] + 6 x %1 + x - 7 x X[2, 1, 2, 1, 2] 6 7 6 5 5 - 4 x %1 + x + 5 x X[2, 1, 2, 1, 2] + x %1 + 2 x X[2, 1, 2, 1, 2] 5 4 4 3 3 - 9 x - 6 x X[2, 1, 2, 1, 2] + 21 x + 4 x X[2, 1, 2, 1, 2] - 24 x 2 2 / 5 5 - x X[2, 1, 2, 1, 2] + 16 x - 6 x + 1) / ((x X[2, 1, 2, 1, 2] - x / 4 4 3 3 - x X[2, 1, 2, 1, 2] + x + 2 x X[2, 1, 2, 1, 2] - 2 x 2 2 10 10 - x X[2, 1, 2, 1, 2] + 3 x - 3 x + 1) (x %1 - 2 x X[2, 1, 2, 1, 2] 10 8 8 7 8 7 + x - x %1 + 2 x X[2, 1, 2, 1, 2] + x %1 - x - 3 x X[2, 1, 2, 1, 2] 7 5 5 4 4 + 2 x + 3 x X[2, 1, 2, 1, 2] - 3 x - 3 x X[2, 1, 2, 1, 2] + 2 x 3 3 2 + x X[2, 1, 2, 1, 2] + 3 x - 6 x + 4 x - 1)) 2 %1 := X[2, 1, 2, 1, 2] 3 %2 := X[2, 1, 2, 1, 2] and in Maple format -(x^14*X[2,1,2,1,2]^3-3*x^14*X[2,1,2,1,2]^2+3*x^14*X[2,1,2,1,2]-x^14+2*x^11*X[2 ,1,2,1,2]^3-5*x^11*X[2,1,2,1,2]^2-2*x^10*X[2,1,2,1,2]^3+4*x^11*X[2,1,2,1,2]+5*x ^10*X[2,1,2,1,2]^2+x^9*X[2,1,2,1,2]^3-x^11-4*x^10*X[2,1,2,1,2]-3*x^9*X[2,1,2,1, 2]^2+x^10+4*x^9*X[2,1,2,1,2]-3*x^8*X[2,1,2,1,2]^2-2*x^9+2*x^8*X[2,1,2,1,2]+6*x^ 7*X[2,1,2,1,2]^2+x^8-7*x^7*X[2,1,2,1,2]-4*x^6*X[2,1,2,1,2]^2+x^7+5*x^6*X[2,1,2, 1,2]+x^5*X[2,1,2,1,2]^2+2*x^5*X[2,1,2,1,2]-9*x^5-6*x^4*X[2,1,2,1,2]+21*x^4+4*x^ 3*X[2,1,2,1,2]-24*x^3-x^2*X[2,1,2,1,2]+16*x^2-6*x+1)/(x^5*X[2,1,2,1,2]-x^5-x^4* X[2,1,2,1,2]+x^4+2*x^3*X[2,1,2,1,2]-2*x^3-x^2*X[2,1,2,1,2]+3*x^2-3*x+1)/(x^10*X [2,1,2,1,2]^2-2*x^10*X[2,1,2,1,2]+x^10-x^8*X[2,1,2,1,2]^2+2*x^8*X[2,1,2,1,2]+x^ 7*X[2,1,2,1,2]^2-x^8-3*x^7*X[2,1,2,1,2]+2*x^7+3*x^5*X[2,1,2,1,2]-3*x^5-3*x^4*X[ 2,1,2,1,2]+2*x^4+x^3*X[2,1,2,1,2]+3*x^3-6*x^2+4*x-1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [2, 1, 2, 1, 2], equals , - 5/8 + ---- 16 25 n The variance equals , - 5/4 + ---- 256 2919 351 n The , 3, -th moment about the mean is , - ---- + ----- 1024 2048 229 1875 2 14581 The , 4, -th moment about the mean is , - --- + ----- n - ----- n 128 65536 32768 The compositions of, 8, that yield the, 9, -th largest growth, that is, 1.9275619754829253043, are , [1, 1, 1, 5], [1, 1, 5, 1], [1, 5, 1, 1], [5, 1, 1, 1] Theorem Number, 9, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 1, 1, 5] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 7 6 5 3 2 \ n x - x + x - x + 3 x - 3 x + 1 ) a(n) x = ---------------------------------- / 4 3 2 3 ----- (x + x + x + x - 1) (-1 + x) n = 0 and in Maple format (x^7-x^6+x^5-x^3+3*x^2-3*x+1)/(x^4+x^3+x^2+x-1)/(-1+x)^3 The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 127, 250, 489, 952, 1847, 3575, 6909, 13339, 25737, 49639, 95716, 184537, 355750, 685779, 1321936, 2548174, 4911830, 9467930, 18250102, 35178290, 67808429, 130705052, 251942199] The limit of a(n+1)/a(n) as n goes to infinity is 1.92756197548 a(n) is asymptotic to .709660512200*1.92756197548^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [1, 1, 1, 5], denoted by the variable, X[1, 1, 1, 5], is 7 7 6 6 5 5 3 (x X[1, 1, 1, 5] - x - x X[1, 1, 1, 5] + x + x X[1, 1, 1, 5] - x + x 2 / 2 5 5 - 3 x + 3 x - 1) / ((-1 + x) (x X[1, 1, 1, 5] - x + 2 x - 1)) / and in Maple format (x^7*X[1,1,1,5]-x^7-x^6*X[1,1,1,5]+x^6+x^5*X[1,1,1,5]-x^5+x^3-3*x^2+3*x-1)/(-1+ x)^2/(x^5*X[1,1,1,5]-x^5+2*x-1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [1, 1, 1, 5], equals , - 9/32 + ---- 32 183 23 n The variance equals , - ---- + ---- 1024 1024 807 171 n The , 3, -th moment about the mean is , - ----- + ----- 16384 16384 170889 1587 2 6967 The , 4, -th moment about the mean is , ------- + ------- n - ------ n 1048576 1048576 262144 The compositions of, 8, that yield the, 10, -th largest growth, that is, 1.9331849818995204468, are , [2, 1, 1, 4], [4, 1, 1, 2] Theorem Number, 10, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 1, 4] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 19 15 14 13 11 10 9 8 7 ) a(n) x = - (x + 2 x - x + x + 2 x - 3 x + x - x + 4 x / ----- n = 0 6 5 4 3 2 / - 5 x + 3 x - 2 x + 4 x - 6 x + 4 x - 1) / ( / 15 10 9 7 6 3 2 5 4 (x - x + x + x - x + x - 3 x + 3 x - 1) (x - x + 2 x - 1)) and in Maple format -(x^19+2*x^15-x^14+x^13+2*x^11-3*x^10+x^9-x^8+4*x^7-5*x^6+3*x^5-2*x^4+4*x^3-6*x ^2+4*x-1)/(x^15-x^10+x^9+x^7-x^6+x^3-3*x^2+3*x-1)/(x^5-x^4+2*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 127, 250, 489, 952, 1847, 3575, 6910, 13347, 25775, 49779, 96157, 185787, 359037, 693957, 1341448, 2593249, 5013366, 9692144, 18737428, 36224057, 70029397, 135381750, 261720071] The limit of a(n+1)/a(n) as n goes to infinity is 1.93318498190 a(n) is asymptotic to .675512458314*1.93318498190^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [2, 1, 1, 4], denoted by the variable, X[2, 1, 1, 4], is 19 4 19 3 19 2 - (x X[2, 1, 1, 4] - 4 x X[2, 1, 1, 4] + 6 x X[2, 1, 1, 4] 19 19 15 3 15 2 - 4 x X[2, 1, 1, 4] + x - 2 x X[2, 1, 1, 4] + 6 x X[2, 1, 1, 4] 14 3 15 14 2 + x X[2, 1, 1, 4] - 6 x X[2, 1, 1, 4] - 3 x X[2, 1, 1, 4] 13 3 15 14 13 2 - x X[2, 1, 1, 4] + 2 x + 3 x X[2, 1, 1, 4] + 3 x X[2, 1, 1, 4] 14 13 13 11 2 - x - 3 x X[2, 1, 1, 4] + x + 2 x X[2, 1, 1, 4] 11 10 2 11 10 - 4 x X[2, 1, 1, 4] - 3 x X[2, 1, 1, 4] + 2 x + 6 x X[2, 1, 1, 4] 9 2 10 9 9 8 + x X[2, 1, 1, 4] - 3 x - 2 x X[2, 1, 1, 4] + x + x X[2, 1, 1, 4] 8 7 7 6 6 - x - 4 x X[2, 1, 1, 4] + 4 x + 5 x X[2, 1, 1, 4] - 5 x 5 5 4 4 3 2 - 3 x X[2, 1, 1, 4] + 3 x + x X[2, 1, 1, 4] - 2 x + 4 x - 6 x + 4 x / 5 5 4 4 - 1) / ((x X[2, 1, 1, 4] - x - x X[2, 1, 1, 4] + x - 2 x + 1) ( / 15 3 15 2 15 15 x X[2, 1, 1, 4] - 3 x X[2, 1, 1, 4] + 3 x X[2, 1, 1, 4] - x 10 2 10 9 2 10 + x X[2, 1, 1, 4] - 2 x X[2, 1, 1, 4] - x X[2, 1, 1, 4] + x 9 9 7 7 6 6 + 2 x X[2, 1, 1, 4] - x + x X[2, 1, 1, 4] - x - x X[2, 1, 1, 4] + x 3 2 - x + 3 x - 3 x + 1)) and in Maple format -(x^19*X[2,1,1,4]^4-4*x^19*X[2,1,1,4]^3+6*x^19*X[2,1,1,4]^2-4*x^19*X[2,1,1,4]+x ^19-2*x^15*X[2,1,1,4]^3+6*x^15*X[2,1,1,4]^2+x^14*X[2,1,1,4]^3-6*x^15*X[2,1,1,4] -3*x^14*X[2,1,1,4]^2-x^13*X[2,1,1,4]^3+2*x^15+3*x^14*X[2,1,1,4]+3*x^13*X[2,1,1, 4]^2-x^14-3*x^13*X[2,1,1,4]+x^13+2*x^11*X[2,1,1,4]^2-4*x^11*X[2,1,1,4]-3*x^10*X [2,1,1,4]^2+2*x^11+6*x^10*X[2,1,1,4]+x^9*X[2,1,1,4]^2-3*x^10-2*x^9*X[2,1,1,4]+x ^9+x^8*X[2,1,1,4]-x^8-4*x^7*X[2,1,1,4]+4*x^7+5*x^6*X[2,1,1,4]-5*x^6-3*x^5*X[2,1 ,1,4]+3*x^5+x^4*X[2,1,1,4]-2*x^4+4*x^3-6*x^2+4*x-1)/(x^5*X[2,1,1,4]-x^5-x^4*X[2 ,1,1,4]+x^4-2*x+1)/(x^15*X[2,1,1,4]^3-3*x^15*X[2,1,1,4]^2+3*x^15*X[2,1,1,4]-x^ 15+x^10*X[2,1,1,4]^2-2*x^10*X[2,1,1,4]-x^9*X[2,1,1,4]^2+x^10+2*x^9*X[2,1,1,4]-x ^9+x^7*X[2,1,1,4]-x^7-x^6*X[2,1,1,4]+x^6-x^3+3*x^2-3*x+1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [2, 1, 1, 4], equals , - 9/32 + ---- 32 251 27 n The variance equals , - ---- + ---- 1024 1024 2973 297 n The , 3, -th moment about the mean is , - ----- + ----- 16384 16384 107905 2187 2 8715 The , 4, -th moment about the mean is , ------- + ------- n - ------ n 1048576 1048576 262144 The compositions of, 8, that yield the, 11, -th largest growth, that is, 1.9407101328380924652, are , [2, 1, 2, 3], [2, 1, 3, 2], [2, 2, 1, 3], [2, 3, 1, 2], [3, 1, 2, 2], [3, 2, 1, 2] Theorem Number, 11, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 2, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 9 5 4 2 \ n x + 2 x - 2 x + 3 x - 3 x + 1 ) a(n) x = - ------------------------------------------------------ / 10 9 7 6 5 4 3 2 ----- x - x + x + x - 3 x + 3 x + x - 5 x + 4 x - 1 n = 0 and in Maple format -(x^9+2*x^5-2*x^4+3*x^2-3*x+1)/(x^10-x^9+x^7+x^6-3*x^5+3*x^4+x^3-5*x^2+4*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 127, 250, 489, 952, 1848, 3582, 6940, 13448, 26069, 50556, 98078, 190316, 369352, 716859, 1391341, 2700402, 5240996, 10171609, 19740485, 38310763, 74349980, 144291136, 280025926] The limit of a(n+1)/a(n) as n goes to infinity is 1.94071013284 a(n) is asymptotic to .643241925447*1.94071013284^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [2, 1, 2, 3], denoted by the variable, X[2, 1, 2, 3], is 9 2 9 9 5 5 - (x X[2, 1, 2, 3] - 2 x X[2, 1, 2, 3] + x - 2 x X[2, 1, 2, 3] + 2 x 4 4 3 2 / + 2 x X[2, 1, 2, 3] - 2 x - x X[2, 1, 2, 3] + 3 x - 3 x + 1) / ( / 10 2 10 9 2 10 x X[2, 1, 2, 3] - 2 x X[2, 1, 2, 3] - x X[2, 1, 2, 3] + x 9 9 7 7 6 6 + 2 x X[2, 1, 2, 3] - x - x X[2, 1, 2, 3] + x - x X[2, 1, 2, 3] + x 5 5 4 4 3 + 3 x X[2, 1, 2, 3] - 3 x - 3 x X[2, 1, 2, 3] + 3 x + x X[2, 1, 2, 3] 3 2 + x - 5 x + 4 x - 1) and in Maple format -(x^9*X[2,1,2,3]^2-2*x^9*X[2,1,2,3]+x^9-2*x^5*X[2,1,2,3]+2*x^5+2*x^4*X[2,1,2,3] -2*x^4-x^3*X[2,1,2,3]+3*x^2-3*x+1)/(x^10*X[2,1,2,3]^2-2*x^10*X[2,1,2,3]-x^9*X[2 ,1,2,3]^2+x^10+2*x^9*X[2,1,2,3]-x^9-x^7*X[2,1,2,3]+x^7-x^6*X[2,1,2,3]+x^6+3*x^5 *X[2,1,2,3]-3*x^5-3*x^4*X[2,1,2,3]+3*x^4+x^3*X[2,1,2,3]+x^3-5*x^2+4*x-1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [2, 1, 2, 3], equals , - 9/32 + ---- 32 363 35 n The variance equals , - ---- + ---- 1024 1024 8877 681 n The , 3, -th moment about the mean is , - ----- + ----- 16384 16384 672543 3675 2 3511 The , 4, -th moment about the mean is , - ------- + ------- n - ------ n 1048576 1048576 262144 The compositions of, 8, that yield the, 12, -th largest growth, that is, 1.9417130342786384772, are , [3, 1, 1, 3] Theorem Number, 12, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [3, 1, 1, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 18 17 16 15 14 13 10 9 ) a(n) x = - (x + 3 x + 3 x + x + 2 x + 3 x + 3 x - x / ----- n = 0 8 7 6 5 4 3 2 / 2 - 3 x + 3 x + x - 4 x + 2 x + 3 x - 6 x + 4 x - 1) / ((x + 1) / 5 4 3 13 12 11 10 9 8 7 (x + x - x + 2 x - 1) (x + 3 x + 2 x - 2 x - 2 x + 2 x + x 6 5 4 3 2 - 2 x + x + 2 x - 2 x - 2 x + 3 x - 1)) and in Maple format -(x^18+3*x^17+3*x^16+x^15+2*x^14+3*x^13+3*x^10-x^9-3*x^8+3*x^7+x^6-4*x^5+2*x^4+ 3*x^3-6*x^2+4*x-1)/(x^2+1)/(x^5+x^4-x^3+2*x-1)/(x^13+3*x^12+2*x^11-2*x^10-2*x^9 +2*x^8+x^7-2*x^6+x^5+2*x^4-2*x^3-2*x^2+3*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 127, 250, 489, 952, 1847, 3576, 6918, 13385, 25915, 50220, 97406, 189063, 367149, 713178, 1385465, 2691421, 5227890, 10153586, 19718017, 38288440, 74343896, 144347147, 280263162] The limit of a(n+1)/a(n) as n goes to infinity is 1.94171303428 a(n) is asymptotic to .633809828964*1.94171303428^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [3, 1, 1, 3], denoted by the variable, X[3, 1, 1, 3], is 18 4 18 3 17 4 - (x X[3, 1, 1, 3] - 4 x X[3, 1, 1, 3] + 3 x X[3, 1, 1, 3] 18 2 17 3 16 4 + 6 x X[3, 1, 1, 3] - 12 x X[3, 1, 1, 3] + 3 x X[3, 1, 1, 3] 18 17 2 16 3 - 4 x X[3, 1, 1, 3] + 18 x X[3, 1, 1, 3] - 12 x X[3, 1, 1, 3] 15 4 18 17 16 2 + x X[3, 1, 1, 3] + x - 12 x X[3, 1, 1, 3] + 18 x X[3, 1, 1, 3] 15 3 17 16 - 4 x X[3, 1, 1, 3] + 3 x - 12 x X[3, 1, 1, 3] 15 2 14 3 16 + 6 x X[3, 1, 1, 3] - 2 x X[3, 1, 1, 3] + 3 x 15 14 2 13 3 15 - 4 x X[3, 1, 1, 3] + 6 x X[3, 1, 1, 3] - 3 x X[3, 1, 1, 3] + x 14 13 2 14 13 - 6 x X[3, 1, 1, 3] + 9 x X[3, 1, 1, 3] + 2 x - 9 x X[3, 1, 1, 3] 13 10 3 10 2 10 + 3 x - x X[3, 1, 1, 3] + 5 x X[3, 1, 1, 3] - 7 x X[3, 1, 1, 3] 9 2 10 9 8 2 - x X[3, 1, 1, 3] + 3 x + 2 x X[3, 1, 1, 3] - 2 x X[3, 1, 1, 3] 9 8 7 2 8 7 - x + 5 x X[3, 1, 1, 3] + x X[3, 1, 1, 3] - 3 x - 4 x X[3, 1, 1, 3] 7 6 6 5 5 + 3 x - x X[3, 1, 1, 3] + x + 4 x X[3, 1, 1, 3] - 4 x 4 4 3 3 2 - 3 x X[3, 1, 1, 3] + 2 x + x X[3, 1, 1, 3] + 3 x - 6 x + 4 x - 1) / 5 5 4 4 3 / ((x X[3, 1, 1, 3] - x + x X[3, 1, 1, 3] - x - x X[3, 1, 1, 3] / 3 15 3 15 2 + x - 2 x + 1) (x X[3, 1, 1, 3] - 3 x X[3, 1, 1, 3] 14 3 15 14 2 + 3 x X[3, 1, 1, 3] + 3 x X[3, 1, 1, 3] - 9 x X[3, 1, 1, 3] 13 3 15 14 13 2 + 3 x X[3, 1, 1, 3] - x + 9 x X[3, 1, 1, 3] - 9 x X[3, 1, 1, 3] 12 3 14 13 12 2 + x X[3, 1, 1, 3] - 3 x + 9 x X[3, 1, 1, 3] - 3 x X[3, 1, 1, 3] 13 12 12 9 2 - 3 x + 3 x X[3, 1, 1, 3] - x + x X[3, 1, 1, 3] 9 9 7 2 7 7 - 2 x X[3, 1, 1, 3] + x - x X[3, 1, 1, 3] + 3 x X[3, 1, 1, 3] - 2 x 5 5 3 2 - x X[3, 1, 1, 3] + x - x + 3 x - 3 x + 1)) and in Maple format -(x^18*X[3,1,1,3]^4-4*x^18*X[3,1,1,3]^3+3*x^17*X[3,1,1,3]^4+6*x^18*X[3,1,1,3]^2 -12*x^17*X[3,1,1,3]^3+3*x^16*X[3,1,1,3]^4-4*x^18*X[3,1,1,3]+18*x^17*X[3,1,1,3]^ 2-12*x^16*X[3,1,1,3]^3+x^15*X[3,1,1,3]^4+x^18-12*x^17*X[3,1,1,3]+18*x^16*X[3,1, 1,3]^2-4*x^15*X[3,1,1,3]^3+3*x^17-12*x^16*X[3,1,1,3]+6*x^15*X[3,1,1,3]^2-2*x^14 *X[3,1,1,3]^3+3*x^16-4*x^15*X[3,1,1,3]+6*x^14*X[3,1,1,3]^2-3*x^13*X[3,1,1,3]^3+ x^15-6*x^14*X[3,1,1,3]+9*x^13*X[3,1,1,3]^2+2*x^14-9*x^13*X[3,1,1,3]+3*x^13-x^10 *X[3,1,1,3]^3+5*x^10*X[3,1,1,3]^2-7*x^10*X[3,1,1,3]-x^9*X[3,1,1,3]^2+3*x^10+2*x ^9*X[3,1,1,3]-2*x^8*X[3,1,1,3]^2-x^9+5*x^8*X[3,1,1,3]+x^7*X[3,1,1,3]^2-3*x^8-4* x^7*X[3,1,1,3]+3*x^7-x^6*X[3,1,1,3]+x^6+4*x^5*X[3,1,1,3]-4*x^5-3*x^4*X[3,1,1,3] +2*x^4+x^3*X[3,1,1,3]+3*x^3-6*x^2+4*x-1)/(x^5*X[3,1,1,3]-x^5+x^4*X[3,1,1,3]-x^4 -x^3*X[3,1,1,3]+x^3-2*x+1)/(x^15*X[3,1,1,3]^3-3*x^15*X[3,1,1,3]^2+3*x^14*X[3,1, 1,3]^3+3*x^15*X[3,1,1,3]-9*x^14*X[3,1,1,3]^2+3*x^13*X[3,1,1,3]^3-x^15+9*x^14*X[ 3,1,1,3]-9*x^13*X[3,1,1,3]^2+x^12*X[3,1,1,3]^3-3*x^14+9*x^13*X[3,1,1,3]-3*x^12* X[3,1,1,3]^2-3*x^13+3*x^12*X[3,1,1,3]-x^12+x^9*X[3,1,1,3]^2-2*x^9*X[3,1,1,3]+x^ 9-x^7*X[3,1,1,3]^2+3*x^7*X[3,1,1,3]-2*x^7-x^5*X[3,1,1,3]+x^5-x^3+3*x^2-3*x+1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [3, 1, 1, 3], equals , - 9/32 + ---- 32 379 35 n The variance equals , - ---- + ---- 1024 1024 8925 633 n The , 3, -th moment about the mean is , - ----- + ----- 16384 16384 474751 3675 2 8671 The , 4, -th moment about the mean is , - ------- + ------- n - ------ n 1048576 1048576 262144 The compositions of, 8, that yield the, 13, -th largest growth, that is, 1.9454365275632690792, are , [1, 2, 2, 3], [1, 2, 3, 2], [1, 3, 2, 2], [2, 2, 3, 1], [2, 3, 2, 1], [3, 2, 2, 1] Theorem Number, 13, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 2, 2, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 7 6 5 4 2 \ n x - x + 2 x - 2 x + 3 x - 3 x + 1 ) a(n) x = ----------------------------------------------- / 6 5 4 3 2 ----- (-1 + x) (x - x + 2 x - x - 2 x + 3 x - 1) n = 0 and in Maple format (x^7-x^6+2*x^5-2*x^4+3*x^2-3*x+1)/(-1+x)/(x^6-x^5+2*x^4-x^3-2*x^2+3*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 127, 250, 489, 953, 1854, 3605, 7010, 13634, 26522, 51598, 100386, 195304, 379963, 739204, 1438079, 2797687, 5442708, 10588421, 20599078, 40074184, 77961784, 151669728, 295063876] The limit of a(n+1)/a(n) as n goes to infinity is 1.94543652756 a(n) is asymptotic to .630092222605*1.94543652756^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [1, 2, 2, 3], denoted by the variable, X[1, 2, 2, 3], is 7 7 6 6 5 5 (x X[1, 2, 2, 3] - x - x X[1, 2, 2, 3] + x + 2 x X[1, 2, 2, 3] - 2 x 4 4 3 2 / - 2 x X[1, 2, 2, 3] + 2 x + x X[1, 2, 2, 3] - 3 x + 3 x - 1) / ( / 6 6 5 5 (-1 + x) (x X[1, 2, 2, 3] - x - x X[1, 2, 2, 3] + x 4 4 3 3 2 + 2 x X[1, 2, 2, 3] - 2 x - x X[1, 2, 2, 3] + x + 2 x - 3 x + 1)) and in Maple format (x^7*X[1,2,2,3]-x^7-x^6*X[1,2,2,3]+x^6+2*x^5*X[1,2,2,3]-2*x^5-2*x^4*X[1,2,2,3]+ 2*x^4+x^3*X[1,2,2,3]-3*x^2+3*x-1)/(-1+x)/(x^6*X[1,2,2,3]-x^6-x^5*X[1,2,2,3]+x^5 +2*x^4*X[1,2,2,3]-2*x^4-x^3*X[1,2,2,3]+x^3+2*x^2-3*x+1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [1, 2, 2, 3], equals , - 9/32 + ---- 32 391 39 n The variance equals , - ---- + ---- 1024 1024 9375 819 n The , 3, -th moment about the mean is , - ----- + ----- 16384 16384 4563 2 448855 5703 The , 4, -th moment about the mean is , ------- n - ------- - ------ n 1048576 1048576 262144 The compositions of, 8, that yield the, 14, -th largest growth, that is, 1.9611865309023902347, are , [2, 2, 2, 2] Theorem Number, 14, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 2, 2, 2] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 2 4 2 \ n (x - x + 1) (x + x - 2 x + 1) ) a(n) x = - --------------------------------------------- / 7 6 5 4 3 2 ----- x - x + 2 x - 3 x + 5 x - 6 x + 4 x - 1 n = 0 and in Maple format -(x^2-x+1)*(x^4+x^2-2*x+1)/(x^7-x^6+2*x^5-3*x^4+5*x^3-6*x^2+4*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 127, 250, 490, 959, 1877, 3676, 7204, 14125, 27703, 54339, 106585, 209055, 410018, 804135, 1577052, 3092857, 6065601, 11895683, 23329562, 45753571, 89731321, 175980016, 345129929] The limit of a(n+1)/a(n) as n goes to infinity is 1.96118653090 a(n) is asymptotic to .578649889688*1.96118653090^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [2, 2, 2, 2], denoted by the variable, X[2, 2, 2, 2], is 6 6 5 5 4 4 - (x X[2, 2, 2, 2] - x - x X[2, 2, 2, 2] + x + 2 x X[2, 2, 2, 2] - 2 x 3 3 2 2 / - 2 x X[2, 2, 2, 2] + 3 x + x X[2, 2, 2, 2] - 4 x + 3 x - 1) / ( / 7 7 6 6 5 5 x X[2, 2, 2, 2] - x - x X[2, 2, 2, 2] + x + 2 x X[2, 2, 2, 2] - 2 x 4 4 3 3 2 - 3 x X[2, 2, 2, 2] + 3 x + 3 x X[2, 2, 2, 2] - 5 x - x X[2, 2, 2, 2] 2 + 6 x - 4 x + 1) and in Maple format -(x^6*X[2,2,2,2]-x^6-x^5*X[2,2,2,2]+x^5+2*x^4*X[2,2,2,2]-2*x^4-2*x^3*X[2,2,2,2] +3*x^3+x^2*X[2,2,2,2]-4*x^2+3*x-1)/(x^7*X[2,2,2,2]-x^7-x^6*X[2,2,2,2]+x^6+2*x^5 *X[2,2,2,2]-2*x^5-3*x^4*X[2,2,2,2]+3*x^4+3*x^3*X[2,2,2,2]-5*x^3-x^2*X[2,2,2,2]+ 6*x^2-4*x+1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [2, 2, 2, 2], equals , - 9/32 + ---- 32 747 67 n The variance equals , - ---- + ---- 1024 1024 39309 2889 n The , 3, -th moment about the mean is , - ----- + ------ 16384 16384 7946271 13467 2 72793 The , 4, -th moment about the mean is , - ------- + ------- n + ------ n 1048576 1048576 262144 The compositions of, 8, that yield the, 15, -th largest growth, that is, 1.9659482366454853372, are , [1, 1, 6], [1, 6, 1], [6, 1, 1] Theorem Number, 15, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 1, 6] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 6 2 \ n x + x - 2 x + 1 ) a(n) x = - ------------------------------------- / 5 4 3 2 2 ----- (x + x + x + x + x - 1) (-1 + x) n = 0 and in Maple format -(x^6+x^2-2*x+1)/(x^5+x^4+x^3+x^2+x-1)/(-1+x)^2 The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 127, 251, 495, 975, 1919, 3775, 7424, 14598, 28702, 56430, 110942, 218110, 428797, 842997, 1657293, 3258157, 6405373, 12592637, 24756478, 48669960, 95682628, 188107100, 369808828] The limit of a(n+1)/a(n) as n goes to infinity is 1.96594823665 a(n) is asymptotic to .576520722438*1.96594823665^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [1, 1, 6], denoted by the variable, X[1, 1, 6], is 6 6 2 x X[1, 1, 6] - x - x + 2 x - 1 - --------------------------------------- 6 6 (-1 + x) (x X[1, 1, 6] - x + 2 x - 1) and in Maple format -(x^6*X[1,1,6]-x^6-x^2+2*x-1)/(-1+x)/(x^6*X[1,1,6]-x^6+2*x-1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [1, 1, 6], equals , - 1/8 + ---- 64 49 53 n The variance equals , - --- + ---- 512 4096 831 141 n The , 3, -th moment about the mean is , - ----- + ----- 16384 16384 18545 8427 2 42041 The , 4, -th moment about the mean is , ------ + -------- n - ------- n 524288 16777216 8388608 The compositions of, 8, that yield the, 16, -th largest growth, that is, 1.9671682128139660358, are , [2, 1, 5], [5, 1, 2] Theorem Number, 16, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 5] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 11 5 2 \ n x + x + x - 2 x + 1 ) a(n) x = - -------------------------------- / 6 5 6 ----- (x - x + 2 x - 1) (x - x + 1) n = 0 and in Maple format -(x^11+x^5+x^2-2*x+1)/(x^6-x^5+2*x-1)/(x^6-x+1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 127, 251, 495, 975, 1919, 3775, 7425, 14604, 28725, 56502, 111143, 218631, 430079, 846035, 1664294, 3273952, 6440426, 12669419, 24922901, 49027564, 96445490, 189724521, 373220054] The limit of a(n+1)/a(n) as n goes to infinity is 1.96716821281 a(n) is asymptotic to .571110296331*1.96716821281^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [2, 1, 5], denoted by the variable, X[2, 1, 5], is 11 2 11 11 5 5 2 - (x X[2, 1, 5] - 2 x X[2, 1, 5] + x - x X[2, 1, 5] + x + x - 2 x + 1 / 6 6 ) / ((x X[2, 1, 5] - x + x - 1) / 6 6 5 5 (x X[2, 1, 5] - x - x X[2, 1, 5] + x - 2 x + 1)) and in Maple format -(x^11*X[2,1,5]^2-2*x^11*X[2,1,5]+x^11-x^5*X[2,1,5]+x^5+x^2-2*x+1)/(x^6*X[2,1,5 ]-x^6+x-1)/(x^6*X[2,1,5]-x^6-x^5*X[2,1,5]+x^5-2*x+1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [2, 1, 5], equals , - 1/8 + ---- 64 113 57 n The variance equals , - ---- + ---- 1024 4096 693 357 n The , 3, -th moment about the mean is , - ---- + ----- 8192 32768 6155 9747 2 27789 The , 4, -th moment about the mean is , - ------- + -------- n - ------- n 1048576 16777216 8388608 The compositions of, 8, that yield the, 17, -th largest growth, that is, 1.9693144732632464526, are , [3, 1, 4], [4, 1, 3] Theorem Number, 17, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [3, 1, 4] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 10 9 4 2 \ n x + x + x + x - 2 x + 1 ) a(n) x = - -------------------------------------------------------- / 2 4 3 2 6 5 4 ----- (x + 1) (x + x - x - x + 1) (x + x - x + 2 x - 1) n = 0 and in Maple format -(x^10+x^9+x^4+x^2-2*x+1)/(x^2+1)/(x^4+x^3-x^2-x+1)/(x^6+x^5-x^4+2*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 127, 251, 495, 975, 1919, 3776, 7431, 14627, 28797, 56703, 111663, 219905, 433080, 852904, 1679685, 3307887, 6514326, 12828789, 25263901, 49752465, 97978041, 192949251, 379977341] The limit of a(n+1)/a(n) as n goes to infinity is 1.96931447326 a(n) is asymptotic to .562736849592*1.96931447326^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [3, 1, 4], denoted by the variable, X[3, 1, 4], is 10 2 10 9 2 10 9 - (x X[3, 1, 4] - 2 x X[3, 1, 4] + x X[3, 1, 4] + x - 2 x X[3, 1, 4] 9 4 4 2 / + x - x X[3, 1, 4] + x + x - 2 x + 1) / ( / 6 6 5 5 (x X[3, 1, 4] - x + x X[3, 1, 4] - x + x - 1) 6 6 5 5 4 4 (x X[3, 1, 4] - x + x X[3, 1, 4] - x - x X[3, 1, 4] + x - 2 x + 1)) and in Maple format -(x^10*X[3,1,4]^2-2*x^10*X[3,1,4]+x^9*X[3,1,4]^2+x^10-2*x^9*X[3,1,4]+x^9-x^4*X[ 3,1,4]+x^4+x^2-2*x+1)/(x^6*X[3,1,4]-x^6+x^5*X[3,1,4]-x^5+x-1)/(x^6*X[3,1,4]-x^6 +x^5*X[3,1,4]-x^5-x^4*X[3,1,4]+x^4-2*x+1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [3, 1, 4], equals , - 1/8 + ---- 64 141 65 n The variance equals , - ---- + ---- 1024 4096 2625 33 n The , 3, -th moment about the mean is , - ----- + ---- 16384 2048 145923 12675 2 23299 The , 4, -th moment about the mean is , - ------- + -------- n + ------- n 1048576 16777216 8388608 The compositions of, 8, that yield the, 18, -th largest growth, that is, 1.9703230372932668084, are , [2, 2, 4], [2, 4, 2], [4, 2, 2] Theorem Number, 18, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 2, 4] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 6 5 4 2 \ n x - x + x + x - 2 x + 1 ) a(n) x = - ------------------------------------ / 7 6 5 4 2 ----- x - x + 2 x - x - 2 x + 3 x - 1 n = 0 and in Maple format -(x^6-x^5+x^4+x^2-2*x+1)/(x^7-x^6+2*x^5-x^4-2*x^2+3*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 127, 251, 495, 975, 1920, 3781, 7447, 14670, 28902, 56945, 112201, 221076, 435598, 858278, 1691095, 3332012, 6565144, 12935451, 25487004, 50217608, 98944879, 194953341, 384121031] The limit of a(n+1)/a(n) as n goes to infinity is 1.97032303729 a(n) is asymptotic to .560202503508*1.97032303729^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [2, 2, 4], denoted by the variable, X[2, 2, 4], is 6 6 5 5 4 4 2 - (x X[2, 2, 4] - x - x X[2, 2, 4] + x + x X[2, 2, 4] - x - x + 2 x - 1) / 7 7 6 6 5 5 / (x X[2, 2, 4] - x - x X[2, 2, 4] + x + 2 x X[2, 2, 4] - 2 x / 4 4 2 - x X[2, 2, 4] + x + 2 x - 3 x + 1) and in Maple format -(x^6*X[2,2,4]-x^6-x^5*X[2,2,4]+x^5+x^4*X[2,2,4]-x^4-x^2+2*x-1)/(x^7*X[2,2,4]-x ^7-x^6*X[2,2,4]+x^6+2*x^5*X[2,2,4]-2*x^5-x^4*X[2,2,4]+x^4+2*x^2-3*x+1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [2, 2, 4], equals , - 1/8 + ---- 64 75 69 n The variance equals , - --- + ---- 512 4096 3033 309 n The , 3, -th moment about the mean is , - ----- + ----- 16384 16384 95163 14283 2 55959 The , 4, -th moment about the mean is , - ------ + -------- n + ------- n 524288 16777216 8388608 The compositions of, 8, that yield the, 19, -th largest growth, that is, 1.9717270001741243154, are , [3, 2, 3] Theorem Number, 19, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [3, 2, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 8 5 4 3 2 \ n x + x - x + x + x - 2 x + 1 ) a(n) x = - --------------------------------------------------------- / 10 9 8 7 6 5 4 3 2 ----- x + x - x + x + x - x + 2 x - x - 2 x + 3 x - 1 n = 0 and in Maple format -(x^8+x^5-x^4+x^3+x^2-2*x+1)/(x^10+x^9-x^8+x^7+x^6-x^5+2*x^4-x^3-2*x^2+3*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 127, 251, 495, 975, 1920, 3782, 7453, 14692, 28968, 57121, 112637, 222105, 437949, 863532, 1702655, 3357157, 6619357, 13051498, 25733907, 50740161, 100045708, 197262868, 388948683] The limit of a(n+1)/a(n) as n goes to infinity is 1.97172700017 a(n) is asymptotic to .555250679255*1.97172700017^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [3, 2, 3], denoted by the variable, X[3, 2, 3], is 8 2 8 8 5 5 4 - (x X[3, 2, 3] - 2 x X[3, 2, 3] + x - x X[3, 2, 3] + x + x X[3, 2, 3] 4 3 3 2 / 10 2 - x - x X[3, 2, 3] + x + x - 2 x + 1) / (x X[3, 2, 3] / 10 9 2 10 9 - 2 x X[3, 2, 3] + x X[3, 2, 3] + x - 2 x X[3, 2, 3] 8 2 9 8 8 7 7 - x X[3, 2, 3] + x + 2 x X[3, 2, 3] - x - x X[3, 2, 3] + x 6 6 5 5 4 4 - x X[3, 2, 3] + x + x X[3, 2, 3] - x - 2 x X[3, 2, 3] + 2 x 3 3 2 + x X[3, 2, 3] - x - 2 x + 3 x - 1) and in Maple format -(x^8*X[3,2,3]^2-2*x^8*X[3,2,3]+x^8-x^5*X[3,2,3]+x^5+x^4*X[3,2,3]-x^4-x^3*X[3,2 ,3]+x^3+x^2-2*x+1)/(x^10*X[3,2,3]^2-2*x^10*X[3,2,3]+x^9*X[3,2,3]^2+x^10-2*x^9*X [3,2,3]-x^8*X[3,2,3]^2+x^9+2*x^8*X[3,2,3]-x^8-x^7*X[3,2,3]+x^7-x^6*X[3,2,3]+x^6 +x^5*X[3,2,3]-x^5-2*x^4*X[3,2,3]+2*x^4+x^3*X[3,2,3]-x^3-2*x^2+3*x-1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [3, 2, 3], equals , - 1/8 + ---- 64 89 77 n The variance equals , - --- + ---- 512 4096 4929 867 n The , 3, -th moment about the mean is , - ----- + ----- 16384 32768 298439 17787 2 222727 The , 4, -th moment about the mean is , - ------ + -------- n + ------- n 524288 16777216 8388608 The compositions of, 8, that yield the, 20, -th largest growth, that is, 1.9735704833094816886, are , [2, 3, 3], [3, 3, 2] Theorem Number, 20, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 3, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 2 4 3 2 \ n (x - x + 1) (x + x - x - x + 1) ) a(n) x = - ------------------------------------ / 7 6 4 3 2 ----- x - x + 2 x - x - 2 x + 3 x - 1 n = 0 and in Maple format -(x^2-x+1)*(x^4+x^3-x^2-x+1)/(x^7-x^6+2*x^4-x^3-2*x^2+3*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 127, 251, 495, 976, 1925, 3798, 7495, 14792, 29194, 57618, 113715, 224426, 442921, 874135, 1725165, 3404732, 6719476, 13261358, 26172225, 51652733, 101940313, 201186397, 397055538] The limit of a(n+1)/a(n) as n goes to infinity is 1.97357048331 a(n) is asymptotic to .551152801880*1.97357048331^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [2, 3, 3], denoted by the variable, X[2, 3, 3], is 6 6 4 4 3 3 2 - (x X[2, 3, 3] - x - x X[2, 3, 3] + x + x X[2, 3, 3] - x - x + 2 x - 1) / 7 7 6 6 4 4 / (x X[2, 3, 3] - x - x X[2, 3, 3] + x + 2 x X[2, 3, 3] - 2 x / 3 3 2 - x X[2, 3, 3] + x + 2 x - 3 x + 1) and in Maple format -(x^6*X[2,3,3]-x^6-x^4*X[2,3,3]+x^4+x^3*X[2,3,3]-x^3-x^2+2*x-1)/(x^7*X[2,3,3]-x ^7-x^6*X[2,3,3]+x^6+2*x^4*X[2,3,3]-2*x^4-x^3*X[2,3,3]+x^3+2*x^2-3*x+1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [2, 3, 3], equals , - 1/8 + ---- 64 97 85 n The variance equals , - --- + ---- 512 4096 5691 531 n The , 3, -th moment about the mean is , - ----- + ----- 16384 16384 337711 21675 2 301703 The , 4, -th moment about the mean is , - ------ + -------- n + ------- n 524288 16777216 8388608 The compositions of, 8, that yield the, 21, -th largest growth, that is, 1.9835828434243263304, are , [1, 7], [7, 1] Theorem Number, 21, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 7] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 7 \ n x - x + 1 ) a(n) x = ----------------------------------------- / 6 5 4 3 2 ----- (-1 + x) (x + x + x + x + x + x - 1) n = 0 and in Maple format (x^7-x+1)/(-1+x)/(x^6+x^5+x^4+x^3+x^2+x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 127, 252, 500, 992, 1968, 3904, 7744, 15361, 30470, 60440, 119888, 237808, 471712, 935680, 1855999, 3681528, 7302616, 14485344, 28732880, 56994048, 113052416, 224248833, 444816138] The limit of a(n+1)/a(n) as n goes to infinity is 1.98358284342 a(n) is asymptotic to .530481492001*1.98358284342^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [1, 7], denoted by the variable, X[1, 7], is 7 7 x X[1, 7] - x + x - 1 ------------------------- 7 7 x X[1, 7] - x + 2 x - 1 and in Maple format (x^7*X[1,7]-x^7+x-1)/(x^7*X[1,7]-x^7+2*x-1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [1, 7], equals , - 7/128 + --- 128 763 115 n The variance equals , - ----- + ----- 16384 16384 17115 2943 n The , 3, -th moment about the mean is , - ------ + ------ 524288 524288 1139047 39675 2 88507 The , 4, -th moment about the mean is , - --------- + --------- n + -------- n 268435456 268435456 67108864 The compositions of, 8, that yield the, 22, -th largest growth, that is, 1.9838613961621262283, are , [2, 6], [6, 2] Theorem Number, 22, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 6] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 6 \ n x - x + 1 ) a(n) x = - ----------------- / 7 6 ----- x - x + 2 x - 1 n = 0 and in Maple format -(x^6-x+1)/(x^7-x^6+2*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 127, 252, 500, 992, 1968, 3904, 7745, 15365, 30482, 60472, 119968, 238000, 472159, 936698, 1858279, 3686568, 7313640, 14509248, 28784337, 57104135, 113286689, 224745089, 445863106] The limit of a(n+1)/a(n) as n goes to infinity is 1.98386139616 a(n) is asymptotic to .529494848582*1.98386139616^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [2, 6], denoted by the variable, X[2, 6], is 6 6 x X[2, 6] - x + x - 1 - ------------------------------------------- 7 7 6 6 x X[2, 6] - x - x X[2, 6] + x - 2 x + 1 and in Maple format -(x^6*X[2,6]-x^6+x-1)/(x^7*X[2,6]-x^7-x^6*X[2,6]+x^6-2*x+1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [2, 6], equals , - 7/128 + --- 128 815 119 n The variance equals , - ----- + ----- 16384 16384 10677 819 n The , 3, -th moment about the mean is , - ------ + ------ 262144 131072 4742831 42483 2 153215 The , 4, -th moment about the mean is , - --------- + --------- n + -------- n 268435456 268435456 67108864 The compositions of, 8, that yield the, 23, -th largest growth, that is, 1.9843858253440954550, are , [3, 5], [5, 3] Theorem Number, 23, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [3, 5] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 5 \ n x - x + 1 ) a(n) x = - ---------------------- / 7 6 5 ----- x + x - x + 2 x - 1 n = 0 and in Maple format -(x^5-x+1)/(x^7+x^6-x^5+2*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 127, 252, 500, 992, 1968, 3905, 7749, 15377, 30514, 60552, 120159, 238442, 473161, 938934, 1863207, 3697321, 7336911, 14559262, 28891193, 57331274, 113767368, 225758353, 447991676] The limit of a(n+1)/a(n) as n goes to infinity is 1.98438582534 a(n) is asymptotic to .527820750127*1.98438582534^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [3, 5], denoted by the variable, X[3, 5], is 5 5 x X[3, 5] - x + x - 1 - ------------------------------------------------------------- 7 7 6 6 5 5 x X[3, 5] - x + x X[3, 5] - x - x X[3, 5] + x - 2 x + 1 and in Maple format -(x^5*X[3,5]-x^5+x-1)/(x^7*X[3,5]-x^7+x^6*X[3,5]-x^6-x^5*X[3,5]+x^5-2*x+1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [3, 5], equals , - 7/128 + --- 128 911 127 n The variance equals , - ----- + ----- 16384 16384 14931 249 n The , 3, -th moment about the mean is , - ------ + ----- 262144 32768 13153967 48387 2 310123 The , 4, -th moment about the mean is , - --------- + --------- n + -------- n 268435456 268435456 67108864 The compositions of, 8, that yield the, 24, -th largest growth, that is, 1.9853288885629234253, are , [4, 4] Theorem Number, 24, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [4, 4] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 4 \ n x - x + 1 ) a(n) x = - --------------------------- / 7 6 5 4 ----- x + x + x - x + 2 x - 1 n = 0 and in Maple format -(x^4-x+1)/(x^7+x^6+x^5-x^4+2*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 127, 252, 500, 992, 1969, 3909, 7761, 15409, 30593, 60738, 120585, 239400, 475286, 943597, 1873349, 3719214, 7383865, 14659404, 29103742, 57780502, 114713299, 227743622, 452145985] The limit of a(n+1)/a(n) as n goes to infinity is 1.98532888856 a(n) is asymptotic to .525175931290*1.98532888856^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [4, 4], denoted by the variable, X[4, 4], is 4 4 / 7 7 6 6 5 - (x X[4, 4] - x + x - 1) / (x X[4, 4] - x + x X[4, 4] - x + x X[4, 4] / 5 4 4 - x - x X[4, 4] + x - 2 x + 1) and in Maple format -(x^4*X[4,4]-x^4+x-1)/(x^7*X[4,4]-x^7+x^6*X[4,4]-x^6+x^5*X[4,4]-x^5-x^4*X[4,4]+ x^4-2*x+1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [4, 4], equals , - 7/128 + --- 128 1087 143 n The variance equals , - ----- + ----- 16384 16384 23901 1389 n The , 3, -th moment about the mean is , - ------ + ------ 262144 131072 35368783 61347 2 733931 The , 4, -th moment about the mean is , - --------- + --------- n + -------- n 268435456 268435456 67108864 The compositions of, 8, that yield the, 25, -th largest growth, that is, 1.9919641966050350211, are , [8] Theorem Number, 25, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [8] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 1 ) a(n) x = - ----------------------------------- / 7 6 5 4 3 2 ----- x + x + x + x + x + x + x - 1 n = 0 and in Maple format -1/(x^7+x^6+x^5+x^4+x^3+x^2+x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 127, 253, 504, 1004, 2000, 3984, 7936, 15808, 31489, 62725, 124946, 248888, 495776, 987568, 1967200, 3918592, 7805695, 15548665, 30972384, 61695880, 122895984, 244804400, 487641600] The limit of a(n+1)/a(n) as n goes to infinity is 1.99196419661 a(n) is asymptotic to .512454017228*1.99196419661^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [8], denoted by the variable, X[8], is -1 + x ---------------------- 8 8 x X[8] - x + 2 x - 1 and in Maple format (-1+x)/(x^8*X[8]-x^8+2*x-1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [8], equals , - 3/128 + --- 256 87 241 n The variance equals , - ---- + ----- 4096 65536 72231 27261 n The , 3, -th moment about the mean is , - ------- + ------- 4194304 8388608 The , 4, -th moment about the mean is , 594771 174243 2 4316675 - -------- + ---------- n + ---------- n 67108864 4294967296 2147483648 This ends this article, that took, 3.439, seconds to generate. ---------------------------------------------------------- ---------------------------------------- Generating functions and Growth rates for Weight-Enumerating Sequences Accor\ ding to the Number of occurrences of compositions of, 9 By Shalosh B. Ekhad The compositions of, 9, that yield polynomial growth (hence growth ratio 1, that is the lowest) are , [2, 1, 1, 1, 1, 1, 2] Theorem Number, 1, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 1, 1, 1, 1, 2] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 47 45 44 43 42 41 40 39 ) a(n) x = - (x + 2 x + x + x + x + x - 3 x + 6 x / ----- n = 0 38 37 36 35 34 33 32 31 30 - 8 x + 5 x + x - 3 x - x + 8 x - 12 x + 7 x + 9 x 29 28 27 26 25 24 23 22 - 33 x + 55 x - 82 x + 126 x - 151 x + 100 x + 16 x - 87 x 21 20 19 18 17 16 15 - 52 x + 547 x - 1424 x + 2487 x - 3311 x + 3455 x - 2910 x 14 13 12 11 10 9 + 2437 x - 3293 x + 6353 x - 11267 x + 16328 x - 19272 x 8 7 6 5 4 3 2 + 18634 x - 14739 x + 9452 x - 4836 x + 1926 x - 575 x + 121 x / 3 3 2 - 16 x + 1) / ((x - x + 1) (x - x + 2 x - 1) / 9 6 5 4 3 2 6 4 3 2 (x - x + 2 x - x + x - 3 x + 3 x - 1) (x + x - x + x - 2 x + 1) 9 6 5 4 3 2 18 12 11 10 (x + x - 2 x + x + x - 3 x + 3 x - 1) (x + x - 5 x + 8 x 9 8 7 6 5 4 3 2 - 4 x - x + x + x - 6 x + 15 x - 20 x + 15 x - 6 x + 1)) and in Maple format -(x^47+2*x^45+x^44+x^43+x^42+x^41-3*x^40+6*x^39-8*x^38+5*x^37+x^36-3*x^35-x^34+ 8*x^33-12*x^32+7*x^31+9*x^30-33*x^29+55*x^28-82*x^27+126*x^26-151*x^25+100*x^24 +16*x^23-87*x^22-52*x^21+547*x^20-1424*x^19+2487*x^18-3311*x^17+3455*x^16-2910* x^15+2437*x^14-3293*x^13+6353*x^12-11267*x^11+16328*x^10-19272*x^9+18634*x^8-\ 14739*x^7+9452*x^6-4836*x^5+1926*x^4-575*x^3+121*x^2-16*x+1)/(x^3-x+1)/(x^3-x^2 +2*x-1)/(x^9-x^6+2*x^5-x^4+x^3-3*x^2+3*x-1)/(x^6+x^4-x^3+x^2-2*x+1)/(x^9+x^6-2* x^5+x^4+x^3-3*x^2+3*x-1)/(x^18+x^12-5*x^11+8*x^10-4*x^9-x^8+x^7+x^6-6*x^5+15*x^ 4-20*x^3+15*x^2-6*x+1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 503, 977, 1862, 3477, 6363, 11426, 20170, 35082, 60283, 102654, 173816, 293648, 496576, 842854, 1438829, 2473334, 4282923, 7468452, 13102648, 23098217, 40858122] ---------------------------------------------------------------------------- The compositions of, 9, that yield the, 2, -th largest growth, that is, 1.6180339887498948482, are , [1, 1, 1, 1, 1, 1, 3], [1, 1, 1, 1, 1, 3, 1], [1, 1, 1, 1, 3, 1, 1], [1, 1, 1, 3, 1, 1, 1], [1, 1, 3, 1, 1, 1, 1], [1, 3, 1, 1, 1, 1, 1], [3, 1, 1, 1, 1, 1, 1] Theorem Number, 2, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 1, 1, 1, 1, 1, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 7 6 5 4 3 2 \ n 3 x - 5 x + x + 11 x - 19 x + 15 x - 6 x + 1 ) a(n) x = - -------------------------------------------------- / 2 6 ----- (x + x - 1) (-1 + x) n = 0 and in Maple format -(3*x^7-5*x^6+x^5+11*x^4-19*x^3+15*x^2-6*x+1)/(x^2+x-1)/(-1+x)^6 The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 503, 977, 1862, 3477, 6363, 11426, 20169, 35068, 60181, 102134, 171717, 286467, 474848, 783015, 1285759, 2104217, 3434528, 5594200, 9097134, 14775016, 23973734] The limit of a(n+1)/a(n) as n goes to infinity is 1.61803398875 a(n) is asymptotic to 12.9845971347*1.61803398875^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [1, 1, 1, 1, 1, 1, 3], denoted by the variable, X[1, 1, 1, 1, 1, 1, 3], is 7 7 6 6 5 5 4 4 3 - (3 x %1 - 3 x - 6 x %1 + 5 x + 7 x %1 - x - 4 x %1 - 11 x + x %1 3 2 / 5 3 3 + 19 x - 15 x + 6 x - 1) / ((-1 + x) (x %1 - x + 2 x - 1)) / %1 := X[1, 1, 1, 1, 1, 1, 3] and in Maple format -(3*x^7*X[1,1,1,1,1,1,3]-3*x^7-6*x^6*X[1,1,1,1,1,1,3]+5*x^6+7*x^5*X[1,1,1,1,1,1 ,3]-x^5-4*x^4*X[1,1,1,1,1,1,3]-11*x^4+x^3*X[1,1,1,1,1,1,3]+19*x^3-15*x^2+6*x-1) /(-1+x)^5/(x^3*X[1,1,1,1,1,1,3]-x^3+2*x-1) Furthermore, the expectation of the random variable number of occurences (by\ containment) of , [1, 1, 1, 1, 1, 1, 3], equals , - 13/8 + n/8 23 3 n The variance equals , - -- + --- 64 64 21 The , 3, -th moment about the mean is , --- 128 1681 27 2 111 The , 4, -th moment about the mean is , ---- + ---- n - ---- n 4096 4096 1024 The compositions of, 9, that yield the, 3, -th largest growth, that is, 1.7548776662466927601, are , [1, 1, 2, 1, 1, 1, 2], [1, 2, 1, 1, 1, 2, 1], [2, 1, 1, 1, 2, 1, 1] Theorem Number, 3, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 1, 2, 1, 1, 1, 2] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 22 21 20 19 18 17 16 ) a(n) x = - (2 x - 2 x + 5 x - 2 x + x + 3 x - 7 x / ----- n = 0 15 14 13 12 11 10 9 8 + 11 x - 14 x + 8 x + 7 x - 26 x + 39 x - 29 x - 18 x 7 6 5 4 3 2 / + 94 x - 163 x + 187 x - 155 x + 92 x - 37 x + 9 x - 1) / ( / 3 3 2 (x - x + 1) (x - x + 2 x - 1) 12 8 7 6 5 4 3 2 3 (x + x - x - x + x + x - 4 x + 6 x - 4 x + 1) (x + x - 1) 2 (-1 + x) ) and in Maple format -(2*x^22-2*x^21+5*x^20-2*x^19+x^18+3*x^17-7*x^16+11*x^15-14*x^14+8*x^13+7*x^12-\ 26*x^11+39*x^10-29*x^9-18*x^8+94*x^7-163*x^6+187*x^5-155*x^4+92*x^3-37*x^2+9*x-\ 1)/(x^3-x+1)/(x^3-x^2+2*x-1)/(x^12+x^8-x^7-x^6+x^5+x^4-4*x^3+6*x^2-4*x+1)/(x^3+ x-1)/(-1+x)^2 The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 503, 977, 1862, 3477, 6364, 11438, 20246, 35422, 61495, 106332, 183718, 317959, 552063, 962217, 1683298, 2953668, 5193624, 9142572, 16098739, 28338148, 49847442] The limit of a(n+1)/a(n) as n goes to infinity is 1.75487766625 a(n) is asymptotic to 2.34123928046*1.75487766625^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [1, 1, 2, 1, 1, 1, 2], denoted by the variable, X[1, 1, 2, 1, 1, 1, 2], is 22 7 22 6 21 7 22 5 21 6 20 7 - (2 x %1 - 14 x %1 - 2 x %1 + 42 x %1 + 14 x %1 + x %1 22 4 21 5 20 6 22 3 21 4 - 70 x %1 - 42 x %1 - 11 x %1 + 70 x %1 + 70 x %1 20 5 19 6 22 2 21 3 20 4 + 45 x %1 + 2 x %1 - 42 x %1 - 70 x %1 - 95 x %1 19 5 18 6 22 21 2 20 3 19 4 - 12 x %1 + x %1 + 14 x %1 + 42 x %1 + 115 x %1 + 30 x %1 18 5 17 6 22 21 20 2 19 3 - 4 x %1 - 4 x %1 - 2 x - 14 x %1 - 81 x %1 - 40 x %1 18 4 17 5 16 6 21 20 19 2 + 5 x %1 + 23 x %1 + 3 x %1 + 2 x + 31 x %1 + 30 x %1 17 4 16 5 15 6 20 19 18 2 - 55 x %1 - 23 x %1 - x %1 - 5 x - 12 x %1 - 5 x %1 17 3 16 4 15 5 19 18 17 2 + 70 x %1 + 69 x %1 + 14 x %1 + 2 x + 4 x %1 - 50 x %1 16 3 15 4 14 5 18 17 16 2 - 106 x %1 - 56 x %1 - 5 x %1 - x + 19 x %1 + 89 x %1 15 3 14 4 13 5 17 16 15 2 + 105 x %1 + 27 x %1 + x %1 - 3 x - 39 x %1 - 104 x %1 14 3 16 15 14 2 13 3 12 4 - 65 x %1 + 7 x + 53 x %1 + 83 x %1 + x %1 - 12 x %1 15 14 13 2 12 3 11 4 14 - 11 x - 54 x %1 - 14 x %1 + 46 x %1 + 11 x %1 + 14 x 13 12 2 11 3 10 4 13 12 + 20 x %1 - 63 x %1 - 60 x %1 - 5 x %1 - 8 x + 36 x %1 11 2 10 3 9 4 12 11 10 2 + 113 x %1 + 46 x %1 + x %1 - 7 x - 90 x %1 - 116 x %1 9 3 11 10 9 2 8 3 10 - 23 x %1 + 26 x + 114 x %1 + 79 x %1 + 7 x %1 - 39 x 9 8 2 7 3 9 8 7 2 8 - 87 x %1 - 35 x %1 - x %1 + 29 x + 19 x %1 + 9 x %1 + 18 x 7 6 2 7 6 6 5 5 + 50 x %1 - x %1 - 94 x - 78 x %1 + 163 x + 61 x %1 - 187 x 4 4 3 3 2 2 / - 29 x %1 + 155 x + 8 x %1 - 92 x - x %1 + 37 x - 9 x + 1) / ( / 2 3 3 2 2 3 3 (-1 + x) (x %1 - x - x %1 + x - 2 x + 1) (x %1 - x + x - 1) 3 3 12 4 12 3 12 2 12 12 (x %1 - x - x + 1) (x %1 - 4 x %1 + 6 x %1 - 4 x %1 + x 8 3 8 2 7 3 8 7 2 8 7 7 6 6 + x %1 - x %1 - x %1 - x %1 + x %1 + x + x %1 - x + x %1 - x 5 5 4 3 2 - x %1 + x + x - 4 x + 6 x - 4 x + 1)) %1 := X[1, 1, 2, 1, 1, 1, 2] and in Maple format -(2*x^22*X[1,1,2,1,1,1,2]^7-14*x^22*X[1,1,2,1,1,1,2]^6-2*x^21*X[1,1,2,1,1,1,2]^ 7+42*x^22*X[1,1,2,1,1,1,2]^5+14*x^21*X[1,1,2,1,1,1,2]^6+x^20*X[1,1,2,1,1,1,2]^7 -70*x^22*X[1,1,2,1,1,1,2]^4-42*x^21*X[1,1,2,1,1,1,2]^5-11*x^20*X[1,1,2,1,1,1,2] ^6+70*x^22*X[1,1,2,1,1,1,2]^3+70*x^21*X[1,1,2,1,1,1,2]^4+45*x^20*X[1,1,2,1,1,1, 2]^5+2*x^19*X[1,1,2,1,1,1,2]^6-42*x^22*X[1,1,2,1,1,1,2]^2-70*x^21*X[1,1,2,1,1,1 ,2]^3-95*x^20*X[1,1,2,1,1,1,2]^4-12*x^19*X[1,1,2,1,1,1,2]^5+x^18*X[1,1,2,1,1,1, 2]^6+14*x^22*X[1,1,2,1,1,1,2]+42*x^21*X[1,1,2,1,1,1,2]^2+115*x^20*X[1,1,2,1,1,1 ,2]^3+30*x^19*X[1,1,2,1,1,1,2]^4-4*x^18*X[1,1,2,1,1,1,2]^5-4*x^17*X[1,1,2,1,1,1 ,2]^6-2*x^22-14*x^21*X[1,1,2,1,1,1,2]-81*x^20*X[1,1,2,1,1,1,2]^2-40*x^19*X[1,1, 2,1,1,1,2]^3+5*x^18*X[1,1,2,1,1,1,2]^4+23*x^17*X[1,1,2,1,1,1,2]^5+3*x^16*X[1,1, 2,1,1,1,2]^6+2*x^21+31*x^20*X[1,1,2,1,1,1,2]+30*x^19*X[1,1,2,1,1,1,2]^2-55*x^17 *X[1,1,2,1,1,1,2]^4-23*x^16*X[1,1,2,1,1,1,2]^5-x^15*X[1,1,2,1,1,1,2]^6-5*x^20-\ 12*x^19*X[1,1,2,1,1,1,2]-5*x^18*X[1,1,2,1,1,1,2]^2+70*x^17*X[1,1,2,1,1,1,2]^3+ 69*x^16*X[1,1,2,1,1,1,2]^4+14*x^15*X[1,1,2,1,1,1,2]^5+2*x^19+4*x^18*X[1,1,2,1,1 ,1,2]-50*x^17*X[1,1,2,1,1,1,2]^2-106*x^16*X[1,1,2,1,1,1,2]^3-56*x^15*X[1,1,2,1, 1,1,2]^4-5*x^14*X[1,1,2,1,1,1,2]^5-x^18+19*x^17*X[1,1,2,1,1,1,2]+89*x^16*X[1,1, 2,1,1,1,2]^2+105*x^15*X[1,1,2,1,1,1,2]^3+27*x^14*X[1,1,2,1,1,1,2]^4+x^13*X[1,1, 2,1,1,1,2]^5-3*x^17-39*x^16*X[1,1,2,1,1,1,2]-104*x^15*X[1,1,2,1,1,1,2]^2-65*x^ 14*X[1,1,2,1,1,1,2]^3+7*x^16+53*x^15*X[1,1,2,1,1,1,2]+83*x^14*X[1,1,2,1,1,1,2]^ 2+x^13*X[1,1,2,1,1,1,2]^3-12*x^12*X[1,1,2,1,1,1,2]^4-11*x^15-54*x^14*X[1,1,2,1, 1,1,2]-14*x^13*X[1,1,2,1,1,1,2]^2+46*x^12*X[1,1,2,1,1,1,2]^3+11*x^11*X[1,1,2,1, 1,1,2]^4+14*x^14+20*x^13*X[1,1,2,1,1,1,2]-63*x^12*X[1,1,2,1,1,1,2]^2-60*x^11*X[ 1,1,2,1,1,1,2]^3-5*x^10*X[1,1,2,1,1,1,2]^4-8*x^13+36*x^12*X[1,1,2,1,1,1,2]+113* x^11*X[1,1,2,1,1,1,2]^2+46*x^10*X[1,1,2,1,1,1,2]^3+x^9*X[1,1,2,1,1,1,2]^4-7*x^ 12-90*x^11*X[1,1,2,1,1,1,2]-116*x^10*X[1,1,2,1,1,1,2]^2-23*x^9*X[1,1,2,1,1,1,2] ^3+26*x^11+114*x^10*X[1,1,2,1,1,1,2]+79*x^9*X[1,1,2,1,1,1,2]^2+7*x^8*X[1,1,2,1, 1,1,2]^3-39*x^10-87*x^9*X[1,1,2,1,1,1,2]-35*x^8*X[1,1,2,1,1,1,2]^2-x^7*X[1,1,2, 1,1,1,2]^3+29*x^9+19*x^8*X[1,1,2,1,1,1,2]+9*x^7*X[1,1,2,1,1,1,2]^2+18*x^8+50*x^ 7*X[1,1,2,1,1,1,2]-x^6*X[1,1,2,1,1,1,2]^2-94*x^7-78*x^6*X[1,1,2,1,1,1,2]+163*x^ 6+61*x^5*X[1,1,2,1,1,1,2]-187*x^5-29*x^4*X[1,1,2,1,1,1,2]+155*x^4+8*x^3*X[1,1,2 ,1,1,1,2]-92*x^3-x^2*X[1,1,2,1,1,1,2]+37*x^2-9*x+1)/(-1+x)^2/(x^3*X[1,1,2,1,1,1 ,2]-x^3-x^2*X[1,1,2,1,1,1,2]+x^2-2*x+1)/(x^3*X[1,1,2,1,1,1,2]-x^3+x-1)/(x^3*X[1 ,1,2,1,1,1,2]-x^3-x+1)/(x^12*X[1,1,2,1,1,1,2]^4-4*x^12*X[1,1,2,1,1,1,2]^3+6*x^ 12*X[1,1,2,1,1,1,2]^2-4*x^12*X[1,1,2,1,1,1,2]+x^12+x^8*X[1,1,2,1,1,1,2]^3-x^8*X [1,1,2,1,1,1,2]^2-x^7*X[1,1,2,1,1,1,2]^3-x^8*X[1,1,2,1,1,1,2]+x^7*X[1,1,2,1,1,1 ,2]^2+x^8+x^7*X[1,1,2,1,1,1,2]-x^7+x^6*X[1,1,2,1,1,1,2]-x^6-x^5*X[1,1,2,1,1,1,2 ]+x^5+x^4-4*x^3+6*x^2-4*x+1) Furthermore, the expectation of the random variable number of occurences (by\ containment) of , [1, 1, 2, 1, 1, 1, 2], equals , - 13/8 + n/8 107 7 n The variance equals , - --- + --- 64 64 93 9 n The , 3, -th moment about the mean is , - -- + --- 64 128 33193 147 2 1139 The , 4, -th moment about the mean is , ----- + ---- n - ---- n 4096 4096 1024 The compositions of, 9, that yield the, 4, -th largest growth, that is, 1.8392867552141611326, are , [1, 1, 1, 1, 1, 4], [1, 1, 1, 1, 4, 1], [1, 1, 1, 4, 1, 1], [1, 1, 4, 1, 1, 1], [1, 4, 1, 1, 1, 1], [4, 1, 1, 1, 1, 1] Theorem Number, 4, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 1, 1, 1, 1, 4] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 8 7 6 5 4 3 2 \ n x - 2 x + 4 x - 4 x + 6 x - 10 x + 10 x - 5 x + 1 ) a(n) x = -------------------------------------------------------- / 3 2 5 ----- (x + x + x - 1) (-1 + x) n = 0 and in Maple format (x^8-2*x^7+4*x^6-4*x^5+6*x^4-10*x^3+10*x^2-5*x+1)/(x^3+x^2+x-1)/(-1+x)^5 The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 504, 986, 1908, 3654, 6934, 13058, 24440, 45525, 84494, 156400, 288936, 533044, 982428, 1809444, 3331112, 6130531, 11280196, 20752742, 38176420, 70224634, 129171698] The limit of a(n+1)/a(n) as n goes to infinity is 1.83928675521 a(n) is asymptotic to 1.48501767945*1.83928675521^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [1, 1, 1, 1, 1, 4], denoted by the variable, X[1, 1, 1, 1, 1, 4], is 8 8 7 7 6 6 5 5 4 4 (x %1 - x - 2 x %1 + 2 x + 4 x %1 - 4 x - 3 x %1 + 4 x + x %1 - 6 x 3 2 / 4 4 4 + 10 x - 10 x + 5 x - 1) / ((-1 + x) (x %1 - x + 2 x - 1)) / %1 := X[1, 1, 1, 1, 1, 4] and in Maple format (x^8*X[1,1,1,1,1,4]-x^8-2*x^7*X[1,1,1,1,1,4]+2*x^7+4*x^6*X[1,1,1,1,1,4]-4*x^6-3 *x^5*X[1,1,1,1,1,4]+4*x^5+x^4*X[1,1,1,1,1,4]-6*x^4+10*x^3-10*x^2+5*x-1)/(-1+x)^ 4/(x^4*X[1,1,1,1,1,4]-x^4+2*x-1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [1, 1, 1, 1, 1, 4], equals , - 3/4 + ---- 16 11 9 n The variance equals , - -- + --- 32 256 15 n The , 3, -th moment about the mean is , 9/512 + ---- 2048 1033 243 2 2685 The , 4, -th moment about the mean is , ---- + ----- n - ----- n 2048 65536 32768 The compositions of, 9, that yield the, 5, -th largest growth, that is, 1.8667603991738620930, are , [2, 1, 1, 1, 1, 3], [3, 1, 1, 1, 1, 2] Theorem Number, 5, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 1, 1, 1, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 43 40 39 38 37 36 34 33 ) a(n) x = - (x + 2 x + x + x - x + 3 x - 5 x + 5 x / ----- n = 0 31 30 29 28 27 26 25 23 - 4 x + 3 x - 10 x + 17 x - 4 x - 16 x + 17 x - 4 x 22 21 20 19 18 17 16 15 - 15 x + 36 x - 47 x + 39 x + 11 x - 84 x + 101 x - 39 x 14 13 12 11 10 9 8 7 - 24 x + 35 x - 43 x + 110 x - 214 x + 260 x - 156 x - 90 x 6 5 4 3 2 / + 334 x - 416 x + 320 x - 164 x + 55 x - 11 x + 1) / ( / 4 3 (x - x + 2 x - 1) 20 13 12 10 9 5 4 3 2 20 (x - x + 2 x - 2 x + x + x - 5 x + 10 x - 10 x + 5 x - 1) (x 14 13 12 11 10 8 7 5 4 3 - x + 4 x - 3 x - 3 x + 3 x + x - x + x - 5 x + 10 x 2 - 10 x + 5 x - 1)) and in Maple format -(x^43+2*x^40+x^39+x^38-x^37+3*x^36-5*x^34+5*x^33-4*x^31+3*x^30-10*x^29+17*x^28 -4*x^27-16*x^26+17*x^25-4*x^23-15*x^22+36*x^21-47*x^20+39*x^19+11*x^18-84*x^17+ 101*x^16-39*x^15-24*x^14+35*x^13-43*x^12+110*x^11-214*x^10+260*x^9-156*x^8-90*x ^7+334*x^6-416*x^5+320*x^4-164*x^3+55*x^2-11*x+1)/(x^4-x^3+2*x-1)/(x^20-x^13+2* x^12-2*x^10+x^9+x^5-5*x^4+10*x^3-10*x^2+5*x-1)/(x^20-x^14+4*x^13-3*x^12-3*x^11+ 3*x^10+x^8-x^7+x^5-5*x^4+10*x^3-10*x^2+5*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 504, 986, 1908, 3654, 6934, 13058, 24441, 45537, 84574, 156788, 290464, 538232, 998199, 1853488, 3446216, 6415882, 11958159, 22308646, 41646747, 77783915, 145315977] The limit of a(n+1)/a(n) as n goes to infinity is 1.86676039917 a(n) is asymptotic to 1.07461638455*1.86676039917^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [2, 1, 1, 1, 1, 3], denoted by the variable, X[2, 1, 1, 1, 1, 3], is 43 11 43 10 43 9 43 8 43 7 40 10 - (x %1 - 11 x %1 + 55 x %1 - 165 x %1 + 330 x %1 - 2 x %1 43 6 40 9 39 10 43 5 40 8 - 462 x %1 + 20 x %1 - x %1 + 462 x %1 - 90 x %1 39 9 38 10 43 4 40 7 39 8 + 10 x %1 - x %1 - 330 x %1 + 240 x %1 - 45 x %1 38 9 37 10 43 3 40 6 39 7 + 10 x %1 + x %1 + 165 x %1 - 420 x %1 + 120 x %1 38 8 37 9 36 10 43 2 40 5 - 45 x %1 - 10 x %1 - x %1 - 55 x %1 + 504 x %1 39 6 38 7 37 8 36 9 43 - 210 x %1 + 120 x %1 + 45 x %1 + 12 x %1 + 11 x %1 40 4 39 5 38 6 37 7 36 8 43 - 420 x %1 + 252 x %1 - 210 x %1 - 120 x %1 - 63 x %1 - x 40 3 39 4 38 5 37 6 36 7 + 240 x %1 - 210 x %1 + 252 x %1 + 210 x %1 + 192 x %1 34 9 40 2 39 3 38 4 37 5 - 3 x %1 - 90 x %1 + 120 x %1 - 210 x %1 - 252 x %1 36 6 34 8 33 9 40 39 2 - 378 x %1 + 29 x %1 + x %1 + 20 x %1 - 45 x %1 38 3 37 4 36 5 34 7 33 8 + 120 x %1 + 210 x %1 + 504 x %1 - 124 x %1 - 13 x %1 40 39 38 2 37 3 36 4 34 6 - 2 x + 10 x %1 - 45 x %1 - 120 x %1 - 462 x %1 + 308 x %1 33 7 39 38 37 2 36 3 34 5 + 68 x %1 - x + 10 x %1 + 45 x %1 + 288 x %1 - 490 x %1 33 6 31 8 38 37 36 2 34 4 - 196 x %1 + 2 x %1 - x - 10 x %1 - 117 x %1 + 518 x %1 33 5 31 7 30 8 37 36 34 3 + 350 x %1 - 18 x %1 - 2 x %1 + x + 28 x %1 - 364 x %1 33 4 31 6 30 7 29 8 36 34 2 - 406 x %1 + 70 x %1 + 17 x %1 + 4 x %1 - 3 x + 164 x %1 33 3 31 5 30 6 29 7 28 8 + 308 x %1 - 154 x %1 - 63 x %1 - 37 x %1 - 3 x %1 34 33 2 31 4 30 5 29 6 - 43 x %1 - 148 x %1 + 210 x %1 + 133 x %1 + 148 x %1 28 7 27 8 34 33 31 3 30 4 + 33 x %1 + x %1 + 5 x + 41 x %1 - 182 x %1 - 175 x %1 29 5 28 6 27 7 33 31 2 30 3 - 335 x %1 - 152 x %1 - 9 x %1 - 5 x + 98 x %1 + 147 x %1 29 4 28 5 27 6 26 7 31 + 470 x %1 + 387 x %1 + 37 x %1 - 6 x %1 - 30 x %1 30 2 29 3 28 4 27 5 26 6 - 77 x %1 - 419 x %1 - 600 x %1 - 89 x %1 + 51 x %1 25 7 31 30 29 2 28 3 27 4 + 5 x %1 + 4 x + 23 x %1 + 232 x %1 + 583 x %1 + 135 x %1 26 5 25 6 24 7 30 29 28 2 - 181 x %1 - 46 x %1 - x %1 - 3 x - 73 x %1 - 348 x %1 27 3 26 4 25 5 24 6 29 - 131 x %1 + 350 x %1 + 172 x %1 + 9 x %1 + 10 x 28 27 2 26 3 25 4 24 5 + 117 x %1 + 79 x %1 - 400 x %1 - 345 x %1 - 30 x %1 23 6 28 27 26 2 25 3 24 4 + 3 x %1 - 17 x - 27 x %1 + 271 x %1 + 405 x %1 + 50 x %1 23 5 22 6 27 26 25 2 24 3 - 20 x %1 - x %1 + 4 x - 101 x %1 - 280 x %1 - 45 x %1 23 4 22 5 26 25 24 2 23 3 + 54 x %1 - 3 x %1 + 16 x + 106 x %1 + 21 x %1 - 76 x %1 22 4 21 5 25 24 23 2 22 3 + 37 x %1 + 19 x %1 - 17 x - 4 x %1 + 59 x %1 - 98 x %1 21 4 20 5 23 22 2 21 3 - 112 x %1 - 24 x %1 - 24 x %1 + 117 x %1 + 258 x %1 20 4 19 5 23 22 21 2 20 3 + 144 x %1 + 24 x %1 + 4 x - 67 x %1 - 292 x %1 - 335 x %1 19 4 18 5 22 21 20 2 - 143 x %1 - 16 x %1 + 15 x + 163 x %1 + 381 x %1 19 3 18 4 17 5 21 20 19 2 + 324 x %1 + 84 x %1 + 6 x %1 - 36 x - 213 x %1 - 354 x %1 18 3 17 4 16 5 20 19 18 2 - 145 x %1 - 3 x %1 - x %1 + 47 x + 188 x %1 + 91 x %1 17 3 16 4 19 18 17 2 16 3 - 113 x %1 - 34 x %1 - 39 x - 3 x %1 + 295 x %1 + 221 x %1 15 4 18 17 16 2 15 3 14 4 + 25 x %1 - 11 x - 269 x %1 - 437 x %1 - 143 x %1 - 8 x %1 17 16 15 2 14 3 13 4 16 + 84 x + 352 x %1 + 250 x %1 + 35 x %1 + x %1 - 101 x 15 14 2 13 3 15 14 13 2 - 171 x %1 - 23 x %1 + 7 x %1 + 39 x - 28 x %1 - 42 x %1 12 3 14 13 12 2 11 3 13 - 6 x %1 + 24 x + 69 x %1 + 10 x %1 + x %1 - 35 x 12 11 2 12 11 10 2 11 - 47 x %1 + 10 x %1 + 43 x + 100 x %1 - 6 x %1 - 110 x 10 9 2 10 9 9 8 8 - 219 x %1 + x %1 + 214 x + 314 x %1 - 260 x - 321 x %1 + 156 x 7 7 6 6 5 5 4 + 240 x %1 + 90 x - 128 x %1 - 334 x + 46 x %1 + 416 x - 10 x %1 4 3 3 2 / - 320 x + x %1 + 164 x - 55 x + 11 x - 1) / ( / 4 4 3 3 20 5 20 4 20 3 (x %1 - x - x %1 + x - 2 x + 1) (x %1 - 5 x %1 + 10 x %1 20 2 20 20 13 3 13 2 12 3 13 - 10 x %1 + 5 x %1 - x - x %1 + 3 x %1 + 2 x %1 - 3 x %1 12 2 11 3 13 12 11 2 12 11 - 6 x %1 - x %1 + x + 6 x %1 + 2 x %1 - 2 x - x %1 10 2 10 9 2 10 9 9 5 4 3 + 2 x %1 - 4 x %1 - x %1 + 2 x + 2 x %1 - x - x + 5 x - 10 x 2 20 5 20 4 20 3 20 2 + 10 x - 5 x + 1) (x %1 - 5 x %1 + 10 x %1 - 10 x %1 20 20 14 4 14 3 13 4 14 2 13 3 + 5 x %1 - x + x %1 - 4 x %1 - x %1 + 6 x %1 + 7 x %1 14 13 2 12 3 14 13 12 2 13 - 4 x %1 - 15 x %1 - 3 x %1 + x + 13 x %1 + 9 x %1 - 4 x 12 11 2 12 11 10 2 11 10 - 9 x %1 + 3 x %1 + 3 x - 6 x %1 - 3 x %1 + 3 x + 6 x %1 10 8 8 7 7 5 4 3 2 - 3 x + x %1 - x - x %1 + x - x + 5 x - 10 x + 10 x - 5 x + 1)) %1 := X[2, 1, 1, 1, 1, 3] and in Maple format -(x^43*X[2,1,1,1,1,3]^11-11*x^43*X[2,1,1,1,1,3]^10+55*x^43*X[2,1,1,1,1,3]^9-165 *x^43*X[2,1,1,1,1,3]^8+330*x^43*X[2,1,1,1,1,3]^7-2*x^40*X[2,1,1,1,1,3]^10-462*x ^43*X[2,1,1,1,1,3]^6+20*x^40*X[2,1,1,1,1,3]^9-x^39*X[2,1,1,1,1,3]^10+462*x^43*X [2,1,1,1,1,3]^5-90*x^40*X[2,1,1,1,1,3]^8+10*x^39*X[2,1,1,1,1,3]^9-x^38*X[2,1,1, 1,1,3]^10-330*x^43*X[2,1,1,1,1,3]^4+240*x^40*X[2,1,1,1,1,3]^7-45*x^39*X[2,1,1,1 ,1,3]^8+10*x^38*X[2,1,1,1,1,3]^9+x^37*X[2,1,1,1,1,3]^10+165*x^43*X[2,1,1,1,1,3] ^3-420*x^40*X[2,1,1,1,1,3]^6+120*x^39*X[2,1,1,1,1,3]^7-45*x^38*X[2,1,1,1,1,3]^8 -10*x^37*X[2,1,1,1,1,3]^9-x^36*X[2,1,1,1,1,3]^10-55*x^43*X[2,1,1,1,1,3]^2+504*x ^40*X[2,1,1,1,1,3]^5-210*x^39*X[2,1,1,1,1,3]^6+120*x^38*X[2,1,1,1,1,3]^7+45*x^ 37*X[2,1,1,1,1,3]^8+12*x^36*X[2,1,1,1,1,3]^9+11*x^43*X[2,1,1,1,1,3]-420*x^40*X[ 2,1,1,1,1,3]^4+252*x^39*X[2,1,1,1,1,3]^5-210*x^38*X[2,1,1,1,1,3]^6-120*x^37*X[2 ,1,1,1,1,3]^7-63*x^36*X[2,1,1,1,1,3]^8-x^43+240*x^40*X[2,1,1,1,1,3]^3-210*x^39* X[2,1,1,1,1,3]^4+252*x^38*X[2,1,1,1,1,3]^5+210*x^37*X[2,1,1,1,1,3]^6+192*x^36*X [2,1,1,1,1,3]^7-3*x^34*X[2,1,1,1,1,3]^9-90*x^40*X[2,1,1,1,1,3]^2+120*x^39*X[2,1 ,1,1,1,3]^3-210*x^38*X[2,1,1,1,1,3]^4-252*x^37*X[2,1,1,1,1,3]^5-378*x^36*X[2,1, 1,1,1,3]^6+29*x^34*X[2,1,1,1,1,3]^8+x^33*X[2,1,1,1,1,3]^9+20*x^40*X[2,1,1,1,1,3 ]-45*x^39*X[2,1,1,1,1,3]^2+120*x^38*X[2,1,1,1,1,3]^3+210*x^37*X[2,1,1,1,1,3]^4+ 504*x^36*X[2,1,1,1,1,3]^5-124*x^34*X[2,1,1,1,1,3]^7-13*x^33*X[2,1,1,1,1,3]^8-2* x^40+10*x^39*X[2,1,1,1,1,3]-45*x^38*X[2,1,1,1,1,3]^2-120*x^37*X[2,1,1,1,1,3]^3-\ 462*x^36*X[2,1,1,1,1,3]^4+308*x^34*X[2,1,1,1,1,3]^6+68*x^33*X[2,1,1,1,1,3]^7-x^ 39+10*x^38*X[2,1,1,1,1,3]+45*x^37*X[2,1,1,1,1,3]^2+288*x^36*X[2,1,1,1,1,3]^3-\ 490*x^34*X[2,1,1,1,1,3]^5-196*x^33*X[2,1,1,1,1,3]^6+2*x^31*X[2,1,1,1,1,3]^8-x^ 38-10*x^37*X[2,1,1,1,1,3]-117*x^36*X[2,1,1,1,1,3]^2+518*x^34*X[2,1,1,1,1,3]^4+ 350*x^33*X[2,1,1,1,1,3]^5-18*x^31*X[2,1,1,1,1,3]^7-2*x^30*X[2,1,1,1,1,3]^8+x^37 +28*x^36*X[2,1,1,1,1,3]-364*x^34*X[2,1,1,1,1,3]^3-406*x^33*X[2,1,1,1,1,3]^4+70* x^31*X[2,1,1,1,1,3]^6+17*x^30*X[2,1,1,1,1,3]^7+4*x^29*X[2,1,1,1,1,3]^8-3*x^36+ 164*x^34*X[2,1,1,1,1,3]^2+308*x^33*X[2,1,1,1,1,3]^3-154*x^31*X[2,1,1,1,1,3]^5-\ 63*x^30*X[2,1,1,1,1,3]^6-37*x^29*X[2,1,1,1,1,3]^7-3*x^28*X[2,1,1,1,1,3]^8-43*x^ 34*X[2,1,1,1,1,3]-148*x^33*X[2,1,1,1,1,3]^2+210*x^31*X[2,1,1,1,1,3]^4+133*x^30* X[2,1,1,1,1,3]^5+148*x^29*X[2,1,1,1,1,3]^6+33*x^28*X[2,1,1,1,1,3]^7+x^27*X[2,1, 1,1,1,3]^8+5*x^34+41*x^33*X[2,1,1,1,1,3]-182*x^31*X[2,1,1,1,1,3]^3-175*x^30*X[2 ,1,1,1,1,3]^4-335*x^29*X[2,1,1,1,1,3]^5-152*x^28*X[2,1,1,1,1,3]^6-9*x^27*X[2,1, 1,1,1,3]^7-5*x^33+98*x^31*X[2,1,1,1,1,3]^2+147*x^30*X[2,1,1,1,1,3]^3+470*x^29*X [2,1,1,1,1,3]^4+387*x^28*X[2,1,1,1,1,3]^5+37*x^27*X[2,1,1,1,1,3]^6-6*x^26*X[2,1 ,1,1,1,3]^7-30*x^31*X[2,1,1,1,1,3]-77*x^30*X[2,1,1,1,1,3]^2-419*x^29*X[2,1,1,1, 1,3]^3-600*x^28*X[2,1,1,1,1,3]^4-89*x^27*X[2,1,1,1,1,3]^5+51*x^26*X[2,1,1,1,1,3 ]^6+5*x^25*X[2,1,1,1,1,3]^7+4*x^31+23*x^30*X[2,1,1,1,1,3]+232*x^29*X[2,1,1,1,1, 3]^2+583*x^28*X[2,1,1,1,1,3]^3+135*x^27*X[2,1,1,1,1,3]^4-181*x^26*X[2,1,1,1,1,3 ]^5-46*x^25*X[2,1,1,1,1,3]^6-x^24*X[2,1,1,1,1,3]^7-3*x^30-73*x^29*X[2,1,1,1,1,3 ]-348*x^28*X[2,1,1,1,1,3]^2-131*x^27*X[2,1,1,1,1,3]^3+350*x^26*X[2,1,1,1,1,3]^4 +172*x^25*X[2,1,1,1,1,3]^5+9*x^24*X[2,1,1,1,1,3]^6+10*x^29+117*x^28*X[2,1,1,1,1 ,3]+79*x^27*X[2,1,1,1,1,3]^2-400*x^26*X[2,1,1,1,1,3]^3-345*x^25*X[2,1,1,1,1,3]^ 4-30*x^24*X[2,1,1,1,1,3]^5+3*x^23*X[2,1,1,1,1,3]^6-17*x^28-27*x^27*X[2,1,1,1,1, 3]+271*x^26*X[2,1,1,1,1,3]^2+405*x^25*X[2,1,1,1,1,3]^3+50*x^24*X[2,1,1,1,1,3]^4 -20*x^23*X[2,1,1,1,1,3]^5-x^22*X[2,1,1,1,1,3]^6+4*x^27-101*x^26*X[2,1,1,1,1,3]-\ 280*x^25*X[2,1,1,1,1,3]^2-45*x^24*X[2,1,1,1,1,3]^3+54*x^23*X[2,1,1,1,1,3]^4-3*x ^22*X[2,1,1,1,1,3]^5+16*x^26+106*x^25*X[2,1,1,1,1,3]+21*x^24*X[2,1,1,1,1,3]^2-\ 76*x^23*X[2,1,1,1,1,3]^3+37*x^22*X[2,1,1,1,1,3]^4+19*x^21*X[2,1,1,1,1,3]^5-17*x ^25-4*x^24*X[2,1,1,1,1,3]+59*x^23*X[2,1,1,1,1,3]^2-98*x^22*X[2,1,1,1,1,3]^3-112 *x^21*X[2,1,1,1,1,3]^4-24*x^20*X[2,1,1,1,1,3]^5-24*x^23*X[2,1,1,1,1,3]+117*x^22 *X[2,1,1,1,1,3]^2+258*x^21*X[2,1,1,1,1,3]^3+144*x^20*X[2,1,1,1,1,3]^4+24*x^19*X [2,1,1,1,1,3]^5+4*x^23-67*x^22*X[2,1,1,1,1,3]-292*x^21*X[2,1,1,1,1,3]^2-335*x^ 20*X[2,1,1,1,1,3]^3-143*x^19*X[2,1,1,1,1,3]^4-16*x^18*X[2,1,1,1,1,3]^5+15*x^22+ 163*x^21*X[2,1,1,1,1,3]+381*x^20*X[2,1,1,1,1,3]^2+324*x^19*X[2,1,1,1,1,3]^3+84* x^18*X[2,1,1,1,1,3]^4+6*x^17*X[2,1,1,1,1,3]^5-36*x^21-213*x^20*X[2,1,1,1,1,3]-\ 354*x^19*X[2,1,1,1,1,3]^2-145*x^18*X[2,1,1,1,1,3]^3-3*x^17*X[2,1,1,1,1,3]^4-x^ 16*X[2,1,1,1,1,3]^5+47*x^20+188*x^19*X[2,1,1,1,1,3]+91*x^18*X[2,1,1,1,1,3]^2-\ 113*x^17*X[2,1,1,1,1,3]^3-34*x^16*X[2,1,1,1,1,3]^4-39*x^19-3*x^18*X[2,1,1,1,1,3 ]+295*x^17*X[2,1,1,1,1,3]^2+221*x^16*X[2,1,1,1,1,3]^3+25*x^15*X[2,1,1,1,1,3]^4-\ 11*x^18-269*x^17*X[2,1,1,1,1,3]-437*x^16*X[2,1,1,1,1,3]^2-143*x^15*X[2,1,1,1,1, 3]^3-8*x^14*X[2,1,1,1,1,3]^4+84*x^17+352*x^16*X[2,1,1,1,1,3]+250*x^15*X[2,1,1,1 ,1,3]^2+35*x^14*X[2,1,1,1,1,3]^3+x^13*X[2,1,1,1,1,3]^4-101*x^16-171*x^15*X[2,1, 1,1,1,3]-23*x^14*X[2,1,1,1,1,3]^2+7*x^13*X[2,1,1,1,1,3]^3+39*x^15-28*x^14*X[2,1 ,1,1,1,3]-42*x^13*X[2,1,1,1,1,3]^2-6*x^12*X[2,1,1,1,1,3]^3+24*x^14+69*x^13*X[2, 1,1,1,1,3]+10*x^12*X[2,1,1,1,1,3]^2+x^11*X[2,1,1,1,1,3]^3-35*x^13-47*x^12*X[2,1 ,1,1,1,3]+10*x^11*X[2,1,1,1,1,3]^2+43*x^12+100*x^11*X[2,1,1,1,1,3]-6*x^10*X[2,1 ,1,1,1,3]^2-110*x^11-219*x^10*X[2,1,1,1,1,3]+x^9*X[2,1,1,1,1,3]^2+214*x^10+314* x^9*X[2,1,1,1,1,3]-260*x^9-321*x^8*X[2,1,1,1,1,3]+156*x^8+240*x^7*X[2,1,1,1,1,3 ]+90*x^7-128*x^6*X[2,1,1,1,1,3]-334*x^6+46*x^5*X[2,1,1,1,1,3]+416*x^5-10*x^4*X[ 2,1,1,1,1,3]-320*x^4+x^3*X[2,1,1,1,1,3]+164*x^3-55*x^2+11*x-1)/(x^4*X[2,1,1,1,1 ,3]-x^4-x^3*X[2,1,1,1,1,3]+x^3-2*x+1)/(x^20*X[2,1,1,1,1,3]^5-5*x^20*X[2,1,1,1,1 ,3]^4+10*x^20*X[2,1,1,1,1,3]^3-10*x^20*X[2,1,1,1,1,3]^2+5*x^20*X[2,1,1,1,1,3]-x ^20-x^13*X[2,1,1,1,1,3]^3+3*x^13*X[2,1,1,1,1,3]^2+2*x^12*X[2,1,1,1,1,3]^3-3*x^ 13*X[2,1,1,1,1,3]-6*x^12*X[2,1,1,1,1,3]^2-x^11*X[2,1,1,1,1,3]^3+x^13+6*x^12*X[2 ,1,1,1,1,3]+2*x^11*X[2,1,1,1,1,3]^2-2*x^12-x^11*X[2,1,1,1,1,3]+2*x^10*X[2,1,1,1 ,1,3]^2-4*x^10*X[2,1,1,1,1,3]-x^9*X[2,1,1,1,1,3]^2+2*x^10+2*x^9*X[2,1,1,1,1,3]- x^9-x^5+5*x^4-10*x^3+10*x^2-5*x+1)/(x^20*X[2,1,1,1,1,3]^5-5*x^20*X[2,1,1,1,1,3] ^4+10*x^20*X[2,1,1,1,1,3]^3-10*x^20*X[2,1,1,1,1,3]^2+5*x^20*X[2,1,1,1,1,3]-x^20 +x^14*X[2,1,1,1,1,3]^4-4*x^14*X[2,1,1,1,1,3]^3-x^13*X[2,1,1,1,1,3]^4+6*x^14*X[2 ,1,1,1,1,3]^2+7*x^13*X[2,1,1,1,1,3]^3-4*x^14*X[2,1,1,1,1,3]-15*x^13*X[2,1,1,1,1 ,3]^2-3*x^12*X[2,1,1,1,1,3]^3+x^14+13*x^13*X[2,1,1,1,1,3]+9*x^12*X[2,1,1,1,1,3] ^2-4*x^13-9*x^12*X[2,1,1,1,1,3]+3*x^11*X[2,1,1,1,1,3]^2+3*x^12-6*x^11*X[2,1,1,1 ,1,3]-3*x^10*X[2,1,1,1,1,3]^2+3*x^11+6*x^10*X[2,1,1,1,1,3]-3*x^10+x^8*X[2,1,1,1 ,1,3]-x^8-x^7*X[2,1,1,1,1,3]+x^7-x^5+5*x^4-10*x^3+10*x^2-5*x+1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [2, 1, 1, 1, 1, 3], equals , - 3/4 + ---- 16 45 13 n The variance equals , - -- + ---- 64 256 147 63 n The , 3, -th moment about the mean is , - --- + ---- 256 2048 5013 507 2 7009 The , 4, -th moment about the mean is , ---- + ----- n - ----- n 4096 65536 32768 The compositions of, 9, that yield the, 6, -th largest growth, that is, 1.8908049048980778372, are , [1, 1, 2, 1, 2, 2], [1, 1, 2, 2, 1, 2], [1, 2, 1, 2, 2, 1], [1, 2, 2, 1, 2, 1], [2, 1, 2, 2, 1, 1], [2, 2, 1, 2, 1, 1] Theorem Number, 6, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 1, 2, 1, 2, 2] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n ) a(n) x = / ----- n = 0 9 8 7 6 5 4 3 2 2 x - 2 x + 2 x + 2 x - 7 x + 12 x - 14 x + 11 x - 5 x + 1 - ------------------------------------------------------------------ 8 7 6 5 4 3 2 2 (x - x + x + x - 3 x + 5 x - 6 x + 4 x - 1) (-1 + x) and in Maple format -(2*x^9-2*x^8+2*x^7+2*x^6-7*x^5+12*x^4-14*x^3+11*x^2-5*x+1)/(x^8-x^7+x^6+x^5-3* x^4+5*x^3-6*x^2+4*x-1)/(-1+x)^2 The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 504, 986, 1909, 3663, 6981, 13244, 25059, 47354, 89451, 168988, 319336, 603622, 1141239, 2157962, 4080694, 7716596, 14591772, 27591649, 52171819, 98647537, 186522776] The limit of a(n+1)/a(n) as n goes to infinity is 1.89080490490 a(n) is asymptotic to .936094451478*1.89080490490^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [1, 1, 2, 1, 2, 2], denoted by the variable, X[1, 1, 2, 1, 2, 2], is 9 2 9 8 2 9 8 7 2 8 7 - (2 x %1 - 4 x %1 - 2 x %1 + 2 x + 4 x %1 + x %1 - 2 x - 3 x %1 7 6 6 5 5 4 4 3 + 2 x - 2 x %1 + 2 x + 6 x %1 - 7 x - 7 x %1 + 12 x + 4 x %1 3 2 2 / 2 8 2 8 - 14 x - x %1 + 11 x - 5 x + 1) / ((-1 + x) (x %1 - 2 x %1 / 7 2 8 7 7 6 6 5 5 4 4 - x %1 + x + 2 x %1 - x - x %1 + x - x %1 + x + 3 x %1 - 3 x 3 3 2 2 - 3 x %1 + 5 x + x %1 - 6 x + 4 x - 1)) %1 := X[1, 1, 2, 1, 2, 2] and in Maple format -(2*x^9*X[1,1,2,1,2,2]^2-4*x^9*X[1,1,2,1,2,2]-2*x^8*X[1,1,2,1,2,2]^2+2*x^9+4*x^ 8*X[1,1,2,1,2,2]+x^7*X[1,1,2,1,2,2]^2-2*x^8-3*x^7*X[1,1,2,1,2,2]+2*x^7-2*x^6*X[ 1,1,2,1,2,2]+2*x^6+6*x^5*X[1,1,2,1,2,2]-7*x^5-7*x^4*X[1,1,2,1,2,2]+12*x^4+4*x^3 *X[1,1,2,1,2,2]-14*x^3-x^2*X[1,1,2,1,2,2]+11*x^2-5*x+1)/(-1+x)^2/(x^8*X[1,1,2,1 ,2,2]^2-2*x^8*X[1,1,2,1,2,2]-x^7*X[1,1,2,1,2,2]^2+x^8+2*x^7*X[1,1,2,1,2,2]-x^7- x^6*X[1,1,2,1,2,2]+x^6-x^5*X[1,1,2,1,2,2]+x^5+3*x^4*X[1,1,2,1,2,2]-3*x^4-3*x^3* X[1,1,2,1,2,2]+5*x^3+x^2*X[1,1,2,1,2,2]-6*x^2+4*x-1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [1, 1, 2, 1, 2, 2], equals , - 3/4 + ---- 16 73 21 n The variance equals , - -- + ---- 64 256 309 291 n The , 3, -th moment about the mean is , - --- + ----- 128 2048 10651 1323 2 7953 The , 4, -th moment about the mean is , - ----- + ----- n - ----- n 4096 65536 32768 The compositions of, 9, that yield the, 7, -th largest growth, that is, 1.8922218871524161071, are , [2, 1, 1, 2, 1, 2], [2, 1, 2, 1, 1, 2] Theorem Number, 7, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 1, 2, 1, 2] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 15 13 11 10 9 8 7 6 ) a(n) x = - (x + 2 x - x + 5 x - 8 x + 7 x - x - 9 x / ----- n = 0 5 4 3 2 / 16 15 14 13 + 19 x - 26 x + 25 x - 16 x + 6 x - 1) / (x - x + 2 x - 2 x / 12 11 10 9 8 7 6 5 4 3 + x + 3 x - 9 x + 13 x - 10 x - x + 16 x - 31 x + 40 x - 36 x 2 + 21 x - 7 x + 1) and in Maple format -(x^15+2*x^13-x^11+5*x^10-8*x^9+7*x^8-x^7-9*x^6+19*x^5-26*x^4+25*x^3-16*x^2+6*x -1)/(x^16-x^15+2*x^14-2*x^13+x^12+3*x^11-9*x^10+13*x^9-10*x^8-x^7+16*x^6-31*x^5 +40*x^4-36*x^3+21*x^2-7*x+1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 504, 986, 1908, 3655, 6944, 13115, 24682, 46378, 87138, 163863, 308564, 581916, 1098958, 2077791, 3931797, 7444185, 14098240, 26702030, 50569995, 95757810, 181289613] The limit of a(n+1)/a(n) as n goes to infinity is 1.89222188715 a(n) is asymptotic to .889598891994*1.89222188715^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [2, 1, 1, 2, 1, 2], denoted by the variable, X[2, 1, 1, 2, 1, 2], is 18 5 18 4 18 3 17 4 18 2 17 3 - (x %1 - 4 x %1 + 6 x %1 + x %1 - 4 x %1 - 4 x %1 16 4 18 17 2 16 3 17 16 2 - 3 x %1 + x %1 + 6 x %1 + 11 x %1 - 4 x %1 - 15 x %1 15 3 14 4 17 16 15 2 14 3 - 3 x %1 + 2 x %1 + x + 9 x %1 + 9 x %1 - 2 x %1 13 4 16 15 14 2 13 3 12 4 - 4 x %1 - 2 x - 9 x %1 - 6 x %1 + 12 x %1 + 3 x %1 15 14 13 2 12 3 11 4 14 13 + 3 x + 10 x %1 - 11 x %1 - 16 x %1 - x %1 - 4 x + 2 x %1 12 2 11 3 13 12 11 2 10 3 + 30 x %1 + 12 x %1 + x - 24 x %1 - 39 x %1 - 5 x %1 12 11 10 2 9 3 11 10 9 2 + 7 x + 47 x %1 + 33 x %1 + x %1 - 19 x - 56 x %1 - 18 x %1 10 9 8 2 9 8 7 2 7 + 28 x + 40 x %1 + 6 x %1 - 23 x - 7 x %1 - x %1 - 27 x %1 7 6 6 5 5 4 4 3 + 36 x + 45 x %1 - 73 x - 40 x %1 + 96 x + 22 x %1 - 92 x - 7 x %1 3 2 2 / 19 5 19 4 18 5 + 63 x + x %1 - 29 x + 8 x - 1) / (x %1 - 4 x %1 - x %1 / 19 3 18 4 19 2 18 3 17 4 19 + 6 x %1 + 5 x %1 - 4 x %1 - 10 x %1 - 4 x %1 + x %1 18 2 17 3 16 4 18 17 2 16 3 + 10 x %1 + 15 x %1 + 3 x %1 - 5 x %1 - 21 x %1 - 14 x %1 18 17 16 2 15 3 14 4 17 16 + x + 13 x %1 + 24 x %1 + 7 x %1 - 4 x %1 - 3 x - 18 x %1 15 2 14 3 13 4 16 15 14 2 - 21 x %1 + 6 x %1 + 6 x %1 + 5 x + 21 x %1 + 7 x %1 13 3 12 4 15 14 13 2 12 3 - 20 x %1 - 4 x %1 - 7 x - 16 x %1 + 21 x %1 + 24 x %1 11 4 14 13 12 2 11 3 13 12 + x %1 + 7 x - 6 x %1 - 50 x %1 - 16 x %1 - x + 44 x %1 11 2 10 3 12 11 10 2 9 3 + 60 x %1 + 6 x %1 - 14 x - 79 x %1 - 46 x %1 - x %1 11 10 9 2 10 9 8 2 9 + 34 x + 85 x %1 + 23 x %1 - 45 x - 54 x %1 - 7 x %1 + 32 x 8 7 2 8 7 7 6 6 5 + x %1 + x %1 + 8 x + 48 x %1 - 64 x - 69 x %1 + 118 x + 56 x %1 5 4 4 3 3 2 2 - 147 x - 28 x %1 + 133 x + 8 x %1 - 85 x - x %1 + 36 x - 9 x + 1) %1 := X[2, 1, 1, 2, 1, 2] and in Maple format -(x^18*X[2,1,1,2,1,2]^5-4*x^18*X[2,1,1,2,1,2]^4+6*x^18*X[2,1,1,2,1,2]^3+x^17*X[ 2,1,1,2,1,2]^4-4*x^18*X[2,1,1,2,1,2]^2-4*x^17*X[2,1,1,2,1,2]^3-3*x^16*X[2,1,1,2 ,1,2]^4+x^18*X[2,1,1,2,1,2]+6*x^17*X[2,1,1,2,1,2]^2+11*x^16*X[2,1,1,2,1,2]^3-4* x^17*X[2,1,1,2,1,2]-15*x^16*X[2,1,1,2,1,2]^2-3*x^15*X[2,1,1,2,1,2]^3+2*x^14*X[2 ,1,1,2,1,2]^4+x^17+9*x^16*X[2,1,1,2,1,2]+9*x^15*X[2,1,1,2,1,2]^2-2*x^14*X[2,1,1 ,2,1,2]^3-4*x^13*X[2,1,1,2,1,2]^4-2*x^16-9*x^15*X[2,1,1,2,1,2]-6*x^14*X[2,1,1,2 ,1,2]^2+12*x^13*X[2,1,1,2,1,2]^3+3*x^12*X[2,1,1,2,1,2]^4+3*x^15+10*x^14*X[2,1,1 ,2,1,2]-11*x^13*X[2,1,1,2,1,2]^2-16*x^12*X[2,1,1,2,1,2]^3-x^11*X[2,1,1,2,1,2]^4 -4*x^14+2*x^13*X[2,1,1,2,1,2]+30*x^12*X[2,1,1,2,1,2]^2+12*x^11*X[2,1,1,2,1,2]^3 +x^13-24*x^12*X[2,1,1,2,1,2]-39*x^11*X[2,1,1,2,1,2]^2-5*x^10*X[2,1,1,2,1,2]^3+7 *x^12+47*x^11*X[2,1,1,2,1,2]+33*x^10*X[2,1,1,2,1,2]^2+x^9*X[2,1,1,2,1,2]^3-19*x ^11-56*x^10*X[2,1,1,2,1,2]-18*x^9*X[2,1,1,2,1,2]^2+28*x^10+40*x^9*X[2,1,1,2,1,2 ]+6*x^8*X[2,1,1,2,1,2]^2-23*x^9-7*x^8*X[2,1,1,2,1,2]-x^7*X[2,1,1,2,1,2]^2-27*x^ 7*X[2,1,1,2,1,2]+36*x^7+45*x^6*X[2,1,1,2,1,2]-73*x^6-40*x^5*X[2,1,1,2,1,2]+96*x ^5+22*x^4*X[2,1,1,2,1,2]-92*x^4-7*x^3*X[2,1,1,2,1,2]+63*x^3+x^2*X[2,1,1,2,1,2]-\ 29*x^2+8*x-1)/(x^19*X[2,1,1,2,1,2]^5-4*x^19*X[2,1,1,2,1,2]^4-x^18*X[2,1,1,2,1,2 ]^5+6*x^19*X[2,1,1,2,1,2]^3+5*x^18*X[2,1,1,2,1,2]^4-4*x^19*X[2,1,1,2,1,2]^2-10* x^18*X[2,1,1,2,1,2]^3-4*x^17*X[2,1,1,2,1,2]^4+x^19*X[2,1,1,2,1,2]+10*x^18*X[2,1 ,1,2,1,2]^2+15*x^17*X[2,1,1,2,1,2]^3+3*x^16*X[2,1,1,2,1,2]^4-5*x^18*X[2,1,1,2,1 ,2]-21*x^17*X[2,1,1,2,1,2]^2-14*x^16*X[2,1,1,2,1,2]^3+x^18+13*x^17*X[2,1,1,2,1, 2]+24*x^16*X[2,1,1,2,1,2]^2+7*x^15*X[2,1,1,2,1,2]^3-4*x^14*X[2,1,1,2,1,2]^4-3*x ^17-18*x^16*X[2,1,1,2,1,2]-21*x^15*X[2,1,1,2,1,2]^2+6*x^14*X[2,1,1,2,1,2]^3+6*x ^13*X[2,1,1,2,1,2]^4+5*x^16+21*x^15*X[2,1,1,2,1,2]+7*x^14*X[2,1,1,2,1,2]^2-20*x ^13*X[2,1,1,2,1,2]^3-4*x^12*X[2,1,1,2,1,2]^4-7*x^15-16*x^14*X[2,1,1,2,1,2]+21*x ^13*X[2,1,1,2,1,2]^2+24*x^12*X[2,1,1,2,1,2]^3+x^11*X[2,1,1,2,1,2]^4+7*x^14-6*x^ 13*X[2,1,1,2,1,2]-50*x^12*X[2,1,1,2,1,2]^2-16*x^11*X[2,1,1,2,1,2]^3-x^13+44*x^ 12*X[2,1,1,2,1,2]+60*x^11*X[2,1,1,2,1,2]^2+6*x^10*X[2,1,1,2,1,2]^3-14*x^12-79*x ^11*X[2,1,1,2,1,2]-46*x^10*X[2,1,1,2,1,2]^2-x^9*X[2,1,1,2,1,2]^3+34*x^11+85*x^ 10*X[2,1,1,2,1,2]+23*x^9*X[2,1,1,2,1,2]^2-45*x^10-54*x^9*X[2,1,1,2,1,2]-7*x^8*X [2,1,1,2,1,2]^2+32*x^9+x^8*X[2,1,1,2,1,2]+x^7*X[2,1,1,2,1,2]^2+8*x^8+48*x^7*X[2 ,1,1,2,1,2]-64*x^7-69*x^6*X[2,1,1,2,1,2]+118*x^6+56*x^5*X[2,1,1,2,1,2]-147*x^5-\ 28*x^4*X[2,1,1,2,1,2]+133*x^4+8*x^3*X[2,1,1,2,1,2]-85*x^3-x^2*X[2,1,1,2,1,2]+36 *x^2-9*x+1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [2, 1, 1, 2, 1, 2], equals , - 3/4 + ---- 16 81 21 n The variance equals , - -- + ---- 64 256 183 279 n The , 3, -th moment about the mean is , - --- + ----- 64 2048 12939 1323 2 11073 The , 4, -th moment about the mean is , - ----- + ----- n - ----- n 4096 65536 32768 The compositions of, 9, that yield the, 8, -th largest growth, that is, 1.8922578866301683686, are , [2, 1, 1, 1, 2, 2], [2, 2, 1, 1, 1, 2] Theorem Number, 8, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 1, 1, 2, 2] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 18 16 15 14 13 11 10 9 ) a(n) x = - (x + x - x + 3 x - x + 3 x - 6 x + 11 x / ----- n = 0 8 7 6 5 4 3 2 / - 17 x + 24 x - 34 x + 46 x - 51 x + 41 x - 22 x + 7 x - 1) / ( / 3 16 15 12 11 9 8 7 6 5 (x + x - 1) (x - x + 3 x - 4 x + 7 x - 8 x + 2 x - 2 x + 17 x 4 3 2 - 34 x + 35 x - 21 x + 7 x - 1)) and in Maple format -(x^18+x^16-x^15+3*x^14-x^13+3*x^11-6*x^10+11*x^9-17*x^8+24*x^7-34*x^6+46*x^5-\ 51*x^4+41*x^3-22*x^2+7*x-1)/(x^3+x-1)/(x^16-x^15+3*x^12-4*x^11+7*x^9-8*x^8+2*x^ 7-2*x^6+17*x^5-34*x^4+35*x^3-21*x^2+7*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 504, 986, 1909, 3662, 6972, 13198, 24885, 46812, 87978, 165368, 311096, 585953, 1105098, 2086758, 3944502, 7461996, 14123834, 26741645, 50638763, 95891469, 181568857] The limit of a(n+1)/a(n) as n goes to infinity is 1.89225788663 a(n) is asymptotic to .891218879754*1.89225788663^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [2, 1, 1, 1, 2, 2], denoted by the variable, X[2, 1, 1, 1, 2, 2], is 19 5 19 4 19 3 18 4 17 5 19 2 - (x %1 - 5 x %1 + 10 x %1 + x %1 + x %1 - 10 x %1 18 3 17 4 19 18 2 17 3 16 4 - 4 x %1 - 5 x %1 + 5 x %1 + 6 x %1 + 10 x %1 + 2 x %1 19 18 17 2 16 3 15 4 18 17 - x - 4 x %1 - 10 x %1 - 8 x %1 - 3 x %1 + x + 5 x %1 16 2 15 3 14 4 17 16 15 2 + 12 x %1 + 13 x %1 + 3 x %1 - x - 8 x %1 - 21 x %1 14 3 13 4 16 15 14 2 13 3 - 13 x %1 - 4 x %1 + 2 x + 15 x %1 + 21 x %1 + 14 x %1 12 4 15 14 13 2 12 3 11 4 + 3 x %1 - 4 x - 15 x %1 - 17 x %1 - 11 x %1 - x %1 14 13 12 2 11 3 13 12 11 2 + 4 x + 8 x %1 + 10 x %1 + 5 x %1 - x + x %1 + x %1 10 3 12 11 10 2 11 10 9 2 - x %1 - 3 x - 14 x %1 - 9 x %1 + 9 x + 27 x %1 + 10 x %1 10 9 8 2 9 8 7 2 8 - 17 x - 38 x %1 - 5 x %1 + 28 x + 45 x %1 + x %1 - 41 x 7 7 6 6 5 5 4 - 51 x %1 + 58 x + 52 x %1 - 80 x - 41 x %1 + 97 x + 22 x %1 4 3 3 2 2 / 20 5 - 92 x - 7 x %1 + 63 x + x %1 - 29 x + 8 x - 1) / (x %1 / 20 4 20 3 19 4 18 5 20 2 19 3 - 5 x %1 + 10 x %1 + 2 x %1 + x %1 - 10 x %1 - 8 x %1 18 4 17 5 20 19 2 18 3 17 4 - 6 x %1 - x %1 + 5 x %1 + 12 x %1 + 14 x %1 + 6 x %1 20 19 18 2 17 3 16 4 19 18 - x - 8 x %1 - 16 x %1 - 15 x %1 - 4 x %1 + 2 x + 9 x %1 17 2 16 3 15 4 18 17 16 2 + 19 x %1 + 18 x %1 + 4 x %1 - 2 x - 12 x %1 - 30 x %1 15 3 14 4 17 16 15 2 14 3 - 20 x %1 - 5 x %1 + 3 x + 22 x %1 + 36 x %1 + 21 x %1 13 4 16 15 14 2 13 3 12 4 + 6 x %1 - 6 x - 28 x %1 - 34 x %1 - 21 x %1 - 4 x %1 15 14 13 2 12 3 11 4 14 + 8 x + 25 x %1 + 27 x %1 + 15 x %1 + x %1 - 7 x 13 12 2 11 3 13 12 11 2 - 15 x %1 - 14 x %1 - 6 x %1 + 3 x - x %1 - 3 x %1 10 3 12 11 10 2 11 10 + x %1 + 4 x + 21 x %1 + 15 x %1 - 13 x - 42 x %1 9 2 10 9 8 2 9 8 7 2 - 14 x %1 + 26 x + 58 x %1 + 6 x %1 - 44 x - 69 x %1 - x %1 8 7 7 6 6 5 5 + 65 x + 78 x %1 - 92 x - 77 x %1 + 126 x + 57 x %1 - 148 x 4 4 3 3 2 2 - 28 x %1 + 133 x + 8 x %1 - 85 x - x %1 + 36 x - 9 x + 1) %1 := X[2, 1, 1, 1, 2, 2] and in Maple format -(x^19*X[2,1,1,1,2,2]^5-5*x^19*X[2,1,1,1,2,2]^4+10*x^19*X[2,1,1,1,2,2]^3+x^18*X [2,1,1,1,2,2]^4+x^17*X[2,1,1,1,2,2]^5-10*x^19*X[2,1,1,1,2,2]^2-4*x^18*X[2,1,1,1 ,2,2]^3-5*x^17*X[2,1,1,1,2,2]^4+5*x^19*X[2,1,1,1,2,2]+6*x^18*X[2,1,1,1,2,2]^2+ 10*x^17*X[2,1,1,1,2,2]^3+2*x^16*X[2,1,1,1,2,2]^4-x^19-4*x^18*X[2,1,1,1,2,2]-10* x^17*X[2,1,1,1,2,2]^2-8*x^16*X[2,1,1,1,2,2]^3-3*x^15*X[2,1,1,1,2,2]^4+x^18+5*x^ 17*X[2,1,1,1,2,2]+12*x^16*X[2,1,1,1,2,2]^2+13*x^15*X[2,1,1,1,2,2]^3+3*x^14*X[2, 1,1,1,2,2]^4-x^17-8*x^16*X[2,1,1,1,2,2]-21*x^15*X[2,1,1,1,2,2]^2-13*x^14*X[2,1, 1,1,2,2]^3-4*x^13*X[2,1,1,1,2,2]^4+2*x^16+15*x^15*X[2,1,1,1,2,2]+21*x^14*X[2,1, 1,1,2,2]^2+14*x^13*X[2,1,1,1,2,2]^3+3*x^12*X[2,1,1,1,2,2]^4-4*x^15-15*x^14*X[2, 1,1,1,2,2]-17*x^13*X[2,1,1,1,2,2]^2-11*x^12*X[2,1,1,1,2,2]^3-x^11*X[2,1,1,1,2,2 ]^4+4*x^14+8*x^13*X[2,1,1,1,2,2]+10*x^12*X[2,1,1,1,2,2]^2+5*x^11*X[2,1,1,1,2,2] ^3-x^13+x^12*X[2,1,1,1,2,2]+x^11*X[2,1,1,1,2,2]^2-x^10*X[2,1,1,1,2,2]^3-3*x^12-\ 14*x^11*X[2,1,1,1,2,2]-9*x^10*X[2,1,1,1,2,2]^2+9*x^11+27*x^10*X[2,1,1,1,2,2]+10 *x^9*X[2,1,1,1,2,2]^2-17*x^10-38*x^9*X[2,1,1,1,2,2]-5*x^8*X[2,1,1,1,2,2]^2+28*x ^9+45*x^8*X[2,1,1,1,2,2]+x^7*X[2,1,1,1,2,2]^2-41*x^8-51*x^7*X[2,1,1,1,2,2]+58*x ^7+52*x^6*X[2,1,1,1,2,2]-80*x^6-41*x^5*X[2,1,1,1,2,2]+97*x^5+22*x^4*X[2,1,1,1,2 ,2]-92*x^4-7*x^3*X[2,1,1,1,2,2]+63*x^3+x^2*X[2,1,1,1,2,2]-29*x^2+8*x-1)/(x^20*X [2,1,1,1,2,2]^5-5*x^20*X[2,1,1,1,2,2]^4+10*x^20*X[2,1,1,1,2,2]^3+2*x^19*X[2,1,1 ,1,2,2]^4+x^18*X[2,1,1,1,2,2]^5-10*x^20*X[2,1,1,1,2,2]^2-8*x^19*X[2,1,1,1,2,2]^ 3-6*x^18*X[2,1,1,1,2,2]^4-x^17*X[2,1,1,1,2,2]^5+5*x^20*X[2,1,1,1,2,2]+12*x^19*X [2,1,1,1,2,2]^2+14*x^18*X[2,1,1,1,2,2]^3+6*x^17*X[2,1,1,1,2,2]^4-x^20-8*x^19*X[ 2,1,1,1,2,2]-16*x^18*X[2,1,1,1,2,2]^2-15*x^17*X[2,1,1,1,2,2]^3-4*x^16*X[2,1,1,1 ,2,2]^4+2*x^19+9*x^18*X[2,1,1,1,2,2]+19*x^17*X[2,1,1,1,2,2]^2+18*x^16*X[2,1,1,1 ,2,2]^3+4*x^15*X[2,1,1,1,2,2]^4-2*x^18-12*x^17*X[2,1,1,1,2,2]-30*x^16*X[2,1,1,1 ,2,2]^2-20*x^15*X[2,1,1,1,2,2]^3-5*x^14*X[2,1,1,1,2,2]^4+3*x^17+22*x^16*X[2,1,1 ,1,2,2]+36*x^15*X[2,1,1,1,2,2]^2+21*x^14*X[2,1,1,1,2,2]^3+6*x^13*X[2,1,1,1,2,2] ^4-6*x^16-28*x^15*X[2,1,1,1,2,2]-34*x^14*X[2,1,1,1,2,2]^2-21*x^13*X[2,1,1,1,2,2 ]^3-4*x^12*X[2,1,1,1,2,2]^4+8*x^15+25*x^14*X[2,1,1,1,2,2]+27*x^13*X[2,1,1,1,2,2 ]^2+15*x^12*X[2,1,1,1,2,2]^3+x^11*X[2,1,1,1,2,2]^4-7*x^14-15*x^13*X[2,1,1,1,2,2 ]-14*x^12*X[2,1,1,1,2,2]^2-6*x^11*X[2,1,1,1,2,2]^3+3*x^13-x^12*X[2,1,1,1,2,2]-3 *x^11*X[2,1,1,1,2,2]^2+x^10*X[2,1,1,1,2,2]^3+4*x^12+21*x^11*X[2,1,1,1,2,2]+15*x ^10*X[2,1,1,1,2,2]^2-13*x^11-42*x^10*X[2,1,1,1,2,2]-14*x^9*X[2,1,1,1,2,2]^2+26* x^10+58*x^9*X[2,1,1,1,2,2]+6*x^8*X[2,1,1,1,2,2]^2-44*x^9-69*x^8*X[2,1,1,1,2,2]- x^7*X[2,1,1,1,2,2]^2+65*x^8+78*x^7*X[2,1,1,1,2,2]-92*x^7-77*x^6*X[2,1,1,1,2,2]+ 126*x^6+57*x^5*X[2,1,1,1,2,2]-148*x^5-28*x^4*X[2,1,1,1,2,2]+133*x^4+8*x^3*X[2,1 ,1,1,2,2]-85*x^3-x^2*X[2,1,1,1,2,2]+36*x^2-9*x+1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [2, 1, 1, 1, 2, 2], equals , - 3/4 + ---- 16 81 21 n The variance equals , - -- + ---- 64 256 369 279 n The , 3, -th moment about the mean is , - --- + ----- 128 2048 13635 1323 2 11073 The , 4, -th moment about the mean is , - ----- + ----- n - ----- n 4096 65536 32768 The compositions of, 9, that yield the, 9, -th largest growth, that is, 1.8923110706522823122, are , [1, 2, 1, 1, 2, 2], [1, 2, 2, 1, 1, 2], [2, 1, 1, 2, 2, 1], [2, 2, 1, 1, 2, 1] Theorem Number, 9, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 2, 1, 1, 2, 2] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 12 11 10 9 8 7 6 5 4 ) a(n) x = (x - x + 2 x - x + 2 x - x + 3 x - 7 x + 12 x / ----- n = 0 3 2 / - 14 x + 11 x - 5 x + 1) / ((-1 + x) / 11 10 9 8 7 6 5 4 3 2 (x - x + x - x + x - x + 4 x - 8 x + 11 x - 10 x + 5 x - 1)) and in Maple format (x^12-x^11+2*x^10-x^9+2*x^8-x^7+3*x^6-7*x^5+12*x^4-14*x^3+11*x^2-5*x+1)/(-1+x)/ (x^11-x^10+x^9-x^8+x^7-x^6+4*x^5-8*x^4+11*x^3-10*x^2+5*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 504, 986, 1909, 3662, 6973, 13208, 24938, 47015, 88614, 167107, 315408, 595889, 1126729, 2131781, 4034921, 7638532, 14461138, 27376121, 51820096, 98079193, 185615082] The limit of a(n+1)/a(n) as n goes to infinity is 1.89231107065 a(n) is asymptotic to .909398372273*1.89231107065^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [1, 2, 1, 1, 2, 2], denoted by the variable, X[1, 2, 1, 1, 2, 2], is 13 3 13 2 12 3 13 12 2 11 3 13 (x %1 - 3 x %1 - x %1 + 3 x %1 + 4 x %1 + 2 x %1 - x 12 11 2 10 3 12 11 10 2 9 3 - 5 x %1 - 7 x %1 - 2 x %1 + 2 x + 8 x %1 + 7 x %1 + x %1 11 10 9 2 10 9 8 2 9 8 - 3 x - 8 x %1 - 4 x %1 + 3 x + 6 x %1 + x %1 - 3 x - 4 x %1 8 7 7 6 6 5 5 4 + 3 x + 4 x %1 - 4 x - 9 x %1 + 10 x + 13 x %1 - 19 x - 11 x %1 4 3 3 2 2 / + 26 x + 5 x %1 - 25 x - x %1 + 16 x - 6 x + 1) / ((-1 + x) ( / 12 3 12 2 11 3 12 11 2 10 3 12 x %1 - 3 x %1 - x %1 + 3 x %1 + 4 x %1 + 2 x %1 - x 11 10 2 9 3 11 10 9 2 10 - 5 x %1 - 6 x %1 - x %1 + 2 x + 6 x %1 + 4 x %1 - 2 x 9 8 2 9 8 8 7 7 6 - 5 x %1 - x %1 + 2 x + 3 x %1 - 2 x - 2 x %1 + 2 x + 5 x %1 6 5 5 4 4 3 3 2 - 5 x - 10 x %1 + 12 x + 10 x %1 - 19 x - 5 x %1 + 21 x + x %1 2 - 15 x + 6 x - 1)) %1 := X[1, 2, 1, 1, 2, 2] and in Maple format (x^13*X[1,2,1,1,2,2]^3-3*x^13*X[1,2,1,1,2,2]^2-x^12*X[1,2,1,1,2,2]^3+3*x^13*X[1 ,2,1,1,2,2]+4*x^12*X[1,2,1,1,2,2]^2+2*x^11*X[1,2,1,1,2,2]^3-x^13-5*x^12*X[1,2,1 ,1,2,2]-7*x^11*X[1,2,1,1,2,2]^2-2*x^10*X[1,2,1,1,2,2]^3+2*x^12+8*x^11*X[1,2,1,1 ,2,2]+7*x^10*X[1,2,1,1,2,2]^2+x^9*X[1,2,1,1,2,2]^3-3*x^11-8*x^10*X[1,2,1,1,2,2] -4*x^9*X[1,2,1,1,2,2]^2+3*x^10+6*x^9*X[1,2,1,1,2,2]+x^8*X[1,2,1,1,2,2]^2-3*x^9-\ 4*x^8*X[1,2,1,1,2,2]+3*x^8+4*x^7*X[1,2,1,1,2,2]-4*x^7-9*x^6*X[1,2,1,1,2,2]+10*x ^6+13*x^5*X[1,2,1,1,2,2]-19*x^5-11*x^4*X[1,2,1,1,2,2]+26*x^4+5*x^3*X[1,2,1,1,2, 2]-25*x^3-x^2*X[1,2,1,1,2,2]+16*x^2-6*x+1)/(-1+x)/(x^12*X[1,2,1,1,2,2]^3-3*x^12 *X[1,2,1,1,2,2]^2-x^11*X[1,2,1,1,2,2]^3+3*x^12*X[1,2,1,1,2,2]+4*x^11*X[1,2,1,1, 2,2]^2+2*x^10*X[1,2,1,1,2,2]^3-x^12-5*x^11*X[1,2,1,1,2,2]-6*x^10*X[1,2,1,1,2,2] ^2-x^9*X[1,2,1,1,2,2]^3+2*x^11+6*x^10*X[1,2,1,1,2,2]+4*x^9*X[1,2,1,1,2,2]^2-2*x ^10-5*x^9*X[1,2,1,1,2,2]-x^8*X[1,2,1,1,2,2]^2+2*x^9+3*x^8*X[1,2,1,1,2,2]-2*x^8-\ 2*x^7*X[1,2,1,1,2,2]+2*x^7+5*x^6*X[1,2,1,1,2,2]-5*x^6-10*x^5*X[1,2,1,1,2,2]+12* x^5+10*x^4*X[1,2,1,1,2,2]-19*x^4-5*x^3*X[1,2,1,1,2,2]+21*x^3+x^2*X[1,2,1,1,2,2] -15*x^2+6*x-1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [1, 2, 1, 1, 2, 2], equals , - 3/4 + ---- 16 77 21 n The variance equals , - -- + ---- 64 256 333 279 n The , 3, -th moment about the mean is , - --- + ----- 128 2048 1323 2 9969 11243 The , 4, -th moment about the mean is , ----- n - ----- n - ----- 65536 32768 4096 The compositions of, 9, that yield the, 10, -th largest growth, that is, 1.9087907387871591034, are , [1, 2, 1, 2, 1, 2], [2, 1, 2, 1, 2, 1] Theorem Number, 10, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 2, 1, 2, 1, 2] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 2 ) a(n) x = (x - x + 1) ( / ----- n = 0 14 12 10 9 8 7 6 4 3 2 x + x - x + 2 x - 2 x + x + 2 x - 11 x + 19 x - 15 x + 6 x - 1 / 5 4 3 2 ) / ((-1 + x) (x - x + 2 x - 3 x + 3 x - 1) / 10 8 7 5 4 3 2 (x - x + 2 x - 3 x + 2 x + 3 x - 6 x + 4 x - 1)) and in Maple format (x^2-x+1)*(x^14+x^12-x^10+2*x^9-2*x^8+x^7+2*x^6-11*x^4+19*x^3-15*x^2+6*x-1)/(-1 +x)/(x^5-x^4+2*x^3-3*x^2+3*x-1)/(x^10-x^8+2*x^7-3*x^5+2*x^4+3*x^3-6*x^2+4*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 504, 986, 1909, 3664, 6990, 13291, 25244, 47964, 91234, 173769, 331361, 632425, 1207657, 2306605, 4405581, 8413543, 16064967, 30669666, 58544033, 111743162, 213276372] The limit of a(n+1)/a(n) as n goes to infinity is 1.90879073879 a(n) is asymptotic to .805564293796*1.90879073879^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [1, 2, 1, 2, 1, 2], denoted by the variable, X[1, 2, 1, 2, 1, 2], is 16 3 16 2 15 3 16 15 2 14 3 16 (x %1 - 3 x %1 - x %1 + 3 x %1 + 3 x %1 + 2 x %1 - x 15 14 2 15 14 13 2 12 3 14 - 3 x %1 - 6 x %1 + x + 6 x %1 + x %1 - 2 x %1 - 2 x 13 12 2 11 3 13 12 11 2 - 2 x %1 + 4 x %1 + 4 x %1 + x - 2 x %1 - 11 x %1 10 3 11 10 2 9 3 11 10 10 - 3 x %1 + 10 x %1 + 9 x %1 + x %1 - 3 x - 11 x %1 + 5 x 9 8 2 9 8 7 2 8 7 + 4 x %1 - 9 x %1 - 5 x + 8 x %1 + 10 x %1 + x - 12 x %1 6 2 7 6 5 2 6 5 5 4 - 5 x %1 + x + 3 x %1 + x %1 + 9 x + 8 x %1 - 30 x - 10 x %1 4 3 3 2 2 / + 45 x + 5 x %1 - 40 x - x %1 + 22 x - 7 x + 1) / ((-1 + x) / 5 5 4 4 3 3 2 2 (x %1 - x - x %1 + x + 2 x %1 - 2 x - x %1 + 3 x - 3 x + 1) ( 10 2 10 10 8 2 8 7 2 8 7 7 x %1 - 2 x %1 + x - x %1 + 2 x %1 + x %1 - x - 3 x %1 + 2 x 5 5 4 4 3 3 2 + 3 x %1 - 3 x - 3 x %1 + 2 x + x %1 + 3 x - 6 x + 4 x - 1)) %1 := X[1, 2, 1, 2, 1, 2] and in Maple format (x^16*X[1,2,1,2,1,2]^3-3*x^16*X[1,2,1,2,1,2]^2-x^15*X[1,2,1,2,1,2]^3+3*x^16*X[1 ,2,1,2,1,2]+3*x^15*X[1,2,1,2,1,2]^2+2*x^14*X[1,2,1,2,1,2]^3-x^16-3*x^15*X[1,2,1 ,2,1,2]-6*x^14*X[1,2,1,2,1,2]^2+x^15+6*x^14*X[1,2,1,2,1,2]+x^13*X[1,2,1,2,1,2]^ 2-2*x^12*X[1,2,1,2,1,2]^3-2*x^14-2*x^13*X[1,2,1,2,1,2]+4*x^12*X[1,2,1,2,1,2]^2+ 4*x^11*X[1,2,1,2,1,2]^3+x^13-2*x^12*X[1,2,1,2,1,2]-11*x^11*X[1,2,1,2,1,2]^2-3*x ^10*X[1,2,1,2,1,2]^3+10*x^11*X[1,2,1,2,1,2]+9*x^10*X[1,2,1,2,1,2]^2+x^9*X[1,2,1 ,2,1,2]^3-3*x^11-11*x^10*X[1,2,1,2,1,2]+5*x^10+4*x^9*X[1,2,1,2,1,2]-9*x^8*X[1,2 ,1,2,1,2]^2-5*x^9+8*x^8*X[1,2,1,2,1,2]+10*x^7*X[1,2,1,2,1,2]^2+x^8-12*x^7*X[1,2 ,1,2,1,2]-5*x^6*X[1,2,1,2,1,2]^2+x^7+3*x^6*X[1,2,1,2,1,2]+x^5*X[1,2,1,2,1,2]^2+ 9*x^6+8*x^5*X[1,2,1,2,1,2]-30*x^5-10*x^4*X[1,2,1,2,1,2]+45*x^4+5*x^3*X[1,2,1,2, 1,2]-40*x^3-x^2*X[1,2,1,2,1,2]+22*x^2-7*x+1)/(-1+x)/(x^5*X[1,2,1,2,1,2]-x^5-x^4 *X[1,2,1,2,1,2]+x^4+2*x^3*X[1,2,1,2,1,2]-2*x^3-x^2*X[1,2,1,2,1,2]+3*x^2-3*x+1)/ (x^10*X[1,2,1,2,1,2]^2-2*x^10*X[1,2,1,2,1,2]+x^10-x^8*X[1,2,1,2,1,2]^2+2*x^8*X[ 1,2,1,2,1,2]+x^7*X[1,2,1,2,1,2]^2-x^8-3*x^7*X[1,2,1,2,1,2]+2*x^7+3*x^5*X[1,2,1, 2,1,2]-3*x^5-3*x^4*X[1,2,1,2,1,2]+2*x^4+x^3*X[1,2,1,2,1,2]+3*x^3-6*x^2+4*x-1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [1, 2, 1, 2, 1, 2], equals , - 3/4 + ---- 16 23 25 n The variance equals , - -- + ---- 16 256 1617 351 n The , 3, -th moment about the mean is , - ---- + ----- 512 2048 1875 2 185 18181 The , 4, -th moment about the mean is , ----- n - --- - ----- n 65536 256 32768 The compositions of, 9, that yield the, 11, -th largest growth, that is, 1.9275619754829253043, are , [1, 1, 1, 1, 5], [1, 1, 1, 5, 1], [1, 1, 5, 1, 1], [1, 5, 1, 1, 1], [5, 1, 1, 1, 1] Theorem Number, 11, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 1, 1, 1, 5] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 3 4 3 2 \ n (x - x + 1) (2 x - 2 x + 3 x - 3 x + 1) ) a(n) x = - ------------------------------------------- / 4 3 2 4 ----- (x + x + x + x - 1) (-1 + x) n = 0 and in Maple format -(x^3-x+1)*(2*x^4-2*x^3+3*x^2-3*x+1)/(x^4+x^3+x^2+x-1)/(-1+x)^4 The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 505, 994, 1946, 3793, 7368, 14277, 27616, 53353, 102992, 198708, 383245, 738995, 1424774, 2746710, 5294884, 10206714, 19674644, 37924746, 73103036, 140911465, 271616517] The limit of a(n+1)/a(n) as n goes to infinity is 1.92756197548 a(n) is asymptotic to .765081505018*1.92756197548^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [1, 1, 1, 1, 5], denoted by the variable, X[1, 1, 1, 1, 5], is 7 7 6 6 - (2 x X[1, 1, 1, 1, 5] - 2 x - 2 x X[1, 1, 1, 1, 5] + 2 x 5 5 4 3 2 / 3 + x X[1, 1, 1, 1, 5] - x - x + 4 x - 6 x + 4 x - 1) / ((-1 + x) / 5 5 (x X[1, 1, 1, 1, 5] - x + 2 x - 1)) and in Maple format -(2*x^7*X[1,1,1,1,5]-2*x^7-2*x^6*X[1,1,1,1,5]+2*x^6+x^5*X[1,1,1,1,5]-x^5-x^4+4* x^3-6*x^2+4*x-1)/(-1+x)^3/(x^5*X[1,1,1,1,5]-x^5+2*x-1) Furthermore, the expectation of the random variable number of occurences (by\ 11 n containment) of , [1, 1, 1, 1, 5], equals , - -- + ---- 32 32 227 23 n The variance equals , - ---- + ---- 1024 1024 1083 171 n The , 3, -th moment about the mean is , - ----- + ----- 16384 16384 235345 1587 2 8485 The , 4, -th moment about the mean is , ------- + ------- n - ------ n 1048576 1048576 262144 The compositions of, 9, that yield the, 12, -th largest growth, that is, 1.9331849818995204468, are , [2, 1, 1, 1, 4], [4, 1, 1, 1, 2] Theorem Number, 12, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 1, 1, 4] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 34 30 29 28 27 26 25 24 23 ) a(n) x = - (x + 2 x + x - x + x + x + x - 3 x + x / ----- n = 0 21 20 19 18 17 16 15 14 13 12 + x + x - 3 x - x + x + 5 x - 7 x + 3 x - 2 x + 5 x 11 9 8 7 6 5 4 3 2 - 4 x + 7 x - 18 x + 24 x - 23 x + 27 x - 36 x + 35 x - 21 x / 2 5 3 2 + 7 x - 1) / ((x - x + 1) (x - x + 1) (x + x - 1) / 5 4 (x - x + 2 x - 1) 20 14 13 12 11 8 7 4 3 2 (x - x + x + 2 x - 2 x - x + x + x - 4 x + 6 x - 4 x + 1)) and in Maple format -(x^34+2*x^30+x^29-x^28+x^27+x^26+x^25-3*x^24+x^23+x^21+x^20-3*x^19-x^18+x^17+5 *x^16-7*x^15+3*x^14-2*x^13+5*x^12-4*x^11+7*x^9-18*x^8+24*x^7-23*x^6+27*x^5-36*x ^4+35*x^3-21*x^2+7*x-1)/(x^2-x+1)/(x^5-x+1)/(x^3+x^2-1)/(x^5-x^4+2*x-1)/(x^20-x ^14+x^13+2*x^12-2*x^11-x^8+x^7+x^4-4*x^3+6*x^2-4*x+1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 505, 994, 1946, 3793, 7368, 14277, 27617, 53363, 103049, 198952, 384116, 741735, 1432633, 2767735, 5348192, 10336306, 19979327, 38622105, 74664785, 144347422, 279066317] The limit of a(n+1)/a(n) as n goes to infinity is 1.93318498190 a(n) is asymptotic to .720431945925*1.93318498190^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [2, 1, 1, 1, 4], denoted by the variable, X[2, 1, 1, 1, 4], is 34 7 34 34 34 34 - (x X[2, 1, 1, 1, 4] - 7 x %5 + 21 x %4 - 35 x %3 + 35 x %2 34 30 34 30 29 34 - 21 x %1 - 2 x %5 + 7 x X[2, 1, 1, 1, 4] + 12 x %4 - x %5 - x 30 29 28 30 29 28 - 30 x %3 + 6 x %4 + x %5 + 40 x %2 - 15 x %3 - 6 x %4 27 30 29 28 27 - x %5 - 30 x %1 + 20 x %2 + 15 x %3 + 6 x %4 30 29 28 27 26 + 12 x X[2, 1, 1, 1, 4] - 15 x %1 - 20 x %2 - 15 x %3 + x %4 30 29 28 27 26 - 2 x + 6 x X[2, 1, 1, 1, 4] + 15 x %1 + 20 x %2 - 5 x %3 25 29 28 27 26 25 + x %4 - x - 6 x X[2, 1, 1, 1, 4] - 15 x %1 + 10 x %2 - 5 x %3 24 28 27 26 25 - 3 x %4 + x + 6 x X[2, 1, 1, 1, 4] - 10 x %1 + 10 x %2 24 23 27 26 25 + 15 x %3 + x %4 - x + 5 x X[2, 1, 1, 1, 4] - 10 x %1 24 23 26 25 24 - 30 x %2 - 5 x %3 - x + 5 x X[2, 1, 1, 1, 4] + 30 x %1 23 25 24 23 21 24 + 10 x %2 - x - 15 x X[2, 1, 1, 1, 4] - 10 x %1 - x %3 + 3 x 23 21 20 23 21 20 + 5 x X[2, 1, 1, 1, 4] + 4 x %2 - x %3 - x - 6 x %1 + 4 x %2 19 21 20 19 18 + 4 x %3 + 4 x X[2, 1, 1, 1, 4] - 6 x %1 - 15 x %2 - 3 x %3 21 20 19 18 17 20 - x + 4 x X[2, 1, 1, 1, 4] + 21 x %1 + 8 x %2 + x %3 - x 19 18 17 19 16 18 - 13 x X[2, 1, 1, 1, 4] - 6 x %1 - 2 x %2 + 3 x + 5 x %2 + x 17 16 15 17 + 2 x X[2, 1, 1, 1, 4] - 15 x %1 - 8 x %2 - x 16 15 14 16 + 15 x X[2, 1, 1, 1, 4] + 23 x %1 + 5 x %2 - 5 x 15 14 13 15 - 22 x X[2, 1, 1, 1, 4] - 13 x %1 - x %2 + 7 x 14 13 14 13 + 11 x X[2, 1, 1, 1, 4] + 4 x %1 - 3 x - 5 x X[2, 1, 1, 1, 4] 12 13 12 11 12 - 5 x %1 + 2 x + 10 x X[2, 1, 1, 1, 4] + 4 x %1 - 5 x 11 10 11 10 - 8 x X[2, 1, 1, 1, 4] - x %1 + 4 x + x X[2, 1, 1, 1, 4] 9 9 8 8 + 7 x X[2, 1, 1, 1, 4] - 7 x - 18 x X[2, 1, 1, 1, 4] + 18 x 7 7 6 6 + 23 x X[2, 1, 1, 1, 4] - 24 x - 16 x X[2, 1, 1, 1, 4] + 23 x 5 5 4 4 3 + 6 x X[2, 1, 1, 1, 4] - 27 x - x X[2, 1, 1, 1, 4] + 36 x - 35 x 2 / + 21 x - 7 x + 1) / ( / 5 5 4 4 (x X[2, 1, 1, 1, 4] - x - x X[2, 1, 1, 1, 4] + x - 2 x + 1) 5 5 5 5 (x X[2, 1, 1, 1, 4] - x + x - 1) (x X[2, 1, 1, 1, 4] - x - x + 1) ( 20 20 20 20 20 14 x %3 - 4 x %2 + 6 x %1 - 4 x X[2, 1, 1, 1, 4] + x + x %2 14 13 14 13 14 - 3 x %1 - x %2 + 3 x X[2, 1, 1, 1, 4] + 3 x %1 - x 13 12 13 12 - 3 x X[2, 1, 1, 1, 4] + 2 x %1 + x - 4 x X[2, 1, 1, 1, 4] 11 12 11 11 8 - 2 x %1 + 2 x + 4 x X[2, 1, 1, 1, 4] - 2 x + x X[2, 1, 1, 1, 4] 8 7 7 4 3 2 - x - x X[2, 1, 1, 1, 4] + x + x - 4 x + 6 x - 4 x + 1)) 2 %1 := X[2, 1, 1, 1, 4] 3 %2 := X[2, 1, 1, 1, 4] 4 %3 := X[2, 1, 1, 1, 4] 5 %4 := X[2, 1, 1, 1, 4] 6 %5 := X[2, 1, 1, 1, 4] and in Maple format -(x^34*X[2,1,1,1,4]^7-7*x^34*X[2,1,1,1,4]^6+21*x^34*X[2,1,1,1,4]^5-35*x^34*X[2, 1,1,1,4]^4+35*x^34*X[2,1,1,1,4]^3-21*x^34*X[2,1,1,1,4]^2-2*x^30*X[2,1,1,1,4]^6+ 7*x^34*X[2,1,1,1,4]+12*x^30*X[2,1,1,1,4]^5-x^29*X[2,1,1,1,4]^6-x^34-30*x^30*X[2 ,1,1,1,4]^4+6*x^29*X[2,1,1,1,4]^5+x^28*X[2,1,1,1,4]^6+40*x^30*X[2,1,1,1,4]^3-15 *x^29*X[2,1,1,1,4]^4-6*x^28*X[2,1,1,1,4]^5-x^27*X[2,1,1,1,4]^6-30*x^30*X[2,1,1, 1,4]^2+20*x^29*X[2,1,1,1,4]^3+15*x^28*X[2,1,1,1,4]^4+6*x^27*X[2,1,1,1,4]^5+12*x ^30*X[2,1,1,1,4]-15*x^29*X[2,1,1,1,4]^2-20*x^28*X[2,1,1,1,4]^3-15*x^27*X[2,1,1, 1,4]^4+x^26*X[2,1,1,1,4]^5-2*x^30+6*x^29*X[2,1,1,1,4]+15*x^28*X[2,1,1,1,4]^2+20 *x^27*X[2,1,1,1,4]^3-5*x^26*X[2,1,1,1,4]^4+x^25*X[2,1,1,1,4]^5-x^29-6*x^28*X[2, 1,1,1,4]-15*x^27*X[2,1,1,1,4]^2+10*x^26*X[2,1,1,1,4]^3-5*x^25*X[2,1,1,1,4]^4-3* x^24*X[2,1,1,1,4]^5+x^28+6*x^27*X[2,1,1,1,4]-10*x^26*X[2,1,1,1,4]^2+10*x^25*X[2 ,1,1,1,4]^3+15*x^24*X[2,1,1,1,4]^4+x^23*X[2,1,1,1,4]^5-x^27+5*x^26*X[2,1,1,1,4] -10*x^25*X[2,1,1,1,4]^2-30*x^24*X[2,1,1,1,4]^3-5*x^23*X[2,1,1,1,4]^4-x^26+5*x^ 25*X[2,1,1,1,4]+30*x^24*X[2,1,1,1,4]^2+10*x^23*X[2,1,1,1,4]^3-x^25-15*x^24*X[2, 1,1,1,4]-10*x^23*X[2,1,1,1,4]^2-x^21*X[2,1,1,1,4]^4+3*x^24+5*x^23*X[2,1,1,1,4]+ 4*x^21*X[2,1,1,1,4]^3-x^20*X[2,1,1,1,4]^4-x^23-6*x^21*X[2,1,1,1,4]^2+4*x^20*X[2 ,1,1,1,4]^3+4*x^19*X[2,1,1,1,4]^4+4*x^21*X[2,1,1,1,4]-6*x^20*X[2,1,1,1,4]^2-15* x^19*X[2,1,1,1,4]^3-3*x^18*X[2,1,1,1,4]^4-x^21+4*x^20*X[2,1,1,1,4]+21*x^19*X[2, 1,1,1,4]^2+8*x^18*X[2,1,1,1,4]^3+x^17*X[2,1,1,1,4]^4-x^20-13*x^19*X[2,1,1,1,4]-\ 6*x^18*X[2,1,1,1,4]^2-2*x^17*X[2,1,1,1,4]^3+3*x^19+5*x^16*X[2,1,1,1,4]^3+x^18+2 *x^17*X[2,1,1,1,4]-15*x^16*X[2,1,1,1,4]^2-8*x^15*X[2,1,1,1,4]^3-x^17+15*x^16*X[ 2,1,1,1,4]+23*x^15*X[2,1,1,1,4]^2+5*x^14*X[2,1,1,1,4]^3-5*x^16-22*x^15*X[2,1,1, 1,4]-13*x^14*X[2,1,1,1,4]^2-x^13*X[2,1,1,1,4]^3+7*x^15+11*x^14*X[2,1,1,1,4]+4*x ^13*X[2,1,1,1,4]^2-3*x^14-5*x^13*X[2,1,1,1,4]-5*x^12*X[2,1,1,1,4]^2+2*x^13+10*x ^12*X[2,1,1,1,4]+4*x^11*X[2,1,1,1,4]^2-5*x^12-8*x^11*X[2,1,1,1,4]-x^10*X[2,1,1, 1,4]^2+4*x^11+x^10*X[2,1,1,1,4]+7*x^9*X[2,1,1,1,4]-7*x^9-18*x^8*X[2,1,1,1,4]+18 *x^8+23*x^7*X[2,1,1,1,4]-24*x^7-16*x^6*X[2,1,1,1,4]+23*x^6+6*x^5*X[2,1,1,1,4]-\ 27*x^5-x^4*X[2,1,1,1,4]+36*x^4-35*x^3+21*x^2-7*x+1)/(x^5*X[2,1,1,1,4]-x^5-x^4*X [2,1,1,1,4]+x^4-2*x+1)/(x^5*X[2,1,1,1,4]-x^5+x-1)/(x^5*X[2,1,1,1,4]-x^5-x+1)/(x ^20*X[2,1,1,1,4]^4-4*x^20*X[2,1,1,1,4]^3+6*x^20*X[2,1,1,1,4]^2-4*x^20*X[2,1,1,1 ,4]+x^20+x^14*X[2,1,1,1,4]^3-3*x^14*X[2,1,1,1,4]^2-x^13*X[2,1,1,1,4]^3+3*x^14*X [2,1,1,1,4]+3*x^13*X[2,1,1,1,4]^2-x^14-3*x^13*X[2,1,1,1,4]+2*x^12*X[2,1,1,1,4]^ 2+x^13-4*x^12*X[2,1,1,1,4]-2*x^11*X[2,1,1,1,4]^2+2*x^12+4*x^11*X[2,1,1,1,4]-2*x ^11+x^8*X[2,1,1,1,4]-x^8-x^7*X[2,1,1,1,4]+x^7+x^4-4*x^3+6*x^2-4*x+1) Furthermore, the expectation of the random variable number of occurences (by\ 11 n containment) of , [2, 1, 1, 1, 4], equals , - -- + ---- 32 32 311 27 n The variance equals , - ---- + ---- 1024 1024 3801 297 n The , 3, -th moment about the mean is , - ----- + ----- 16384 16384 2187 2 173065 11145 The , 4, -th moment about the mean is , ------- n + ------- - ------ n 1048576 1048576 262144 The compositions of, 9, that yield the, 13, -th largest growth, that is, 1.9407101328380924652, are , [1, 2, 1, 2, 3], [1, 2, 1, 3, 2], [1, 2, 2, 1, 3], [1, 2, 3, 1, 2], [1, 3, 1, 2, 2], [1, 3, 2, 1, 2], [2, 1, 2, 3, 1], [2, 1, 3, 2, 1], [2, 2, 1, 3, 1], [2, 3, 1, 2, 1], [3, 1, 2, 2, 1], [3, 2, 1, 2, 1] Theorem Number, 13, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 2, 1, 2, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n ) a(n) x = / ----- n = 0 11 10 9 8 6 5 4 3 2 x - x + x + x - 2 x + 4 x - 2 x - 3 x + 6 x - 4 x + 1 ----------------------------------------------------------------- 10 9 7 6 5 4 3 2 (-1 + x) (x - x + x + x - 3 x + 3 x + x - 5 x + 4 x - 1) and in Maple format (x^11-x^10+x^9+x^8-2*x^6+4*x^5-2*x^4-3*x^3+6*x^2-4*x+1)/(-1+x)/(x^10-x^9+x^7+x^ 6-3*x^5+3*x^4+x^3-5*x^2+4*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 505, 994, 1946, 3794, 7376, 14316, 27764, 53833, 104389, 202467, 392783, 762135, 1478994, 2870335, 5570737, 10811733, 20983342, 40723827, 79034590, 153384570, 297675706] The limit of a(n+1)/a(n) as n goes to infinity is 1.94071013284 a(n) is asymptotic to .683783349401*1.94071013284^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [1, 2, 1, 2, 3], denoted by the variable, X[1, 2, 1, 2, 3], is 11 11 10 11 10 (x %1 - 2 x X[1, 2, 1, 2, 3] - x %1 + x + 2 x X[1, 2, 1, 2, 3] 9 10 9 9 8 8 + x %1 - x - 2 x X[1, 2, 1, 2, 3] + x - x X[1, 2, 1, 2, 3] + x 6 6 5 5 + 2 x X[1, 2, 1, 2, 3] - 2 x - 4 x X[1, 2, 1, 2, 3] + 4 x 4 4 3 3 2 + 3 x X[1, 2, 1, 2, 3] - 2 x - x X[1, 2, 1, 2, 3] - 3 x + 6 x - 4 x / 10 10 9 10 + 1) / ((-1 + x) (x %1 - 2 x X[1, 2, 1, 2, 3] - x %1 + x / 9 9 7 7 + 2 x X[1, 2, 1, 2, 3] - x - x X[1, 2, 1, 2, 3] + x 6 6 5 5 - x X[1, 2, 1, 2, 3] + x + 3 x X[1, 2, 1, 2, 3] - 3 x 4 4 3 3 2 - 3 x X[1, 2, 1, 2, 3] + 3 x + x X[1, 2, 1, 2, 3] + x - 5 x + 4 x - 1 )) 2 %1 := X[1, 2, 1, 2, 3] and in Maple format (x^11*X[1,2,1,2,3]^2-2*x^11*X[1,2,1,2,3]-x^10*X[1,2,1,2,3]^2+x^11+2*x^10*X[1,2, 1,2,3]+x^9*X[1,2,1,2,3]^2-x^10-2*x^9*X[1,2,1,2,3]+x^9-x^8*X[1,2,1,2,3]+x^8+2*x^ 6*X[1,2,1,2,3]-2*x^6-4*x^5*X[1,2,1,2,3]+4*x^5+3*x^4*X[1,2,1,2,3]-2*x^4-x^3*X[1, 2,1,2,3]-3*x^3+6*x^2-4*x+1)/(-1+x)/(x^10*X[1,2,1,2,3]^2-2*x^10*X[1,2,1,2,3]-x^9 *X[1,2,1,2,3]^2+x^10+2*x^9*X[1,2,1,2,3]-x^9-x^7*X[1,2,1,2,3]+x^7-x^6*X[1,2,1,2, 3]+x^6+3*x^5*X[1,2,1,2,3]-3*x^5-3*x^4*X[1,2,1,2,3]+3*x^4+x^3*X[1,2,1,2,3]+x^3-5 *x^2+4*x-1) Furthermore, the expectation of the random variable number of occurences (by\ 11 n containment) of , [1, 2, 1, 2, 3], equals , - -- + ---- 32 32 431 35 n The variance equals , - ---- + ---- 1024 1024 10137 681 n The , 3, -th moment about the mean is , - ----- + ----- 16384 16384 617927 3675 2 7081 The , 4, -th moment about the mean is , - ------- + ------- n - ------ n 1048576 1048576 262144 The compositions of, 9, that yield the, 14, -th largest growth, that is, 1.9409751179367153000, are , [2, 1, 1, 2, 3], [2, 1, 1, 3, 2], [2, 2, 1, 1, 3], [2, 3, 1, 1, 2], [3, 1, 1, 2, 2], [3, 2, 1, 1, 2] Theorem Number, 14, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 1, 2, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n ) a(n) x = / ----- n = 0 13 11 10 7 6 5 4 3 2 x + x + x + x - 2 x + 4 x - 2 x - 3 x + 6 x - 4 x + 1 - ---------------------------------------------------------------------- 14 12 10 8 7 6 5 4 3 2 x + x - x + x - x + 4 x - 6 x + 2 x + 6 x - 9 x + 5 x - 1 and in Maple format -(x^13+x^11+x^10+x^7-2*x^6+4*x^5-2*x^4-3*x^3+6*x^2-4*x+1)/(x^14+x^12-x^10+x^8-x ^7+4*x^6-6*x^5+2*x^4+6*x^3-9*x^2+5*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 505, 994, 1946, 3794, 7375, 14308, 27728, 53712, 104049, 201620, 390846, 757976, 1470478, 2853511, 5538369, 10750613, 20869316, 40512499, 78643837, 152661544, 296334509] The limit of a(n+1)/a(n) as n goes to infinity is 1.94097511794 a(n) is asymptotic to .677953839844*1.94097511794^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [2, 1, 1, 2, 3], denoted by the variable, X[2, 1, 1, 2, 3], is 14 14 13 14 13 12 - (x %2 - 3 x %1 - x %2 + 3 x X[2, 1, 1, 2, 3] + 3 x %1 + x %2 14 13 12 13 12 - x - 3 x X[2, 1, 1, 2, 3] - 3 x %1 + x + 3 x X[2, 1, 1, 2, 3] 12 10 10 10 8 8 - x + x %1 - 2 x X[2, 1, 1, 2, 3] + x + x X[2, 1, 1, 2, 3] - x 7 7 6 6 - 3 x X[2, 1, 1, 2, 3] + 3 x + 6 x X[2, 1, 1, 2, 3] - 6 x 5 5 4 4 - 7 x X[2, 1, 1, 2, 3] + 6 x + 4 x X[2, 1, 1, 2, 3] + x 3 3 2 / 15 15 - x X[2, 1, 1, 2, 3] - 9 x + 10 x - 5 x + 1) / (x %2 - 3 x %1 / 14 15 14 13 15 - x %2 + 3 x X[2, 1, 1, 2, 3] + 3 x %1 + 2 x %2 - x 14 13 12 14 - 3 x X[2, 1, 1, 2, 3] - 5 x %1 - x %2 + x 13 12 13 12 + 4 x X[2, 1, 1, 2, 3] + 3 x %1 - x - 3 x X[2, 1, 1, 2, 3] 11 12 11 10 11 + x %1 + x - 2 x X[2, 1, 1, 2, 3] - x %1 + x 10 10 9 9 + 2 x X[2, 1, 1, 2, 3] - x + x X[2, 1, 1, 2, 3] - x 8 8 7 7 - 2 x X[2, 1, 1, 2, 3] + 2 x + 5 x X[2, 1, 1, 2, 3] - 5 x 6 6 5 5 - 10 x X[2, 1, 1, 2, 3] + 10 x + 10 x X[2, 1, 1, 2, 3] - 8 x 4 4 3 3 2 - 5 x X[2, 1, 1, 2, 3] - 4 x + x X[2, 1, 1, 2, 3] + 15 x - 14 x + 6 x - 1) 2 %1 := X[2, 1, 1, 2, 3] 3 %2 := X[2, 1, 1, 2, 3] and in Maple format -(x^14*X[2,1,1,2,3]^3-3*x^14*X[2,1,1,2,3]^2-x^13*X[2,1,1,2,3]^3+3*x^14*X[2,1,1, 2,3]+3*x^13*X[2,1,1,2,3]^2+x^12*X[2,1,1,2,3]^3-x^14-3*x^13*X[2,1,1,2,3]-3*x^12* X[2,1,1,2,3]^2+x^13+3*x^12*X[2,1,1,2,3]-x^12+x^10*X[2,1,1,2,3]^2-2*x^10*X[2,1,1 ,2,3]+x^10+x^8*X[2,1,1,2,3]-x^8-3*x^7*X[2,1,1,2,3]+3*x^7+6*x^6*X[2,1,1,2,3]-6*x ^6-7*x^5*X[2,1,1,2,3]+6*x^5+4*x^4*X[2,1,1,2,3]+x^4-x^3*X[2,1,1,2,3]-9*x^3+10*x^ 2-5*x+1)/(x^15*X[2,1,1,2,3]^3-3*x^15*X[2,1,1,2,3]^2-x^14*X[2,1,1,2,3]^3+3*x^15* X[2,1,1,2,3]+3*x^14*X[2,1,1,2,3]^2+2*x^13*X[2,1,1,2,3]^3-x^15-3*x^14*X[2,1,1,2, 3]-5*x^13*X[2,1,1,2,3]^2-x^12*X[2,1,1,2,3]^3+x^14+4*x^13*X[2,1,1,2,3]+3*x^12*X[ 2,1,1,2,3]^2-x^13-3*x^12*X[2,1,1,2,3]+x^11*X[2,1,1,2,3]^2+x^12-2*x^11*X[2,1,1,2 ,3]-x^10*X[2,1,1,2,3]^2+x^11+2*x^10*X[2,1,1,2,3]-x^10+x^9*X[2,1,1,2,3]-x^9-2*x^ 8*X[2,1,1,2,3]+2*x^8+5*x^7*X[2,1,1,2,3]-5*x^7-10*x^6*X[2,1,1,2,3]+10*x^6+10*x^5 *X[2,1,1,2,3]-8*x^5-5*x^4*X[2,1,1,2,3]-4*x^4+x^3*X[2,1,1,2,3]+15*x^3-14*x^2+6*x -1) Furthermore, the expectation of the random variable number of occurences (by\ 11 n containment) of , [2, 1, 1, 2, 3], equals , - -- + ---- 32 32 447 35 n The variance equals , - ---- + ---- 1024 1024 10845 669 n The , 3, -th moment about the mean is , - ----- + ----- 16384 16384 678855 3675 2 8905 The , 4, -th moment about the mean is , - ------- + ------- n - ------ n 1048576 1048576 262144 The compositions of, 9, that yield the, 15, -th largest growth, that is, 1.9417130342786384772, are , [3, 1, 1, 1, 3] Theorem Number, 15, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [3, 1, 1, 1, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 33 32 31 30 29 28 27 ) a(n) x = - (x + 6 x + 15 x + 20 x + 17 x + 15 x + 18 x / ----- n = 0 26 25 24 23 22 21 20 19 18 + 17 x + 9 x + 2 x + 2 x + 3 x - 2 x - 5 x + x + 3 x 17 16 14 13 12 11 10 8 7 - 2 x - 3 x + 4 x + 2 x - 7 x + 3 x + 3 x - 2 x - 7 x 6 5 4 3 2 / 5 4 + 11 x + 6 x - 29 x + 34 x - 21 x + 7 x - 1) / ((x + x - x + 1) / 5 4 5 4 3 20 19 18 17 (x + x + x - 1) (x + x - x + 2 x - 1) (x + 4 x + 6 x + 4 x 16 12 10 8 6 4 3 2 + x + x - x - x + x + x - 4 x + 6 x - 4 x + 1)) and in Maple format -(x^33+6*x^32+15*x^31+20*x^30+17*x^29+15*x^28+18*x^27+17*x^26+9*x^25+2*x^24+2*x ^23+3*x^22-2*x^21-5*x^20+x^19+3*x^18-2*x^17-3*x^16+4*x^14+2*x^13-7*x^12+3*x^11+ 3*x^10-2*x^8-7*x^7+11*x^6+6*x^5-29*x^4+34*x^3-21*x^2+7*x-1)/(x^5+x^4-x+1)/(x^5+ x^4+x-1)/(x^5+x^4-x^3+2*x-1)/(x^20+4*x^19+6*x^18+4*x^17+x^16+x^12-x^10-x^8+x^6+ x^4-4*x^3+6*x^2-4*x+1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 505, 994, 1946, 3793, 7368, 14278, 27627, 53420, 103293, 199823, 386856, 749593, 1453645, 2820949, 5477286, 10638840, 20668699, 40157379, 78020560, 151571392, 294425861] The limit of a(n+1)/a(n) as n goes to infinity is 1.94171303428 a(n) is asymptotic to .665971509149*1.94171303428^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [3, 1, 1, 1, 3], denoted by the variable, X[3, 1, 1, 1, 3], is 33 33 32 33 32 31 33 - (x %6 - 7 x %5 + 6 x %6 + 21 x %4 - 42 x %5 + 15 x %6 - 35 x %3 32 31 30 33 32 + 126 x %4 - 105 x %5 + 20 x %6 + 35 x %2 - 210 x %3 31 30 29 33 32 + 315 x %4 - 140 x %5 + 15 x %6 - 21 x %1 + 210 x %2 31 30 29 28 33 - 525 x %3 + 420 x %4 - 107 x %5 + 6 x %6 + 7 x X[3, 1, 1, 1, 3] 32 31 30 29 28 27 - 126 x %1 + 525 x %2 - 700 x %3 + 327 x %4 - 51 x %5 + x %6 33 32 31 30 29 - x + 42 x X[3, 1, 1, 1, 3] - 315 x %1 + 700 x %2 - 555 x %3 28 27 32 31 30 + 180 x %4 - 24 x %5 - 6 x + 105 x X[3, 1, 1, 1, 3] - 420 x %1 29 28 27 26 31 + 565 x %2 - 345 x %3 + 123 x %4 - 17 x %5 - 15 x 30 29 28 27 + 140 x X[3, 1, 1, 1, 3] - 345 x %1 + 390 x %2 - 290 x %3 26 25 30 29 28 + 102 x %4 - 8 x %5 - 20 x + 117 x X[3, 1, 1, 1, 3] - 261 x %1 27 26 25 29 28 + 375 x %2 - 255 x %3 + 49 x %4 - 17 x + 96 x X[3, 1, 1, 1, 3] 27 26 25 24 28 - 276 x %1 + 340 x %2 - 125 x %3 + 2 x %4 - 15 x 27 26 25 24 + 109 x X[3, 1, 1, 1, 3] - 255 x %1 + 170 x %2 - 10 x %3 23 22 27 26 25 + 2 x %4 - 2 x %5 - 18 x + 102 x X[3, 1, 1, 1, 3] - 130 x %1 24 23 22 21 26 + 20 x %2 - 10 x %3 + 13 x %4 - x %5 - 17 x 25 24 23 22 21 + 53 x X[3, 1, 1, 1, 3] - 20 x %1 + 20 x %2 - 35 x %3 + 3 x %4 25 24 23 22 20 - 9 x + 10 x X[3, 1, 1, 1, 3] - 20 x %1 + 50 x %2 - 4 x %4 24 23 22 21 20 - 2 x + 10 x X[3, 1, 1, 1, 3] - 40 x %1 - 10 x %2 + 21 x %3 23 22 21 20 18 22 - 2 x + 17 x X[3, 1, 1, 1, 3] + 15 x %1 - 44 x %2 + x %4 - 3 x 21 20 19 18 21 - 9 x X[3, 1, 1, 1, 3] + 46 x %1 + x %2 - 8 x %3 + 2 x 20 19 18 17 20 - 24 x X[3, 1, 1, 1, 3] - 3 x %1 + 21 x %2 - x %3 + 5 x 19 18 17 16 19 + 3 x X[3, 1, 1, 1, 3] - 25 x %1 + x %2 + 4 x %3 - x 18 17 16 18 + 14 x X[3, 1, 1, 1, 3] + 3 x %1 - 15 x %2 - 3 x 17 16 15 14 17 - 5 x X[3, 1, 1, 1, 3] + 21 x %1 - x %2 - 2 x %3 + 2 x 16 15 14 13 16 - 13 x X[3, 1, 1, 1, 3] + 2 x %1 + 13 x %2 + x %3 + 3 x 15 14 13 14 - x X[3, 1, 1, 1, 3] - 24 x %1 - 5 x %2 + 17 x X[3, 1, 1, 1, 3] 13 12 14 13 12 + 5 x %1 - 4 x %2 - 4 x + x X[3, 1, 1, 1, 3] + 15 x %1 11 13 12 11 10 12 + 4 x %2 - 2 x - 18 x X[3, 1, 1, 1, 3] - 11 x %1 - x %2 + 7 x 11 10 11 10 + 10 x X[3, 1, 1, 1, 3] - 2 x %1 - 3 x + 6 x X[3, 1, 1, 1, 3] 9 10 9 8 8 + 4 x %1 - 3 x - 4 x X[3, 1, 1, 1, 3] - x %1 - x X[3, 1, 1, 1, 3] 8 7 7 6 6 + 2 x - 8 x X[3, 1, 1, 1, 3] + 7 x + 18 x X[3, 1, 1, 1, 3] - 11 x 5 5 4 4 - 15 x X[3, 1, 1, 1, 3] - 6 x + 6 x X[3, 1, 1, 1, 3] + 29 x 3 3 2 / - x X[3, 1, 1, 1, 3] - 34 x + 21 x - 7 x + 1) / ( / 5 5 4 4 (x X[3, 1, 1, 1, 3] - x + x X[3, 1, 1, 1, 3] - x + x - 1) 5 5 4 4 (x X[3, 1, 1, 1, 3] - x + x X[3, 1, 1, 1, 3] - x - x + 1) ( 5 5 4 4 3 x X[3, 1, 1, 1, 3] - x + x X[3, 1, 1, 1, 3] - x - x X[3, 1, 1, 1, 3] 3 20 20 19 20 19 + x - 2 x + 1) (x %3 - 4 x %2 + 4 x %3 + 6 x %1 - 16 x %2 18 20 19 18 17 + 6 x %3 - 4 x X[3, 1, 1, 1, 3] + 24 x %1 - 24 x %2 + 4 x %3 20 19 18 17 16 19 + x - 16 x X[3, 1, 1, 1, 3] + 36 x %1 - 16 x %2 + x %3 + 4 x 18 17 16 18 - 24 x X[3, 1, 1, 1, 3] + 24 x %1 - 4 x %2 + 6 x 17 16 17 16 - 16 x X[3, 1, 1, 1, 3] + 6 x %1 + 4 x - 4 x X[3, 1, 1, 1, 3] 16 12 12 12 10 12 10 + x + x %2 - x %1 - x X[3, 1, 1, 1, 3] - x %2 + x + x %1 10 10 8 8 + x X[3, 1, 1, 1, 3] - x + x X[3, 1, 1, 1, 3] - x 6 6 4 3 2 - x X[3, 1, 1, 1, 3] + x + x - 4 x + 6 x - 4 x + 1)) 2 %1 := X[3, 1, 1, 1, 3] 3 %2 := X[3, 1, 1, 1, 3] 4 %3 := X[3, 1, 1, 1, 3] 5 %4 := X[3, 1, 1, 1, 3] 6 %5 := X[3, 1, 1, 1, 3] 7 %6 := X[3, 1, 1, 1, 3] and in Maple format -(x^33*X[3,1,1,1,3]^7-7*x^33*X[3,1,1,1,3]^6+6*x^32*X[3,1,1,1,3]^7+21*x^33*X[3,1 ,1,1,3]^5-42*x^32*X[3,1,1,1,3]^6+15*x^31*X[3,1,1,1,3]^7-35*x^33*X[3,1,1,1,3]^4+ 126*x^32*X[3,1,1,1,3]^5-105*x^31*X[3,1,1,1,3]^6+20*x^30*X[3,1,1,1,3]^7+35*x^33* X[3,1,1,1,3]^3-210*x^32*X[3,1,1,1,3]^4+315*x^31*X[3,1,1,1,3]^5-140*x^30*X[3,1,1 ,1,3]^6+15*x^29*X[3,1,1,1,3]^7-21*x^33*X[3,1,1,1,3]^2+210*x^32*X[3,1,1,1,3]^3-\ 525*x^31*X[3,1,1,1,3]^4+420*x^30*X[3,1,1,1,3]^5-107*x^29*X[3,1,1,1,3]^6+6*x^28* X[3,1,1,1,3]^7+7*x^33*X[3,1,1,1,3]-126*x^32*X[3,1,1,1,3]^2+525*x^31*X[3,1,1,1,3 ]^3-700*x^30*X[3,1,1,1,3]^4+327*x^29*X[3,1,1,1,3]^5-51*x^28*X[3,1,1,1,3]^6+x^27 *X[3,1,1,1,3]^7-x^33+42*x^32*X[3,1,1,1,3]-315*x^31*X[3,1,1,1,3]^2+700*x^30*X[3, 1,1,1,3]^3-555*x^29*X[3,1,1,1,3]^4+180*x^28*X[3,1,1,1,3]^5-24*x^27*X[3,1,1,1,3] ^6-6*x^32+105*x^31*X[3,1,1,1,3]-420*x^30*X[3,1,1,1,3]^2+565*x^29*X[3,1,1,1,3]^3 -345*x^28*X[3,1,1,1,3]^4+123*x^27*X[3,1,1,1,3]^5-17*x^26*X[3,1,1,1,3]^6-15*x^31 +140*x^30*X[3,1,1,1,3]-345*x^29*X[3,1,1,1,3]^2+390*x^28*X[3,1,1,1,3]^3-290*x^27 *X[3,1,1,1,3]^4+102*x^26*X[3,1,1,1,3]^5-8*x^25*X[3,1,1,1,3]^6-20*x^30+117*x^29* X[3,1,1,1,3]-261*x^28*X[3,1,1,1,3]^2+375*x^27*X[3,1,1,1,3]^3-255*x^26*X[3,1,1,1 ,3]^4+49*x^25*X[3,1,1,1,3]^5-17*x^29+96*x^28*X[3,1,1,1,3]-276*x^27*X[3,1,1,1,3] ^2+340*x^26*X[3,1,1,1,3]^3-125*x^25*X[3,1,1,1,3]^4+2*x^24*X[3,1,1,1,3]^5-15*x^ 28+109*x^27*X[3,1,1,1,3]-255*x^26*X[3,1,1,1,3]^2+170*x^25*X[3,1,1,1,3]^3-10*x^ 24*X[3,1,1,1,3]^4+2*x^23*X[3,1,1,1,3]^5-2*x^22*X[3,1,1,1,3]^6-18*x^27+102*x^26* X[3,1,1,1,3]-130*x^25*X[3,1,1,1,3]^2+20*x^24*X[3,1,1,1,3]^3-10*x^23*X[3,1,1,1,3 ]^4+13*x^22*X[3,1,1,1,3]^5-x^21*X[3,1,1,1,3]^6-17*x^26+53*x^25*X[3,1,1,1,3]-20* x^24*X[3,1,1,1,3]^2+20*x^23*X[3,1,1,1,3]^3-35*x^22*X[3,1,1,1,3]^4+3*x^21*X[3,1, 1,1,3]^5-9*x^25+10*x^24*X[3,1,1,1,3]-20*x^23*X[3,1,1,1,3]^2+50*x^22*X[3,1,1,1,3 ]^3-4*x^20*X[3,1,1,1,3]^5-2*x^24+10*x^23*X[3,1,1,1,3]-40*x^22*X[3,1,1,1,3]^2-10 *x^21*X[3,1,1,1,3]^3+21*x^20*X[3,1,1,1,3]^4-2*x^23+17*x^22*X[3,1,1,1,3]+15*x^21 *X[3,1,1,1,3]^2-44*x^20*X[3,1,1,1,3]^3+x^18*X[3,1,1,1,3]^5-3*x^22-9*x^21*X[3,1, 1,1,3]+46*x^20*X[3,1,1,1,3]^2+x^19*X[3,1,1,1,3]^3-8*x^18*X[3,1,1,1,3]^4+2*x^21-\ 24*x^20*X[3,1,1,1,3]-3*x^19*X[3,1,1,1,3]^2+21*x^18*X[3,1,1,1,3]^3-x^17*X[3,1,1, 1,3]^4+5*x^20+3*x^19*X[3,1,1,1,3]-25*x^18*X[3,1,1,1,3]^2+x^17*X[3,1,1,1,3]^3+4* x^16*X[3,1,1,1,3]^4-x^19+14*x^18*X[3,1,1,1,3]+3*x^17*X[3,1,1,1,3]^2-15*x^16*X[3 ,1,1,1,3]^3-3*x^18-5*x^17*X[3,1,1,1,3]+21*x^16*X[3,1,1,1,3]^2-x^15*X[3,1,1,1,3] ^3-2*x^14*X[3,1,1,1,3]^4+2*x^17-13*x^16*X[3,1,1,1,3]+2*x^15*X[3,1,1,1,3]^2+13*x ^14*X[3,1,1,1,3]^3+x^13*X[3,1,1,1,3]^4+3*x^16-x^15*X[3,1,1,1,3]-24*x^14*X[3,1,1 ,1,3]^2-5*x^13*X[3,1,1,1,3]^3+17*x^14*X[3,1,1,1,3]+5*x^13*X[3,1,1,1,3]^2-4*x^12 *X[3,1,1,1,3]^3-4*x^14+x^13*X[3,1,1,1,3]+15*x^12*X[3,1,1,1,3]^2+4*x^11*X[3,1,1, 1,3]^3-2*x^13-18*x^12*X[3,1,1,1,3]-11*x^11*X[3,1,1,1,3]^2-x^10*X[3,1,1,1,3]^3+7 *x^12+10*x^11*X[3,1,1,1,3]-2*x^10*X[3,1,1,1,3]^2-3*x^11+6*x^10*X[3,1,1,1,3]+4*x ^9*X[3,1,1,1,3]^2-3*x^10-4*x^9*X[3,1,1,1,3]-x^8*X[3,1,1,1,3]^2-x^8*X[3,1,1,1,3] +2*x^8-8*x^7*X[3,1,1,1,3]+7*x^7+18*x^6*X[3,1,1,1,3]-11*x^6-15*x^5*X[3,1,1,1,3]-\ 6*x^5+6*x^4*X[3,1,1,1,3]+29*x^4-x^3*X[3,1,1,1,3]-34*x^3+21*x^2-7*x+1)/(x^5*X[3, 1,1,1,3]-x^5+x^4*X[3,1,1,1,3]-x^4+x-1)/(x^5*X[3,1,1,1,3]-x^5+x^4*X[3,1,1,1,3]-x ^4-x+1)/(x^5*X[3,1,1,1,3]-x^5+x^4*X[3,1,1,1,3]-x^4-x^3*X[3,1,1,1,3]+x^3-2*x+1)/ (x^20*X[3,1,1,1,3]^4-4*x^20*X[3,1,1,1,3]^3+4*x^19*X[3,1,1,1,3]^4+6*x^20*X[3,1,1 ,1,3]^2-16*x^19*X[3,1,1,1,3]^3+6*x^18*X[3,1,1,1,3]^4-4*x^20*X[3,1,1,1,3]+24*x^ 19*X[3,1,1,1,3]^2-24*x^18*X[3,1,1,1,3]^3+4*x^17*X[3,1,1,1,3]^4+x^20-16*x^19*X[3 ,1,1,1,3]+36*x^18*X[3,1,1,1,3]^2-16*x^17*X[3,1,1,1,3]^3+x^16*X[3,1,1,1,3]^4+4*x ^19-24*x^18*X[3,1,1,1,3]+24*x^17*X[3,1,1,1,3]^2-4*x^16*X[3,1,1,1,3]^3+6*x^18-16 *x^17*X[3,1,1,1,3]+6*x^16*X[3,1,1,1,3]^2+4*x^17-4*x^16*X[3,1,1,1,3]+x^16+x^12*X [3,1,1,1,3]^3-x^12*X[3,1,1,1,3]^2-x^12*X[3,1,1,1,3]-x^10*X[3,1,1,1,3]^3+x^12+x^ 10*X[3,1,1,1,3]^2+x^10*X[3,1,1,1,3]-x^10+x^8*X[3,1,1,1,3]-x^8-x^6*X[3,1,1,1,3]+ x^6+x^4-4*x^3+6*x^2-4*x+1) Furthermore, the expectation of the random variable number of occurences (by\ 11 n containment) of , [3, 1, 1, 1, 3], equals , - -- + ---- 32 32 471 35 n The variance equals , - ---- + ---- 1024 1024 11337 633 n The , 3, -th moment about the mean is , - ----- + ----- 16384 16384 517047 3675 2 13501 The , 4, -th moment about the mean is , - ------- + ------- n - ------ n 1048576 1048576 262144 The compositions of, 9, that yield the, 16, -th largest growth, that is, 1.9454365275632690792, are , [2, 1, 2, 1, 3], [2, 1, 3, 1, 2], [3, 1, 2, 1, 2] Theorem Number, 16, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 2, 1, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 17 14 13 12 10 9 8 7 ) a(n) x = - (x + 2 x - 3 x + 2 x + 4 x - 10 x + 9 x - 4 x / ----- n = 0 6 4 3 2 / + 3 x - 11 x + 19 x - 15 x + 6 x - 1) / ( / 6 5 4 3 2 (x - x + 2 x - x - 2 x + 3 x - 1) 12 10 9 8 6 5 4 3 2 (x - x + x + x - 3 x + 3 x - 2 x + 4 x - 6 x + 4 x - 1)) and in Maple format -(x^17+2*x^14-3*x^13+2*x^12+4*x^10-10*x^9+9*x^8-4*x^7+3*x^6-11*x^4+19*x^3-15*x^ 2+6*x-1)/(x^6-x^5+2*x^4-x^3-2*x^2+3*x-1)/(x^12-x^10+x^9+x^8-3*x^6+3*x^5-2*x^4+4 *x^3-6*x^2+4*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 505, 994, 1946, 3794, 7377, 14323, 27795, 53944, 104738, 203470, 395480, 769021, 1495859, 2910260, 5662646, 11018497, 21439793, 41716082, 81164766, 157911440, 307216819] The limit of a(n+1)/a(n) as n goes to infinity is 1.94543652756 a(n) is asymptotic to .656001947608*1.94543652756^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [2, 1, 2, 1, 3], denoted by the variable, X[2, 1, 2, 1, 3], is 17 17 17 17 14 14 - (x %2 - 3 x %1 + 3 x X[2, 1, 2, 1, 3] - x + 2 x %2 - 6 x %1 13 14 13 12 14 - 2 x %2 + 6 x X[2, 1, 2, 1, 3] + 7 x %1 + x %2 - 2 x 13 12 13 12 - 8 x X[2, 1, 2, 1, 3] - 4 x %1 + 3 x + 5 x X[2, 1, 2, 1, 3] 12 10 10 9 10 - 2 x - 3 x %1 + 7 x X[2, 1, 2, 1, 3] + 6 x %1 - 4 x 9 8 9 8 - 16 x X[2, 1, 2, 1, 3] - 4 x %1 + 10 x + 13 x X[2, 1, 2, 1, 3] 7 8 7 7 6 + x %1 - 9 x - 5 x X[2, 1, 2, 1, 3] + 4 x + 4 x X[2, 1, 2, 1, 3] 6 5 4 4 - 3 x - 6 x X[2, 1, 2, 1, 3] + 4 x X[2, 1, 2, 1, 3] + 11 x 3 3 2 / 6 - x X[2, 1, 2, 1, 3] - 19 x + 15 x - 6 x + 1) / ((x X[2, 1, 2, 1, 3] / 6 5 5 4 4 - x - x X[2, 1, 2, 1, 3] + x + 2 x X[2, 1, 2, 1, 3] - 2 x 3 3 2 12 - x X[2, 1, 2, 1, 3] + x + 2 x - 3 x + 1) (x %1 12 12 10 10 9 - 2 x X[2, 1, 2, 1, 3] + x - x %1 + 2 x X[2, 1, 2, 1, 3] + x %1 10 9 9 8 8 - x - 2 x X[2, 1, 2, 1, 3] + x - x X[2, 1, 2, 1, 3] + x 6 6 5 5 + 3 x X[2, 1, 2, 1, 3] - 3 x - 3 x X[2, 1, 2, 1, 3] + 3 x 4 4 3 2 + x X[2, 1, 2, 1, 3] - 2 x + 4 x - 6 x + 4 x - 1)) 2 %1 := X[2, 1, 2, 1, 3] 3 %2 := X[2, 1, 2, 1, 3] and in Maple format -(x^17*X[2,1,2,1,3]^3-3*x^17*X[2,1,2,1,3]^2+3*x^17*X[2,1,2,1,3]-x^17+2*x^14*X[2 ,1,2,1,3]^3-6*x^14*X[2,1,2,1,3]^2-2*x^13*X[2,1,2,1,3]^3+6*x^14*X[2,1,2,1,3]+7*x ^13*X[2,1,2,1,3]^2+x^12*X[2,1,2,1,3]^3-2*x^14-8*x^13*X[2,1,2,1,3]-4*x^12*X[2,1, 2,1,3]^2+3*x^13+5*x^12*X[2,1,2,1,3]-2*x^12-3*x^10*X[2,1,2,1,3]^2+7*x^10*X[2,1,2 ,1,3]+6*x^9*X[2,1,2,1,3]^2-4*x^10-16*x^9*X[2,1,2,1,3]-4*x^8*X[2,1,2,1,3]^2+10*x ^9+13*x^8*X[2,1,2,1,3]+x^7*X[2,1,2,1,3]^2-9*x^8-5*x^7*X[2,1,2,1,3]+4*x^7+4*x^6* X[2,1,2,1,3]-3*x^6-6*x^5*X[2,1,2,1,3]+4*x^4*X[2,1,2,1,3]+11*x^4-x^3*X[2,1,2,1,3 ]-19*x^3+15*x^2-6*x+1)/(x^6*X[2,1,2,1,3]-x^6-x^5*X[2,1,2,1,3]+x^5+2*x^4*X[2,1,2 ,1,3]-2*x^4-x^3*X[2,1,2,1,3]+x^3+2*x^2-3*x+1)/(x^12*X[2,1,2,1,3]^2-2*x^12*X[2,1 ,2,1,3]+x^12-x^10*X[2,1,2,1,3]^2+2*x^10*X[2,1,2,1,3]+x^9*X[2,1,2,1,3]^2-x^10-2* x^9*X[2,1,2,1,3]+x^9-x^8*X[2,1,2,1,3]+x^8+3*x^6*X[2,1,2,1,3]-3*x^6-3*x^5*X[2,1, 2,1,3]+3*x^5+x^4*X[2,1,2,1,3]-2*x^4+4*x^3-6*x^2+4*x-1) Furthermore, the expectation of the random variable number of occurences (by\ 11 n containment) of , [2, 1, 2, 1, 3], equals , - -- + ---- 32 32 507 39 n The variance equals , - ---- + ---- 1024 1024 13155 819 n The , 3, -th moment about the mean is , - ----- + ----- 16384 16384 654975 4563 2 12489 The , 4, -th moment about the mean is , - ------- + ------- n - ------ n 1048576 1048576 262144 The compositions of, 9, that yield the, 17, -th largest growth, that is, 1.9515637714286765859, are , [2, 1, 2, 2, 2], [2, 2, 2, 1, 2] Theorem Number, 17, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 2, 2, 2] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n ) a(n) x = / ----- n = 0 5 3 2 4 2 (x - x + 3 x - 3 x + 1) (x + x - x + 1) - ----------------------------------------------------------------- 10 9 8 7 6 5 4 3 2 x - x + x + x - 3 x + 5 x - 8 x + 11 x - 10 x + 5 x - 1 and in Maple format -(x^5-x^3+3*x^2-3*x+1)*(x^4+x^2-x+1)/(x^10-x^9+x^8+x^7-3*x^6+5*x^5-8*x^4+11*x^3 -10*x^2+5*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 505, 994, 1947, 3802, 7413, 14446, 28152, 54876, 107003, 208709, 407186, 794549, 1550588, 3026198, 5906192, 11527032, 22496889, 43905714, 85686751, 167225164, 326352059] The limit of a(n+1)/a(n) as n goes to infinity is 1.95156377143 a(n) is asymptotic to .634159562704*1.95156377143^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [2, 1, 2, 2, 2], denoted by the variable, X[2, 1, 2, 2, 2], is 9 2 9 9 6 - (x X[2, 1, 2, 2, 2] - 2 x X[2, 1, 2, 2, 2] + x - 2 x X[2, 1, 2, 2, 2] 6 5 5 4 4 + 2 x + 3 x X[2, 1, 2, 2, 2] - 3 x - 4 x X[2, 1, 2, 2, 2] + 5 x 3 3 2 2 + 3 x X[2, 1, 2, 2, 2] - 7 x - x X[2, 1, 2, 2, 2] + 7 x - 4 x + 1) / 10 2 10 9 2 / (x X[2, 1, 2, 2, 2] - 2 x X[2, 1, 2, 2, 2] - x X[2, 1, 2, 2, 2] / 10 9 9 8 8 + x + 2 x X[2, 1, 2, 2, 2] - x - x X[2, 1, 2, 2, 2] + x 7 7 6 6 - x X[2, 1, 2, 2, 2] + x + 3 x X[2, 1, 2, 2, 2] - 3 x 5 5 4 4 - 5 x X[2, 1, 2, 2, 2] + 5 x + 6 x X[2, 1, 2, 2, 2] - 8 x 3 3 2 2 - 4 x X[2, 1, 2, 2, 2] + 11 x + x X[2, 1, 2, 2, 2] - 10 x + 5 x - 1) and in Maple format -(x^9*X[2,1,2,2,2]^2-2*x^9*X[2,1,2,2,2]+x^9-2*x^6*X[2,1,2,2,2]+2*x^6+3*x^5*X[2, 1,2,2,2]-3*x^5-4*x^4*X[2,1,2,2,2]+5*x^4+3*x^3*X[2,1,2,2,2]-7*x^3-x^2*X[2,1,2,2, 2]+7*x^2-4*x+1)/(x^10*X[2,1,2,2,2]^2-2*x^10*X[2,1,2,2,2]-x^9*X[2,1,2,2,2]^2+x^ 10+2*x^9*X[2,1,2,2,2]-x^9-x^8*X[2,1,2,2,2]+x^8-x^7*X[2,1,2,2,2]+x^7+3*x^6*X[2,1 ,2,2,2]-3*x^6-5*x^5*X[2,1,2,2,2]+5*x^5+6*x^4*X[2,1,2,2,2]-8*x^4-4*x^3*X[2,1,2,2 ,2]+11*x^3+x^2*X[2,1,2,2,2]-10*x^2+5*x-1) Furthermore, the expectation of the random variable number of occurences (by\ 11 n containment) of , [2, 1, 2, 2, 2], equals , - -- + ---- 32 32 755 55 n The variance equals , - ---- + ---- 1024 1024 39267 2259 n The , 3, -th moment about the mean is , - ----- + ------ 16384 16384 9029903 9075 2 65827 The , 4, -th moment about the mean is , - ------- + ------- n + ------ n 1048576 1048576 262144 The compositions of, 9, that yield the, 18, -th largest growth, that is, 1.9527971478516900544, are , [2, 2, 1, 2, 2] Theorem Number, 18, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 2, 1, 2, 2] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 3 ) a(n) x = - (x - x + 1) / ----- n = 0 8 7 6 5 4 3 2 / 12 11 (x - 2 x + 4 x - 6 x + 8 x - 8 x + 7 x - 4 x + 1) / (x - 2 x / 10 9 8 7 6 5 4 3 2 + 4 x - 5 x + 5 x - 2 x - 4 x + 12 x - 19 x + 21 x - 15 x + 6 x - 1) and in Maple format -(x^3-x+1)*(x^8-2*x^7+4*x^6-6*x^5+8*x^4-8*x^3+7*x^2-4*x+1)/(x^12-2*x^11+4*x^10-\ 5*x^9+5*x^8-2*x^7-4*x^6+12*x^5-19*x^4+21*x^3-15*x^2+6*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 505, 994, 1947, 3801, 7406, 14418, 28068, 54666, 106542, 207795, 405527, 791788, 1546434, 3020814, 5901168, 11527651, 22517214, 43979794, 85893053, 167739492, 327561567] The limit of a(n+1)/a(n) as n goes to infinity is 1.95279714785 a(n) is asymptotic to .624456866523*1.95279714785^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [2, 2, 1, 2, 2], denoted by the variable, X[2, 2, 1, 2, 2], is 11 11 10 11 10 - (x %1 - 2 x X[2, 2, 1, 2, 2] - 2 x %1 + x + 4 x X[2, 2, 1, 2, 2] 9 10 9 8 9 + 3 x %1 - 2 x - 6 x X[2, 2, 1, 2, 2] - 2 x %1 + 3 x 8 7 8 7 7 + 5 x X[2, 2, 1, 2, 2] + x %1 - 3 x - 3 x X[2, 2, 1, 2, 2] + 2 x 6 6 5 5 - 2 x X[2, 2, 1, 2, 2] + 2 x + 6 x X[2, 2, 1, 2, 2] - 7 x 4 4 3 3 - 7 x X[2, 2, 1, 2, 2] + 12 x + 4 x X[2, 2, 1, 2, 2] - 14 x 2 2 / 12 - x X[2, 2, 1, 2, 2] + 11 x - 5 x + 1) / (x %1 / 12 11 12 11 - 2 x X[2, 2, 1, 2, 2] - 2 x %1 + x + 4 x X[2, 2, 1, 2, 2] 10 11 10 9 10 + 4 x %1 - 2 x - 8 x X[2, 2, 1, 2, 2] - 4 x %1 + 4 x 9 8 9 8 7 + 9 x X[2, 2, 1, 2, 2] + 3 x %1 - 5 x - 8 x X[2, 2, 1, 2, 2] - x %1 8 7 7 6 6 + 5 x + 3 x X[2, 2, 1, 2, 2] - 2 x + 4 x X[2, 2, 1, 2, 2] - 4 x 5 5 4 4 - 10 x X[2, 2, 1, 2, 2] + 12 x + 10 x X[2, 2, 1, 2, 2] - 19 x 3 3 2 2 - 5 x X[2, 2, 1, 2, 2] + 21 x + x X[2, 2, 1, 2, 2] - 15 x + 6 x - 1) 2 %1 := X[2, 2, 1, 2, 2] and in Maple format -(x^11*X[2,2,1,2,2]^2-2*x^11*X[2,2,1,2,2]-2*x^10*X[2,2,1,2,2]^2+x^11+4*x^10*X[2 ,2,1,2,2]+3*x^9*X[2,2,1,2,2]^2-2*x^10-6*x^9*X[2,2,1,2,2]-2*x^8*X[2,2,1,2,2]^2+3 *x^9+5*x^8*X[2,2,1,2,2]+x^7*X[2,2,1,2,2]^2-3*x^8-3*x^7*X[2,2,1,2,2]+2*x^7-2*x^6 *X[2,2,1,2,2]+2*x^6+6*x^5*X[2,2,1,2,2]-7*x^5-7*x^4*X[2,2,1,2,2]+12*x^4+4*x^3*X[ 2,2,1,2,2]-14*x^3-x^2*X[2,2,1,2,2]+11*x^2-5*x+1)/(x^12*X[2,2,1,2,2]^2-2*x^12*X[ 2,2,1,2,2]-2*x^11*X[2,2,1,2,2]^2+x^12+4*x^11*X[2,2,1,2,2]+4*x^10*X[2,2,1,2,2]^2 -2*x^11-8*x^10*X[2,2,1,2,2]-4*x^9*X[2,2,1,2,2]^2+4*x^10+9*x^9*X[2,2,1,2,2]+3*x^ 8*X[2,2,1,2,2]^2-5*x^9-8*x^8*X[2,2,1,2,2]-x^7*X[2,2,1,2,2]^2+5*x^8+3*x^7*X[2,2, 1,2,2]-2*x^7+4*x^6*X[2,2,1,2,2]-4*x^6-10*x^5*X[2,2,1,2,2]+12*x^5+10*x^4*X[2,2,1 ,2,2]-19*x^4-5*x^3*X[2,2,1,2,2]+21*x^3+x^2*X[2,2,1,2,2]-15*x^2+6*x-1) Furthermore, the expectation of the random variable number of occurences (by\ 11 n containment) of , [2, 2, 1, 2, 2], equals , - -- + ---- 32 32 771 55 n The variance equals , - ---- + ---- 1024 1024 39195 2187 n The , 3, -th moment about the mean is , - ----- + ------ 16384 16384 8775663 9075 2 57307 The , 4, -th moment about the mean is , - ------- + ------- n + ------ n 1048576 1048576 262144 The compositions of, 9, that yield the, 19, -th largest growth, that is, 1.9611865309023902347, are , [1, 2, 2, 2, 2], [2, 2, 2, 2, 1] Theorem Number, 19, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 2, 2, 2, 2] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 8 7 6 5 4 3 2 \ n x - x + 2 x - 3 x + 5 x - 7 x + 7 x - 4 x + 1 ) a(n) x = -------------------------------------------------------- / 7 6 5 4 3 2 ----- (-1 + x) (x - x + 2 x - 3 x + 5 x - 6 x + 4 x - 1) n = 0 and in Maple format (x^8-x^7+2*x^6-3*x^5+5*x^4-7*x^3+7*x^2-4*x+1)/(-1+x)/(x^7-x^6+2*x^5-3*x^4+5*x^3 -6*x^2+4*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 505, 995, 1954, 3831, 7507, 14711, 28836, 56539, 110878, 217463, 426518, 836536, 1640671, 3217723, 6310580, 12376181, 24271864, 47601426, 93354997, 183086318, 359066334] The limit of a(n+1)/a(n) as n goes to infinity is 1.96118653090 a(n) is asymptotic to .602016227947*1.96118653090^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [1, 2, 2, 2, 2], denoted by the variable, X[1, 2, 2, 2, 2], is 8 8 7 7 6 (x X[1, 2, 2, 2, 2] - x - x X[1, 2, 2, 2, 2] + x + 2 x X[1, 2, 2, 2, 2] 6 5 5 4 4 - 2 x - 3 x X[1, 2, 2, 2, 2] + 3 x + 4 x X[1, 2, 2, 2, 2] - 5 x 3 3 2 2 - 3 x X[1, 2, 2, 2, 2] + 7 x + x X[1, 2, 2, 2, 2] - 7 x + 4 x - 1) / 7 7 6 6 / ((-1 + x) (x X[1, 2, 2, 2, 2] - x - x X[1, 2, 2, 2, 2] + x / 5 5 4 4 + 2 x X[1, 2, 2, 2, 2] - 2 x - 3 x X[1, 2, 2, 2, 2] + 3 x 3 3 2 2 + 3 x X[1, 2, 2, 2, 2] - 5 x - x X[1, 2, 2, 2, 2] + 6 x - 4 x + 1)) and in Maple format (x^8*X[1,2,2,2,2]-x^8-x^7*X[1,2,2,2,2]+x^7+2*x^6*X[1,2,2,2,2]-2*x^6-3*x^5*X[1,2 ,2,2,2]+3*x^5+4*x^4*X[1,2,2,2,2]-5*x^4-3*x^3*X[1,2,2,2,2]+7*x^3+x^2*X[1,2,2,2,2 ]-7*x^2+4*x-1)/(-1+x)/(x^7*X[1,2,2,2,2]-x^7-x^6*X[1,2,2,2,2]+x^6+2*x^5*X[1,2,2, 2,2]-2*x^5-3*x^4*X[1,2,2,2,2]+3*x^4+3*x^3*X[1,2,2,2,2]-5*x^3-x^2*X[1,2,2,2,2]+6 *x^2-4*x+1) Furthermore, the expectation of the random variable number of occurences (by\ 11 n containment) of , [1, 2, 2, 2, 2], equals , - -- + ---- 32 32 879 67 n The variance equals , - ---- + ---- 1024 1024 44889 2889 n The , 3, -th moment about the mean is , - ----- + ------ 16384 16384 8414535 13467 2 59527 The , 4, -th moment about the mean is , - ------- + ------- n + ------ n 1048576 1048576 262144 The compositions of, 9, that yield the, 20, -th largest growth, that is, 1.9659482366454853372, are , [1, 1, 1, 6], [1, 1, 6, 1], [1, 6, 1, 1], [6, 1, 1, 1] Theorem Number, 20, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 1, 1, 6] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 8 7 6 3 2 \ n x - x + x - x + 3 x - 3 x + 1 ) a(n) x = ------------------------------------- / 5 4 3 2 3 ----- (x + x + x + x + x - 1) (-1 + x) n = 0 and in Maple format (x^8-x^7+x^6-x^3+3*x^2-3*x+1)/(x^5+x^4+x^3+x^2+x-1)/(-1+x)^3 The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 506, 1001, 1976, 3895, 7670, 15094, 29692, 58394, 114824, 225766, 443876, 872673, 1715670, 3372963, 6631120, 13036493, 25629130, 50385608, 99055568, 194738196, 382845296] The limit of a(n+1)/a(n) as n goes to infinity is 1.96594823665 a(n) is asymptotic to .596844324123*1.96594823665^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [1, 1, 1, 6], denoted by the variable, X[1, 1, 1, 6], is 8 8 7 7 6 6 3 (x X[1, 1, 1, 6] - x - x X[1, 1, 1, 6] + x + x X[1, 1, 1, 6] - x + x 2 / 2 6 6 - 3 x + 3 x - 1) / ((-1 + x) (x X[1, 1, 1, 6] - x + 2 x - 1)) / and in Maple format (x^8*X[1,1,1,6]-x^8-x^7*X[1,1,1,6]+x^7+x^6*X[1,1,1,6]-x^6+x^3-3*x^2+3*x-1)/(-1+ x)^2/(x^6*X[1,1,1,6]-x^6+2*x-1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [1, 1, 1, 6], equals , - 5/32 + ---- 64 31 53 n The variance equals , - --- + ---- 256 4096 2187 141 n The , 3, -th moment about the mean is , - ----- + ----- 32768 16384 25699 8427 2 58577 The , 4, -th moment about the mean is , ------ + -------- n - ------- n 524288 16777216 8388608 The compositions of, 9, that yield the, 21, -th largest growth, that is, 1.9671682128139660358, are , [2, 1, 1, 5], [5, 1, 1, 2] Theorem Number, 21, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 1, 5] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 23 18 17 16 13 12 11 9 8 ) a(n) x = - (x + 2 x - x + x + 2 x - 3 x + x - x + 4 x / ----- n = 0 7 6 5 4 3 2 / 6 5 - 5 x + 3 x - x - x + 4 x - 6 x + 4 x - 1) / ((x - x + 2 x - 1) / 18 12 11 8 7 3 2 (x - x + x + x - x + x - 3 x + 3 x - 1)) and in Maple format -(x^23+2*x^18-x^17+x^16+2*x^13-3*x^12+x^11-x^9+4*x^8-5*x^7+3*x^6-x^5-x^4+4*x^3-\ 6*x^2+4*x-1)/(x^6-x^5+2*x-1)/(x^18-x^12+x^11+x^8-x^7+x^3-3*x^2+3*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 506, 1001, 1976, 3895, 7670, 15094, 29693, 58402, 114862, 225906, 444319, 873939, 1719036, 3381450, 6651679, 13084782, 25739844, 50634627, 99607111, 195944622, 385457266] The limit of a(n+1)/a(n) as n goes to infinity is 1.96716821281 a(n) is asymptotic to .589839993458*1.96716821281^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [2, 1, 1, 5], denoted by the variable, X[2, 1, 1, 5], is 23 4 23 3 23 2 - (x X[2, 1, 1, 5] - 4 x X[2, 1, 1, 5] + 6 x X[2, 1, 1, 5] 23 23 18 3 18 2 - 4 x X[2, 1, 1, 5] + x - 2 x X[2, 1, 1, 5] + 6 x X[2, 1, 1, 5] 17 3 18 17 2 + x X[2, 1, 1, 5] - 6 x X[2, 1, 1, 5] - 3 x X[2, 1, 1, 5] 16 3 18 17 16 2 - x X[2, 1, 1, 5] + 2 x + 3 x X[2, 1, 1, 5] + 3 x X[2, 1, 1, 5] 17 16 16 13 2 - x - 3 x X[2, 1, 1, 5] + x + 2 x X[2, 1, 1, 5] 13 12 2 13 12 - 4 x X[2, 1, 1, 5] - 3 x X[2, 1, 1, 5] + 2 x + 6 x X[2, 1, 1, 5] 11 2 12 11 11 + x X[2, 1, 1, 5] - 3 x - 2 x X[2, 1, 1, 5] + x 9 9 8 8 7 + x X[2, 1, 1, 5] - x - 4 x X[2, 1, 1, 5] + 4 x + 5 x X[2, 1, 1, 5] 7 6 6 5 5 4 3 - 5 x - 3 x X[2, 1, 1, 5] + 3 x + x X[2, 1, 1, 5] - x - x + 4 x 2 / - 6 x + 4 x - 1) / ( / 6 6 5 5 (x X[2, 1, 1, 5] - x - x X[2, 1, 1, 5] + x - 2 x + 1) ( 18 3 18 2 18 18 x X[2, 1, 1, 5] - 3 x X[2, 1, 1, 5] + 3 x X[2, 1, 1, 5] - x 12 2 12 11 2 12 + x X[2, 1, 1, 5] - 2 x X[2, 1, 1, 5] - x X[2, 1, 1, 5] + x 11 11 8 8 7 + 2 x X[2, 1, 1, 5] - x + x X[2, 1, 1, 5] - x - x X[2, 1, 1, 5] 7 3 2 + x - x + 3 x - 3 x + 1)) and in Maple format -(x^23*X[2,1,1,5]^4-4*x^23*X[2,1,1,5]^3+6*x^23*X[2,1,1,5]^2-4*x^23*X[2,1,1,5]+x ^23-2*x^18*X[2,1,1,5]^3+6*x^18*X[2,1,1,5]^2+x^17*X[2,1,1,5]^3-6*x^18*X[2,1,1,5] -3*x^17*X[2,1,1,5]^2-x^16*X[2,1,1,5]^3+2*x^18+3*x^17*X[2,1,1,5]+3*x^16*X[2,1,1, 5]^2-x^17-3*x^16*X[2,1,1,5]+x^16+2*x^13*X[2,1,1,5]^2-4*x^13*X[2,1,1,5]-3*x^12*X [2,1,1,5]^2+2*x^13+6*x^12*X[2,1,1,5]+x^11*X[2,1,1,5]^2-3*x^12-2*x^11*X[2,1,1,5] +x^11+x^9*X[2,1,1,5]-x^9-4*x^8*X[2,1,1,5]+4*x^8+5*x^7*X[2,1,1,5]-5*x^7-3*x^6*X[ 2,1,1,5]+3*x^6+x^5*X[2,1,1,5]-x^5-x^4+4*x^3-6*x^2+4*x-1)/(x^6*X[2,1,1,5]-x^6-x^ 5*X[2,1,1,5]+x^5-2*x+1)/(x^18*X[2,1,1,5]^3-3*x^18*X[2,1,1,5]^2+3*x^18*X[2,1,1,5 ]-x^18+x^12*X[2,1,1,5]^2-2*x^12*X[2,1,1,5]-x^11*X[2,1,1,5]^2+x^12+2*x^11*X[2,1, 1,5]-x^11+x^8*X[2,1,1,5]-x^8-x^7*X[2,1,1,5]+x^7-x^3+3*x^2-3*x+1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [2, 1, 1, 5], equals , - 5/32 + ---- 64 143 57 n The variance equals , - ---- + ---- 1024 4096 903 357 n The , 3, -th moment about the mean is , - ---- + ----- 8192 32768 3041 9747 2 48309 The , 4, -th moment about the mean is , - ------- + -------- n - ------- n 1048576 16777216 8388608 The compositions of, 9, that yield the, 22, -th largest growth, that is, 1.9691817825046685829, are , [2, 1, 2, 4], [2, 1, 4, 2], [2, 2, 1, 4], [2, 4, 1, 2], [4, 1, 2, 2], [4, 2, 1, 2] Theorem Number, 22, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 2, 4] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n ) a(n) x = / ----- n = 0 2 9 8 6 5 4 3 (x - x + 1) (x + x - x - x + 2 x + x - 2 x + 1) - -------------------------------------------------------------- 12 11 8 7 6 5 4 3 2 x - x + x + x - 3 x + 3 x - x + 2 x - 5 x + 4 x - 1 and in Maple format -(x^2-x+1)*(x^9+x^8-x^6-x^5+2*x^4+x^3-2*x+1)/(x^12-x^11+x^8+x^7-3*x^6+3*x^5-x^4 +2*x^3-5*x^2+4*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 506, 1001, 1976, 3895, 7671, 15101, 29723, 58504, 115164, 226722, 446387, 878943, 1730737, 3408114, 6711252, 13215856, 26024779, 51248095, 100917625, 198726161, 391329035] The limit of a(n+1)/a(n) as n goes to infinity is 1.96918178250 a(n) is asymptotic to .580721127112*1.96918178250^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [2, 1, 2, 4], denoted by the variable, X[2, 1, 2, 4], is 11 2 11 11 6 6 - (x X[2, 1, 2, 4] - 2 x X[2, 1, 2, 4] + x - 2 x X[2, 1, 2, 4] + 2 x 5 5 4 4 3 2 + 2 x X[2, 1, 2, 4] - 2 x - x X[2, 1, 2, 4] + x - x + 3 x - 3 x + 1) / 12 2 12 11 2 12 / (x X[2, 1, 2, 4] - 2 x X[2, 1, 2, 4] - x X[2, 1, 2, 4] + x / 11 11 8 8 7 + 2 x X[2, 1, 2, 4] - x - x X[2, 1, 2, 4] + x - x X[2, 1, 2, 4] 7 6 6 5 5 + x + 3 x X[2, 1, 2, 4] - 3 x - 3 x X[2, 1, 2, 4] + 3 x 4 4 3 2 + x X[2, 1, 2, 4] - x + 2 x - 5 x + 4 x - 1) and in Maple format -(x^11*X[2,1,2,4]^2-2*x^11*X[2,1,2,4]+x^11-2*x^6*X[2,1,2,4]+2*x^6+2*x^5*X[2,1,2 ,4]-2*x^5-x^4*X[2,1,2,4]+x^4-x^3+3*x^2-3*x+1)/(x^12*X[2,1,2,4]^2-2*x^12*X[2,1,2 ,4]-x^11*X[2,1,2,4]^2+x^12+2*x^11*X[2,1,2,4]-x^11-x^8*X[2,1,2,4]+x^8-x^7*X[2,1, 2,4]+x^7+3*x^6*X[2,1,2,4]-3*x^6-3*x^5*X[2,1,2,4]+3*x^5+x^4*X[2,1,2,4]-x^4+2*x^3 -5*x^2+4*x-1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [2, 1, 2, 4], equals , - 5/32 + ---- 64 175 65 n The variance equals , - ---- + ---- 1024 4096 1671 135 n The , 3, -th moment about the mean is , - ---- + ----- 8192 8192 200365 12675 2 13867 The , 4, -th moment about the mean is , - ------- + -------- n + ------- n 1048576 16777216 8388608 The compositions of, 9, that yield the, 23, -th largest growth, that is, 1.9693144732632464526, are , [3, 1, 1, 4], [4, 1, 1, 3] Theorem Number, 23, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [3, 1, 1, 4] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 22 21 20 19 17 16 13 12 ) a(n) x = - (x + 3 x + 3 x + x + 2 x + 3 x + x + 2 x / ----- n = 0 11 10 8 7 6 5 4 3 2 - x - 2 x + 2 x + x - 4 x + 3 x - 2 x + 4 x - 6 x + 4 x - 1) / / ( / 18 17 16 15 11 9 8 6 3 2 (x + 3 x + 3 x + x - x + x + x - x + x - 3 x + 3 x - 1) 6 5 4 (x + x - x + 2 x - 1)) and in Maple format -(x^22+3*x^21+3*x^20+x^19+2*x^17+3*x^16+x^13+2*x^12-x^11-2*x^10+2*x^8+x^7-4*x^6 +3*x^5-2*x^4+4*x^3-6*x^2+4*x-1)/(x^18+3*x^17+3*x^16+x^15-x^11+x^9+x^8-x^6+x^3-3 *x^2+3*x-1)/(x^6+x^5-x^4+2*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 506, 1001, 1976, 3895, 7670, 15095, 29701, 58440, 115002, 226349, 445585, 877304, 1727512, 3401942, 6699668, 13194388, 25985277, 51175626, 100784702, 198482047, 390880020] The limit of a(n+1)/a(n) as n goes to infinity is 1.96931447326 a(n) is asymptotic to .578881635909*1.96931447326^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [3, 1, 1, 4], denoted by the variable, X[3, 1, 1, 4], is 22 4 22 3 21 4 - (x X[3, 1, 1, 4] - 4 x X[3, 1, 1, 4] + 3 x X[3, 1, 1, 4] 22 2 21 3 20 4 + 6 x X[3, 1, 1, 4] - 12 x X[3, 1, 1, 4] + 3 x X[3, 1, 1, 4] 22 21 2 20 3 - 4 x X[3, 1, 1, 4] + 18 x X[3, 1, 1, 4] - 12 x X[3, 1, 1, 4] 19 4 22 21 20 2 + x X[3, 1, 1, 4] + x - 12 x X[3, 1, 1, 4] + 18 x X[3, 1, 1, 4] 19 3 21 20 - 4 x X[3, 1, 1, 4] + 3 x - 12 x X[3, 1, 1, 4] 19 2 20 19 + 6 x X[3, 1, 1, 4] + 3 x - 4 x X[3, 1, 1, 4] 17 3 19 17 2 16 3 - 2 x X[3, 1, 1, 4] + x + 6 x X[3, 1, 1, 4] - 3 x X[3, 1, 1, 4] 17 16 2 17 16 - 6 x X[3, 1, 1, 4] + 9 x X[3, 1, 1, 4] + 2 x - 9 x X[3, 1, 1, 4] 16 13 3 13 2 13 + 3 x - x X[3, 1, 1, 4] + 3 x X[3, 1, 1, 4] - 3 x X[3, 1, 1, 4] 12 2 13 12 11 2 + 2 x X[3, 1, 1, 4] + x - 4 x X[3, 1, 1, 4] - x X[3, 1, 1, 4] 12 11 10 2 11 + 2 x + 2 x X[3, 1, 1, 4] - 2 x X[3, 1, 1, 4] - x 10 9 2 10 9 + 4 x X[3, 1, 1, 4] + x X[3, 1, 1, 4] - 2 x - x X[3, 1, 1, 4] 8 8 7 7 6 - 2 x X[3, 1, 1, 4] + 2 x - x X[3, 1, 1, 4] + x + 4 x X[3, 1, 1, 4] 6 5 5 4 4 3 2 - 4 x - 3 x X[3, 1, 1, 4] + 3 x + x X[3, 1, 1, 4] - 2 x + 4 x - 6 x / 6 6 5 5 + 4 x - 1) / ((x X[3, 1, 1, 4] - x + x X[3, 1, 1, 4] - x / 4 4 18 3 - x X[3, 1, 1, 4] + x - 2 x + 1) (x X[3, 1, 1, 4] 18 2 17 3 18 - 3 x X[3, 1, 1, 4] + 3 x X[3, 1, 1, 4] + 3 x X[3, 1, 1, 4] 17 2 16 3 18 17 - 9 x X[3, 1, 1, 4] + 3 x X[3, 1, 1, 4] - x + 9 x X[3, 1, 1, 4] 16 2 15 3 17 16 - 9 x X[3, 1, 1, 4] + x X[3, 1, 1, 4] - 3 x + 9 x X[3, 1, 1, 4] 15 2 16 15 15 - 3 x X[3, 1, 1, 4] - 3 x + 3 x X[3, 1, 1, 4] - x 11 2 11 11 9 2 + x X[3, 1, 1, 4] - 2 x X[3, 1, 1, 4] + x - x X[3, 1, 1, 4] 9 9 8 8 6 6 + 2 x X[3, 1, 1, 4] - x + x X[3, 1, 1, 4] - x - x X[3, 1, 1, 4] + x 3 2 - x + 3 x - 3 x + 1)) and in Maple format -(x^22*X[3,1,1,4]^4-4*x^22*X[3,1,1,4]^3+3*x^21*X[3,1,1,4]^4+6*x^22*X[3,1,1,4]^2 -12*x^21*X[3,1,1,4]^3+3*x^20*X[3,1,1,4]^4-4*x^22*X[3,1,1,4]+18*x^21*X[3,1,1,4]^ 2-12*x^20*X[3,1,1,4]^3+x^19*X[3,1,1,4]^4+x^22-12*x^21*X[3,1,1,4]+18*x^20*X[3,1, 1,4]^2-4*x^19*X[3,1,1,4]^3+3*x^21-12*x^20*X[3,1,1,4]+6*x^19*X[3,1,1,4]^2+3*x^20 -4*x^19*X[3,1,1,4]-2*x^17*X[3,1,1,4]^3+x^19+6*x^17*X[3,1,1,4]^2-3*x^16*X[3,1,1, 4]^3-6*x^17*X[3,1,1,4]+9*x^16*X[3,1,1,4]^2+2*x^17-9*x^16*X[3,1,1,4]+3*x^16-x^13 *X[3,1,1,4]^3+3*x^13*X[3,1,1,4]^2-3*x^13*X[3,1,1,4]+2*x^12*X[3,1,1,4]^2+x^13-4* x^12*X[3,1,1,4]-x^11*X[3,1,1,4]^2+2*x^12+2*x^11*X[3,1,1,4]-2*x^10*X[3,1,1,4]^2- x^11+4*x^10*X[3,1,1,4]+x^9*X[3,1,1,4]^2-2*x^10-x^9*X[3,1,1,4]-2*x^8*X[3,1,1,4]+ 2*x^8-x^7*X[3,1,1,4]+x^7+4*x^6*X[3,1,1,4]-4*x^6-3*x^5*X[3,1,1,4]+3*x^5+x^4*X[3, 1,1,4]-2*x^4+4*x^3-6*x^2+4*x-1)/(x^6*X[3,1,1,4]-x^6+x^5*X[3,1,1,4]-x^5-x^4*X[3, 1,1,4]+x^4-2*x+1)/(x^18*X[3,1,1,4]^3-3*x^18*X[3,1,1,4]^2+3*x^17*X[3,1,1,4]^3+3* x^18*X[3,1,1,4]-9*x^17*X[3,1,1,4]^2+3*x^16*X[3,1,1,4]^3-x^18+9*x^17*X[3,1,1,4]-\ 9*x^16*X[3,1,1,4]^2+x^15*X[3,1,1,4]^3-3*x^17+9*x^16*X[3,1,1,4]-3*x^15*X[3,1,1,4 ]^2-3*x^16+3*x^15*X[3,1,1,4]-x^15+x^11*X[3,1,1,4]^2-2*x^11*X[3,1,1,4]+x^11-x^9* X[3,1,1,4]^2+2*x^9*X[3,1,1,4]-x^9+x^8*X[3,1,1,4]-x^8-x^6*X[3,1,1,4]+x^6-x^3+3*x ^2-3*x+1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [3, 1, 1, 4], equals , - 5/32 + ---- 64 179 65 n The variance equals , - ---- + ---- 1024 4096 855 33 n The , 3, -th moment about the mean is , - ---- + ---- 4096 2048 182897 12675 2 6341 The , 4, -th moment about the mean is , - ------- + -------- n - ------- n 1048576 16777216 8388608 The compositions of, 9, that yield the, 24, -th largest growth, that is, 1.9703230372932668084, are , [1, 2, 2, 4], [1, 2, 4, 2], [1, 4, 2, 2], [2, 2, 4, 1], [2, 4, 2, 1], [4, 2, 2, 1] Theorem Number, 24, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 2, 2, 4] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 8 7 6 5 4 3 2 \ n x - x + 2 x - 2 x + x - x + 3 x - 3 x + 1 ) a(n) x = ------------------------------------------------ / 7 6 5 4 2 ----- (-1 + x) (x - x + 2 x - x - 2 x + 3 x - 1) n = 0 and in Maple format (x^8-x^7+2*x^6-2*x^5+x^4-x^3+3*x^2-3*x+1)/(-1+x)/(x^7-x^6+2*x^5-x^4-2*x^2+3*x-1 ) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 506, 1001, 1976, 3896, 7677, 15124, 29794, 58696, 115641, 227842, 448918, 884516, 1742794, 3433889, 6765901, 13331045, 26266496, 51753500, 101971108, 200915987, 395869328] The limit of a(n+1)/a(n) as n goes to infinity is 1.97032303729 a(n) is asymptotic to .577336084970*1.97032303729^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [1, 2, 2, 4], denoted by the variable, X[1, 2, 2, 4], is 8 8 7 7 6 6 (x X[1, 2, 2, 4] - x - x X[1, 2, 2, 4] + x + 2 x X[1, 2, 2, 4] - 2 x 5 5 4 4 3 2 - 2 x X[1, 2, 2, 4] + 2 x + x X[1, 2, 2, 4] - x + x - 3 x + 3 x - 1) / 7 7 6 6 / ((-1 + x) (x X[1, 2, 2, 4] - x - x X[1, 2, 2, 4] + x / 5 5 4 4 2 + 2 x X[1, 2, 2, 4] - 2 x - x X[1, 2, 2, 4] + x + 2 x - 3 x + 1)) and in Maple format (x^8*X[1,2,2,4]-x^8-x^7*X[1,2,2,4]+x^7+2*x^6*X[1,2,2,4]-2*x^6-2*x^5*X[1,2,2,4]+ 2*x^5+x^4*X[1,2,2,4]-x^4+x^3-3*x^2+3*x-1)/(-1+x)/(x^7*X[1,2,2,4]-x^7-x^6*X[1,2, 2,4]+x^6+2*x^5*X[1,2,2,4]-2*x^5-x^4*X[1,2,2,4]+x^4+2*x^2-3*x+1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [1, 2, 2, 4], equals , - 5/32 + ---- 64 23 69 n The variance equals , - --- + ---- 128 4096 7251 309 n The , 3, -th moment about the mean is , - ----- + ----- 32768 16384 98597 14283 2 27807 The , 4, -th moment about the mean is , - ------ + -------- n + ------- n 524288 16777216 8388608 The compositions of, 9, that yield the, 25, -th largest growth, that is, 1.9706560177668563263, are , [2, 1, 3, 3], [3, 3, 1, 2] Theorem Number, 25, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 3, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 8 7 5 3 \ n x + x - x + 2 x - 2 x + 1 ) a(n) x = - ----------------------------------------- / 9 7 5 4 3 2 ----- x - x + 2 x + x - 3 x - x + 3 x - 1 n = 0 and in Maple format -(x^8+x^7-x^5+2*x^3-2*x+1)/(x^9-x^7+2*x^5+x^4-3*x^3-x^2+3*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 506, 1001, 1976, 3896, 7677, 15124, 29794, 58697, 115648, 227873, 449027, 884848, 1743712, 3436259, 6771721, 13344816, 26298182, 51824871, 102129247, 201261897, 396618234] The limit of a(n+1)/a(n) as n goes to infinity is 1.97065601777 a(n) is asymptotic to .575503355863*1.97065601777^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [2, 1, 3, 3], denoted by the variable, X[2, 1, 3, 3], is 10 2 10 10 6 6 - (x X[2, 1, 3, 3] - 2 x X[2, 1, 3, 3] + x - x X[2, 1, 3, 3] + x 5 5 4 4 3 - x X[2, 1, 3, 3] + x + 2 x X[2, 1, 3, 3] - 2 x - x X[2, 1, 3, 3] 2 / 11 2 11 + 3 x - 3 x + 1) / (x X[2, 1, 3, 3] - 2 x X[2, 1, 3, 3] / 10 2 11 10 10 8 - x X[2, 1, 3, 3] + x + 2 x X[2, 1, 3, 3] - x - x X[2, 1, 3, 3] 8 7 7 6 6 5 + x - x X[2, 1, 3, 3] + x + x X[2, 1, 3, 3] - x + 2 x X[2, 1, 3, 3] 5 4 4 3 3 2 - 2 x - 3 x X[2, 1, 3, 3] + 3 x + x X[2, 1, 3, 3] + x - 5 x + 4 x - 1) and in Maple format -(x^10*X[2,1,3,3]^2-2*x^10*X[2,1,3,3]+x^10-x^6*X[2,1,3,3]+x^6-x^5*X[2,1,3,3]+x^ 5+2*x^4*X[2,1,3,3]-2*x^4-x^3*X[2,1,3,3]+3*x^2-3*x+1)/(x^11*X[2,1,3,3]^2-2*x^11* X[2,1,3,3]-x^10*X[2,1,3,3]^2+x^11+2*x^10*X[2,1,3,3]-x^10-x^8*X[2,1,3,3]+x^8-x^7 *X[2,1,3,3]+x^7+x^6*X[2,1,3,3]-x^6+2*x^5*X[2,1,3,3]-2*x^5-3*x^4*X[2,1,3,3]+3*x^ 4+x^3*X[2,1,3,3]+x^3-5*x^2+4*x-1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [2, 1, 3, 3], equals , - 5/32 + ---- 64 203 73 n The variance equals , - ---- + ---- 1024 4096 2619 783 n The , 3, -th moment about the mean is , - ---- + ----- 8192 32768 15987 2 608333 172091 The , 4, -th moment about the mean is , -------- n - ------- + ------- n 16777216 1048576 8388608 The compositions of, 9, that yield the, 26, -th largest growth, that is, 1.9708395870474530685, are , [2, 3, 1, 3], [3, 1, 3, 2] Theorem Number, 26, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 3, 1, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n ) a(n) x = / ----- n = 0 2 3 6 5 4 2 (x - x + 1) (x - x + 1) (x + 2 x + x - x - x + 1) - ---------------------------------------------------------------- 12 10 9 7 6 5 4 3 2 x - x + 2 x - x + 2 x - 3 x + 3 x + x - 5 x + 4 x - 1 and in Maple format -(x^2-x+1)*(x^3-x+1)*(x^6+2*x^5+x^4-x^2-x+1)/(x^12-x^10+2*x^9-x^7+2*x^6-3*x^5+3 *x^4+x^3-5*x^2+4*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 506, 1001, 1976, 3895, 7672, 15108, 29753, 58605, 115459, 227507, 448346, 883614, 1741513, 3432379, 6764915, 13332930, 26277535, 51789306, 102068807, 201161313, 396456130] The limit of a(n+1)/a(n) as n goes to infinity is 1.97083958705 a(n) is asymptotic to .573659392112*1.97083958705^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [2, 3, 1, 3], denoted by the variable, X[2, 3, 1, 3], is 11 2 11 10 2 11 - (x X[2, 3, 1, 3] - 2 x X[2, 3, 1, 3] + x X[2, 3, 1, 3] + x 10 9 2 10 9 - 2 x X[2, 3, 1, 3] - x X[2, 3, 1, 3] + x + 2 x X[2, 3, 1, 3] 8 2 9 8 8 7 7 + x X[2, 3, 1, 3] - x - 2 x X[2, 3, 1, 3] + x - x X[2, 3, 1, 3] + x 6 6 5 5 4 + x X[2, 3, 1, 3] - x - 2 x X[2, 3, 1, 3] + 2 x + 2 x X[2, 3, 1, 3] 4 3 2 / 12 2 - 2 x - x X[2, 3, 1, 3] + 3 x - 3 x + 1) / (x X[2, 3, 1, 3] / 12 12 10 2 10 - 2 x X[2, 3, 1, 3] + x - x X[2, 3, 1, 3] + 2 x X[2, 3, 1, 3] 9 2 10 9 8 2 + 2 x X[2, 3, 1, 3] - x - 4 x X[2, 3, 1, 3] - x X[2, 3, 1, 3] 9 8 7 7 6 + 2 x + x X[2, 3, 1, 3] + x X[2, 3, 1, 3] - x - 2 x X[2, 3, 1, 3] 6 5 5 4 4 + 2 x + 3 x X[2, 3, 1, 3] - 3 x - 3 x X[2, 3, 1, 3] + 3 x 3 3 2 + x X[2, 3, 1, 3] + x - 5 x + 4 x - 1) and in Maple format -(x^11*X[2,3,1,3]^2-2*x^11*X[2,3,1,3]+x^10*X[2,3,1,3]^2+x^11-2*x^10*X[2,3,1,3]- x^9*X[2,3,1,3]^2+x^10+2*x^9*X[2,3,1,3]+x^8*X[2,3,1,3]^2-x^9-2*x^8*X[2,3,1,3]+x^ 8-x^7*X[2,3,1,3]+x^7+x^6*X[2,3,1,3]-x^6-2*x^5*X[2,3,1,3]+2*x^5+2*x^4*X[2,3,1,3] -2*x^4-x^3*X[2,3,1,3]+3*x^2-3*x+1)/(x^12*X[2,3,1,3]^2-2*x^12*X[2,3,1,3]+x^12-x^ 10*X[2,3,1,3]^2+2*x^10*X[2,3,1,3]+2*x^9*X[2,3,1,3]^2-x^10-4*x^9*X[2,3,1,3]-x^8* X[2,3,1,3]^2+2*x^9+x^8*X[2,3,1,3]+x^7*X[2,3,1,3]-x^7-2*x^6*X[2,3,1,3]+2*x^6+3*x ^5*X[2,3,1,3]-3*x^5-3*x^4*X[2,3,1,3]+3*x^4+x^3*X[2,3,1,3]+x^3-5*x^2+4*x-1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [2, 3, 1, 3], equals , - 5/32 + ---- 64 207 73 n The variance equals , - ---- + ---- 1024 4096 2655 765 n The , 3, -th moment about the mean is , - ---- + ----- 8192 32768 587853 15987 2 141515 The , 4, -th moment about the mean is , - ------- + -------- n + ------- n 1048576 16777216 8388608 The compositions of, 9, that yield the, 27, -th largest growth, that is, 1.9708817901785482789, are , [3, 1, 2, 3], [3, 2, 1, 3] Theorem Number, 27, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [3, 1, 2, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 18 17 15 14 12 11 10 9 7 ) a(n) x = - (x + x + x + 2 x - x + 2 x + x - 3 x + 3 x / ----- n = 0 6 5 4 3 2 / 20 19 16 15 - x - 3 x + 2 x + 3 x - 6 x + 4 x - 1) / (x + 2 x + 3 x + x / 14 13 12 11 10 9 8 7 5 4 - 2 x + x + 3 x - 3 x - 3 x + 5 x + x - 5 x + 5 x - 2 x 3 2 - 6 x + 9 x - 5 x + 1) and in Maple format -(x^18+x^17+x^15+2*x^14-x^12+2*x^11+x^10-3*x^9+3*x^7-x^6-3*x^5+2*x^4+3*x^3-6*x^ 2+4*x-1)/(x^20+2*x^19+3*x^16+x^15-2*x^14+x^13+3*x^12-3*x^11-3*x^10+5*x^9+x^8-5* x^7+5*x^5-2*x^4-6*x^3+9*x^2-5*x+1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 506, 1001, 1976, 3895, 7671, 15102, 29731, 58542, 115303, 227156, 447607, 882129, 1738622, 3426860, 6754481, 13313247, 26240296, 51718450, 101933109, 200899907, 395950321] The limit of a(n+1)/a(n) as n goes to infinity is 1.97088179018 a(n) is asymptotic to .572549404359*1.97088179018^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [3, 1, 2, 3], denoted by the variable, X[3, 1, 2, 3], is 18 4 18 3 17 4 - (x X[3, 1, 2, 3] - 4 x X[3, 1, 2, 3] + x X[3, 1, 2, 3] 18 2 17 3 18 + 6 x X[3, 1, 2, 3] - 4 x X[3, 1, 2, 3] - 4 x X[3, 1, 2, 3] 17 2 18 17 15 3 + 6 x X[3, 1, 2, 3] + x - 4 x X[3, 1, 2, 3] - x X[3, 1, 2, 3] 17 15 2 14 3 15 + x + 3 x X[3, 1, 2, 3] - 2 x X[3, 1, 2, 3] - 3 x X[3, 1, 2, 3] 14 2 15 14 12 3 + 6 x X[3, 1, 2, 3] + x - 6 x X[3, 1, 2, 3] + x X[3, 1, 2, 3] 14 12 2 11 3 12 + 2 x - 3 x X[3, 1, 2, 3] - x X[3, 1, 2, 3] + 3 x X[3, 1, 2, 3] 11 2 12 11 10 2 + 4 x X[3, 1, 2, 3] - x - 5 x X[3, 1, 2, 3] + x X[3, 1, 2, 3] 11 10 9 2 10 + 2 x - 2 x X[3, 1, 2, 3] - 3 x X[3, 1, 2, 3] + x 9 8 2 9 8 + 6 x X[3, 1, 2, 3] + x X[3, 1, 2, 3] - 3 x - x X[3, 1, 2, 3] 7 7 6 6 5 - 3 x X[3, 1, 2, 3] + 3 x + x X[3, 1, 2, 3] - x + 3 x X[3, 1, 2, 3] 5 4 4 3 3 2 - 3 x - 3 x X[3, 1, 2, 3] + 2 x + x X[3, 1, 2, 3] + 3 x - 6 x + 4 x / 20 4 20 3 19 4 - 1) / (x X[3, 1, 2, 3] - 4 x X[3, 1, 2, 3] + 2 x X[3, 1, 2, 3] / 20 2 19 3 20 + 6 x X[3, 1, 2, 3] - 8 x X[3, 1, 2, 3] - 4 x X[3, 1, 2, 3] 19 2 17 4 20 19 + 12 x X[3, 1, 2, 3] - x X[3, 1, 2, 3] + x - 8 x X[3, 1, 2, 3] 17 3 19 17 2 + 3 x X[3, 1, 2, 3] + 2 x - 3 x X[3, 1, 2, 3] 16 3 17 16 2 - 3 x X[3, 1, 2, 3] + x X[3, 1, 2, 3] + 9 x X[3, 1, 2, 3] 15 3 16 15 2 - x X[3, 1, 2, 3] - 9 x X[3, 1, 2, 3] + 3 x X[3, 1, 2, 3] 14 3 16 15 + 2 x X[3, 1, 2, 3] + 3 x - 3 x X[3, 1, 2, 3] 14 2 15 14 13 2 - 6 x X[3, 1, 2, 3] + x + 6 x X[3, 1, 2, 3] + x X[3, 1, 2, 3] 12 3 14 13 - 2 x X[3, 1, 2, 3] - 2 x - 2 x X[3, 1, 2, 3] 12 2 11 3 13 12 + 7 x X[3, 1, 2, 3] + x X[3, 1, 2, 3] + x - 8 x X[3, 1, 2, 3] 11 2 12 11 - 5 x X[3, 1, 2, 3] + 3 x + 7 x X[3, 1, 2, 3] 10 2 11 10 9 2 - 3 x X[3, 1, 2, 3] - 3 x + 6 x X[3, 1, 2, 3] + 4 x X[3, 1, 2, 3] 10 9 8 2 9 8 - 3 x - 9 x X[3, 1, 2, 3] - x X[3, 1, 2, 3] + 5 x + x 7 7 5 5 + 5 x X[3, 1, 2, 3] - 5 x - 5 x X[3, 1, 2, 3] + 5 x 4 4 3 3 2 + 4 x X[3, 1, 2, 3] - 2 x - x X[3, 1, 2, 3] - 6 x + 9 x - 5 x + 1) and in Maple format -(x^18*X[3,1,2,3]^4-4*x^18*X[3,1,2,3]^3+x^17*X[3,1,2,3]^4+6*x^18*X[3,1,2,3]^2-4 *x^17*X[3,1,2,3]^3-4*x^18*X[3,1,2,3]+6*x^17*X[3,1,2,3]^2+x^18-4*x^17*X[3,1,2,3] -x^15*X[3,1,2,3]^3+x^17+3*x^15*X[3,1,2,3]^2-2*x^14*X[3,1,2,3]^3-3*x^15*X[3,1,2, 3]+6*x^14*X[3,1,2,3]^2+x^15-6*x^14*X[3,1,2,3]+x^12*X[3,1,2,3]^3+2*x^14-3*x^12*X [3,1,2,3]^2-x^11*X[3,1,2,3]^3+3*x^12*X[3,1,2,3]+4*x^11*X[3,1,2,3]^2-x^12-5*x^11 *X[3,1,2,3]+x^10*X[3,1,2,3]^2+2*x^11-2*x^10*X[3,1,2,3]-3*x^9*X[3,1,2,3]^2+x^10+ 6*x^9*X[3,1,2,3]+x^8*X[3,1,2,3]^2-3*x^9-x^8*X[3,1,2,3]-3*x^7*X[3,1,2,3]+3*x^7+x ^6*X[3,1,2,3]-x^6+3*x^5*X[3,1,2,3]-3*x^5-3*x^4*X[3,1,2,3]+2*x^4+x^3*X[3,1,2,3]+ 3*x^3-6*x^2+4*x-1)/(x^20*X[3,1,2,3]^4-4*x^20*X[3,1,2,3]^3+2*x^19*X[3,1,2,3]^4+6 *x^20*X[3,1,2,3]^2-8*x^19*X[3,1,2,3]^3-4*x^20*X[3,1,2,3]+12*x^19*X[3,1,2,3]^2-x ^17*X[3,1,2,3]^4+x^20-8*x^19*X[3,1,2,3]+3*x^17*X[3,1,2,3]^3+2*x^19-3*x^17*X[3,1 ,2,3]^2-3*x^16*X[3,1,2,3]^3+x^17*X[3,1,2,3]+9*x^16*X[3,1,2,3]^2-x^15*X[3,1,2,3] ^3-9*x^16*X[3,1,2,3]+3*x^15*X[3,1,2,3]^2+2*x^14*X[3,1,2,3]^3+3*x^16-3*x^15*X[3, 1,2,3]-6*x^14*X[3,1,2,3]^2+x^15+6*x^14*X[3,1,2,3]+x^13*X[3,1,2,3]^2-2*x^12*X[3, 1,2,3]^3-2*x^14-2*x^13*X[3,1,2,3]+7*x^12*X[3,1,2,3]^2+x^11*X[3,1,2,3]^3+x^13-8* x^12*X[3,1,2,3]-5*x^11*X[3,1,2,3]^2+3*x^12+7*x^11*X[3,1,2,3]-3*x^10*X[3,1,2,3]^ 2-3*x^11+6*x^10*X[3,1,2,3]+4*x^9*X[3,1,2,3]^2-3*x^10-9*x^9*X[3,1,2,3]-x^8*X[3,1 ,2,3]^2+5*x^9+x^8+5*x^7*X[3,1,2,3]-5*x^7-5*x^5*X[3,1,2,3]+5*x^5+4*x^4*X[3,1,2,3 ]-2*x^4-x^3*X[3,1,2,3]-6*x^3+9*x^2-5*x+1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [3, 1, 2, 3], equals , - 5/32 + ---- 64 211 73 n The variance equals , - ---- + ---- 1024 4096 2745 759 n The , 3, -th moment about the mean is , - ---- + ----- 8192 32768 603853 15987 2 127067 The , 4, -th moment about the mean is , - ------- + -------- n + ------- n 1048576 16777216 8388608 The compositions of, 9, that yield the, 28, -th largest growth, that is, 1.9717270001741243154, are , [1, 3, 2, 3], [3, 2, 3, 1] Theorem Number, 28, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 3, 2, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n ) a(n) x = / ----- n = 0 3 8 5 2 (x - x + 1) (x + x + x - 2 x + 1) -------------------------------------------------------------------- 10 9 8 7 6 5 4 3 2 (-1 + x) (x + x - x + x + x - x + 2 x - x - 2 x + 3 x - 1) and in Maple format (x^3-x+1)*(x^8+x^5+x^2-2*x+1)/(-1+x)/(x^10+x^9-x^8+x^7+x^6-x^5+2*x^4-x^3-2*x^2+ 3*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 506, 1001, 1976, 3896, 7678, 15131, 29823, 58791, 115912, 228549, 450654, 888603, 1752135, 3454790, 6811947, 13431304, 26482802, 52216709, 102956870, 203002578, 400265446] The limit of a(n+1)/a(n) as n goes to infinity is 1.97172700017 a(n) is asymptotic to .571406042186*1.97172700017^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [1, 3, 2, 3], denoted by the variable, X[1, 3, 2, 3], is 11 2 11 11 9 2 (x X[1, 3, 2, 3] - 2 x X[1, 3, 2, 3] + x - x X[1, 3, 2, 3] 9 8 2 9 8 8 + 2 x X[1, 3, 2, 3] + x X[1, 3, 2, 3] - x - 3 x X[1, 3, 2, 3] + 2 x 6 6 5 5 4 + x X[1, 3, 2, 3] - x - 2 x X[1, 3, 2, 3] + 2 x + 2 x X[1, 3, 2, 3] 4 3 2 / - 2 x - x X[1, 3, 2, 3] + 3 x - 3 x + 1) / ((-1 + x) ( / 10 2 10 9 2 10 x X[1, 3, 2, 3] - 2 x X[1, 3, 2, 3] + x X[1, 3, 2, 3] + x 9 8 2 9 8 8 - 2 x X[1, 3, 2, 3] - x X[1, 3, 2, 3] + x + 2 x X[1, 3, 2, 3] - x 7 7 6 6 5 5 - x X[1, 3, 2, 3] + x - x X[1, 3, 2, 3] + x + x X[1, 3, 2, 3] - x 4 4 3 3 2 - 2 x X[1, 3, 2, 3] + 2 x + x X[1, 3, 2, 3] - x - 2 x + 3 x - 1)) and in Maple format (x^11*X[1,3,2,3]^2-2*x^11*X[1,3,2,3]+x^11-x^9*X[1,3,2,3]^2+2*x^9*X[1,3,2,3]+x^8 *X[1,3,2,3]^2-x^9-3*x^8*X[1,3,2,3]+2*x^8+x^6*X[1,3,2,3]-x^6-2*x^5*X[1,3,2,3]+2* x^5+2*x^4*X[1,3,2,3]-2*x^4-x^3*X[1,3,2,3]+3*x^2-3*x+1)/(-1+x)/(x^10*X[1,3,2,3]^ 2-2*x^10*X[1,3,2,3]+x^9*X[1,3,2,3]^2+x^10-2*x^9*X[1,3,2,3]-x^8*X[1,3,2,3]^2+x^9 +2*x^8*X[1,3,2,3]-x^8-x^7*X[1,3,2,3]+x^7-x^6*X[1,3,2,3]+x^6+x^5*X[1,3,2,3]-x^5-\ 2*x^4*X[1,3,2,3]+2*x^4+x^3*X[1,3,2,3]-x^3-2*x^2+3*x-1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [1, 3, 2, 3], equals , - 5/32 + ---- 64 27 77 n The variance equals , - --- + ---- 128 4096 11535 867 n The , 3, -th moment about the mean is , - ----- + ----- 32768 32768 321621 17787 2 187615 The , 4, -th moment about the mean is , - ------ + -------- n + ------- n 524288 16777216 8388608 The compositions of, 9, that yield the, 29, -th largest growth, that is, 1.9735704833094816886, are , [1, 2, 3, 3], [1, 3, 3, 2], [2, 3, 3, 1], [3, 3, 2, 1] Theorem Number, 29, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 2, 3, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 8 7 6 5 4 2 \ n x - x + x + x - 2 x + 3 x - 3 x + 1 ) a(n) x = ----------------------------------------------- / 7 6 4 3 2 ----- (-1 + x) (x - x + 2 x - x - 2 x + 3 x - 1) n = 0 and in Maple format (x^8-x^7+x^6+x^5-2*x^4+3*x^2-3*x+1)/(-1+x)/(x^7-x^6+2*x^4-x^3-2*x^2+3*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 506, 1001, 1977, 3902, 7700, 15195, 29987, 59181, 116799, 230514, 454940, 897861, 1771996, 3497161, 6901893, 13621369, 26882727, 53054952, 104707685, 206647998, 407834395] The limit of a(n+1)/a(n) as n goes to infinity is 1.97357048331 a(n) is asymptotic to .566114946302*1.97357048331^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [1, 2, 3, 3], denoted by the variable, X[1, 2, 3, 3], is 8 8 7 7 6 6 (x X[1, 2, 3, 3] - x - x X[1, 2, 3, 3] + x + x X[1, 2, 3, 3] - x 5 5 4 4 3 + x X[1, 2, 3, 3] - x - 2 x X[1, 2, 3, 3] + 2 x + x X[1, 2, 3, 3] 2 / 7 7 6 - 3 x + 3 x - 1) / ((-1 + x) (x X[1, 2, 3, 3] - x - x X[1, 2, 3, 3] / 6 4 4 3 3 2 + x + 2 x X[1, 2, 3, 3] - 2 x - x X[1, 2, 3, 3] + x + 2 x - 3 x + 1) ) and in Maple format (x^8*X[1,2,3,3]-x^8-x^7*X[1,2,3,3]+x^7+x^6*X[1,2,3,3]-x^6+x^5*X[1,2,3,3]-x^5-2* x^4*X[1,2,3,3]+2*x^4+x^3*X[1,2,3,3]-3*x^2+3*x-1)/(-1+x)/(x^7*X[1,2,3,3]-x^7-x^6 *X[1,2,3,3]+x^6+2*x^4*X[1,2,3,3]-2*x^4-x^3*X[1,2,3,3]+x^3+2*x^2-3*x+1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [1, 2, 3, 3], equals , - 5/32 + ---- 64 59 85 n The variance equals , - --- + ---- 256 4096 13443 531 n The , 3, -th moment about the mean is , - ----- + ----- 32768 16384 369685 21675 2 258863 The , 4, -th moment about the mean is , - ------ + -------- n + ------- n 524288 16777216 8388608 The compositions of, 9, that yield the, 30, -th largest growth, that is, 1.9756564557792322769, are , [2, 2, 2, 3], [2, 2, 3, 2], [2, 3, 2, 2], [3, 2, 2, 2] Theorem Number, 30, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 2, 2, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 7 6 5 4 2 \ n x - x + 2 x - 2 x + 3 x - 3 x + 1 ) a(n) x = - -------------------------------------------------- / 8 7 6 5 4 3 2 ----- x - x + 2 x - 3 x + 3 x + x - 5 x + 4 x - 1 n = 0 and in Maple format -(x^7-x^6+2*x^5-2*x^4+3*x^2-3*x+1)/(x^8-x^7+2*x^6-3*x^5+3*x^4+x^3-5*x^2+4*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 506, 1001, 1977, 3902, 7701, 15202, 30018, 59289, 117124, 231400, 457194, 903318, 1784742, 3526162, 6966615, 13763729, 27192387, 53722620, 106136993, 209689501, 414273443] The limit of a(n+1)/a(n) as n goes to infinity is 1.97565645578 a(n) is asymptotic to .557113786055*1.97565645578^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [2, 2, 2, 3], denoted by the variable, X[2, 2, 2, 3], is 7 7 6 6 5 5 - (x X[2, 2, 2, 3] - x - x X[2, 2, 2, 3] + x + 2 x X[2, 2, 2, 3] - 2 x 4 4 3 2 / - 2 x X[2, 2, 2, 3] + 2 x + x X[2, 2, 2, 3] - 3 x + 3 x - 1) / ( / 8 8 7 7 6 6 x X[2, 2, 2, 3] - x - x X[2, 2, 2, 3] + x + 2 x X[2, 2, 2, 3] - 2 x 5 5 4 4 3 - 3 x X[2, 2, 2, 3] + 3 x + 3 x X[2, 2, 2, 3] - 3 x - x X[2, 2, 2, 3] 3 2 - x + 5 x - 4 x + 1) and in Maple format -(x^7*X[2,2,2,3]-x^7-x^6*X[2,2,2,3]+x^6+2*x^5*X[2,2,2,3]-2*x^5-2*x^4*X[2,2,2,3] +2*x^4+x^3*X[2,2,2,3]-3*x^2+3*x-1)/(x^8*X[2,2,2,3]-x^8-x^7*X[2,2,2,3]+x^7+2*x^6 *X[2,2,2,3]-2*x^6-3*x^5*X[2,2,2,3]+3*x^5+3*x^4*X[2,2,2,3]-3*x^4-x^3*X[2,2,2,3]- x^3+5*x^2-4*x+1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [2, 2, 2, 3], equals , - 5/32 + ---- 64 287 97 n The variance equals , - ---- + ---- 1024 4096 5115 45 n The , 3, -th moment about the mean is , - ---- + ---- 8192 1024 1476029 28227 2 482219 The , 4, -th moment about the mean is , - ------- + -------- n + ------- n 1048576 16777216 8388608 The compositions of, 9, that yield the, 31, -th largest growth, that is, 1.9835828434243263304, are , [1, 1, 7], [1, 7, 1], [7, 1, 1] Theorem Number, 31, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 1, 7] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 2 5 4 2 \ n (x - x + 1) (x + x - x - x + 1) ) a(n) x = - ------------------------------------------ / 6 5 4 3 2 2 ----- (x + x + x + x + x + x - 1) (-1 + x) n = 0 and in Maple format -(x^2-x+1)*(x^5+x^4-x^2-x+1)/(x^6+x^5+x^4+x^3+x^2+x-1)/(-1+x)^2 The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 507, 1007, 1999, 3967, 7871, 15615, 30976, 61446, 121886, 241774, 479582, 951294, 1886974, 3742973, 7424501, 14727117, 29212461, 57945341, 114939389, 227991805, 452240638] The limit of a(n+1)/a(n) as n goes to infinity is 1.98358284342 a(n) is asymptotic to .539335853149*1.98358284342^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [1, 1, 7], denoted by the variable, X[1, 1, 7], is 7 7 2 x X[1, 1, 7] - x - x + 2 x - 1 - --------------------------------------- 7 7 (-1 + x) (x X[1, 1, 7] - x + 2 x - 1) and in Maple format -(x^7*X[1,1,7]-x^7-x^2+2*x-1)/(-1+x)/(x^7*X[1,1,7]-x^7+2*x-1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [1, 1, 7], equals , - 9/128 + --- 128 991 115 n The variance equals , - ----- + ----- 16384 16384 11415 2943 n The , 3, -th moment about the mean is , - ------ + ------ 262144 524288 1530895 39675 2 49177 The , 4, -th moment about the mean is , - --------- + --------- n + -------- n 268435456 268435456 67108864 The compositions of, 9, that yield the, 32, -th largest growth, that is, 1.9838613961621262283, are , [2, 1, 6], [6, 1, 2] Theorem Number, 32, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 6] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 13 6 2 \ n x + x + x - 2 x + 1 ) a(n) x = - -------------------------------- / 7 6 7 ----- (x - x + 2 x - 1) (x - x + 1) n = 0 and in Maple format -(x^13+x^6+x^2-2*x+1)/(x^7-x^6+2*x-1)/(x^7-x+1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 507, 1007, 1999, 3967, 7871, 15615, 30977, 61452, 121909, 241846, 479783, 951816, 1888264, 3746048, 7431636, 14743335, 29248737, 58025450, 115114467, 228371171, 453056780] The limit of a(n+1)/a(n) as n goes to infinity is 1.98386139616 a(n) is asymptotic to .538037894604*1.98386139616^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [2, 1, 6], denoted by the variable, X[2, 1, 6], is 13 2 13 13 6 6 2 - (x X[2, 1, 6] - 2 x X[2, 1, 6] + x - x X[2, 1, 6] + x + x - 2 x + 1 / 7 7 ) / ((x X[2, 1, 6] - x + x - 1) / 7 7 6 6 (x X[2, 1, 6] - x - x X[2, 1, 6] + x - 2 x + 1)) and in Maple format -(x^13*X[2,1,6]^2-2*x^13*X[2,1,6]+x^13-x^6*X[2,1,6]+x^6+x^2-2*x+1)/(x^7*X[2,1,6 ]-x^7+x-1)/(x^7*X[2,1,6]-x^7-x^6*X[2,1,6]+x^6-2*x+1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [2, 1, 6], equals , - 9/128 + --- 128 1059 119 n The variance equals , - ----- + ----- 16384 16384 28437 819 n The , 3, -th moment about the mean is , - ------ + ------ 524288 131072 6317079 42483 2 109661 The , 4, -th moment about the mean is , - --------- + --------- n + -------- n 268435456 268435456 67108864 The compositions of, 9, that yield the, 33, -th largest growth, that is, 1.9843858253440954550, are , [3, 1, 5], [5, 1, 3] Theorem Number, 33, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [3, 1, 5] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 12 11 5 2 \ n x + x + x + x - 2 x + 1 ) a(n) x = - ------------------------------------------ / 7 6 7 6 5 ----- (x + x - x + 1) (x + x - x + 2 x - 1) n = 0 and in Maple format -(x^12+x^11+x^5+x^2-2*x+1)/(x^7+x^6-x+1)/(x^7+x^6-x^5+2*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 507, 1007, 1999, 3967, 7871, 15616, 30983, 61475, 121981, 242047, 480305, 953105, 1891331, 3753146, 7447722, 14779206, 29327719, 58197597, 115486576, 229169995, 454761712] The limit of a(n+1)/a(n) as n goes to infinity is 1.98438582534 a(n) is asymptotic to .535796930829*1.98438582534^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [3, 1, 5], denoted by the variable, X[3, 1, 5], is 12 2 12 11 2 12 - (x X[3, 1, 5] - 2 x X[3, 1, 5] + x X[3, 1, 5] + x 11 11 5 5 2 / - 2 x X[3, 1, 5] + x - x X[3, 1, 5] + x + x - 2 x + 1) / ( / 7 7 6 6 (x X[3, 1, 5] - x + x X[3, 1, 5] - x + x - 1) 7 7 6 6 5 5 (x X[3, 1, 5] - x + x X[3, 1, 5] - x - x X[3, 1, 5] + x - 2 x + 1)) and in Maple format -(x^12*X[3,1,5]^2-2*x^12*X[3,1,5]+x^11*X[3,1,5]^2+x^12-2*x^11*X[3,1,5]+x^11-x^5 *X[3,1,5]+x^5+x^2-2*x+1)/(x^7*X[3,1,5]-x^7+x^6*X[3,1,5]-x^6+x-1)/(x^7*X[3,1,5]- x^7+x^6*X[3,1,5]-x^6-x^5*X[3,1,5]+x^5-2*x+1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [3, 1, 5], equals , - 9/128 + --- 128 1187 127 n The variance equals , - ----- + ----- 16384 16384 39921 249 n The , 3, -th moment about the mean is , - ------ + ----- 524288 32768 17737943 48387 2 257545 The , 4, -th moment about the mean is , - --------- + --------- n + -------- n 268435456 268435456 67108864 The compositions of, 9, that yield the, 34, -th largest growth, that is, 1.9846407398915826487, are , [2, 2, 5], [2, 5, 2], [5, 2, 2] Theorem Number, 34, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 2, 5] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 7 6 5 2 \ n x - x + x + x - 2 x + 1 ) a(n) x = - ------------------------------------ / 8 7 6 5 2 ----- x - x + 2 x - x - 2 x + 3 x - 1 n = 0 and in Maple format -(x^7-x^6+x^5+x^2-2*x+1)/(x^8-x^7+2*x^6-x^5-2*x^2+3*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 507, 1007, 1999, 3967, 7872, 15621, 30999, 61518, 122087, 242295, 480866, 954346, 1894037, 3758990, 7460256, 14805943, 29384495, 58317683, 115739663, 229701656, 455875258] The limit of a(n+1)/a(n) as n goes to infinity is 1.98464073989 a(n) is asymptotic to .535043291827*1.98464073989^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [2, 2, 5], denoted by the variable, X[2, 2, 5], is 7 7 6 6 5 5 2 - (x X[2, 2, 5] - x - x X[2, 2, 5] + x + x X[2, 2, 5] - x - x + 2 x - 1) / 8 8 7 7 6 6 / (x X[2, 2, 5] - x - x X[2, 2, 5] + x + 2 x X[2, 2, 5] - 2 x / 5 5 2 - x X[2, 2, 5] + x + 2 x - 3 x + 1) and in Maple format -(x^7*X[2,2,5]-x^7-x^6*X[2,2,5]+x^6+x^5*X[2,2,5]-x^5-x^2+2*x-1)/(x^8*X[2,2,5]-x ^8-x^7*X[2,2,5]+x^7+2*x^6*X[2,2,5]-2*x^6-x^5*X[2,2,5]+x^5+2*x^2-3*x+1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [2, 2, 5], equals , - 9/128 + --- 128 1231 131 n The variance equals , - ----- + ----- 16384 16384 21957 4347 n The , 3, -th moment about the mean is , - ------ + ------ 262144 524288 21704239 51483 2 339929 The , 4, -th moment about the mean is , - --------- + --------- n + -------- n 268435456 268435456 67108864 The compositions of, 9, that yield the, 35, -th largest growth, that is, 1.9850654703526320630, are , [3, 2, 4], [4, 2, 3] Theorem Number, 35, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [3, 2, 4] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n ) a(n) x = / ----- n = 0 10 6 5 4 2 x + x - x + x + x - 2 x + 1 - ----------------------------------------------------------- 12 11 10 8 7 6 5 4 2 x + x - x + x + x - x + 2 x - x - 2 x + 3 x - 1 and in Maple format -(x^10+x^6-x^5+x^4+x^2-2*x+1)/(x^12+x^11-x^10+x^8+x^7-x^6+2*x^5-x^4-2*x^2+3*x-1 ) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 507, 1007, 1999, 3967, 7872, 15622, 31005, 61541, 122158, 242490, 481363, 955550, 1896852, 3765407, 7474616, 14837636, 29453698, 58467508, 116061775, 230390109, 457339279] The limit of a(n+1)/a(n) as n goes to infinity is 1.98506547035 a(n) is asymptotic to .533326675988*1.98506547035^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [3, 2, 4], denoted by the variable, X[3, 2, 4], is 10 2 10 10 6 6 - (x X[3, 2, 4] - 2 x X[3, 2, 4] + x - x X[3, 2, 4] + x 5 5 4 4 2 / + x X[3, 2, 4] - x - x X[3, 2, 4] + x + x - 2 x + 1) / ( / 12 2 12 11 2 12 x X[3, 2, 4] - 2 x X[3, 2, 4] + x X[3, 2, 4] + x 11 10 2 11 10 10 - 2 x X[3, 2, 4] - x X[3, 2, 4] + x + 2 x X[3, 2, 4] - x 8 8 7 7 6 6 - x X[3, 2, 4] + x - x X[3, 2, 4] + x + x X[3, 2, 4] - x 5 5 4 4 2 - 2 x X[3, 2, 4] + 2 x + x X[3, 2, 4] - x - 2 x + 3 x - 1) and in Maple format -(x^10*X[3,2,4]^2-2*x^10*X[3,2,4]+x^10-x^6*X[3,2,4]+x^6+x^5*X[3,2,4]-x^5-x^4*X[ 3,2,4]+x^4+x^2-2*x+1)/(x^12*X[3,2,4]^2-2*x^12*X[3,2,4]+x^11*X[3,2,4]^2+x^12-2*x ^11*X[3,2,4]-x^10*X[3,2,4]^2+x^11+2*x^10*X[3,2,4]-x^10-x^8*X[3,2,4]+x^8-x^7*X[3 ,2,4]+x^7+x^6*X[3,2,4]-x^6-2*x^5*X[3,2,4]+2*x^5+x^4*X[3,2,4]-x^4-2*x^2+3*x-1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [3, 2, 4], equals , - 9/128 + --- 128 1359 139 n The variance equals , - ----- + ----- 16384 16384 29247 5211 n The , 3, -th moment about the mean is , - ------ + ------ 262144 524288 42647151 57963 2 601909 The , 4, -th moment about the mean is , - --------- + --------- n + -------- n 268435456 268435456 67108864 The compositions of, 9, that yield the, 36, -th largest growth, that is, 1.9853288885629234253, are , [4, 1, 4] Theorem Number, 36, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [4, 1, 4] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 11 10 9 4 2 \ n x + x + x + x + x - 2 x + 1 ) a(n) x = - ---------------------------------------------------- / 7 6 5 4 7 6 5 ----- (x + x + x - x + 2 x - 1) (x + x + x - x + 1) n = 0 and in Maple format -(x^11+x^10+x^9+x^4+x^2-2*x+1)/(x^7+x^6+x^5-x^4+2*x-1)/(x^7+x^6+x^5-x+1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 507, 1007, 1999, 3967, 7872, 15622, 31006, 61547, 122182, 242568, 481586, 956135, 1898297, 3768827, 7482467, 14855267, 29492665, 58552621, 116246024, 230786160, 458185704] The limit of a(n+1)/a(n) as n goes to infinity is 1.98532888856 a(n) is asymptotic to .532190286380*1.98532888856^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [4, 1, 4], denoted by the variable, X[4, 1, 4], is 11 2 11 10 2 11 - (x X[4, 1, 4] - 2 x X[4, 1, 4] + x X[4, 1, 4] + x 10 9 2 10 9 9 - 2 x X[4, 1, 4] + x X[4, 1, 4] + x - 2 x X[4, 1, 4] + x 4 4 2 / - x X[4, 1, 4] + x + x - 2 x + 1) / ( / 7 7 6 6 5 5 (x X[4, 1, 4] - x + x X[4, 1, 4] - x + x X[4, 1, 4] - x + x - 1) ( 7 7 6 6 5 5 x X[4, 1, 4] - x + x X[4, 1, 4] - x + x X[4, 1, 4] - x 4 4 - x X[4, 1, 4] + x - 2 x + 1)) and in Maple format -(x^11*X[4,1,4]^2-2*x^11*X[4,1,4]+x^10*X[4,1,4]^2+x^11-2*x^10*X[4,1,4]+x^9*X[4, 1,4]^2+x^10-2*x^9*X[4,1,4]+x^9-x^4*X[4,1,4]+x^4+x^2-2*x+1)/(x^7*X[4,1,4]-x^7+x^ 6*X[4,1,4]-x^6+x^5*X[4,1,4]-x^5+x-1)/(x^7*X[4,1,4]-x^7+x^6*X[4,1,4]-x^6+x^5*X[4 ,1,4]-x^5-x^4*X[4,1,4]+x^4-2*x+1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [4, 1, 4], equals , - 9/128 + --- 128 1427 143 n The variance equals , - ----- + ----- 16384 16384 64725 1389 n The , 3, -th moment about the mean is , - ------ + ------ 524288 131072 48712823 61347 2 661001 The , 4, -th moment about the mean is , - --------- + --------- n + -------- n 268435456 268435456 67108864 The compositions of, 9, that yield the, 37, -th largest growth, that is, 1.9855529777414181545, are , [2, 3, 4], [2, 4, 3], [3, 4, 2], [4, 3, 2] Theorem Number, 37, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 3, 4] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 7 5 4 2 \ n x - x + x + x - 2 x + 1 ) a(n) x = - ------------------------------------ / 8 7 5 4 2 ----- x - x + 2 x - x - 2 x + 3 x - 1 n = 0 and in Maple format -(x^7-x^5+x^4+x^2-2*x+1)/(x^8-x^7+2*x^5-x^4-2*x^2+3*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 507, 1007, 1999, 3968, 7877, 15638, 31048, 61646, 122401, 243035, 482562, 958157, 1902477, 3777474, 7500378, 14892398, 29569641, 58712080, 116575933, 231467677, 459591323] The limit of a(n+1)/a(n) as n goes to infinity is 1.98555297774 a(n) is asymptotic to .532019371341*1.98555297774^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [2, 3, 4], denoted by the variable, X[2, 3, 4], is 7 7 5 5 4 4 2 - (x X[2, 3, 4] - x - x X[2, 3, 4] + x + x X[2, 3, 4] - x - x + 2 x - 1) / 8 8 7 7 5 5 / (x X[2, 3, 4] - x - x X[2, 3, 4] + x + 2 x X[2, 3, 4] - 2 x / 4 4 2 - x X[2, 3, 4] + x + 2 x - 3 x + 1) and in Maple format -(x^7*X[2,3,4]-x^7-x^5*X[2,3,4]+x^5+x^4*X[2,3,4]-x^4-x^2+2*x-1)/(x^8*X[2,3,4]-x ^8-x^7*X[2,3,4]+x^7+2*x^5*X[2,3,4]-2*x^5-x^4*X[2,3,4]+x^4+2*x^2-3*x+1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [2, 3, 4], equals , - 9/128 + --- 128 1439 147 n The variance equals , - ----- + ----- 16384 16384 32991 5967 n The , 3, -th moment about the mean is , - ------ + ------ 262144 524288 50211983 64827 2 783945 The , 4, -th moment about the mean is , - --------- + --------- n + -------- n 268435456 268435456 67108864 The compositions of, 9, that yield the, 38, -th largest growth, that is, 1.9874108030247649893, are , [3, 3, 3] Theorem Number, 38, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [3, 3, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 2 4 3 2 \ n (x - x + 1) (x + x - x - x + 1) ) a(n) x = - ----------------------------------------- / 8 7 6 4 3 2 ----- x + x - x + 2 x - x - 2 x + 3 x - 1 n = 0 and in Maple format -(x^2-x+1)*(x^4+x^3-x^2-x+1)/(x^8+x^7-x^6+2*x^4-x^3-2*x^2+3*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 507, 1007, 2000, 3973, 7894, 15687, 31176, 61961, 123146, 244748, 486422, 966726, 1921283, 3818373, 7588662, 15081768, 29973645, 59569927, 118389912, 235289409, 467616760] The limit of a(n+1)/a(n) as n goes to infinity is 1.98741080302 a(n) is asymptotic to .526333163485*1.98741080302^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [3, 3, 3], denoted by the variable, X[3, 3, 3], is 6 6 4 4 3 3 2 - (x X[3, 3, 3] - x - x X[3, 3, 3] + x + x X[3, 3, 3] - x - x + 2 x - 1) / 8 8 7 7 6 6 / (x X[3, 3, 3] - x + x X[3, 3, 3] - x - x X[3, 3, 3] + x / 4 4 3 3 2 + 2 x X[3, 3, 3] - 2 x - x X[3, 3, 3] + x + 2 x - 3 x + 1) and in Maple format -(x^6*X[3,3,3]-x^6-x^4*X[3,3,3]+x^4+x^3*X[3,3,3]-x^3-x^2+2*x-1)/(x^8*X[3,3,3]-x ^8+x^7*X[3,3,3]-x^7-x^6*X[3,3,3]+x^6+2*x^4*X[3,3,3]-2*x^4-x^3*X[3,3,3]+x^3+2*x^ 2-3*x+1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [3, 3, 3], equals , - 9/128 + --- 128 1951 187 n The variance equals , - ----- + ----- 16384 16384 67161 10863 n The , 3, -th moment about the mean is , - ------ + ------- 262144 524288 171785935 104907 2 2600245 The , 4, -th moment about the mean is , - --------- + --------- n + -------- n 268435456 268435456 67108864 The compositions of, 9, that yield the, 39, -th largest growth, that is, 1.9919641966050350211, are , [1, 8], [8, 1] Theorem Number, 39, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 8] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 2 6 5 3 2 \ n (x - x + 1) (x + x - x - x + 1) ) a(n) x = ---------------------------------------------- / 7 6 5 4 3 2 ----- (-1 + x) (x + x + x + x + x + x + x - 1) n = 0 and in Maple format (x^2-x+1)*(x^6+x^5-x^3-x^2+1)/(-1+x)/(x^7+x^6+x^5+x^4+x^3+x^2+x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 508, 1012, 2016, 4016, 8000, 15936, 31744, 63233, 125958, 250904, 499792, 995568, 1983136, 3950336, 7868928, 15674623, 31223288, 62195672, 123891552, 246787536, 491591936] The limit of a(n+1)/a(n) as n goes to infinity is 1.99196419661 a(n) is asymptotic to .516605356304*1.99196419661^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [1, 8], denoted by the variable, X[1, 8], is 8 8 x X[1, 8] - x + x - 1 ------------------------- 8 8 x X[1, 8] - x + 2 x - 1 and in Maple format (x^8*X[1,8]-x^8+x-1)/(x^8*X[1,8]-x^8+2*x-1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [1, 8], equals , - 1/32 + --- 256 117 241 n The variance equals , - ---- + ----- 4096 65536 24783 27261 n The , 3, -th moment about the mean is , - ------- + ------- 1048576 8388608 The , 4, -th moment about the mean is , 841899 174243 2 3969635 - -------- + ---------- n + ---------- n 67108864 4294967296 2147483648 The compositions of, 9, that yield the, 40, -th largest growth, that is, 1.9920300868462484222, are , [2, 7], [7, 2] Theorem Number, 40, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 7] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 7 \ n x - x + 1 ) a(n) x = - ----------------- / 8 7 ----- x - x + 2 x - 1 n = 0 and in Maple format -(x^7-x+1)/(x^8-x^7+2*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 508, 1012, 2016, 4016, 8000, 15936, 31745, 63237, 125970, 250936, 499872, 995760, 1983584, 3951359, 7871226, 15679719, 31234472, 62220008, 123944128, 246900432, 491833089] The limit of a(n+1)/a(n) as n goes to infinity is 1.99203008685 a(n) is asymptotic to .516346142451*1.99203008685^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [2, 7], denoted by the variable, X[2, 7], is 7 7 x X[2, 7] - x + x - 1 - ------------------------------------------- 8 8 7 7 x X[2, 7] - x - x X[2, 7] + x - 2 x + 1 and in Maple format -(x^7*X[2,7]-x^7+x-1)/(x^8*X[2,7]-x^8-x^7*X[2,7]+x^7-2*x+1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [2, 7], equals , - 1/32 + --- 256 483 245 n The variance equals , - ----- + ----- 16384 65536 54813 28677 n The , 3, -th moment about the mean is , - ------- + ------- 2097152 8388608 The , 4, -th moment about the mean is , 4671177 180075 2 4644703 - --------- + ---------- n + ---------- n 268435456 4294967296 2147483648 The compositions of, 9, that yield the, 41, -th largest growth, that is, 1.9921580953553798820, are , [3, 6], [6, 3] Theorem Number, 41, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [3, 6] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 6 \ n x - x + 1 ) a(n) x = - ---------------------- / 8 7 6 ----- x + x - x + 2 x - 1 n = 0 and in Maple format -(x^6-x+1)/(x^8+x^7-x^6+2*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 508, 1012, 2016, 4016, 8000, 15937, 31749, 63249, 126002, 251016, 500064, 996207, 1984602, 3953641, 7876278, 15690791, 31258536, 62271945, 124055559, 247138286, 492338537] The limit of a(n+1)/a(n) as n goes to infinity is 1.99215809536 a(n) is asymptotic to .515881333788*1.99215809536^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [3, 6], denoted by the variable, X[3, 6], is 6 6 x X[3, 6] - x + x - 1 - ------------------------------------------------------------- 8 8 7 7 6 6 x X[3, 6] - x + x X[3, 6] - x - x X[3, 6] + x - 2 x + 1 and in Maple format -(x^6*X[3,6]-x^6+x-1)/(x^8*X[3,6]-x^8+x^7*X[3,6]-x^7-x^6*X[3,6]+x^6-2*x+1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [3, 6], equals , - 1/32 + --- 256 511 253 n The variance equals , - ----- + ----- 16384 65536 65007 31593 n The , 3, -th moment about the mean is , - ------- + ------- 2097152 8388608 The , 4, -th moment about the mean is , 7381297 192027 2 6114863 - --------- + ---------- n + ---------- n 268435456 4294967296 2147483648 The compositions of, 9, that yield the, 42, -th largest growth, that is, 1.9924010004614550874, are , [4, 5], [5, 4] Theorem Number, 42, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [4, 5] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 5 \ n x - x + 1 ) a(n) x = - --------------------------- / 8 7 6 5 ----- x + x + x - x + 2 x - 1 n = 0 and in Maple format -(x^5-x+1)/(x^8+x^7+x^6-x^5+2*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 508, 1012, 2016, 4016, 8001, 15941, 31761, 63281, 126082, 251207, 500506, 997209, 1986840, 3958581, 7887079, 15714222, 31309030, 62380142, 124286258, 247628067, 493374412] The limit of a(n+1)/a(n) as n goes to infinity is 1.99240100046 a(n) is asymptotic to .515079291766*1.99240100046^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [4, 5], denoted by the variable, X[4, 5], is 5 5 / 8 8 7 7 6 - (x X[4, 5] - x + x - 1) / (x X[4, 5] - x + x X[4, 5] - x + x X[4, 5] / 6 5 5 - x - x X[4, 5] + x - 2 x + 1) and in Maple format -(x^5*X[4,5]-x^5+x-1)/(x^8*X[4,5]-x^8+x^7*X[4,5]-x^7+x^6*X[4,5]-x^6-x^5*X[4,5]+ x^5-2*x+1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [4, 5], equals , - 1/32 + --- 256 563 269 n The variance equals , - ----- + ----- 16384 65536 85335 37737 n The , 3, -th moment about the mean is , - ------- + ------- 2097152 8388608 The , 4, -th moment about the mean is , 13437977 217083 2 9511231 - --------- + ---------- n + ---------- n 268435456 4294967296 2147483648 The compositions of, 9, that yield the, 43, -th largest growth, that is, 1.9960311797354145898, are , [9] Theorem Number, 43, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [9] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 1 ) a(n) x = - ---------------------------------------- / 8 7 6 5 4 3 2 ----- x + x + x + x + x + x + x + x - 1 n = 0 and in Maple format -1/(x^8+x^7+x^6+x^5+x^4+x^3+x^2+x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 509, 1016, 2028, 4048, 8080, 16128, 32192, 64256, 128257, 256005, 510994, 1019960, 2035872, 4063664, 8111200, 16190208, 32316160, 64504063, 128752121, 256993248, 512966536] The limit of a(n+1)/a(n) as n goes to infinity is 1.99603117974 a(n) is asymptotic to .507071734457*1.99603117974^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [9], denoted by the variable, X[9], is -1 + x ---------------------- 9 9 x X[9] - x + 2 x - 1 and in Maple format (-1+x)/(x^9*X[9]-x^9+2*x-1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [9], equals , - 7/512 + --- 512 3395 495 n The variance equals , - ------ + ------ 262144 262144 777021 118341 n The , 3, -th moment about the mean is , - -------- + -------- 67108864 67108864 The , 4, -th moment about the mean is , 586142879 735075 2 23724243 - ----------- + ----------- n + ----------- n 68719476736 68719476736 17179869184 This ends this article, that took, 84.359, seconds to generate. ---------------------------------------------------------- ---------------------------------------- Generating functions and Growth rates for Weight-Enumerating Sequences Accor\ ding to the Number of occurrences of compositions of, 10 By Shalosh B. Ekhad The compositions of, 10, that yield polynomial growth (hence growth ratio 1, that is the lowest) are , [2, 1, 1, 1, 1, 1, 1, 2] Theorem Number, 1, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 1, 1, 1, 1, 1, 2] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 65 63 62 61 60 59 57 56 ) a(n) x = - (x + 2 x + x + x + x + x - x + 6 x / ----- n = 0 55 54 53 51 50 49 48 47 - 12 x + 14 x - 9 x + 3 x - x + 8 x - 38 x + 85 x 46 45 44 43 42 41 40 - 119 x + 101 x - 19 x - 84 x + 177 x - 297 x + 435 x 39 38 37 36 35 34 33 - 432 x + 128 x + 393 x - 791 x + 814 x - 802 x + 1746 x 32 31 30 29 28 27 - 4517 x + 8872 x - 13127 x + 14839 x - 12325 x + 5846 x 26 25 24 23 22 21 + 3334 x - 15840 x + 35735 x - 67411 x + 108916 x - 147678 x 20 19 18 17 16 + 163952 x - 140026 x + 66452 x + 60424 x - 243380 x 15 14 13 12 11 + 478967 x - 740707 x + 969352 x - 1090740 x + 1057036 x 10 9 8 7 6 5 - 879887 x + 626040 x - 378048 x + 191879 x - 80770 x + 27683 x 4 3 2 / 3 2 21 - 7526 x + 1561 x - 232 x + 22 x - 1) / ((x - x + 2 x - 1) (x / 14 13 12 11 10 9 8 7 6 5 4 - x + x + 5 x - 10 x + 5 x + x - x + x - 7 x + 21 x - 35 x 3 2 21 14 13 12 11 10 + 35 x - 21 x + 7 x - 1) (x + 2 x - 7 x + 8 x - 2 x - 2 x 9 7 6 5 4 3 2 21 14 + x + x - 7 x + 21 x - 35 x + 35 x - 21 x + 7 x - 1) (x - x 13 12 11 10 7 6 5 4 3 2 + 4 x - 6 x + 4 x - x + x - 7 x + 21 x - 35 x + 35 x - 21 x + 7 x - 1)) and in Maple format -(x^65+2*x^63+x^62+x^61+x^60+x^59-x^57+6*x^56-12*x^55+14*x^54-9*x^53+3*x^51-x^ 50+8*x^49-38*x^48+85*x^47-119*x^46+101*x^45-19*x^44-84*x^43+177*x^42-297*x^41+ 435*x^40-432*x^39+128*x^38+393*x^37-791*x^36+814*x^35-802*x^34+1746*x^33-4517*x ^32+8872*x^31-13127*x^30+14839*x^29-12325*x^28+5846*x^27+3334*x^26-15840*x^25+ 35735*x^24-67411*x^23+108916*x^22-147678*x^21+163952*x^20-140026*x^19+66452*x^ 18+60424*x^17-243380*x^16+478967*x^15-740707*x^14+969352*x^13-1090740*x^12+ 1057036*x^11-879887*x^10+626040*x^9-378048*x^8+191879*x^7-80770*x^6+27683*x^5-\ 7526*x^4+1561*x^3-232*x^2+22*x-1)/(x^3-x^2+2*x-1)/(x^21-x^14+x^13+5*x^12-10*x^ 11+5*x^10+x^9-x^8+x^7-7*x^6+21*x^5-35*x^4+35*x^3-21*x^2+7*x-1)/(x^21+2*x^14-7*x ^13+8*x^12-2*x^11-2*x^10+x^9+x^7-7*x^6+21*x^5-35*x^4+35*x^3-21*x^2+7*x-1)/(x^21 -x^14+4*x^13-6*x^12+4*x^11-x^10+x^7-7*x^6+21*x^5-35*x^4+35*x^3-21*x^2+7*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1014, 1991, 3853, 7330, 13693, 25119, 45288, 80357, 140553, 242803, 415142, 704228, 1188356, 2000280, 3367607, 5684752, 9641648, 16454525, 28280058, 48958244] ---------------------------------------------------------------------------- The compositions of, 10, that yield the, 2, -th largest growth, that is, 1.6180339887498948482, are , [1, 1, 1, 1, 1, 1, 1, 3], [1, 1, 1, 1, 1, 1, 3, 1], [1, 1, 1, 1, 1, 3, 1, 1], [1, 1, 1, 1, 3, 1, 1, 1], [1, 1, 1, 3, 1, 1, 1, 1], [1, 1, 3, 1, 1, 1, 1, 1], [1, 3, 1, 1, 1, 1, 1, 1], [3, 1, 1, 1, 1, 1, 1, 1] Theorem Number, 2, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 1, 1, 1, 1, 1, 1, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n ) a(n) x = / ----- n = 0 9 8 7 6 5 4 3 2 x - 3 x + 8 x - 6 x - 10 x + 30 x - 34 x + 21 x - 7 x + 1 ----------------------------------------------------------------- 2 7 (x + x - 1) (-1 + x) and in Maple format (x^9-3*x^8+8*x^7-6*x^6-10*x^5+30*x^4-34*x^3+21*x^2-7*x+1)/(x^2+x-1)/(-1+x)^7 The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1014, 1991, 3853, 7330, 13693, 25119, 45288, 80356, 140537, 242671, 414388, 700855, 1175703, 1958718, 3244477, 5348694, 8783222, 14377422, 23474556, 38249572] The limit of a(n+1)/a(n) as n goes to infinity is 1.61803398875 a(n) is asymptotic to 21.0095194942*1.61803398875^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [1, 1, 1, 1, 1, 1, 1, 3], denoted by the variable, X[1, 1, 1, 1, 1, 1, 1, 3], is 9 9 8 8 7 7 6 6 5 (x %1 - x - 3 x %1 + 3 x + 9 x %1 - 8 x - 13 x %1 + 6 x + 11 x %1 5 4 4 3 3 2 / + 10 x - 5 x %1 - 30 x + x %1 + 34 x - 21 x + 7 x - 1) / ( / 6 3 3 (-1 + x) (x %1 - x + 2 x - 1)) %1 := X[1, 1, 1, 1, 1, 1, 1, 3] and in Maple format (x^9*X[1,1,1,1,1,1,1,3]-x^9-3*x^8*X[1,1,1,1,1,1,1,3]+3*x^8+9*x^7*X[1,1,1,1,1,1, 1,3]-8*x^7-13*x^6*X[1,1,1,1,1,1,1,3]+6*x^6+11*x^5*X[1,1,1,1,1,1,1,3]+10*x^5-5*x ^4*X[1,1,1,1,1,1,1,3]-30*x^4+x^3*X[1,1,1,1,1,1,1,3]+34*x^3-21*x^2+7*x-1)/(-1+x) ^6/(x^3*X[1,1,1,1,1,1,1,3]-x^3+2*x-1) Furthermore, the expectation of the random variable number of occurences (by\ containment) of , [1, 1, 1, 1, 1, 1, 1, 3], equals , - 15/8 + n/8 27 3 n The variance equals , - -- + --- 64 64 The , 3, -th moment about the mean is , 3/16 2313 27 2 129 The , 4, -th moment about the mean is , ---- + ---- n - ---- n 4096 4096 1024 The compositions of, 10, that yield the, 3, -th largest growth, that is, 1.7548776662466927601, are , [1, 1, 1, 1, 2, 1, 1, 2], [1, 1, 1, 2, 1, 1, 2, 1], [1, 1, 2, 1, 1, 2, 1, 1], [1, 2, 1, 1, 2, 1, 1, 1], [2, 1, 1, 2, 1, 1, 1, 1] Theorem Number, 3, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 1, 1, 1, 2, 1, 1, 2] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 15 14 13 12 11 10 9 ) a(n) x = - (3 x - 6 x + 13 x - 17 x + 24 x - 34 x + 50 x / ----- n = 0 8 7 6 5 4 3 2 / - 68 x + 83 x - 96 x + 103 x - 93 x + 63 x - 29 x + 8 x - 1) / ( / 3 2 9 6 5 4 3 2 4 (x - x + 2 x - 1) (x - x + 2 x - x + x - 3 x + 3 x - 1) (-1 + x) ) and in Maple format -(3*x^15-6*x^14+13*x^13-17*x^12+24*x^11-34*x^10+50*x^9-68*x^8+83*x^7-96*x^6+103 *x^5-93*x^4+63*x^3-29*x^2+8*x-1)/(x^3-x^2+2*x-1)/(x^9-x^6+2*x^5-x^4+x^3-3*x^2+3 *x-1)/(-1+x)^4 The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1014, 1991, 3853, 7331, 13705, 25197, 45654, 81747, 145088, 255975, 450012, 789786, 1385413, 2430613, 4265962, 7489722, 13152028, 23095240, 40550770, 71186142] The limit of a(n+1)/a(n) as n goes to infinity is 1.75487766625 a(n) is asymptotic to 3.34834608804*1.75487766625^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [1, 1, 1, 1, 2, 1, 1, 2], denoted by the variable, X[1, 1, 1, 1, 2, 1, 1, 2], is 15 4 15 3 14 4 15 2 14 3 13 4 - (3 x %1 - 12 x %1 - 6 x %1 + 18 x %1 + 24 x %1 + 7 x %1 15 14 2 13 3 12 4 15 14 - 12 x %1 - 36 x %1 - 34 x %1 - 4 x %1 + 3 x + 24 x %1 13 2 12 3 11 4 14 13 12 2 + 60 x %1 + 29 x %1 + x %1 - 6 x - 46 x %1 - 63 x %1 11 3 13 12 11 2 10 3 12 - 23 x %1 + 13 x + 55 x %1 + 67 x %1 + 18 x %1 - 17 x 11 10 2 9 3 11 10 9 2 - 69 x %1 - 69 x %1 - 12 x %1 + 24 x + 85 x %1 + 64 x %1 8 3 10 9 8 2 7 3 9 8 + 5 x %1 - 34 x - 102 x %1 - 46 x %1 - x %1 + 50 x + 108 x %1 7 2 8 7 6 2 7 6 5 2 + 23 x %1 - 68 x - 97 x %1 - 7 x %1 + 83 x + 75 x %1 + x %1 6 5 5 4 4 3 3 2 - 96 x - 48 x %1 + 103 x + 23 x %1 - 93 x - 7 x %1 + 63 x + x %1 2 / 4 3 3 2 2 - 29 x + 8 x - 1) / ((-1 + x) (x %1 - x - x %1 + x - 2 x + 1) ( / 9 3 9 2 9 9 6 2 6 5 2 6 5 x %1 - 3 x %1 + 3 x %1 - x + x %1 - 2 x %1 - x %1 + x + 3 x %1 5 4 4 3 2 - 2 x - x %1 + x - x + 3 x - 3 x + 1)) %1 := X[1, 1, 1, 1, 2, 1, 1, 2] and in Maple format -(3*x^15*X[1,1,1,1,2,1,1,2]^4-12*x^15*X[1,1,1,1,2,1,1,2]^3-6*x^14*X[1,1,1,1,2,1 ,1,2]^4+18*x^15*X[1,1,1,1,2,1,1,2]^2+24*x^14*X[1,1,1,1,2,1,1,2]^3+7*x^13*X[1,1, 1,1,2,1,1,2]^4-12*x^15*X[1,1,1,1,2,1,1,2]-36*x^14*X[1,1,1,1,2,1,1,2]^2-34*x^13* X[1,1,1,1,2,1,1,2]^3-4*x^12*X[1,1,1,1,2,1,1,2]^4+3*x^15+24*x^14*X[1,1,1,1,2,1,1 ,2]+60*x^13*X[1,1,1,1,2,1,1,2]^2+29*x^12*X[1,1,1,1,2,1,1,2]^3+x^11*X[1,1,1,1,2, 1,1,2]^4-6*x^14-46*x^13*X[1,1,1,1,2,1,1,2]-63*x^12*X[1,1,1,1,2,1,1,2]^2-23*x^11 *X[1,1,1,1,2,1,1,2]^3+13*x^13+55*x^12*X[1,1,1,1,2,1,1,2]+67*x^11*X[1,1,1,1,2,1, 1,2]^2+18*x^10*X[1,1,1,1,2,1,1,2]^3-17*x^12-69*x^11*X[1,1,1,1,2,1,1,2]-69*x^10* X[1,1,1,1,2,1,1,2]^2-12*x^9*X[1,1,1,1,2,1,1,2]^3+24*x^11+85*x^10*X[1,1,1,1,2,1, 1,2]+64*x^9*X[1,1,1,1,2,1,1,2]^2+5*x^8*X[1,1,1,1,2,1,1,2]^3-34*x^10-102*x^9*X[1 ,1,1,1,2,1,1,2]-46*x^8*X[1,1,1,1,2,1,1,2]^2-x^7*X[1,1,1,1,2,1,1,2]^3+50*x^9+108 *x^8*X[1,1,1,1,2,1,1,2]+23*x^7*X[1,1,1,1,2,1,1,2]^2-68*x^8-97*x^7*X[1,1,1,1,2,1 ,1,2]-7*x^6*X[1,1,1,1,2,1,1,2]^2+83*x^7+75*x^6*X[1,1,1,1,2,1,1,2]+x^5*X[1,1,1,1 ,2,1,1,2]^2-96*x^6-48*x^5*X[1,1,1,1,2,1,1,2]+103*x^5+23*x^4*X[1,1,1,1,2,1,1,2]-\ 93*x^4-7*x^3*X[1,1,1,1,2,1,1,2]+63*x^3+x^2*X[1,1,1,1,2,1,1,2]-29*x^2+8*x-1)/(-1 +x)^4/(x^3*X[1,1,1,1,2,1,1,2]-x^3-x^2*X[1,1,1,1,2,1,1,2]+x^2-2*x+1)/(x^9*X[1,1, 1,1,2,1,1,2]^3-3*x^9*X[1,1,1,1,2,1,1,2]^2+3*x^9*X[1,1,1,1,2,1,1,2]-x^9+x^6*X[1, 1,1,1,2,1,1,2]^2-2*x^6*X[1,1,1,1,2,1,1,2]-x^5*X[1,1,1,1,2,1,1,2]^2+x^6+3*x^5*X[ 1,1,1,1,2,1,1,2]-2*x^5-x^4*X[1,1,1,1,2,1,1,2]+x^4-x^3+3*x^2-3*x+1) Furthermore, the expectation of the random variable number of occurences (by\ containment) of , [1, 1, 1, 1, 2, 1, 1, 2], equals , - 15/8 + n/8 111 7 n The variance equals , - --- + --- 64 64 159 9 n The , 3, -th moment about the mean is , - --- + --- 128 128 37569 147 2 1181 The , 4, -th moment about the mean is , ----- + ---- n - ---- n 4096 4096 1024 The compositions of, 10, that yield the, 4, -th largest growth, that is, 1.8392867552141611326, are , [1, 1, 1, 1, 1, 1, 4], [1, 1, 1, 1, 1, 4, 1], [1, 1, 1, 1, 4, 1, 1], [1, 1, 1, 4, 1, 1, 1], [1, 1, 4, 1, 1, 1, 1], [1, 4, 1, 1, 1, 1, 1], [4, 1, 1, 1, 1, 1, 1] Theorem Number, 4, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 1, 1, 1, 1, 1, 4] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n ) a(n) x = / ----- n = 0 8 7 6 5 4 3 2 3 x - 6 x + 8 x - 10 x + 16 x - 20 x + 15 x - 6 x + 1 - ------------------------------------------------------------ 3 2 6 (x + x + x - 1) (-1 + x) and in Maple format -(3*x^8-6*x^7+8*x^6-10*x^5+16*x^4-20*x^3+15*x^2-6*x+1)/(x^3+x^2+x-1)/(-1+x)^6 The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1015, 2001, 3909, 7563, 14497, 27555, 51995, 97520, 182014, 338414, 627350, 1160394, 2142822, 3952266, 7283378, 13413909, 24694105, 45446847, 83623267, 153847901] The limit of a(n+1)/a(n) as n goes to infinity is 1.83928675521 a(n) is asymptotic to 1.76938057252*1.83928675521^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [1, 1, 1, 1, 1, 1, 4], denoted by the variable, X[1, 1, 1, 1, 1, 1, 4], is 8 8 7 7 6 6 5 5 4 - (3 x %1 - 3 x - 6 x %1 + 6 x + 7 x %1 - 8 x - 4 x %1 + 10 x + x %1 4 3 2 / 5 4 4 - 16 x + 20 x - 15 x + 6 x - 1) / ((-1 + x) (x %1 - x + 2 x - 1)) / %1 := X[1, 1, 1, 1, 1, 1, 4] and in Maple format -(3*x^8*X[1,1,1,1,1,1,4]-3*x^8-6*x^7*X[1,1,1,1,1,1,4]+6*x^7+7*x^6*X[1,1,1,1,1,1 ,4]-8*x^6-4*x^5*X[1,1,1,1,1,1,4]+10*x^5+x^4*X[1,1,1,1,1,1,4]-16*x^4+20*x^3-15*x ^2+6*x-1)/(-1+x)^5/(x^4*X[1,1,1,1,1,1,4]-x^4+2*x-1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [1, 1, 1, 1, 1, 1, 4], equals , - 7/8 + ---- 16 13 9 n The variance equals , - -- + --- 32 256 15 15 n The , 3, -th moment about the mean is , ---- + ---- 1024 2048 1373 243 2 3117 The , 4, -th moment about the mean is , ---- + ----- n - ----- n 2048 65536 32768 The compositions of, 10, that yield the, 5, -th largest growth, that is, 1.8667603991738620930, are , [2, 1, 1, 1, 1, 1, 3], [3, 1, 1, 1, 1, 1, 2] Theorem Number, 5, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 1, 1, 1, 1, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 63 60 59 58 57 56 55 54 ) a(n) x = - (x + 2 x + x + x + x - x + 4 x - 2 x / ----- n = 0 53 52 50 49 48 47 46 45 - 6 x + 9 x - 10 x + 2 x + 10 x - 17 x + 31 x - 23 x 44 43 42 41 40 39 38 37 - 26 x + 62 x - 30 x - 27 x + 24 x + 15 x + 12 x - 112 x 36 35 34 33 32 31 30 + 176 x - 108 x - 102 x + 300 x - 243 x - 16 x + 122 x 29 28 27 26 25 24 23 + 27 x - 175 x + 248 x - 462 x + 760 x - 646 x - 146 x 22 21 20 19 18 17 16 + 1077 x - 1243 x + 464 x + 396 x - 490 x + 17 x + 168 x 15 14 13 12 11 10 9 + 489 x - 2017 x + 3977 x - 5184 x + 4002 x + 165 x - 5675 x 8 7 6 5 4 3 2 + 9571 x - 9996 x + 7540 x - 4262 x + 1805 x - 559 x + 120 x - 16 x / 2 12 8 7 6 5 3 2 + 1) / ((x - x + 1) (x + x - x - x + x + x - 3 x + 3 x - 1) / 4 3 12 8 7 6 5 3 2 (x - x + 2 x - 1) (x - x + x + x - x + x - 3 x + 3 x - 1) 6 5 2 24 17 16 15 14 13 12 (x + x - x - x + 1) (x - x + 5 x - 6 x - 2 x + 4 x + 4 x 11 9 8 6 5 4 3 2 - 4 x - x + x + x - 6 x + 15 x - 20 x + 15 x - 6 x + 1) 4 (x - x + 1)) and in Maple format -(x^63+2*x^60+x^59+x^58+x^57-x^56+4*x^55-2*x^54-6*x^53+9*x^52-10*x^50+2*x^49+10 *x^48-17*x^47+31*x^46-23*x^45-26*x^44+62*x^43-30*x^42-27*x^41+24*x^40+15*x^39+ 12*x^38-112*x^37+176*x^36-108*x^35-102*x^34+300*x^33-243*x^32-16*x^31+122*x^30+ 27*x^29-175*x^28+248*x^27-462*x^26+760*x^25-646*x^24-146*x^23+1077*x^22-1243*x^ 21+464*x^20+396*x^19-490*x^18+17*x^17+168*x^16+489*x^15-2017*x^14+3977*x^13-\ 5184*x^12+4002*x^11+165*x^10-5675*x^9+9571*x^8-9996*x^7+7540*x^6-4262*x^5+1805* x^4-559*x^3+120*x^2-16*x+1)/(x^2-x+1)/(x^12+x^8-x^7-x^6+x^5+x^3-3*x^2+3*x-1)/(x ^4-x^3+2*x-1)/(x^12-x^8+x^7+x^6-x^5+x^3-3*x^2+3*x-1)/(x^6+x^5-x^2-x+1)/(x^24-x^ 17+5*x^16-6*x^15-2*x^14+4*x^13+4*x^12-4*x^11-x^9+x^8+x^6-6*x^5+15*x^4-20*x^3+15 *x^2-6*x+1)/(x^4-x+1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1015, 2001, 3909, 7563, 14497, 27555, 51995, 97521, 182028, 338521, 627937, 1162977, 2152509, 3984440, 7380556, 13686022, 25410854, 47242840, 87942397, 163886779] The limit of a(n+1)/a(n) as n goes to infinity is 1.86676039917 a(n) is asymptotic to 1.21492409182*1.86676039917^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [2, 1, 1, 1, 1, 1, 3], denoted by the variable, X[2, 1, 1, 1, 1, 1, 3], is 63 16 63 15 63 14 63 13 63 12 - (x %1 - 16 x %1 + 120 x %1 - 560 x %1 + 1820 x %1 60 15 63 11 60 14 59 15 63 10 - 2 x %1 - 4368 x %1 + 30 x %1 - x %1 + 8008 x %1 60 13 59 14 58 15 63 9 60 12 - 210 x %1 + 15 x %1 - x %1 - 11440 x %1 + 910 x %1 59 13 58 14 57 15 63 8 60 11 - 105 x %1 + 15 x %1 - x %1 + 12870 x %1 - 2730 x %1 59 12 58 13 57 14 56 15 63 7 + 455 x %1 - 105 x %1 + 15 x %1 + x %1 - 11440 x %1 60 10 59 11 58 12 57 13 + 6006 x %1 - 1365 x %1 + 455 x %1 - 105 x %1 56 14 55 15 63 6 60 9 59 10 - 15 x %1 - x %1 + 8008 x %1 - 10010 x %1 + 3003 x %1 58 11 57 12 56 13 55 14 63 5 - 1365 x %1 + 455 x %1 + 105 x %1 + 18 x %1 - 4368 x %1 60 8 59 9 58 10 57 11 + 12870 x %1 - 5005 x %1 + 3003 x %1 - 1365 x %1 56 12 55 13 54 14 63 4 60 7 - 455 x %1 - 147 x %1 - x %1 + 1820 x %1 - 12870 x %1 59 8 58 9 57 10 56 11 + 6435 x %1 - 5005 x %1 + 3003 x %1 + 1365 x %1 55 12 54 13 53 14 63 3 60 6 + 728 x %1 + 15 x %1 - 3 x %1 - 560 x %1 + 10010 x %1 59 7 58 8 57 9 56 10 - 6435 x %1 + 6435 x %1 - 5005 x %1 - 3003 x %1 55 11 54 12 53 13 52 14 63 2 - 2457 x %1 - 104 x %1 + 45 x %1 + x %1 + 120 x %1 60 5 59 6 58 7 57 8 56 9 - 6006 x %1 + 5005 x %1 - 6435 x %1 + 6435 x %1 + 5005 x %1 55 10 54 11 53 12 52 13 63 + 6006 x %1 + 442 x %1 - 312 x %1 - 22 x %1 - 16 x %1 60 4 59 5 58 6 57 7 56 8 + 2730 x %1 - 3003 x %1 + 5005 x %1 - 6435 x %1 - 6435 x %1 55 9 54 10 53 11 52 12 51 13 - 11011 x %1 - 1287 x %1 + 1326 x %1 + 195 x %1 + x %1 63 60 3 59 4 58 5 57 6 + x - 910 x %1 + 1365 x %1 - 3003 x %1 + 5005 x %1 56 7 55 8 54 9 53 10 + 6435 x %1 + 15444 x %1 + 2717 x %1 - 3861 x %1 52 11 51 12 50 13 60 2 59 3 - 988 x %1 - 12 x %1 + 2 x %1 + 210 x %1 - 455 x %1 58 4 57 5 56 6 55 7 + 1365 x %1 - 3003 x %1 - 5005 x %1 - 16731 x %1 54 8 53 9 52 10 51 11 50 12 - 4290 x %1 + 8151 x %1 + 3289 x %1 + 66 x %1 - 34 x %1 49 13 60 59 2 58 3 57 4 + x %1 - 30 x %1 + 105 x %1 - 455 x %1 + 1365 x %1 56 5 55 6 54 7 53 8 + 3003 x %1 + 14014 x %1 + 5148 x %1 - 12870 x %1 52 9 51 10 50 11 49 12 48 13 - 7722 x %1 - 220 x %1 + 252 x %1 - 10 x %1 - 2 x %1 60 59 58 2 57 3 56 4 + 2 x - 15 x %1 + 105 x %1 - 455 x %1 - 1365 x %1 55 5 54 6 53 7 52 8 - 9009 x %1 - 4719 x %1 + 15444 x %1 + 13299 x %1 51 9 50 10 49 11 48 12 47 13 + 495 x %1 - 1100 x %1 + 42 x %1 + 30 x %1 + 4 x %1 59 58 57 2 56 3 55 4 + x - 15 x %1 + 105 x %1 + 455 x %1 + 4368 x %1 54 5 53 6 52 7 51 8 + 3289 x %1 - 14157 x %1 - 17160 x %1 - 792 x %1 50 9 49 10 48 11 47 12 46 13 + 3190 x %1 - 88 x %1 - 208 x %1 - 62 x %1 - 3 x %1 58 57 56 2 55 3 54 4 + x - 15 x %1 - 105 x %1 - 1547 x %1 - 1716 x %1 53 5 52 6 51 7 50 8 49 9 + 9867 x %1 + 16731 x %1 + 924 x %1 - 6534 x %1 + 55 x %1 48 10 47 11 46 12 45 13 57 56 + 880 x %1 + 435 x %1 + 56 x %1 + x %1 + x + 15 x %1 55 2 54 3 53 4 52 5 51 6 + 378 x %1 + 650 x %1 - 5148 x %1 - 12298 x %1 - 792 x %1 50 7 49 8 48 9 47 10 46 11 + 9768 x %1 + 198 x %1 - 2530 x %1 - 1837 x %1 - 448 x %1 45 12 56 55 54 2 53 3 - 20 x %1 - x - 57 x %1 - 169 x %1 + 1950 x %1 52 4 51 5 50 6 49 7 48 8 + 6721 x %1 + 495 x %1 - 10824 x %1 - 660 x %1 + 5214 x %1 47 9 46 10 45 11 44 12 55 + 5225 x %1 + 2091 x %1 + 172 x %1 - 4 x %1 + 4 x 54 53 2 52 3 51 4 50 5 + 27 x %1 - 507 x %1 - 2652 x %1 - 220 x %1 + 8910 x %1 49 6 48 7 47 8 46 9 + 1056 x %1 - 7920 x %1 - 10593 x %1 - 6445 x %1 45 10 44 11 43 12 54 53 - 863 x %1 + 65 x %1 + 5 x %1 - 2 x + 81 x %1 52 2 51 3 50 4 49 5 48 6 + 715 x %1 + 66 x %1 - 5390 x %1 - 1089 x %1 + 8976 x %1 47 7 46 8 45 9 44 10 + 15774 x %1 + 13971 x %1 + 2855 x %1 - 456 x %1 43 11 42 12 53 52 51 2 50 3 - 90 x %1 - x %1 - 6 x - 118 x %1 - 12 x %1 + 2332 x %1 49 4 48 5 47 6 46 7 + 770 x %1 - 7590 x %1 - 17490 x %1 - 22032 x %1 45 8 44 9 43 10 42 11 52 - 6627 x %1 + 1865 x %1 + 686 x %1 + 26 x %1 + 9 x 51 50 2 49 3 48 4 47 5 + x %1 - 684 x %1 - 374 x %1 + 4730 x %1 + 14454 x %1 46 6 45 7 44 8 43 9 + 25686 x %1 + 11160 x %1 - 5010 x %1 - 3011 x %1 42 10 41 11 50 49 2 48 3 - 238 x %1 + x %1 + 122 x %1 + 120 x %1 - 2112 x %1 47 4 46 5 45 6 44 7 - 8800 x %1 - 22197 x %1 - 13854 x %1 + 9378 x %1 43 8 42 9 41 10 40 11 50 + 8604 x %1 + 1167 x %1 - 19 x %1 - x %1 - 10 x 49 48 2 47 3 46 4 45 5 - 23 x %1 + 640 x %1 + 3839 x %1 + 14070 x %1 + 12711 x %1 44 6 43 7 42 8 41 9 40 10 - 12600 x %1 - 17016 x %1 - 3588 x %1 + 153 x %1 + 19 x %1 49 48 47 2 46 3 45 4 + 2 x - 118 x %1 - 1137 x %1 - 6368 x %1 - 8530 x %1 44 5 43 6 42 7 41 8 40 9 + 12282 x %1 + 24024 x %1 + 7464 x %1 - 687 x %1 - 151 x %1 48 47 46 2 45 3 44 4 + 10 x + 205 x %1 + 1951 x %1 + 4076 x %1 - 8640 x %1 43 5 42 6 41 7 40 8 39 9 - 24498 x %1 - 10920 x %1 + 1938 x %1 + 668 x %1 - 7 x %1 38 10 47 46 45 2 44 3 + 10 x %1 - 17 x - 363 x %1 - 1315 x %1 + 4285 x %1 43 4 42 5 41 6 40 7 39 8 + 17961 x %1 + 11418 x %1 - 3654 x %1 - 1858 x %1 + 71 x %1 38 9 37 10 46 45 44 2 - 106 x %1 - 22 x %1 + 31 x + 257 x %1 - 1424 x %1 43 3 42 4 41 5 40 6 39 7 - 9254 x %1 - 8517 x %1 + 4746 x %1 + 3458 x %1 - 316 x %1 38 8 37 9 36 10 45 44 + 500 x %1 + 278 x %1 + 24 x %1 - 23 x + 285 x %1 43 2 42 3 41 4 40 5 39 6 + 3186 x %1 + 4438 x %1 - 4278 x %1 - 4438 x %1 + 812 x %1 38 7 37 8 36 9 35 10 44 - 1384 x %1 - 1544 x %1 - 333 x %1 - 16 x %1 - 26 x 43 42 2 41 3 40 4 39 5 - 659 x %1 - 1538 x %1 + 2637 x %1 + 3956 x %1 - 1330 x %1 38 6 37 7 36 8 35 9 34 10 + 2492 x %1 + 4984 x %1 + 1984 x %1 + 237 x %1 + 6 x %1 43 42 41 2 40 3 39 4 + 62 x + 319 x %1 - 1063 x %1 - 2413 x %1 + 1442 x %1 38 5 37 6 36 7 35 8 34 9 - 3052 x %1 - 10388 x %1 - 6756 x %1 - 1463 x %1 - 81 x %1 33 10 42 41 40 2 39 3 - x %1 - 30 x + 253 x %1 + 963 x %1 - 1036 x %1 38 4 37 5 36 6 35 7 + 2576 x %1 + 14644 x %1 + 14672 x %1 + 5057 x %1 34 8 33 9 41 40 39 2 + 418 x %1 - 11 x %1 - 27 x - 227 x %1 + 476 x %1 38 3 37 4 36 5 35 6 - 1480 x %1 - 14168 x %1 - 21350 x %1 - 10983 x %1 34 7 33 8 32 9 40 39 - 1059 x %1 + 287 x %1 + 22 x %1 + 24 x - 127 x %1 38 2 37 3 36 4 35 5 + 554 x %1 + 9304 x %1 + 21168 x %1 + 15799 x %1 34 6 33 7 32 8 31 9 39 + 1254 x %1 - 1931 x %1 - 344 x %1 - 8 x %1 + 15 x 38 37 2 36 3 35 4 34 5 - 122 x %1 - 3974 x %1 - 14164 x %1 - 15337 x %1 + x %1 33 6 32 7 31 8 30 9 38 + 6601 x %1 + 2100 x %1 + 139 x %1 + x %1 + 12 x 37 36 2 35 3 34 4 33 5 + 998 x %1 + 6136 x %1 + 9963 x %1 - 2064 x %1 - 13489 x %1 32 6 31 7 30 8 37 36 - 6851 x %1 - 864 x %1 - 17 x %1 - 112 x - 1557 x %1 35 2 34 3 33 4 32 5 - 4157 x %1 + 2907 x %1 + 17521 x %1 + 13498 x %1 31 6 30 7 29 8 36 35 + 2693 x %1 + 70 x %1 - 4 x %1 + 176 x + 1008 x %1 34 2 33 3 32 4 31 5 30 6 - 1976 x %1 - 14657 x %1 - 16917 x %1 - 4794 x %1 + 27 x %1 29 7 28 8 35 34 33 2 + 48 x %1 + x %1 - 108 x + 696 x %1 + 7676 x %1 32 3 31 4 30 5 29 6 28 7 + 13624 x %1 + 5093 x %1 - 782 x %1 - 233 x %1 + 3 x %1 34 33 32 2 31 3 30 4 - 102 x - 2296 x %1 - 6845 x %1 - 3164 x %1 + 2119 x %1 29 5 28 6 27 7 33 32 + 558 x %1 - 145 x %1 - 10 x %1 + 300 x + 1956 x %1 31 2 30 3 29 4 28 5 27 6 + 1019 x %1 - 2778 x %1 - 695 x %1 + 818 x %1 + 210 x %1 26 7 32 31 30 2 29 3 + 2 x %1 - 243 x - 98 x %1 + 2005 x %1 + 404 x %1 28 4 27 5 26 6 31 30 - 2092 x %1 - 1130 x %1 - 215 x %1 - 16 x - 767 x %1 29 2 28 3 27 4 26 5 25 6 - 23 x %1 + 2927 x %1 + 2868 x %1 + 1451 x %1 + 288 x %1 30 29 28 2 27 3 26 4 + 122 x - 82 x %1 - 2325 x %1 - 4022 x %1 - 4156 x %1 25 5 24 6 29 28 27 2 - 2157 x %1 - 312 x %1 + 27 x + 988 x %1 + 3218 x %1 26 3 25 4 24 5 23 6 28 + 6344 x %1 + 6507 x %1 + 2402 x %1 + 239 x %1 - 175 x 27 26 2 25 3 24 4 23 5 - 1382 x %1 - 5423 x %1 - 10219 x %1 - 7119 x %1 - 1759 x %1 22 6 27 26 25 2 24 3 - 128 x %1 + 248 x + 2459 x %1 + 8865 x %1 + 10711 x %1 23 4 22 5 21 6 26 25 + 4406 x %1 + 711 x %1 + 46 x %1 - 462 x - 4044 x %1 24 2 23 3 22 4 21 5 20 6 - 8759 x %1 - 4908 x %1 - 145 x %1 + 41 x %1 - 10 x %1 25 24 23 2 22 3 21 4 + 760 x + 3723 x %1 + 2293 x %1 - 3625 x %1 - 2676 x %1 20 5 19 6 24 23 22 2 - 260 x %1 + x %1 - 646 x - 125 x %1 + 6558 x %1 21 3 20 4 19 5 23 22 + 8534 x %1 + 2889 x %1 + 175 x %1 - 146 x - 4448 x %1 21 2 20 3 19 4 18 5 22 - 10677 x %1 - 7327 x %1 - 1645 x %1 - 62 x %1 + 1077 x 21 20 2 19 3 18 4 17 5 + 5975 x %1 + 7541 x %1 + 3084 x %1 + 521 x %1 + 12 x %1 21 20 19 2 18 3 17 4 16 5 - 1243 x - 3297 x %1 - 1542 x %1 + 2 x %1 - 53 x %1 - x %1 20 19 18 2 17 3 16 4 19 + 464 x - 469 x %1 - 1807 x %1 - 747 x %1 - 23 x %1 + 396 x 18 17 2 16 3 15 4 18 + 1836 x %1 + 1593 x %1 + 387 x %1 + 9 x %1 - 490 x 17 16 2 15 3 14 4 17 16 - 822 x %1 - 341 x %1 - 64 x %1 - x %1 + 17 x - 189 x %1 15 2 14 3 16 15 14 2 13 3 - 180 x %1 - 15 x %1 + 168 x - 270 x %1 + 96 x %1 + 8 x %1 15 14 13 2 12 3 14 13 + 489 x + 2057 x %1 + 21 x %1 - x %1 - 2017 x - 4566 x %1 12 2 13 12 11 2 12 11 - 29 x %1 + 3977 x + 7034 x %1 + 9 x %1 - 5184 x - 8379 x %1 10 2 11 10 10 9 9 - x %1 + 4002 x + 7844 x %1 + 165 x - 5765 x %1 - 5675 x 8 8 7 7 6 6 + 3299 x %1 + 9571 x - 1444 x %1 - 9996 x + 468 x %1 + 7540 x 5 5 4 4 3 3 2 - 106 x %1 - 4262 x + 15 x %1 + 1805 x - x %1 - 559 x + 120 x / 4 4 4 4 3 3 - 16 x + 1) / ((x %1 - x + x - 1) (x %1 - x - x %1 + x - 2 x + 1) / 8 2 8 8 5 5 4 4 2 12 3 (x %1 - 2 x %1 + x - x %1 + x + x %1 - x + x - 2 x + 1) (x %1 12 2 12 12 8 2 8 7 2 8 7 - 3 x %1 + 3 x %1 - x - x %1 + 2 x %1 + x %1 - x - 2 x %1 7 6 6 5 5 3 2 12 3 12 2 + x - x %1 + x + x %1 - x - x + 3 x - 3 x + 1) (x %1 - 3 x %1 12 12 8 2 8 7 2 8 7 7 6 + 3 x %1 - x + x %1 - 2 x %1 - x %1 + x + 2 x %1 - x + x %1 6 5 5 3 2 24 6 24 5 24 4 - x - x %1 + x - x + 3 x - 3 x + 1) (x %1 - 6 x %1 + 15 x %1 24 3 24 2 24 24 17 5 17 4 16 5 - 20 x %1 + 15 x %1 - 6 x %1 + x + x %1 - 5 x %1 - x %1 17 3 16 4 17 2 16 3 15 4 17 + 10 x %1 + 9 x %1 - 10 x %1 - 26 x %1 - 4 x %1 + 5 x %1 16 2 15 3 17 16 15 2 14 3 + 34 x %1 + 18 x %1 - x - 21 x %1 - 30 x %1 + 2 x %1 16 15 14 2 13 3 15 14 + 5 x + 22 x %1 - 6 x %1 - 4 x %1 - 6 x + 6 x %1 13 2 14 13 12 2 13 12 + 12 x %1 - 2 x - 12 x %1 + 4 x %1 + 4 x - 8 x %1 11 2 12 11 11 9 9 8 8 6 - 4 x %1 + 4 x + 8 x %1 - 4 x + x %1 - x - x %1 + x + x 5 4 3 2 - 6 x + 15 x - 20 x + 15 x - 6 x + 1)) %1 := X[2, 1, 1, 1, 1, 1, 3] and in Maple format -(x^63*X[2,1,1,1,1,1,3]^16-16*x^63*X[2,1,1,1,1,1,3]^15+120*x^63*X[2,1,1,1,1,1,3 ]^14-560*x^63*X[2,1,1,1,1,1,3]^13+1820*x^63*X[2,1,1,1,1,1,3]^12-2*x^60*X[2,1,1, 1,1,1,3]^15-4368*x^63*X[2,1,1,1,1,1,3]^11+30*x^60*X[2,1,1,1,1,1,3]^14-x^59*X[2, 1,1,1,1,1,3]^15+8008*x^63*X[2,1,1,1,1,1,3]^10-210*x^60*X[2,1,1,1,1,1,3]^13+15*x ^59*X[2,1,1,1,1,1,3]^14-x^58*X[2,1,1,1,1,1,3]^15-11440*x^63*X[2,1,1,1,1,1,3]^9+ 910*x^60*X[2,1,1,1,1,1,3]^12-105*x^59*X[2,1,1,1,1,1,3]^13+15*x^58*X[2,1,1,1,1,1 ,3]^14-x^57*X[2,1,1,1,1,1,3]^15+12870*x^63*X[2,1,1,1,1,1,3]^8-2730*x^60*X[2,1,1 ,1,1,1,3]^11+455*x^59*X[2,1,1,1,1,1,3]^12-105*x^58*X[2,1,1,1,1,1,3]^13+15*x^57* X[2,1,1,1,1,1,3]^14+x^56*X[2,1,1,1,1,1,3]^15-11440*x^63*X[2,1,1,1,1,1,3]^7+6006 *x^60*X[2,1,1,1,1,1,3]^10-1365*x^59*X[2,1,1,1,1,1,3]^11+455*x^58*X[2,1,1,1,1,1, 3]^12-105*x^57*X[2,1,1,1,1,1,3]^13-15*x^56*X[2,1,1,1,1,1,3]^14-x^55*X[2,1,1,1,1 ,1,3]^15+8008*x^63*X[2,1,1,1,1,1,3]^6-10010*x^60*X[2,1,1,1,1,1,3]^9+3003*x^59*X [2,1,1,1,1,1,3]^10-1365*x^58*X[2,1,1,1,1,1,3]^11+455*x^57*X[2,1,1,1,1,1,3]^12+ 105*x^56*X[2,1,1,1,1,1,3]^13+18*x^55*X[2,1,1,1,1,1,3]^14-4368*x^63*X[2,1,1,1,1, 1,3]^5+12870*x^60*X[2,1,1,1,1,1,3]^8-5005*x^59*X[2,1,1,1,1,1,3]^9+3003*x^58*X[2 ,1,1,1,1,1,3]^10-1365*x^57*X[2,1,1,1,1,1,3]^11-455*x^56*X[2,1,1,1,1,1,3]^12-147 *x^55*X[2,1,1,1,1,1,3]^13-x^54*X[2,1,1,1,1,1,3]^14+1820*x^63*X[2,1,1,1,1,1,3]^4 -12870*x^60*X[2,1,1,1,1,1,3]^7+6435*x^59*X[2,1,1,1,1,1,3]^8-5005*x^58*X[2,1,1,1 ,1,1,3]^9+3003*x^57*X[2,1,1,1,1,1,3]^10+1365*x^56*X[2,1,1,1,1,1,3]^11+728*x^55* X[2,1,1,1,1,1,3]^12+15*x^54*X[2,1,1,1,1,1,3]^13-3*x^53*X[2,1,1,1,1,1,3]^14-560* x^63*X[2,1,1,1,1,1,3]^3+10010*x^60*X[2,1,1,1,1,1,3]^6-6435*x^59*X[2,1,1,1,1,1,3 ]^7+6435*x^58*X[2,1,1,1,1,1,3]^8-5005*x^57*X[2,1,1,1,1,1,3]^9-3003*x^56*X[2,1,1 ,1,1,1,3]^10-2457*x^55*X[2,1,1,1,1,1,3]^11-104*x^54*X[2,1,1,1,1,1,3]^12+45*x^53 *X[2,1,1,1,1,1,3]^13+x^52*X[2,1,1,1,1,1,3]^14+120*x^63*X[2,1,1,1,1,1,3]^2-6006* x^60*X[2,1,1,1,1,1,3]^5+5005*x^59*X[2,1,1,1,1,1,3]^6-6435*x^58*X[2,1,1,1,1,1,3] ^7+6435*x^57*X[2,1,1,1,1,1,3]^8+5005*x^56*X[2,1,1,1,1,1,3]^9+6006*x^55*X[2,1,1, 1,1,1,3]^10+442*x^54*X[2,1,1,1,1,1,3]^11-312*x^53*X[2,1,1,1,1,1,3]^12-22*x^52*X [2,1,1,1,1,1,3]^13-16*x^63*X[2,1,1,1,1,1,3]+2730*x^60*X[2,1,1,1,1,1,3]^4-3003*x ^59*X[2,1,1,1,1,1,3]^5+5005*x^58*X[2,1,1,1,1,1,3]^6-6435*x^57*X[2,1,1,1,1,1,3]^ 7-6435*x^56*X[2,1,1,1,1,1,3]^8-11011*x^55*X[2,1,1,1,1,1,3]^9-1287*x^54*X[2,1,1, 1,1,1,3]^10+1326*x^53*X[2,1,1,1,1,1,3]^11+195*x^52*X[2,1,1,1,1,1,3]^12+x^51*X[2 ,1,1,1,1,1,3]^13+x^63-910*x^60*X[2,1,1,1,1,1,3]^3+1365*x^59*X[2,1,1,1,1,1,3]^4-\ 3003*x^58*X[2,1,1,1,1,1,3]^5+5005*x^57*X[2,1,1,1,1,1,3]^6+6435*x^56*X[2,1,1,1,1 ,1,3]^7+15444*x^55*X[2,1,1,1,1,1,3]^8+2717*x^54*X[2,1,1,1,1,1,3]^9-3861*x^53*X[ 2,1,1,1,1,1,3]^10-988*x^52*X[2,1,1,1,1,1,3]^11-12*x^51*X[2,1,1,1,1,1,3]^12+2*x^ 50*X[2,1,1,1,1,1,3]^13+210*x^60*X[2,1,1,1,1,1,3]^2-455*x^59*X[2,1,1,1,1,1,3]^3+ 1365*x^58*X[2,1,1,1,1,1,3]^4-3003*x^57*X[2,1,1,1,1,1,3]^5-5005*x^56*X[2,1,1,1,1 ,1,3]^6-16731*x^55*X[2,1,1,1,1,1,3]^7-4290*x^54*X[2,1,1,1,1,1,3]^8+8151*x^53*X[ 2,1,1,1,1,1,3]^9+3289*x^52*X[2,1,1,1,1,1,3]^10+66*x^51*X[2,1,1,1,1,1,3]^11-34*x ^50*X[2,1,1,1,1,1,3]^12+x^49*X[2,1,1,1,1,1,3]^13-30*x^60*X[2,1,1,1,1,1,3]+105*x ^59*X[2,1,1,1,1,1,3]^2-455*x^58*X[2,1,1,1,1,1,3]^3+1365*x^57*X[2,1,1,1,1,1,3]^4 +3003*x^56*X[2,1,1,1,1,1,3]^5+14014*x^55*X[2,1,1,1,1,1,3]^6+5148*x^54*X[2,1,1,1 ,1,1,3]^7-12870*x^53*X[2,1,1,1,1,1,3]^8-7722*x^52*X[2,1,1,1,1,1,3]^9-220*x^51*X [2,1,1,1,1,1,3]^10+252*x^50*X[2,1,1,1,1,1,3]^11-10*x^49*X[2,1,1,1,1,1,3]^12-2*x ^48*X[2,1,1,1,1,1,3]^13+2*x^60-15*x^59*X[2,1,1,1,1,1,3]+105*x^58*X[2,1,1,1,1,1, 3]^2-455*x^57*X[2,1,1,1,1,1,3]^3-1365*x^56*X[2,1,1,1,1,1,3]^4-9009*x^55*X[2,1,1 ,1,1,1,3]^5-4719*x^54*X[2,1,1,1,1,1,3]^6+15444*x^53*X[2,1,1,1,1,1,3]^7+13299*x^ 52*X[2,1,1,1,1,1,3]^8+495*x^51*X[2,1,1,1,1,1,3]^9-1100*x^50*X[2,1,1,1,1,1,3]^10 +42*x^49*X[2,1,1,1,1,1,3]^11+30*x^48*X[2,1,1,1,1,1,3]^12+4*x^47*X[2,1,1,1,1,1,3 ]^13+x^59-15*x^58*X[2,1,1,1,1,1,3]+105*x^57*X[2,1,1,1,1,1,3]^2+455*x^56*X[2,1,1 ,1,1,1,3]^3+4368*x^55*X[2,1,1,1,1,1,3]^4+3289*x^54*X[2,1,1,1,1,1,3]^5-14157*x^ 53*X[2,1,1,1,1,1,3]^6-17160*x^52*X[2,1,1,1,1,1,3]^7-792*x^51*X[2,1,1,1,1,1,3]^8 +3190*x^50*X[2,1,1,1,1,1,3]^9-88*x^49*X[2,1,1,1,1,1,3]^10-208*x^48*X[2,1,1,1,1, 1,3]^11-62*x^47*X[2,1,1,1,1,1,3]^12-3*x^46*X[2,1,1,1,1,1,3]^13+x^58-15*x^57*X[2 ,1,1,1,1,1,3]-105*x^56*X[2,1,1,1,1,1,3]^2-1547*x^55*X[2,1,1,1,1,1,3]^3-1716*x^ 54*X[2,1,1,1,1,1,3]^4+9867*x^53*X[2,1,1,1,1,1,3]^5+16731*x^52*X[2,1,1,1,1,1,3]^ 6+924*x^51*X[2,1,1,1,1,1,3]^7-6534*x^50*X[2,1,1,1,1,1,3]^8+55*x^49*X[2,1,1,1,1, 1,3]^9+880*x^48*X[2,1,1,1,1,1,3]^10+435*x^47*X[2,1,1,1,1,1,3]^11+56*x^46*X[2,1, 1,1,1,1,3]^12+x^45*X[2,1,1,1,1,1,3]^13+x^57+15*x^56*X[2,1,1,1,1,1,3]+378*x^55*X [2,1,1,1,1,1,3]^2+650*x^54*X[2,1,1,1,1,1,3]^3-5148*x^53*X[2,1,1,1,1,1,3]^4-\ 12298*x^52*X[2,1,1,1,1,1,3]^5-792*x^51*X[2,1,1,1,1,1,3]^6+9768*x^50*X[2,1,1,1,1 ,1,3]^7+198*x^49*X[2,1,1,1,1,1,3]^8-2530*x^48*X[2,1,1,1,1,1,3]^9-1837*x^47*X[2, 1,1,1,1,1,3]^10-448*x^46*X[2,1,1,1,1,1,3]^11-20*x^45*X[2,1,1,1,1,1,3]^12-x^56-\ 57*x^55*X[2,1,1,1,1,1,3]-169*x^54*X[2,1,1,1,1,1,3]^2+1950*x^53*X[2,1,1,1,1,1,3] ^3+6721*x^52*X[2,1,1,1,1,1,3]^4+495*x^51*X[2,1,1,1,1,1,3]^5-10824*x^50*X[2,1,1, 1,1,1,3]^6-660*x^49*X[2,1,1,1,1,1,3]^7+5214*x^48*X[2,1,1,1,1,1,3]^8+5225*x^47*X [2,1,1,1,1,1,3]^9+2091*x^46*X[2,1,1,1,1,1,3]^10+172*x^45*X[2,1,1,1,1,1,3]^11-4* x^44*X[2,1,1,1,1,1,3]^12+4*x^55+27*x^54*X[2,1,1,1,1,1,3]-507*x^53*X[2,1,1,1,1,1 ,3]^2-2652*x^52*X[2,1,1,1,1,1,3]^3-220*x^51*X[2,1,1,1,1,1,3]^4+8910*x^50*X[2,1, 1,1,1,1,3]^5+1056*x^49*X[2,1,1,1,1,1,3]^6-7920*x^48*X[2,1,1,1,1,1,3]^7-10593*x^ 47*X[2,1,1,1,1,1,3]^8-6445*x^46*X[2,1,1,1,1,1,3]^9-863*x^45*X[2,1,1,1,1,1,3]^10 +65*x^44*X[2,1,1,1,1,1,3]^11+5*x^43*X[2,1,1,1,1,1,3]^12-2*x^54+81*x^53*X[2,1,1, 1,1,1,3]+715*x^52*X[2,1,1,1,1,1,3]^2+66*x^51*X[2,1,1,1,1,1,3]^3-5390*x^50*X[2,1 ,1,1,1,1,3]^4-1089*x^49*X[2,1,1,1,1,1,3]^5+8976*x^48*X[2,1,1,1,1,1,3]^6+15774*x ^47*X[2,1,1,1,1,1,3]^7+13971*x^46*X[2,1,1,1,1,1,3]^8+2855*x^45*X[2,1,1,1,1,1,3] ^9-456*x^44*X[2,1,1,1,1,1,3]^10-90*x^43*X[2,1,1,1,1,1,3]^11-x^42*X[2,1,1,1,1,1, 3]^12-6*x^53-118*x^52*X[2,1,1,1,1,1,3]-12*x^51*X[2,1,1,1,1,1,3]^2+2332*x^50*X[2 ,1,1,1,1,1,3]^3+770*x^49*X[2,1,1,1,1,1,3]^4-7590*x^48*X[2,1,1,1,1,1,3]^5-17490* x^47*X[2,1,1,1,1,1,3]^6-22032*x^46*X[2,1,1,1,1,1,3]^7-6627*x^45*X[2,1,1,1,1,1,3 ]^8+1865*x^44*X[2,1,1,1,1,1,3]^9+686*x^43*X[2,1,1,1,1,1,3]^10+26*x^42*X[2,1,1,1 ,1,1,3]^11+9*x^52+x^51*X[2,1,1,1,1,1,3]-684*x^50*X[2,1,1,1,1,1,3]^2-374*x^49*X[ 2,1,1,1,1,1,3]^3+4730*x^48*X[2,1,1,1,1,1,3]^4+14454*x^47*X[2,1,1,1,1,1,3]^5+ 25686*x^46*X[2,1,1,1,1,1,3]^6+11160*x^45*X[2,1,1,1,1,1,3]^7-5010*x^44*X[2,1,1,1 ,1,1,3]^8-3011*x^43*X[2,1,1,1,1,1,3]^9-238*x^42*X[2,1,1,1,1,1,3]^10+x^41*X[2,1, 1,1,1,1,3]^11+122*x^50*X[2,1,1,1,1,1,3]+120*x^49*X[2,1,1,1,1,1,3]^2-2112*x^48*X [2,1,1,1,1,1,3]^3-8800*x^47*X[2,1,1,1,1,1,3]^4-22197*x^46*X[2,1,1,1,1,1,3]^5-\ 13854*x^45*X[2,1,1,1,1,1,3]^6+9378*x^44*X[2,1,1,1,1,1,3]^7+8604*x^43*X[2,1,1,1, 1,1,3]^8+1167*x^42*X[2,1,1,1,1,1,3]^9-19*x^41*X[2,1,1,1,1,1,3]^10-x^40*X[2,1,1, 1,1,1,3]^11-10*x^50-23*x^49*X[2,1,1,1,1,1,3]+640*x^48*X[2,1,1,1,1,1,3]^2+3839*x ^47*X[2,1,1,1,1,1,3]^3+14070*x^46*X[2,1,1,1,1,1,3]^4+12711*x^45*X[2,1,1,1,1,1,3 ]^5-12600*x^44*X[2,1,1,1,1,1,3]^6-17016*x^43*X[2,1,1,1,1,1,3]^7-3588*x^42*X[2,1 ,1,1,1,1,3]^8+153*x^41*X[2,1,1,1,1,1,3]^9+19*x^40*X[2,1,1,1,1,1,3]^10+2*x^49-\ 118*x^48*X[2,1,1,1,1,1,3]-1137*x^47*X[2,1,1,1,1,1,3]^2-6368*x^46*X[2,1,1,1,1,1, 3]^3-8530*x^45*X[2,1,1,1,1,1,3]^4+12282*x^44*X[2,1,1,1,1,1,3]^5+24024*x^43*X[2, 1,1,1,1,1,3]^6+7464*x^42*X[2,1,1,1,1,1,3]^7-687*x^41*X[2,1,1,1,1,1,3]^8-151*x^ 40*X[2,1,1,1,1,1,3]^9+10*x^48+205*x^47*X[2,1,1,1,1,1,3]+1951*x^46*X[2,1,1,1,1,1 ,3]^2+4076*x^45*X[2,1,1,1,1,1,3]^3-8640*x^44*X[2,1,1,1,1,1,3]^4-24498*x^43*X[2, 1,1,1,1,1,3]^5-10920*x^42*X[2,1,1,1,1,1,3]^6+1938*x^41*X[2,1,1,1,1,1,3]^7+668*x ^40*X[2,1,1,1,1,1,3]^8-7*x^39*X[2,1,1,1,1,1,3]^9+10*x^38*X[2,1,1,1,1,1,3]^10-17 *x^47-363*x^46*X[2,1,1,1,1,1,3]-1315*x^45*X[2,1,1,1,1,1,3]^2+4285*x^44*X[2,1,1, 1,1,1,3]^3+17961*x^43*X[2,1,1,1,1,1,3]^4+11418*x^42*X[2,1,1,1,1,1,3]^5-3654*x^ 41*X[2,1,1,1,1,1,3]^6-1858*x^40*X[2,1,1,1,1,1,3]^7+71*x^39*X[2,1,1,1,1,1,3]^8-\ 106*x^38*X[2,1,1,1,1,1,3]^9-22*x^37*X[2,1,1,1,1,1,3]^10+31*x^46+257*x^45*X[2,1, 1,1,1,1,3]-1424*x^44*X[2,1,1,1,1,1,3]^2-9254*x^43*X[2,1,1,1,1,1,3]^3-8517*x^42* X[2,1,1,1,1,1,3]^4+4746*x^41*X[2,1,1,1,1,1,3]^5+3458*x^40*X[2,1,1,1,1,1,3]^6-\ 316*x^39*X[2,1,1,1,1,1,3]^7+500*x^38*X[2,1,1,1,1,1,3]^8+278*x^37*X[2,1,1,1,1,1, 3]^9+24*x^36*X[2,1,1,1,1,1,3]^10-23*x^45+285*x^44*X[2,1,1,1,1,1,3]+3186*x^43*X[ 2,1,1,1,1,1,3]^2+4438*x^42*X[2,1,1,1,1,1,3]^3-4278*x^41*X[2,1,1,1,1,1,3]^4-4438 *x^40*X[2,1,1,1,1,1,3]^5+812*x^39*X[2,1,1,1,1,1,3]^6-1384*x^38*X[2,1,1,1,1,1,3] ^7-1544*x^37*X[2,1,1,1,1,1,3]^8-333*x^36*X[2,1,1,1,1,1,3]^9-16*x^35*X[2,1,1,1,1 ,1,3]^10-26*x^44-659*x^43*X[2,1,1,1,1,1,3]-1538*x^42*X[2,1,1,1,1,1,3]^2+2637*x^ 41*X[2,1,1,1,1,1,3]^3+3956*x^40*X[2,1,1,1,1,1,3]^4-1330*x^39*X[2,1,1,1,1,1,3]^5 +2492*x^38*X[2,1,1,1,1,1,3]^6+4984*x^37*X[2,1,1,1,1,1,3]^7+1984*x^36*X[2,1,1,1, 1,1,3]^8+237*x^35*X[2,1,1,1,1,1,3]^9+6*x^34*X[2,1,1,1,1,1,3]^10+62*x^43+319*x^ 42*X[2,1,1,1,1,1,3]-1063*x^41*X[2,1,1,1,1,1,3]^2-2413*x^40*X[2,1,1,1,1,1,3]^3+ 1442*x^39*X[2,1,1,1,1,1,3]^4-3052*x^38*X[2,1,1,1,1,1,3]^5-10388*x^37*X[2,1,1,1, 1,1,3]^6-6756*x^36*X[2,1,1,1,1,1,3]^7-1463*x^35*X[2,1,1,1,1,1,3]^8-81*x^34*X[2, 1,1,1,1,1,3]^9-x^33*X[2,1,1,1,1,1,3]^10-30*x^42+253*x^41*X[2,1,1,1,1,1,3]+963*x ^40*X[2,1,1,1,1,1,3]^2-1036*x^39*X[2,1,1,1,1,1,3]^3+2576*x^38*X[2,1,1,1,1,1,3]^ 4+14644*x^37*X[2,1,1,1,1,1,3]^5+14672*x^36*X[2,1,1,1,1,1,3]^6+5057*x^35*X[2,1,1 ,1,1,1,3]^7+418*x^34*X[2,1,1,1,1,1,3]^8-11*x^33*X[2,1,1,1,1,1,3]^9-27*x^41-227* x^40*X[2,1,1,1,1,1,3]+476*x^39*X[2,1,1,1,1,1,3]^2-1480*x^38*X[2,1,1,1,1,1,3]^3-\ 14168*x^37*X[2,1,1,1,1,1,3]^4-21350*x^36*X[2,1,1,1,1,1,3]^5-10983*x^35*X[2,1,1, 1,1,1,3]^6-1059*x^34*X[2,1,1,1,1,1,3]^7+287*x^33*X[2,1,1,1,1,1,3]^8+22*x^32*X[2 ,1,1,1,1,1,3]^9+24*x^40-127*x^39*X[2,1,1,1,1,1,3]+554*x^38*X[2,1,1,1,1,1,3]^2+ 9304*x^37*X[2,1,1,1,1,1,3]^3+21168*x^36*X[2,1,1,1,1,1,3]^4+15799*x^35*X[2,1,1,1 ,1,1,3]^5+1254*x^34*X[2,1,1,1,1,1,3]^6-1931*x^33*X[2,1,1,1,1,1,3]^7-344*x^32*X[ 2,1,1,1,1,1,3]^8-8*x^31*X[2,1,1,1,1,1,3]^9+15*x^39-122*x^38*X[2,1,1,1,1,1,3]-\ 3974*x^37*X[2,1,1,1,1,1,3]^2-14164*x^36*X[2,1,1,1,1,1,3]^3-15337*x^35*X[2,1,1,1 ,1,1,3]^4+x^34*X[2,1,1,1,1,1,3]^5+6601*x^33*X[2,1,1,1,1,1,3]^6+2100*x^32*X[2,1, 1,1,1,1,3]^7+139*x^31*X[2,1,1,1,1,1,3]^8+x^30*X[2,1,1,1,1,1,3]^9+12*x^38+998*x^ 37*X[2,1,1,1,1,1,3]+6136*x^36*X[2,1,1,1,1,1,3]^2+9963*x^35*X[2,1,1,1,1,1,3]^3-\ 2064*x^34*X[2,1,1,1,1,1,3]^4-13489*x^33*X[2,1,1,1,1,1,3]^5-6851*x^32*X[2,1,1,1, 1,1,3]^6-864*x^31*X[2,1,1,1,1,1,3]^7-17*x^30*X[2,1,1,1,1,1,3]^8-112*x^37-1557*x ^36*X[2,1,1,1,1,1,3]-4157*x^35*X[2,1,1,1,1,1,3]^2+2907*x^34*X[2,1,1,1,1,1,3]^3+ 17521*x^33*X[2,1,1,1,1,1,3]^4+13498*x^32*X[2,1,1,1,1,1,3]^5+2693*x^31*X[2,1,1,1 ,1,1,3]^6+70*x^30*X[2,1,1,1,1,1,3]^7-4*x^29*X[2,1,1,1,1,1,3]^8+176*x^36+1008*x^ 35*X[2,1,1,1,1,1,3]-1976*x^34*X[2,1,1,1,1,1,3]^2-14657*x^33*X[2,1,1,1,1,1,3]^3-\ 16917*x^32*X[2,1,1,1,1,1,3]^4-4794*x^31*X[2,1,1,1,1,1,3]^5+27*x^30*X[2,1,1,1,1, 1,3]^6+48*x^29*X[2,1,1,1,1,1,3]^7+x^28*X[2,1,1,1,1,1,3]^8-108*x^35+696*x^34*X[2 ,1,1,1,1,1,3]+7676*x^33*X[2,1,1,1,1,1,3]^2+13624*x^32*X[2,1,1,1,1,1,3]^3+5093*x ^31*X[2,1,1,1,1,1,3]^4-782*x^30*X[2,1,1,1,1,1,3]^5-233*x^29*X[2,1,1,1,1,1,3]^6+ 3*x^28*X[2,1,1,1,1,1,3]^7-102*x^34-2296*x^33*X[2,1,1,1,1,1,3]-6845*x^32*X[2,1,1 ,1,1,1,3]^2-3164*x^31*X[2,1,1,1,1,1,3]^3+2119*x^30*X[2,1,1,1,1,1,3]^4+558*x^29* X[2,1,1,1,1,1,3]^5-145*x^28*X[2,1,1,1,1,1,3]^6-10*x^27*X[2,1,1,1,1,1,3]^7+300*x ^33+1956*x^32*X[2,1,1,1,1,1,3]+1019*x^31*X[2,1,1,1,1,1,3]^2-2778*x^30*X[2,1,1,1 ,1,1,3]^3-695*x^29*X[2,1,1,1,1,1,3]^4+818*x^28*X[2,1,1,1,1,1,3]^5+210*x^27*X[2, 1,1,1,1,1,3]^6+2*x^26*X[2,1,1,1,1,1,3]^7-243*x^32-98*x^31*X[2,1,1,1,1,1,3]+2005 *x^30*X[2,1,1,1,1,1,3]^2+404*x^29*X[2,1,1,1,1,1,3]^3-2092*x^28*X[2,1,1,1,1,1,3] ^4-1130*x^27*X[2,1,1,1,1,1,3]^5-215*x^26*X[2,1,1,1,1,1,3]^6-16*x^31-767*x^30*X[ 2,1,1,1,1,1,3]-23*x^29*X[2,1,1,1,1,1,3]^2+2927*x^28*X[2,1,1,1,1,1,3]^3+2868*x^ 27*X[2,1,1,1,1,1,3]^4+1451*x^26*X[2,1,1,1,1,1,3]^5+288*x^25*X[2,1,1,1,1,1,3]^6+ 122*x^30-82*x^29*X[2,1,1,1,1,1,3]-2325*x^28*X[2,1,1,1,1,1,3]^2-4022*x^27*X[2,1, 1,1,1,1,3]^3-4156*x^26*X[2,1,1,1,1,1,3]^4-2157*x^25*X[2,1,1,1,1,1,3]^5-312*x^24 *X[2,1,1,1,1,1,3]^6+27*x^29+988*x^28*X[2,1,1,1,1,1,3]+3218*x^27*X[2,1,1,1,1,1,3 ]^2+6344*x^26*X[2,1,1,1,1,1,3]^3+6507*x^25*X[2,1,1,1,1,1,3]^4+2402*x^24*X[2,1,1 ,1,1,1,3]^5+239*x^23*X[2,1,1,1,1,1,3]^6-175*x^28-1382*x^27*X[2,1,1,1,1,1,3]-\ 5423*x^26*X[2,1,1,1,1,1,3]^2-10219*x^25*X[2,1,1,1,1,1,3]^3-7119*x^24*X[2,1,1,1, 1,1,3]^4-1759*x^23*X[2,1,1,1,1,1,3]^5-128*x^22*X[2,1,1,1,1,1,3]^6+248*x^27+2459 *x^26*X[2,1,1,1,1,1,3]+8865*x^25*X[2,1,1,1,1,1,3]^2+10711*x^24*X[2,1,1,1,1,1,3] ^3+4406*x^23*X[2,1,1,1,1,1,3]^4+711*x^22*X[2,1,1,1,1,1,3]^5+46*x^21*X[2,1,1,1,1 ,1,3]^6-462*x^26-4044*x^25*X[2,1,1,1,1,1,3]-8759*x^24*X[2,1,1,1,1,1,3]^2-4908*x ^23*X[2,1,1,1,1,1,3]^3-145*x^22*X[2,1,1,1,1,1,3]^4+41*x^21*X[2,1,1,1,1,1,3]^5-\ 10*x^20*X[2,1,1,1,1,1,3]^6+760*x^25+3723*x^24*X[2,1,1,1,1,1,3]+2293*x^23*X[2,1, 1,1,1,1,3]^2-3625*x^22*X[2,1,1,1,1,1,3]^3-2676*x^21*X[2,1,1,1,1,1,3]^4-260*x^20 *X[2,1,1,1,1,1,3]^5+x^19*X[2,1,1,1,1,1,3]^6-646*x^24-125*x^23*X[2,1,1,1,1,1,3]+ 6558*x^22*X[2,1,1,1,1,1,3]^2+8534*x^21*X[2,1,1,1,1,1,3]^3+2889*x^20*X[2,1,1,1,1 ,1,3]^4+175*x^19*X[2,1,1,1,1,1,3]^5-146*x^23-4448*x^22*X[2,1,1,1,1,1,3]-10677*x ^21*X[2,1,1,1,1,1,3]^2-7327*x^20*X[2,1,1,1,1,1,3]^3-1645*x^19*X[2,1,1,1,1,1,3]^ 4-62*x^18*X[2,1,1,1,1,1,3]^5+1077*x^22+5975*x^21*X[2,1,1,1,1,1,3]+7541*x^20*X[2 ,1,1,1,1,1,3]^2+3084*x^19*X[2,1,1,1,1,1,3]^3+521*x^18*X[2,1,1,1,1,1,3]^4+12*x^ 17*X[2,1,1,1,1,1,3]^5-1243*x^21-3297*x^20*X[2,1,1,1,1,1,3]-1542*x^19*X[2,1,1,1, 1,1,3]^2+2*x^18*X[2,1,1,1,1,1,3]^3-53*x^17*X[2,1,1,1,1,1,3]^4-x^16*X[2,1,1,1,1, 1,3]^5+464*x^20-469*x^19*X[2,1,1,1,1,1,3]-1807*x^18*X[2,1,1,1,1,1,3]^2-747*x^17 *X[2,1,1,1,1,1,3]^3-23*x^16*X[2,1,1,1,1,1,3]^4+396*x^19+1836*x^18*X[2,1,1,1,1,1 ,3]+1593*x^17*X[2,1,1,1,1,1,3]^2+387*x^16*X[2,1,1,1,1,1,3]^3+9*x^15*X[2,1,1,1,1 ,1,3]^4-490*x^18-822*x^17*X[2,1,1,1,1,1,3]-341*x^16*X[2,1,1,1,1,1,3]^2-64*x^15* X[2,1,1,1,1,1,3]^3-x^14*X[2,1,1,1,1,1,3]^4+17*x^17-189*x^16*X[2,1,1,1,1,1,3]-\ 180*x^15*X[2,1,1,1,1,1,3]^2-15*x^14*X[2,1,1,1,1,1,3]^3+168*x^16-270*x^15*X[2,1, 1,1,1,1,3]+96*x^14*X[2,1,1,1,1,1,3]^2+8*x^13*X[2,1,1,1,1,1,3]^3+489*x^15+2057*x ^14*X[2,1,1,1,1,1,3]+21*x^13*X[2,1,1,1,1,1,3]^2-x^12*X[2,1,1,1,1,1,3]^3-2017*x^ 14-4566*x^13*X[2,1,1,1,1,1,3]-29*x^12*X[2,1,1,1,1,1,3]^2+3977*x^13+7034*x^12*X[ 2,1,1,1,1,1,3]+9*x^11*X[2,1,1,1,1,1,3]^2-5184*x^12-8379*x^11*X[2,1,1,1,1,1,3]-x ^10*X[2,1,1,1,1,1,3]^2+4002*x^11+7844*x^10*X[2,1,1,1,1,1,3]+165*x^10-5765*x^9*X [2,1,1,1,1,1,3]-5675*x^9+3299*x^8*X[2,1,1,1,1,1,3]+9571*x^8-1444*x^7*X[2,1,1,1, 1,1,3]-9996*x^7+468*x^6*X[2,1,1,1,1,1,3]+7540*x^6-106*x^5*X[2,1,1,1,1,1,3]-4262 *x^5+15*x^4*X[2,1,1,1,1,1,3]+1805*x^4-x^3*X[2,1,1,1,1,1,3]-559*x^3+120*x^2-16*x +1)/(x^4*X[2,1,1,1,1,1,3]-x^4+x-1)/(x^4*X[2,1,1,1,1,1,3]-x^4-x^3*X[2,1,1,1,1,1, 3]+x^3-2*x+1)/(x^8*X[2,1,1,1,1,1,3]^2-2*x^8*X[2,1,1,1,1,1,3]+x^8-x^5*X[2,1,1,1, 1,1,3]+x^5+x^4*X[2,1,1,1,1,1,3]-x^4+x^2-2*x+1)/(x^12*X[2,1,1,1,1,1,3]^3-3*x^12* X[2,1,1,1,1,1,3]^2+3*x^12*X[2,1,1,1,1,1,3]-x^12-x^8*X[2,1,1,1,1,1,3]^2+2*x^8*X[ 2,1,1,1,1,1,3]+x^7*X[2,1,1,1,1,1,3]^2-x^8-2*x^7*X[2,1,1,1,1,1,3]+x^7-x^6*X[2,1, 1,1,1,1,3]+x^6+x^5*X[2,1,1,1,1,1,3]-x^5-x^3+3*x^2-3*x+1)/(x^12*X[2,1,1,1,1,1,3] ^3-3*x^12*X[2,1,1,1,1,1,3]^2+3*x^12*X[2,1,1,1,1,1,3]-x^12+x^8*X[2,1,1,1,1,1,3]^ 2-2*x^8*X[2,1,1,1,1,1,3]-x^7*X[2,1,1,1,1,1,3]^2+x^8+2*x^7*X[2,1,1,1,1,1,3]-x^7+ x^6*X[2,1,1,1,1,1,3]-x^6-x^5*X[2,1,1,1,1,1,3]+x^5-x^3+3*x^2-3*x+1)/(x^24*X[2,1, 1,1,1,1,3]^6-6*x^24*X[2,1,1,1,1,1,3]^5+15*x^24*X[2,1,1,1,1,1,3]^4-20*x^24*X[2,1 ,1,1,1,1,3]^3+15*x^24*X[2,1,1,1,1,1,3]^2-6*x^24*X[2,1,1,1,1,1,3]+x^24+x^17*X[2, 1,1,1,1,1,3]^5-5*x^17*X[2,1,1,1,1,1,3]^4-x^16*X[2,1,1,1,1,1,3]^5+10*x^17*X[2,1, 1,1,1,1,3]^3+9*x^16*X[2,1,1,1,1,1,3]^4-10*x^17*X[2,1,1,1,1,1,3]^2-26*x^16*X[2,1 ,1,1,1,1,3]^3-4*x^15*X[2,1,1,1,1,1,3]^4+5*x^17*X[2,1,1,1,1,1,3]+34*x^16*X[2,1,1 ,1,1,1,3]^2+18*x^15*X[2,1,1,1,1,1,3]^3-x^17-21*x^16*X[2,1,1,1,1,1,3]-30*x^15*X[ 2,1,1,1,1,1,3]^2+2*x^14*X[2,1,1,1,1,1,3]^3+5*x^16+22*x^15*X[2,1,1,1,1,1,3]-6*x^ 14*X[2,1,1,1,1,1,3]^2-4*x^13*X[2,1,1,1,1,1,3]^3-6*x^15+6*x^14*X[2,1,1,1,1,1,3]+ 12*x^13*X[2,1,1,1,1,1,3]^2-2*x^14-12*x^13*X[2,1,1,1,1,1,3]+4*x^12*X[2,1,1,1,1,1 ,3]^2+4*x^13-8*x^12*X[2,1,1,1,1,1,3]-4*x^11*X[2,1,1,1,1,1,3]^2+4*x^12+8*x^11*X[ 2,1,1,1,1,1,3]-4*x^11+x^9*X[2,1,1,1,1,1,3]-x^9-x^8*X[2,1,1,1,1,1,3]+x^8+x^6-6*x ^5+15*x^4-20*x^3+15*x^2-6*x+1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [2, 1, 1, 1, 1, 1, 3], equals , - 7/8 + ---- 16 53 13 n The variance equals , - -- + ---- 64 256 705 63 n The , 3, -th moment about the mean is , - ---- + ---- 1024 2048 7075 507 2 8257 The , 4, -th moment about the mean is , ---- + ----- n - ----- n 4096 65536 32768 The compositions of, 10, that yield the, 6, -th largest growth, that is, 1.8908049048980778372, are , [2, 1, 1, 1, 2, 1, 2], [2, 1, 2, 1, 1, 1, 2] Theorem Number, 6, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 1, 1, 2, 1, 2] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 37 36 35 34 33 32 31 29 28 ) a(n) x = - (x + x + x + 2 x - x + x - 2 x + x - x / ----- n = 0 26 25 24 23 22 21 20 19 + 5 x - x - 19 x + 36 x - 22 x - 38 x + 105 x - 77 x 18 17 16 15 14 13 12 - 98 x + 324 x - 428 x + 318 x - 105 x + 53 x - 283 x 11 10 9 8 7 6 5 + 519 x - 256 x - 731 x + 2039 x - 2881 x + 2772 x - 1935 x 4 3 2 / + 989 x - 363 x + 91 x - 14 x + 1) / ( / 8 7 6 5 4 3 2 30 29 27 26 (x - x + x + x - 3 x + 5 x - 6 x + 4 x - 1) (x + x - x - x 23 22 21 20 19 18 17 16 15 + 3 x + x - 3 x - 7 x + 10 x - x - 8 x + 15 x + 3 x 14 13 12 11 10 9 8 7 - 55 x + 77 x - 20 x - 60 x + 67 x + 17 x - 86 x + 12 x 6 5 4 3 2 + 177 x - 307 x + 276 x - 154 x + 54 x - 11 x + 1)) and in Maple format -(x^37+x^36+x^35+2*x^34-x^33+x^32-2*x^31+x^29-x^28+5*x^26-x^25-19*x^24+36*x^23-\ 22*x^22-38*x^21+105*x^20-77*x^19-98*x^18+324*x^17-428*x^16+318*x^15-105*x^14+53 *x^13-283*x^12+519*x^11-256*x^10-731*x^9+2039*x^8-2881*x^7+2772*x^6-1935*x^5+ 989*x^4-363*x^3+91*x^2-14*x+1)/(x^8-x^7+x^6+x^5-3*x^4+5*x^3-6*x^2+4*x-1)/(x^30+ x^29-x^27-x^26+3*x^23+x^22-3*x^21-7*x^20+10*x^19-x^18-8*x^17+15*x^16+3*x^15-55* x^14+77*x^13-20*x^12-60*x^11+67*x^10+17*x^9-86*x^8+12*x^7+177*x^6-307*x^5+276*x ^4-154*x^3+54*x^2-11*x+1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1015, 2001, 3909, 7564, 14507, 27613, 52249, 98450, 185017, 347249, 651554, 1223092, 2298116, 4323124, 8142720, 15355531, 28988376, 54772869, 103561954, 195902705] The limit of a(n+1)/a(n) as n goes to infinity is 1.89080490490 a(n) is asymptotic to .988014882613*1.89080490490^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [2, 1, 1, 1, 2, 1, 2], denoted by the variable, X[2, 1, 1, 1, 2, 1, 2], is 39 10 39 9 39 8 39 7 37 9 39 6 - (x %1 - 10 x %1 + 45 x %1 - 120 x %1 - x %1 + 210 x %1 37 8 36 9 39 5 37 7 36 8 + 9 x %1 - 2 x %1 - 252 x %1 - 36 x %1 + 18 x %1 35 9 39 4 37 6 36 7 35 8 34 9 + 2 x %1 + 210 x %1 + 84 x %1 - 72 x %1 - 18 x %1 - x %1 39 3 37 5 36 6 35 7 34 8 - 120 x %1 - 126 x %1 + 168 x %1 + 72 x %1 + 12 x %1 33 9 39 2 37 4 36 5 35 6 - 2 x %1 + 45 x %1 + 126 x %1 - 252 x %1 - 168 x %1 34 7 33 8 32 9 39 37 3 - 60 x %1 + 12 x %1 + 2 x %1 - 10 x %1 - 84 x %1 36 4 35 5 34 6 33 7 32 8 + 252 x %1 + 252 x %1 + 168 x %1 - 24 x %1 - 14 x %1 31 9 39 37 2 36 3 35 4 34 5 - x %1 + x + 36 x %1 - 168 x %1 - 252 x %1 - 294 x %1 32 7 31 8 37 36 2 35 3 + 39 x %1 + 6 x %1 - 9 x %1 + 72 x %1 + 168 x %1 34 4 33 5 32 6 31 7 30 8 37 + 336 x %1 + 84 x %1 - 49 x %1 - 13 x %1 + 7 x %1 + x 36 35 2 34 3 33 4 32 5 - 18 x %1 - 72 x %1 - 252 x %1 - 168 x %1 + 7 x %1 31 6 30 7 29 8 36 35 34 2 + 7 x %1 - 47 x %1 - 10 x %1 + 2 x + 18 x %1 + 120 x %1 33 3 32 4 31 5 30 6 29 7 + 168 x %1 + 63 x %1 + 21 x %1 + 133 x %1 + 65 x %1 28 8 35 34 33 2 32 3 31 4 + 8 x %1 - 2 x - 33 x %1 - 96 x %1 - 91 x %1 - 49 x %1 30 5 29 6 28 7 27 8 34 33 - 203 x %1 - 178 x %1 - 45 x %1 - 4 x %1 + 4 x + 30 x %1 32 2 31 3 30 4 29 5 28 6 + 61 x %1 + 49 x %1 + 175 x %1 + 263 x %1 + 105 x %1 27 7 26 8 33 32 31 2 30 3 + 11 x %1 + x %1 - 4 x - 21 x %1 - 27 x %1 - 77 x %1 29 4 28 5 27 6 26 7 32 31 - 220 x %1 - 134 x %1 + 19 x %1 + 3 x %1 + 3 x + 8 x %1 30 2 29 3 28 4 27 5 26 6 + 7 x %1 + 95 x %1 + 110 x %1 - 114 x %1 - 61 x %1 25 7 31 30 29 2 28 3 27 4 + 7 x %1 - x + 7 x %1 - 10 x %1 - 73 x %1 + 180 x %1 26 5 25 6 24 7 30 29 28 2 + 239 x %1 - 3 x %1 - 18 x %1 - 2 x - 7 x %1 + 45 x %1 27 3 26 4 25 5 24 6 23 7 - 121 x %1 - 454 x %1 - 146 x %1 + 75 x %1 + 15 x %1 29 28 27 2 26 3 25 4 24 5 + 2 x - 20 x %1 + 19 x %1 + 485 x %1 + 528 x %1 - 31 x %1 23 6 22 7 28 27 26 2 25 3 - 63 x %1 - 6 x %1 + 4 x + 16 x %1 - 297 x %1 - 837 x %1 24 4 23 5 22 6 21 7 27 26 - 343 x %1 + 77 x %1 - 18 x %1 + x %1 - 6 x + 97 x %1 25 2 24 3 23 4 22 5 21 6 + 701 x %1 + 812 x %1 + 42 x %1 + 78 x %1 + 88 x %1 26 25 24 2 23 3 22 4 - 13 x - 304 x %1 - 815 x %1 - 203 x %1 + 108 x %1 21 5 20 6 25 24 23 2 - 286 x %1 - 100 x %1 + 54 x + 397 x %1 + 217 x %1 22 3 21 4 20 5 19 6 24 - 643 x %1 + 16 x %1 + 356 x %1 + 68 x %1 - 77 x 23 22 2 21 3 20 4 19 5 - 105 x %1 + 897 x %1 + 994 x %1 - 249 x %1 - 248 x %1 18 6 23 22 21 2 20 3 - 30 x %1 + 20 x - 537 x %1 - 1558 x %1 - 486 x %1 19 4 18 5 17 6 22 21 + 311 x %1 + 77 x %1 + 8 x %1 + 121 x + 965 x %1 20 2 19 3 18 4 17 5 16 6 21 + 879 x %1 - 398 x %1 - 116 x %1 + 32 x %1 - x %1 - 220 x 20 19 2 18 3 17 4 16 5 - 484 x %1 + 860 x %1 + 807 x %1 - 157 x %1 - 51 x %1 20 19 18 2 17 3 16 4 + 84 x - 938 x %1 - 2275 x %1 - 399 x %1 + 292 x %1 15 5 19 18 17 2 16 3 + 28 x %1 + 345 x + 2387 x %1 + 2182 x %1 - 395 x %1 15 4 14 5 18 17 16 2 - 249 x %1 - 8 x %1 - 850 x - 2736 x %1 - 781 x %1 15 3 14 4 13 5 17 16 + 910 x %1 + 134 x %1 + x %1 + 1070 x + 1787 x %1 15 2 14 3 13 4 16 15 - 562 x %1 - 888 x %1 - 47 x %1 - 851 x - 604 x %1 14 2 13 3 12 4 15 14 + 698 x %1 + 555 x %1 + 10 x %1 + 476 x + 520 x %1 13 2 12 3 11 4 14 13 + 278 x %1 - 237 x %1 - x %1 - 441 x - 1747 x %1 12 2 11 3 13 12 11 2 - 1361 x %1 + 68 x %1 + 855 x + 3101 x %1 + 1733 x %1 10 3 12 11 10 2 9 3 11 - 12 x %1 - 1058 x - 3209 x %1 - 1371 x %1 + x %1 + 44 x 10 9 2 10 9 8 2 9 + 1872 x %1 + 757 x %1 + 2514 x - 112 x %1 - 296 x %1 - 5651 x 8 7 2 8 7 6 2 7 - 961 x %1 + 79 x %1 + 7692 x + 1074 x %1 - 13 x %1 - 7588 x 6 5 2 6 5 5 4 4 - 678 x %1 + x %1 + 5696 x + 283 x %1 - 3287 x - 78 x %1 + 1443 x 3 3 2 2 / 8 2 8 + 13 x %1 - 468 x - x %1 + 106 x - 15 x + 1) / ((x %1 - 2 x %1 / 7 2 8 7 7 6 6 5 5 4 4 - x %1 + x + 2 x %1 - x - x %1 + x - x %1 + x + 3 x %1 - 3 x 3 3 2 2 32 8 32 7 - 3 x %1 + 5 x + x %1 - 6 x + 4 x - 1) (x %1 - 8 x %1 32 6 32 5 32 4 32 3 32 2 + 28 x %1 - 56 x %1 + 70 x %1 - 56 x %1 + 28 x %1 32 26 7 32 26 6 25 7 26 5 - 8 x %1 - x %1 + x + 5 x %1 + 2 x %1 - 9 x %1 25 6 24 7 26 4 25 5 24 6 26 3 - 10 x %1 - x %1 + 5 x %1 + 17 x %1 + 3 x %1 + 5 x %1 25 4 24 5 23 6 26 2 25 3 24 4 - 5 x %1 + 2 x %1 + 5 x %1 - 9 x %1 - 20 x %1 - 20 x %1 23 5 22 6 26 25 2 24 3 23 4 - 25 x %1 - 4 x %1 + 5 x %1 + 28 x %1 + 35 x %1 + 49 x %1 22 5 21 6 26 25 24 2 23 3 + 20 x %1 + x %1 - x - 15 x %1 - 29 x %1 - 46 x %1 22 4 25 24 23 2 22 3 21 4 - 43 x %1 + 3 x + 12 x %1 + 19 x %1 + 52 x %1 + 3 x %1 20 5 24 23 22 2 21 3 20 4 - 11 x %1 - 2 x - x %1 - 38 x %1 - 33 x %1 + 28 x %1 19 5 23 22 21 2 19 4 18 5 + 5 x %1 - x + 16 x %1 + 66 x %1 - 14 x %1 + 10 x %1 22 21 20 2 19 3 18 4 17 5 - 3 x - 51 x %1 - 58 x %1 + 9 x %1 - 29 x %1 - 19 x %1 21 20 19 2 18 3 17 4 16 5 + 14 x + 59 x %1 + 7 x %1 + 4 x %1 + 50 x %1 + 15 x %1 20 19 18 2 17 3 16 4 15 5 - 18 x - 10 x %1 + 61 x %1 - 6 x %1 - 24 x %1 - 6 x %1 19 18 17 2 16 3 15 4 14 5 + 3 x - 68 x %1 - 82 x %1 - 16 x %1 - 17 x %1 + x %1 18 17 16 2 15 3 14 4 17 + 22 x + 77 x %1 + x %1 + 39 x %1 + 32 x %1 - 20 x 16 15 2 14 3 13 4 16 15 + 67 x %1 + 134 x %1 - 39 x %1 - 21 x %1 - 43 x - 285 x %1 14 2 13 3 12 4 15 14 - 189 x %1 + 22 x %1 + 7 x %1 + 135 x + 347 x %1 13 2 12 3 11 4 14 13 12 2 + 102 x %1 - 7 x %1 - x %1 - 152 x - 140 x %1 + 35 x %1 11 3 13 12 11 2 12 11 + x %1 + 37 x - 141 x %1 - 101 x %1 + 107 x + 199 x %1 10 2 11 10 9 2 10 9 + 81 x %1 - 110 x + 21 x %1 - 36 x %1 - 36 x - 299 x %1 8 2 9 8 7 2 8 7 7 + 9 x %1 + 115 x + 407 x %1 - x %1 + 79 x - 319 x %1 - 472 x 6 6 5 5 4 4 3 + 164 x %1 + 760 x - 55 x %1 - 737 x + 11 x %1 + 484 x - x %1 3 2 - 219 x + 66 x - 12 x + 1)) %1 := X[2, 1, 1, 1, 2, 1, 2] and in Maple format -(x^39*X[2,1,1,1,2,1,2]^10-10*x^39*X[2,1,1,1,2,1,2]^9+45*x^39*X[2,1,1,1,2,1,2]^ 8-120*x^39*X[2,1,1,1,2,1,2]^7-x^37*X[2,1,1,1,2,1,2]^9+210*x^39*X[2,1,1,1,2,1,2] ^6+9*x^37*X[2,1,1,1,2,1,2]^8-2*x^36*X[2,1,1,1,2,1,2]^9-252*x^39*X[2,1,1,1,2,1,2 ]^5-36*x^37*X[2,1,1,1,2,1,2]^7+18*x^36*X[2,1,1,1,2,1,2]^8+2*x^35*X[2,1,1,1,2,1, 2]^9+210*x^39*X[2,1,1,1,2,1,2]^4+84*x^37*X[2,1,1,1,2,1,2]^6-72*x^36*X[2,1,1,1,2 ,1,2]^7-18*x^35*X[2,1,1,1,2,1,2]^8-x^34*X[2,1,1,1,2,1,2]^9-120*x^39*X[2,1,1,1,2 ,1,2]^3-126*x^37*X[2,1,1,1,2,1,2]^5+168*x^36*X[2,1,1,1,2,1,2]^6+72*x^35*X[2,1,1 ,1,2,1,2]^7+12*x^34*X[2,1,1,1,2,1,2]^8-2*x^33*X[2,1,1,1,2,1,2]^9+45*x^39*X[2,1, 1,1,2,1,2]^2+126*x^37*X[2,1,1,1,2,1,2]^4-252*x^36*X[2,1,1,1,2,1,2]^5-168*x^35*X [2,1,1,1,2,1,2]^6-60*x^34*X[2,1,1,1,2,1,2]^7+12*x^33*X[2,1,1,1,2,1,2]^8+2*x^32* X[2,1,1,1,2,1,2]^9-10*x^39*X[2,1,1,1,2,1,2]-84*x^37*X[2,1,1,1,2,1,2]^3+252*x^36 *X[2,1,1,1,2,1,2]^4+252*x^35*X[2,1,1,1,2,1,2]^5+168*x^34*X[2,1,1,1,2,1,2]^6-24* x^33*X[2,1,1,1,2,1,2]^7-14*x^32*X[2,1,1,1,2,1,2]^8-x^31*X[2,1,1,1,2,1,2]^9+x^39 +36*x^37*X[2,1,1,1,2,1,2]^2-168*x^36*X[2,1,1,1,2,1,2]^3-252*x^35*X[2,1,1,1,2,1, 2]^4-294*x^34*X[2,1,1,1,2,1,2]^5+39*x^32*X[2,1,1,1,2,1,2]^7+6*x^31*X[2,1,1,1,2, 1,2]^8-9*x^37*X[2,1,1,1,2,1,2]+72*x^36*X[2,1,1,1,2,1,2]^2+168*x^35*X[2,1,1,1,2, 1,2]^3+336*x^34*X[2,1,1,1,2,1,2]^4+84*x^33*X[2,1,1,1,2,1,2]^5-49*x^32*X[2,1,1,1 ,2,1,2]^6-13*x^31*X[2,1,1,1,2,1,2]^7+7*x^30*X[2,1,1,1,2,1,2]^8+x^37-18*x^36*X[2 ,1,1,1,2,1,2]-72*x^35*X[2,1,1,1,2,1,2]^2-252*x^34*X[2,1,1,1,2,1,2]^3-168*x^33*X [2,1,1,1,2,1,2]^4+7*x^32*X[2,1,1,1,2,1,2]^5+7*x^31*X[2,1,1,1,2,1,2]^6-47*x^30*X [2,1,1,1,2,1,2]^7-10*x^29*X[2,1,1,1,2,1,2]^8+2*x^36+18*x^35*X[2,1,1,1,2,1,2]+ 120*x^34*X[2,1,1,1,2,1,2]^2+168*x^33*X[2,1,1,1,2,1,2]^3+63*x^32*X[2,1,1,1,2,1,2 ]^4+21*x^31*X[2,1,1,1,2,1,2]^5+133*x^30*X[2,1,1,1,2,1,2]^6+65*x^29*X[2,1,1,1,2, 1,2]^7+8*x^28*X[2,1,1,1,2,1,2]^8-2*x^35-33*x^34*X[2,1,1,1,2,1,2]-96*x^33*X[2,1, 1,1,2,1,2]^2-91*x^32*X[2,1,1,1,2,1,2]^3-49*x^31*X[2,1,1,1,2,1,2]^4-203*x^30*X[2 ,1,1,1,2,1,2]^5-178*x^29*X[2,1,1,1,2,1,2]^6-45*x^28*X[2,1,1,1,2,1,2]^7-4*x^27*X [2,1,1,1,2,1,2]^8+4*x^34+30*x^33*X[2,1,1,1,2,1,2]+61*x^32*X[2,1,1,1,2,1,2]^2+49 *x^31*X[2,1,1,1,2,1,2]^3+175*x^30*X[2,1,1,1,2,1,2]^4+263*x^29*X[2,1,1,1,2,1,2]^ 5+105*x^28*X[2,1,1,1,2,1,2]^6+11*x^27*X[2,1,1,1,2,1,2]^7+x^26*X[2,1,1,1,2,1,2]^ 8-4*x^33-21*x^32*X[2,1,1,1,2,1,2]-27*x^31*X[2,1,1,1,2,1,2]^2-77*x^30*X[2,1,1,1, 2,1,2]^3-220*x^29*X[2,1,1,1,2,1,2]^4-134*x^28*X[2,1,1,1,2,1,2]^5+19*x^27*X[2,1, 1,1,2,1,2]^6+3*x^26*X[2,1,1,1,2,1,2]^7+3*x^32+8*x^31*X[2,1,1,1,2,1,2]+7*x^30*X[ 2,1,1,1,2,1,2]^2+95*x^29*X[2,1,1,1,2,1,2]^3+110*x^28*X[2,1,1,1,2,1,2]^4-114*x^ 27*X[2,1,1,1,2,1,2]^5-61*x^26*X[2,1,1,1,2,1,2]^6+7*x^25*X[2,1,1,1,2,1,2]^7-x^31 +7*x^30*X[2,1,1,1,2,1,2]-10*x^29*X[2,1,1,1,2,1,2]^2-73*x^28*X[2,1,1,1,2,1,2]^3+ 180*x^27*X[2,1,1,1,2,1,2]^4+239*x^26*X[2,1,1,1,2,1,2]^5-3*x^25*X[2,1,1,1,2,1,2] ^6-18*x^24*X[2,1,1,1,2,1,2]^7-2*x^30-7*x^29*X[2,1,1,1,2,1,2]+45*x^28*X[2,1,1,1, 2,1,2]^2-121*x^27*X[2,1,1,1,2,1,2]^3-454*x^26*X[2,1,1,1,2,1,2]^4-146*x^25*X[2,1 ,1,1,2,1,2]^5+75*x^24*X[2,1,1,1,2,1,2]^6+15*x^23*X[2,1,1,1,2,1,2]^7+2*x^29-20*x ^28*X[2,1,1,1,2,1,2]+19*x^27*X[2,1,1,1,2,1,2]^2+485*x^26*X[2,1,1,1,2,1,2]^3+528 *x^25*X[2,1,1,1,2,1,2]^4-31*x^24*X[2,1,1,1,2,1,2]^5-63*x^23*X[2,1,1,1,2,1,2]^6-\ 6*x^22*X[2,1,1,1,2,1,2]^7+4*x^28+16*x^27*X[2,1,1,1,2,1,2]-297*x^26*X[2,1,1,1,2, 1,2]^2-837*x^25*X[2,1,1,1,2,1,2]^3-343*x^24*X[2,1,1,1,2,1,2]^4+77*x^23*X[2,1,1, 1,2,1,2]^5-18*x^22*X[2,1,1,1,2,1,2]^6+x^21*X[2,1,1,1,2,1,2]^7-6*x^27+97*x^26*X[ 2,1,1,1,2,1,2]+701*x^25*X[2,1,1,1,2,1,2]^2+812*x^24*X[2,1,1,1,2,1,2]^3+42*x^23* X[2,1,1,1,2,1,2]^4+78*x^22*X[2,1,1,1,2,1,2]^5+88*x^21*X[2,1,1,1,2,1,2]^6-13*x^ 26-304*x^25*X[2,1,1,1,2,1,2]-815*x^24*X[2,1,1,1,2,1,2]^2-203*x^23*X[2,1,1,1,2,1 ,2]^3+108*x^22*X[2,1,1,1,2,1,2]^4-286*x^21*X[2,1,1,1,2,1,2]^5-100*x^20*X[2,1,1, 1,2,1,2]^6+54*x^25+397*x^24*X[2,1,1,1,2,1,2]+217*x^23*X[2,1,1,1,2,1,2]^2-643*x^ 22*X[2,1,1,1,2,1,2]^3+16*x^21*X[2,1,1,1,2,1,2]^4+356*x^20*X[2,1,1,1,2,1,2]^5+68 *x^19*X[2,1,1,1,2,1,2]^6-77*x^24-105*x^23*X[2,1,1,1,2,1,2]+897*x^22*X[2,1,1,1,2 ,1,2]^2+994*x^21*X[2,1,1,1,2,1,2]^3-249*x^20*X[2,1,1,1,2,1,2]^4-248*x^19*X[2,1, 1,1,2,1,2]^5-30*x^18*X[2,1,1,1,2,1,2]^6+20*x^23-537*x^22*X[2,1,1,1,2,1,2]-1558* x^21*X[2,1,1,1,2,1,2]^2-486*x^20*X[2,1,1,1,2,1,2]^3+311*x^19*X[2,1,1,1,2,1,2]^4 +77*x^18*X[2,1,1,1,2,1,2]^5+8*x^17*X[2,1,1,1,2,1,2]^6+121*x^22+965*x^21*X[2,1,1 ,1,2,1,2]+879*x^20*X[2,1,1,1,2,1,2]^2-398*x^19*X[2,1,1,1,2,1,2]^3-116*x^18*X[2, 1,1,1,2,1,2]^4+32*x^17*X[2,1,1,1,2,1,2]^5-x^16*X[2,1,1,1,2,1,2]^6-220*x^21-484* x^20*X[2,1,1,1,2,1,2]+860*x^19*X[2,1,1,1,2,1,2]^2+807*x^18*X[2,1,1,1,2,1,2]^3-\ 157*x^17*X[2,1,1,1,2,1,2]^4-51*x^16*X[2,1,1,1,2,1,2]^5+84*x^20-938*x^19*X[2,1,1 ,1,2,1,2]-2275*x^18*X[2,1,1,1,2,1,2]^2-399*x^17*X[2,1,1,1,2,1,2]^3+292*x^16*X[2 ,1,1,1,2,1,2]^4+28*x^15*X[2,1,1,1,2,1,2]^5+345*x^19+2387*x^18*X[2,1,1,1,2,1,2]+ 2182*x^17*X[2,1,1,1,2,1,2]^2-395*x^16*X[2,1,1,1,2,1,2]^3-249*x^15*X[2,1,1,1,2,1 ,2]^4-8*x^14*X[2,1,1,1,2,1,2]^5-850*x^18-2736*x^17*X[2,1,1,1,2,1,2]-781*x^16*X[ 2,1,1,1,2,1,2]^2+910*x^15*X[2,1,1,1,2,1,2]^3+134*x^14*X[2,1,1,1,2,1,2]^4+x^13*X [2,1,1,1,2,1,2]^5+1070*x^17+1787*x^16*X[2,1,1,1,2,1,2]-562*x^15*X[2,1,1,1,2,1,2 ]^2-888*x^14*X[2,1,1,1,2,1,2]^3-47*x^13*X[2,1,1,1,2,1,2]^4-851*x^16-604*x^15*X[ 2,1,1,1,2,1,2]+698*x^14*X[2,1,1,1,2,1,2]^2+555*x^13*X[2,1,1,1,2,1,2]^3+10*x^12* X[2,1,1,1,2,1,2]^4+476*x^15+520*x^14*X[2,1,1,1,2,1,2]+278*x^13*X[2,1,1,1,2,1,2] ^2-237*x^12*X[2,1,1,1,2,1,2]^3-x^11*X[2,1,1,1,2,1,2]^4-441*x^14-1747*x^13*X[2,1 ,1,1,2,1,2]-1361*x^12*X[2,1,1,1,2,1,2]^2+68*x^11*X[2,1,1,1,2,1,2]^3+855*x^13+ 3101*x^12*X[2,1,1,1,2,1,2]+1733*x^11*X[2,1,1,1,2,1,2]^2-12*x^10*X[2,1,1,1,2,1,2 ]^3-1058*x^12-3209*x^11*X[2,1,1,1,2,1,2]-1371*x^10*X[2,1,1,1,2,1,2]^2+x^9*X[2,1 ,1,1,2,1,2]^3+44*x^11+1872*x^10*X[2,1,1,1,2,1,2]+757*x^9*X[2,1,1,1,2,1,2]^2+ 2514*x^10-112*x^9*X[2,1,1,1,2,1,2]-296*x^8*X[2,1,1,1,2,1,2]^2-5651*x^9-961*x^8* X[2,1,1,1,2,1,2]+79*x^7*X[2,1,1,1,2,1,2]^2+7692*x^8+1074*x^7*X[2,1,1,1,2,1,2]-\ 13*x^6*X[2,1,1,1,2,1,2]^2-7588*x^7-678*x^6*X[2,1,1,1,2,1,2]+x^5*X[2,1,1,1,2,1,2 ]^2+5696*x^6+283*x^5*X[2,1,1,1,2,1,2]-3287*x^5-78*x^4*X[2,1,1,1,2,1,2]+1443*x^4 +13*x^3*X[2,1,1,1,2,1,2]-468*x^3-x^2*X[2,1,1,1,2,1,2]+106*x^2-15*x+1)/(x^8*X[2, 1,1,1,2,1,2]^2-2*x^8*X[2,1,1,1,2,1,2]-x^7*X[2,1,1,1,2,1,2]^2+x^8+2*x^7*X[2,1,1, 1,2,1,2]-x^7-x^6*X[2,1,1,1,2,1,2]+x^6-x^5*X[2,1,1,1,2,1,2]+x^5+3*x^4*X[2,1,1,1, 2,1,2]-3*x^4-3*x^3*X[2,1,1,1,2,1,2]+5*x^3+x^2*X[2,1,1,1,2,1,2]-6*x^2+4*x-1)/(x^ 32*X[2,1,1,1,2,1,2]^8-8*x^32*X[2,1,1,1,2,1,2]^7+28*x^32*X[2,1,1,1,2,1,2]^6-56*x ^32*X[2,1,1,1,2,1,2]^5+70*x^32*X[2,1,1,1,2,1,2]^4-56*x^32*X[2,1,1,1,2,1,2]^3+28 *x^32*X[2,1,1,1,2,1,2]^2-8*x^32*X[2,1,1,1,2,1,2]-x^26*X[2,1,1,1,2,1,2]^7+x^32+5 *x^26*X[2,1,1,1,2,1,2]^6+2*x^25*X[2,1,1,1,2,1,2]^7-9*x^26*X[2,1,1,1,2,1,2]^5-10 *x^25*X[2,1,1,1,2,1,2]^6-x^24*X[2,1,1,1,2,1,2]^7+5*x^26*X[2,1,1,1,2,1,2]^4+17*x ^25*X[2,1,1,1,2,1,2]^5+3*x^24*X[2,1,1,1,2,1,2]^6+5*x^26*X[2,1,1,1,2,1,2]^3-5*x^ 25*X[2,1,1,1,2,1,2]^4+2*x^24*X[2,1,1,1,2,1,2]^5+5*x^23*X[2,1,1,1,2,1,2]^6-9*x^ 26*X[2,1,1,1,2,1,2]^2-20*x^25*X[2,1,1,1,2,1,2]^3-20*x^24*X[2,1,1,1,2,1,2]^4-25* x^23*X[2,1,1,1,2,1,2]^5-4*x^22*X[2,1,1,1,2,1,2]^6+5*x^26*X[2,1,1,1,2,1,2]+28*x^ 25*X[2,1,1,1,2,1,2]^2+35*x^24*X[2,1,1,1,2,1,2]^3+49*x^23*X[2,1,1,1,2,1,2]^4+20* x^22*X[2,1,1,1,2,1,2]^5+x^21*X[2,1,1,1,2,1,2]^6-x^26-15*x^25*X[2,1,1,1,2,1,2]-\ 29*x^24*X[2,1,1,1,2,1,2]^2-46*x^23*X[2,1,1,1,2,1,2]^3-43*x^22*X[2,1,1,1,2,1,2]^ 4+3*x^25+12*x^24*X[2,1,1,1,2,1,2]+19*x^23*X[2,1,1,1,2,1,2]^2+52*x^22*X[2,1,1,1, 2,1,2]^3+3*x^21*X[2,1,1,1,2,1,2]^4-11*x^20*X[2,1,1,1,2,1,2]^5-2*x^24-x^23*X[2,1 ,1,1,2,1,2]-38*x^22*X[2,1,1,1,2,1,2]^2-33*x^21*X[2,1,1,1,2,1,2]^3+28*x^20*X[2,1 ,1,1,2,1,2]^4+5*x^19*X[2,1,1,1,2,1,2]^5-x^23+16*x^22*X[2,1,1,1,2,1,2]+66*x^21*X [2,1,1,1,2,1,2]^2-14*x^19*X[2,1,1,1,2,1,2]^4+10*x^18*X[2,1,1,1,2,1,2]^5-3*x^22-\ 51*x^21*X[2,1,1,1,2,1,2]-58*x^20*X[2,1,1,1,2,1,2]^2+9*x^19*X[2,1,1,1,2,1,2]^3-\ 29*x^18*X[2,1,1,1,2,1,2]^4-19*x^17*X[2,1,1,1,2,1,2]^5+14*x^21+59*x^20*X[2,1,1,1 ,2,1,2]+7*x^19*X[2,1,1,1,2,1,2]^2+4*x^18*X[2,1,1,1,2,1,2]^3+50*x^17*X[2,1,1,1,2 ,1,2]^4+15*x^16*X[2,1,1,1,2,1,2]^5-18*x^20-10*x^19*X[2,1,1,1,2,1,2]+61*x^18*X[2 ,1,1,1,2,1,2]^2-6*x^17*X[2,1,1,1,2,1,2]^3-24*x^16*X[2,1,1,1,2,1,2]^4-6*x^15*X[2 ,1,1,1,2,1,2]^5+3*x^19-68*x^18*X[2,1,1,1,2,1,2]-82*x^17*X[2,1,1,1,2,1,2]^2-16*x ^16*X[2,1,1,1,2,1,2]^3-17*x^15*X[2,1,1,1,2,1,2]^4+x^14*X[2,1,1,1,2,1,2]^5+22*x^ 18+77*x^17*X[2,1,1,1,2,1,2]+x^16*X[2,1,1,1,2,1,2]^2+39*x^15*X[2,1,1,1,2,1,2]^3+ 32*x^14*X[2,1,1,1,2,1,2]^4-20*x^17+67*x^16*X[2,1,1,1,2,1,2]+134*x^15*X[2,1,1,1, 2,1,2]^2-39*x^14*X[2,1,1,1,2,1,2]^3-21*x^13*X[2,1,1,1,2,1,2]^4-43*x^16-285*x^15 *X[2,1,1,1,2,1,2]-189*x^14*X[2,1,1,1,2,1,2]^2+22*x^13*X[2,1,1,1,2,1,2]^3+7*x^12 *X[2,1,1,1,2,1,2]^4+135*x^15+347*x^14*X[2,1,1,1,2,1,2]+102*x^13*X[2,1,1,1,2,1,2 ]^2-7*x^12*X[2,1,1,1,2,1,2]^3-x^11*X[2,1,1,1,2,1,2]^4-152*x^14-140*x^13*X[2,1,1 ,1,2,1,2]+35*x^12*X[2,1,1,1,2,1,2]^2+x^11*X[2,1,1,1,2,1,2]^3+37*x^13-141*x^12*X [2,1,1,1,2,1,2]-101*x^11*X[2,1,1,1,2,1,2]^2+107*x^12+199*x^11*X[2,1,1,1,2,1,2]+ 81*x^10*X[2,1,1,1,2,1,2]^2-110*x^11+21*x^10*X[2,1,1,1,2,1,2]-36*x^9*X[2,1,1,1,2 ,1,2]^2-36*x^10-299*x^9*X[2,1,1,1,2,1,2]+9*x^8*X[2,1,1,1,2,1,2]^2+115*x^9+407*x ^8*X[2,1,1,1,2,1,2]-x^7*X[2,1,1,1,2,1,2]^2+79*x^8-319*x^7*X[2,1,1,1,2,1,2]-472* x^7+164*x^6*X[2,1,1,1,2,1,2]+760*x^6-55*x^5*X[2,1,1,1,2,1,2]-737*x^5+11*x^4*X[2 ,1,1,1,2,1,2]+484*x^4-x^3*X[2,1,1,1,2,1,2]-219*x^3+66*x^2-12*x+1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [2, 1, 1, 1, 2, 1, 2], equals , - 7/8 + ---- 16 95 21 n The variance equals , - -- + ---- 64 256 3597 291 n The , 3, -th moment about the mean is , - ---- + ----- 1024 2048 15551 1323 2 13497 The , 4, -th moment about the mean is , - ----- + ----- n - ----- n 4096 65536 32768 The compositions of, 10, that yield the, 7, -th largest growth, that is, 1.8922218871524161071, are , [1, 2, 1, 1, 2, 1, 2], [1, 2, 1, 2, 1, 1, 2], [2, 1, 1, 2, 1, 2, 1], [2, 1, 2, 1, 1, 2, 1] Theorem Number, 7, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 2, 1, 1, 2, 1, 2] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 17 16 15 14 13 12 11 10 ) a(n) x = (x - x + 3 x - 2 x + 3 x + x - 7 x + 14 x / ----- n = 0 9 8 7 6 5 4 3 2 - 16 x + 8 x + 8 x - 28 x + 45 x - 51 x + 41 x - 22 x + 7 x - 1) / 16 15 14 13 12 11 10 9 / ((-1 + x) (x - x + 2 x - 2 x + x + 3 x - 9 x + 13 x / 8 7 6 5 4 3 2 - 10 x - x + 16 x - 31 x + 40 x - 36 x + 21 x - 7 x + 1)) and in Maple format (x^17-x^16+3*x^15-2*x^14+3*x^13+x^12-7*x^11+14*x^10-16*x^9+8*x^8+8*x^7-28*x^6+ 45*x^5-51*x^4+41*x^3-22*x^2+7*x-1)/(-1+x)/(x^16-x^15+2*x^14-2*x^13+x^12+3*x^11-\ 9*x^10+13*x^9-10*x^8-x^7+16*x^6-31*x^5+40*x^4-36*x^3+21*x^2-7*x+1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1015, 2001, 3909, 7564, 14508, 27623, 52305, 98683, 185821, 349684, 658248, 1240164, 2339122, 4416913, 8348710, 15792895, 29891135, 56593165, 107163160, 202920970] The limit of a(n+1)/a(n) as n goes to infinity is 1.89222188715 a(n) is asymptotic to .997060153762*1.89222188715^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [1, 2, 1, 1, 2, 1, 2], denoted by the variable, X[1, 2, 1, 1, 2, 1, 2], is 20 5 20 4 19 5 20 3 19 4 18 5 20 2 (x %1 - 4 x %1 - x %1 + 6 x %1 + 5 x %1 + x %1 - 4 x %1 19 3 18 4 20 19 2 18 3 17 4 - 10 x %1 - 8 x %1 + x %1 + 10 x %1 + 21 x %1 + 4 x %1 19 18 2 17 3 16 4 19 18 - 5 x %1 - 25 x %1 - 18 x %1 - 3 x %1 + x + 14 x %1 17 2 16 3 15 4 18 17 16 2 + 30 x %1 + 18 x %1 - 2 x %1 - 3 x - 22 x %1 - 36 x %1 15 3 14 4 17 16 15 2 14 3 - 3 x %1 + 6 x %1 + 6 x + 30 x %1 + 22 x %1 - 14 x %1 13 4 16 15 14 2 13 3 12 4 - 7 x %1 - 9 x - 27 x %1 + 3 x %1 + 28 x %1 + 4 x %1 15 14 13 2 12 3 11 4 14 + 10 x + 12 x %1 - 41 x %1 - 28 x %1 - x %1 - 7 x 13 12 2 11 3 13 12 11 2 + 26 x %1 + 71 x %1 + 17 x %1 - 6 x - 76 x %1 - 73 x %1 10 3 12 11 10 2 9 3 11 - 6 x %1 + 29 x + 108 x %1 + 51 x %1 + x %1 - 51 x 10 9 2 10 9 8 2 9 8 - 99 x %1 - 24 x %1 + 54 x + 48 x %1 + 7 x %1 - 24 x + 20 x %1 7 2 8 7 7 6 6 5 - x %1 - 36 x - 72 x %1 + 109 x + 85 x %1 - 169 x - 62 x %1 5 4 4 3 3 2 2 + 188 x + 29 x %1 - 155 x - 8 x %1 + 92 x + x %1 - 37 x + 9 x - 1) / 19 5 19 4 18 5 19 3 18 4 / ((-1 + x) (x %1 - 4 x %1 - x %1 + 6 x %1 + 5 x %1 / 19 2 18 3 17 4 19 18 2 17 3 - 4 x %1 - 10 x %1 - 4 x %1 + x %1 + 10 x %1 + 15 x %1 16 4 18 17 2 16 3 18 17 + 3 x %1 - 5 x %1 - 21 x %1 - 14 x %1 + x + 13 x %1 16 2 15 3 14 4 17 16 15 2 + 24 x %1 + 7 x %1 - 4 x %1 - 3 x - 18 x %1 - 21 x %1 14 3 13 4 16 15 14 2 13 3 + 6 x %1 + 6 x %1 + 5 x + 21 x %1 + 7 x %1 - 20 x %1 12 4 15 14 13 2 12 3 11 4 - 4 x %1 - 7 x - 16 x %1 + 21 x %1 + 24 x %1 + x %1 14 13 12 2 11 3 13 12 + 7 x - 6 x %1 - 50 x %1 - 16 x %1 - x + 44 x %1 11 2 10 3 12 11 10 2 9 3 + 60 x %1 + 6 x %1 - 14 x - 79 x %1 - 46 x %1 - x %1 11 10 9 2 10 9 8 2 9 + 34 x + 85 x %1 + 23 x %1 - 45 x - 54 x %1 - 7 x %1 + 32 x 8 7 2 8 7 7 6 6 5 + x %1 + x %1 + 8 x + 48 x %1 - 64 x - 69 x %1 + 118 x + 56 x %1 5 4 4 3 3 2 2 - 147 x - 28 x %1 + 133 x + 8 x %1 - 85 x - x %1 + 36 x - 9 x + 1)) %1 := X[1, 2, 1, 1, 2, 1, 2] and in Maple format (x^20*X[1,2,1,1,2,1,2]^5-4*x^20*X[1,2,1,1,2,1,2]^4-x^19*X[1,2,1,1,2,1,2]^5+6*x^ 20*X[1,2,1,1,2,1,2]^3+5*x^19*X[1,2,1,1,2,1,2]^4+x^18*X[1,2,1,1,2,1,2]^5-4*x^20* X[1,2,1,1,2,1,2]^2-10*x^19*X[1,2,1,1,2,1,2]^3-8*x^18*X[1,2,1,1,2,1,2]^4+x^20*X[ 1,2,1,1,2,1,2]+10*x^19*X[1,2,1,1,2,1,2]^2+21*x^18*X[1,2,1,1,2,1,2]^3+4*x^17*X[1 ,2,1,1,2,1,2]^4-5*x^19*X[1,2,1,1,2,1,2]-25*x^18*X[1,2,1,1,2,1,2]^2-18*x^17*X[1, 2,1,1,2,1,2]^3-3*x^16*X[1,2,1,1,2,1,2]^4+x^19+14*x^18*X[1,2,1,1,2,1,2]+30*x^17* X[1,2,1,1,2,1,2]^2+18*x^16*X[1,2,1,1,2,1,2]^3-2*x^15*X[1,2,1,1,2,1,2]^4-3*x^18-\ 22*x^17*X[1,2,1,1,2,1,2]-36*x^16*X[1,2,1,1,2,1,2]^2-3*x^15*X[1,2,1,1,2,1,2]^3+6 *x^14*X[1,2,1,1,2,1,2]^4+6*x^17+30*x^16*X[1,2,1,1,2,1,2]+22*x^15*X[1,2,1,1,2,1, 2]^2-14*x^14*X[1,2,1,1,2,1,2]^3-7*x^13*X[1,2,1,1,2,1,2]^4-9*x^16-27*x^15*X[1,2, 1,1,2,1,2]+3*x^14*X[1,2,1,1,2,1,2]^2+28*x^13*X[1,2,1,1,2,1,2]^3+4*x^12*X[1,2,1, 1,2,1,2]^4+10*x^15+12*x^14*X[1,2,1,1,2,1,2]-41*x^13*X[1,2,1,1,2,1,2]^2-28*x^12* X[1,2,1,1,2,1,2]^3-x^11*X[1,2,1,1,2,1,2]^4-7*x^14+26*x^13*X[1,2,1,1,2,1,2]+71*x ^12*X[1,2,1,1,2,1,2]^2+17*x^11*X[1,2,1,1,2,1,2]^3-6*x^13-76*x^12*X[1,2,1,1,2,1, 2]-73*x^11*X[1,2,1,1,2,1,2]^2-6*x^10*X[1,2,1,1,2,1,2]^3+29*x^12+108*x^11*X[1,2, 1,1,2,1,2]+51*x^10*X[1,2,1,1,2,1,2]^2+x^9*X[1,2,1,1,2,1,2]^3-51*x^11-99*x^10*X[ 1,2,1,1,2,1,2]-24*x^9*X[1,2,1,1,2,1,2]^2+54*x^10+48*x^9*X[1,2,1,1,2,1,2]+7*x^8* X[1,2,1,1,2,1,2]^2-24*x^9+20*x^8*X[1,2,1,1,2,1,2]-x^7*X[1,2,1,1,2,1,2]^2-36*x^8 -72*x^7*X[1,2,1,1,2,1,2]+109*x^7+85*x^6*X[1,2,1,1,2,1,2]-169*x^6-62*x^5*X[1,2,1 ,1,2,1,2]+188*x^5+29*x^4*X[1,2,1,1,2,1,2]-155*x^4-8*x^3*X[1,2,1,1,2,1,2]+92*x^3 +x^2*X[1,2,1,1,2,1,2]-37*x^2+9*x-1)/(-1+x)/(x^19*X[1,2,1,1,2,1,2]^5-4*x^19*X[1, 2,1,1,2,1,2]^4-x^18*X[1,2,1,1,2,1,2]^5+6*x^19*X[1,2,1,1,2,1,2]^3+5*x^18*X[1,2,1 ,1,2,1,2]^4-4*x^19*X[1,2,1,1,2,1,2]^2-10*x^18*X[1,2,1,1,2,1,2]^3-4*x^17*X[1,2,1 ,1,2,1,2]^4+x^19*X[1,2,1,1,2,1,2]+10*x^18*X[1,2,1,1,2,1,2]^2+15*x^17*X[1,2,1,1, 2,1,2]^3+3*x^16*X[1,2,1,1,2,1,2]^4-5*x^18*X[1,2,1,1,2,1,2]-21*x^17*X[1,2,1,1,2, 1,2]^2-14*x^16*X[1,2,1,1,2,1,2]^3+x^18+13*x^17*X[1,2,1,1,2,1,2]+24*x^16*X[1,2,1 ,1,2,1,2]^2+7*x^15*X[1,2,1,1,2,1,2]^3-4*x^14*X[1,2,1,1,2,1,2]^4-3*x^17-18*x^16* X[1,2,1,1,2,1,2]-21*x^15*X[1,2,1,1,2,1,2]^2+6*x^14*X[1,2,1,1,2,1,2]^3+6*x^13*X[ 1,2,1,1,2,1,2]^4+5*x^16+21*x^15*X[1,2,1,1,2,1,2]+7*x^14*X[1,2,1,1,2,1,2]^2-20*x ^13*X[1,2,1,1,2,1,2]^3-4*x^12*X[1,2,1,1,2,1,2]^4-7*x^15-16*x^14*X[1,2,1,1,2,1,2 ]+21*x^13*X[1,2,1,1,2,1,2]^2+24*x^12*X[1,2,1,1,2,1,2]^3+x^11*X[1,2,1,1,2,1,2]^4 +7*x^14-6*x^13*X[1,2,1,1,2,1,2]-50*x^12*X[1,2,1,1,2,1,2]^2-16*x^11*X[1,2,1,1,2, 1,2]^3-x^13+44*x^12*X[1,2,1,1,2,1,2]+60*x^11*X[1,2,1,1,2,1,2]^2+6*x^10*X[1,2,1, 1,2,1,2]^3-14*x^12-79*x^11*X[1,2,1,1,2,1,2]-46*x^10*X[1,2,1,1,2,1,2]^2-x^9*X[1, 2,1,1,2,1,2]^3+34*x^11+85*x^10*X[1,2,1,1,2,1,2]+23*x^9*X[1,2,1,1,2,1,2]^2-45*x^ 10-54*x^9*X[1,2,1,1,2,1,2]-7*x^8*X[1,2,1,1,2,1,2]^2+32*x^9+x^8*X[1,2,1,1,2,1,2] +x^7*X[1,2,1,1,2,1,2]^2+8*x^8+48*x^7*X[1,2,1,1,2,1,2]-64*x^7-69*x^6*X[1,2,1,1,2 ,1,2]+118*x^6+56*x^5*X[1,2,1,1,2,1,2]-147*x^5-28*x^4*X[1,2,1,1,2,1,2]+133*x^4+8 *x^3*X[1,2,1,1,2,1,2]-85*x^3-x^2*X[1,2,1,1,2,1,2]+36*x^2-9*x+1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [1, 2, 1, 1, 2, 1, 2], equals , - 7/8 + ---- 16 91 21 n The variance equals , - -- + ---- 64 256 3177 279 n The , 3, -th moment about the mean is , - ---- + ----- 1024 2048 9715 1323 2 13593 The , 4, -th moment about the mean is , - ---- + ----- n - ----- n 4096 65536 32768 The compositions of, 10, that yield the, 8, -th largest growth, that is, 1.8922512945970379670, are , [2, 1, 1, 1, 1, 2, 2], [2, 2, 1, 1, 1, 1, 2] Theorem Number, 8, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 1, 1, 1, 2, 2] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 34 32 29 28 27 26 25 24 ) a(n) x = - (x + x + 4 x - 3 x + x + 6 x - 14 x + 18 x / ----- n = 0 23 22 21 20 19 18 17 16 - 13 x - 4 x + 34 x - 67 x + 89 x - 95 x + 70 x + 21 x 15 14 13 12 11 10 9 - 195 x + 412 x - 565 x + 498 x - 75 x - 707 x + 1647 x 8 7 6 5 4 3 2 - 2394 x + 2625 x - 2256 x + 1517 x - 782 x + 298 x - 79 x + 13 x / 35 34 33 32 30 29 28 27 26 - 1) / (x - x + x - x + 4 x - 7 x + 6 x + 2 x - 15 x / 25 24 23 22 21 20 19 18 + 26 x - 29 x + 17 x + 18 x - 69 x + 115 x - 146 x + 157 x 17 16 15 14 13 12 11 - 114 x - 38 x + 315 x - 642 x + 843 x - 676 x - 50 x 10 9 8 7 6 5 4 + 1296 x - 2706 x + 3729 x - 3915 x + 3222 x - 2068 x + 1013 x 3 2 - 365 x + 91 x - 14 x + 1) and in Maple format -(x^34+x^32+4*x^29-3*x^28+x^27+6*x^26-14*x^25+18*x^24-13*x^23-4*x^22+34*x^21-67 *x^20+89*x^19-95*x^18+70*x^17+21*x^16-195*x^15+412*x^14-565*x^13+498*x^12-75*x^ 11-707*x^10+1647*x^9-2394*x^8+2625*x^7-2256*x^6+1517*x^5-782*x^4+298*x^3-79*x^2 +13*x-1)/(x^35-x^34+x^33-x^32+4*x^30-7*x^29+6*x^28+2*x^27-15*x^26+26*x^25-29*x^ 24+17*x^23+18*x^22-69*x^21+115*x^20-146*x^19+157*x^18-114*x^17-38*x^16+315*x^15 -642*x^14+843*x^13-676*x^12-50*x^11+1296*x^10-2706*x^9+3729*x^8-3915*x^7+3222*x ^6-2068*x^5+1013*x^4-365*x^3+91*x^2-14*x+1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1015, 2001, 3910, 7572, 14544, 27741, 52615, 99363, 187076, 351560, 660114, 1239498, 2328916, 4380445, 8249451, 15555646, 29367633, 55500462, 104974373, 198672263] The limit of a(n+1)/a(n) as n goes to infinity is 1.89225129460 a(n) is asymptotic to .979774130645*1.89225129460^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [2, 1, 1, 1, 1, 2, 2], denoted by the variable, X[2, 1, 1, 1, 1, 2, 2], is 36 9 36 8 36 7 35 8 36 6 35 7 - (x %1 - 9 x %1 + 36 x %1 + 2 x %1 - 84 x %1 - 16 x %1 34 8 36 5 35 6 34 7 33 8 - 2 x %1 + 126 x %1 + 56 x %1 + 16 x %1 + 2 x %1 36 4 35 5 34 6 33 7 36 3 - 126 x %1 - 112 x %1 - 56 x %1 - 16 x %1 + 84 x %1 35 4 34 5 33 6 32 7 36 2 + 140 x %1 + 112 x %1 + 56 x %1 + x %1 - 36 x %1 35 3 34 4 33 5 32 6 31 7 - 112 x %1 - 140 x %1 - 112 x %1 - 7 x %1 + 4 x %1 36 35 2 34 3 33 4 32 5 + 9 x %1 + 56 x %1 + 112 x %1 + 140 x %1 + 21 x %1 31 6 30 7 29 8 36 35 34 2 - 28 x %1 - 9 x %1 - 2 x %1 - x - 16 x %1 - 56 x %1 33 3 32 4 31 5 30 6 29 7 - 112 x %1 - 35 x %1 + 84 x %1 + 65 x %1 + 23 x %1 28 8 35 34 33 2 32 3 31 4 + 4 x %1 + 2 x + 16 x %1 + 56 x %1 + 35 x %1 - 140 x %1 30 5 29 6 28 7 27 8 34 33 - 201 x %1 - 107 x %1 - 36 x %1 - 3 x %1 - 2 x - 16 x %1 32 2 31 3 30 4 29 5 28 6 - 21 x %1 + 140 x %1 + 345 x %1 + 271 x %1 + 132 x %1 27 7 26 8 33 32 31 2 30 3 + 29 x %1 + x %1 + 2 x + 7 x %1 - 84 x %1 - 355 x %1 29 4 28 5 27 6 26 7 32 31 - 415 x %1 - 259 x %1 - 93 x %1 - 11 x %1 - x + 28 x %1 30 2 29 3 28 4 27 5 26 6 + 219 x %1 + 397 x %1 + 295 x %1 + 110 x %1 + 13 x %1 25 7 31 30 29 2 28 3 27 4 - 6 x %1 - 4 x - 75 x %1 - 233 x %1 - 194 x %1 + 40 x %1 26 5 25 6 24 7 30 29 28 2 + 116 x %1 + 77 x %1 + 13 x %1 + 11 x + 77 x %1 + 66 x %1 27 3 26 4 25 5 24 6 23 7 - 247 x %1 - 461 x %1 - 353 x %1 - 137 x %1 - 11 x %1 29 28 27 2 26 3 25 4 - 11 x - 7 x %1 + 271 x %1 + 753 x %1 + 825 x %1 24 5 23 6 22 7 28 27 26 2 + 524 x %1 + 149 x %1 + 5 x %1 - x - 132 x %1 - 645 x %1 25 3 24 4 23 5 22 6 21 7 - 1100 x %1 - 1026 x %1 - 572 x %1 - 119 x %1 - x %1 27 26 25 2 24 3 23 4 + 25 x + 286 x %1 + 851 x %1 + 1149 x %1 + 990 x %1 22 5 21 6 26 25 24 2 + 479 x %1 + 71 x %1 - 52 x - 357 x %1 - 749 x %1 23 3 22 4 21 5 20 6 25 - 834 x %1 - 695 x %1 - 277 x %1 - 30 x %1 + 63 x 24 23 2 22 3 21 4 20 5 + 266 x %1 + 278 x %1 + 187 x %1 + 193 x %1 + 49 x %1 19 6 24 23 22 2 21 3 20 4 + 8 x %1 - 40 x + 29 x %1 + 499 x %1 + 644 x %1 + 361 x %1 19 5 18 6 23 22 21 2 + 115 x %1 - x %1 - 29 x - 495 x %1 - 1365 x %1 20 3 19 4 18 5 22 21 - 1409 x %1 - 753 x %1 - 177 x %1 + 139 x + 992 x %1 20 2 19 3 18 4 17 5 21 + 2029 x %1 + 1813 x %1 + 850 x %1 + 154 x %1 - 257 x 20 19 2 18 3 17 4 16 5 - 1340 x %1 - 2216 x %1 - 1669 x %1 - 686 x %1 - 92 x %1 20 19 18 2 17 3 16 4 + 340 x + 1382 x %1 + 1716 x %1 + 1064 x %1 + 414 x %1 15 5 19 18 17 2 16 3 + 37 x %1 - 349 x - 933 x %1 - 530 x %1 - 327 x %1 15 4 14 5 18 17 16 2 - 184 x %1 - 9 x %1 + 214 x - 169 x %1 - 989 x %1 15 3 14 4 13 5 17 16 - 205 x %1 + 57 x %1 + x %1 + 167 x + 1817 x %1 15 2 14 3 13 4 16 15 + 2251 x %1 + 392 x %1 - 11 x %1 - 823 x - 3484 x %1 14 2 13 3 12 4 15 14 - 2765 x %1 - 318 x %1 + x %1 + 1584 x + 4380 x %1 13 2 12 3 14 13 12 2 + 2444 x %1 + 164 x %1 - 2040 x - 3857 x %1 - 1635 x %1 11 3 13 12 11 2 10 3 12 - 55 x %1 + 1636 x + 1866 x %1 + 827 x %1 + 11 x %1 + 59 x 11 10 2 9 3 11 10 9 2 + 849 x %1 - 308 x %1 - x %1 - 2986 x - 3095 x %1 + 80 x %1 10 9 8 2 9 8 7 2 + 6395 x + 3976 x %1 - 13 x %1 - 9060 x - 3452 x %1 + x %1 8 7 7 6 6 5 + 9900 x + 2218 x %1 - 8654 x - 1067 x %1 + 6072 x + 376 x %1 5 4 4 3 3 2 2 - 3379 x - 92 x %1 + 1457 x + 14 x %1 - 469 x - x %1 + 106 x - 15 x / 37 9 37 8 37 7 36 8 37 6 + 1) / (x %1 - 9 x %1 + 36 x %1 + 3 x %1 - 84 x %1 / 36 7 35 8 37 5 36 6 35 7 - 24 x %1 - 3 x %1 + 126 x %1 + 84 x %1 + 25 x %1 34 8 37 4 36 5 35 6 34 7 + 3 x %1 - 126 x %1 - 168 x %1 - 91 x %1 - 25 x %1 33 8 37 3 36 4 35 5 34 6 - x %1 + 84 x %1 + 210 x %1 + 189 x %1 + 91 x %1 33 7 32 8 37 2 36 3 35 4 + 10 x %1 + x %1 - 36 x %1 - 168 x %1 - 245 x %1 34 5 33 6 32 7 31 8 37 36 2 - 189 x %1 - 42 x %1 - 3 x %1 - 2 x %1 + 9 x %1 + 84 x %1 35 3 34 4 33 5 32 6 31 7 37 + 203 x %1 + 245 x %1 + 98 x %1 - 6 x %1 + 2 x %1 - x 36 35 2 34 3 33 4 32 5 - 24 x %1 - 105 x %1 - 203 x %1 - 140 x %1 + 43 x %1 31 6 30 7 29 8 36 35 34 2 + 45 x %1 + 12 x %1 + 4 x %1 + 3 x + 31 x %1 + 105 x %1 33 3 32 4 31 5 30 6 29 7 + 126 x %1 - 90 x %1 - 200 x %1 - 95 x %1 - 37 x %1 28 8 35 34 33 2 32 3 31 4 - 6 x %1 - 4 x - 31 x %1 - 70 x %1 + 99 x %1 + 395 x %1 30 5 29 6 28 7 27 8 34 33 + 319 x %1 + 152 x %1 + 53 x %1 + 4 x %1 + 4 x + 22 x %1 32 2 31 3 30 4 29 5 28 6 - 62 x %1 - 438 x %1 - 590 x %1 - 362 x %1 - 178 x %1 27 7 26 8 33 32 31 2 30 3 - 42 x %1 - x %1 - 3 x + 21 x %1 + 283 x %1 + 650 x %1 29 4 28 5 27 6 26 7 32 + 545 x %1 + 292 x %1 + 125 x %1 + 12 x %1 - 3 x 31 30 2 29 3 28 4 27 5 - 100 x %1 - 427 x %1 - 529 x %1 - 225 x %1 - 77 x %1 26 6 25 7 31 30 29 2 28 3 + 4 x %1 + 12 x %1 + 15 x + 155 x %1 + 322 x %1 + 21 x %1 27 4 26 5 25 6 24 7 30 - 300 x %1 - 259 x %1 - 136 x %1 - 20 x %1 - 24 x 29 28 2 27 3 26 4 25 5 - 112 x %1 + 92 x %1 + 716 x %1 + 887 x %1 + 592 x %1 24 6 23 7 29 28 27 2 + 213 x %1 + 15 x %1 + 17 x - 62 x %1 - 683 x %1 26 3 25 4 24 5 23 6 22 7 - 1402 x %1 - 1349 x %1 - 805 x %1 - 219 x %1 - 6 x %1 28 27 26 2 25 3 24 4 + 13 x + 315 x %1 + 1190 x %1 + 1776 x %1 + 1535 x %1 23 5 22 6 21 7 27 26 25 2 + 840 x %1 + 167 x %1 + x %1 - 58 x - 527 x %1 - 1366 x %1 24 3 23 4 22 5 21 6 26 - 1650 x %1 - 1384 x %1 - 673 x %1 - 94 x %1 + 96 x 25 24 2 23 3 22 4 21 5 + 572 x %1 + 1015 x %1 + 989 x %1 + 887 x %1 + 354 x %1 20 6 25 24 23 2 22 3 + 37 x %1 - 101 x - 333 x %1 - 93 x %1 + 68 x %1 21 4 20 5 19 6 24 23 - 116 x %1 - 20 x %1 - 9 x %1 + 45 x - 236 x %1 22 2 21 3 20 4 19 5 18 6 - 1175 x %1 - 1294 x %1 - 686 x %1 - 201 x %1 + x %1 23 22 21 2 20 3 19 4 + 88 x + 1003 x %1 + 2429 x %1 + 2355 x %1 + 1199 x %1 18 5 22 21 20 2 19 3 + 268 x %1 - 271 x - 1725 x %1 - 3321 x %1 - 2818 x %1 18 4 17 5 21 20 19 2 - 1260 x %1 - 217 x %1 + 445 x + 2199 x %1 + 3462 x %1 18 3 17 4 16 5 20 19 + 2437 x %1 + 963 x %1 + 121 x %1 - 564 x - 2207 x %1 18 2 17 3 16 4 15 5 19 - 2519 x %1 - 1425 x %1 - 551 x %1 - 45 x %1 + 574 x 18 17 2 16 3 15 4 14 5 + 1420 x %1 + 565 x %1 + 322 x %1 + 231 x %1 + 10 x %1 18 17 16 2 15 3 14 4 13 5 - 347 x + 391 x %1 + 1765 x %1 + 397 x %1 - 67 x %1 - x %1 17 16 15 2 14 3 13 4 - 277 x - 2967 x %1 - 3555 x %1 - 591 x %1 + 12 x %1 16 15 14 2 13 3 12 4 15 + 1310 x + 5416 x %1 + 4132 x %1 + 437 x %1 - x %1 - 2442 x 14 13 2 12 3 14 13 - 6517 x %1 - 3484 x %1 - 209 x %1 + 3004 x + 5377 x %1 12 2 11 3 13 12 11 2 + 2221 x %1 + 65 x %1 - 2145 x - 2110 x %1 - 1067 x %1 10 3 12 11 10 2 9 3 11 - 12 x %1 - 720 x - 1980 x %1 + 376 x %1 + x %1 + 5348 x 10 9 2 10 9 8 2 9 + 5068 x %1 - 92 x %1 - 10437 x - 5980 x %1 + 14 x %1 + 14079 x 8 7 2 8 7 7 6 + 4900 x %1 - x %1 - 14781 x - 2988 x %1 + 12427 x + 1364 x %1 6 5 5 4 4 3 3 - 8371 x - 455 x %1 + 4459 x + 105 x %1 - 1834 x - 15 x %1 + 561 x 2 2 + x %1 - 120 x + 16 x - 1) %1 := X[2, 1, 1, 1, 1, 2, 2] and in Maple format -(x^36*X[2,1,1,1,1,2,2]^9-9*x^36*X[2,1,1,1,1,2,2]^8+36*x^36*X[2,1,1,1,1,2,2]^7+ 2*x^35*X[2,1,1,1,1,2,2]^8-84*x^36*X[2,1,1,1,1,2,2]^6-16*x^35*X[2,1,1,1,1,2,2]^7 -2*x^34*X[2,1,1,1,1,2,2]^8+126*x^36*X[2,1,1,1,1,2,2]^5+56*x^35*X[2,1,1,1,1,2,2] ^6+16*x^34*X[2,1,1,1,1,2,2]^7+2*x^33*X[2,1,1,1,1,2,2]^8-126*x^36*X[2,1,1,1,1,2, 2]^4-112*x^35*X[2,1,1,1,1,2,2]^5-56*x^34*X[2,1,1,1,1,2,2]^6-16*x^33*X[2,1,1,1,1 ,2,2]^7+84*x^36*X[2,1,1,1,1,2,2]^3+140*x^35*X[2,1,1,1,1,2,2]^4+112*x^34*X[2,1,1 ,1,1,2,2]^5+56*x^33*X[2,1,1,1,1,2,2]^6+x^32*X[2,1,1,1,1,2,2]^7-36*x^36*X[2,1,1, 1,1,2,2]^2-112*x^35*X[2,1,1,1,1,2,2]^3-140*x^34*X[2,1,1,1,1,2,2]^4-112*x^33*X[2 ,1,1,1,1,2,2]^5-7*x^32*X[2,1,1,1,1,2,2]^6+4*x^31*X[2,1,1,1,1,2,2]^7+9*x^36*X[2, 1,1,1,1,2,2]+56*x^35*X[2,1,1,1,1,2,2]^2+112*x^34*X[2,1,1,1,1,2,2]^3+140*x^33*X[ 2,1,1,1,1,2,2]^4+21*x^32*X[2,1,1,1,1,2,2]^5-28*x^31*X[2,1,1,1,1,2,2]^6-9*x^30*X [2,1,1,1,1,2,2]^7-2*x^29*X[2,1,1,1,1,2,2]^8-x^36-16*x^35*X[2,1,1,1,1,2,2]-56*x^ 34*X[2,1,1,1,1,2,2]^2-112*x^33*X[2,1,1,1,1,2,2]^3-35*x^32*X[2,1,1,1,1,2,2]^4+84 *x^31*X[2,1,1,1,1,2,2]^5+65*x^30*X[2,1,1,1,1,2,2]^6+23*x^29*X[2,1,1,1,1,2,2]^7+ 4*x^28*X[2,1,1,1,1,2,2]^8+2*x^35+16*x^34*X[2,1,1,1,1,2,2]+56*x^33*X[2,1,1,1,1,2 ,2]^2+35*x^32*X[2,1,1,1,1,2,2]^3-140*x^31*X[2,1,1,1,1,2,2]^4-201*x^30*X[2,1,1,1 ,1,2,2]^5-107*x^29*X[2,1,1,1,1,2,2]^6-36*x^28*X[2,1,1,1,1,2,2]^7-3*x^27*X[2,1,1 ,1,1,2,2]^8-2*x^34-16*x^33*X[2,1,1,1,1,2,2]-21*x^32*X[2,1,1,1,1,2,2]^2+140*x^31 *X[2,1,1,1,1,2,2]^3+345*x^30*X[2,1,1,1,1,2,2]^4+271*x^29*X[2,1,1,1,1,2,2]^5+132 *x^28*X[2,1,1,1,1,2,2]^6+29*x^27*X[2,1,1,1,1,2,2]^7+x^26*X[2,1,1,1,1,2,2]^8+2*x ^33+7*x^32*X[2,1,1,1,1,2,2]-84*x^31*X[2,1,1,1,1,2,2]^2-355*x^30*X[2,1,1,1,1,2,2 ]^3-415*x^29*X[2,1,1,1,1,2,2]^4-259*x^28*X[2,1,1,1,1,2,2]^5-93*x^27*X[2,1,1,1,1 ,2,2]^6-11*x^26*X[2,1,1,1,1,2,2]^7-x^32+28*x^31*X[2,1,1,1,1,2,2]+219*x^30*X[2,1 ,1,1,1,2,2]^2+397*x^29*X[2,1,1,1,1,2,2]^3+295*x^28*X[2,1,1,1,1,2,2]^4+110*x^27* X[2,1,1,1,1,2,2]^5+13*x^26*X[2,1,1,1,1,2,2]^6-6*x^25*X[2,1,1,1,1,2,2]^7-4*x^31-\ 75*x^30*X[2,1,1,1,1,2,2]-233*x^29*X[2,1,1,1,1,2,2]^2-194*x^28*X[2,1,1,1,1,2,2]^ 3+40*x^27*X[2,1,1,1,1,2,2]^4+116*x^26*X[2,1,1,1,1,2,2]^5+77*x^25*X[2,1,1,1,1,2, 2]^6+13*x^24*X[2,1,1,1,1,2,2]^7+11*x^30+77*x^29*X[2,1,1,1,1,2,2]+66*x^28*X[2,1, 1,1,1,2,2]^2-247*x^27*X[2,1,1,1,1,2,2]^3-461*x^26*X[2,1,1,1,1,2,2]^4-353*x^25*X [2,1,1,1,1,2,2]^5-137*x^24*X[2,1,1,1,1,2,2]^6-11*x^23*X[2,1,1,1,1,2,2]^7-11*x^ 29-7*x^28*X[2,1,1,1,1,2,2]+271*x^27*X[2,1,1,1,1,2,2]^2+753*x^26*X[2,1,1,1,1,2,2 ]^3+825*x^25*X[2,1,1,1,1,2,2]^4+524*x^24*X[2,1,1,1,1,2,2]^5+149*x^23*X[2,1,1,1, 1,2,2]^6+5*x^22*X[2,1,1,1,1,2,2]^7-x^28-132*x^27*X[2,1,1,1,1,2,2]-645*x^26*X[2, 1,1,1,1,2,2]^2-1100*x^25*X[2,1,1,1,1,2,2]^3-1026*x^24*X[2,1,1,1,1,2,2]^4-572*x^ 23*X[2,1,1,1,1,2,2]^5-119*x^22*X[2,1,1,1,1,2,2]^6-x^21*X[2,1,1,1,1,2,2]^7+25*x^ 27+286*x^26*X[2,1,1,1,1,2,2]+851*x^25*X[2,1,1,1,1,2,2]^2+1149*x^24*X[2,1,1,1,1, 2,2]^3+990*x^23*X[2,1,1,1,1,2,2]^4+479*x^22*X[2,1,1,1,1,2,2]^5+71*x^21*X[2,1,1, 1,1,2,2]^6-52*x^26-357*x^25*X[2,1,1,1,1,2,2]-749*x^24*X[2,1,1,1,1,2,2]^2-834*x^ 23*X[2,1,1,1,1,2,2]^3-695*x^22*X[2,1,1,1,1,2,2]^4-277*x^21*X[2,1,1,1,1,2,2]^5-\ 30*x^20*X[2,1,1,1,1,2,2]^6+63*x^25+266*x^24*X[2,1,1,1,1,2,2]+278*x^23*X[2,1,1,1 ,1,2,2]^2+187*x^22*X[2,1,1,1,1,2,2]^3+193*x^21*X[2,1,1,1,1,2,2]^4+49*x^20*X[2,1 ,1,1,1,2,2]^5+8*x^19*X[2,1,1,1,1,2,2]^6-40*x^24+29*x^23*X[2,1,1,1,1,2,2]+499*x^ 22*X[2,1,1,1,1,2,2]^2+644*x^21*X[2,1,1,1,1,2,2]^3+361*x^20*X[2,1,1,1,1,2,2]^4+ 115*x^19*X[2,1,1,1,1,2,2]^5-x^18*X[2,1,1,1,1,2,2]^6-29*x^23-495*x^22*X[2,1,1,1, 1,2,2]-1365*x^21*X[2,1,1,1,1,2,2]^2-1409*x^20*X[2,1,1,1,1,2,2]^3-753*x^19*X[2,1 ,1,1,1,2,2]^4-177*x^18*X[2,1,1,1,1,2,2]^5+139*x^22+992*x^21*X[2,1,1,1,1,2,2]+ 2029*x^20*X[2,1,1,1,1,2,2]^2+1813*x^19*X[2,1,1,1,1,2,2]^3+850*x^18*X[2,1,1,1,1, 2,2]^4+154*x^17*X[2,1,1,1,1,2,2]^5-257*x^21-1340*x^20*X[2,1,1,1,1,2,2]-2216*x^ 19*X[2,1,1,1,1,2,2]^2-1669*x^18*X[2,1,1,1,1,2,2]^3-686*x^17*X[2,1,1,1,1,2,2]^4-\ 92*x^16*X[2,1,1,1,1,2,2]^5+340*x^20+1382*x^19*X[2,1,1,1,1,2,2]+1716*x^18*X[2,1, 1,1,1,2,2]^2+1064*x^17*X[2,1,1,1,1,2,2]^3+414*x^16*X[2,1,1,1,1,2,2]^4+37*x^15*X [2,1,1,1,1,2,2]^5-349*x^19-933*x^18*X[2,1,1,1,1,2,2]-530*x^17*X[2,1,1,1,1,2,2]^ 2-327*x^16*X[2,1,1,1,1,2,2]^3-184*x^15*X[2,1,1,1,1,2,2]^4-9*x^14*X[2,1,1,1,1,2, 2]^5+214*x^18-169*x^17*X[2,1,1,1,1,2,2]-989*x^16*X[2,1,1,1,1,2,2]^2-205*x^15*X[ 2,1,1,1,1,2,2]^3+57*x^14*X[2,1,1,1,1,2,2]^4+x^13*X[2,1,1,1,1,2,2]^5+167*x^17+ 1817*x^16*X[2,1,1,1,1,2,2]+2251*x^15*X[2,1,1,1,1,2,2]^2+392*x^14*X[2,1,1,1,1,2, 2]^3-11*x^13*X[2,1,1,1,1,2,2]^4-823*x^16-3484*x^15*X[2,1,1,1,1,2,2]-2765*x^14*X [2,1,1,1,1,2,2]^2-318*x^13*X[2,1,1,1,1,2,2]^3+x^12*X[2,1,1,1,1,2,2]^4+1584*x^15 +4380*x^14*X[2,1,1,1,1,2,2]+2444*x^13*X[2,1,1,1,1,2,2]^2+164*x^12*X[2,1,1,1,1,2 ,2]^3-2040*x^14-3857*x^13*X[2,1,1,1,1,2,2]-1635*x^12*X[2,1,1,1,1,2,2]^2-55*x^11 *X[2,1,1,1,1,2,2]^3+1636*x^13+1866*x^12*X[2,1,1,1,1,2,2]+827*x^11*X[2,1,1,1,1,2 ,2]^2+11*x^10*X[2,1,1,1,1,2,2]^3+59*x^12+849*x^11*X[2,1,1,1,1,2,2]-308*x^10*X[2 ,1,1,1,1,2,2]^2-x^9*X[2,1,1,1,1,2,2]^3-2986*x^11-3095*x^10*X[2,1,1,1,1,2,2]+80* x^9*X[2,1,1,1,1,2,2]^2+6395*x^10+3976*x^9*X[2,1,1,1,1,2,2]-13*x^8*X[2,1,1,1,1,2 ,2]^2-9060*x^9-3452*x^8*X[2,1,1,1,1,2,2]+x^7*X[2,1,1,1,1,2,2]^2+9900*x^8+2218*x ^7*X[2,1,1,1,1,2,2]-8654*x^7-1067*x^6*X[2,1,1,1,1,2,2]+6072*x^6+376*x^5*X[2,1,1 ,1,1,2,2]-3379*x^5-92*x^4*X[2,1,1,1,1,2,2]+1457*x^4+14*x^3*X[2,1,1,1,1,2,2]-469 *x^3-x^2*X[2,1,1,1,1,2,2]+106*x^2-15*x+1)/(x^37*X[2,1,1,1,1,2,2]^9-9*x^37*X[2,1 ,1,1,1,2,2]^8+36*x^37*X[2,1,1,1,1,2,2]^7+3*x^36*X[2,1,1,1,1,2,2]^8-84*x^37*X[2, 1,1,1,1,2,2]^6-24*x^36*X[2,1,1,1,1,2,2]^7-3*x^35*X[2,1,1,1,1,2,2]^8+126*x^37*X[ 2,1,1,1,1,2,2]^5+84*x^36*X[2,1,1,1,1,2,2]^6+25*x^35*X[2,1,1,1,1,2,2]^7+3*x^34*X [2,1,1,1,1,2,2]^8-126*x^37*X[2,1,1,1,1,2,2]^4-168*x^36*X[2,1,1,1,1,2,2]^5-91*x^ 35*X[2,1,1,1,1,2,2]^6-25*x^34*X[2,1,1,1,1,2,2]^7-x^33*X[2,1,1,1,1,2,2]^8+84*x^ 37*X[2,1,1,1,1,2,2]^3+210*x^36*X[2,1,1,1,1,2,2]^4+189*x^35*X[2,1,1,1,1,2,2]^5+ 91*x^34*X[2,1,1,1,1,2,2]^6+10*x^33*X[2,1,1,1,1,2,2]^7+x^32*X[2,1,1,1,1,2,2]^8-\ 36*x^37*X[2,1,1,1,1,2,2]^2-168*x^36*X[2,1,1,1,1,2,2]^3-245*x^35*X[2,1,1,1,1,2,2 ]^4-189*x^34*X[2,1,1,1,1,2,2]^5-42*x^33*X[2,1,1,1,1,2,2]^6-3*x^32*X[2,1,1,1,1,2 ,2]^7-2*x^31*X[2,1,1,1,1,2,2]^8+9*x^37*X[2,1,1,1,1,2,2]+84*x^36*X[2,1,1,1,1,2,2 ]^2+203*x^35*X[2,1,1,1,1,2,2]^3+245*x^34*X[2,1,1,1,1,2,2]^4+98*x^33*X[2,1,1,1,1 ,2,2]^5-6*x^32*X[2,1,1,1,1,2,2]^6+2*x^31*X[2,1,1,1,1,2,2]^7-x^37-24*x^36*X[2,1, 1,1,1,2,2]-105*x^35*X[2,1,1,1,1,2,2]^2-203*x^34*X[2,1,1,1,1,2,2]^3-140*x^33*X[2 ,1,1,1,1,2,2]^4+43*x^32*X[2,1,1,1,1,2,2]^5+45*x^31*X[2,1,1,1,1,2,2]^6+12*x^30*X [2,1,1,1,1,2,2]^7+4*x^29*X[2,1,1,1,1,2,2]^8+3*x^36+31*x^35*X[2,1,1,1,1,2,2]+105 *x^34*X[2,1,1,1,1,2,2]^2+126*x^33*X[2,1,1,1,1,2,2]^3-90*x^32*X[2,1,1,1,1,2,2]^4 -200*x^31*X[2,1,1,1,1,2,2]^5-95*x^30*X[2,1,1,1,1,2,2]^6-37*x^29*X[2,1,1,1,1,2,2 ]^7-6*x^28*X[2,1,1,1,1,2,2]^8-4*x^35-31*x^34*X[2,1,1,1,1,2,2]-70*x^33*X[2,1,1,1 ,1,2,2]^2+99*x^32*X[2,1,1,1,1,2,2]^3+395*x^31*X[2,1,1,1,1,2,2]^4+319*x^30*X[2,1 ,1,1,1,2,2]^5+152*x^29*X[2,1,1,1,1,2,2]^6+53*x^28*X[2,1,1,1,1,2,2]^7+4*x^27*X[2 ,1,1,1,1,2,2]^8+4*x^34+22*x^33*X[2,1,1,1,1,2,2]-62*x^32*X[2,1,1,1,1,2,2]^2-438* x^31*X[2,1,1,1,1,2,2]^3-590*x^30*X[2,1,1,1,1,2,2]^4-362*x^29*X[2,1,1,1,1,2,2]^5 -178*x^28*X[2,1,1,1,1,2,2]^6-42*x^27*X[2,1,1,1,1,2,2]^7-x^26*X[2,1,1,1,1,2,2]^8 -3*x^33+21*x^32*X[2,1,1,1,1,2,2]+283*x^31*X[2,1,1,1,1,2,2]^2+650*x^30*X[2,1,1,1 ,1,2,2]^3+545*x^29*X[2,1,1,1,1,2,2]^4+292*x^28*X[2,1,1,1,1,2,2]^5+125*x^27*X[2, 1,1,1,1,2,2]^6+12*x^26*X[2,1,1,1,1,2,2]^7-3*x^32-100*x^31*X[2,1,1,1,1,2,2]-427* x^30*X[2,1,1,1,1,2,2]^2-529*x^29*X[2,1,1,1,1,2,2]^3-225*x^28*X[2,1,1,1,1,2,2]^4 -77*x^27*X[2,1,1,1,1,2,2]^5+4*x^26*X[2,1,1,1,1,2,2]^6+12*x^25*X[2,1,1,1,1,2,2]^ 7+15*x^31+155*x^30*X[2,1,1,1,1,2,2]+322*x^29*X[2,1,1,1,1,2,2]^2+21*x^28*X[2,1,1 ,1,1,2,2]^3-300*x^27*X[2,1,1,1,1,2,2]^4-259*x^26*X[2,1,1,1,1,2,2]^5-136*x^25*X[ 2,1,1,1,1,2,2]^6-20*x^24*X[2,1,1,1,1,2,2]^7-24*x^30-112*x^29*X[2,1,1,1,1,2,2]+ 92*x^28*X[2,1,1,1,1,2,2]^2+716*x^27*X[2,1,1,1,1,2,2]^3+887*x^26*X[2,1,1,1,1,2,2 ]^4+592*x^25*X[2,1,1,1,1,2,2]^5+213*x^24*X[2,1,1,1,1,2,2]^6+15*x^23*X[2,1,1,1,1 ,2,2]^7+17*x^29-62*x^28*X[2,1,1,1,1,2,2]-683*x^27*X[2,1,1,1,1,2,2]^2-1402*x^26* X[2,1,1,1,1,2,2]^3-1349*x^25*X[2,1,1,1,1,2,2]^4-805*x^24*X[2,1,1,1,1,2,2]^5-219 *x^23*X[2,1,1,1,1,2,2]^6-6*x^22*X[2,1,1,1,1,2,2]^7+13*x^28+315*x^27*X[2,1,1,1,1 ,2,2]+1190*x^26*X[2,1,1,1,1,2,2]^2+1776*x^25*X[2,1,1,1,1,2,2]^3+1535*x^24*X[2,1 ,1,1,1,2,2]^4+840*x^23*X[2,1,1,1,1,2,2]^5+167*x^22*X[2,1,1,1,1,2,2]^6+x^21*X[2, 1,1,1,1,2,2]^7-58*x^27-527*x^26*X[2,1,1,1,1,2,2]-1366*x^25*X[2,1,1,1,1,2,2]^2-\ 1650*x^24*X[2,1,1,1,1,2,2]^3-1384*x^23*X[2,1,1,1,1,2,2]^4-673*x^22*X[2,1,1,1,1, 2,2]^5-94*x^21*X[2,1,1,1,1,2,2]^6+96*x^26+572*x^25*X[2,1,1,1,1,2,2]+1015*x^24*X [2,1,1,1,1,2,2]^2+989*x^23*X[2,1,1,1,1,2,2]^3+887*x^22*X[2,1,1,1,1,2,2]^4+354*x ^21*X[2,1,1,1,1,2,2]^5+37*x^20*X[2,1,1,1,1,2,2]^6-101*x^25-333*x^24*X[2,1,1,1,1 ,2,2]-93*x^23*X[2,1,1,1,1,2,2]^2+68*x^22*X[2,1,1,1,1,2,2]^3-116*x^21*X[2,1,1,1, 1,2,2]^4-20*x^20*X[2,1,1,1,1,2,2]^5-9*x^19*X[2,1,1,1,1,2,2]^6+45*x^24-236*x^23* X[2,1,1,1,1,2,2]-1175*x^22*X[2,1,1,1,1,2,2]^2-1294*x^21*X[2,1,1,1,1,2,2]^3-686* x^20*X[2,1,1,1,1,2,2]^4-201*x^19*X[2,1,1,1,1,2,2]^5+x^18*X[2,1,1,1,1,2,2]^6+88* x^23+1003*x^22*X[2,1,1,1,1,2,2]+2429*x^21*X[2,1,1,1,1,2,2]^2+2355*x^20*X[2,1,1, 1,1,2,2]^3+1199*x^19*X[2,1,1,1,1,2,2]^4+268*x^18*X[2,1,1,1,1,2,2]^5-271*x^22-\ 1725*x^21*X[2,1,1,1,1,2,2]-3321*x^20*X[2,1,1,1,1,2,2]^2-2818*x^19*X[2,1,1,1,1,2 ,2]^3-1260*x^18*X[2,1,1,1,1,2,2]^4-217*x^17*X[2,1,1,1,1,2,2]^5+445*x^21+2199*x^ 20*X[2,1,1,1,1,2,2]+3462*x^19*X[2,1,1,1,1,2,2]^2+2437*x^18*X[2,1,1,1,1,2,2]^3+ 963*x^17*X[2,1,1,1,1,2,2]^4+121*x^16*X[2,1,1,1,1,2,2]^5-564*x^20-2207*x^19*X[2, 1,1,1,1,2,2]-2519*x^18*X[2,1,1,1,1,2,2]^2-1425*x^17*X[2,1,1,1,1,2,2]^3-551*x^16 *X[2,1,1,1,1,2,2]^4-45*x^15*X[2,1,1,1,1,2,2]^5+574*x^19+1420*x^18*X[2,1,1,1,1,2 ,2]+565*x^17*X[2,1,1,1,1,2,2]^2+322*x^16*X[2,1,1,1,1,2,2]^3+231*x^15*X[2,1,1,1, 1,2,2]^4+10*x^14*X[2,1,1,1,1,2,2]^5-347*x^18+391*x^17*X[2,1,1,1,1,2,2]+1765*x^ 16*X[2,1,1,1,1,2,2]^2+397*x^15*X[2,1,1,1,1,2,2]^3-67*x^14*X[2,1,1,1,1,2,2]^4-x^ 13*X[2,1,1,1,1,2,2]^5-277*x^17-2967*x^16*X[2,1,1,1,1,2,2]-3555*x^15*X[2,1,1,1,1 ,2,2]^2-591*x^14*X[2,1,1,1,1,2,2]^3+12*x^13*X[2,1,1,1,1,2,2]^4+1310*x^16+5416*x ^15*X[2,1,1,1,1,2,2]+4132*x^14*X[2,1,1,1,1,2,2]^2+437*x^13*X[2,1,1,1,1,2,2]^3-x ^12*X[2,1,1,1,1,2,2]^4-2442*x^15-6517*x^14*X[2,1,1,1,1,2,2]-3484*x^13*X[2,1,1,1 ,1,2,2]^2-209*x^12*X[2,1,1,1,1,2,2]^3+3004*x^14+5377*x^13*X[2,1,1,1,1,2,2]+2221 *x^12*X[2,1,1,1,1,2,2]^2+65*x^11*X[2,1,1,1,1,2,2]^3-2145*x^13-2110*x^12*X[2,1,1 ,1,1,2,2]-1067*x^11*X[2,1,1,1,1,2,2]^2-12*x^10*X[2,1,1,1,1,2,2]^3-720*x^12-1980 *x^11*X[2,1,1,1,1,2,2]+376*x^10*X[2,1,1,1,1,2,2]^2+x^9*X[2,1,1,1,1,2,2]^3+5348* x^11+5068*x^10*X[2,1,1,1,1,2,2]-92*x^9*X[2,1,1,1,1,2,2]^2-10437*x^10-5980*x^9*X [2,1,1,1,1,2,2]+14*x^8*X[2,1,1,1,1,2,2]^2+14079*x^9+4900*x^8*X[2,1,1,1,1,2,2]-x ^7*X[2,1,1,1,1,2,2]^2-14781*x^8-2988*x^7*X[2,1,1,1,1,2,2]+12427*x^7+1364*x^6*X[ 2,1,1,1,1,2,2]-8371*x^6-455*x^5*X[2,1,1,1,1,2,2]+4459*x^5+105*x^4*X[2,1,1,1,1,2 ,2]-1834*x^4-15*x^3*X[2,1,1,1,1,2,2]+561*x^3+x^2*X[2,1,1,1,1,2,2]-120*x^2+16*x-\ 1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [2, 1, 1, 1, 1, 2, 2], equals , - 7/8 + ---- 16 95 21 n The variance equals , - -- + ---- 64 256 3489 279 n The , 3, -th moment about the mean is , - ---- + ----- 1024 2048 12695 1323 2 14601 The , 4, -th moment about the mean is , - ----- + ----- n - ----- n 4096 65536 32768 The compositions of, 10, that yield the, 9, -th largest growth, that is, 1.8922578866301683686, are , [1, 2, 1, 1, 1, 2, 2], [1, 2, 2, 1, 1, 1, 2], [2, 1, 1, 1, 2, 2, 1], [2, 2, 1, 1, 1, 2, 1] Theorem Number, 9, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 2, 1, 1, 1, 2, 2] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 20 19 18 17 16 15 14 13 ) a(n) x = (x - x + 2 x - 2 x + 5 x - 5 x + 6 x - 3 x / ----- n = 0 12 11 10 9 8 7 6 5 4 - x + 7 x - 15 x + 27 x - 41 x + 58 x - 80 x + 97 x - 92 x 3 2 / 3 16 15 12 + 63 x - 29 x + 8 x - 1) / ((-1 + x) (x + x - 1) (x - x + 3 x / 11 9 8 7 6 5 4 3 2 - 4 x + 7 x - 8 x + 2 x - 2 x + 17 x - 34 x + 35 x - 21 x + 7 x - 1)) and in Maple format (x^20-x^19+2*x^18-2*x^17+5*x^16-5*x^15+6*x^14-3*x^13-x^12+7*x^11-15*x^10+27*x^9 -41*x^8+58*x^7-80*x^6+97*x^5-92*x^4+63*x^3-29*x^2+8*x-1)/(-1+x)/(x^3+x-1)/(x^16 -x^15+3*x^12-4*x^11+7*x^9-8*x^8+2*x^7-2*x^6+17*x^5-34*x^4+35*x^3-21*x^2+7*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1015, 2001, 3910, 7572, 14544, 27742, 52627, 99439, 187417, 352785, 663881, 1249834, 2354932, 4441690, 8386192, 15848188, 29972022, 56713667, 107352430, 203243899] The limit of a(n+1)/a(n) as n goes to infinity is 1.89225788663 a(n) is asymptotic to .998835530746*1.89225788663^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [1, 2, 1, 1, 1, 2, 2], denoted by the variable, X[1, 2, 1, 1, 1, 2, 2], is 21 5 21 4 21 3 20 4 19 5 21 2 (x %1 - 5 x %1 + 10 x %1 + 2 x %1 + x %1 - 10 x %1 20 3 19 4 18 5 21 20 2 19 3 - 8 x %1 - 7 x %1 - x %1 + 5 x %1 + 12 x %1 + 18 x %1 18 4 17 5 21 20 19 2 18 3 + 7 x %1 + x %1 - x - 8 x %1 - 22 x %1 - 19 x %1 17 4 20 19 18 2 17 3 16 4 - 8 x %1 + 2 x + 13 x %1 + 25 x %1 + 25 x %1 + 5 x %1 19 18 17 2 16 3 15 4 18 - 3 x - 16 x %1 - 37 x %1 - 25 x %1 - 6 x %1 + 4 x 17 16 2 15 3 14 4 17 16 + 26 x %1 + 45 x %1 + 28 x %1 + 7 x %1 - 7 x - 35 x %1 15 2 14 3 13 4 16 15 14 2 - 49 x %1 - 28 x %1 - 7 x %1 + 10 x + 38 x %1 + 44 x %1 13 3 12 4 15 14 13 2 12 3 + 25 x %1 + 4 x %1 - 11 x - 32 x %1 - 31 x %1 - 16 x %1 11 4 14 13 12 2 11 3 13 12 - x %1 + 9 x + 15 x %1 + 12 x %1 + 6 x %1 - 2 x + 8 x %1 11 2 10 3 12 11 10 2 11 + 9 x %1 - x %1 - 8 x - 36 x %1 - 19 x %1 + 22 x 10 9 2 10 9 8 2 9 8 + 62 x %1 + 15 x %1 - 42 x - 82 x %1 - 6 x %1 + 68 x + 96 x %1 7 2 8 7 7 6 6 5 + x %1 - 99 x - 103 x %1 + 138 x + 93 x %1 - 177 x - 63 x %1 5 4 4 3 3 2 2 + 189 x + 29 x %1 - 155 x - 8 x %1 + 92 x + x %1 - 37 x + 9 x - 1) / 20 5 20 4 20 3 19 4 18 5 / ((-1 + x) (x %1 - 5 x %1 + 10 x %1 + 2 x %1 + x %1 / 20 2 19 3 18 4 17 5 20 19 2 - 10 x %1 - 8 x %1 - 6 x %1 - x %1 + 5 x %1 + 12 x %1 18 3 17 4 20 19 18 2 17 3 + 14 x %1 + 6 x %1 - x - 8 x %1 - 16 x %1 - 15 x %1 16 4 19 18 17 2 16 3 15 4 - 4 x %1 + 2 x + 9 x %1 + 19 x %1 + 18 x %1 + 4 x %1 18 17 16 2 15 3 14 4 17 - 2 x - 12 x %1 - 30 x %1 - 20 x %1 - 5 x %1 + 3 x 16 15 2 14 3 13 4 16 15 + 22 x %1 + 36 x %1 + 21 x %1 + 6 x %1 - 6 x - 28 x %1 14 2 13 3 12 4 15 14 13 2 - 34 x %1 - 21 x %1 - 4 x %1 + 8 x + 25 x %1 + 27 x %1 12 3 11 4 14 13 12 2 11 3 + 15 x %1 + x %1 - 7 x - 15 x %1 - 14 x %1 - 6 x %1 13 12 11 2 10 3 12 11 10 2 + 3 x - x %1 - 3 x %1 + x %1 + 4 x + 21 x %1 + 15 x %1 11 10 9 2 10 9 8 2 9 - 13 x - 42 x %1 - 14 x %1 + 26 x + 58 x %1 + 6 x %1 - 44 x 8 7 2 8 7 7 6 6 - 69 x %1 - x %1 + 65 x + 78 x %1 - 92 x - 77 x %1 + 126 x 5 5 4 4 3 3 2 2 + 57 x %1 - 148 x - 28 x %1 + 133 x + 8 x %1 - 85 x - x %1 + 36 x - 9 x + 1)) %1 := X[1, 2, 1, 1, 1, 2, 2] and in Maple format (x^21*X[1,2,1,1,1,2,2]^5-5*x^21*X[1,2,1,1,1,2,2]^4+10*x^21*X[1,2,1,1,1,2,2]^3+2 *x^20*X[1,2,1,1,1,2,2]^4+x^19*X[1,2,1,1,1,2,2]^5-10*x^21*X[1,2,1,1,1,2,2]^2-8*x ^20*X[1,2,1,1,1,2,2]^3-7*x^19*X[1,2,1,1,1,2,2]^4-x^18*X[1,2,1,1,1,2,2]^5+5*x^21 *X[1,2,1,1,1,2,2]+12*x^20*X[1,2,1,1,1,2,2]^2+18*x^19*X[1,2,1,1,1,2,2]^3+7*x^18* X[1,2,1,1,1,2,2]^4+x^17*X[1,2,1,1,1,2,2]^5-x^21-8*x^20*X[1,2,1,1,1,2,2]-22*x^19 *X[1,2,1,1,1,2,2]^2-19*x^18*X[1,2,1,1,1,2,2]^3-8*x^17*X[1,2,1,1,1,2,2]^4+2*x^20 +13*x^19*X[1,2,1,1,1,2,2]+25*x^18*X[1,2,1,1,1,2,2]^2+25*x^17*X[1,2,1,1,1,2,2]^3 +5*x^16*X[1,2,1,1,1,2,2]^4-3*x^19-16*x^18*X[1,2,1,1,1,2,2]-37*x^17*X[1,2,1,1,1, 2,2]^2-25*x^16*X[1,2,1,1,1,2,2]^3-6*x^15*X[1,2,1,1,1,2,2]^4+4*x^18+26*x^17*X[1, 2,1,1,1,2,2]+45*x^16*X[1,2,1,1,1,2,2]^2+28*x^15*X[1,2,1,1,1,2,2]^3+7*x^14*X[1,2 ,1,1,1,2,2]^4-7*x^17-35*x^16*X[1,2,1,1,1,2,2]-49*x^15*X[1,2,1,1,1,2,2]^2-28*x^ 14*X[1,2,1,1,1,2,2]^3-7*x^13*X[1,2,1,1,1,2,2]^4+10*x^16+38*x^15*X[1,2,1,1,1,2,2 ]+44*x^14*X[1,2,1,1,1,2,2]^2+25*x^13*X[1,2,1,1,1,2,2]^3+4*x^12*X[1,2,1,1,1,2,2] ^4-11*x^15-32*x^14*X[1,2,1,1,1,2,2]-31*x^13*X[1,2,1,1,1,2,2]^2-16*x^12*X[1,2,1, 1,1,2,2]^3-x^11*X[1,2,1,1,1,2,2]^4+9*x^14+15*x^13*X[1,2,1,1,1,2,2]+12*x^12*X[1, 2,1,1,1,2,2]^2+6*x^11*X[1,2,1,1,1,2,2]^3-2*x^13+8*x^12*X[1,2,1,1,1,2,2]+9*x^11* X[1,2,1,1,1,2,2]^2-x^10*X[1,2,1,1,1,2,2]^3-8*x^12-36*x^11*X[1,2,1,1,1,2,2]-19*x ^10*X[1,2,1,1,1,2,2]^2+22*x^11+62*x^10*X[1,2,1,1,1,2,2]+15*x^9*X[1,2,1,1,1,2,2] ^2-42*x^10-82*x^9*X[1,2,1,1,1,2,2]-6*x^8*X[1,2,1,1,1,2,2]^2+68*x^9+96*x^8*X[1,2 ,1,1,1,2,2]+x^7*X[1,2,1,1,1,2,2]^2-99*x^8-103*x^7*X[1,2,1,1,1,2,2]+138*x^7+93*x ^6*X[1,2,1,1,1,2,2]-177*x^6-63*x^5*X[1,2,1,1,1,2,2]+189*x^5+29*x^4*X[1,2,1,1,1, 2,2]-155*x^4-8*x^3*X[1,2,1,1,1,2,2]+92*x^3+x^2*X[1,2,1,1,1,2,2]-37*x^2+9*x-1)/( -1+x)/(x^20*X[1,2,1,1,1,2,2]^5-5*x^20*X[1,2,1,1,1,2,2]^4+10*x^20*X[1,2,1,1,1,2, 2]^3+2*x^19*X[1,2,1,1,1,2,2]^4+x^18*X[1,2,1,1,1,2,2]^5-10*x^20*X[1,2,1,1,1,2,2] ^2-8*x^19*X[1,2,1,1,1,2,2]^3-6*x^18*X[1,2,1,1,1,2,2]^4-x^17*X[1,2,1,1,1,2,2]^5+ 5*x^20*X[1,2,1,1,1,2,2]+12*x^19*X[1,2,1,1,1,2,2]^2+14*x^18*X[1,2,1,1,1,2,2]^3+6 *x^17*X[1,2,1,1,1,2,2]^4-x^20-8*x^19*X[1,2,1,1,1,2,2]-16*x^18*X[1,2,1,1,1,2,2]^ 2-15*x^17*X[1,2,1,1,1,2,2]^3-4*x^16*X[1,2,1,1,1,2,2]^4+2*x^19+9*x^18*X[1,2,1,1, 1,2,2]+19*x^17*X[1,2,1,1,1,2,2]^2+18*x^16*X[1,2,1,1,1,2,2]^3+4*x^15*X[1,2,1,1,1 ,2,2]^4-2*x^18-12*x^17*X[1,2,1,1,1,2,2]-30*x^16*X[1,2,1,1,1,2,2]^2-20*x^15*X[1, 2,1,1,1,2,2]^3-5*x^14*X[1,2,1,1,1,2,2]^4+3*x^17+22*x^16*X[1,2,1,1,1,2,2]+36*x^ 15*X[1,2,1,1,1,2,2]^2+21*x^14*X[1,2,1,1,1,2,2]^3+6*x^13*X[1,2,1,1,1,2,2]^4-6*x^ 16-28*x^15*X[1,2,1,1,1,2,2]-34*x^14*X[1,2,1,1,1,2,2]^2-21*x^13*X[1,2,1,1,1,2,2] ^3-4*x^12*X[1,2,1,1,1,2,2]^4+8*x^15+25*x^14*X[1,2,1,1,1,2,2]+27*x^13*X[1,2,1,1, 1,2,2]^2+15*x^12*X[1,2,1,1,1,2,2]^3+x^11*X[1,2,1,1,1,2,2]^4-7*x^14-15*x^13*X[1, 2,1,1,1,2,2]-14*x^12*X[1,2,1,1,1,2,2]^2-6*x^11*X[1,2,1,1,1,2,2]^3+3*x^13-x^12*X [1,2,1,1,1,2,2]-3*x^11*X[1,2,1,1,1,2,2]^2+x^10*X[1,2,1,1,1,2,2]^3+4*x^12+21*x^ 11*X[1,2,1,1,1,2,2]+15*x^10*X[1,2,1,1,1,2,2]^2-13*x^11-42*x^10*X[1,2,1,1,1,2,2] -14*x^9*X[1,2,1,1,1,2,2]^2+26*x^10+58*x^9*X[1,2,1,1,1,2,2]+6*x^8*X[1,2,1,1,1,2, 2]^2-44*x^9-69*x^8*X[1,2,1,1,1,2,2]-x^7*X[1,2,1,1,1,2,2]^2+65*x^8+78*x^7*X[1,2, 1,1,1,2,2]-92*x^7-77*x^6*X[1,2,1,1,1,2,2]+126*x^6+57*x^5*X[1,2,1,1,1,2,2]-148*x ^5-28*x^4*X[1,2,1,1,1,2,2]+133*x^4+8*x^3*X[1,2,1,1,1,2,2]-85*x^3-x^2*X[1,2,1,1, 1,2,2]+36*x^2-9*x+1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [1, 2, 1, 1, 1, 2, 2], equals , - 7/8 + ---- 16 91 21 n The variance equals , - -- + ---- 64 256 3201 279 n The , 3, -th moment about the mean is , - ---- + ----- 1024 2048 10411 1323 2 13593 The , 4, -th moment about the mean is , - ----- + ----- n - ----- n 4096 65536 32768 The compositions of, 10, that yield the, 10, -th largest growth, that is, 1.8923110706522823122, are , [1, 1, 2, 1, 1, 2, 2], [1, 1, 2, 2, 1, 1, 2], [1, 2, 1, 1, 2, 2, 1], [1, 2, 2, 1, 1, 2, 1], [2, 1, 1, 2, 2, 1, 1], [2, 2, 1, 1, 2, 1, 1] Theorem Number, 10, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 1, 2, 1, 1, 2, 2] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 12 11 10 9 8 7 6 ) a(n) x = - (2 x - 2 x + 3 x - 2 x + 3 x - 4 x + 10 x / ----- n = 0 5 4 3 2 / - 19 x + 26 x - 25 x + 16 x - 6 x + 1) / ( / 11 10 9 8 7 6 5 4 3 2 (x - x + x - x + x - x + 4 x - 8 x + 11 x - 10 x + 5 x - 1) 2 (-1 + x) ) and in Maple format -(2*x^12-2*x^11+3*x^10-2*x^9+3*x^8-4*x^7+10*x^6-19*x^5+26*x^4-25*x^3+16*x^2-6*x +1)/(x^11-x^10+x^9-x^8+x^7-x^6+4*x^5-8*x^4+11*x^3-10*x^2+5*x-1)/(-1+x)^2 The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1015, 2001, 3910, 7572, 14545, 27753, 52691, 99706, 188320, 355427, 670835, 1266724, 2393453, 4525234, 8560155, 16198687, 30659825, 58035946, 109856042, 207935235] The limit of a(n+1)/a(n) as n goes to infinity is 1.89231107065 a(n) is asymptotic to 1.01914948966*1.89231107065^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [1, 1, 2, 1, 1, 2, 2], denoted by the variable, X[1, 1, 2, 1, 1, 2, 2], is 13 3 13 2 12 3 13 12 2 11 3 - (2 x %1 - 6 x %1 - 3 x %1 + 6 x %1 + 10 x %1 + 4 x %1 13 12 11 2 10 3 12 11 - 2 x - 11 x %1 - 13 x %1 - 3 x %1 + 4 x + 14 x %1 10 2 9 3 11 10 9 2 10 9 + 11 x %1 + x %1 - 5 x - 13 x %1 - 5 x %1 + 5 x + 9 x %1 8 2 9 8 8 7 7 6 6 + x %1 - 5 x - 8 x %1 + 7 x + 13 x %1 - 14 x - 22 x %1 + 29 x 5 5 4 4 3 3 2 2 + 24 x %1 - 45 x - 16 x %1 + 51 x + 6 x %1 - 41 x - x %1 + 22 x / 2 12 3 12 2 11 3 12 - 7 x + 1) / ((-1 + x) (x %1 - 3 x %1 - x %1 + 3 x %1 / 11 2 10 3 12 11 10 2 9 3 11 + 4 x %1 + 2 x %1 - x - 5 x %1 - 6 x %1 - x %1 + 2 x 10 9 2 10 9 8 2 9 8 8 + 6 x %1 + 4 x %1 - 2 x - 5 x %1 - x %1 + 2 x + 3 x %1 - 2 x 7 7 6 6 5 5 4 4 - 2 x %1 + 2 x + 5 x %1 - 5 x - 10 x %1 + 12 x + 10 x %1 - 19 x 3 3 2 2 - 5 x %1 + 21 x + x %1 - 15 x + 6 x - 1)) %1 := X[1, 1, 2, 1, 1, 2, 2] and in Maple format -(2*x^13*X[1,1,2,1,1,2,2]^3-6*x^13*X[1,1,2,1,1,2,2]^2-3*x^12*X[1,1,2,1,1,2,2]^3 +6*x^13*X[1,1,2,1,1,2,2]+10*x^12*X[1,1,2,1,1,2,2]^2+4*x^11*X[1,1,2,1,1,2,2]^3-2 *x^13-11*x^12*X[1,1,2,1,1,2,2]-13*x^11*X[1,1,2,1,1,2,2]^2-3*x^10*X[1,1,2,1,1,2, 2]^3+4*x^12+14*x^11*X[1,1,2,1,1,2,2]+11*x^10*X[1,1,2,1,1,2,2]^2+x^9*X[1,1,2,1,1 ,2,2]^3-5*x^11-13*x^10*X[1,1,2,1,1,2,2]-5*x^9*X[1,1,2,1,1,2,2]^2+5*x^10+9*x^9*X [1,1,2,1,1,2,2]+x^8*X[1,1,2,1,1,2,2]^2-5*x^9-8*x^8*X[1,1,2,1,1,2,2]+7*x^8+13*x^ 7*X[1,1,2,1,1,2,2]-14*x^7-22*x^6*X[1,1,2,1,1,2,2]+29*x^6+24*x^5*X[1,1,2,1,1,2,2 ]-45*x^5-16*x^4*X[1,1,2,1,1,2,2]+51*x^4+6*x^3*X[1,1,2,1,1,2,2]-41*x^3-x^2*X[1,1 ,2,1,1,2,2]+22*x^2-7*x+1)/(-1+x)^2/(x^12*X[1,1,2,1,1,2,2]^3-3*x^12*X[1,1,2,1,1, 2,2]^2-x^11*X[1,1,2,1,1,2,2]^3+3*x^12*X[1,1,2,1,1,2,2]+4*x^11*X[1,1,2,1,1,2,2]^ 2+2*x^10*X[1,1,2,1,1,2,2]^3-x^12-5*x^11*X[1,1,2,1,1,2,2]-6*x^10*X[1,1,2,1,1,2,2 ]^2-x^9*X[1,1,2,1,1,2,2]^3+2*x^11+6*x^10*X[1,1,2,1,1,2,2]+4*x^9*X[1,1,2,1,1,2,2 ]^2-2*x^10-5*x^9*X[1,1,2,1,1,2,2]-x^8*X[1,1,2,1,1,2,2]^2+2*x^9+3*x^8*X[1,1,2,1, 1,2,2]-2*x^8-2*x^7*X[1,1,2,1,1,2,2]+2*x^7+5*x^6*X[1,1,2,1,1,2,2]-5*x^6-10*x^5*X [1,1,2,1,1,2,2]+12*x^5+10*x^4*X[1,1,2,1,1,2,2]-19*x^4-5*x^3*X[1,1,2,1,1,2,2]+21 *x^3+x^2*X[1,1,2,1,1,2,2]-15*x^2+6*x-1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [1, 1, 2, 1, 1, 2, 2], equals , - 7/8 + ---- 16 87 21 n The variance equals , - -- + ---- 64 256 2913 279 n The , 3, -th moment about the mean is , - ---- + ----- 1024 2048 8283 1323 2 12489 The , 4, -th moment about the mean is , - ---- + ----- n - ----- n 4096 65536 32768 The compositions of, 10, that yield the, 11, -th largest growth, that is, 1.9087907387871591034, are , [2, 1, 1, 2, 1, 1, 2] Theorem Number, 11, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 1, 2, 1, 1, 2] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 39 37 36 35 32 31 30 29 ) a(n) x = - (x + 2 x - x + x - 2 x + 6 x - 10 x + 11 x / ----- n = 0 28 27 26 25 24 23 22 - 4 x - 16 x + 56 x - 121 x + 198 x - 256 x + 262 x 21 20 19 18 17 16 15 - 178 x - 15 x + 255 x - 401 x + 358 x - 260 x + 567 x 14 13 12 11 10 9 - 1956 x + 4989 x - 9657 x + 15057 x - 19527 x + 21338 x 8 7 6 5 4 3 2 - 19647 x + 15104 x - 9543 x + 4850 x - 1927 x + 575 x - 121 x / 5 4 3 2 30 27 26 + 16 x - 1) / ((x - x + 2 x - 3 x + 3 x - 1) (x + 2 x - 3 x / 24 23 22 20 19 18 17 16 + 4 x - 10 x + 11 x - 21 x + 47 x - 52 x + 13 x + 55 x 15 14 13 12 11 10 9 8 - 111 x + 103 x - 3 x - 152 x + 264 x - 213 x - 58 x + 466 x 7 6 5 4 3 2 - 822 x + 949 x - 800 x + 496 x - 220 x + 66 x - 12 x + 1) 5 2 (x - x + 2 x - 1)) and in Maple format -(x^39+2*x^37-x^36+x^35-2*x^32+6*x^31-10*x^30+11*x^29-4*x^28-16*x^27+56*x^26-\ 121*x^25+198*x^24-256*x^23+262*x^22-178*x^21-15*x^20+255*x^19-401*x^18+358*x^17 -260*x^16+567*x^15-1956*x^14+4989*x^13-9657*x^12+15057*x^11-19527*x^10+21338*x^ 9-19647*x^8+15104*x^7-9543*x^6+4850*x^5-1927*x^4+575*x^3-121*x^2+16*x-1)/(x^5-x ^4+2*x^3-3*x^2+3*x-1)/(x^30+2*x^27-3*x^26+4*x^24-10*x^23+11*x^22-21*x^20+47*x^ 19-52*x^18+13*x^17+55*x^16-111*x^15+103*x^14-3*x^13-152*x^12+264*x^11-213*x^10-\ 58*x^9+466*x^8-822*x^7+949*x^6-800*x^5+496*x^4-220*x^3+66*x^2-12*x+1)/(x^5-x^2+ 2*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1015, 2001, 3909, 7564, 14509, 27634, 52373, 98992, 186972, 353412, 669139, 1269583, 2413920, 4598297, 8772449, 16753831, 32018822, 61213409, 117036204, 223738591] The limit of a(n+1)/a(n) as n goes to infinity is 1.90879073879 a(n) is asymptotic to .846805878447*1.90879073879^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [2, 1, 1, 2, 1, 1, 2], denoted by the variable, X[2, 1, 1, 2, 1, 1, 2], is 39 8 39 7 39 6 37 8 39 5 37 7 - (x %1 - 8 x %1 + 28 x %1 + 2 x %1 - 56 x %1 - 16 x %1 36 8 39 4 37 6 36 7 35 8 39 3 - x %1 + 70 x %1 + 56 x %1 + 8 x %1 + 3 x %1 - 56 x %1 37 5 36 6 35 7 34 8 39 2 - 112 x %1 - 28 x %1 - 22 x %1 - 4 x %1 + 28 x %1 37 4 36 5 35 6 34 7 33 8 + 140 x %1 + 56 x %1 + 70 x %1 + 28 x %1 + 5 x %1 39 37 3 36 4 35 5 34 6 - 8 x %1 - 112 x %1 - 70 x %1 - 126 x %1 - 84 x %1 33 7 32 8 39 37 2 36 3 35 4 - 36 x %1 - 6 x %1 + x + 56 x %1 + 56 x %1 + 140 x %1 34 5 33 6 32 7 31 8 37 + 140 x %1 + 111 x %1 + 44 x %1 + 7 x %1 - 16 x %1 36 2 35 3 34 4 33 5 32 6 - 28 x %1 - 98 x %1 - 140 x %1 - 190 x %1 - 140 x %1 31 7 30 8 37 36 35 2 34 3 - 49 x %1 - 7 x %1 + 2 x + 8 x %1 + 42 x %1 + 84 x %1 33 4 32 5 31 6 30 7 29 8 36 + 195 x %1 + 252 x %1 + 152 x %1 + 53 x %1 + 4 x %1 - x 35 34 2 33 3 32 4 31 5 - 10 x %1 - 28 x %1 - 120 x %1 - 280 x %1 - 276 x %1 30 6 29 7 28 8 35 34 33 2 - 173 x %1 - 46 x %1 - x %1 + x + 4 x %1 + 41 x %1 32 3 31 4 30 5 29 6 28 7 + 196 x %1 + 325 x %1 + 325 x %1 + 184 x %1 + 35 x %1 33 32 2 31 3 30 4 29 5 - 6 x %1 - 84 x %1 - 257 x %1 - 395 x %1 - 381 x %1 28 6 27 7 32 31 2 30 3 - 179 x %1 - 29 x %1 + 20 x %1 + 134 x %1 + 327 x %1 29 4 28 5 27 6 26 7 32 31 + 475 x %1 + 409 x %1 + 176 x %1 + 28 x %1 - 2 x - 42 x %1 30 2 29 3 28 4 27 5 26 6 - 183 x %1 - 384 x %1 - 515 x %1 - 433 x %1 - 186 x %1 25 7 31 30 29 2 28 3 27 4 - 25 x %1 + 6 x + 63 x %1 + 206 x %1 + 381 x %1 + 536 x %1 26 5 25 6 24 7 30 29 + 482 x %1 + 194 x %1 + 16 x %1 - 10 x - 69 x %1 28 2 27 3 26 4 25 5 24 6 - 165 x %1 - 319 x %1 - 572 x %1 - 562 x %1 - 181 x %1 23 7 29 28 27 2 26 3 25 4 - 6 x %1 + 11 x + 39 x %1 + 40 x %1 + 208 x %1 + 688 x %1 24 5 23 6 22 7 28 27 26 2 + 643 x %1 + 154 x %1 + x %1 - 4 x + 45 x %1 + 190 x %1 25 3 24 4 23 5 22 6 27 - 136 x %1 - 890 x %1 - 700 x %1 - 128 x %1 - 16 x 26 25 2 24 3 23 4 22 5 - 206 x %1 - 485 x %1 + 184 x %1 + 1131 x %1 + 745 x %1 21 6 26 25 24 2 23 3 + 100 x %1 + 56 x + 447 x %1 + 737 x %1 - 387 x %1 22 4 21 5 20 6 25 24 - 1421 x %1 - 782 x %1 - 64 x %1 - 121 x - 707 x %1 23 2 22 3 21 4 20 5 19 6 - 809 x %1 + 794 x %1 + 1787 x %1 + 768 x %1 + 29 x %1 24 23 22 2 21 3 20 4 + 198 x + 873 x %1 + 591 x %1 - 1472 x %1 - 2142 x %1 19 5 18 6 23 22 21 2 - 664 x %1 - 8 x %1 - 256 x - 844 x %1 + 33 x %1 20 3 19 4 18 5 17 6 22 + 2398 x %1 + 2328 x %1 + 488 x %1 + x %1 + 262 x 21 20 2 19 3 18 4 17 5 + 512 x %1 - 1137 x %1 - 3407 x %1 - 2252 x %1 - 295 x %1 21 20 19 2 18 3 17 4 - 178 x + 192 x %1 + 2597 x %1 + 4303 x %1 + 1914 x %1 16 5 20 19 18 2 17 3 + 139 x %1 - 15 x - 1138 x %1 - 4176 x %1 - 4948 x %1 16 4 15 5 19 18 17 2 - 1385 x %1 - 47 x %1 + 255 x + 2046 x %1 + 5763 x %1 16 3 15 4 14 5 18 17 + 5208 x %1 + 811 x %1 + 10 x %1 - 401 x - 2793 x %1 16 2 15 3 14 4 13 5 17 - 7331 x %1 - 4945 x %1 - 361 x %1 - x %1 + 358 x 16 15 2 14 3 13 4 16 + 3628 x %1 + 8695 x %1 + 4140 x %1 + 113 x %1 - 260 x 15 14 2 13 3 12 4 15 - 5065 x %1 - 9453 x %1 - 2977 x %1 - 22 x %1 + 567 x 14 13 2 12 3 11 4 14 + 7500 x %1 + 9179 x %1 + 1781 x %1 + 2 x %1 - 1956 x 13 12 2 11 3 13 12 - 10743 x %1 - 7724 x %1 - 852 x %1 + 4989 x + 13802 x %1 11 2 10 3 12 11 10 2 + 5447 x %1 + 310 x %1 - 9657 x - 15286 x %1 - 3107 x %1 9 3 11 10 9 2 8 3 - 80 x %1 + 15057 x + 14316 x %1 + 1380 x %1 + 13 x %1 10 9 8 2 7 3 9 8 - 19527 x - 11198 x %1 - 456 x %1 - x %1 + 21338 x + 7220 x %1 7 2 8 7 6 2 7 6 + 105 x %1 - 19647 x - 3768 x %1 - 15 x %1 + 15104 x + 1550 x %1 5 2 6 5 5 4 4 3 + x %1 - 9543 x - 483 x %1 + 4850 x + 107 x %1 - 1927 x - 15 x %1 3 2 2 / + 575 x + x %1 - 121 x + 16 x - 1) / ( / 5 5 4 4 3 3 2 2 (x %1 - x - x %1 + x + 2 x %1 - 2 x - x %1 + 3 x - 3 x + 1) 5 5 2 30 6 30 5 30 4 30 3 (x %1 - x + x - 2 x + 1) (x %1 - 6 x %1 + 15 x %1 - 20 x %1 27 6 30 2 27 5 26 6 30 26 5 - x %1 + 15 x %1 + 3 x %1 + 2 x %1 - 6 x %1 - 7 x %1 25 6 30 27 3 26 4 25 5 24 6 - x %1 + x - 10 x %1 + 5 x %1 + 5 x %1 + x %1 27 2 26 3 25 4 24 5 23 6 27 + 15 x %1 + 10 x %1 - 10 x %1 - 3 x %1 - 3 x %1 - 9 x %1 26 2 25 3 24 4 23 5 22 6 27 - 20 x %1 + 10 x %1 + 6 x %1 + 6 x %1 + 3 x %1 + 2 x 26 25 2 24 3 23 4 22 5 21 6 + 13 x %1 - 5 x %1 - 14 x %1 - 4 x %1 - 7 x %1 - x %1 26 25 24 2 23 3 22 4 21 5 - 3 x + x %1 + 21 x %1 + 16 x %1 + 9 x %1 + 4 x %1 24 23 2 22 3 21 4 20 5 24 - 15 x %1 - 39 x %1 - 26 x %1 - 12 x %1 + x %1 + 4 x 23 22 2 21 3 20 4 19 5 23 + 34 x %1 + 49 x %1 + 22 x %1 + 8 x %1 - 4 x %1 - 10 x 22 21 2 20 3 19 4 22 21 - 39 x %1 - 19 x %1 - 9 x %1 - 8 x %1 + 11 x + 6 x %1 20 2 19 3 18 4 17 5 20 19 2 - 31 x %1 + x %1 + 21 x %1 + 5 x %1 + 52 x %1 + 85 x %1 18 3 17 4 16 5 20 19 18 2 - 7 x %1 - 32 x %1 - 4 x %1 - 21 x - 121 x %1 - 101 x %1 17 3 16 4 15 5 19 18 17 2 + 20 x %1 + 35 x %1 + x %1 + 47 x + 139 x %1 + 49 x %1 16 3 15 4 18 17 16 2 15 3 - 25 x %1 - 38 x %1 - 52 x - 55 x %1 + 16 x %1 + 23 x %1 14 4 17 16 15 2 14 3 13 4 + 35 x %1 + 13 x - 77 x %1 - 49 x %1 - 27 x %1 - 21 x %1 16 15 14 2 13 3 12 4 15 + 55 x + 174 x %1 + 74 x %1 + 36 x %1 + 7 x %1 - 111 x 14 13 2 12 3 11 4 14 13 - 185 x %1 - 116 x %1 - 35 x %1 - x %1 + 103 x + 104 x %1 12 2 11 3 13 12 11 2 10 3 + 176 x %1 + 21 x %1 - 3 x + 5 x %1 - 238 x %1 - 7 x %1 12 11 10 2 9 3 11 10 - 152 x - 58 x %1 + 260 x %1 + x %1 + 264 x + 26 x %1 9 2 10 9 8 2 9 8 - 211 x %1 - 213 x + 48 x %1 + 120 x %1 - 58 x - 91 x %1 7 2 8 7 6 2 7 6 5 2 - 45 x %1 + 466 x + 75 x %1 + 10 x %1 - 822 x - 35 x %1 - x %1 6 5 5 4 4 3 2 + 949 x + 9 x %1 - 800 x - x %1 + 496 x - 220 x + 66 x - 12 x + 1)) %1 := X[2, 1, 1, 2, 1, 1, 2] and in Maple format -(x^39*X[2,1,1,2,1,1,2]^8-8*x^39*X[2,1,1,2,1,1,2]^7+28*x^39*X[2,1,1,2,1,1,2]^6+ 2*x^37*X[2,1,1,2,1,1,2]^8-56*x^39*X[2,1,1,2,1,1,2]^5-16*x^37*X[2,1,1,2,1,1,2]^7 -x^36*X[2,1,1,2,1,1,2]^8+70*x^39*X[2,1,1,2,1,1,2]^4+56*x^37*X[2,1,1,2,1,1,2]^6+ 8*x^36*X[2,1,1,2,1,1,2]^7+3*x^35*X[2,1,1,2,1,1,2]^8-56*x^39*X[2,1,1,2,1,1,2]^3-\ 112*x^37*X[2,1,1,2,1,1,2]^5-28*x^36*X[2,1,1,2,1,1,2]^6-22*x^35*X[2,1,1,2,1,1,2] ^7-4*x^34*X[2,1,1,2,1,1,2]^8+28*x^39*X[2,1,1,2,1,1,2]^2+140*x^37*X[2,1,1,2,1,1, 2]^4+56*x^36*X[2,1,1,2,1,1,2]^5+70*x^35*X[2,1,1,2,1,1,2]^6+28*x^34*X[2,1,1,2,1, 1,2]^7+5*x^33*X[2,1,1,2,1,1,2]^8-8*x^39*X[2,1,1,2,1,1,2]-112*x^37*X[2,1,1,2,1,1 ,2]^3-70*x^36*X[2,1,1,2,1,1,2]^4-126*x^35*X[2,1,1,2,1,1,2]^5-84*x^34*X[2,1,1,2, 1,1,2]^6-36*x^33*X[2,1,1,2,1,1,2]^7-6*x^32*X[2,1,1,2,1,1,2]^8+x^39+56*x^37*X[2, 1,1,2,1,1,2]^2+56*x^36*X[2,1,1,2,1,1,2]^3+140*x^35*X[2,1,1,2,1,1,2]^4+140*x^34* X[2,1,1,2,1,1,2]^5+111*x^33*X[2,1,1,2,1,1,2]^6+44*x^32*X[2,1,1,2,1,1,2]^7+7*x^ 31*X[2,1,1,2,1,1,2]^8-16*x^37*X[2,1,1,2,1,1,2]-28*x^36*X[2,1,1,2,1,1,2]^2-98*x^ 35*X[2,1,1,2,1,1,2]^3-140*x^34*X[2,1,1,2,1,1,2]^4-190*x^33*X[2,1,1,2,1,1,2]^5-\ 140*x^32*X[2,1,1,2,1,1,2]^6-49*x^31*X[2,1,1,2,1,1,2]^7-7*x^30*X[2,1,1,2,1,1,2]^ 8+2*x^37+8*x^36*X[2,1,1,2,1,1,2]+42*x^35*X[2,1,1,2,1,1,2]^2+84*x^34*X[2,1,1,2,1 ,1,2]^3+195*x^33*X[2,1,1,2,1,1,2]^4+252*x^32*X[2,1,1,2,1,1,2]^5+152*x^31*X[2,1, 1,2,1,1,2]^6+53*x^30*X[2,1,1,2,1,1,2]^7+4*x^29*X[2,1,1,2,1,1,2]^8-x^36-10*x^35* X[2,1,1,2,1,1,2]-28*x^34*X[2,1,1,2,1,1,2]^2-120*x^33*X[2,1,1,2,1,1,2]^3-280*x^ 32*X[2,1,1,2,1,1,2]^4-276*x^31*X[2,1,1,2,1,1,2]^5-173*x^30*X[2,1,1,2,1,1,2]^6-\ 46*x^29*X[2,1,1,2,1,1,2]^7-x^28*X[2,1,1,2,1,1,2]^8+x^35+4*x^34*X[2,1,1,2,1,1,2] +41*x^33*X[2,1,1,2,1,1,2]^2+196*x^32*X[2,1,1,2,1,1,2]^3+325*x^31*X[2,1,1,2,1,1, 2]^4+325*x^30*X[2,1,1,2,1,1,2]^5+184*x^29*X[2,1,1,2,1,1,2]^6+35*x^28*X[2,1,1,2, 1,1,2]^7-6*x^33*X[2,1,1,2,1,1,2]-84*x^32*X[2,1,1,2,1,1,2]^2-257*x^31*X[2,1,1,2, 1,1,2]^3-395*x^30*X[2,1,1,2,1,1,2]^4-381*x^29*X[2,1,1,2,1,1,2]^5-179*x^28*X[2,1 ,1,2,1,1,2]^6-29*x^27*X[2,1,1,2,1,1,2]^7+20*x^32*X[2,1,1,2,1,1,2]+134*x^31*X[2, 1,1,2,1,1,2]^2+327*x^30*X[2,1,1,2,1,1,2]^3+475*x^29*X[2,1,1,2,1,1,2]^4+409*x^28 *X[2,1,1,2,1,1,2]^5+176*x^27*X[2,1,1,2,1,1,2]^6+28*x^26*X[2,1,1,2,1,1,2]^7-2*x^ 32-42*x^31*X[2,1,1,2,1,1,2]-183*x^30*X[2,1,1,2,1,1,2]^2-384*x^29*X[2,1,1,2,1,1, 2]^3-515*x^28*X[2,1,1,2,1,1,2]^4-433*x^27*X[2,1,1,2,1,1,2]^5-186*x^26*X[2,1,1,2 ,1,1,2]^6-25*x^25*X[2,1,1,2,1,1,2]^7+6*x^31+63*x^30*X[2,1,1,2,1,1,2]+206*x^29*X [2,1,1,2,1,1,2]^2+381*x^28*X[2,1,1,2,1,1,2]^3+536*x^27*X[2,1,1,2,1,1,2]^4+482*x ^26*X[2,1,1,2,1,1,2]^5+194*x^25*X[2,1,1,2,1,1,2]^6+16*x^24*X[2,1,1,2,1,1,2]^7-\ 10*x^30-69*x^29*X[2,1,1,2,1,1,2]-165*x^28*X[2,1,1,2,1,1,2]^2-319*x^27*X[2,1,1,2 ,1,1,2]^3-572*x^26*X[2,1,1,2,1,1,2]^4-562*x^25*X[2,1,1,2,1,1,2]^5-181*x^24*X[2, 1,1,2,1,1,2]^6-6*x^23*X[2,1,1,2,1,1,2]^7+11*x^29+39*x^28*X[2,1,1,2,1,1,2]+40*x^ 27*X[2,1,1,2,1,1,2]^2+208*x^26*X[2,1,1,2,1,1,2]^3+688*x^25*X[2,1,1,2,1,1,2]^4+ 643*x^24*X[2,1,1,2,1,1,2]^5+154*x^23*X[2,1,1,2,1,1,2]^6+x^22*X[2,1,1,2,1,1,2]^7 -4*x^28+45*x^27*X[2,1,1,2,1,1,2]+190*x^26*X[2,1,1,2,1,1,2]^2-136*x^25*X[2,1,1,2 ,1,1,2]^3-890*x^24*X[2,1,1,2,1,1,2]^4-700*x^23*X[2,1,1,2,1,1,2]^5-128*x^22*X[2, 1,1,2,1,1,2]^6-16*x^27-206*x^26*X[2,1,1,2,1,1,2]-485*x^25*X[2,1,1,2,1,1,2]^2+ 184*x^24*X[2,1,1,2,1,1,2]^3+1131*x^23*X[2,1,1,2,1,1,2]^4+745*x^22*X[2,1,1,2,1,1 ,2]^5+100*x^21*X[2,1,1,2,1,1,2]^6+56*x^26+447*x^25*X[2,1,1,2,1,1,2]+737*x^24*X[ 2,1,1,2,1,1,2]^2-387*x^23*X[2,1,1,2,1,1,2]^3-1421*x^22*X[2,1,1,2,1,1,2]^4-782*x ^21*X[2,1,1,2,1,1,2]^5-64*x^20*X[2,1,1,2,1,1,2]^6-121*x^25-707*x^24*X[2,1,1,2,1 ,1,2]-809*x^23*X[2,1,1,2,1,1,2]^2+794*x^22*X[2,1,1,2,1,1,2]^3+1787*x^21*X[2,1,1 ,2,1,1,2]^4+768*x^20*X[2,1,1,2,1,1,2]^5+29*x^19*X[2,1,1,2,1,1,2]^6+198*x^24+873 *x^23*X[2,1,1,2,1,1,2]+591*x^22*X[2,1,1,2,1,1,2]^2-1472*x^21*X[2,1,1,2,1,1,2]^3 -2142*x^20*X[2,1,1,2,1,1,2]^4-664*x^19*X[2,1,1,2,1,1,2]^5-8*x^18*X[2,1,1,2,1,1, 2]^6-256*x^23-844*x^22*X[2,1,1,2,1,1,2]+33*x^21*X[2,1,1,2,1,1,2]^2+2398*x^20*X[ 2,1,1,2,1,1,2]^3+2328*x^19*X[2,1,1,2,1,1,2]^4+488*x^18*X[2,1,1,2,1,1,2]^5+x^17* X[2,1,1,2,1,1,2]^6+262*x^22+512*x^21*X[2,1,1,2,1,1,2]-1137*x^20*X[2,1,1,2,1,1,2 ]^2-3407*x^19*X[2,1,1,2,1,1,2]^3-2252*x^18*X[2,1,1,2,1,1,2]^4-295*x^17*X[2,1,1, 2,1,1,2]^5-178*x^21+192*x^20*X[2,1,1,2,1,1,2]+2597*x^19*X[2,1,1,2,1,1,2]^2+4303 *x^18*X[2,1,1,2,1,1,2]^3+1914*x^17*X[2,1,1,2,1,1,2]^4+139*x^16*X[2,1,1,2,1,1,2] ^5-15*x^20-1138*x^19*X[2,1,1,2,1,1,2]-4176*x^18*X[2,1,1,2,1,1,2]^2-4948*x^17*X[ 2,1,1,2,1,1,2]^3-1385*x^16*X[2,1,1,2,1,1,2]^4-47*x^15*X[2,1,1,2,1,1,2]^5+255*x^ 19+2046*x^18*X[2,1,1,2,1,1,2]+5763*x^17*X[2,1,1,2,1,1,2]^2+5208*x^16*X[2,1,1,2, 1,1,2]^3+811*x^15*X[2,1,1,2,1,1,2]^4+10*x^14*X[2,1,1,2,1,1,2]^5-401*x^18-2793*x ^17*X[2,1,1,2,1,1,2]-7331*x^16*X[2,1,1,2,1,1,2]^2-4945*x^15*X[2,1,1,2,1,1,2]^3-\ 361*x^14*X[2,1,1,2,1,1,2]^4-x^13*X[2,1,1,2,1,1,2]^5+358*x^17+3628*x^16*X[2,1,1, 2,1,1,2]+8695*x^15*X[2,1,1,2,1,1,2]^2+4140*x^14*X[2,1,1,2,1,1,2]^3+113*x^13*X[2 ,1,1,2,1,1,2]^4-260*x^16-5065*x^15*X[2,1,1,2,1,1,2]-9453*x^14*X[2,1,1,2,1,1,2]^ 2-2977*x^13*X[2,1,1,2,1,1,2]^3-22*x^12*X[2,1,1,2,1,1,2]^4+567*x^15+7500*x^14*X[ 2,1,1,2,1,1,2]+9179*x^13*X[2,1,1,2,1,1,2]^2+1781*x^12*X[2,1,1,2,1,1,2]^3+2*x^11 *X[2,1,1,2,1,1,2]^4-1956*x^14-10743*x^13*X[2,1,1,2,1,1,2]-7724*x^12*X[2,1,1,2,1 ,1,2]^2-852*x^11*X[2,1,1,2,1,1,2]^3+4989*x^13+13802*x^12*X[2,1,1,2,1,1,2]+5447* x^11*X[2,1,1,2,1,1,2]^2+310*x^10*X[2,1,1,2,1,1,2]^3-9657*x^12-15286*x^11*X[2,1, 1,2,1,1,2]-3107*x^10*X[2,1,1,2,1,1,2]^2-80*x^9*X[2,1,1,2,1,1,2]^3+15057*x^11+ 14316*x^10*X[2,1,1,2,1,1,2]+1380*x^9*X[2,1,1,2,1,1,2]^2+13*x^8*X[2,1,1,2,1,1,2] ^3-19527*x^10-11198*x^9*X[2,1,1,2,1,1,2]-456*x^8*X[2,1,1,2,1,1,2]^2-x^7*X[2,1,1 ,2,1,1,2]^3+21338*x^9+7220*x^8*X[2,1,1,2,1,1,2]+105*x^7*X[2,1,1,2,1,1,2]^2-\ 19647*x^8-3768*x^7*X[2,1,1,2,1,1,2]-15*x^6*X[2,1,1,2,1,1,2]^2+15104*x^7+1550*x^ 6*X[2,1,1,2,1,1,2]+x^5*X[2,1,1,2,1,1,2]^2-9543*x^6-483*x^5*X[2,1,1,2,1,1,2]+ 4850*x^5+107*x^4*X[2,1,1,2,1,1,2]-1927*x^4-15*x^3*X[2,1,1,2,1,1,2]+575*x^3+x^2* X[2,1,1,2,1,1,2]-121*x^2+16*x-1)/(x^5*X[2,1,1,2,1,1,2]-x^5-x^4*X[2,1,1,2,1,1,2] +x^4+2*x^3*X[2,1,1,2,1,1,2]-2*x^3-x^2*X[2,1,1,2,1,1,2]+3*x^2-3*x+1)/(x^5*X[2,1, 1,2,1,1,2]-x^5+x^2-2*x+1)/(x^30*X[2,1,1,2,1,1,2]^6-6*x^30*X[2,1,1,2,1,1,2]^5+15 *x^30*X[2,1,1,2,1,1,2]^4-20*x^30*X[2,1,1,2,1,1,2]^3-x^27*X[2,1,1,2,1,1,2]^6+15* x^30*X[2,1,1,2,1,1,2]^2+3*x^27*X[2,1,1,2,1,1,2]^5+2*x^26*X[2,1,1,2,1,1,2]^6-6*x ^30*X[2,1,1,2,1,1,2]-7*x^26*X[2,1,1,2,1,1,2]^5-x^25*X[2,1,1,2,1,1,2]^6+x^30-10* x^27*X[2,1,1,2,1,1,2]^3+5*x^26*X[2,1,1,2,1,1,2]^4+5*x^25*X[2,1,1,2,1,1,2]^5+x^ 24*X[2,1,1,2,1,1,2]^6+15*x^27*X[2,1,1,2,1,1,2]^2+10*x^26*X[2,1,1,2,1,1,2]^3-10* x^25*X[2,1,1,2,1,1,2]^4-3*x^24*X[2,1,1,2,1,1,2]^5-3*x^23*X[2,1,1,2,1,1,2]^6-9*x ^27*X[2,1,1,2,1,1,2]-20*x^26*X[2,1,1,2,1,1,2]^2+10*x^25*X[2,1,1,2,1,1,2]^3+6*x^ 24*X[2,1,1,2,1,1,2]^4+6*x^23*X[2,1,1,2,1,1,2]^5+3*x^22*X[2,1,1,2,1,1,2]^6+2*x^ 27+13*x^26*X[2,1,1,2,1,1,2]-5*x^25*X[2,1,1,2,1,1,2]^2-14*x^24*X[2,1,1,2,1,1,2]^ 3-4*x^23*X[2,1,1,2,1,1,2]^4-7*x^22*X[2,1,1,2,1,1,2]^5-x^21*X[2,1,1,2,1,1,2]^6-3 *x^26+x^25*X[2,1,1,2,1,1,2]+21*x^24*X[2,1,1,2,1,1,2]^2+16*x^23*X[2,1,1,2,1,1,2] ^3+9*x^22*X[2,1,1,2,1,1,2]^4+4*x^21*X[2,1,1,2,1,1,2]^5-15*x^24*X[2,1,1,2,1,1,2] -39*x^23*X[2,1,1,2,1,1,2]^2-26*x^22*X[2,1,1,2,1,1,2]^3-12*x^21*X[2,1,1,2,1,1,2] ^4+x^20*X[2,1,1,2,1,1,2]^5+4*x^24+34*x^23*X[2,1,1,2,1,1,2]+49*x^22*X[2,1,1,2,1, 1,2]^2+22*x^21*X[2,1,1,2,1,1,2]^3+8*x^20*X[2,1,1,2,1,1,2]^4-4*x^19*X[2,1,1,2,1, 1,2]^5-10*x^23-39*x^22*X[2,1,1,2,1,1,2]-19*x^21*X[2,1,1,2,1,1,2]^2-9*x^20*X[2,1 ,1,2,1,1,2]^3-8*x^19*X[2,1,1,2,1,1,2]^4+11*x^22+6*x^21*X[2,1,1,2,1,1,2]-31*x^20 *X[2,1,1,2,1,1,2]^2+x^19*X[2,1,1,2,1,1,2]^3+21*x^18*X[2,1,1,2,1,1,2]^4+5*x^17*X [2,1,1,2,1,1,2]^5+52*x^20*X[2,1,1,2,1,1,2]+85*x^19*X[2,1,1,2,1,1,2]^2-7*x^18*X[ 2,1,1,2,1,1,2]^3-32*x^17*X[2,1,1,2,1,1,2]^4-4*x^16*X[2,1,1,2,1,1,2]^5-21*x^20-\ 121*x^19*X[2,1,1,2,1,1,2]-101*x^18*X[2,1,1,2,1,1,2]^2+20*x^17*X[2,1,1,2,1,1,2]^ 3+35*x^16*X[2,1,1,2,1,1,2]^4+x^15*X[2,1,1,2,1,1,2]^5+47*x^19+139*x^18*X[2,1,1,2 ,1,1,2]+49*x^17*X[2,1,1,2,1,1,2]^2-25*x^16*X[2,1,1,2,1,1,2]^3-38*x^15*X[2,1,1,2 ,1,1,2]^4-52*x^18-55*x^17*X[2,1,1,2,1,1,2]+16*x^16*X[2,1,1,2,1,1,2]^2+23*x^15*X [2,1,1,2,1,1,2]^3+35*x^14*X[2,1,1,2,1,1,2]^4+13*x^17-77*x^16*X[2,1,1,2,1,1,2]-\ 49*x^15*X[2,1,1,2,1,1,2]^2-27*x^14*X[2,1,1,2,1,1,2]^3-21*x^13*X[2,1,1,2,1,1,2]^ 4+55*x^16+174*x^15*X[2,1,1,2,1,1,2]+74*x^14*X[2,1,1,2,1,1,2]^2+36*x^13*X[2,1,1, 2,1,1,2]^3+7*x^12*X[2,1,1,2,1,1,2]^4-111*x^15-185*x^14*X[2,1,1,2,1,1,2]-116*x^ 13*X[2,1,1,2,1,1,2]^2-35*x^12*X[2,1,1,2,1,1,2]^3-x^11*X[2,1,1,2,1,1,2]^4+103*x^ 14+104*x^13*X[2,1,1,2,1,1,2]+176*x^12*X[2,1,1,2,1,1,2]^2+21*x^11*X[2,1,1,2,1,1, 2]^3-3*x^13+5*x^12*X[2,1,1,2,1,1,2]-238*x^11*X[2,1,1,2,1,1,2]^2-7*x^10*X[2,1,1, 2,1,1,2]^3-152*x^12-58*x^11*X[2,1,1,2,1,1,2]+260*x^10*X[2,1,1,2,1,1,2]^2+x^9*X[ 2,1,1,2,1,1,2]^3+264*x^11+26*x^10*X[2,1,1,2,1,1,2]-211*x^9*X[2,1,1,2,1,1,2]^2-\ 213*x^10+48*x^9*X[2,1,1,2,1,1,2]+120*x^8*X[2,1,1,2,1,1,2]^2-58*x^9-91*x^8*X[2,1 ,1,2,1,1,2]-45*x^7*X[2,1,1,2,1,1,2]^2+466*x^8+75*x^7*X[2,1,1,2,1,1,2]+10*x^6*X[ 2,1,1,2,1,1,2]^2-822*x^7-35*x^6*X[2,1,1,2,1,1,2]-x^5*X[2,1,1,2,1,1,2]^2+949*x^6 +9*x^5*X[2,1,1,2,1,1,2]-800*x^5-x^4*X[2,1,1,2,1,1,2]+496*x^4-220*x^3+66*x^2-12* x+1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [2, 1, 1, 2, 1, 1, 2], equals , - 7/8 + ---- 16 57 25 n The variance equals , - -- + ---- 32 256 4257 351 n The , 3, -th moment about the mean is , - ---- + ----- 1024 2048 561 1875 2 24781 The , 4, -th moment about the mean is , - ---- + ----- n - ----- n 2048 65536 32768 The compositions of, 10, that yield the, 12, -th largest growth, that is, 1.9275619754829253043, are , [1, 1, 1, 1, 1, 5], [1, 1, 1, 1, 5, 1], [1, 1, 1, 5, 1, 1], [1, 1, 5, 1, 1, 1], [1, 5, 1, 1, 1, 1], [5, 1, 1, 1, 1, 1] Theorem Number, 12, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 1, 1, 1, 1, 5] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 4 3 2 5 4 3 2 \ n (x - x + 2 x - 2 x + 1) (x - x + x + 2 x - 3 x + 1) ) a(n) x = ---------------------------------------------------------- / 4 3 2 5 ----- (x + x + x + x - 1) (-1 + x) n = 0 and in Maple format (x^4-x^3+2*x^2-2*x+1)*(x^5-x^4+x^3+2*x^2-3*x+1)/(x^4+x^3+x^2+x-1)/(-1+x)^5 The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1016, 2010, 3956, 7749, 15117, 29394, 57010, 110363, 213355, 412063, 795308, 1534303, 2959077, 5705787, 11000671, 21207385, 40882029, 78806775, 151909811, 292821276] The limit of a(n+1)/a(n) as n goes to infinity is 1.92756197548 a(n) is asymptotic to .824830604574*1.92756197548^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [1, 1, 1, 1, 1, 5], denoted by the variable, X[1, 1, 1, 1, 1, 5], is 9 9 8 8 7 7 6 6 5 4 (x %1 - x - 2 x %1 + 2 x + 4 x %1 - 4 x - 3 x %1 + 3 x + x %1 - 5 x 3 2 / 4 5 5 + 10 x - 10 x + 5 x - 1) / ((-1 + x) (x %1 - x + 2 x - 1)) / %1 := X[1, 1, 1, 1, 1, 5] and in Maple format (x^9*X[1,1,1,1,1,5]-x^9-2*x^8*X[1,1,1,1,1,5]+2*x^8+4*x^7*X[1,1,1,1,1,5]-4*x^7-3 *x^6*X[1,1,1,1,1,5]+3*x^6+x^5*X[1,1,1,1,1,5]-5*x^4+10*x^3-10*x^2+5*x-1)/(-1+x)^ 4/(x^5*X[1,1,1,1,1,5]-x^5+2*x-1) Furthermore, the expectation of the random variable number of occurences (by\ 13 n containment) of , [1, 1, 1, 1, 1, 5], equals , - -- + ---- 32 32 271 23 n The variance equals , - ---- + ---- 1024 1024 1359 171 n The , 3, -th moment about the mean is , - ----- + ----- 16384 16384 311417 1587 2 10003 The , 4, -th moment about the mean is , ------- + ------- n - ------ n 1048576 1048576 262144 The compositions of, 10, that yield the, 13, -th largest growth, that is, 1.9331849818995204468, are , [2, 1, 1, 1, 1, 4], [4, 1, 1, 1, 1, 2] Theorem Number, 13, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 1, 1, 1, 4] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 54 50 49 48 47 46 45 43 42 ) a(n) x = - (x + 2 x + x + x - x + x + 2 x - 3 x - x / ----- n = 0 41 39 37 36 35 34 33 32 31 + 4 x - 2 x - 3 x - 2 x + 7 x + 4 x - 8 x - 4 x + 9 x 30 28 27 26 25 23 22 21 20 - x - x - 8 x + 12 x - 7 x + 8 x - 6 x + 3 x - 26 x 19 18 17 16 15 14 13 12 + 52 x - 47 x + 31 x - 23 x + 8 x + 19 x - 58 x + 129 x 11 10 9 8 7 6 5 4 - 233 x + 327 x - 376 x + 405 x - 458 x + 508 x - 472 x + 331 x 3 2 / 5 4 25 16 15 - 165 x + 55 x - 11 x + 1) / ((x - x + 2 x - 1) (x - x + 2 x / 14 13 12 11 5 4 3 2 25 - x + x - 2 x + x + x - 5 x + 10 x - 10 x + 5 x - 1) (x 18 17 16 15 13 12 9 8 5 4 3 - x + x + 3 x - 3 x - 3 x + 3 x + x - x + x - 5 x + 10 x 2 - 10 x + 5 x - 1)) and in Maple format -(x^54+2*x^50+x^49+x^48-x^47+x^46+2*x^45-3*x^43-x^42+4*x^41-2*x^39-3*x^37-2*x^ 36+7*x^35+4*x^34-8*x^33-4*x^32+9*x^31-x^30-x^28-8*x^27+12*x^26-7*x^25+8*x^23-6* x^22+3*x^21-26*x^20+52*x^19-47*x^18+31*x^17-23*x^16+8*x^15+19*x^14-58*x^13+129* x^12-233*x^11+327*x^10-376*x^9+405*x^8-458*x^7+508*x^6-472*x^5+331*x^4-165*x^3+ 55*x^2-11*x+1)/(x^5-x^4+2*x-1)/(x^25-x^16+2*x^15-x^14+x^13-2*x^12+x^11+x^5-5*x^ 4+10*x^3-10*x^2+5*x-1)/(x^25-x^18+x^17+3*x^16-3*x^15-3*x^13+3*x^12+x^9-x^8+x^5-\ 5*x^4+10*x^3-10*x^2+5*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1016, 2010, 3956, 7749, 15117, 29394, 57010, 110364, 213367, 412143, 795700, 1535877, 2964563, 5722987, 11050367, 21342026, 41228546, 79662370, 153952283, 297563937] The limit of a(n+1)/a(n) as n goes to infinity is 1.93318498190 a(n) is asymptotic to .768338440425*1.93318498190^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [2, 1, 1, 1, 1, 4], denoted by the variable, X[2, 1, 1, 1, 1, 4], is 54 11 54 10 54 9 54 8 54 7 - (x %1 - 11 x %1 + 55 x %1 - 165 x %1 + 330 x %1 54 6 50 10 54 5 50 9 49 10 - 462 x %1 - 2 x %1 + 462 x %1 + 20 x %1 - x %1 54 4 50 8 49 9 48 10 54 3 - 330 x %1 - 90 x %1 + 10 x %1 - x %1 + 165 x %1 50 7 49 8 48 9 47 10 54 2 + 240 x %1 - 45 x %1 + 10 x %1 + x %1 - 55 x %1 50 6 49 7 48 8 47 9 46 10 - 420 x %1 + 120 x %1 - 45 x %1 - 10 x %1 - x %1 54 50 5 49 6 48 7 47 8 + 11 x %1 + 504 x %1 - 210 x %1 + 120 x %1 + 45 x %1 46 9 54 50 4 49 5 48 6 47 7 + 10 x %1 - x - 420 x %1 + 252 x %1 - 210 x %1 - 120 x %1 46 8 45 9 50 3 49 4 48 5 - 45 x %1 + 2 x %1 + 240 x %1 - 210 x %1 + 252 x %1 47 6 46 7 45 8 50 2 49 3 + 210 x %1 + 120 x %1 - 18 x %1 - 90 x %1 + 120 x %1 48 4 47 5 46 6 45 7 43 9 - 210 x %1 - 252 x %1 - 210 x %1 + 72 x %1 - 3 x %1 50 49 2 48 3 47 4 46 5 + 20 x %1 - 45 x %1 + 120 x %1 + 210 x %1 + 252 x %1 45 6 43 8 42 9 50 49 48 2 - 168 x %1 + 27 x %1 + x %1 - 2 x + 10 x %1 - 45 x %1 47 3 46 4 45 5 43 7 42 8 49 - 120 x %1 - 210 x %1 + 252 x %1 - 108 x %1 - 7 x %1 - x 48 47 2 46 3 45 4 43 6 + 10 x %1 + 45 x %1 + 120 x %1 - 252 x %1 + 252 x %1 42 7 41 8 48 47 46 2 45 3 + 20 x %1 - 4 x %1 - x - 10 x %1 - 45 x %1 + 168 x %1 43 5 42 6 41 7 47 46 45 2 - 378 x %1 - 28 x %1 + 32 x %1 + x + 10 x %1 - 72 x %1 43 4 42 5 41 6 39 8 46 45 + 378 x %1 + 14 x %1 - 112 x %1 + 2 x %1 - x + 18 x %1 43 3 42 4 41 5 39 7 38 8 45 - 252 x %1 + 14 x %1 + 224 x %1 - 16 x %1 - 2 x %1 - 2 x 43 2 42 3 41 4 39 6 38 7 + 108 x %1 - 28 x %1 - 280 x %1 + 56 x %1 + 14 x %1 37 8 43 42 2 41 3 39 5 + 4 x %1 - 27 x %1 + 20 x %1 + 224 x %1 - 112 x %1 38 6 37 7 36 8 43 42 41 2 - 42 x %1 - 31 x %1 - 3 x %1 + 3 x - 7 x %1 - 112 x %1 39 4 38 5 37 6 36 7 35 8 42 + 140 x %1 + 70 x %1 + 105 x %1 + 19 x %1 + x %1 + x 41 39 3 38 4 37 5 36 6 + 32 x %1 - 112 x %1 - 70 x %1 - 203 x %1 - 49 x %1 35 7 41 39 2 38 3 37 4 36 5 + x %1 - 4 x + 56 x %1 + 42 x %1 + 245 x %1 + 63 x %1 35 6 34 7 39 38 2 37 3 36 4 - 34 x %1 - x %1 - 16 x %1 - 14 x %1 - 189 x %1 - 35 x %1 35 5 34 6 33 7 39 38 37 2 + 127 x %1 + 2 x %1 - 6 x %1 + 2 x + 2 x %1 + 91 x %1 36 3 35 4 34 5 33 6 32 7 37 - 7 x %1 - 230 x %1 + 9 x %1 + 44 x %1 + 5 x %1 - 25 x %1 36 2 35 3 34 4 33 5 32 6 + 21 x %1 + 239 x %1 - 40 x %1 - 138 x %1 - 26 x %1 31 7 37 36 35 2 34 3 33 4 - x %1 + 3 x - 11 x %1 - 146 x %1 + 65 x %1 + 240 x %1 32 5 31 6 36 35 34 2 33 3 + 51 x %1 - 4 x %1 + 2 x + 49 x %1 - 54 x %1 - 250 x %1 32 4 31 5 30 6 35 34 33 2 - 40 x %1 + 44 x %1 + 2 x %1 - 7 x + 23 x %1 + 156 x %1 32 3 31 4 30 5 29 6 34 33 - 5 x %1 - 125 x %1 - 11 x %1 + 3 x %1 - 4 x - 54 x %1 32 2 31 3 30 4 29 5 28 6 33 + 30 x %1 + 175 x %1 + 25 x %1 - 15 x %1 - x %1 + 8 x 32 31 2 30 3 29 4 28 5 32 - 19 x %1 - 134 x %1 - 30 x %1 + 30 x %1 + 4 x %1 + 4 x 31 30 2 29 3 28 4 27 5 31 + 54 x %1 + 20 x %1 - 30 x %1 - 5 x %1 - 9 x %1 - 9 x 30 29 2 27 4 26 5 30 29 - 7 x %1 + 15 x %1 + 44 x %1 + 19 x %1 + x - 3 x %1 28 2 27 3 26 4 25 5 28 27 2 + 5 x %1 - 86 x %1 - 88 x %1 - 24 x %1 - 4 x %1 + 84 x %1 26 3 25 4 24 5 28 27 26 2 + 162 x %1 + 103 x %1 + 24 x %1 + x - 41 x %1 - 148 x %1 25 3 24 4 23 5 27 26 25 2 - 172 x %1 - 96 x %1 - 16 x %1 + 8 x + 67 x %1 + 138 x %1 24 3 23 4 22 5 26 25 24 2 + 144 x %1 + 57 x %1 + 6 x %1 - 12 x - 52 x %1 - 96 x %1 23 3 22 4 21 5 25 24 23 2 - 67 x %1 - 26 x %1 - x %1 + 7 x + 24 x %1 + 19 x %1 22 3 21 4 23 22 2 21 3 + 36 x %1 + 32 x %1 + 15 x %1 - 12 x %1 - 87 x %1 20 4 23 22 21 2 20 3 19 4 - 39 x %1 - 8 x - 10 x %1 + 79 x %1 + 91 x %1 + 25 x %1 22 21 20 2 19 3 18 4 21 + 6 x - 20 x %1 - 39 x %1 - 25 x %1 - 8 x %1 - 3 x 20 19 2 18 3 17 4 20 19 - 39 x %1 - 77 x %1 - 11 x %1 + x %1 + 26 x + 129 x %1 18 2 17 3 19 18 17 2 16 3 + 93 x %1 - x %1 - 52 x - 121 x %1 - 32 x %1 + 11 x %1 18 17 16 2 15 3 17 16 + 47 x + 63 x %1 + x %1 - 6 x %1 - 31 x - 35 x %1 15 2 14 3 16 15 14 2 15 + 3 x %1 + x %1 + 23 x + 11 x %1 - 11 x %1 - 8 x 14 13 2 14 13 12 2 13 + 29 x %1 + 13 x %1 - 19 x - 71 x %1 - 6 x %1 + 58 x 12 11 2 12 11 11 10 + 135 x %1 + x %1 - 129 x - 233 x %1 + 233 x + 316 x %1 10 9 9 8 8 7 7 - 327 x - 321 x %1 + 376 x + 240 x %1 - 405 x - 128 x %1 + 458 x 6 6 5 5 4 4 3 2 + 46 x %1 - 508 x - 10 x %1 + 472 x + x %1 - 331 x + 165 x - 55 x / 5 5 4 4 25 5 25 4 + 11 x - 1) / ((x %1 - x - x %1 + x - 2 x + 1) (x %1 - 5 x %1 / 25 3 25 2 25 25 16 3 16 2 + 10 x %1 - 10 x %1 + 5 x %1 - x - x %1 + 3 x %1 15 3 16 15 2 14 3 16 15 14 2 + 2 x %1 - 3 x %1 - 6 x %1 - x %1 + x + 6 x %1 + 3 x %1 15 14 13 2 14 13 12 2 13 12 - 2 x - 3 x %1 - x %1 + x + 2 x %1 + 2 x %1 - x - 4 x %1 11 2 12 11 11 5 4 3 2 - x %1 + 2 x + 2 x %1 - x - x + 5 x - 10 x + 10 x - 5 x + 1) 25 5 25 4 25 3 25 2 25 25 18 4 (x %1 - 5 x %1 + 10 x %1 - 10 x %1 + 5 x %1 - x + x %1 18 3 17 4 18 2 17 3 18 17 2 - 4 x %1 - x %1 + 6 x %1 + 4 x %1 - 4 x %1 - 6 x %1 16 3 18 17 16 2 15 3 17 16 + 3 x %1 + x + 4 x %1 - 9 x %1 - 3 x %1 - x + 9 x %1 15 2 16 15 15 13 2 13 12 2 + 9 x %1 - 3 x - 9 x %1 + 3 x + 3 x %1 - 6 x %1 - 3 x %1 13 12 12 9 9 8 8 5 4 3 + 3 x + 6 x %1 - 3 x + x %1 - x - x %1 + x - x + 5 x - 10 x 2 + 10 x - 5 x + 1)) %1 := X[2, 1, 1, 1, 1, 4] and in Maple format -(x^54*X[2,1,1,1,1,4]^11-11*x^54*X[2,1,1,1,1,4]^10+55*x^54*X[2,1,1,1,1,4]^9-165 *x^54*X[2,1,1,1,1,4]^8+330*x^54*X[2,1,1,1,1,4]^7-462*x^54*X[2,1,1,1,1,4]^6-2*x^ 50*X[2,1,1,1,1,4]^10+462*x^54*X[2,1,1,1,1,4]^5+20*x^50*X[2,1,1,1,1,4]^9-x^49*X[ 2,1,1,1,1,4]^10-330*x^54*X[2,1,1,1,1,4]^4-90*x^50*X[2,1,1,1,1,4]^8+10*x^49*X[2, 1,1,1,1,4]^9-x^48*X[2,1,1,1,1,4]^10+165*x^54*X[2,1,1,1,1,4]^3+240*x^50*X[2,1,1, 1,1,4]^7-45*x^49*X[2,1,1,1,1,4]^8+10*x^48*X[2,1,1,1,1,4]^9+x^47*X[2,1,1,1,1,4]^ 10-55*x^54*X[2,1,1,1,1,4]^2-420*x^50*X[2,1,1,1,1,4]^6+120*x^49*X[2,1,1,1,1,4]^7 -45*x^48*X[2,1,1,1,1,4]^8-10*x^47*X[2,1,1,1,1,4]^9-x^46*X[2,1,1,1,1,4]^10+11*x^ 54*X[2,1,1,1,1,4]+504*x^50*X[2,1,1,1,1,4]^5-210*x^49*X[2,1,1,1,1,4]^6+120*x^48* X[2,1,1,1,1,4]^7+45*x^47*X[2,1,1,1,1,4]^8+10*x^46*X[2,1,1,1,1,4]^9-x^54-420*x^ 50*X[2,1,1,1,1,4]^4+252*x^49*X[2,1,1,1,1,4]^5-210*x^48*X[2,1,1,1,1,4]^6-120*x^ 47*X[2,1,1,1,1,4]^7-45*x^46*X[2,1,1,1,1,4]^8+2*x^45*X[2,1,1,1,1,4]^9+240*x^50*X [2,1,1,1,1,4]^3-210*x^49*X[2,1,1,1,1,4]^4+252*x^48*X[2,1,1,1,1,4]^5+210*x^47*X[ 2,1,1,1,1,4]^6+120*x^46*X[2,1,1,1,1,4]^7-18*x^45*X[2,1,1,1,1,4]^8-90*x^50*X[2,1 ,1,1,1,4]^2+120*x^49*X[2,1,1,1,1,4]^3-210*x^48*X[2,1,1,1,1,4]^4-252*x^47*X[2,1, 1,1,1,4]^5-210*x^46*X[2,1,1,1,1,4]^6+72*x^45*X[2,1,1,1,1,4]^7-3*x^43*X[2,1,1,1, 1,4]^9+20*x^50*X[2,1,1,1,1,4]-45*x^49*X[2,1,1,1,1,4]^2+120*x^48*X[2,1,1,1,1,4]^ 3+210*x^47*X[2,1,1,1,1,4]^4+252*x^46*X[2,1,1,1,1,4]^5-168*x^45*X[2,1,1,1,1,4]^6 +27*x^43*X[2,1,1,1,1,4]^8+x^42*X[2,1,1,1,1,4]^9-2*x^50+10*x^49*X[2,1,1,1,1,4]-\ 45*x^48*X[2,1,1,1,1,4]^2-120*x^47*X[2,1,1,1,1,4]^3-210*x^46*X[2,1,1,1,1,4]^4+ 252*x^45*X[2,1,1,1,1,4]^5-108*x^43*X[2,1,1,1,1,4]^7-7*x^42*X[2,1,1,1,1,4]^8-x^ 49+10*x^48*X[2,1,1,1,1,4]+45*x^47*X[2,1,1,1,1,4]^2+120*x^46*X[2,1,1,1,1,4]^3-\ 252*x^45*X[2,1,1,1,1,4]^4+252*x^43*X[2,1,1,1,1,4]^6+20*x^42*X[2,1,1,1,1,4]^7-4* x^41*X[2,1,1,1,1,4]^8-x^48-10*x^47*X[2,1,1,1,1,4]-45*x^46*X[2,1,1,1,1,4]^2+168* x^45*X[2,1,1,1,1,4]^3-378*x^43*X[2,1,1,1,1,4]^5-28*x^42*X[2,1,1,1,1,4]^6+32*x^ 41*X[2,1,1,1,1,4]^7+x^47+10*x^46*X[2,1,1,1,1,4]-72*x^45*X[2,1,1,1,1,4]^2+378*x^ 43*X[2,1,1,1,1,4]^4+14*x^42*X[2,1,1,1,1,4]^5-112*x^41*X[2,1,1,1,1,4]^6+2*x^39*X [2,1,1,1,1,4]^8-x^46+18*x^45*X[2,1,1,1,1,4]-252*x^43*X[2,1,1,1,1,4]^3+14*x^42*X [2,1,1,1,1,4]^4+224*x^41*X[2,1,1,1,1,4]^5-16*x^39*X[2,1,1,1,1,4]^7-2*x^38*X[2,1 ,1,1,1,4]^8-2*x^45+108*x^43*X[2,1,1,1,1,4]^2-28*x^42*X[2,1,1,1,1,4]^3-280*x^41* X[2,1,1,1,1,4]^4+56*x^39*X[2,1,1,1,1,4]^6+14*x^38*X[2,1,1,1,1,4]^7+4*x^37*X[2,1 ,1,1,1,4]^8-27*x^43*X[2,1,1,1,1,4]+20*x^42*X[2,1,1,1,1,4]^2+224*x^41*X[2,1,1,1, 1,4]^3-112*x^39*X[2,1,1,1,1,4]^5-42*x^38*X[2,1,1,1,1,4]^6-31*x^37*X[2,1,1,1,1,4 ]^7-3*x^36*X[2,1,1,1,1,4]^8+3*x^43-7*x^42*X[2,1,1,1,1,4]-112*x^41*X[2,1,1,1,1,4 ]^2+140*x^39*X[2,1,1,1,1,4]^4+70*x^38*X[2,1,1,1,1,4]^5+105*x^37*X[2,1,1,1,1,4]^ 6+19*x^36*X[2,1,1,1,1,4]^7+x^35*X[2,1,1,1,1,4]^8+x^42+32*x^41*X[2,1,1,1,1,4]-\ 112*x^39*X[2,1,1,1,1,4]^3-70*x^38*X[2,1,1,1,1,4]^4-203*x^37*X[2,1,1,1,1,4]^5-49 *x^36*X[2,1,1,1,1,4]^6+x^35*X[2,1,1,1,1,4]^7-4*x^41+56*x^39*X[2,1,1,1,1,4]^2+42 *x^38*X[2,1,1,1,1,4]^3+245*x^37*X[2,1,1,1,1,4]^4+63*x^36*X[2,1,1,1,1,4]^5-34*x^ 35*X[2,1,1,1,1,4]^6-x^34*X[2,1,1,1,1,4]^7-16*x^39*X[2,1,1,1,1,4]-14*x^38*X[2,1, 1,1,1,4]^2-189*x^37*X[2,1,1,1,1,4]^3-35*x^36*X[2,1,1,1,1,4]^4+127*x^35*X[2,1,1, 1,1,4]^5+2*x^34*X[2,1,1,1,1,4]^6-6*x^33*X[2,1,1,1,1,4]^7+2*x^39+2*x^38*X[2,1,1, 1,1,4]+91*x^37*X[2,1,1,1,1,4]^2-7*x^36*X[2,1,1,1,1,4]^3-230*x^35*X[2,1,1,1,1,4] ^4+9*x^34*X[2,1,1,1,1,4]^5+44*x^33*X[2,1,1,1,1,4]^6+5*x^32*X[2,1,1,1,1,4]^7-25* x^37*X[2,1,1,1,1,4]+21*x^36*X[2,1,1,1,1,4]^2+239*x^35*X[2,1,1,1,1,4]^3-40*x^34* X[2,1,1,1,1,4]^4-138*x^33*X[2,1,1,1,1,4]^5-26*x^32*X[2,1,1,1,1,4]^6-x^31*X[2,1, 1,1,1,4]^7+3*x^37-11*x^36*X[2,1,1,1,1,4]-146*x^35*X[2,1,1,1,1,4]^2+65*x^34*X[2, 1,1,1,1,4]^3+240*x^33*X[2,1,1,1,1,4]^4+51*x^32*X[2,1,1,1,1,4]^5-4*x^31*X[2,1,1, 1,1,4]^6+2*x^36+49*x^35*X[2,1,1,1,1,4]-54*x^34*X[2,1,1,1,1,4]^2-250*x^33*X[2,1, 1,1,1,4]^3-40*x^32*X[2,1,1,1,1,4]^4+44*x^31*X[2,1,1,1,1,4]^5+2*x^30*X[2,1,1,1,1 ,4]^6-7*x^35+23*x^34*X[2,1,1,1,1,4]+156*x^33*X[2,1,1,1,1,4]^2-5*x^32*X[2,1,1,1, 1,4]^3-125*x^31*X[2,1,1,1,1,4]^4-11*x^30*X[2,1,1,1,1,4]^5+3*x^29*X[2,1,1,1,1,4] ^6-4*x^34-54*x^33*X[2,1,1,1,1,4]+30*x^32*X[2,1,1,1,1,4]^2+175*x^31*X[2,1,1,1,1, 4]^3+25*x^30*X[2,1,1,1,1,4]^4-15*x^29*X[2,1,1,1,1,4]^5-x^28*X[2,1,1,1,1,4]^6+8* x^33-19*x^32*X[2,1,1,1,1,4]-134*x^31*X[2,1,1,1,1,4]^2-30*x^30*X[2,1,1,1,1,4]^3+ 30*x^29*X[2,1,1,1,1,4]^4+4*x^28*X[2,1,1,1,1,4]^5+4*x^32+54*x^31*X[2,1,1,1,1,4]+ 20*x^30*X[2,1,1,1,1,4]^2-30*x^29*X[2,1,1,1,1,4]^3-5*x^28*X[2,1,1,1,1,4]^4-9*x^ 27*X[2,1,1,1,1,4]^5-9*x^31-7*x^30*X[2,1,1,1,1,4]+15*x^29*X[2,1,1,1,1,4]^2+44*x^ 27*X[2,1,1,1,1,4]^4+19*x^26*X[2,1,1,1,1,4]^5+x^30-3*x^29*X[2,1,1,1,1,4]+5*x^28* X[2,1,1,1,1,4]^2-86*x^27*X[2,1,1,1,1,4]^3-88*x^26*X[2,1,1,1,1,4]^4-24*x^25*X[2, 1,1,1,1,4]^5-4*x^28*X[2,1,1,1,1,4]+84*x^27*X[2,1,1,1,1,4]^2+162*x^26*X[2,1,1,1, 1,4]^3+103*x^25*X[2,1,1,1,1,4]^4+24*x^24*X[2,1,1,1,1,4]^5+x^28-41*x^27*X[2,1,1, 1,1,4]-148*x^26*X[2,1,1,1,1,4]^2-172*x^25*X[2,1,1,1,1,4]^3-96*x^24*X[2,1,1,1,1, 4]^4-16*x^23*X[2,1,1,1,1,4]^5+8*x^27+67*x^26*X[2,1,1,1,1,4]+138*x^25*X[2,1,1,1, 1,4]^2+144*x^24*X[2,1,1,1,1,4]^3+57*x^23*X[2,1,1,1,1,4]^4+6*x^22*X[2,1,1,1,1,4] ^5-12*x^26-52*x^25*X[2,1,1,1,1,4]-96*x^24*X[2,1,1,1,1,4]^2-67*x^23*X[2,1,1,1,1, 4]^3-26*x^22*X[2,1,1,1,1,4]^4-x^21*X[2,1,1,1,1,4]^5+7*x^25+24*x^24*X[2,1,1,1,1, 4]+19*x^23*X[2,1,1,1,1,4]^2+36*x^22*X[2,1,1,1,1,4]^3+32*x^21*X[2,1,1,1,1,4]^4+ 15*x^23*X[2,1,1,1,1,4]-12*x^22*X[2,1,1,1,1,4]^2-87*x^21*X[2,1,1,1,1,4]^3-39*x^ 20*X[2,1,1,1,1,4]^4-8*x^23-10*x^22*X[2,1,1,1,1,4]+79*x^21*X[2,1,1,1,1,4]^2+91*x ^20*X[2,1,1,1,1,4]^3+25*x^19*X[2,1,1,1,1,4]^4+6*x^22-20*x^21*X[2,1,1,1,1,4]-39* x^20*X[2,1,1,1,1,4]^2-25*x^19*X[2,1,1,1,1,4]^3-8*x^18*X[2,1,1,1,1,4]^4-3*x^21-\ 39*x^20*X[2,1,1,1,1,4]-77*x^19*X[2,1,1,1,1,4]^2-11*x^18*X[2,1,1,1,1,4]^3+x^17*X [2,1,1,1,1,4]^4+26*x^20+129*x^19*X[2,1,1,1,1,4]+93*x^18*X[2,1,1,1,1,4]^2-x^17*X [2,1,1,1,1,4]^3-52*x^19-121*x^18*X[2,1,1,1,1,4]-32*x^17*X[2,1,1,1,1,4]^2+11*x^ 16*X[2,1,1,1,1,4]^3+47*x^18+63*x^17*X[2,1,1,1,1,4]+x^16*X[2,1,1,1,1,4]^2-6*x^15 *X[2,1,1,1,1,4]^3-31*x^17-35*x^16*X[2,1,1,1,1,4]+3*x^15*X[2,1,1,1,1,4]^2+x^14*X [2,1,1,1,1,4]^3+23*x^16+11*x^15*X[2,1,1,1,1,4]-11*x^14*X[2,1,1,1,1,4]^2-8*x^15+ 29*x^14*X[2,1,1,1,1,4]+13*x^13*X[2,1,1,1,1,4]^2-19*x^14-71*x^13*X[2,1,1,1,1,4]-\ 6*x^12*X[2,1,1,1,1,4]^2+58*x^13+135*x^12*X[2,1,1,1,1,4]+x^11*X[2,1,1,1,1,4]^2-\ 129*x^12-233*x^11*X[2,1,1,1,1,4]+233*x^11+316*x^10*X[2,1,1,1,1,4]-327*x^10-321* x^9*X[2,1,1,1,1,4]+376*x^9+240*x^8*X[2,1,1,1,1,4]-405*x^8-128*x^7*X[2,1,1,1,1,4 ]+458*x^7+46*x^6*X[2,1,1,1,1,4]-508*x^6-10*x^5*X[2,1,1,1,1,4]+472*x^5+x^4*X[2,1 ,1,1,1,4]-331*x^4+165*x^3-55*x^2+11*x-1)/(x^5*X[2,1,1,1,1,4]-x^5-x^4*X[2,1,1,1, 1,4]+x^4-2*x+1)/(x^25*X[2,1,1,1,1,4]^5-5*x^25*X[2,1,1,1,1,4]^4+10*x^25*X[2,1,1, 1,1,4]^3-10*x^25*X[2,1,1,1,1,4]^2+5*x^25*X[2,1,1,1,1,4]-x^25-x^16*X[2,1,1,1,1,4 ]^3+3*x^16*X[2,1,1,1,1,4]^2+2*x^15*X[2,1,1,1,1,4]^3-3*x^16*X[2,1,1,1,1,4]-6*x^ 15*X[2,1,1,1,1,4]^2-x^14*X[2,1,1,1,1,4]^3+x^16+6*x^15*X[2,1,1,1,1,4]+3*x^14*X[2 ,1,1,1,1,4]^2-2*x^15-3*x^14*X[2,1,1,1,1,4]-x^13*X[2,1,1,1,1,4]^2+x^14+2*x^13*X[ 2,1,1,1,1,4]+2*x^12*X[2,1,1,1,1,4]^2-x^13-4*x^12*X[2,1,1,1,1,4]-x^11*X[2,1,1,1, 1,4]^2+2*x^12+2*x^11*X[2,1,1,1,1,4]-x^11-x^5+5*x^4-10*x^3+10*x^2-5*x+1)/(x^25*X [2,1,1,1,1,4]^5-5*x^25*X[2,1,1,1,1,4]^4+10*x^25*X[2,1,1,1,1,4]^3-10*x^25*X[2,1, 1,1,1,4]^2+5*x^25*X[2,1,1,1,1,4]-x^25+x^18*X[2,1,1,1,1,4]^4-4*x^18*X[2,1,1,1,1, 4]^3-x^17*X[2,1,1,1,1,4]^4+6*x^18*X[2,1,1,1,1,4]^2+4*x^17*X[2,1,1,1,1,4]^3-4*x^ 18*X[2,1,1,1,1,4]-6*x^17*X[2,1,1,1,1,4]^2+3*x^16*X[2,1,1,1,1,4]^3+x^18+4*x^17*X [2,1,1,1,1,4]-9*x^16*X[2,1,1,1,1,4]^2-3*x^15*X[2,1,1,1,1,4]^3-x^17+9*x^16*X[2,1 ,1,1,1,4]+9*x^15*X[2,1,1,1,1,4]^2-3*x^16-9*x^15*X[2,1,1,1,1,4]+3*x^15+3*x^13*X[ 2,1,1,1,1,4]^2-6*x^13*X[2,1,1,1,1,4]-3*x^12*X[2,1,1,1,1,4]^2+3*x^13+6*x^12*X[2, 1,1,1,1,4]-3*x^12+x^9*X[2,1,1,1,1,4]-x^9-x^8*X[2,1,1,1,1,4]+x^8-x^5+5*x^4-10*x^ 3+10*x^2-5*x+1) Furthermore, the expectation of the random variable number of occurences (by\ 13 n containment) of , [2, 1, 1, 1, 1, 4], equals , - -- + ---- 32 32 371 27 n The variance equals , - ---- + ---- 1024 1024 4629 297 n The , 3, -th moment about the mean is , - ----- + ----- 16384 16384 259825 2187 2 13575 The , 4, -th moment about the mean is , ------- + ------- n - ------ n 1048576 1048576 262144 The compositions of, 10, that yield the, 14, -th largest growth, that is, 1.9407101328380924652, are , [1, 1, 2, 1, 2, 3], [1, 1, 2, 1, 3, 2], [1, 1, 2, 2, 1, 3], [1, 1, 2, 3, 1, 2], [1, 1, 3, 1, 2, 2], [1, 1, 3, 2, 1, 2], [1, 2, 1, 2, 3, 1], [1, 2, 1, 3, 2, 1], [1, 2, 2, 1, 3, 1], [1, 2, 3, 1, 2, 1], [1, 3, 1, 2, 2, 1], [1, 3, 2, 1, 2, 1], [2, 1, 2, 3, 1, 1], [2, 1, 3, 2, 1, 1], [2, 2, 1, 3, 1, 1], [2, 3, 1, 2, 1, 1], [3, 1, 2, 2, 1, 1], [3, 2, 1, 2, 1, 1] Theorem Number, 14, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 1, 2, 1, 2, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n ) a(n) x = - ( / ----- n = 0 11 10 9 8 7 6 5 4 3 2 2 x - 2 x + x + x + 2 x - 6 x + 6 x + x - 9 x + 10 x - 5 x + 1) / 10 9 7 6 5 4 3 2 2 / ((x - x + x + x - 3 x + 3 x + x - 5 x + 4 x - 1) (-1 + x) ) / and in Maple format -(2*x^11-2*x^10+x^9+x^8+2*x^7-6*x^6+6*x^5+x^4-9*x^3+10*x^2-5*x+1)/(x^10-x^9+x^7 +x^6-3*x^5+3*x^4+x^3-5*x^2+4*x-1)/(-1+x)^2 The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1016, 2010, 3956, 7750, 15126, 29442, 57206, 111039, 215428, 417895, 810678, 1572813, 3051807, 5922142, 11492879, 22304612, 43287954, 84011781, 163046371, 316430941] The limit of a(n+1)/a(n) as n goes to infinity is 1.94071013284 a(n) is asymptotic to .726879966030*1.94071013284^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [1, 1, 2, 1, 2, 3], denoted by the variable, X[1, 1, 2, 1, 2, 3], is 11 2 11 10 2 11 10 9 2 10 - (2 x %1 - 4 x %1 - 2 x %1 + 2 x + 4 x %1 + x %1 - 2 x 9 9 8 8 7 7 6 6 5 - 2 x %1 + x - x %1 + x - 2 x %1 + 2 x + 6 x %1 - 6 x - 7 x %1 5 4 4 3 3 2 / 2 + 6 x + 4 x %1 + x - x %1 - 9 x + 10 x - 5 x + 1) / ((-1 + x) ( / 10 2 10 9 2 10 9 9 7 7 6 6 x %1 - 2 x %1 - x %1 + x + 2 x %1 - x - x %1 + x - x %1 + x 5 5 4 4 3 3 2 + 3 x %1 - 3 x - 3 x %1 + 3 x + x %1 + x - 5 x + 4 x - 1)) %1 := X[1, 1, 2, 1, 2, 3] and in Maple format -(2*x^11*X[1,1,2,1,2,3]^2-4*x^11*X[1,1,2,1,2,3]-2*x^10*X[1,1,2,1,2,3]^2+2*x^11+ 4*x^10*X[1,1,2,1,2,3]+x^9*X[1,1,2,1,2,3]^2-2*x^10-2*x^9*X[1,1,2,1,2,3]+x^9-x^8* X[1,1,2,1,2,3]+x^8-2*x^7*X[1,1,2,1,2,3]+2*x^7+6*x^6*X[1,1,2,1,2,3]-6*x^6-7*x^5* X[1,1,2,1,2,3]+6*x^5+4*x^4*X[1,1,2,1,2,3]+x^4-x^3*X[1,1,2,1,2,3]-9*x^3+10*x^2-5 *x+1)/(-1+x)^2/(x^10*X[1,1,2,1,2,3]^2-2*x^10*X[1,1,2,1,2,3]-x^9*X[1,1,2,1,2,3]^ 2+x^10+2*x^9*X[1,1,2,1,2,3]-x^9-x^7*X[1,1,2,1,2,3]+x^7-x^6*X[1,1,2,1,2,3]+x^6+3 *x^5*X[1,1,2,1,2,3]-3*x^5-3*x^4*X[1,1,2,1,2,3]+3*x^4+x^3*X[1,1,2,1,2,3]+x^3-5*x ^2+4*x-1) Furthermore, the expectation of the random variable number of occurences (by\ 13 n containment) of , [1, 1, 2, 1, 2, 3], equals , - -- + ---- 32 32 499 35 n The variance equals , - ---- + ---- 1024 1024 11397 681 n The , 3, -th moment about the mean is , - ----- + ----- 16384 16384 535567 3675 2 10651 The , 4, -th moment about the mean is , - ------- + ------- n - ------ n 1048576 1048576 262144 The compositions of, 10, that yield the, 15, -th largest growth, that is, 1.9409607910644739216, are , [2, 1, 1, 2, 1, 3], [2, 1, 1, 3, 1, 2], [2, 1, 2, 1, 1, 3], [2, 1, 3, 1, 1, 2], [3, 1, 1, 2, 1, 2], [3, 1, 2, 1, 1, 2] Theorem Number, 15, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 1, 2, 1, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 22 20 19 18 17 16 15 14 ) a(n) x = - (x - x + 2 x + x - 4 x + 6 x - x - 5 x / ----- n = 0 13 12 11 10 9 8 7 6 5 4 + 3 x + 7 x - 9 x + x + 2 x + 8 x - 20 x + 17 x + 5 x - 29 x 3 2 / 23 22 21 20 19 18 + 34 x - 21 x + 7 x - 1) / (x - x - x + 3 x - x - 3 x / 17 16 15 14 13 12 11 10 9 + 8 x - 8 x - x + 9 x - x - 13 x + 11 x + 2 x - 2 x 8 7 6 5 4 3 2 - 16 x + 32 x - 22 x - 15 x + 48 x - 49 x + 27 x - 8 x + 1) and in Maple format -(x^22-x^20+2*x^19+x^18-4*x^17+6*x^16-x^15-5*x^14+3*x^13+7*x^12-9*x^11+x^10+2*x ^9+8*x^8-20*x^7+17*x^6+5*x^5-29*x^4+34*x^3-21*x^2+7*x-1)/(x^23-x^22-x^21+3*x^20 -x^19-3*x^18+8*x^17-8*x^16-x^15+9*x^14-x^13-13*x^12+11*x^11+2*x^10-2*x^9-16*x^8 +32*x^7-22*x^6-15*x^5+48*x^4-49*x^3+27*x^2-8*x+1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1016, 2010, 3956, 7749, 15118, 29404, 57067, 110608, 214238, 414882, 803545, 1556803, 3017378, 5850608, 11348275, 22018512, 42731049, 82940412, 161001402, 312545508] The limit of a(n+1)/a(n) as n goes to infinity is 1.94096079106 a(n) is asymptotic to .715631497849*1.94096079106^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [2, 1, 1, 2, 1, 3], denoted by the variable, X[2, 1, 1, 2, 1, 3], is 23 5 23 4 23 3 22 4 23 2 22 3 - (x %1 - 5 x %1 + 10 x %1 + x %1 - 10 x %1 - 4 x %1 21 4 23 22 2 21 3 20 4 23 22 + x %1 + 5 x %1 + 6 x %1 - 4 x %1 - 3 x %1 - x - 4 x %1 21 2 20 3 22 21 20 2 19 3 + 6 x %1 + 12 x %1 + x - 4 x %1 - 18 x %1 - x %1 18 4 21 20 19 2 18 3 17 4 20 + 2 x %1 + x + 12 x %1 + 3 x %1 - 11 x %1 - 4 x %1 - 3 x 19 18 2 17 3 16 4 19 18 - 3 x %1 + 21 x %1 + 22 x %1 + 3 x %1 + x - 17 x %1 17 2 16 3 15 4 18 17 16 2 - 42 x %1 - 16 x %1 - x %1 + 5 x + 34 x %1 + 30 x %1 17 16 15 2 14 3 16 15 - 10 x - 24 x %1 + 7 x %1 + 8 x %1 + 7 x - 10 x %1 14 2 13 3 15 14 13 2 12 3 14 - 24 x %1 - 5 x %1 + 4 x + 24 x %1 + 6 x %1 + x %1 - 8 x 13 12 2 13 12 11 2 12 + 3 x %1 + 15 x %1 - 4 x - 32 x %1 - 15 x %1 + 16 x 11 10 2 11 10 9 2 10 9 + 25 x %1 + 6 x %1 - 10 x - 5 x %1 - x %1 - x + 7 x %1 9 8 8 7 7 6 6 5 - 6 x - 29 x %1 + 28 x + 45 x %1 - 37 x - 40 x %1 + 12 x + 22 x %1 5 4 4 3 3 2 / 24 5 + 34 x - 7 x %1 - 63 x + x %1 + 55 x - 28 x + 8 x - 1) / (x %1 / 24 4 23 5 24 3 23 4 24 2 23 3 - 5 x %1 - x %1 + 10 x %1 + 6 x %1 - 10 x %1 - 14 x %1 24 23 2 21 4 24 23 21 3 + 5 x %1 + 16 x %1 - 4 x %1 - x - 9 x %1 + 16 x %1 20 4 23 21 2 20 3 21 20 2 + 3 x %1 + 2 x - 24 x %1 - 13 x %1 + 16 x %1 + 21 x %1 19 3 18 4 21 20 19 2 18 3 - 2 x %1 - 4 x %1 - 4 x - 15 x %1 + 6 x %1 + 23 x %1 17 4 20 19 18 2 17 3 16 4 + 6 x %1 + 4 x - 6 x %1 - 45 x %1 - 34 x %1 - 4 x %1 19 18 17 2 16 3 15 4 18 + 2 x + 37 x %1 + 66 x %1 + 20 x %1 + x %1 - 11 x 17 16 2 15 3 17 16 15 2 - 54 x %1 - 35 x %1 + 4 x %1 + 16 x + 26 x %1 - 21 x %1 14 3 16 15 14 2 13 3 15 - 12 x %1 - 7 x + 26 x %1 + 34 x %1 + 6 x %1 - 10 x 14 12 3 14 13 12 2 13 - 32 x %1 - x %1 + 10 x - 18 x %1 - 25 x %1 + 12 x 12 11 2 12 11 10 2 11 + 50 x %1 + 20 x %1 - 24 x - 29 x %1 - 7 x %1 + 9 x 10 9 2 10 9 9 8 8 + 3 x %1 + x %1 + 4 x - 15 x %1 + 14 x + 50 x %1 - 48 x 7 7 6 6 5 5 4 4 - 69 x %1 + 54 x + 56 x %1 - 7 x - 28 x %1 - 63 x + 8 x %1 + 97 x 3 3 2 - x %1 - 76 x + 35 x - 9 x + 1) %1 := X[2, 1, 1, 2, 1, 3] and in Maple format -(x^23*X[2,1,1,2,1,3]^5-5*x^23*X[2,1,1,2,1,3]^4+10*x^23*X[2,1,1,2,1,3]^3+x^22*X [2,1,1,2,1,3]^4-10*x^23*X[2,1,1,2,1,3]^2-4*x^22*X[2,1,1,2,1,3]^3+x^21*X[2,1,1,2 ,1,3]^4+5*x^23*X[2,1,1,2,1,3]+6*x^22*X[2,1,1,2,1,3]^2-4*x^21*X[2,1,1,2,1,3]^3-3 *x^20*X[2,1,1,2,1,3]^4-x^23-4*x^22*X[2,1,1,2,1,3]+6*x^21*X[2,1,1,2,1,3]^2+12*x^ 20*X[2,1,1,2,1,3]^3+x^22-4*x^21*X[2,1,1,2,1,3]-18*x^20*X[2,1,1,2,1,3]^2-x^19*X[ 2,1,1,2,1,3]^3+2*x^18*X[2,1,1,2,1,3]^4+x^21+12*x^20*X[2,1,1,2,1,3]+3*x^19*X[2,1 ,1,2,1,3]^2-11*x^18*X[2,1,1,2,1,3]^3-4*x^17*X[2,1,1,2,1,3]^4-3*x^20-3*x^19*X[2, 1,1,2,1,3]+21*x^18*X[2,1,1,2,1,3]^2+22*x^17*X[2,1,1,2,1,3]^3+3*x^16*X[2,1,1,2,1 ,3]^4+x^19-17*x^18*X[2,1,1,2,1,3]-42*x^17*X[2,1,1,2,1,3]^2-16*x^16*X[2,1,1,2,1, 3]^3-x^15*X[2,1,1,2,1,3]^4+5*x^18+34*x^17*X[2,1,1,2,1,3]+30*x^16*X[2,1,1,2,1,3] ^2-10*x^17-24*x^16*X[2,1,1,2,1,3]+7*x^15*X[2,1,1,2,1,3]^2+8*x^14*X[2,1,1,2,1,3] ^3+7*x^16-10*x^15*X[2,1,1,2,1,3]-24*x^14*X[2,1,1,2,1,3]^2-5*x^13*X[2,1,1,2,1,3] ^3+4*x^15+24*x^14*X[2,1,1,2,1,3]+6*x^13*X[2,1,1,2,1,3]^2+x^12*X[2,1,1,2,1,3]^3-\ 8*x^14+3*x^13*X[2,1,1,2,1,3]+15*x^12*X[2,1,1,2,1,3]^2-4*x^13-32*x^12*X[2,1,1,2, 1,3]-15*x^11*X[2,1,1,2,1,3]^2+16*x^12+25*x^11*X[2,1,1,2,1,3]+6*x^10*X[2,1,1,2,1 ,3]^2-10*x^11-5*x^10*X[2,1,1,2,1,3]-x^9*X[2,1,1,2,1,3]^2-x^10+7*x^9*X[2,1,1,2,1 ,3]-6*x^9-29*x^8*X[2,1,1,2,1,3]+28*x^8+45*x^7*X[2,1,1,2,1,3]-37*x^7-40*x^6*X[2, 1,1,2,1,3]+12*x^6+22*x^5*X[2,1,1,2,1,3]+34*x^5-7*x^4*X[2,1,1,2,1,3]-63*x^4+x^3* X[2,1,1,2,1,3]+55*x^3-28*x^2+8*x-1)/(x^24*X[2,1,1,2,1,3]^5-5*x^24*X[2,1,1,2,1,3 ]^4-x^23*X[2,1,1,2,1,3]^5+10*x^24*X[2,1,1,2,1,3]^3+6*x^23*X[2,1,1,2,1,3]^4-10*x ^24*X[2,1,1,2,1,3]^2-14*x^23*X[2,1,1,2,1,3]^3+5*x^24*X[2,1,1,2,1,3]+16*x^23*X[2 ,1,1,2,1,3]^2-4*x^21*X[2,1,1,2,1,3]^4-x^24-9*x^23*X[2,1,1,2,1,3]+16*x^21*X[2,1, 1,2,1,3]^3+3*x^20*X[2,1,1,2,1,3]^4+2*x^23-24*x^21*X[2,1,1,2,1,3]^2-13*x^20*X[2, 1,1,2,1,3]^3+16*x^21*X[2,1,1,2,1,3]+21*x^20*X[2,1,1,2,1,3]^2-2*x^19*X[2,1,1,2,1 ,3]^3-4*x^18*X[2,1,1,2,1,3]^4-4*x^21-15*x^20*X[2,1,1,2,1,3]+6*x^19*X[2,1,1,2,1, 3]^2+23*x^18*X[2,1,1,2,1,3]^3+6*x^17*X[2,1,1,2,1,3]^4+4*x^20-6*x^19*X[2,1,1,2,1 ,3]-45*x^18*X[2,1,1,2,1,3]^2-34*x^17*X[2,1,1,2,1,3]^3-4*x^16*X[2,1,1,2,1,3]^4+2 *x^19+37*x^18*X[2,1,1,2,1,3]+66*x^17*X[2,1,1,2,1,3]^2+20*x^16*X[2,1,1,2,1,3]^3+ x^15*X[2,1,1,2,1,3]^4-11*x^18-54*x^17*X[2,1,1,2,1,3]-35*x^16*X[2,1,1,2,1,3]^2+4 *x^15*X[2,1,1,2,1,3]^3+16*x^17+26*x^16*X[2,1,1,2,1,3]-21*x^15*X[2,1,1,2,1,3]^2-\ 12*x^14*X[2,1,1,2,1,3]^3-7*x^16+26*x^15*X[2,1,1,2,1,3]+34*x^14*X[2,1,1,2,1,3]^2 +6*x^13*X[2,1,1,2,1,3]^3-10*x^15-32*x^14*X[2,1,1,2,1,3]-x^12*X[2,1,1,2,1,3]^3+ 10*x^14-18*x^13*X[2,1,1,2,1,3]-25*x^12*X[2,1,1,2,1,3]^2+12*x^13+50*x^12*X[2,1,1 ,2,1,3]+20*x^11*X[2,1,1,2,1,3]^2-24*x^12-29*x^11*X[2,1,1,2,1,3]-7*x^10*X[2,1,1, 2,1,3]^2+9*x^11+3*x^10*X[2,1,1,2,1,3]+x^9*X[2,1,1,2,1,3]^2+4*x^10-15*x^9*X[2,1, 1,2,1,3]+14*x^9+50*x^8*X[2,1,1,2,1,3]-48*x^8-69*x^7*X[2,1,1,2,1,3]+54*x^7+56*x^ 6*X[2,1,1,2,1,3]-7*x^6-28*x^5*X[2,1,1,2,1,3]-63*x^5+8*x^4*X[2,1,1,2,1,3]+97*x^4 -x^3*X[2,1,1,2,1,3]-76*x^3+35*x^2-9*x+1) Furthermore, the expectation of the random variable number of occurences (by\ 13 n containment) of , [2, 1, 1, 2, 1, 3], equals , - -- + ---- 32 32 531 35 n The variance equals , - ---- + ---- 1024 1024 12945 669 n The , 3, -th moment about the mean is , - ----- + ----- 16384 16384 680367 3675 2 13363 The , 4, -th moment about the mean is , - ------- + ------- n - ------ n 1048576 1048576 262144 The compositions of, 10, that yield the, 16, -th largest growth, that is, 1.9409620467295949022, are , [2, 1, 1, 1, 2, 3], [2, 1, 1, 1, 3, 2], [2, 2, 1, 1, 1, 3], [2, 3, 1, 1, 1, 2], [3, 1, 1, 1, 2, 2], [3, 2, 1, 1, 1, 2] Theorem Number, 16, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 1, 1, 2, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 23 22 21 20 19 18 16 15 14 ) a(n) x = - (x + x + x + x - x + 2 x + 3 x - 2 x - x / ----- n = 0 13 12 11 10 9 8 7 6 5 + 5 x - 6 x + 5 x + x - 12 x + 22 x - 26 x + 18 x + 5 x 4 3 2 / 24 23 20 19 17 - 29 x + 34 x - 21 x + 7 x - 1) / (x + x - x + 2 x + x / 16 15 14 13 12 11 10 9 8 - 4 x + 4 x + 2 x - 7 x + 9 x - 8 x - 3 x + 21 x - 35 x 7 6 5 4 3 2 + 39 x - 23 x - 15 x + 48 x - 49 x + 27 x - 8 x + 1) and in Maple format -(x^23+x^22+x^21+x^20-x^19+2*x^18+3*x^16-2*x^15-x^14+5*x^13-6*x^12+5*x^11+x^10-\ 12*x^9+22*x^8-26*x^7+18*x^6+5*x^5-29*x^4+34*x^3-21*x^2+7*x-1)/(x^24+x^23-x^20+2 *x^19+x^17-4*x^16+4*x^15+2*x^14-7*x^13+9*x^12-8*x^11-3*x^10+21*x^9-35*x^8+39*x^ 7-23*x^6-15*x^5+48*x^4-49*x^3+27*x^2-8*x+1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1016, 2010, 3956, 7750, 15125, 29432, 57151, 110817, 214693, 415778, 805177, 1559595, 3021911, 5857640, 11358751, 22033593, 42752282, 82970386, 161045718, 312617885] The limit of a(n+1)/a(n) as n goes to infinity is 1.94096204673 a(n) is asymptotic to .715931280313*1.94096204673^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [2, 1, 1, 1, 2, 3], denoted by the variable, X[2, 1, 1, 1, 2, 3], is 24 5 24 4 24 3 22 5 24 2 22 4 - (x %1 - 5 x %1 + 10 x %1 + x %1 - 10 x %1 - 4 x %1 24 22 3 24 22 2 20 4 22 20 3 + 5 x %1 + 6 x %1 - x - 4 x %1 + 2 x %1 + x %1 - 8 x %1 19 4 20 2 19 3 18 4 20 19 2 - 3 x %1 + 12 x %1 + 12 x %1 + 3 x %1 - 8 x %1 - 18 x %1 18 3 17 4 20 19 18 2 17 3 - 11 x %1 - 4 x %1 + 2 x + 12 x %1 + 15 x %1 + 15 x %1 16 4 19 18 17 2 16 3 15 4 18 + 3 x %1 - 3 x - 9 x %1 - 21 x %1 - 14 x %1 - x %1 + 2 x 17 16 2 15 3 17 16 15 2 + 13 x %1 + 24 x %1 + 5 x %1 - 3 x - 18 x %1 - 8 x %1 14 3 16 15 14 2 13 3 15 14 + x %1 + 5 x + 5 x %1 - 8 x %1 - x %1 - x + 13 x %1 13 2 14 13 12 2 13 12 + 13 x %1 - 6 x - 23 x %1 - 12 x %1 + 11 x + 23 x %1 11 2 12 11 10 2 11 10 9 2 + 10 x %1 - 11 x - 14 x %1 - 5 x %1 + 4 x - 8 x %1 + x %1 10 9 9 8 8 7 7 + 13 x + 33 x %1 - 34 x - 49 x %1 + 48 x + 52 x %1 - 44 x 6 6 5 5 4 4 3 3 - 41 x %1 + 13 x + 22 x %1 + 34 x - 7 x %1 - 63 x + x %1 + 55 x 2 / 25 5 25 4 25 3 23 5 - 28 x + 8 x - 1) / (x %1 - 5 x %1 + 10 x %1 + x %1 / 25 2 23 4 22 5 25 23 3 22 4 - 10 x %1 - 3 x %1 - x %1 + 5 x %1 + 2 x %1 + 4 x %1 25 23 2 22 3 21 4 23 22 2 21 3 - x + 2 x %1 - 6 x %1 + x %1 - 3 x %1 + 4 x %1 - 4 x %1 20 4 23 22 21 2 20 3 19 4 21 - 4 x %1 + x - x %1 + 6 x %1 + 15 x %1 + 4 x %1 - 4 x %1 20 2 19 3 18 4 21 20 19 2 - 21 x %1 - 14 x %1 - 5 x %1 + x + 13 x %1 + 18 x %1 18 3 17 4 20 19 18 2 17 3 + 16 x %1 + 6 x %1 - 3 x - 10 x %1 - 18 x %1 - 23 x %1 16 4 19 18 17 2 16 3 15 4 18 - 4 x %1 + 2 x + 8 x %1 + 33 x %1 + 19 x %1 + x %1 - x 17 16 2 15 3 17 16 15 2 - 21 x %1 - 34 x %1 - 5 x %1 + 5 x + 27 x %1 + 9 x %1 14 3 16 15 14 2 13 3 15 14 - 2 x %1 - 8 x - 7 x %1 + 13 x %1 + x %1 + 2 x - 20 x %1 13 2 14 13 12 2 13 12 - 18 x %1 + 9 x + 33 x %1 + 18 x %1 - 16 x - 35 x %1 11 2 12 11 10 2 11 10 9 2 - 14 x %1 + 17 x + 19 x %1 + 6 x %1 - 5 x + 18 x %1 - x %1 10 9 9 8 8 7 7 - 24 x - 55 x %1 + 56 x + 76 x %1 - 74 x - 77 x %1 + 62 x 6 6 5 5 4 4 3 3 + 57 x %1 - 8 x - 28 x %1 - 63 x + 8 x %1 + 97 x - x %1 - 76 x 2 + 35 x - 9 x + 1) %1 := X[2, 1, 1, 1, 2, 3] and in Maple format -(x^24*X[2,1,1,1,2,3]^5-5*x^24*X[2,1,1,1,2,3]^4+10*x^24*X[2,1,1,1,2,3]^3+x^22*X [2,1,1,1,2,3]^5-10*x^24*X[2,1,1,1,2,3]^2-4*x^22*X[2,1,1,1,2,3]^4+5*x^24*X[2,1,1 ,1,2,3]+6*x^22*X[2,1,1,1,2,3]^3-x^24-4*x^22*X[2,1,1,1,2,3]^2+2*x^20*X[2,1,1,1,2 ,3]^4+x^22*X[2,1,1,1,2,3]-8*x^20*X[2,1,1,1,2,3]^3-3*x^19*X[2,1,1,1,2,3]^4+12*x^ 20*X[2,1,1,1,2,3]^2+12*x^19*X[2,1,1,1,2,3]^3+3*x^18*X[2,1,1,1,2,3]^4-8*x^20*X[2 ,1,1,1,2,3]-18*x^19*X[2,1,1,1,2,3]^2-11*x^18*X[2,1,1,1,2,3]^3-4*x^17*X[2,1,1,1, 2,3]^4+2*x^20+12*x^19*X[2,1,1,1,2,3]+15*x^18*X[2,1,1,1,2,3]^2+15*x^17*X[2,1,1,1 ,2,3]^3+3*x^16*X[2,1,1,1,2,3]^4-3*x^19-9*x^18*X[2,1,1,1,2,3]-21*x^17*X[2,1,1,1, 2,3]^2-14*x^16*X[2,1,1,1,2,3]^3-x^15*X[2,1,1,1,2,3]^4+2*x^18+13*x^17*X[2,1,1,1, 2,3]+24*x^16*X[2,1,1,1,2,3]^2+5*x^15*X[2,1,1,1,2,3]^3-3*x^17-18*x^16*X[2,1,1,1, 2,3]-8*x^15*X[2,1,1,1,2,3]^2+x^14*X[2,1,1,1,2,3]^3+5*x^16+5*x^15*X[2,1,1,1,2,3] -8*x^14*X[2,1,1,1,2,3]^2-x^13*X[2,1,1,1,2,3]^3-x^15+13*x^14*X[2,1,1,1,2,3]+13*x ^13*X[2,1,1,1,2,3]^2-6*x^14-23*x^13*X[2,1,1,1,2,3]-12*x^12*X[2,1,1,1,2,3]^2+11* x^13+23*x^12*X[2,1,1,1,2,3]+10*x^11*X[2,1,1,1,2,3]^2-11*x^12-14*x^11*X[2,1,1,1, 2,3]-5*x^10*X[2,1,1,1,2,3]^2+4*x^11-8*x^10*X[2,1,1,1,2,3]+x^9*X[2,1,1,1,2,3]^2+ 13*x^10+33*x^9*X[2,1,1,1,2,3]-34*x^9-49*x^8*X[2,1,1,1,2,3]+48*x^8+52*x^7*X[2,1, 1,1,2,3]-44*x^7-41*x^6*X[2,1,1,1,2,3]+13*x^6+22*x^5*X[2,1,1,1,2,3]+34*x^5-7*x^4 *X[2,1,1,1,2,3]-63*x^4+x^3*X[2,1,1,1,2,3]+55*x^3-28*x^2+8*x-1)/(x^25*X[2,1,1,1, 2,3]^5-5*x^25*X[2,1,1,1,2,3]^4+10*x^25*X[2,1,1,1,2,3]^3+x^23*X[2,1,1,1,2,3]^5-\ 10*x^25*X[2,1,1,1,2,3]^2-3*x^23*X[2,1,1,1,2,3]^4-x^22*X[2,1,1,1,2,3]^5+5*x^25*X [2,1,1,1,2,3]+2*x^23*X[2,1,1,1,2,3]^3+4*x^22*X[2,1,1,1,2,3]^4-x^25+2*x^23*X[2,1 ,1,1,2,3]^2-6*x^22*X[2,1,1,1,2,3]^3+x^21*X[2,1,1,1,2,3]^4-3*x^23*X[2,1,1,1,2,3] +4*x^22*X[2,1,1,1,2,3]^2-4*x^21*X[2,1,1,1,2,3]^3-4*x^20*X[2,1,1,1,2,3]^4+x^23-x ^22*X[2,1,1,1,2,3]+6*x^21*X[2,1,1,1,2,3]^2+15*x^20*X[2,1,1,1,2,3]^3+4*x^19*X[2, 1,1,1,2,3]^4-4*x^21*X[2,1,1,1,2,3]-21*x^20*X[2,1,1,1,2,3]^2-14*x^19*X[2,1,1,1,2 ,3]^3-5*x^18*X[2,1,1,1,2,3]^4+x^21+13*x^20*X[2,1,1,1,2,3]+18*x^19*X[2,1,1,1,2,3 ]^2+16*x^18*X[2,1,1,1,2,3]^3+6*x^17*X[2,1,1,1,2,3]^4-3*x^20-10*x^19*X[2,1,1,1,2 ,3]-18*x^18*X[2,1,1,1,2,3]^2-23*x^17*X[2,1,1,1,2,3]^3-4*x^16*X[2,1,1,1,2,3]^4+2 *x^19+8*x^18*X[2,1,1,1,2,3]+33*x^17*X[2,1,1,1,2,3]^2+19*x^16*X[2,1,1,1,2,3]^3+x ^15*X[2,1,1,1,2,3]^4-x^18-21*x^17*X[2,1,1,1,2,3]-34*x^16*X[2,1,1,1,2,3]^2-5*x^ 15*X[2,1,1,1,2,3]^3+5*x^17+27*x^16*X[2,1,1,1,2,3]+9*x^15*X[2,1,1,1,2,3]^2-2*x^ 14*X[2,1,1,1,2,3]^3-8*x^16-7*x^15*X[2,1,1,1,2,3]+13*x^14*X[2,1,1,1,2,3]^2+x^13* X[2,1,1,1,2,3]^3+2*x^15-20*x^14*X[2,1,1,1,2,3]-18*x^13*X[2,1,1,1,2,3]^2+9*x^14+ 33*x^13*X[2,1,1,1,2,3]+18*x^12*X[2,1,1,1,2,3]^2-16*x^13-35*x^12*X[2,1,1,1,2,3]-\ 14*x^11*X[2,1,1,1,2,3]^2+17*x^12+19*x^11*X[2,1,1,1,2,3]+6*x^10*X[2,1,1,1,2,3]^2 -5*x^11+18*x^10*X[2,1,1,1,2,3]-x^9*X[2,1,1,1,2,3]^2-24*x^10-55*x^9*X[2,1,1,1,2, 3]+56*x^9+76*x^8*X[2,1,1,1,2,3]-74*x^8-77*x^7*X[2,1,1,1,2,3]+62*x^7+57*x^6*X[2, 1,1,1,2,3]-8*x^6-28*x^5*X[2,1,1,1,2,3]-63*x^5+8*x^4*X[2,1,1,1,2,3]+97*x^4-x^3*X [2,1,1,1,2,3]-76*x^3+35*x^2-9*x+1) Furthermore, the expectation of the random variable number of occurences (by\ 13 n containment) of , [2, 1, 1, 1, 2, 3], equals , - -- + ---- 32 32 531 35 n The variance equals , - ---- + ---- 1024 1024 12993 669 n The , 3, -th moment about the mean is , - ----- + ----- 16384 16384 699567 3675 2 13363 The , 4, -th moment about the mean is , - ------- + ------- n - ------ n 1048576 1048576 262144 The compositions of, 10, that yield the, 17, -th largest growth, that is, 1.9409751179367153000, are , [1, 2, 1, 1, 2, 3], [1, 2, 1, 1, 3, 2], [1, 2, 2, 1, 1, 3], [1, 2, 3, 1, 1, 2], [1, 3, 1, 1, 2, 2], [1, 3, 2, 1, 1, 2], [2, 1, 1, 2, 3, 1], [2, 1, 1, 3, 2, 1], [2, 2, 1, 1, 3, 1], [2, 3, 1, 1, 2, 1], [3, 1, 1, 2, 2, 1], [3, 2, 1, 1, 2, 1] Theorem Number, 17, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 2, 1, 1, 2, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 15 13 10 9 8 7 6 5 4 3 ) a(n) x = (x + x + x + x - x + 3 x - 6 x + 6 x + x - 9 x / ----- n = 0 2 / + 10 x - 5 x + 1) / ((-1 + x) / 14 12 10 8 7 6 5 4 3 2 (x + x - x + x - x + 4 x - 6 x + 2 x + 6 x - 9 x + 5 x - 1)) and in Maple format (x^15+x^13+x^10+x^9-x^8+3*x^7-6*x^6+6*x^5+x^4-9*x^3+10*x^2-5*x+1)/(-1+x)/(x^14+ x^12-x^10+x^8-x^7+4*x^6-6*x^5+2*x^4+6*x^3-9*x^2+5*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1016, 2010, 3956, 7750, 15125, 29433, 57161, 110873, 214922, 416542, 807388, 1565364, 3035842, 5889353, 11427722, 22178335, 43047651, 83560150, 162203987, 314865531] The limit of a(n+1)/a(n) as n goes to infinity is 1.94097511794 a(n) is asymptotic to .720480092322*1.94097511794^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [1, 2, 1, 1, 2, 3], denoted by the variable, X[1, 2, 1, 1, 2, 3], is 16 3 16 2 15 3 16 15 2 14 3 16 (x %1 - 3 x %1 - x %1 + 3 x %1 + 3 x %1 + 2 x %1 - x 15 14 2 13 3 15 14 13 2 12 3 - 3 x %1 - 5 x %1 - 2 x %1 + x + 4 x %1 + 5 x %1 + x %1 14 13 12 2 13 12 11 2 11 10 2 - x - 4 x %1 - 2 x %1 + x + x %1 - x %1 + 2 x %1 + x %1 11 10 9 9 8 8 7 7 - x - x %1 - 2 x %1 + 2 x + 4 x %1 - 4 x - 9 x %1 + 9 x 6 6 5 5 4 4 3 3 + 13 x %1 - 12 x - 11 x %1 + 5 x + 5 x %1 + 10 x - x %1 - 19 x 2 / 15 3 15 2 14 3 15 + 15 x - 6 x + 1) / ((-1 + x) (x %1 - 3 x %1 - x %1 + 3 x %1 / 14 2 13 3 15 14 13 2 12 3 14 + 3 x %1 + 2 x %1 - x - 3 x %1 - 5 x %1 - x %1 + x 13 12 2 13 12 11 2 12 11 + 4 x %1 + 3 x %1 - x - 3 x %1 + x %1 + x - 2 x %1 10 2 11 10 10 9 9 8 8 7 - x %1 + x + 2 x %1 - x + x %1 - x - 2 x %1 + 2 x + 5 x %1 7 6 6 5 5 4 4 3 - 5 x - 10 x %1 + 10 x + 10 x %1 - 8 x - 5 x %1 - 4 x + x %1 3 2 + 15 x - 14 x + 6 x - 1)) %1 := X[1, 2, 1, 1, 2, 3] and in Maple format (x^16*X[1,2,1,1,2,3]^3-3*x^16*X[1,2,1,1,2,3]^2-x^15*X[1,2,1,1,2,3]^3+3*x^16*X[1 ,2,1,1,2,3]+3*x^15*X[1,2,1,1,2,3]^2+2*x^14*X[1,2,1,1,2,3]^3-x^16-3*x^15*X[1,2,1 ,1,2,3]-5*x^14*X[1,2,1,1,2,3]^2-2*x^13*X[1,2,1,1,2,3]^3+x^15+4*x^14*X[1,2,1,1,2 ,3]+5*x^13*X[1,2,1,1,2,3]^2+x^12*X[1,2,1,1,2,3]^3-x^14-4*x^13*X[1,2,1,1,2,3]-2* x^12*X[1,2,1,1,2,3]^2+x^13+x^12*X[1,2,1,1,2,3]-x^11*X[1,2,1,1,2,3]^2+2*x^11*X[1 ,2,1,1,2,3]+x^10*X[1,2,1,1,2,3]^2-x^11-x^10*X[1,2,1,1,2,3]-2*x^9*X[1,2,1,1,2,3] +2*x^9+4*x^8*X[1,2,1,1,2,3]-4*x^8-9*x^7*X[1,2,1,1,2,3]+9*x^7+13*x^6*X[1,2,1,1,2 ,3]-12*x^6-11*x^5*X[1,2,1,1,2,3]+5*x^5+5*x^4*X[1,2,1,1,2,3]+10*x^4-x^3*X[1,2,1, 1,2,3]-19*x^3+15*x^2-6*x+1)/(-1+x)/(x^15*X[1,2,1,1,2,3]^3-3*x^15*X[1,2,1,1,2,3] ^2-x^14*X[1,2,1,1,2,3]^3+3*x^15*X[1,2,1,1,2,3]+3*x^14*X[1,2,1,1,2,3]^2+2*x^13*X [1,2,1,1,2,3]^3-x^15-3*x^14*X[1,2,1,1,2,3]-5*x^13*X[1,2,1,1,2,3]^2-x^12*X[1,2,1 ,1,2,3]^3+x^14+4*x^13*X[1,2,1,1,2,3]+3*x^12*X[1,2,1,1,2,3]^2-x^13-3*x^12*X[1,2, 1,1,2,3]+x^11*X[1,2,1,1,2,3]^2+x^12-2*x^11*X[1,2,1,1,2,3]-x^10*X[1,2,1,1,2,3]^2 +x^11+2*x^10*X[1,2,1,1,2,3]-x^10+x^9*X[1,2,1,1,2,3]-x^9-2*x^8*X[1,2,1,1,2,3]+2* x^8+5*x^7*X[1,2,1,1,2,3]-5*x^7-10*x^6*X[1,2,1,1,2,3]+10*x^6+10*x^5*X[1,2,1,1,2, 3]-8*x^5-5*x^4*X[1,2,1,1,2,3]-4*x^4+x^3*X[1,2,1,1,2,3]+15*x^3-14*x^2+6*x-1) Furthermore, the expectation of the random variable number of occurences (by\ 13 n containment) of , [1, 2, 1, 1, 2, 3], equals , - -- + ---- 32 32 515 35 n The variance equals , - ---- + ---- 1024 1024 12081 669 n The , 3, -th moment about the mean is , - ----- + ----- 16384 16384 582287 3675 2 12475 The , 4, -th moment about the mean is , - ------- + ------- n - ------ n 1048576 1048576 262144 The compositions of, 10, that yield the, 18, -th largest growth, that is, 1.9417130342786384772, are , [3, 1, 1, 1, 1, 3] Theorem Number, 18, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [3, 1, 1, 1, 1, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 53 52 51 50 49 48 ) a(n) x = - (x + 10 x + 45 x + 120 x + 212 x + 269 x / ----- n = 0 47 46 45 44 43 42 41 + 275 x + 267 x + 263 x + 233 x + 164 x + 90 x + 41 x 40 39 38 37 36 35 34 33 32 + 12 x - 7 x - 12 x - 7 x - 3 x + x + 2 x - 2 x + 4 x 31 30 29 28 27 26 25 24 23 + 15 x + 5 x - 10 x - 3 x + 4 x - 4 x - 2 x + 6 x - 7 x 22 21 20 19 18 17 16 15 - 8 x + 22 x - 4 x - 32 x + 34 x + 4 x - 40 x + 45 x 14 13 12 11 10 9 8 7 - 13 x - 40 x + 75 x - 52 x - 31 x + 106 x - 64 x - 127 x 6 5 4 3 2 / 2 + 343 x - 417 x + 320 x - 164 x + 55 x - 11 x + 1) / ((x + 1) / 5 4 3 25 24 23 22 21 20 (x + x - x + 2 x - 1) (x + 5 x + 10 x + 10 x + 5 x + x 15 13 11 9 7 5 4 3 2 + 2 x - 5 x + 3 x + x - x + x - 5 x + 10 x - 10 x + 5 x - 1) ( 23 22 21 20 19 18 17 16 15 14 x + 5 x + 9 x + 5 x - 4 x - 4 x + 4 x + 4 x - 4 x - 4 x 13 12 10 9 8 7 6 5 4 3 + 3 x + 4 x - 4 x - 3 x + 4 x + 4 x - 4 x - 4 x + 4 x + 5 x 2 - 9 x + 5 x - 1)) and in Maple format -(x^53+10*x^52+45*x^51+120*x^50+212*x^49+269*x^48+275*x^47+267*x^46+263*x^45+ 233*x^44+164*x^43+90*x^42+41*x^41+12*x^40-7*x^39-12*x^38-7*x^37-3*x^36+x^35+2*x ^34-2*x^33+4*x^32+15*x^31+5*x^30-10*x^29-3*x^28+4*x^27-4*x^26-2*x^25+6*x^24-7*x ^23-8*x^22+22*x^21-4*x^20-32*x^19+34*x^18+4*x^17-40*x^16+45*x^15-13*x^14-40*x^ 13+75*x^12-52*x^11-31*x^10+106*x^9-64*x^8-127*x^7+343*x^6-417*x^5+320*x^4-164*x ^3+55*x^2-11*x+1)/(x^2+1)/(x^5+x^4-x^3+2*x-1)/(x^25+5*x^24+10*x^23+10*x^22+5*x^ 21+x^20+2*x^15-5*x^13+3*x^11+x^9-x^7+x^5-5*x^4+10*x^3-10*x^2+5*x-1)/(x^23+5*x^ 22+9*x^21+5*x^20-4*x^19-4*x^18+4*x^17+4*x^16-4*x^15-4*x^14+3*x^13+4*x^12-4*x^10 -3*x^9+4*x^8+4*x^7-4*x^6-4*x^5+4*x^4+5*x^3-9*x^2+5*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1016, 2010, 3956, 7749, 15117, 29394, 57011, 110376, 213447, 412535, 797274, 1541363, 2981763, 5772683, 11185005, 21688496, 42083745, 81702459, 158683471, 308280590] The limit of a(n+1)/a(n) as n goes to infinity is 1.94171303428 a(n) is asymptotic to .699765183073*1.94171303428^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [3, 1, 1, 1, 1, 3], denoted by the variable, X[3, 1, 1, 1, 1, 3], is 53 11 53 10 52 11 53 9 52 10 - (x %1 - 11 x %1 + 10 x %1 + 55 x %1 - 110 x %1 51 11 53 8 52 9 51 10 50 11 + 45 x %1 - 165 x %1 + 550 x %1 - 495 x %1 + 120 x %1 53 7 52 8 51 9 50 10 49 11 + 330 x %1 - 1650 x %1 + 2475 x %1 - 1320 x %1 + 210 x %1 53 6 52 7 51 8 50 9 49 10 - 462 x %1 + 3300 x %1 - 7425 x %1 + 6600 x %1 - 2312 x %1 48 11 53 5 52 6 51 7 + 252 x %1 + 462 x %1 - 4620 x %1 + 14850 x %1 50 8 49 9 48 10 47 11 - 19800 x %1 + 11570 x %1 - 2789 x %1 + 210 x %1 53 4 52 5 51 6 50 7 - 330 x %1 + 4620 x %1 - 20790 x %1 + 39600 x %1 49 8 48 9 47 10 46 11 - 34740 x %1 + 14030 x %1 - 2375 x %1 + 120 x %1 53 3 52 4 51 5 50 6 + 165 x %1 - 3300 x %1 + 20790 x %1 - 55440 x %1 49 7 48 8 47 9 46 10 + 69540 x %1 - 42345 x %1 + 12200 x %1 - 1467 x %1 45 11 53 2 52 3 51 4 50 5 + 45 x %1 - 55 x %1 + 1650 x %1 - 14850 x %1 + 55440 x %1 49 6 48 7 47 8 46 9 - 97440 x %1 + 85200 x %1 - 37575 x %1 + 8070 x %1 45 10 44 11 53 52 2 51 3 - 713 x %1 + 10 x %1 + 11 x %1 - 550 x %1 + 7425 x %1 50 4 49 5 48 6 47 7 - 39600 x %1 + 97524 x %1 - 119994 x %1 + 77100 x %1 46 8 45 9 44 10 43 11 53 - 26415 x %1 + 4655 x %1 - 333 x %1 + x %1 - x 52 51 2 50 3 49 4 + 110 x %1 - 2475 x %1 + 19800 x %1 - 69720 x %1 48 5 47 6 46 7 45 8 + 120708 x %1 - 110670 x %1 + 57240 x %1 - 17235 x %1 44 9 43 10 52 51 50 2 + 2780 x %1 - 172 x %1 - 10 x + 495 x %1 - 6600 x %1 49 3 48 4 47 5 46 6 + 34890 x %1 - 86730 x %1 + 113400 x %1 - 86310 x %1 45 7 44 8 43 9 42 10 51 + 41010 x %1 - 11685 x %1 + 1667 x %1 - 80 x %1 - 45 x 50 49 2 48 3 47 4 + 1320 x %1 - 11640 x %1 + 43620 x %1 - 82950 x %1 46 5 45 6 44 7 43 8 + 92484 x %1 - 66570 x %1 + 30060 x %1 - 7428 x %1 42 9 41 10 50 49 48 2 + 810 x %1 - 23 x %1 - 120 x + 2330 x %1 - 14625 x %1 47 3 46 4 45 5 44 6 + 42450 x %1 - 70470 x %1 + 75726 x %1 - 51450 x %1 43 7 42 8 41 9 40 10 49 + 19722 x %1 - 3690 x %1 + 250 x %1 - 3 x %1 - 212 x 48 47 2 46 3 45 4 + 2942 x %1 - 14475 x %1 + 37440 x %1 - 60630 x %1 44 5 43 6 42 7 41 8 40 9 + 60816 x %1 - 34440 x %1 + 9960 x %1 - 1213 x %1 + 49 x %1 39 10 48 47 46 2 45 3 - 6 x %1 - 269 x + 2960 x %1 - 13215 x %1 + 33585 x %1 44 4 43 5 42 6 41 7 - 50130 x %1 + 41286 x %1 - 17640 x %1 + 3464 x %1 40 8 39 9 38 10 47 46 - 296 x %1 + 63 x %1 - 9 x %1 - 275 x + 2790 x %1 45 2 44 3 43 4 42 5 - 12285 x %1 + 28410 x %1 - 34392 x %1 + 21420 x %1 41 6 40 7 39 8 38 9 37 10 - 6454 x %1 + 964 x %1 - 281 x %1 + 73 x %1 - 5 x %1 46 45 44 2 43 3 42 4 - 267 x + 2675 x %1 - 10585 x %1 + 19653 x %1 - 18060 x %1 41 5 40 6 39 7 38 8 37 9 + 8204 x %1 - 1946 x %1 + 700 x %1 - 248 x %1 + 25 x %1 36 10 45 44 43 2 42 3 - x %1 - 263 x + 2340 x %1 - 7372 x %1 + 10440 x %1 41 4 40 5 39 6 38 7 37 8 - 7210 x %1 + 2590 x %1 - 1064 x %1 + 436 x %1 - 15 x %1 36 9 44 43 42 2 41 3 - 5 x %1 - 233 x + 1639 x %1 - 3960 x %1 + 4328 x %1 40 4 39 5 38 6 37 7 36 8 - 2324 x %1 + 994 x %1 - 350 x %1 - 182 x %1 + 72 x %1 35 9 43 42 41 2 40 3 - x %1 - 164 x + 890 x %1 - 1699 x %1 + 1396 x %1 39 4 38 5 37 6 36 7 34 9 - 518 x %1 - 98 x %1 + 644 x %1 - 283 x %1 + 3 x %1 42 41 40 2 39 3 38 4 - 90 x + 394 x %1 - 539 x %1 + 76 x %1 + 532 x %1 37 5 36 6 35 7 34 8 33 9 41 - 1092 x %1 + 595 x %1 + 30 x %1 - 28 x %1 + x %1 - 41 x 40 39 2 38 3 37 4 36 5 + 121 x %1 + 70 x %1 - 572 x %1 + 1120 x %1 - 777 x %1 35 6 34 7 33 8 40 39 38 2 - 125 x %1 + 115 x %1 - 7 x %1 - 12 x - 41 x %1 + 319 x %1 37 3 36 4 35 5 34 6 33 7 - 730 x %1 + 665 x %1 + 246 x %1 - 272 x %1 + 14 x %1 32 8 39 38 37 2 36 3 35 4 + 4 x %1 + 7 x - 95 x %1 + 297 x %1 - 377 x %1 - 279 x %1 34 5 33 6 32 7 31 8 38 37 + 407 x %1 + 9 x %1 - 32 x %1 + 6 x %1 + 12 x - 69 x %1 36 2 35 3 34 4 33 5 32 6 + 138 x %1 + 190 x %1 - 398 x %1 - 82 x %1 + 104 x %1 31 7 30 8 37 36 35 2 34 3 - 33 x %1 + x %1 + 7 x - 30 x %1 - 75 x %1 + 253 x %1 33 4 32 5 31 6 29 8 36 35 + 149 x %1 - 176 x %1 + 56 x %1 - 3 x %1 + 3 x + 15 x %1 34 2 33 3 32 4 31 5 30 6 - 100 x %1 - 138 x %1 + 160 x %1 + 20 x %1 - 25 x %1 29 7 35 34 33 2 32 3 31 4 + 24 x %1 - x + 22 x %1 + 71 x %1 - 64 x %1 - 205 x %1 30 5 29 6 28 7 27 8 34 33 + 95 x %1 - 71 x %1 - 2 x %1 + x %1 - 2 x - 19 x %1 32 2 31 3 30 4 29 5 28 6 - 8 x %1 + 319 x %1 - 170 x %1 + 90 x %1 + 12 x %1 27 7 33 32 31 2 30 3 29 4 - 11 x %1 + 2 x + 16 x %1 - 242 x %1 + 174 x %1 - 15 x %1 28 5 27 6 26 7 32 31 30 2 - 33 x %1 + 48 x %1 + 3 x %1 - 4 x + 94 x %1 - 105 x %1 29 3 28 4 27 5 26 6 25 7 31 - 92 x %1 + 55 x %1 - 107 x %1 - 14 x %1 + 2 x %1 - 15 x 30 29 2 28 3 27 4 26 5 + 35 x %1 + 111 x %1 - 60 x %1 + 129 x %1 + 23 x %1 25 6 24 7 30 29 28 2 27 3 - 18 x %1 - x %1 - 5 x - 54 x %1 + 42 x %1 - 77 x %1 26 4 25 5 24 6 29 28 27 2 - 8 x %1 + 62 x %1 + 7 x %1 + 10 x - 17 x %1 + 10 x %1 26 3 25 4 24 5 23 6 28 27 - 23 x %1 - 106 x %1 - 25 x %1 + 3 x %1 + 3 x + 11 x %1 26 2 25 3 24 4 23 5 22 6 27 + 34 x %1 + 94 x %1 + 44 x %1 - 19 x %1 - x %1 - 4 x 26 25 2 24 3 23 4 22 5 26 - 19 x %1 - 38 x %1 - 31 x %1 + 54 x %1 + 14 x %1 + 4 x 25 24 2 23 3 22 4 21 5 25 + 2 x %1 - 5 x %1 - 85 x %1 - 43 x %1 - 9 x %1 + 2 x 24 23 2 22 3 21 4 20 5 24 + 17 x %1 + 76 x %1 + 47 x %1 + 20 x %1 + 6 x %1 - 6 x 23 22 2 21 3 20 4 19 5 23 - 36 x %1 - 8 x %1 + 16 x %1 - 4 x %1 + 4 x %1 + 7 x 22 21 2 20 3 19 4 18 5 22 - 17 x %1 - 78 x %1 - 28 x %1 - 36 x %1 - 9 x %1 + 8 x 21 20 2 19 3 18 4 17 5 21 + 73 x %1 + 48 x %1 + 50 x %1 + 48 x %1 + 5 x %1 - 22 x 20 19 2 18 3 17 4 16 5 20 - 26 x %1 + 24 x %1 - 44 x %1 - 11 x %1 - x %1 + 4 x 19 18 2 17 3 16 4 19 18 - 74 x %1 - 54 x %1 - 23 x %1 - 20 x %1 + 32 x + 93 x %1 17 2 16 3 15 4 18 17 16 2 + 55 x %1 + 65 x %1 + 19 x %1 - 34 x - 22 x %1 - 26 x %1 15 3 14 4 17 16 15 2 14 3 - 34 x %1 - 7 x %1 - 4 x - 58 x %1 - 35 x %1 - 9 x %1 13 4 16 15 14 2 13 3 15 + x %1 + 40 x + 95 x %1 + 60 x %1 + 17 x %1 - 45 x 14 13 2 12 3 14 13 12 2 - 57 x %1 - 24 x %1 - 7 x %1 + 13 x - 34 x %1 - 14 x %1 11 3 13 12 11 2 12 11 + x %1 + 40 x + 96 x %1 + 18 x %1 - 75 x - 70 x %1 10 2 11 10 9 2 10 9 9 - 7 x %1 + 52 x - 35 x %1 + x %1 + 31 x + 160 x %1 - 106 x 8 8 7 7 6 6 5 - 229 x %1 + 64 x + 203 x %1 + 127 x - 119 x %1 - 343 x + 45 x %1 5 4 4 3 3 2 / + 417 x - 10 x %1 - 320 x + x %1 + 164 x - 55 x + 11 x - 1) / ( / 5 5 4 4 3 3 25 5 25 4 (x %1 - x + x %1 - x - x %1 + x - 2 x + 1) (x %1 - 5 x %1 24 5 25 3 24 4 23 5 25 2 + 5 x %1 + 10 x %1 - 25 x %1 + 10 x %1 - 10 x %1 24 3 23 4 22 5 25 24 2 + 50 x %1 - 50 x %1 + 10 x %1 + 5 x %1 - 50 x %1 23 3 22 4 21 5 25 24 23 2 + 100 x %1 - 50 x %1 + 5 x %1 - x + 25 x %1 - 100 x %1 22 3 21 4 20 5 24 23 22 2 + 100 x %1 - 25 x %1 + x %1 - 5 x + 50 x %1 - 100 x %1 21 3 20 4 23 22 21 2 20 3 + 50 x %1 - 5 x %1 - 10 x + 50 x %1 - 50 x %1 + 10 x %1 22 21 20 2 21 20 20 15 4 - 10 x + 25 x %1 - 10 x %1 - 5 x + 5 x %1 - x + x %1 15 3 15 2 13 4 15 13 3 15 13 2 - x %1 - 3 x %1 - x %1 + 5 x %1 + x %1 - 2 x + 6 x %1 13 13 11 2 11 11 9 9 7 - 11 x %1 + 5 x - 3 x %1 + 6 x %1 - 3 x + x %1 - x - x %1 7 5 4 3 2 25 5 25 4 + x - x + 5 x - 10 x + 10 x - 5 x + 1) (x %1 - 5 x %1 24 5 25 3 24 4 23 5 25 2 + 5 x %1 + 10 x %1 - 25 x %1 + 10 x %1 - 10 x %1 24 3 23 4 22 5 25 24 2 + 50 x %1 - 50 x %1 + 10 x %1 + 5 x %1 - 50 x %1 23 3 22 4 21 5 25 24 23 2 + 100 x %1 - 50 x %1 + 5 x %1 - x + 25 x %1 - 100 x %1 22 3 21 4 20 5 24 23 22 2 + 100 x %1 - 25 x %1 + x %1 - 5 x + 50 x %1 - 100 x %1 21 3 20 4 23 22 21 2 20 3 + 50 x %1 - 5 x %1 - 10 x + 50 x %1 - 50 x %1 + 10 x %1 22 21 20 2 21 20 20 15 3 - 10 x + 25 x %1 - 10 x %1 - 5 x + 5 x %1 - x - x %1 15 2 15 13 3 15 13 2 13 11 3 + 3 x %1 - 3 x %1 + 2 x %1 + x - 7 x %1 + 8 x %1 - x %1 13 11 2 11 11 9 2 9 9 5 4 - 3 x + 5 x %1 - 7 x %1 + 3 x - x %1 + 2 x %1 - x - x + 5 x 3 2 - 10 x + 10 x - 5 x + 1)) %1 := X[3, 1, 1, 1, 1, 3] and in Maple format -(x^53*X[3,1,1,1,1,3]^11-11*x^53*X[3,1,1,1,1,3]^10+10*x^52*X[3,1,1,1,1,3]^11+55 *x^53*X[3,1,1,1,1,3]^9-110*x^52*X[3,1,1,1,1,3]^10+45*x^51*X[3,1,1,1,1,3]^11-165 *x^53*X[3,1,1,1,1,3]^8+550*x^52*X[3,1,1,1,1,3]^9-495*x^51*X[3,1,1,1,1,3]^10+120 *x^50*X[3,1,1,1,1,3]^11+330*x^53*X[3,1,1,1,1,3]^7-1650*x^52*X[3,1,1,1,1,3]^8+ 2475*x^51*X[3,1,1,1,1,3]^9-1320*x^50*X[3,1,1,1,1,3]^10+210*x^49*X[3,1,1,1,1,3]^ 11-462*x^53*X[3,1,1,1,1,3]^6+3300*x^52*X[3,1,1,1,1,3]^7-7425*x^51*X[3,1,1,1,1,3 ]^8+6600*x^50*X[3,1,1,1,1,3]^9-2312*x^49*X[3,1,1,1,1,3]^10+252*x^48*X[3,1,1,1,1 ,3]^11+462*x^53*X[3,1,1,1,1,3]^5-4620*x^52*X[3,1,1,1,1,3]^6+14850*x^51*X[3,1,1, 1,1,3]^7-19800*x^50*X[3,1,1,1,1,3]^8+11570*x^49*X[3,1,1,1,1,3]^9-2789*x^48*X[3, 1,1,1,1,3]^10+210*x^47*X[3,1,1,1,1,3]^11-330*x^53*X[3,1,1,1,1,3]^4+4620*x^52*X[ 3,1,1,1,1,3]^5-20790*x^51*X[3,1,1,1,1,3]^6+39600*x^50*X[3,1,1,1,1,3]^7-34740*x^ 49*X[3,1,1,1,1,3]^8+14030*x^48*X[3,1,1,1,1,3]^9-2375*x^47*X[3,1,1,1,1,3]^10+120 *x^46*X[3,1,1,1,1,3]^11+165*x^53*X[3,1,1,1,1,3]^3-3300*x^52*X[3,1,1,1,1,3]^4+ 20790*x^51*X[3,1,1,1,1,3]^5-55440*x^50*X[3,1,1,1,1,3]^6+69540*x^49*X[3,1,1,1,1, 3]^7-42345*x^48*X[3,1,1,1,1,3]^8+12200*x^47*X[3,1,1,1,1,3]^9-1467*x^46*X[3,1,1, 1,1,3]^10+45*x^45*X[3,1,1,1,1,3]^11-55*x^53*X[3,1,1,1,1,3]^2+1650*x^52*X[3,1,1, 1,1,3]^3-14850*x^51*X[3,1,1,1,1,3]^4+55440*x^50*X[3,1,1,1,1,3]^5-97440*x^49*X[3 ,1,1,1,1,3]^6+85200*x^48*X[3,1,1,1,1,3]^7-37575*x^47*X[3,1,1,1,1,3]^8+8070*x^46 *X[3,1,1,1,1,3]^9-713*x^45*X[3,1,1,1,1,3]^10+10*x^44*X[3,1,1,1,1,3]^11+11*x^53* X[3,1,1,1,1,3]-550*x^52*X[3,1,1,1,1,3]^2+7425*x^51*X[3,1,1,1,1,3]^3-39600*x^50* X[3,1,1,1,1,3]^4+97524*x^49*X[3,1,1,1,1,3]^5-119994*x^48*X[3,1,1,1,1,3]^6+77100 *x^47*X[3,1,1,1,1,3]^7-26415*x^46*X[3,1,1,1,1,3]^8+4655*x^45*X[3,1,1,1,1,3]^9-\ 333*x^44*X[3,1,1,1,1,3]^10+x^43*X[3,1,1,1,1,3]^11-x^53+110*x^52*X[3,1,1,1,1,3]-\ 2475*x^51*X[3,1,1,1,1,3]^2+19800*x^50*X[3,1,1,1,1,3]^3-69720*x^49*X[3,1,1,1,1,3 ]^4+120708*x^48*X[3,1,1,1,1,3]^5-110670*x^47*X[3,1,1,1,1,3]^6+57240*x^46*X[3,1, 1,1,1,3]^7-17235*x^45*X[3,1,1,1,1,3]^8+2780*x^44*X[3,1,1,1,1,3]^9-172*x^43*X[3, 1,1,1,1,3]^10-10*x^52+495*x^51*X[3,1,1,1,1,3]-6600*x^50*X[3,1,1,1,1,3]^2+34890* x^49*X[3,1,1,1,1,3]^3-86730*x^48*X[3,1,1,1,1,3]^4+113400*x^47*X[3,1,1,1,1,3]^5-\ 86310*x^46*X[3,1,1,1,1,3]^6+41010*x^45*X[3,1,1,1,1,3]^7-11685*x^44*X[3,1,1,1,1, 3]^8+1667*x^43*X[3,1,1,1,1,3]^9-80*x^42*X[3,1,1,1,1,3]^10-45*x^51+1320*x^50*X[3 ,1,1,1,1,3]-11640*x^49*X[3,1,1,1,1,3]^2+43620*x^48*X[3,1,1,1,1,3]^3-82950*x^47* X[3,1,1,1,1,3]^4+92484*x^46*X[3,1,1,1,1,3]^5-66570*x^45*X[3,1,1,1,1,3]^6+30060* x^44*X[3,1,1,1,1,3]^7-7428*x^43*X[3,1,1,1,1,3]^8+810*x^42*X[3,1,1,1,1,3]^9-23*x ^41*X[3,1,1,1,1,3]^10-120*x^50+2330*x^49*X[3,1,1,1,1,3]-14625*x^48*X[3,1,1,1,1, 3]^2+42450*x^47*X[3,1,1,1,1,3]^3-70470*x^46*X[3,1,1,1,1,3]^4+75726*x^45*X[3,1,1 ,1,1,3]^5-51450*x^44*X[3,1,1,1,1,3]^6+19722*x^43*X[3,1,1,1,1,3]^7-3690*x^42*X[3 ,1,1,1,1,3]^8+250*x^41*X[3,1,1,1,1,3]^9-3*x^40*X[3,1,1,1,1,3]^10-212*x^49+2942* x^48*X[3,1,1,1,1,3]-14475*x^47*X[3,1,1,1,1,3]^2+37440*x^46*X[3,1,1,1,1,3]^3-\ 60630*x^45*X[3,1,1,1,1,3]^4+60816*x^44*X[3,1,1,1,1,3]^5-34440*x^43*X[3,1,1,1,1, 3]^6+9960*x^42*X[3,1,1,1,1,3]^7-1213*x^41*X[3,1,1,1,1,3]^8+49*x^40*X[3,1,1,1,1, 3]^9-6*x^39*X[3,1,1,1,1,3]^10-269*x^48+2960*x^47*X[3,1,1,1,1,3]-13215*x^46*X[3, 1,1,1,1,3]^2+33585*x^45*X[3,1,1,1,1,3]^3-50130*x^44*X[3,1,1,1,1,3]^4+41286*x^43 *X[3,1,1,1,1,3]^5-17640*x^42*X[3,1,1,1,1,3]^6+3464*x^41*X[3,1,1,1,1,3]^7-296*x^ 40*X[3,1,1,1,1,3]^8+63*x^39*X[3,1,1,1,1,3]^9-9*x^38*X[3,1,1,1,1,3]^10-275*x^47+ 2790*x^46*X[3,1,1,1,1,3]-12285*x^45*X[3,1,1,1,1,3]^2+28410*x^44*X[3,1,1,1,1,3]^ 3-34392*x^43*X[3,1,1,1,1,3]^4+21420*x^42*X[3,1,1,1,1,3]^5-6454*x^41*X[3,1,1,1,1 ,3]^6+964*x^40*X[3,1,1,1,1,3]^7-281*x^39*X[3,1,1,1,1,3]^8+73*x^38*X[3,1,1,1,1,3 ]^9-5*x^37*X[3,1,1,1,1,3]^10-267*x^46+2675*x^45*X[3,1,1,1,1,3]-10585*x^44*X[3,1 ,1,1,1,3]^2+19653*x^43*X[3,1,1,1,1,3]^3-18060*x^42*X[3,1,1,1,1,3]^4+8204*x^41*X [3,1,1,1,1,3]^5-1946*x^40*X[3,1,1,1,1,3]^6+700*x^39*X[3,1,1,1,1,3]^7-248*x^38*X [3,1,1,1,1,3]^8+25*x^37*X[3,1,1,1,1,3]^9-x^36*X[3,1,1,1,1,3]^10-263*x^45+2340*x ^44*X[3,1,1,1,1,3]-7372*x^43*X[3,1,1,1,1,3]^2+10440*x^42*X[3,1,1,1,1,3]^3-7210* x^41*X[3,1,1,1,1,3]^4+2590*x^40*X[3,1,1,1,1,3]^5-1064*x^39*X[3,1,1,1,1,3]^6+436 *x^38*X[3,1,1,1,1,3]^7-15*x^37*X[3,1,1,1,1,3]^8-5*x^36*X[3,1,1,1,1,3]^9-233*x^ 44+1639*x^43*X[3,1,1,1,1,3]-3960*x^42*X[3,1,1,1,1,3]^2+4328*x^41*X[3,1,1,1,1,3] ^3-2324*x^40*X[3,1,1,1,1,3]^4+994*x^39*X[3,1,1,1,1,3]^5-350*x^38*X[3,1,1,1,1,3] ^6-182*x^37*X[3,1,1,1,1,3]^7+72*x^36*X[3,1,1,1,1,3]^8-x^35*X[3,1,1,1,1,3]^9-164 *x^43+890*x^42*X[3,1,1,1,1,3]-1699*x^41*X[3,1,1,1,1,3]^2+1396*x^40*X[3,1,1,1,1, 3]^3-518*x^39*X[3,1,1,1,1,3]^4-98*x^38*X[3,1,1,1,1,3]^5+644*x^37*X[3,1,1,1,1,3] ^6-283*x^36*X[3,1,1,1,1,3]^7+3*x^34*X[3,1,1,1,1,3]^9-90*x^42+394*x^41*X[3,1,1,1 ,1,3]-539*x^40*X[3,1,1,1,1,3]^2+76*x^39*X[3,1,1,1,1,3]^3+532*x^38*X[3,1,1,1,1,3 ]^4-1092*x^37*X[3,1,1,1,1,3]^5+595*x^36*X[3,1,1,1,1,3]^6+30*x^35*X[3,1,1,1,1,3] ^7-28*x^34*X[3,1,1,1,1,3]^8+x^33*X[3,1,1,1,1,3]^9-41*x^41+121*x^40*X[3,1,1,1,1, 3]+70*x^39*X[3,1,1,1,1,3]^2-572*x^38*X[3,1,1,1,1,3]^3+1120*x^37*X[3,1,1,1,1,3]^ 4-777*x^36*X[3,1,1,1,1,3]^5-125*x^35*X[3,1,1,1,1,3]^6+115*x^34*X[3,1,1,1,1,3]^7 -7*x^33*X[3,1,1,1,1,3]^8-12*x^40-41*x^39*X[3,1,1,1,1,3]+319*x^38*X[3,1,1,1,1,3] ^2-730*x^37*X[3,1,1,1,1,3]^3+665*x^36*X[3,1,1,1,1,3]^4+246*x^35*X[3,1,1,1,1,3]^ 5-272*x^34*X[3,1,1,1,1,3]^6+14*x^33*X[3,1,1,1,1,3]^7+4*x^32*X[3,1,1,1,1,3]^8+7* x^39-95*x^38*X[3,1,1,1,1,3]+297*x^37*X[3,1,1,1,1,3]^2-377*x^36*X[3,1,1,1,1,3]^3 -279*x^35*X[3,1,1,1,1,3]^4+407*x^34*X[3,1,1,1,1,3]^5+9*x^33*X[3,1,1,1,1,3]^6-32 *x^32*X[3,1,1,1,1,3]^7+6*x^31*X[3,1,1,1,1,3]^8+12*x^38-69*x^37*X[3,1,1,1,1,3]+ 138*x^36*X[3,1,1,1,1,3]^2+190*x^35*X[3,1,1,1,1,3]^3-398*x^34*X[3,1,1,1,1,3]^4-\ 82*x^33*X[3,1,1,1,1,3]^5+104*x^32*X[3,1,1,1,1,3]^6-33*x^31*X[3,1,1,1,1,3]^7+x^ 30*X[3,1,1,1,1,3]^8+7*x^37-30*x^36*X[3,1,1,1,1,3]-75*x^35*X[3,1,1,1,1,3]^2+253* x^34*X[3,1,1,1,1,3]^3+149*x^33*X[3,1,1,1,1,3]^4-176*x^32*X[3,1,1,1,1,3]^5+56*x^ 31*X[3,1,1,1,1,3]^6-3*x^29*X[3,1,1,1,1,3]^8+3*x^36+15*x^35*X[3,1,1,1,1,3]-100*x ^34*X[3,1,1,1,1,3]^2-138*x^33*X[3,1,1,1,1,3]^3+160*x^32*X[3,1,1,1,1,3]^4+20*x^ 31*X[3,1,1,1,1,3]^5-25*x^30*X[3,1,1,1,1,3]^6+24*x^29*X[3,1,1,1,1,3]^7-x^35+22*x ^34*X[3,1,1,1,1,3]+71*x^33*X[3,1,1,1,1,3]^2-64*x^32*X[3,1,1,1,1,3]^3-205*x^31*X [3,1,1,1,1,3]^4+95*x^30*X[3,1,1,1,1,3]^5-71*x^29*X[3,1,1,1,1,3]^6-2*x^28*X[3,1, 1,1,1,3]^7+x^27*X[3,1,1,1,1,3]^8-2*x^34-19*x^33*X[3,1,1,1,1,3]-8*x^32*X[3,1,1,1 ,1,3]^2+319*x^31*X[3,1,1,1,1,3]^3-170*x^30*X[3,1,1,1,1,3]^4+90*x^29*X[3,1,1,1,1 ,3]^5+12*x^28*X[3,1,1,1,1,3]^6-11*x^27*X[3,1,1,1,1,3]^7+2*x^33+16*x^32*X[3,1,1, 1,1,3]-242*x^31*X[3,1,1,1,1,3]^2+174*x^30*X[3,1,1,1,1,3]^3-15*x^29*X[3,1,1,1,1, 3]^4-33*x^28*X[3,1,1,1,1,3]^5+48*x^27*X[3,1,1,1,1,3]^6+3*x^26*X[3,1,1,1,1,3]^7-\ 4*x^32+94*x^31*X[3,1,1,1,1,3]-105*x^30*X[3,1,1,1,1,3]^2-92*x^29*X[3,1,1,1,1,3]^ 3+55*x^28*X[3,1,1,1,1,3]^4-107*x^27*X[3,1,1,1,1,3]^5-14*x^26*X[3,1,1,1,1,3]^6+2 *x^25*X[3,1,1,1,1,3]^7-15*x^31+35*x^30*X[3,1,1,1,1,3]+111*x^29*X[3,1,1,1,1,3]^2 -60*x^28*X[3,1,1,1,1,3]^3+129*x^27*X[3,1,1,1,1,3]^4+23*x^26*X[3,1,1,1,1,3]^5-18 *x^25*X[3,1,1,1,1,3]^6-x^24*X[3,1,1,1,1,3]^7-5*x^30-54*x^29*X[3,1,1,1,1,3]+42*x ^28*X[3,1,1,1,1,3]^2-77*x^27*X[3,1,1,1,1,3]^3-8*x^26*X[3,1,1,1,1,3]^4+62*x^25*X [3,1,1,1,1,3]^5+7*x^24*X[3,1,1,1,1,3]^6+10*x^29-17*x^28*X[3,1,1,1,1,3]+10*x^27* X[3,1,1,1,1,3]^2-23*x^26*X[3,1,1,1,1,3]^3-106*x^25*X[3,1,1,1,1,3]^4-25*x^24*X[3 ,1,1,1,1,3]^5+3*x^23*X[3,1,1,1,1,3]^6+3*x^28+11*x^27*X[3,1,1,1,1,3]+34*x^26*X[3 ,1,1,1,1,3]^2+94*x^25*X[3,1,1,1,1,3]^3+44*x^24*X[3,1,1,1,1,3]^4-19*x^23*X[3,1,1 ,1,1,3]^5-x^22*X[3,1,1,1,1,3]^6-4*x^27-19*x^26*X[3,1,1,1,1,3]-38*x^25*X[3,1,1,1 ,1,3]^2-31*x^24*X[3,1,1,1,1,3]^3+54*x^23*X[3,1,1,1,1,3]^4+14*x^22*X[3,1,1,1,1,3 ]^5+4*x^26+2*x^25*X[3,1,1,1,1,3]-5*x^24*X[3,1,1,1,1,3]^2-85*x^23*X[3,1,1,1,1,3] ^3-43*x^22*X[3,1,1,1,1,3]^4-9*x^21*X[3,1,1,1,1,3]^5+2*x^25+17*x^24*X[3,1,1,1,1, 3]+76*x^23*X[3,1,1,1,1,3]^2+47*x^22*X[3,1,1,1,1,3]^3+20*x^21*X[3,1,1,1,1,3]^4+6 *x^20*X[3,1,1,1,1,3]^5-6*x^24-36*x^23*X[3,1,1,1,1,3]-8*x^22*X[3,1,1,1,1,3]^2+16 *x^21*X[3,1,1,1,1,3]^3-4*x^20*X[3,1,1,1,1,3]^4+4*x^19*X[3,1,1,1,1,3]^5+7*x^23-\ 17*x^22*X[3,1,1,1,1,3]-78*x^21*X[3,1,1,1,1,3]^2-28*x^20*X[3,1,1,1,1,3]^3-36*x^ 19*X[3,1,1,1,1,3]^4-9*x^18*X[3,1,1,1,1,3]^5+8*x^22+73*x^21*X[3,1,1,1,1,3]+48*x^ 20*X[3,1,1,1,1,3]^2+50*x^19*X[3,1,1,1,1,3]^3+48*x^18*X[3,1,1,1,1,3]^4+5*x^17*X[ 3,1,1,1,1,3]^5-22*x^21-26*x^20*X[3,1,1,1,1,3]+24*x^19*X[3,1,1,1,1,3]^2-44*x^18* X[3,1,1,1,1,3]^3-11*x^17*X[3,1,1,1,1,3]^4-x^16*X[3,1,1,1,1,3]^5+4*x^20-74*x^19* X[3,1,1,1,1,3]-54*x^18*X[3,1,1,1,1,3]^2-23*x^17*X[3,1,1,1,1,3]^3-20*x^16*X[3,1, 1,1,1,3]^4+32*x^19+93*x^18*X[3,1,1,1,1,3]+55*x^17*X[3,1,1,1,1,3]^2+65*x^16*X[3, 1,1,1,1,3]^3+19*x^15*X[3,1,1,1,1,3]^4-34*x^18-22*x^17*X[3,1,1,1,1,3]-26*x^16*X[ 3,1,1,1,1,3]^2-34*x^15*X[3,1,1,1,1,3]^3-7*x^14*X[3,1,1,1,1,3]^4-4*x^17-58*x^16* X[3,1,1,1,1,3]-35*x^15*X[3,1,1,1,1,3]^2-9*x^14*X[3,1,1,1,1,3]^3+x^13*X[3,1,1,1, 1,3]^4+40*x^16+95*x^15*X[3,1,1,1,1,3]+60*x^14*X[3,1,1,1,1,3]^2+17*x^13*X[3,1,1, 1,1,3]^3-45*x^15-57*x^14*X[3,1,1,1,1,3]-24*x^13*X[3,1,1,1,1,3]^2-7*x^12*X[3,1,1 ,1,1,3]^3+13*x^14-34*x^13*X[3,1,1,1,1,3]-14*x^12*X[3,1,1,1,1,3]^2+x^11*X[3,1,1, 1,1,3]^3+40*x^13+96*x^12*X[3,1,1,1,1,3]+18*x^11*X[3,1,1,1,1,3]^2-75*x^12-70*x^ 11*X[3,1,1,1,1,3]-7*x^10*X[3,1,1,1,1,3]^2+52*x^11-35*x^10*X[3,1,1,1,1,3]+x^9*X[ 3,1,1,1,1,3]^2+31*x^10+160*x^9*X[3,1,1,1,1,3]-106*x^9-229*x^8*X[3,1,1,1,1,3]+64 *x^8+203*x^7*X[3,1,1,1,1,3]+127*x^7-119*x^6*X[3,1,1,1,1,3]-343*x^6+45*x^5*X[3,1 ,1,1,1,3]+417*x^5-10*x^4*X[3,1,1,1,1,3]-320*x^4+x^3*X[3,1,1,1,1,3]+164*x^3-55*x ^2+11*x-1)/(x^5*X[3,1,1,1,1,3]-x^5+x^4*X[3,1,1,1,1,3]-x^4-x^3*X[3,1,1,1,1,3]+x^ 3-2*x+1)/(x^25*X[3,1,1,1,1,3]^5-5*x^25*X[3,1,1,1,1,3]^4+5*x^24*X[3,1,1,1,1,3]^5 +10*x^25*X[3,1,1,1,1,3]^3-25*x^24*X[3,1,1,1,1,3]^4+10*x^23*X[3,1,1,1,1,3]^5-10* x^25*X[3,1,1,1,1,3]^2+50*x^24*X[3,1,1,1,1,3]^3-50*x^23*X[3,1,1,1,1,3]^4+10*x^22 *X[3,1,1,1,1,3]^5+5*x^25*X[3,1,1,1,1,3]-50*x^24*X[3,1,1,1,1,3]^2+100*x^23*X[3,1 ,1,1,1,3]^3-50*x^22*X[3,1,1,1,1,3]^4+5*x^21*X[3,1,1,1,1,3]^5-x^25+25*x^24*X[3,1 ,1,1,1,3]-100*x^23*X[3,1,1,1,1,3]^2+100*x^22*X[3,1,1,1,1,3]^3-25*x^21*X[3,1,1,1 ,1,3]^4+x^20*X[3,1,1,1,1,3]^5-5*x^24+50*x^23*X[3,1,1,1,1,3]-100*x^22*X[3,1,1,1, 1,3]^2+50*x^21*X[3,1,1,1,1,3]^3-5*x^20*X[3,1,1,1,1,3]^4-10*x^23+50*x^22*X[3,1,1 ,1,1,3]-50*x^21*X[3,1,1,1,1,3]^2+10*x^20*X[3,1,1,1,1,3]^3-10*x^22+25*x^21*X[3,1 ,1,1,1,3]-10*x^20*X[3,1,1,1,1,3]^2-5*x^21+5*x^20*X[3,1,1,1,1,3]-x^20+x^15*X[3,1 ,1,1,1,3]^4-x^15*X[3,1,1,1,1,3]^3-3*x^15*X[3,1,1,1,1,3]^2-x^13*X[3,1,1,1,1,3]^4 +5*x^15*X[3,1,1,1,1,3]+x^13*X[3,1,1,1,1,3]^3-2*x^15+6*x^13*X[3,1,1,1,1,3]^2-11* x^13*X[3,1,1,1,1,3]+5*x^13-3*x^11*X[3,1,1,1,1,3]^2+6*x^11*X[3,1,1,1,1,3]-3*x^11 +x^9*X[3,1,1,1,1,3]-x^9-x^7*X[3,1,1,1,1,3]+x^7-x^5+5*x^4-10*x^3+10*x^2-5*x+1)/( x^25*X[3,1,1,1,1,3]^5-5*x^25*X[3,1,1,1,1,3]^4+5*x^24*X[3,1,1,1,1,3]^5+10*x^25*X [3,1,1,1,1,3]^3-25*x^24*X[3,1,1,1,1,3]^4+10*x^23*X[3,1,1,1,1,3]^5-10*x^25*X[3,1 ,1,1,1,3]^2+50*x^24*X[3,1,1,1,1,3]^3-50*x^23*X[3,1,1,1,1,3]^4+10*x^22*X[3,1,1,1 ,1,3]^5+5*x^25*X[3,1,1,1,1,3]-50*x^24*X[3,1,1,1,1,3]^2+100*x^23*X[3,1,1,1,1,3]^ 3-50*x^22*X[3,1,1,1,1,3]^4+5*x^21*X[3,1,1,1,1,3]^5-x^25+25*x^24*X[3,1,1,1,1,3]-\ 100*x^23*X[3,1,1,1,1,3]^2+100*x^22*X[3,1,1,1,1,3]^3-25*x^21*X[3,1,1,1,1,3]^4+x^ 20*X[3,1,1,1,1,3]^5-5*x^24+50*x^23*X[3,1,1,1,1,3]-100*x^22*X[3,1,1,1,1,3]^2+50* x^21*X[3,1,1,1,1,3]^3-5*x^20*X[3,1,1,1,1,3]^4-10*x^23+50*x^22*X[3,1,1,1,1,3]-50 *x^21*X[3,1,1,1,1,3]^2+10*x^20*X[3,1,1,1,1,3]^3-10*x^22+25*x^21*X[3,1,1,1,1,3]-\ 10*x^20*X[3,1,1,1,1,3]^2-5*x^21+5*x^20*X[3,1,1,1,1,3]-x^20-x^15*X[3,1,1,1,1,3]^ 3+3*x^15*X[3,1,1,1,1,3]^2-3*x^15*X[3,1,1,1,1,3]+2*x^13*X[3,1,1,1,1,3]^3+x^15-7* x^13*X[3,1,1,1,1,3]^2+8*x^13*X[3,1,1,1,1,3]-x^11*X[3,1,1,1,1,3]^3-3*x^13+5*x^11 *X[3,1,1,1,1,3]^2-7*x^11*X[3,1,1,1,1,3]+3*x^11-x^9*X[3,1,1,1,1,3]^2+2*x^9*X[3,1 ,1,1,1,3]-x^9-x^5+5*x^4-10*x^3+10*x^2-5*x+1) Furthermore, the expectation of the random variable number of occurences (by\ 13 n containment) of , [3, 1, 1, 1, 1, 3], equals , - -- + ---- 32 32 563 35 n The variance equals , - ---- + ---- 1024 1024 13749 633 n The , 3, -th moment about the mean is , - ----- + ----- 16384 16384 508559 3675 2 18331 The , 4, -th moment about the mean is , - ------- + ------- n - ------ n 1048576 1048576 262144 The compositions of, 10, that yield the, 19, -th largest growth, that is, 1.9454365275632690792, are , [1, 2, 1, 2, 1, 3], [1, 2, 1, 3, 1, 2], [1, 3, 1, 2, 1, 2], [2, 1, 2, 1, 3, 1], [2, 1, 3, 1, 2, 1], [3, 1, 2, 1, 2, 1] Theorem Number, 19, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 2, 1, 2, 1, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 19 18 17 15 14 13 12 11 ) a(n) x = (x - x + 2 x - 3 x + 6 x - 4 x + x - 5 x / ----- n = 0 10 9 8 7 6 5 4 3 2 + 16 x - 20 x + 13 x - 7 x + 3 x + 11 x - 30 x + 34 x - 21 x / 6 5 4 3 2 + 7 x - 1) / ((-1 + x) (x - x + 2 x - x - 2 x + 3 x - 1) / 12 10 9 8 6 5 4 3 2 (x - x + x + x - 3 x + 3 x - 2 x + 4 x - 6 x + 4 x - 1)) and in Maple format (x^19-x^18+2*x^17-3*x^15+6*x^14-4*x^13+x^12-5*x^11+16*x^10-20*x^9+13*x^8-7*x^7+ 3*x^6+11*x^5-30*x^4+34*x^3-21*x^2+7*x-1)/(-1+x)/(x^6-x^5+2*x^4-x^3-2*x^2+3*x-1) /(x^12-x^10+x^9+x^8-3*x^6+3*x^5-2*x^4+4*x^3-6*x^2+4*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1016, 2010, 3956, 7750, 15127, 29450, 57245, 111189, 215927, 419397, 814877, 1583898, 3079757, 5990017, 11652663, 22671160, 44110953, 85827035, 166991801, 324903241] The limit of a(n+1)/a(n) as n goes to infinity is 1.94543652756 a(n) is asymptotic to .693861437001*1.94543652756^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [1, 2, 1, 2, 1, 3], denoted by the variable, X[1, 2, 1, 2, 1, 3], is 19 3 19 2 18 3 19 18 2 17 3 19 (x %1 - 3 x %1 - x %1 + 3 x %1 + 3 x %1 + 2 x %1 - x 18 17 2 18 17 15 3 17 15 2 - 3 x %1 - 6 x %1 + x + 6 x %1 - 2 x %1 - 2 x + 7 x %1 14 3 15 14 2 13 3 15 14 + 4 x %1 - 8 x %1 - 14 x %1 - 3 x %1 + 3 x + 16 x %1 13 2 12 3 14 13 12 2 13 12 + 10 x %1 + x %1 - 6 x - 11 x %1 - 3 x %1 + 4 x + 3 x %1 11 2 12 11 10 2 11 10 9 2 + 3 x %1 - x - 8 x %1 - 9 x %1 + 5 x + 25 x %1 + 10 x %1 10 9 8 2 9 8 7 2 8 - 16 x - 30 x %1 - 5 x %1 + 20 x + 18 x %1 + x %1 - 13 x 7 7 6 6 5 5 4 4 - 9 x %1 + 7 x + 10 x %1 - 3 x - 10 x %1 - 11 x + 5 x %1 + 30 x 3 3 2 / - x %1 - 34 x + 21 x - 7 x + 1) / ((-1 + x) / 6 6 5 5 4 4 3 3 2 (x %1 - x - x %1 + x + 2 x %1 - 2 x - x %1 + x + 2 x - 3 x + 1) ( 12 2 12 12 10 2 10 9 2 10 9 9 x %1 - 2 x %1 + x - x %1 + 2 x %1 + x %1 - x - 2 x %1 + x 8 8 6 6 5 5 4 4 3 - x %1 + x + 3 x %1 - 3 x - 3 x %1 + 3 x + x %1 - 2 x + 4 x 2 - 6 x + 4 x - 1)) %1 := X[1, 2, 1, 2, 1, 3] and in Maple format (x^19*X[1,2,1,2,1,3]^3-3*x^19*X[1,2,1,2,1,3]^2-x^18*X[1,2,1,2,1,3]^3+3*x^19*X[1 ,2,1,2,1,3]+3*x^18*X[1,2,1,2,1,3]^2+2*x^17*X[1,2,1,2,1,3]^3-x^19-3*x^18*X[1,2,1 ,2,1,3]-6*x^17*X[1,2,1,2,1,3]^2+x^18+6*x^17*X[1,2,1,2,1,3]-2*x^15*X[1,2,1,2,1,3 ]^3-2*x^17+7*x^15*X[1,2,1,2,1,3]^2+4*x^14*X[1,2,1,2,1,3]^3-8*x^15*X[1,2,1,2,1,3 ]-14*x^14*X[1,2,1,2,1,3]^2-3*x^13*X[1,2,1,2,1,3]^3+3*x^15+16*x^14*X[1,2,1,2,1,3 ]+10*x^13*X[1,2,1,2,1,3]^2+x^12*X[1,2,1,2,1,3]^3-6*x^14-11*x^13*X[1,2,1,2,1,3]-\ 3*x^12*X[1,2,1,2,1,3]^2+4*x^13+3*x^12*X[1,2,1,2,1,3]+3*x^11*X[1,2,1,2,1,3]^2-x^ 12-8*x^11*X[1,2,1,2,1,3]-9*x^10*X[1,2,1,2,1,3]^2+5*x^11+25*x^10*X[1,2,1,2,1,3]+ 10*x^9*X[1,2,1,2,1,3]^2-16*x^10-30*x^9*X[1,2,1,2,1,3]-5*x^8*X[1,2,1,2,1,3]^2+20 *x^9+18*x^8*X[1,2,1,2,1,3]+x^7*X[1,2,1,2,1,3]^2-13*x^8-9*x^7*X[1,2,1,2,1,3]+7*x ^7+10*x^6*X[1,2,1,2,1,3]-3*x^6-10*x^5*X[1,2,1,2,1,3]-11*x^5+5*x^4*X[1,2,1,2,1,3 ]+30*x^4-x^3*X[1,2,1,2,1,3]-34*x^3+21*x^2-7*x+1)/(-1+x)/(x^6*X[1,2,1,2,1,3]-x^6 -x^5*X[1,2,1,2,1,3]+x^5+2*x^4*X[1,2,1,2,1,3]-2*x^4-x^3*X[1,2,1,2,1,3]+x^3+2*x^2 -3*x+1)/(x^12*X[1,2,1,2,1,3]^2-2*x^12*X[1,2,1,2,1,3]+x^12-x^10*X[1,2,1,2,1,3]^2 +2*x^10*X[1,2,1,2,1,3]+x^9*X[1,2,1,2,1,3]^2-x^10-2*x^9*X[1,2,1,2,1,3]+x^9-x^8*X [1,2,1,2,1,3]+x^8+3*x^6*X[1,2,1,2,1,3]-3*x^6-3*x^5*X[1,2,1,2,1,3]+3*x^5+x^4*X[1 ,2,1,2,1,3]-2*x^4+4*x^3-6*x^2+4*x-1) Furthermore, the expectation of the random variable number of occurences (by\ 13 n containment) of , [1, 2, 1, 2, 1, 3], equals , - -- + ---- 32 32 583 39 n The variance equals , - ---- + ---- 1024 1024 14679 819 n The , 3, -th moment about the mean is , - ----- + ----- 16384 16384 522967 4563 2 16935 The , 4, -th moment about the mean is , - ------- + ------- n - ------ n 1048576 1048576 262144 The compositions of, 10, that yield the, 20, -th largest growth, that is, 1.9501414915693502232, are , [2, 1, 1, 2, 2, 2], [2, 2, 2, 1, 1, 2] Theorem Number, 20, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 1, 2, 2, 2] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n ) a(n) x = - ( / ----- n = 0 12 10 8 7 6 5 4 3 2 / x + x + x - 2 x + 5 x - 8 x + 12 x - 14 x + 11 x - 5 x + 1) / / 13 12 11 10 9 8 7 6 5 4 3 (x - x + x - x + x - x + 4 x - 8 x + 13 x - 19 x + 21 x 2 - 15 x + 6 x - 1) and in Maple format -(x^12+x^10+x^8-2*x^7+5*x^6-8*x^5+12*x^4-14*x^3+11*x^2-5*x+1)/(x^13-x^12+x^11-x ^10+x^9-x^8+4*x^7-8*x^6+13*x^5-19*x^4+21*x^3-15*x^2+6*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1016, 2010, 3957, 7758, 15164, 29581, 57639, 112251, 218579, 425672, 829170, 1615595, 3148748, 6138228, 11968110, 23337963, 45512975, 88761905, 173111414, 337617555] The limit of a(n+1)/a(n) as n goes to infinity is 1.95014149157 a(n) is asymptotic to .670690304686*1.95014149157^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [2, 1, 1, 2, 2, 2], denoted by the variable, X[2, 1, 1, 2, 2, 2], is 13 3 13 2 12 3 13 12 2 11 3 13 - (x %1 - 3 x %1 - x %1 + 3 x %1 + 3 x %1 + x %1 - x 12 11 2 12 11 10 2 11 10 10 - 3 x %1 - 3 x %1 + x + 3 x %1 + x %1 - x - 2 x %1 + x 9 9 8 8 7 7 6 6 + x %1 - x - 3 x %1 + 3 x + 7 x %1 - 7 x - 12 x %1 + 13 x 5 5 4 4 3 3 2 2 + 14 x %1 - 20 x - 11 x %1 + 26 x + 5 x %1 - 25 x - x %1 + 16 x / 14 3 14 2 13 3 14 13 2 - 6 x + 1) / (x %1 - 3 x %1 - x %1 + 3 x %1 + 4 x %1 / 12 3 14 13 12 2 11 3 13 12 + 2 x %1 - x - 5 x %1 - 6 x %1 - x %1 + 2 x + 6 x %1 11 2 12 11 10 2 11 10 10 + 4 x %1 - 2 x - 5 x %1 - x %1 + 2 x + 3 x %1 - 2 x 9 9 8 8 7 7 6 6 - 2 x %1 + 2 x + 5 x %1 - 5 x - 12 x %1 + 12 x + 19 x %1 - 21 x 5 5 4 4 3 3 2 2 - 21 x %1 + 32 x + 15 x %1 - 40 x - 6 x %1 + 36 x + x %1 - 21 x + 7 x - 1) %1 := X[2, 1, 1, 2, 2, 2] and in Maple format -(x^13*X[2,1,1,2,2,2]^3-3*x^13*X[2,1,1,2,2,2]^2-x^12*X[2,1,1,2,2,2]^3+3*x^13*X[ 2,1,1,2,2,2]+3*x^12*X[2,1,1,2,2,2]^2+x^11*X[2,1,1,2,2,2]^3-x^13-3*x^12*X[2,1,1, 2,2,2]-3*x^11*X[2,1,1,2,2,2]^2+x^12+3*x^11*X[2,1,1,2,2,2]+x^10*X[2,1,1,2,2,2]^2 -x^11-2*x^10*X[2,1,1,2,2,2]+x^10+x^9*X[2,1,1,2,2,2]-x^9-3*x^8*X[2,1,1,2,2,2]+3* x^8+7*x^7*X[2,1,1,2,2,2]-7*x^7-12*x^6*X[2,1,1,2,2,2]+13*x^6+14*x^5*X[2,1,1,2,2, 2]-20*x^5-11*x^4*X[2,1,1,2,2,2]+26*x^4+5*x^3*X[2,1,1,2,2,2]-25*x^3-x^2*X[2,1,1, 2,2,2]+16*x^2-6*x+1)/(x^14*X[2,1,1,2,2,2]^3-3*x^14*X[2,1,1,2,2,2]^2-x^13*X[2,1, 1,2,2,2]^3+3*x^14*X[2,1,1,2,2,2]+4*x^13*X[2,1,1,2,2,2]^2+2*x^12*X[2,1,1,2,2,2]^ 3-x^14-5*x^13*X[2,1,1,2,2,2]-6*x^12*X[2,1,1,2,2,2]^2-x^11*X[2,1,1,2,2,2]^3+2*x^ 13+6*x^12*X[2,1,1,2,2,2]+4*x^11*X[2,1,1,2,2,2]^2-2*x^12-5*x^11*X[2,1,1,2,2,2]-x ^10*X[2,1,1,2,2,2]^2+2*x^11+3*x^10*X[2,1,1,2,2,2]-2*x^10-2*x^9*X[2,1,1,2,2,2]+2 *x^9+5*x^8*X[2,1,1,2,2,2]-5*x^8-12*x^7*X[2,1,1,2,2,2]+12*x^7+19*x^6*X[2,1,1,2,2 ,2]-21*x^6-21*x^5*X[2,1,1,2,2,2]+32*x^5+15*x^4*X[2,1,1,2,2,2]-40*x^4-6*x^3*X[2, 1,1,2,2,2]+36*x^3+x^2*X[2,1,1,2,2,2]-21*x^2+7*x-1) Furthermore, the expectation of the random variable number of occurences (by\ 13 n containment) of , [2, 1, 1, 2, 2, 2], equals , - -- + ---- 32 32 819 51 n The variance equals , - ---- + ---- 1024 1024 40005 1929 n The , 3, -th moment about the mean is , - ----- + ------ 16384 16384 9025167 7803 2 45309 The , 4, -th moment about the mean is , - ------- + ------- n + ------ n 1048576 1048576 262144 The compositions of, 10, that yield the, 21, -th largest growth, that is, 1.9505475001908849807, are , [2, 1, 2, 1, 2, 2], [2, 2, 1, 2, 1, 2] Theorem Number, 21, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 2, 1, 2, 2] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 14 13 10 9 7 6 5 4 ) a(n) x = - (x + x - x + x + 3 x - 7 x + 10 x - 16 x / ----- n = 0 3 2 2 / 17 16 15 13 12 + 20 x - 15 x + 6 x - 1) (x - x + 1) / (x - x + x - x + 3 x / 11 10 9 8 7 6 5 4 3 - 4 x + 4 x - 7 x + 17 x - 33 x + 53 x - 72 x + 76 x - 57 x 2 + 28 x - 8 x + 1) and in Maple format -(x^14+x^13-x^10+x^9+3*x^7-7*x^6+10*x^5-16*x^4+20*x^3-15*x^2+6*x-1)*(x^2-x+1)/( x^17-x^16+x^15-x^13+3*x^12-4*x^11+4*x^10-7*x^9+17*x^8-33*x^7+53*x^6-72*x^5+76*x ^4-57*x^3+28*x^2-8*x+1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1016, 2010, 3956, 7751, 15135, 29488, 57384, 111623, 217149, 422603, 822877, 1603140, 3124764, 6092997, 11884101, 23183529, 45230752, 88247263, 172172176, 335898465] The limit of a(n+1)/a(n) as n goes to infinity is 1.95054750019 a(n) is asymptotic to .663150369868*1.95054750019^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [2, 1, 2, 1, 2, 2], denoted by the variable, X[2, 1, 2, 1, 2, 2], is 16 3 16 2 16 16 13 3 13 2 12 3 - (x %1 - 3 x %1 + 3 x %1 - x + 2 x %1 - 5 x %1 - 2 x %1 13 12 2 11 3 13 12 11 2 12 + 4 x %1 + 5 x %1 + x %1 - x - 4 x %1 - 4 x %1 + x 11 10 2 11 10 10 9 9 + 5 x %1 + x %1 - 2 x - 3 x %1 + 2 x + 4 x %1 - 4 x 8 8 7 7 6 6 5 - 10 x %1 + 10 x + 19 x %1 - 20 x - 26 x %1 + 33 x + 25 x %1 5 4 4 3 3 2 2 - 46 x - 16 x %1 + 51 x + 6 x %1 - 41 x - x %1 + 22 x - 7 x + 1) / 17 3 17 2 16 3 17 16 2 15 3 17 / (x %1 - 3 x %1 - x %1 + 3 x %1 + 3 x %1 + x %1 - x / 16 15 2 14 3 16 15 14 2 13 3 - 3 x %1 - 3 x %1 + x %1 + x + 3 x %1 - 2 x %1 - 3 x %1 15 14 13 2 12 3 13 12 2 11 3 - x + x %1 + 7 x %1 + 3 x %1 - 5 x %1 - 9 x %1 - x %1 13 12 11 2 12 11 10 2 11 + x + 9 x %1 + 5 x %1 - 3 x - 8 x %1 - x %1 + 4 x 10 10 9 9 8 8 7 7 + 5 x %1 - 4 x - 7 x %1 + 7 x + 17 x %1 - 17 x - 31 x %1 + 33 x 6 6 5 5 4 4 3 3 + 40 x %1 - 53 x - 36 x %1 + 72 x + 21 x %1 - 76 x - 7 x %1 + 57 x 2 2 + x %1 - 28 x + 8 x - 1) %1 := X[2, 1, 2, 1, 2, 2] and in Maple format -(x^16*X[2,1,2,1,2,2]^3-3*x^16*X[2,1,2,1,2,2]^2+3*x^16*X[2,1,2,1,2,2]-x^16+2*x^ 13*X[2,1,2,1,2,2]^3-5*x^13*X[2,1,2,1,2,2]^2-2*x^12*X[2,1,2,1,2,2]^3+4*x^13*X[2, 1,2,1,2,2]+5*x^12*X[2,1,2,1,2,2]^2+x^11*X[2,1,2,1,2,2]^3-x^13-4*x^12*X[2,1,2,1, 2,2]-4*x^11*X[2,1,2,1,2,2]^2+x^12+5*x^11*X[2,1,2,1,2,2]+x^10*X[2,1,2,1,2,2]^2-2 *x^11-3*x^10*X[2,1,2,1,2,2]+2*x^10+4*x^9*X[2,1,2,1,2,2]-4*x^9-10*x^8*X[2,1,2,1, 2,2]+10*x^8+19*x^7*X[2,1,2,1,2,2]-20*x^7-26*x^6*X[2,1,2,1,2,2]+33*x^6+25*x^5*X[ 2,1,2,1,2,2]-46*x^5-16*x^4*X[2,1,2,1,2,2]+51*x^4+6*x^3*X[2,1,2,1,2,2]-41*x^3-x^ 2*X[2,1,2,1,2,2]+22*x^2-7*x+1)/(x^17*X[2,1,2,1,2,2]^3-3*x^17*X[2,1,2,1,2,2]^2-x ^16*X[2,1,2,1,2,2]^3+3*x^17*X[2,1,2,1,2,2]+3*x^16*X[2,1,2,1,2,2]^2+x^15*X[2,1,2 ,1,2,2]^3-x^17-3*x^16*X[2,1,2,1,2,2]-3*x^15*X[2,1,2,1,2,2]^2+x^14*X[2,1,2,1,2,2 ]^3+x^16+3*x^15*X[2,1,2,1,2,2]-2*x^14*X[2,1,2,1,2,2]^2-3*x^13*X[2,1,2,1,2,2]^3- x^15+x^14*X[2,1,2,1,2,2]+7*x^13*X[2,1,2,1,2,2]^2+3*x^12*X[2,1,2,1,2,2]^3-5*x^13 *X[2,1,2,1,2,2]-9*x^12*X[2,1,2,1,2,2]^2-x^11*X[2,1,2,1,2,2]^3+x^13+9*x^12*X[2,1 ,2,1,2,2]+5*x^11*X[2,1,2,1,2,2]^2-3*x^12-8*x^11*X[2,1,2,1,2,2]-x^10*X[2,1,2,1,2 ,2]^2+4*x^11+5*x^10*X[2,1,2,1,2,2]-4*x^10-7*x^9*X[2,1,2,1,2,2]+7*x^9+17*x^8*X[2 ,1,2,1,2,2]-17*x^8-31*x^7*X[2,1,2,1,2,2]+33*x^7+40*x^6*X[2,1,2,1,2,2]-53*x^6-36 *x^5*X[2,1,2,1,2,2]+72*x^5+21*x^4*X[2,1,2,1,2,2]-76*x^4-7*x^3*X[2,1,2,1,2,2]+57 *x^3+x^2*X[2,1,2,1,2,2]-28*x^2+8*x-1) Furthermore, the expectation of the random variable number of occurences (by\ 13 n containment) of , [2, 1, 2, 1, 2, 2], equals , - -- + ---- 32 32 835 51 n The variance equals , - ---- + ---- 1024 1024 40413 1905 n The , 3, -th moment about the mean is , - ----- + ------ 16384 16384 8931055 7803 2 41589 The , 4, -th moment about the mean is , - ------- + ------- n + ------ n 1048576 1048576 262144 The compositions of, 10, that yield the, 22, -th largest growth, that is, 1.9515637714286765859, are , [1, 2, 1, 2, 2, 2], [1, 2, 2, 2, 1, 2], [2, 1, 2, 2, 2, 1], [2, 2, 2, 1, 2, 1] Theorem Number, 22, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 2, 1, 2, 2, 2] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 2 ) a(n) x = (x - x + 1) / ----- n = 0 9 7 6 5 4 3 2 / (x + x + x - 2 x + 2 x - 4 x + 6 x - 4 x + 1) / ((-1 + x) / 10 9 8 7 6 5 4 3 2 (x - x + x + x - 3 x + 5 x - 8 x + 11 x - 10 x + 5 x - 1)) and in Maple format (x^2-x+1)*(x^9+x^7+x^6-2*x^5+2*x^4-4*x^3+6*x^2-4*x+1)/(-1+x)/(x^10-x^9+x^8+x^7-\ 3*x^6+5*x^5-8*x^4+11*x^3-10*x^2+5*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1016, 2010, 3957, 7759, 15172, 29618, 57770, 112646, 219649, 428358, 835544, 1630093, 3180681, 6206879, 12113071, 23640103, 46136992, 90042706, 175729457, 342954621] The limit of a(n+1)/a(n) as n goes to infinity is 1.95156377143 a(n) is asymptotic to .666439372478*1.95156377143^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [1, 2, 1, 2, 2, 2], denoted by the variable, X[1, 2, 1, 2, 2, 2], is 11 2 11 10 2 11 10 9 2 10 9 9 (x %1 - 2 x %1 - x %1 + x + 2 x %1 + x %1 - x - 3 x %1 + 2 x 7 7 6 6 5 5 4 4 + 2 x %1 - 2 x - 5 x %1 + 5 x + 7 x %1 - 8 x - 7 x %1 + 12 x 3 3 2 2 / 10 2 + 4 x %1 - 14 x - x %1 + 11 x - 5 x + 1) / ((-1 + x) (x %1 / 10 9 2 10 9 9 8 8 7 7 - 2 x %1 - x %1 + x + 2 x %1 - x - x %1 + x - x %1 + x 6 6 5 5 4 4 3 3 + 3 x %1 - 3 x - 5 x %1 + 5 x + 6 x %1 - 8 x - 4 x %1 + 11 x 2 2 + x %1 - 10 x + 5 x - 1)) %1 := X[1, 2, 1, 2, 2, 2] and in Maple format (x^11*X[1,2,1,2,2,2]^2-2*x^11*X[1,2,1,2,2,2]-x^10*X[1,2,1,2,2,2]^2+x^11+2*x^10* X[1,2,1,2,2,2]+x^9*X[1,2,1,2,2,2]^2-x^10-3*x^9*X[1,2,1,2,2,2]+2*x^9+2*x^7*X[1,2 ,1,2,2,2]-2*x^7-5*x^6*X[1,2,1,2,2,2]+5*x^6+7*x^5*X[1,2,1,2,2,2]-8*x^5-7*x^4*X[1 ,2,1,2,2,2]+12*x^4+4*x^3*X[1,2,1,2,2,2]-14*x^3-x^2*X[1,2,1,2,2,2]+11*x^2-5*x+1) /(-1+x)/(x^10*X[1,2,1,2,2,2]^2-2*x^10*X[1,2,1,2,2,2]-x^9*X[1,2,1,2,2,2]^2+x^10+ 2*x^9*X[1,2,1,2,2,2]-x^9-x^8*X[1,2,1,2,2,2]+x^8-x^7*X[1,2,1,2,2,2]+x^7+3*x^6*X[ 1,2,1,2,2,2]-3*x^6-5*x^5*X[1,2,1,2,2,2]+5*x^5+6*x^4*X[1,2,1,2,2,2]-8*x^4-4*x^3* X[1,2,1,2,2,2]+11*x^3+x^2*X[1,2,1,2,2,2]-10*x^2+5*x-1) Furthermore, the expectation of the random variable number of occurences (by\ 13 n containment) of , [1, 2, 1, 2, 2, 2], equals , - -- + ---- 32 32 863 55 n The variance equals , - ---- + ---- 1024 1024 43623 2259 n The , 3, -th moment about the mean is , - ----- + ------ 16384 16384 9478247 9075 2 56917 The , 4, -th moment about the mean is , - ------- + ------- n + ------ n 1048576 1048576 262144 The compositions of, 10, that yield the, 23, -th largest growth, that is, 1.9527971478516900544, are , [1, 2, 2, 1, 2, 2], [2, 2, 1, 2, 2, 1] Theorem Number, 23, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 2, 2, 1, 2, 2] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 4 3 2 ) a(n) x = (x - x + 2 x - 2 x + 1) / ----- n = 0 9 8 7 6 5 4 3 2 / 12 (x - x + x - x + x + x - 4 x + 6 x - 4 x + 1) / ((-1 + x) (x / 11 10 9 8 7 6 5 4 3 - 2 x + 4 x - 5 x + 5 x - 2 x - 4 x + 12 x - 19 x + 21 x 2 - 15 x + 6 x - 1)) and in Maple format (x^4-x^3+2*x^2-2*x+1)*(x^9-x^8+x^7-x^6+x^5+x^4-4*x^3+6*x^2-4*x+1)/(-1+x)/(x^12-\ 2*x^11+4*x^10-5*x^9+5*x^8-2*x^7-4*x^6+12*x^5-19*x^4+21*x^3-15*x^2+6*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1016, 2010, 3957, 7758, 15164, 29582, 57650, 112316, 218858, 426653, 832180, 1623968, 3170402, 6191216, 12092384, 23620035, 46137249, 90117043, 176010096, 343749588] The limit of a(n+1)/a(n) as n goes to infinity is 1.95279714785 a(n) is asymptotic to .655393299540*1.95279714785^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [1, 2, 2, 1, 2, 2], denoted by the variable, X[1, 2, 2, 1, 2, 2], is 13 2 13 12 2 13 12 11 2 12 11 (x %1 - 2 x %1 - 2 x %1 + x + 4 x %1 + 4 x %1 - 2 x - 8 x %1 10 2 11 10 9 2 10 9 8 2 - 5 x %1 + 4 x + 11 x %1 + 5 x %1 - 6 x - 12 x %1 - 3 x %1 9 8 7 2 8 7 6 6 5 + 7 x + 8 x %1 + x %1 - 5 x - x %1 - 8 x %1 + 9 x + 13 x %1 5 4 4 3 3 2 2 - 19 x - 11 x %1 + 26 x + 5 x %1 - 25 x - x %1 + 16 x - 6 x + 1) / 12 2 12 11 2 12 11 10 2 / ((-1 + x) (x %1 - 2 x %1 - 2 x %1 + x + 4 x %1 + 4 x %1 / 11 10 9 2 10 9 8 2 9 - 2 x - 8 x %1 - 4 x %1 + 4 x + 9 x %1 + 3 x %1 - 5 x 8 7 2 8 7 7 6 6 5 - 8 x %1 - x %1 + 5 x + 3 x %1 - 2 x + 4 x %1 - 4 x - 10 x %1 5 4 4 3 3 2 2 + 12 x + 10 x %1 - 19 x - 5 x %1 + 21 x + x %1 - 15 x + 6 x - 1)) %1 := X[1, 2, 2, 1, 2, 2] and in Maple format (x^13*X[1,2,2,1,2,2]^2-2*x^13*X[1,2,2,1,2,2]-2*x^12*X[1,2,2,1,2,2]^2+x^13+4*x^ 12*X[1,2,2,1,2,2]+4*x^11*X[1,2,2,1,2,2]^2-2*x^12-8*x^11*X[1,2,2,1,2,2]-5*x^10*X [1,2,2,1,2,2]^2+4*x^11+11*x^10*X[1,2,2,1,2,2]+5*x^9*X[1,2,2,1,2,2]^2-6*x^10-12* x^9*X[1,2,2,1,2,2]-3*x^8*X[1,2,2,1,2,2]^2+7*x^9+8*x^8*X[1,2,2,1,2,2]+x^7*X[1,2, 2,1,2,2]^2-5*x^8-x^7*X[1,2,2,1,2,2]-8*x^6*X[1,2,2,1,2,2]+9*x^6+13*x^5*X[1,2,2,1 ,2,2]-19*x^5-11*x^4*X[1,2,2,1,2,2]+26*x^4+5*x^3*X[1,2,2,1,2,2]-25*x^3-x^2*X[1,2 ,2,1,2,2]+16*x^2-6*x+1)/(-1+x)/(x^12*X[1,2,2,1,2,2]^2-2*x^12*X[1,2,2,1,2,2]-2*x ^11*X[1,2,2,1,2,2]^2+x^12+4*x^11*X[1,2,2,1,2,2]+4*x^10*X[1,2,2,1,2,2]^2-2*x^11-\ 8*x^10*X[1,2,2,1,2,2]-4*x^9*X[1,2,2,1,2,2]^2+4*x^10+9*x^9*X[1,2,2,1,2,2]+3*x^8* X[1,2,2,1,2,2]^2-5*x^9-8*x^8*X[1,2,2,1,2,2]-x^7*X[1,2,2,1,2,2]^2+5*x^8+3*x^7*X[ 1,2,2,1,2,2]-2*x^7+4*x^6*X[1,2,2,1,2,2]-4*x^6-10*x^5*X[1,2,2,1,2,2]+12*x^5+10*x ^4*X[1,2,2,1,2,2]-19*x^4-5*x^3*X[1,2,2,1,2,2]+21*x^3+x^2*X[1,2,2,1,2,2]-15*x^2+ 6*x-1) Furthermore, the expectation of the random variable number of occurences (by\ 13 n containment) of , [1, 2, 2, 1, 2, 2], equals , - -- + ---- 32 32 879 55 n The variance equals , - ---- + ---- 1024 1024 43407 2187 n The , 3, -th moment about the mean is , - ----- + ------ 16384 16384 9157191 9075 2 48397 The , 4, -th moment about the mean is , - ------- + ------- n + ------ n 1048576 1048576 262144 The compositions of, 10, that yield the, 24, -th largest growth, that is, 1.9539877574581293211, are , [2, 1, 2, 2, 1, 2] Theorem Number, 24, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 2, 2, 1, 2] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 18 16 14 13 12 11 9 8 ) a(n) x = - (x + x - x + 2 x - 3 x + 3 x - x - 10 x / ----- n = 0 7 6 5 4 3 2 / 19 18 + 38 x - 73 x + 96 x - 92 x + 63 x - 29 x + 8 x - 1) / (x - x / 17 16 15 14 13 12 11 9 8 7 + 2 x - 2 x + x + x - 4 x + 7 x - 6 x + x + 20 x - 66 x 6 5 4 3 2 + 118 x - 147 x + 133 x - 85 x + 36 x - 9 x + 1) and in Maple format -(x^18+x^16-x^14+2*x^13-3*x^12+3*x^11-x^9-10*x^8+38*x^7-73*x^6+96*x^5-92*x^4+63 *x^3-29*x^2+8*x-1)/(x^19-x^18+2*x^17-2*x^16+x^15+x^14-4*x^13+7*x^12-6*x^11+x^9+ 20*x^8-66*x^7+118*x^6-147*x^5+133*x^4-85*x^3+36*x^2-9*x+1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1016, 2010, 3956, 7751, 15136, 29498, 57439, 111846, 217898, 424818, 828859, 1618251, 3161068, 6176960, 12072735, 23598041, 46126394, 90157907, 176208878, 344364833] The limit of a(n+1)/a(n) as n goes to infinity is 1.95398775746 a(n) is asymptotic to .644669574312*1.95398775746^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [2, 1, 2, 2, 1, 2], denoted by the variable, X[2, 1, 2, 2, 1, 2], is 18 3 18 2 18 16 3 18 16 2 16 - (x %1 - 3 x %1 + 3 x %1 + x %1 - x - 3 x %1 + 3 x %1 14 3 16 14 2 13 3 14 13 2 - 2 x %1 - x + 5 x %1 + 4 x %1 - 4 x %1 - 10 x %1 12 3 14 13 12 2 11 3 13 12 - 3 x %1 + x + 8 x %1 + 8 x %1 + x %1 - 2 x - 8 x %1 12 11 10 2 11 10 9 2 9 + 3 x + 2 x %1 - 9 x %1 - 3 x + 9 x %1 + 10 x %1 - 11 x %1 8 2 9 8 7 2 8 7 7 6 - 5 x %1 + x - 4 x %1 + x %1 + 10 x + 29 x %1 - 38 x - 45 x %1 6 5 5 4 4 3 3 2 + 73 x + 40 x %1 - 96 x - 22 x %1 + 92 x + 7 x %1 - 63 x - x %1 2 / 19 3 19 2 18 3 19 + 29 x - 8 x + 1) / (x %1 - 3 x %1 - x %1 + 3 x %1 / 18 2 17 3 19 18 17 2 16 3 18 + 3 x %1 + 2 x %1 - x - 3 x %1 - 6 x %1 - 2 x %1 + x 17 16 2 17 16 15 2 14 3 16 + 6 x %1 + 6 x %1 - 2 x - 6 x %1 - x %1 + 4 x %1 + 2 x 15 14 2 13 3 15 14 13 2 + 2 x %1 - 9 x %1 - 6 x %1 - x + 6 x %1 + 16 x %1 12 3 14 13 12 2 11 3 13 12 + 4 x %1 - x - 14 x %1 - 12 x %1 - x %1 + 4 x + 15 x %1 11 2 12 11 10 2 11 10 - 3 x %1 - 7 x - 2 x %1 + 15 x %1 + 6 x - 15 x %1 9 2 9 8 2 9 8 7 2 8 - 14 x %1 + 15 x %1 + 6 x %1 - x + 12 x %1 - x %1 - 20 x 7 7 6 6 5 5 4 - 50 x %1 + 66 x + 69 x %1 - 118 x - 56 x %1 + 147 x + 28 x %1 4 3 3 2 2 - 133 x - 8 x %1 + 85 x + x %1 - 36 x + 9 x - 1) %1 := X[2, 1, 2, 2, 1, 2] and in Maple format -(x^18*X[2,1,2,2,1,2]^3-3*x^18*X[2,1,2,2,1,2]^2+3*x^18*X[2,1,2,2,1,2]+x^16*X[2, 1,2,2,1,2]^3-x^18-3*x^16*X[2,1,2,2,1,2]^2+3*x^16*X[2,1,2,2,1,2]-2*x^14*X[2,1,2, 2,1,2]^3-x^16+5*x^14*X[2,1,2,2,1,2]^2+4*x^13*X[2,1,2,2,1,2]^3-4*x^14*X[2,1,2,2, 1,2]-10*x^13*X[2,1,2,2,1,2]^2-3*x^12*X[2,1,2,2,1,2]^3+x^14+8*x^13*X[2,1,2,2,1,2 ]+8*x^12*X[2,1,2,2,1,2]^2+x^11*X[2,1,2,2,1,2]^3-2*x^13-8*x^12*X[2,1,2,2,1,2]+3* x^12+2*x^11*X[2,1,2,2,1,2]-9*x^10*X[2,1,2,2,1,2]^2-3*x^11+9*x^10*X[2,1,2,2,1,2] +10*x^9*X[2,1,2,2,1,2]^2-11*x^9*X[2,1,2,2,1,2]-5*x^8*X[2,1,2,2,1,2]^2+x^9-4*x^8 *X[2,1,2,2,1,2]+x^7*X[2,1,2,2,1,2]^2+10*x^8+29*x^7*X[2,1,2,2,1,2]-38*x^7-45*x^6 *X[2,1,2,2,1,2]+73*x^6+40*x^5*X[2,1,2,2,1,2]-96*x^5-22*x^4*X[2,1,2,2,1,2]+92*x^ 4+7*x^3*X[2,1,2,2,1,2]-63*x^3-x^2*X[2,1,2,2,1,2]+29*x^2-8*x+1)/(x^19*X[2,1,2,2, 1,2]^3-3*x^19*X[2,1,2,2,1,2]^2-x^18*X[2,1,2,2,1,2]^3+3*x^19*X[2,1,2,2,1,2]+3*x^ 18*X[2,1,2,2,1,2]^2+2*x^17*X[2,1,2,2,1,2]^3-x^19-3*x^18*X[2,1,2,2,1,2]-6*x^17*X [2,1,2,2,1,2]^2-2*x^16*X[2,1,2,2,1,2]^3+x^18+6*x^17*X[2,1,2,2,1,2]+6*x^16*X[2,1 ,2,2,1,2]^2-2*x^17-6*x^16*X[2,1,2,2,1,2]-x^15*X[2,1,2,2,1,2]^2+4*x^14*X[2,1,2,2 ,1,2]^3+2*x^16+2*x^15*X[2,1,2,2,1,2]-9*x^14*X[2,1,2,2,1,2]^2-6*x^13*X[2,1,2,2,1 ,2]^3-x^15+6*x^14*X[2,1,2,2,1,2]+16*x^13*X[2,1,2,2,1,2]^2+4*x^12*X[2,1,2,2,1,2] ^3-x^14-14*x^13*X[2,1,2,2,1,2]-12*x^12*X[2,1,2,2,1,2]^2-x^11*X[2,1,2,2,1,2]^3+4 *x^13+15*x^12*X[2,1,2,2,1,2]-3*x^11*X[2,1,2,2,1,2]^2-7*x^12-2*x^11*X[2,1,2,2,1, 2]+15*x^10*X[2,1,2,2,1,2]^2+6*x^11-15*x^10*X[2,1,2,2,1,2]-14*x^9*X[2,1,2,2,1,2] ^2+15*x^9*X[2,1,2,2,1,2]+6*x^8*X[2,1,2,2,1,2]^2-x^9+12*x^8*X[2,1,2,2,1,2]-x^7*X [2,1,2,2,1,2]^2-20*x^8-50*x^7*X[2,1,2,2,1,2]+66*x^7+69*x^6*X[2,1,2,2,1,2]-118*x ^6-56*x^5*X[2,1,2,2,1,2]+147*x^5+28*x^4*X[2,1,2,2,1,2]-133*x^4-8*x^3*X[2,1,2,2, 1,2]+85*x^3+x^2*X[2,1,2,2,1,2]-36*x^2+9*x-1) Furthermore, the expectation of the random variable number of occurences (by\ 13 n containment) of , [2, 1, 2, 2, 1, 2], equals , - -- + ---- 32 32 903 55 n The variance equals , - ---- + ---- 1024 1024 43851 2103 n The , 3, -th moment about the mean is , - ----- + ------ 16384 16384 8930871 9075 2 37033 The , 4, -th moment about the mean is , - ------- + ------- n + ------ n 1048576 1048576 262144 The compositions of, 10, that yield the, 25, -th largest growth, that is, 1.9543423291595747005, are , [2, 2, 1, 1, 2, 2] Theorem Number, 25, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 2, 1, 1, 2, 2] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 19 18 17 16 15 14 13 ) a(n) x = - (x - x + 2 x - 3 x + 5 x - 6 x + 10 x / ----- n = 0 12 11 10 9 8 7 6 5 4 - 16 x + 24 x - 32 x + 39 x - 46 x + 59 x - 80 x + 97 x - 92 x 3 2 / 20 19 18 17 16 15 + 63 x - 29 x + 8 x - 1) / (x - x + 3 x - 4 x + 6 x - 8 x / 14 13 12 11 10 9 8 7 + 12 x - 18 x + 28 x - 40 x + 51 x - 60 x + 71 x - 93 x 6 5 4 3 2 + 126 x - 148 x + 133 x - 85 x + 36 x - 9 x + 1) and in Maple format -(x^19-x^18+2*x^17-3*x^16+5*x^15-6*x^14+10*x^13-16*x^12+24*x^11-32*x^10+39*x^9-\ 46*x^8+59*x^7-80*x^6+97*x^5-92*x^4+63*x^3-29*x^2+8*x-1)/(x^20-x^19+3*x^18-4*x^ 17+6*x^16-8*x^15+12*x^14-18*x^13+28*x^12-40*x^11+51*x^10-60*x^9+71*x^8-93*x^7+ 126*x^6-148*x^5+133*x^4-85*x^3+36*x^2-9*x+1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1016, 2010, 3957, 7758, 15163, 29573, 57605, 112151, 218362, 425358, 829146, 1617472, 3157585, 6167927, 12053912, 23564443, 46075242, 90096995, 176176663, 344474692] The limit of a(n+1)/a(n) as n goes to infinity is 1.95434232916 a(n) is asymptotic to .641734600337*1.95434232916^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [2, 2, 1, 1, 2, 2], denoted by the variable, X[2, 2, 1, 1, 2, 2], is 19 4 19 3 18 4 19 2 18 3 17 4 - (x %1 - 4 x %1 - 2 x %1 + 6 x %1 + 7 x %1 + 3 x %1 19 18 2 17 3 16 4 19 18 - 4 x %1 - 9 x %1 - 11 x %1 - 2 x %1 + x + 5 x %1 17 2 16 3 15 4 18 17 16 2 + 15 x %1 + 9 x %1 + x %1 - x - 9 x %1 - 15 x %1 15 3 17 16 15 2 14 3 16 - 9 x %1 + 2 x + 11 x %1 + 20 x %1 + 8 x %1 - 3 x 15 14 2 13 3 15 14 13 2 - 17 x %1 - 22 x %1 - 10 x %1 + 5 x + 20 x %1 + 30 x %1 12 3 14 13 12 2 11 3 13 + 10 x %1 - 6 x - 30 x %1 - 36 x %1 - 8 x %1 + 10 x 12 11 2 10 3 12 11 10 2 + 42 x %1 + 38 x %1 + 4 x %1 - 16 x - 54 x %1 - 30 x %1 9 3 11 10 9 2 10 9 8 2 - x %1 + 24 x + 58 x %1 + 17 x %1 - 32 x - 55 x %1 - 6 x %1 9 8 7 2 8 7 7 6 6 + 39 x + 51 x %1 + x %1 - 46 x - 52 x %1 + 59 x + 52 x %1 - 80 x 5 5 4 4 3 3 2 2 - 41 x %1 + 97 x + 22 x %1 - 92 x - 7 x %1 + 63 x + x %1 - 29 x / 20 4 20 3 19 4 20 2 19 3 + 8 x - 1) / (x %1 - 4 x %1 - 2 x %1 + 6 x %1 + 7 x %1 / 18 4 20 19 2 18 3 17 4 20 + 4 x %1 - 4 x %1 - 9 x %1 - 15 x %1 - 4 x %1 + x 19 18 2 17 3 16 4 19 18 + 5 x %1 + 21 x %1 + 16 x %1 + 3 x %1 - x - 13 x %1 17 2 16 3 15 4 18 17 16 2 - 24 x %1 - 16 x %1 - x %1 + 3 x + 16 x %1 + 29 x %1 15 3 17 16 15 2 14 3 16 + 13 x %1 - 4 x - 22 x %1 - 31 x %1 - 13 x %1 + 6 x 15 14 2 13 3 15 14 13 2 + 27 x %1 + 38 x %1 + 15 x %1 - 8 x - 37 x %1 - 48 x %1 12 3 14 13 12 2 11 3 13 - 15 x %1 + 12 x + 51 x %1 + 57 x %1 + 11 x %1 - 18 x 12 11 2 10 3 12 11 10 2 - 70 x %1 - 56 x %1 - 5 x %1 + 28 x + 85 x %1 + 42 x %1 9 3 11 10 9 2 10 9 8 2 + x %1 - 40 x - 88 x %1 - 22 x %1 + 51 x + 81 x %1 + 7 x %1 9 8 7 2 8 7 7 6 6 - 60 x - 76 x %1 - x %1 + 71 x + 79 x %1 - 93 x - 77 x %1 + 126 x 5 5 4 4 3 3 2 2 + 57 x %1 - 148 x - 28 x %1 + 133 x + 8 x %1 - 85 x - x %1 + 36 x - 9 x + 1) %1 := X[2, 2, 1, 1, 2, 2] and in Maple format -(x^19*X[2,2,1,1,2,2]^4-4*x^19*X[2,2,1,1,2,2]^3-2*x^18*X[2,2,1,1,2,2]^4+6*x^19* X[2,2,1,1,2,2]^2+7*x^18*X[2,2,1,1,2,2]^3+3*x^17*X[2,2,1,1,2,2]^4-4*x^19*X[2,2,1 ,1,2,2]-9*x^18*X[2,2,1,1,2,2]^2-11*x^17*X[2,2,1,1,2,2]^3-2*x^16*X[2,2,1,1,2,2]^ 4+x^19+5*x^18*X[2,2,1,1,2,2]+15*x^17*X[2,2,1,1,2,2]^2+9*x^16*X[2,2,1,1,2,2]^3+x ^15*X[2,2,1,1,2,2]^4-x^18-9*x^17*X[2,2,1,1,2,2]-15*x^16*X[2,2,1,1,2,2]^2-9*x^15 *X[2,2,1,1,2,2]^3+2*x^17+11*x^16*X[2,2,1,1,2,2]+20*x^15*X[2,2,1,1,2,2]^2+8*x^14 *X[2,2,1,1,2,2]^3-3*x^16-17*x^15*X[2,2,1,1,2,2]-22*x^14*X[2,2,1,1,2,2]^2-10*x^ 13*X[2,2,1,1,2,2]^3+5*x^15+20*x^14*X[2,2,1,1,2,2]+30*x^13*X[2,2,1,1,2,2]^2+10*x ^12*X[2,2,1,1,2,2]^3-6*x^14-30*x^13*X[2,2,1,1,2,2]-36*x^12*X[2,2,1,1,2,2]^2-8*x ^11*X[2,2,1,1,2,2]^3+10*x^13+42*x^12*X[2,2,1,1,2,2]+38*x^11*X[2,2,1,1,2,2]^2+4* x^10*X[2,2,1,1,2,2]^3-16*x^12-54*x^11*X[2,2,1,1,2,2]-30*x^10*X[2,2,1,1,2,2]^2-x ^9*X[2,2,1,1,2,2]^3+24*x^11+58*x^10*X[2,2,1,1,2,2]+17*x^9*X[2,2,1,1,2,2]^2-32*x ^10-55*x^9*X[2,2,1,1,2,2]-6*x^8*X[2,2,1,1,2,2]^2+39*x^9+51*x^8*X[2,2,1,1,2,2]+x ^7*X[2,2,1,1,2,2]^2-46*x^8-52*x^7*X[2,2,1,1,2,2]+59*x^7+52*x^6*X[2,2,1,1,2,2]-\ 80*x^6-41*x^5*X[2,2,1,1,2,2]+97*x^5+22*x^4*X[2,2,1,1,2,2]-92*x^4-7*x^3*X[2,2,1, 1,2,2]+63*x^3+x^2*X[2,2,1,1,2,2]-29*x^2+8*x-1)/(x^20*X[2,2,1,1,2,2]^4-4*x^20*X[ 2,2,1,1,2,2]^3-2*x^19*X[2,2,1,1,2,2]^4+6*x^20*X[2,2,1,1,2,2]^2+7*x^19*X[2,2,1,1 ,2,2]^3+4*x^18*X[2,2,1,1,2,2]^4-4*x^20*X[2,2,1,1,2,2]-9*x^19*X[2,2,1,1,2,2]^2-\ 15*x^18*X[2,2,1,1,2,2]^3-4*x^17*X[2,2,1,1,2,2]^4+x^20+5*x^19*X[2,2,1,1,2,2]+21* x^18*X[2,2,1,1,2,2]^2+16*x^17*X[2,2,1,1,2,2]^3+3*x^16*X[2,2,1,1,2,2]^4-x^19-13* x^18*X[2,2,1,1,2,2]-24*x^17*X[2,2,1,1,2,2]^2-16*x^16*X[2,2,1,1,2,2]^3-x^15*X[2, 2,1,1,2,2]^4+3*x^18+16*x^17*X[2,2,1,1,2,2]+29*x^16*X[2,2,1,1,2,2]^2+13*x^15*X[2 ,2,1,1,2,2]^3-4*x^17-22*x^16*X[2,2,1,1,2,2]-31*x^15*X[2,2,1,1,2,2]^2-13*x^14*X[ 2,2,1,1,2,2]^3+6*x^16+27*x^15*X[2,2,1,1,2,2]+38*x^14*X[2,2,1,1,2,2]^2+15*x^13*X [2,2,1,1,2,2]^3-8*x^15-37*x^14*X[2,2,1,1,2,2]-48*x^13*X[2,2,1,1,2,2]^2-15*x^12* X[2,2,1,1,2,2]^3+12*x^14+51*x^13*X[2,2,1,1,2,2]+57*x^12*X[2,2,1,1,2,2]^2+11*x^ 11*X[2,2,1,1,2,2]^3-18*x^13-70*x^12*X[2,2,1,1,2,2]-56*x^11*X[2,2,1,1,2,2]^2-5*x ^10*X[2,2,1,1,2,2]^3+28*x^12+85*x^11*X[2,2,1,1,2,2]+42*x^10*X[2,2,1,1,2,2]^2+x^ 9*X[2,2,1,1,2,2]^3-40*x^11-88*x^10*X[2,2,1,1,2,2]-22*x^9*X[2,2,1,1,2,2]^2+51*x^ 10+81*x^9*X[2,2,1,1,2,2]+7*x^8*X[2,2,1,1,2,2]^2-60*x^9-76*x^8*X[2,2,1,1,2,2]-x^ 7*X[2,2,1,1,2,2]^2+71*x^8+79*x^7*X[2,2,1,1,2,2]-93*x^7-77*x^6*X[2,2,1,1,2,2]+ 126*x^6+57*x^5*X[2,2,1,1,2,2]-148*x^5-28*x^4*X[2,2,1,1,2,2]+133*x^4+8*x^3*X[2,2 ,1,1,2,2]-85*x^3-x^2*X[2,2,1,1,2,2]+36*x^2-9*x+1) Furthermore, the expectation of the random variable number of occurences (by\ 13 n containment) of , [2, 2, 1, 1, 2, 2], equals , - -- + ---- 32 32 919 55 n The variance equals , - ---- + ---- 1024 1024 44871 2067 n The , 3, -th moment about the mean is , - ----- + ------ 16384 16384 9042103 9075 2 30697 The , 4, -th moment about the mean is , - ------- + ------- n + ------ n 1048576 1048576 262144 The compositions of, 10, that yield the, 26, -th largest growth, that is, 1.9611865309023902347, are , [1, 1, 2, 2, 2, 2], [1, 2, 2, 2, 2, 1], [2, 2, 2, 2, 1, 1] Theorem Number, 26, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 1, 2, 2, 2, 2] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n ) a(n) x = / ----- n = 0 8 7 6 5 4 3 2 2 x - 3 x + 5 x - 8 x + 12 x - 14 x + 11 x - 5 x + 1 - ----------------------------------------------------------- 7 6 5 4 3 2 2 (x - x + 2 x - 3 x + 5 x - 6 x + 4 x - 1) (-1 + x) and in Maple format -(2*x^8-3*x^7+5*x^6-8*x^5+12*x^4-14*x^3+11*x^2-5*x+1)/(x^7-x^6+2*x^5-3*x^4+5*x^ 3-6*x^2+4*x-1)/(-1+x)^2 The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1016, 2011, 3965, 7796, 15303, 30014, 58850, 115389, 226267, 443730, 870248, 1706784, 3347455, 6565178, 12875758, 25251939, 49523803, 97125229, 190480226, 373566544] The limit of a(n+1)/a(n) as n goes to infinity is 1.96118653090 a(n) is asymptotic to .626326117348*1.96118653090^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [1, 1, 2, 2, 2, 2], denoted by the variable, X[1, 1, 2, 2, 2, 2], is 8 8 7 7 6 6 5 5 4 - (2 x %1 - 2 x - 3 x %1 + 3 x + 5 x %1 - 5 x - 7 x %1 + 8 x + 7 x %1 4 3 3 2 2 / 2 7 - 12 x - 4 x %1 + 14 x + x %1 - 11 x + 5 x - 1) / ((-1 + x) (x %1 / 7 6 6 5 5 4 4 3 3 - x - x %1 + x + 2 x %1 - 2 x - 3 x %1 + 3 x + 3 x %1 - 5 x 2 2 - x %1 + 6 x - 4 x + 1)) %1 := X[1, 1, 2, 2, 2, 2] and in Maple format -(2*x^8*X[1,1,2,2,2,2]-2*x^8-3*x^7*X[1,1,2,2,2,2]+3*x^7+5*x^6*X[1,1,2,2,2,2]-5* x^6-7*x^5*X[1,1,2,2,2,2]+8*x^5+7*x^4*X[1,1,2,2,2,2]-12*x^4-4*x^3*X[1,1,2,2,2,2] +14*x^3+x^2*X[1,1,2,2,2,2]-11*x^2+5*x-1)/(-1+x)^2/(x^7*X[1,1,2,2,2,2]-x^7-x^6*X [1,1,2,2,2,2]+x^6+2*x^5*X[1,1,2,2,2,2]-2*x^5-3*x^4*X[1,1,2,2,2,2]+3*x^4+3*x^3*X [1,1,2,2,2,2]-5*x^3-x^2*X[1,1,2,2,2,2]+6*x^2-4*x+1) Furthermore, the expectation of the random variable number of occurences (by\ 13 n containment) of , [1, 1, 2, 2, 2, 2], equals , - -- + ---- 32 32 1011 67 n The variance equals , - ---- + ---- 1024 1024 50469 2889 n The , 3, -th moment about the mean is , - ----- + ------ 16384 16384 8778255 13467 2 46261 The , 4, -th moment about the mean is , - ------- + ------- n + ------ n 1048576 1048576 262144 The compositions of, 10, that yield the, 27, -th largest growth, that is, 1.9659482366454853372, are , [1, 1, 1, 1, 6], [1, 1, 1, 6, 1], [1, 1, 6, 1, 1], [1, 6, 1, 1, 1], [6, 1, 1, 1, 1] Theorem Number, 27, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 1, 1, 1, 6] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 8 7 6 4 3 2 \ n 2 x - 2 x + x + x - 4 x + 6 x - 4 x + 1 ) a(n) x = - --------------------------------------------- / 5 4 3 2 4 ----- (x + x + x + x + x - 1) (-1 + x) n = 0 and in Maple format -(2*x^8-2*x^7+x^6+x^4-4*x^3+6*x^2-4*x+1)/(x^5+x^4+x^3+x^2+x-1)/(-1+x)^4 The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1017, 2018, 3994, 7889, 15559, 30653, 60345, 118739, 233563, 459329, 903205, 1775878, 3491548, 6864511, 13495631, 26532124, 52161254, 102546862, 201602430, 396340626] The limit of a(n+1)/a(n) as n goes to infinity is 1.96594823665 a(n) is asymptotic to .617884376698*1.96594823665^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [1, 1, 1, 1, 6], denoted by the variable, X[1, 1, 1, 1, 6], is 8 8 7 7 - (2 x X[1, 1, 1, 1, 6] - 2 x - 2 x X[1, 1, 1, 1, 6] + 2 x 6 6 4 3 2 / 3 + x X[1, 1, 1, 1, 6] - x - x + 4 x - 6 x + 4 x - 1) / ((-1 + x) / 6 6 (x X[1, 1, 1, 1, 6] - x + 2 x - 1)) and in Maple format -(2*x^8*X[1,1,1,1,6]-2*x^8-2*x^7*X[1,1,1,1,6]+2*x^7+x^6*X[1,1,1,1,6]-x^6-x^4+4* x^3-6*x^2+4*x-1)/(-1+x)^3/(x^6*X[1,1,1,1,6]-x^6+2*x-1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [1, 1, 1, 1, 6], equals , - 3/16 + ---- 64 75 53 n The variance equals , - --- + ---- 512 4096 339 141 n The , 3, -th moment about the mean is , - ---- + ----- 4096 16384 34881 8427 2 75113 The , 4, -th moment about the mean is , ------ + -------- n - ------- n 524288 16777216 8388608 The compositions of, 10, that yield the, 28, -th largest growth, that is, 1.9671682128139660358, are , [2, 1, 1, 1, 5], [5, 1, 1, 1, 2] Theorem Number, 28, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 1, 1, 5] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 41 36 35 34 33 31 30 29 28 ) a(n) x = - (x + 2 x + x - x + x + x + x - 3 x + x / ----- n = 0 25 24 23 22 21 20 19 18 17 + x + x - 4 x + 4 x - 5 x + 2 x + 5 x - 8 x + 6 x 16 15 14 13 12 11 10 9 8 7 - 3 x - x + 5 x - 4 x + x - x + 7 x - 18 x + 23 x - 15 x 6 5 4 3 2 / 6 - x + 20 x - 35 x + 35 x - 21 x + 7 x - 1) / ((x + x - 1) / 6 5 (x - x + 2 x - 1) 24 17 16 14 13 9 8 4 3 2 (x - x + x + 2 x - 2 x - x + x + x - 4 x + 6 x - 4 x + 1) 6 (x - x + 1)) and in Maple format -(x^41+2*x^36+x^35-x^34+x^33+x^31+x^30-3*x^29+x^28+x^25+x^24-4*x^23+4*x^22-5*x^ 21+2*x^20+5*x^19-8*x^18+6*x^17-3*x^16-x^15+5*x^14-4*x^13+x^12-x^11+7*x^10-18*x^ 9+23*x^8-15*x^7-x^6+20*x^5-35*x^4+35*x^3-21*x^2+7*x-1)/(x^6+x-1)/(x^6-x^5+2*x-1 )/(x^24-x^17+x^16+2*x^14-2*x^13-x^9+x^8+x^4-4*x^3+6*x^2-4*x+1)/(x^6-x+1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1017, 2018, 3994, 7889, 15559, 30653, 60345, 118740, 233573, 459386, 903449, 1776752, 3494317, 6872533, 13517356, 26587980, 52299133, 102876388, 202369818, 398090514] The limit of a(n+1)/a(n) as n goes to infinity is 1.96716821281 a(n) is asymptotic to .609183935429*1.96716821281^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [2, 1, 1, 1, 5], denoted by the variable, X[2, 1, 1, 1, 5], is 41 7 41 41 41 41 - (x X[2, 1, 1, 1, 5] - 7 x %5 + 21 x %4 - 35 x %3 + 35 x %2 41 41 36 41 36 35 - 21 x %1 + 7 x X[2, 1, 1, 1, 5] - 2 x %5 - x + 12 x %4 - x %5 36 35 34 36 35 34 - 30 x %3 + 6 x %4 + x %5 + 40 x %2 - 15 x %3 - 6 x %4 33 36 35 34 33 - x %5 - 30 x %1 + 20 x %2 + 15 x %3 + 6 x %4 36 35 34 33 36 + 12 x X[2, 1, 1, 1, 5] - 15 x %1 - 20 x %2 - 15 x %3 - 2 x 35 34 33 31 35 + 6 x X[2, 1, 1, 1, 5] + 15 x %1 + 20 x %2 + x %4 - x 34 33 31 30 34 - 6 x X[2, 1, 1, 1, 5] - 15 x %1 - 5 x %3 + x %4 + x 33 31 30 29 33 + 6 x X[2, 1, 1, 1, 5] + 10 x %2 - 5 x %3 - 3 x %4 - x 31 30 29 28 31 - 10 x %1 + 10 x %2 + 15 x %3 + x %4 + 5 x X[2, 1, 1, 1, 5] 30 29 28 31 30 - 10 x %1 - 30 x %2 - 5 x %3 - x + 5 x X[2, 1, 1, 1, 5] 29 28 30 29 28 + 30 x %1 + 10 x %2 - x - 15 x X[2, 1, 1, 1, 5] - 10 x %1 29 28 25 28 25 24 + 3 x + 5 x X[2, 1, 1, 1, 5] - x %3 - x + 4 x %2 - x %3 25 24 23 25 24 - 6 x %1 + 4 x %2 + 4 x %3 + 4 x X[2, 1, 1, 1, 5] - 6 x %1 23 22 25 24 23 - 16 x %2 - 3 x %3 - x + 4 x X[2, 1, 1, 1, 5] + 24 x %1 22 21 24 23 22 + 13 x %2 + x %3 - x - 16 x X[2, 1, 1, 1, 5] - 21 x %1 21 23 22 21 20 - 8 x %2 + 4 x + 15 x X[2, 1, 1, 1, 5] + 18 x %1 + 2 x %2 22 21 20 19 21 - 4 x - 16 x X[2, 1, 1, 1, 5] - 6 x %1 + 5 x %2 + 5 x 20 19 18 20 + 6 x X[2, 1, 1, 1, 5] - 15 x %1 - 8 x %2 - 2 x 19 18 17 19 + 15 x X[2, 1, 1, 1, 5] + 24 x %1 + 5 x %2 - 5 x 18 17 16 18 - 24 x X[2, 1, 1, 1, 5] - 16 x %1 - x %2 + 8 x 17 16 17 16 + 17 x X[2, 1, 1, 1, 5] + 5 x %1 - 6 x - 7 x X[2, 1, 1, 1, 5] 15 16 15 14 15 + x %1 + 3 x - 2 x X[2, 1, 1, 1, 5] - 5 x %1 + x 14 13 14 13 + 10 x X[2, 1, 1, 1, 5] + 4 x %1 - 5 x - 8 x X[2, 1, 1, 1, 5] 12 13 12 12 11 - x %1 + 4 x + 2 x X[2, 1, 1, 1, 5] - x - x X[2, 1, 1, 1, 5] 11 10 10 9 9 + x + 7 x X[2, 1, 1, 1, 5] - 7 x - 18 x X[2, 1, 1, 1, 5] + 18 x 8 8 7 7 + 23 x X[2, 1, 1, 1, 5] - 23 x - 16 x X[2, 1, 1, 1, 5] + 15 x 6 6 5 5 4 3 + 6 x X[2, 1, 1, 1, 5] + x - x X[2, 1, 1, 1, 5] - 20 x + 35 x - 35 x 2 / 6 6 + 21 x - 7 x + 1) / ((x X[2, 1, 1, 1, 5] - x - x + 1) / 6 6 (x X[2, 1, 1, 1, 5] - x + x - 1) 6 6 5 5 24 (x X[2, 1, 1, 1, 5] - x - x X[2, 1, 1, 1, 5] + x - 2 x + 1) (x %3 24 24 24 24 17 17 - 4 x %2 + 6 x %1 - 4 x X[2, 1, 1, 1, 5] + x + x %2 - 3 x %1 16 17 16 17 - x %2 + 3 x X[2, 1, 1, 1, 5] + 3 x %1 - x 16 16 14 14 - 3 x X[2, 1, 1, 1, 5] + x + 2 x %1 - 4 x X[2, 1, 1, 1, 5] 13 14 13 13 9 - 2 x %1 + 2 x + 4 x X[2, 1, 1, 1, 5] - 2 x + x X[2, 1, 1, 1, 5] 9 8 8 4 3 2 - x - x X[2, 1, 1, 1, 5] + x + x - 4 x + 6 x - 4 x + 1)) 2 %1 := X[2, 1, 1, 1, 5] 3 %2 := X[2, 1, 1, 1, 5] 4 %3 := X[2, 1, 1, 1, 5] 5 %4 := X[2, 1, 1, 1, 5] 6 %5 := X[2, 1, 1, 1, 5] and in Maple format -(x^41*X[2,1,1,1,5]^7-7*x^41*X[2,1,1,1,5]^6+21*x^41*X[2,1,1,1,5]^5-35*x^41*X[2, 1,1,1,5]^4+35*x^41*X[2,1,1,1,5]^3-21*x^41*X[2,1,1,1,5]^2+7*x^41*X[2,1,1,1,5]-2* x^36*X[2,1,1,1,5]^6-x^41+12*x^36*X[2,1,1,1,5]^5-x^35*X[2,1,1,1,5]^6-30*x^36*X[2 ,1,1,1,5]^4+6*x^35*X[2,1,1,1,5]^5+x^34*X[2,1,1,1,5]^6+40*x^36*X[2,1,1,1,5]^3-15 *x^35*X[2,1,1,1,5]^4-6*x^34*X[2,1,1,1,5]^5-x^33*X[2,1,1,1,5]^6-30*x^36*X[2,1,1, 1,5]^2+20*x^35*X[2,1,1,1,5]^3+15*x^34*X[2,1,1,1,5]^4+6*x^33*X[2,1,1,1,5]^5+12*x ^36*X[2,1,1,1,5]-15*x^35*X[2,1,1,1,5]^2-20*x^34*X[2,1,1,1,5]^3-15*x^33*X[2,1,1, 1,5]^4-2*x^36+6*x^35*X[2,1,1,1,5]+15*x^34*X[2,1,1,1,5]^2+20*x^33*X[2,1,1,1,5]^3 +x^31*X[2,1,1,1,5]^5-x^35-6*x^34*X[2,1,1,1,5]-15*x^33*X[2,1,1,1,5]^2-5*x^31*X[2 ,1,1,1,5]^4+x^30*X[2,1,1,1,5]^5+x^34+6*x^33*X[2,1,1,1,5]+10*x^31*X[2,1,1,1,5]^3 -5*x^30*X[2,1,1,1,5]^4-3*x^29*X[2,1,1,1,5]^5-x^33-10*x^31*X[2,1,1,1,5]^2+10*x^ 30*X[2,1,1,1,5]^3+15*x^29*X[2,1,1,1,5]^4+x^28*X[2,1,1,1,5]^5+5*x^31*X[2,1,1,1,5 ]-10*x^30*X[2,1,1,1,5]^2-30*x^29*X[2,1,1,1,5]^3-5*x^28*X[2,1,1,1,5]^4-x^31+5*x^ 30*X[2,1,1,1,5]+30*x^29*X[2,1,1,1,5]^2+10*x^28*X[2,1,1,1,5]^3-x^30-15*x^29*X[2, 1,1,1,5]-10*x^28*X[2,1,1,1,5]^2+3*x^29+5*x^28*X[2,1,1,1,5]-x^25*X[2,1,1,1,5]^4- x^28+4*x^25*X[2,1,1,1,5]^3-x^24*X[2,1,1,1,5]^4-6*x^25*X[2,1,1,1,5]^2+4*x^24*X[2 ,1,1,1,5]^3+4*x^23*X[2,1,1,1,5]^4+4*x^25*X[2,1,1,1,5]-6*x^24*X[2,1,1,1,5]^2-16* x^23*X[2,1,1,1,5]^3-3*x^22*X[2,1,1,1,5]^4-x^25+4*x^24*X[2,1,1,1,5]+24*x^23*X[2, 1,1,1,5]^2+13*x^22*X[2,1,1,1,5]^3+x^21*X[2,1,1,1,5]^4-x^24-16*x^23*X[2,1,1,1,5] -21*x^22*X[2,1,1,1,5]^2-8*x^21*X[2,1,1,1,5]^3+4*x^23+15*x^22*X[2,1,1,1,5]+18*x^ 21*X[2,1,1,1,5]^2+2*x^20*X[2,1,1,1,5]^3-4*x^22-16*x^21*X[2,1,1,1,5]-6*x^20*X[2, 1,1,1,5]^2+5*x^19*X[2,1,1,1,5]^3+5*x^21+6*x^20*X[2,1,1,1,5]-15*x^19*X[2,1,1,1,5 ]^2-8*x^18*X[2,1,1,1,5]^3-2*x^20+15*x^19*X[2,1,1,1,5]+24*x^18*X[2,1,1,1,5]^2+5* x^17*X[2,1,1,1,5]^3-5*x^19-24*x^18*X[2,1,1,1,5]-16*x^17*X[2,1,1,1,5]^2-x^16*X[2 ,1,1,1,5]^3+8*x^18+17*x^17*X[2,1,1,1,5]+5*x^16*X[2,1,1,1,5]^2-6*x^17-7*x^16*X[2 ,1,1,1,5]+x^15*X[2,1,1,1,5]^2+3*x^16-2*x^15*X[2,1,1,1,5]-5*x^14*X[2,1,1,1,5]^2+ x^15+10*x^14*X[2,1,1,1,5]+4*x^13*X[2,1,1,1,5]^2-5*x^14-8*x^13*X[2,1,1,1,5]-x^12 *X[2,1,1,1,5]^2+4*x^13+2*x^12*X[2,1,1,1,5]-x^12-x^11*X[2,1,1,1,5]+x^11+7*x^10*X [2,1,1,1,5]-7*x^10-18*x^9*X[2,1,1,1,5]+18*x^9+23*x^8*X[2,1,1,1,5]-23*x^8-16*x^7 *X[2,1,1,1,5]+15*x^7+6*x^6*X[2,1,1,1,5]+x^6-x^5*X[2,1,1,1,5]-20*x^5+35*x^4-35*x ^3+21*x^2-7*x+1)/(x^6*X[2,1,1,1,5]-x^6-x+1)/(x^6*X[2,1,1,1,5]-x^6+x-1)/(x^6*X[2 ,1,1,1,5]-x^6-x^5*X[2,1,1,1,5]+x^5-2*x+1)/(x^24*X[2,1,1,1,5]^4-4*x^24*X[2,1,1,1 ,5]^3+6*x^24*X[2,1,1,1,5]^2-4*x^24*X[2,1,1,1,5]+x^24+x^17*X[2,1,1,1,5]^3-3*x^17 *X[2,1,1,1,5]^2-x^16*X[2,1,1,1,5]^3+3*x^17*X[2,1,1,1,5]+3*x^16*X[2,1,1,1,5]^2-x ^17-3*x^16*X[2,1,1,1,5]+x^16+2*x^14*X[2,1,1,1,5]^2-4*x^14*X[2,1,1,1,5]-2*x^13*X [2,1,1,1,5]^2+2*x^14+4*x^13*X[2,1,1,1,5]-2*x^13+x^9*X[2,1,1,1,5]-x^9-x^8*X[2,1, 1,1,5]+x^8+x^4-4*x^3+6*x^2-4*x+1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [2, 1, 1, 1, 5], equals , - 3/16 + ---- 64 173 57 n The variance equals , - ---- + ---- 1024 4096 1113 357 n The , 3, -th moment about the mean is , - ---- + ----- 8192 32768 5473 9747 2 68829 The , 4, -th moment about the mean is , ------- + -------- n - ------- n 1048576 16777216 8388608 The compositions of, 10, that yield the, 29, -th largest growth, that is, 1.9691817825046685829, are , [1, 2, 1, 2, 4], [1, 2, 1, 4, 2], [1, 2, 2, 1, 4], [1, 2, 4, 1, 2], [1, 4, 1, 2, 2], [1, 4, 2, 1, 2], [2, 1, 2, 4, 1], [2, 1, 4, 2, 1], [2, 2, 1, 4, 1], [2, 4, 1, 2, 1], [4, 1, 2, 2, 1], [4, 2, 1, 2, 1] Theorem Number, 29, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 2, 1, 2, 4] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n ) a(n) x = / ----- n = 0 13 12 11 9 7 6 5 4 3 2 x - x + x + x - 2 x + 4 x - 3 x + 2 x - 4 x + 6 x - 4 x + 1 ------------------------------------------------------------------------- 12 11 8 7 6 5 4 3 2 (-1 + x) (x - x + x + x - 3 x + 3 x - x + 2 x - 5 x + 4 x - 1) and in Maple format (x^13-x^12+x^11+x^9-2*x^7+4*x^6-3*x^5+2*x^4-4*x^3+6*x^2-4*x+1)/(-1+x)/(x^12-x^ 11+x^8+x^7-3*x^6+3*x^5-x^4+2*x^3-5*x^2+4*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1017, 2018, 3994, 7889, 15560, 30661, 60384, 118888, 234052, 460774, 907161, 1786104, 3516841, 6924955, 13636207, 26852063, 52876842, 104124937, 205042562, 403768723] The limit of a(n+1)/a(n) as n goes to infinity is 1.96918178250 a(n) is asymptotic to .599187002474*1.96918178250^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [1, 2, 1, 2, 4], denoted by the variable, X[1, 2, 1, 2, 4], is 13 13 12 13 12 (x %1 - 2 x X[1, 2, 1, 2, 4] - x %1 + x + 2 x X[1, 2, 1, 2, 4] 11 12 11 11 9 9 + x %1 - x - 2 x X[1, 2, 1, 2, 4] + x - x X[1, 2, 1, 2, 4] + x 7 7 6 6 + 2 x X[1, 2, 1, 2, 4] - 2 x - 4 x X[1, 2, 1, 2, 4] + 4 x 5 5 4 4 3 2 + 3 x X[1, 2, 1, 2, 4] - 3 x - x X[1, 2, 1, 2, 4] + 2 x - 4 x + 6 x / 12 12 11 12 - 4 x + 1) / ((-1 + x) (x %1 - 2 x X[1, 2, 1, 2, 4] - x %1 + x / 11 11 8 8 + 2 x X[1, 2, 1, 2, 4] - x - x X[1, 2, 1, 2, 4] + x 7 7 6 6 - x X[1, 2, 1, 2, 4] + x + 3 x X[1, 2, 1, 2, 4] - 3 x 5 5 4 4 3 2 - 3 x X[1, 2, 1, 2, 4] + 3 x + x X[1, 2, 1, 2, 4] - x + 2 x - 5 x + 4 x - 1)) 2 %1 := X[1, 2, 1, 2, 4] and in Maple format (x^13*X[1,2,1,2,4]^2-2*x^13*X[1,2,1,2,4]-x^12*X[1,2,1,2,4]^2+x^13+2*x^12*X[1,2, 1,2,4]+x^11*X[1,2,1,2,4]^2-x^12-2*x^11*X[1,2,1,2,4]+x^11-x^9*X[1,2,1,2,4]+x^9+2 *x^7*X[1,2,1,2,4]-2*x^7-4*x^6*X[1,2,1,2,4]+4*x^6+3*x^5*X[1,2,1,2,4]-3*x^5-x^4*X [1,2,1,2,4]+2*x^4-4*x^3+6*x^2-4*x+1)/(-1+x)/(x^12*X[1,2,1,2,4]^2-2*x^12*X[1,2,1 ,2,4]-x^11*X[1,2,1,2,4]^2+x^12+2*x^11*X[1,2,1,2,4]-x^11-x^8*X[1,2,1,2,4]+x^8-x^ 7*X[1,2,1,2,4]+x^7+3*x^6*X[1,2,1,2,4]-3*x^6-3*x^5*X[1,2,1,2,4]+3*x^5+x^4*X[1,2, 1,2,4]-x^4+2*x^3-5*x^2+4*x-1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [1, 2, 1, 2, 4], equals , - 3/16 + ---- 64 207 65 n The variance equals , - ---- + ---- 1024 4096 1929 135 n The , 3, -th moment about the mean is , - ---- + ----- 8192 8192 197685 12675 2 11093 The , 4, -th moment about the mean is , - ------- + -------- n - ------- n 1048576 16777216 8388608 The compositions of, 10, that yield the, 30, -th largest growth, that is, 1.9692165211230598361, are , [2, 1, 1, 2, 4], [2, 1, 1, 4, 2], [2, 2, 1, 1, 4], [2, 4, 1, 1, 2], [4, 1, 1, 2, 2], [4, 2, 1, 1, 2] Theorem Number, 30, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 1, 2, 4] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 16 14 13 12 8 7 6 5 4 ) a(n) x = - (x + x + x + x + x - 2 x + 4 x - 3 x + 2 x / ----- n = 0 3 2 / 17 15 12 9 8 7 6 - 4 x + 6 x - 4 x + 1) / (x + 2 x - x + x - x + 4 x - 6 x / 5 4 3 2 + 4 x - 3 x + 7 x - 9 x + 5 x - 1) and in Maple format -(x^16+x^14+x^13+x^12+x^8-2*x^7+4*x^6-3*x^5+2*x^4-4*x^3+6*x^2-4*x+1)/(x^17+2*x^ 15-x^12+x^9-x^8+4*x^7-6*x^6+4*x^5-3*x^4+7*x^3-9*x^2+5*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1017, 2018, 3994, 7889, 15560, 30660, 60376, 118851, 233923, 460397, 906183, 1783772, 3511611, 6913745, 13612975, 26805099, 52783595, 103942089, 204686894, 403080085] The limit of a(n+1)/a(n) as n goes to infinity is 1.96921652112 a(n) is asymptotic to .597887221874*1.96921652112^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [2, 1, 1, 2, 4], denoted by the variable, X[2, 1, 1, 2, 4], is 17 17 16 17 16 15 - (x %2 - 3 x %1 - x %2 + 3 x X[2, 1, 1, 2, 4] + 3 x %1 + x %2 17 16 15 16 15 - x - 3 x X[2, 1, 1, 2, 4] - 3 x %1 + x + 3 x X[2, 1, 1, 2, 4] 15 12 12 12 9 9 - x + x %1 - 2 x X[2, 1, 1, 2, 4] + x + x X[2, 1, 1, 2, 4] - x 8 8 7 7 - 3 x X[2, 1, 1, 2, 4] + 3 x + 6 x X[2, 1, 1, 2, 4] - 6 x 6 6 5 5 - 7 x X[2, 1, 1, 2, 4] + 7 x + 4 x X[2, 1, 1, 2, 4] - 5 x 4 4 3 2 / 18 - x X[2, 1, 1, 2, 4] + 6 x - 10 x + 10 x - 5 x + 1) / (x %2 / 18 17 18 17 16 18 - 3 x %1 - x %2 + 3 x X[2, 1, 1, 2, 4] + 3 x %1 + 2 x %2 - x 17 16 15 17 - 3 x X[2, 1, 1, 2, 4] - 6 x %1 - x %2 + x 16 15 16 15 + 6 x X[2, 1, 1, 2, 4] + 4 x %1 - 2 x - 5 x X[2, 1, 1, 2, 4] 15 13 13 12 13 + 2 x + x %1 - 2 x X[2, 1, 1, 2, 4] - x %1 + x 12 12 10 10 + 2 x X[2, 1, 1, 2, 4] - x + x X[2, 1, 1, 2, 4] - x 9 9 8 8 - 2 x X[2, 1, 1, 2, 4] + 2 x + 5 x X[2, 1, 1, 2, 4] - 5 x 7 7 6 6 - 10 x X[2, 1, 1, 2, 4] + 10 x + 10 x X[2, 1, 1, 2, 4] - 10 x 5 5 4 4 3 - 5 x X[2, 1, 1, 2, 4] + 7 x + x X[2, 1, 1, 2, 4] - 10 x + 16 x 2 - 14 x + 6 x - 1) 2 %1 := X[2, 1, 1, 2, 4] 3 %2 := X[2, 1, 1, 2, 4] and in Maple format -(x^17*X[2,1,1,2,4]^3-3*x^17*X[2,1,1,2,4]^2-x^16*X[2,1,1,2,4]^3+3*x^17*X[2,1,1, 2,4]+3*x^16*X[2,1,1,2,4]^2+x^15*X[2,1,1,2,4]^3-x^17-3*x^16*X[2,1,1,2,4]-3*x^15* X[2,1,1,2,4]^2+x^16+3*x^15*X[2,1,1,2,4]-x^15+x^12*X[2,1,1,2,4]^2-2*x^12*X[2,1,1 ,2,4]+x^12+x^9*X[2,1,1,2,4]-x^9-3*x^8*X[2,1,1,2,4]+3*x^8+6*x^7*X[2,1,1,2,4]-6*x ^7-7*x^6*X[2,1,1,2,4]+7*x^6+4*x^5*X[2,1,1,2,4]-5*x^5-x^4*X[2,1,1,2,4]+6*x^4-10* x^3+10*x^2-5*x+1)/(x^18*X[2,1,1,2,4]^3-3*x^18*X[2,1,1,2,4]^2-x^17*X[2,1,1,2,4]^ 3+3*x^18*X[2,1,1,2,4]+3*x^17*X[2,1,1,2,4]^2+2*x^16*X[2,1,1,2,4]^3-x^18-3*x^17*X [2,1,1,2,4]-6*x^16*X[2,1,1,2,4]^2-x^15*X[2,1,1,2,4]^3+x^17+6*x^16*X[2,1,1,2,4]+ 4*x^15*X[2,1,1,2,4]^2-2*x^16-5*x^15*X[2,1,1,2,4]+2*x^15+x^13*X[2,1,1,2,4]^2-2*x ^13*X[2,1,1,2,4]-x^12*X[2,1,1,2,4]^2+x^13+2*x^12*X[2,1,1,2,4]-x^12+x^10*X[2,1,1 ,2,4]-x^10-2*x^9*X[2,1,1,2,4]+2*x^9+5*x^8*X[2,1,1,2,4]-5*x^8-10*x^7*X[2,1,1,2,4 ]+10*x^7+10*x^6*X[2,1,1,2,4]-10*x^6-5*x^5*X[2,1,1,2,4]+7*x^5+x^4*X[2,1,1,2,4]-\ 10*x^4+16*x^3-14*x^2+6*x-1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [2, 1, 1, 2, 4], equals , - 3/16 + ---- 64 211 65 n The variance equals , - ---- + ---- 1024 4096 63 537 n The , 3, -th moment about the mean is , - --- + ----- 256 32768 215557 12675 2 18293 The , 4, -th moment about the mean is , - ------- + -------- n - ------- n 1048576 16777216 8388608 The compositions of, 10, that yield the, 31, -th largest growth, that is, 1.9693144732632464526, are , [3, 1, 1, 1, 4], [4, 1, 1, 1, 3] Theorem Number, 31, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [3, 1, 1, 1, 4] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 40 39 38 37 36 35 34 ) a(n) x = - (x + 6 x + 15 x + 20 x + 15 x + 8 x + 10 x / ----- n = 0 33 32 31 30 29 28 27 26 25 + 17 x + 17 x + 8 x + x + 2 x + 4 x + 2 x - 3 x - 4 x 24 23 22 20 19 18 17 16 14 13 - x + x + 4 x - 7 x + x + x + 5 x - 4 x + x - 2 x 12 11 10 9 8 7 6 5 4 + 5 x - 5 x + 5 x - 3 x - 8 x + 19 x - 22 x + 27 x - 36 x 3 2 / 2 24 23 22 21 + 35 x - 21 x + 7 x - 1) / ((x + 1) (x + 4 x + 6 x + 4 x / 20 15 14 13 12 9 7 4 3 2 + x - x + 2 x + x - 2 x - x + x + x - 4 x + 6 x - 4 x + 1) 6 5 4 4 3 2 6 5 (x + x - x + 2 x - 1) (x + x - x - x + 1) (x + x + x - 1)) and in Maple format -(x^40+6*x^39+15*x^38+20*x^37+15*x^36+8*x^35+10*x^34+17*x^33+17*x^32+8*x^31+x^ 30+2*x^29+4*x^28+2*x^27-3*x^26-4*x^25-x^24+x^23+4*x^22-7*x^20+x^19+x^18+5*x^17-\ 4*x^16+x^14-2*x^13+5*x^12-5*x^11+5*x^10-3*x^9-8*x^8+19*x^7-22*x^6+27*x^5-36*x^4 +35*x^3-21*x^2+7*x-1)/(x^2+1)/(x^24+4*x^23+6*x^22+4*x^21+x^20-x^15+2*x^14+x^13-\ 2*x^12-x^9+x^7+x^4-4*x^3+6*x^2-4*x+1)/(x^6+x^5-x^4+2*x-1)/(x^4+x^3-x^2-x+1)/(x^ 6+x^5+x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1017, 2018, 3994, 7889, 15559, 30653, 60346, 118750, 233630, 459630, 904323, 1779521, 3502339, 6894257, 13573198, 26725754, 52628093, 103641308, 204110443, 401981759] The limit of a(n+1)/a(n) as n goes to infinity is 1.96931447326 a(n) is asymptotic to .595489612304*1.96931447326^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [3, 1, 1, 1, 4], denoted by the variable, X[3, 1, 1, 1, 4], is 40 40 39 40 39 38 40 - (x %6 - 7 x %5 + 6 x %6 + 21 x %4 - 42 x %5 + 15 x %6 - 35 x %3 39 38 37 40 39 + 126 x %4 - 105 x %5 + 20 x %6 + 35 x %2 - 210 x %3 38 37 36 40 39 + 315 x %4 - 140 x %5 + 15 x %6 - 21 x %1 + 210 x %2 38 37 36 35 40 - 525 x %3 + 420 x %4 - 105 x %5 + 6 x %6 + 7 x X[3, 1, 1, 1, 4] 39 38 37 36 35 34 - 126 x %1 + 525 x %2 - 700 x %3 + 315 x %4 - 44 x %5 + x %6 40 39 38 37 36 - x + 42 x X[3, 1, 1, 1, 4] - 315 x %1 + 700 x %2 - 525 x %3 35 34 39 38 37 + 138 x %4 - 16 x %5 - 6 x + 105 x X[3, 1, 1, 1, 4] - 420 x %1 36 35 34 33 38 + 525 x %2 - 240 x %3 + 75 x %4 - 17 x %5 - 15 x 37 36 35 34 + 140 x X[3, 1, 1, 1, 4] - 315 x %1 + 250 x %2 - 170 x %3 33 32 37 36 35 + 102 x %4 - 17 x %5 - 20 x + 105 x X[3, 1, 1, 1, 4] - 156 x %1 34 33 32 31 36 + 215 x %2 - 255 x %3 + 102 x %4 - 8 x %5 - 15 x 35 34 33 32 + 54 x X[3, 1, 1, 1, 4] - 156 x %1 + 340 x %2 - 255 x %3 31 35 34 33 32 + 48 x %4 - 8 x + 61 x X[3, 1, 1, 1, 4] - 255 x %1 + 340 x %2 31 30 34 33 32 - 120 x %3 + x %4 - 10 x + 102 x X[3, 1, 1, 1, 4] - 255 x %1 31 30 29 28 33 + 160 x %2 - 5 x %3 + 2 x %4 - 2 x %5 - 17 x 32 31 30 29 + 102 x X[3, 1, 1, 1, 4] - 120 x %1 + 10 x %2 - 10 x %3 28 27 32 31 30 + 14 x %4 - x %5 - 17 x + 48 x X[3, 1, 1, 1, 4] - 10 x %1 29 28 27 31 30 + 20 x %2 - 40 x %3 + 7 x %4 - 8 x + 5 x X[3, 1, 1, 1, 4] 29 28 27 26 30 - 20 x %1 + 60 x %2 - 20 x %3 - 3 x %4 - x 29 28 27 26 25 + 10 x X[3, 1, 1, 1, 4] - 50 x %1 + 30 x %2 + 15 x %3 - 4 x %4 29 28 27 26 25 - 2 x + 22 x X[3, 1, 1, 1, 4] - 25 x %1 - 30 x %2 + 20 x %3 28 27 26 25 24 - 4 x + 11 x X[3, 1, 1, 1, 4] + 30 x %1 - 40 x %2 + x %3 23 27 26 25 24 + x %4 - 2 x - 15 x X[3, 1, 1, 1, 4] + 40 x %1 - 4 x %2 23 26 25 24 23 - 5 x %3 + 3 x - 20 x X[3, 1, 1, 1, 4] + 6 x %1 + 10 x %2 22 25 24 23 22 - 3 x %3 + 4 x - 4 x X[3, 1, 1, 1, 4] - 10 x %1 + 13 x %2 21 24 23 22 21 20 - x %3 + x + 5 x X[3, 1, 1, 1, 4] - 21 x %1 + 3 x %2 + 4 x %3 23 22 21 20 22 - x + 15 x X[3, 1, 1, 1, 4] - 3 x %1 - 19 x %2 - 4 x 21 20 19 18 + x X[3, 1, 1, 1, 4] + 33 x %1 + x %2 - 2 x %3 20 19 18 17 20 - 25 x X[3, 1, 1, 1, 4] - 3 x %1 + 7 x %2 + x %3 + 7 x 19 18 17 19 + 3 x X[3, 1, 1, 1, 4] - 9 x %1 + x %2 - x 18 17 16 18 + 5 x X[3, 1, 1, 1, 4] - 10 x %1 - x %2 - x 17 16 15 17 + 13 x X[3, 1, 1, 1, 4] + 6 x %1 - 4 x %2 - 5 x 16 15 14 16 - 9 x X[3, 1, 1, 1, 4] + 8 x %1 + 4 x %2 + 4 x 15 14 13 14 - 4 x X[3, 1, 1, 1, 4] - 9 x %1 - x %2 + 6 x X[3, 1, 1, 1, 4] 13 14 13 12 13 + 4 x %1 - x - 5 x X[3, 1, 1, 1, 4] - 5 x %1 + 2 x 12 11 12 11 + 10 x X[3, 1, 1, 1, 4] + 4 x %1 - 5 x - 9 x X[3, 1, 1, 1, 4] 10 11 10 10 9 - x %1 + 5 x + 6 x X[3, 1, 1, 1, 4] - 5 x - 3 x X[3, 1, 1, 1, 4] 9 8 8 7 7 + 3 x - 8 x X[3, 1, 1, 1, 4] + 8 x + 18 x X[3, 1, 1, 1, 4] - 19 x 6 6 5 5 - 15 x X[3, 1, 1, 1, 4] + 22 x + 6 x X[3, 1, 1, 1, 4] - 27 x 4 4 3 2 / - x X[3, 1, 1, 1, 4] + 36 x - 35 x + 21 x - 7 x + 1) / ( / 6 6 5 5 (x X[3, 1, 1, 1, 4] - x + x X[3, 1, 1, 1, 4] - x + x - 1) 6 6 5 5 (x X[3, 1, 1, 1, 4] - x + x X[3, 1, 1, 1, 4] - x - x + 1) ( 6 6 5 5 4 x X[3, 1, 1, 1, 4] - x + x X[3, 1, 1, 1, 4] - x - x X[3, 1, 1, 1, 4] 4 24 24 23 24 23 + x - 2 x + 1) (x %3 - 4 x %2 + 4 x %3 + 6 x %1 - 16 x %2 22 24 23 22 21 + 6 x %3 - 4 x X[3, 1, 1, 1, 4] + 24 x %1 - 24 x %2 + 4 x %3 24 23 22 21 20 23 + x - 16 x X[3, 1, 1, 1, 4] + 36 x %1 - 16 x %2 + x %3 + 4 x 22 21 20 22 - 24 x X[3, 1, 1, 1, 4] + 24 x %1 - 4 x %2 + 6 x 21 20 21 20 - 16 x X[3, 1, 1, 1, 4] + 6 x %1 + 4 x - 4 x X[3, 1, 1, 1, 4] 20 15 15 15 14 13 + x + x %2 - 3 x %1 + 3 x X[3, 1, 1, 1, 4] + 2 x %1 - x %2 15 14 13 14 13 - x - 4 x X[3, 1, 1, 1, 4] + 3 x %1 + 2 x - 3 x X[3, 1, 1, 1, 4] 12 13 12 12 9 - 2 x %1 + x + 4 x X[3, 1, 1, 1, 4] - 2 x + x X[3, 1, 1, 1, 4] 9 7 7 4 3 2 - x - x X[3, 1, 1, 1, 4] + x + x - 4 x + 6 x - 4 x + 1)) 2 %1 := X[3, 1, 1, 1, 4] 3 %2 := X[3, 1, 1, 1, 4] 4 %3 := X[3, 1, 1, 1, 4] 5 %4 := X[3, 1, 1, 1, 4] 6 %5 := X[3, 1, 1, 1, 4] 7 %6 := X[3, 1, 1, 1, 4] and in Maple format -(x^40*X[3,1,1,1,4]^7-7*x^40*X[3,1,1,1,4]^6+6*x^39*X[3,1,1,1,4]^7+21*x^40*X[3,1 ,1,1,4]^5-42*x^39*X[3,1,1,1,4]^6+15*x^38*X[3,1,1,1,4]^7-35*x^40*X[3,1,1,1,4]^4+ 126*x^39*X[3,1,1,1,4]^5-105*x^38*X[3,1,1,1,4]^6+20*x^37*X[3,1,1,1,4]^7+35*x^40* X[3,1,1,1,4]^3-210*x^39*X[3,1,1,1,4]^4+315*x^38*X[3,1,1,1,4]^5-140*x^37*X[3,1,1 ,1,4]^6+15*x^36*X[3,1,1,1,4]^7-21*x^40*X[3,1,1,1,4]^2+210*x^39*X[3,1,1,1,4]^3-\ 525*x^38*X[3,1,1,1,4]^4+420*x^37*X[3,1,1,1,4]^5-105*x^36*X[3,1,1,1,4]^6+6*x^35* X[3,1,1,1,4]^7+7*x^40*X[3,1,1,1,4]-126*x^39*X[3,1,1,1,4]^2+525*x^38*X[3,1,1,1,4 ]^3-700*x^37*X[3,1,1,1,4]^4+315*x^36*X[3,1,1,1,4]^5-44*x^35*X[3,1,1,1,4]^6+x^34 *X[3,1,1,1,4]^7-x^40+42*x^39*X[3,1,1,1,4]-315*x^38*X[3,1,1,1,4]^2+700*x^37*X[3, 1,1,1,4]^3-525*x^36*X[3,1,1,1,4]^4+138*x^35*X[3,1,1,1,4]^5-16*x^34*X[3,1,1,1,4] ^6-6*x^39+105*x^38*X[3,1,1,1,4]-420*x^37*X[3,1,1,1,4]^2+525*x^36*X[3,1,1,1,4]^3 -240*x^35*X[3,1,1,1,4]^4+75*x^34*X[3,1,1,1,4]^5-17*x^33*X[3,1,1,1,4]^6-15*x^38+ 140*x^37*X[3,1,1,1,4]-315*x^36*X[3,1,1,1,4]^2+250*x^35*X[3,1,1,1,4]^3-170*x^34* X[3,1,1,1,4]^4+102*x^33*X[3,1,1,1,4]^5-17*x^32*X[3,1,1,1,4]^6-20*x^37+105*x^36* X[3,1,1,1,4]-156*x^35*X[3,1,1,1,4]^2+215*x^34*X[3,1,1,1,4]^3-255*x^33*X[3,1,1,1 ,4]^4+102*x^32*X[3,1,1,1,4]^5-8*x^31*X[3,1,1,1,4]^6-15*x^36+54*x^35*X[3,1,1,1,4 ]-156*x^34*X[3,1,1,1,4]^2+340*x^33*X[3,1,1,1,4]^3-255*x^32*X[3,1,1,1,4]^4+48*x^ 31*X[3,1,1,1,4]^5-8*x^35+61*x^34*X[3,1,1,1,4]-255*x^33*X[3,1,1,1,4]^2+340*x^32* X[3,1,1,1,4]^3-120*x^31*X[3,1,1,1,4]^4+x^30*X[3,1,1,1,4]^5-10*x^34+102*x^33*X[3 ,1,1,1,4]-255*x^32*X[3,1,1,1,4]^2+160*x^31*X[3,1,1,1,4]^3-5*x^30*X[3,1,1,1,4]^4 +2*x^29*X[3,1,1,1,4]^5-2*x^28*X[3,1,1,1,4]^6-17*x^33+102*x^32*X[3,1,1,1,4]-120* x^31*X[3,1,1,1,4]^2+10*x^30*X[3,1,1,1,4]^3-10*x^29*X[3,1,1,1,4]^4+14*x^28*X[3,1 ,1,1,4]^5-x^27*X[3,1,1,1,4]^6-17*x^32+48*x^31*X[3,1,1,1,4]-10*x^30*X[3,1,1,1,4] ^2+20*x^29*X[3,1,1,1,4]^3-40*x^28*X[3,1,1,1,4]^4+7*x^27*X[3,1,1,1,4]^5-8*x^31+5 *x^30*X[3,1,1,1,4]-20*x^29*X[3,1,1,1,4]^2+60*x^28*X[3,1,1,1,4]^3-20*x^27*X[3,1, 1,1,4]^4-3*x^26*X[3,1,1,1,4]^5-x^30+10*x^29*X[3,1,1,1,4]-50*x^28*X[3,1,1,1,4]^2 +30*x^27*X[3,1,1,1,4]^3+15*x^26*X[3,1,1,1,4]^4-4*x^25*X[3,1,1,1,4]^5-2*x^29+22* x^28*X[3,1,1,1,4]-25*x^27*X[3,1,1,1,4]^2-30*x^26*X[3,1,1,1,4]^3+20*x^25*X[3,1,1 ,1,4]^4-4*x^28+11*x^27*X[3,1,1,1,4]+30*x^26*X[3,1,1,1,4]^2-40*x^25*X[3,1,1,1,4] ^3+x^24*X[3,1,1,1,4]^4+x^23*X[3,1,1,1,4]^5-2*x^27-15*x^26*X[3,1,1,1,4]+40*x^25* X[3,1,1,1,4]^2-4*x^24*X[3,1,1,1,4]^3-5*x^23*X[3,1,1,1,4]^4+3*x^26-20*x^25*X[3,1 ,1,1,4]+6*x^24*X[3,1,1,1,4]^2+10*x^23*X[3,1,1,1,4]^3-3*x^22*X[3,1,1,1,4]^4+4*x^ 25-4*x^24*X[3,1,1,1,4]-10*x^23*X[3,1,1,1,4]^2+13*x^22*X[3,1,1,1,4]^3-x^21*X[3,1 ,1,1,4]^4+x^24+5*x^23*X[3,1,1,1,4]-21*x^22*X[3,1,1,1,4]^2+3*x^21*X[3,1,1,1,4]^3 +4*x^20*X[3,1,1,1,4]^4-x^23+15*x^22*X[3,1,1,1,4]-3*x^21*X[3,1,1,1,4]^2-19*x^20* X[3,1,1,1,4]^3-4*x^22+x^21*X[3,1,1,1,4]+33*x^20*X[3,1,1,1,4]^2+x^19*X[3,1,1,1,4 ]^3-2*x^18*X[3,1,1,1,4]^4-25*x^20*X[3,1,1,1,4]-3*x^19*X[3,1,1,1,4]^2+7*x^18*X[3 ,1,1,1,4]^3+x^17*X[3,1,1,1,4]^4+7*x^20+3*x^19*X[3,1,1,1,4]-9*x^18*X[3,1,1,1,4]^ 2+x^17*X[3,1,1,1,4]^3-x^19+5*x^18*X[3,1,1,1,4]-10*x^17*X[3,1,1,1,4]^2-x^16*X[3, 1,1,1,4]^3-x^18+13*x^17*X[3,1,1,1,4]+6*x^16*X[3,1,1,1,4]^2-4*x^15*X[3,1,1,1,4]^ 3-5*x^17-9*x^16*X[3,1,1,1,4]+8*x^15*X[3,1,1,1,4]^2+4*x^14*X[3,1,1,1,4]^3+4*x^16 -4*x^15*X[3,1,1,1,4]-9*x^14*X[3,1,1,1,4]^2-x^13*X[3,1,1,1,4]^3+6*x^14*X[3,1,1,1 ,4]+4*x^13*X[3,1,1,1,4]^2-x^14-5*x^13*X[3,1,1,1,4]-5*x^12*X[3,1,1,1,4]^2+2*x^13 +10*x^12*X[3,1,1,1,4]+4*x^11*X[3,1,1,1,4]^2-5*x^12-9*x^11*X[3,1,1,1,4]-x^10*X[3 ,1,1,1,4]^2+5*x^11+6*x^10*X[3,1,1,1,4]-5*x^10-3*x^9*X[3,1,1,1,4]+3*x^9-8*x^8*X[ 3,1,1,1,4]+8*x^8+18*x^7*X[3,1,1,1,4]-19*x^7-15*x^6*X[3,1,1,1,4]+22*x^6+6*x^5*X[ 3,1,1,1,4]-27*x^5-x^4*X[3,1,1,1,4]+36*x^4-35*x^3+21*x^2-7*x+1)/(x^6*X[3,1,1,1,4 ]-x^6+x^5*X[3,1,1,1,4]-x^5+x-1)/(x^6*X[3,1,1,1,4]-x^6+x^5*X[3,1,1,1,4]-x^5-x+1) /(x^6*X[3,1,1,1,4]-x^6+x^5*X[3,1,1,1,4]-x^5-x^4*X[3,1,1,1,4]+x^4-2*x+1)/(x^24*X [3,1,1,1,4]^4-4*x^24*X[3,1,1,1,4]^3+4*x^23*X[3,1,1,1,4]^4+6*x^24*X[3,1,1,1,4]^2 -16*x^23*X[3,1,1,1,4]^3+6*x^22*X[3,1,1,1,4]^4-4*x^24*X[3,1,1,1,4]+24*x^23*X[3,1 ,1,1,4]^2-24*x^22*X[3,1,1,1,4]^3+4*x^21*X[3,1,1,1,4]^4+x^24-16*x^23*X[3,1,1,1,4 ]+36*x^22*X[3,1,1,1,4]^2-16*x^21*X[3,1,1,1,4]^3+x^20*X[3,1,1,1,4]^4+4*x^23-24*x ^22*X[3,1,1,1,4]+24*x^21*X[3,1,1,1,4]^2-4*x^20*X[3,1,1,1,4]^3+6*x^22-16*x^21*X[ 3,1,1,1,4]+6*x^20*X[3,1,1,1,4]^2+4*x^21-4*x^20*X[3,1,1,1,4]+x^20+x^15*X[3,1,1,1 ,4]^3-3*x^15*X[3,1,1,1,4]^2+3*x^15*X[3,1,1,1,4]+2*x^14*X[3,1,1,1,4]^2-x^13*X[3, 1,1,1,4]^3-x^15-4*x^14*X[3,1,1,1,4]+3*x^13*X[3,1,1,1,4]^2+2*x^14-3*x^13*X[3,1,1 ,1,4]-2*x^12*X[3,1,1,1,4]^2+x^13+4*x^12*X[3,1,1,1,4]-2*x^12+x^9*X[3,1,1,1,4]-x^ 9-x^7*X[3,1,1,1,4]+x^7+x^4-4*x^3+6*x^2-4*x+1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [3, 1, 1, 1, 4], equals , - 3/16 + ---- 64 217 65 n The variance equals , - ---- + ---- 1024 4096 4215 33 n The , 3, -th moment about the mean is , - ----- + ---- 16384 2048 211207 12675 2 35981 The , 4, -th moment about the mean is , - ------- + -------- n - ------- n 1048576 16777216 8388608 The compositions of, 10, that yield the, 32, -th largest growth, that is, 1.9703230372932668084, are , [2, 1, 2, 1, 4], [2, 1, 4, 1, 2], [4, 1, 2, 1, 2] Theorem Number, 32, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 2, 1, 4] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 20 17 16 14 12 11 9 8 ) a(n) x = - (x + 2 x - 2 x + x + 3 x - 5 x + 4 x - 3 x / ----- n = 0 7 6 5 4 3 2 / 3 + 4 x - 7 x + 10 x - 16 x + 20 x - 15 x + 6 x - 1) / ((x - x + 1) / 7 6 5 4 2 (x - x + 2 x - x - 2 x + 3 x - 1) 11 6 4 3 2 (x + x - 2 x + 2 x - 3 x + 3 x - 1)) and in Maple format -(x^20+2*x^17-2*x^16+x^14+3*x^12-5*x^11+4*x^9-3*x^8+4*x^7-7*x^6+10*x^5-16*x^4+ 20*x^3-15*x^2+6*x-1)/(x^3-x+1)/(x^7-x^6+2*x^5-x^4-2*x^2+3*x-1)/(x^11+x^6-2*x^4+ 2*x^3-3*x^2+3*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1017, 2018, 3994, 7889, 15560, 30662, 60391, 118918, 234156, 461095, 908079, 1788586, 3523261, 6940967, 13674936, 26943332, 53087228, 104600923, 206102629, 406098598] The limit of a(n+1)/a(n) as n goes to infinity is 1.97032303729 a(n) is asymptotic to .592288780184*1.97032303729^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [2, 1, 2, 1, 4], denoted by the variable, X[2, 1, 2, 1, 4], is 20 20 20 20 17 17 - (x %2 - 3 x %1 + 3 x X[2, 1, 2, 1, 4] - x + 2 x %2 - 6 x %1 16 17 16 15 17 - 2 x %2 + 6 x X[2, 1, 2, 1, 4] + 6 x %1 + x %2 - 2 x 16 15 16 15 - 6 x X[2, 1, 2, 1, 4] - 2 x %1 + 2 x + x X[2, 1, 2, 1, 4] 14 14 14 12 - x %1 + 2 x X[2, 1, 2, 1, 4] - x - 3 x %1 12 11 12 11 + 6 x X[2, 1, 2, 1, 4] + 6 x %1 - 3 x - 11 x X[2, 1, 2, 1, 4] 10 11 10 9 - 4 x %1 + 5 x + 4 x X[2, 1, 2, 1, 4] + x %1 9 9 8 8 + 3 x X[2, 1, 2, 1, 4] - 4 x - 3 x X[2, 1, 2, 1, 4] + 3 x 7 7 6 6 + 4 x X[2, 1, 2, 1, 4] - 4 x - 6 x X[2, 1, 2, 1, 4] + 7 x 5 5 4 4 3 + 4 x X[2, 1, 2, 1, 4] - 10 x - x X[2, 1, 2, 1, 4] + 16 x - 20 x 2 / 7 7 6 + 15 x - 6 x + 1) / ((x X[2, 1, 2, 1, 4] - x - x X[2, 1, 2, 1, 4] / 6 5 5 4 4 2 + x + 2 x X[2, 1, 2, 1, 4] - 2 x - x X[2, 1, 2, 1, 4] + x + 2 x 14 14 14 12 - 3 x + 1) (x %1 - 2 x X[2, 1, 2, 1, 4] + x - x %1 12 11 12 11 11 + 2 x X[2, 1, 2, 1, 4] + x %1 - x - 2 x X[2, 1, 2, 1, 4] + x 9 9 7 7 - x X[2, 1, 2, 1, 4] + x + 3 x X[2, 1, 2, 1, 4] - 3 x 6 6 5 5 4 3 - 3 x X[2, 1, 2, 1, 4] + 3 x + x X[2, 1, 2, 1, 4] - x - x + 4 x 2 - 6 x + 4 x - 1)) 2 %1 := X[2, 1, 2, 1, 4] 3 %2 := X[2, 1, 2, 1, 4] and in Maple format -(x^20*X[2,1,2,1,4]^3-3*x^20*X[2,1,2,1,4]^2+3*x^20*X[2,1,2,1,4]-x^20+2*x^17*X[2 ,1,2,1,4]^3-6*x^17*X[2,1,2,1,4]^2-2*x^16*X[2,1,2,1,4]^3+6*x^17*X[2,1,2,1,4]+6*x ^16*X[2,1,2,1,4]^2+x^15*X[2,1,2,1,4]^3-2*x^17-6*x^16*X[2,1,2,1,4]-2*x^15*X[2,1, 2,1,4]^2+2*x^16+x^15*X[2,1,2,1,4]-x^14*X[2,1,2,1,4]^2+2*x^14*X[2,1,2,1,4]-x^14-\ 3*x^12*X[2,1,2,1,4]^2+6*x^12*X[2,1,2,1,4]+6*x^11*X[2,1,2,1,4]^2-3*x^12-11*x^11* X[2,1,2,1,4]-4*x^10*X[2,1,2,1,4]^2+5*x^11+4*x^10*X[2,1,2,1,4]+x^9*X[2,1,2,1,4]^ 2+3*x^9*X[2,1,2,1,4]-4*x^9-3*x^8*X[2,1,2,1,4]+3*x^8+4*x^7*X[2,1,2,1,4]-4*x^7-6* x^6*X[2,1,2,1,4]+7*x^6+4*x^5*X[2,1,2,1,4]-10*x^5-x^4*X[2,1,2,1,4]+16*x^4-20*x^3 +15*x^2-6*x+1)/(x^7*X[2,1,2,1,4]-x^7-x^6*X[2,1,2,1,4]+x^6+2*x^5*X[2,1,2,1,4]-2* x^5-x^4*X[2,1,2,1,4]+x^4+2*x^2-3*x+1)/(x^14*X[2,1,2,1,4]^2-2*x^14*X[2,1,2,1,4]+ x^14-x^12*X[2,1,2,1,4]^2+2*x^12*X[2,1,2,1,4]+x^11*X[2,1,2,1,4]^2-x^12-2*x^11*X[ 2,1,2,1,4]+x^11-x^9*X[2,1,2,1,4]+x^9+3*x^7*X[2,1,2,1,4]-3*x^7-3*x^6*X[2,1,2,1,4 ]+3*x^6+x^5*X[2,1,2,1,4]-x^5-x^4+4*x^3-6*x^2+4*x-1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [2, 1, 2, 1, 4], equals , - 3/16 + ---- 64 57 69 n The variance equals , - --- + ---- 256 4096 4725 309 n The , 3, -th moment about the mean is , - ----- + ----- 16384 16384 66087 14283 2 8625 The , 4, -th moment about the mean is , - ------ + -------- n - ------- n 262144 16777216 8388608 The compositions of, 10, that yield the, 33, -th largest growth, that is, 1.9706560177668563263, are , [1, 2, 1, 3, 3], [1, 3, 3, 1, 2], [2, 1, 3, 3, 1], [3, 3, 1, 2, 1] Theorem Number, 33, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 2, 1, 3, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 10 7 6 5 4 3 2 \ n x + x + x - x - 2 x + 2 x + 2 x - 3 x + 1 ) a(n) x = ---------------------------------------------------- / 9 7 5 4 3 2 ----- (-1 + x) (x - x + 2 x + x - 3 x - x + 3 x - 1) n = 0 and in Maple format (x^10+x^7+x^6-x^5-2*x^4+2*x^3+2*x^2-3*x+1)/(-1+x)/(x^9-x^7+2*x^5+x^4-3*x^3-x^2+ 3*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1017, 2018, 3994, 7890, 15567, 30691, 60485, 119182, 234830, 462703, 911730, 1796578, 3540290, 6976549, 13748270, 27093086, 53391268, 105216139, 207345386, 408607283] The limit of a(n+1)/a(n) as n goes to infinity is 1.97065601777 a(n) is asymptotic to .592901445341*1.97065601777^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [1, 2, 1, 3, 3], denoted by the variable, X[1, 2, 1, 3, 3], is 12 12 11 12 11 (x %1 - 2 x X[1, 2, 1, 3, 3] - x %1 + x + 2 x X[1, 2, 1, 3, 3] 10 11 10 10 9 9 + x %1 - x - 2 x X[1, 2, 1, 3, 3] + x - x X[1, 2, 1, 3, 3] + x 7 7 5 5 + x X[1, 2, 1, 3, 3] - x - 3 x X[1, 2, 1, 3, 3] + 3 x 4 4 3 3 2 + 3 x X[1, 2, 1, 3, 3] - 2 x - x X[1, 2, 1, 3, 3] - 3 x + 6 x - 4 x / 11 11 10 11 + 1) / ((-1 + x) (x %1 - 2 x X[1, 2, 1, 3, 3] - x %1 + x / 10 10 8 8 + 2 x X[1, 2, 1, 3, 3] - x - x X[1, 2, 1, 3, 3] + x 7 7 6 6 - x X[1, 2, 1, 3, 3] + x + x X[1, 2, 1, 3, 3] - x 5 5 4 4 + 2 x X[1, 2, 1, 3, 3] - 2 x - 3 x X[1, 2, 1, 3, 3] + 3 x 3 3 2 + x X[1, 2, 1, 3, 3] + x - 5 x + 4 x - 1)) 2 %1 := X[1, 2, 1, 3, 3] and in Maple format (x^12*X[1,2,1,3,3]^2-2*x^12*X[1,2,1,3,3]-x^11*X[1,2,1,3,3]^2+x^12+2*x^11*X[1,2, 1,3,3]+x^10*X[1,2,1,3,3]^2-x^11-2*x^10*X[1,2,1,3,3]+x^10-x^9*X[1,2,1,3,3]+x^9+x ^7*X[1,2,1,3,3]-x^7-3*x^5*X[1,2,1,3,3]+3*x^5+3*x^4*X[1,2,1,3,3]-2*x^4-x^3*X[1,2 ,1,3,3]-3*x^3+6*x^2-4*x+1)/(-1+x)/(x^11*X[1,2,1,3,3]^2-2*x^11*X[1,2,1,3,3]-x^10 *X[1,2,1,3,3]^2+x^11+2*x^10*X[1,2,1,3,3]-x^10-x^8*X[1,2,1,3,3]+x^8-x^7*X[1,2,1, 3,3]+x^7+x^6*X[1,2,1,3,3]-x^6+2*x^5*X[1,2,1,3,3]-2*x^5-3*x^4*X[1,2,1,3,3]+3*x^4 +x^3*X[1,2,1,3,3]+x^3-5*x^2+4*x-1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [1, 2, 1, 3, 3], equals , - 3/16 + ---- 64 239 73 n The variance equals , - ---- + ---- 1024 4096 2997 783 n The , 3, -th moment about the mean is , - ---- + ----- 8192 32768 643109 15987 2 140555 The , 4, -th moment about the mean is , - ------- + -------- n + ------- n 1048576 16777216 8388608 The compositions of, 10, that yield the, 34, -th largest growth, that is, 1.9708395870474530685, are , [1, 2, 3, 1, 3], [1, 3, 1, 3, 2], [2, 3, 1, 3, 1], [3, 1, 3, 2, 1] Theorem Number, 34, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 2, 3, 1, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n ) a(n) x = ( / ----- n = 0 13 10 9 7 6 5 4 3 2 / x + 2 x - x + 2 x - 3 x + 4 x - 2 x - 3 x + 6 x - 4 x + 1) / ( / 12 10 9 7 6 5 4 3 2 (-1 + x) (x - x + 2 x - x + 2 x - 3 x + 3 x + x - 5 x + 4 x - 1) ) and in Maple format (x^13+2*x^10-x^9+2*x^7-3*x^6+4*x^5-2*x^4-3*x^3+6*x^2-4*x+1)/(-1+x)/(x^12-x^10+2 *x^9-x^7+2*x^6-3*x^5+3*x^4+x^3-5*x^2+4*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1017, 2018, 3994, 7889, 15561, 30669, 60422, 119027, 234486, 461993, 910339, 1793953, 3535466, 6967845, 13732760, 27065690, 53343225, 105132531, 207201338, 408362651] The limit of a(n+1)/a(n) as n goes to infinity is 1.97083958705 a(n) is asymptotic to .590889988177*1.97083958705^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [1, 2, 3, 1, 3], denoted by the variable, X[1, 2, 3, 1, 3], is 13 13 13 10 10 (x %1 - 2 x X[1, 2, 3, 1, 3] + x + 2 x %1 - 4 x X[1, 2, 3, 1, 3] 9 10 9 8 9 - 2 x %1 + 2 x + 3 x X[1, 2, 3, 1, 3] + x %1 - x 8 7 7 - x X[1, 2, 3, 1, 3] - 2 x X[1, 2, 3, 1, 3] + 2 x 6 6 5 5 + 3 x X[1, 2, 3, 1, 3] - 3 x - 4 x X[1, 2, 3, 1, 3] + 4 x 4 4 3 3 2 + 3 x X[1, 2, 3, 1, 3] - 2 x - x X[1, 2, 3, 1, 3] - 3 x + 6 x - 4 x / 12 12 12 10 + 1) / ((-1 + x) (x %1 - 2 x X[1, 2, 3, 1, 3] + x - x %1 / 10 9 10 9 8 + 2 x X[1, 2, 3, 1, 3] + 2 x %1 - x - 4 x X[1, 2, 3, 1, 3] - x %1 9 8 7 7 + 2 x + x X[1, 2, 3, 1, 3] + x X[1, 2, 3, 1, 3] - x 6 6 5 5 - 2 x X[1, 2, 3, 1, 3] + 2 x + 3 x X[1, 2, 3, 1, 3] - 3 x 4 4 3 3 2 - 3 x X[1, 2, 3, 1, 3] + 3 x + x X[1, 2, 3, 1, 3] + x - 5 x + 4 x - 1 )) 2 %1 := X[1, 2, 3, 1, 3] and in Maple format (x^13*X[1,2,3,1,3]^2-2*x^13*X[1,2,3,1,3]+x^13+2*x^10*X[1,2,3,1,3]^2-4*x^10*X[1, 2,3,1,3]-2*x^9*X[1,2,3,1,3]^2+2*x^10+3*x^9*X[1,2,3,1,3]+x^8*X[1,2,3,1,3]^2-x^9- x^8*X[1,2,3,1,3]-2*x^7*X[1,2,3,1,3]+2*x^7+3*x^6*X[1,2,3,1,3]-3*x^6-4*x^5*X[1,2, 3,1,3]+4*x^5+3*x^4*X[1,2,3,1,3]-2*x^4-x^3*X[1,2,3,1,3]-3*x^3+6*x^2-4*x+1)/(-1+x )/(x^12*X[1,2,3,1,3]^2-2*x^12*X[1,2,3,1,3]+x^12-x^10*X[1,2,3,1,3]^2+2*x^10*X[1, 2,3,1,3]+2*x^9*X[1,2,3,1,3]^2-x^10-4*x^9*X[1,2,3,1,3]-x^8*X[1,2,3,1,3]^2+2*x^9+ x^8*X[1,2,3,1,3]+x^7*X[1,2,3,1,3]-x^7-2*x^6*X[1,2,3,1,3]+2*x^6+3*x^5*X[1,2,3,1, 3]-3*x^5-3*x^4*X[1,2,3,1,3]+3*x^4+x^3*X[1,2,3,1,3]+x^3-5*x^2+4*x-1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [1, 2, 3, 1, 3], equals , - 3/16 + ---- 64 243 73 n The variance equals , - ---- + ---- 1024 4096 189 765 n The , 3, -th moment about the mean is , - --- + ----- 512 32768 615069 15987 2 109979 The , 4, -th moment about the mean is , - ------- + -------- n + ------- n 1048576 16777216 8388608 The compositions of, 10, that yield the, 35, -th largest growth, that is, 1.9708817901785482789, are , [1, 3, 1, 2, 3], [1, 3, 2, 1, 3], [3, 1, 2, 3, 1], [3, 2, 1, 3, 1] Theorem Number, 35, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 3, 1, 2, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 21 20 19 18 17 16 15 14 13 ) a(n) x = (x + x - x + x + 3 x - x - x + 3 x + x / ----- n = 0 12 11 10 9 8 7 6 5 4 3 - 4 x + x + 5 x - 4 x - 3 x + 4 x + 2 x - 5 x - x + 9 x 2 / 20 19 16 15 14 13 - 10 x + 5 x - 1) / ((-1 + x) (x + 2 x + 3 x + x - 2 x + x / 12 11 10 9 8 7 5 4 3 2 + 3 x - 3 x - 3 x + 5 x + x - 5 x + 5 x - 2 x - 6 x + 9 x - 5 x + 1)) and in Maple format (x^21+x^20-x^19+x^18+3*x^17-x^16-x^15+3*x^14+x^13-4*x^12+x^11+5*x^10-4*x^9-3*x^ 8+4*x^7+2*x^6-5*x^5-x^4+9*x^3-10*x^2+5*x-1)/(-1+x)/(x^20+2*x^19+3*x^16+x^15-2*x ^14+x^13+3*x^12-3*x^11-3*x^10+5*x^9+x^8-5*x^7+5*x^5-2*x^4-6*x^3+9*x^2-5*x+1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1017, 2018, 3994, 7889, 15560, 30662, 60393, 118935, 234238, 461394, 909001, 1791130, 3529752, 6956612, 13711093, 27024340, 53264636, 104983086, 206916195, 407816102] The limit of a(n+1)/a(n) as n goes to infinity is 1.97088179018 a(n) is asymptotic to .589721024898*1.97088179018^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [1, 3, 1, 2, 3], denoted by the variable, X[1, 3, 1, 2, 3], is 21 21 20 21 20 19 (x %3 - 4 x %2 + x %3 + 6 x %1 - 4 x %2 - x %3 21 20 19 21 - 4 x X[1, 3, 1, 2, 3] + 6 x %1 + 4 x %2 + x 20 19 18 17 20 - 4 x X[1, 3, 1, 2, 3] - 6 x %1 - x %2 + x %3 + x 19 18 17 19 + 4 x X[1, 3, 1, 2, 3] + 3 x %1 - 6 x %2 - x 18 17 16 18 - 3 x X[1, 3, 1, 2, 3] + 12 x %1 + x %2 + x 17 16 15 17 - 10 x X[1, 3, 1, 2, 3] - 3 x %1 + x %2 + 3 x 16 15 14 16 + 3 x X[1, 3, 1, 2, 3] - 3 x %1 - 2 x %2 - x 15 14 13 15 + 3 x X[1, 3, 1, 2, 3] + 7 x %1 - x %2 - x 14 13 12 14 - 8 x X[1, 3, 1, 2, 3] + 3 x %1 + 2 x %2 + 3 x 13 12 11 13 - 3 x X[1, 3, 1, 2, 3] - 8 x %1 - x %2 + x 12 11 12 11 + 10 x X[1, 3, 1, 2, 3] + 3 x %1 - 4 x - 3 x X[1, 3, 1, 2, 3] 10 11 10 9 10 + 4 x %1 + x - 9 x X[1, 3, 1, 2, 3] - 4 x %1 + 5 x 9 8 9 8 8 + 8 x X[1, 3, 1, 2, 3] + x %1 - 4 x + 2 x X[1, 3, 1, 2, 3] - 3 x 7 7 6 6 - 4 x X[1, 3, 1, 2, 3] + 4 x - 2 x X[1, 3, 1, 2, 3] + 2 x 5 5 4 4 + 6 x X[1, 3, 1, 2, 3] - 5 x - 4 x X[1, 3, 1, 2, 3] - x 3 3 2 / 20 + x X[1, 3, 1, 2, 3] + 9 x - 10 x + 5 x - 1) / ((-1 + x) (x %3 / 20 19 20 19 20 - 4 x %2 + 2 x %3 + 6 x %1 - 8 x %2 - 4 x X[1, 3, 1, 2, 3] 19 17 20 19 17 19 + 12 x %1 - x %3 + x - 8 x X[1, 3, 1, 2, 3] + 3 x %2 + 2 x 17 16 17 16 15 - 3 x %1 - 3 x %2 + x X[1, 3, 1, 2, 3] + 9 x %1 - x %2 16 15 14 16 - 9 x X[1, 3, 1, 2, 3] + 3 x %1 + 2 x %2 + 3 x 15 14 15 14 - 3 x X[1, 3, 1, 2, 3] - 6 x %1 + x + 6 x X[1, 3, 1, 2, 3] 13 12 14 13 12 11 + x %1 - 2 x %2 - 2 x - 2 x X[1, 3, 1, 2, 3] + 7 x %1 + x %2 13 12 11 12 11 + x - 8 x X[1, 3, 1, 2, 3] - 5 x %1 + 3 x + 7 x X[1, 3, 1, 2, 3] 10 11 10 9 10 - 3 x %1 - 3 x + 6 x X[1, 3, 1, 2, 3] + 4 x %1 - 3 x 9 8 9 8 7 7 - 9 x X[1, 3, 1, 2, 3] - x %1 + 5 x + x + 5 x X[1, 3, 1, 2, 3] - 5 x 5 5 4 4 - 5 x X[1, 3, 1, 2, 3] + 5 x + 4 x X[1, 3, 1, 2, 3] - 2 x 3 3 2 - x X[1, 3, 1, 2, 3] - 6 x + 9 x - 5 x + 1)) 2 %1 := X[1, 3, 1, 2, 3] 3 %2 := X[1, 3, 1, 2, 3] 4 %3 := X[1, 3, 1, 2, 3] and in Maple format (x^21*X[1,3,1,2,3]^4-4*x^21*X[1,3,1,2,3]^3+x^20*X[1,3,1,2,3]^4+6*x^21*X[1,3,1,2 ,3]^2-4*x^20*X[1,3,1,2,3]^3-x^19*X[1,3,1,2,3]^4-4*x^21*X[1,3,1,2,3]+6*x^20*X[1, 3,1,2,3]^2+4*x^19*X[1,3,1,2,3]^3+x^21-4*x^20*X[1,3,1,2,3]-6*x^19*X[1,3,1,2,3]^2 -x^18*X[1,3,1,2,3]^3+x^17*X[1,3,1,2,3]^4+x^20+4*x^19*X[1,3,1,2,3]+3*x^18*X[1,3, 1,2,3]^2-6*x^17*X[1,3,1,2,3]^3-x^19-3*x^18*X[1,3,1,2,3]+12*x^17*X[1,3,1,2,3]^2+ x^16*X[1,3,1,2,3]^3+x^18-10*x^17*X[1,3,1,2,3]-3*x^16*X[1,3,1,2,3]^2+x^15*X[1,3, 1,2,3]^3+3*x^17+3*x^16*X[1,3,1,2,3]-3*x^15*X[1,3,1,2,3]^2-2*x^14*X[1,3,1,2,3]^3 -x^16+3*x^15*X[1,3,1,2,3]+7*x^14*X[1,3,1,2,3]^2-x^13*X[1,3,1,2,3]^3-x^15-8*x^14 *X[1,3,1,2,3]+3*x^13*X[1,3,1,2,3]^2+2*x^12*X[1,3,1,2,3]^3+3*x^14-3*x^13*X[1,3,1 ,2,3]-8*x^12*X[1,3,1,2,3]^2-x^11*X[1,3,1,2,3]^3+x^13+10*x^12*X[1,3,1,2,3]+3*x^ 11*X[1,3,1,2,3]^2-4*x^12-3*x^11*X[1,3,1,2,3]+4*x^10*X[1,3,1,2,3]^2+x^11-9*x^10* X[1,3,1,2,3]-4*x^9*X[1,3,1,2,3]^2+5*x^10+8*x^9*X[1,3,1,2,3]+x^8*X[1,3,1,2,3]^2-\ 4*x^9+2*x^8*X[1,3,1,2,3]-3*x^8-4*x^7*X[1,3,1,2,3]+4*x^7-2*x^6*X[1,3,1,2,3]+2*x^ 6+6*x^5*X[1,3,1,2,3]-5*x^5-4*x^4*X[1,3,1,2,3]-x^4+x^3*X[1,3,1,2,3]+9*x^3-10*x^2 +5*x-1)/(-1+x)/(x^20*X[1,3,1,2,3]^4-4*x^20*X[1,3,1,2,3]^3+2*x^19*X[1,3,1,2,3]^4 +6*x^20*X[1,3,1,2,3]^2-8*x^19*X[1,3,1,2,3]^3-4*x^20*X[1,3,1,2,3]+12*x^19*X[1,3, 1,2,3]^2-x^17*X[1,3,1,2,3]^4+x^20-8*x^19*X[1,3,1,2,3]+3*x^17*X[1,3,1,2,3]^3+2*x ^19-3*x^17*X[1,3,1,2,3]^2-3*x^16*X[1,3,1,2,3]^3+x^17*X[1,3,1,2,3]+9*x^16*X[1,3, 1,2,3]^2-x^15*X[1,3,1,2,3]^3-9*x^16*X[1,3,1,2,3]+3*x^15*X[1,3,1,2,3]^2+2*x^14*X [1,3,1,2,3]^3+3*x^16-3*x^15*X[1,3,1,2,3]-6*x^14*X[1,3,1,2,3]^2+x^15+6*x^14*X[1, 3,1,2,3]+x^13*X[1,3,1,2,3]^2-2*x^12*X[1,3,1,2,3]^3-2*x^14-2*x^13*X[1,3,1,2,3]+7 *x^12*X[1,3,1,2,3]^2+x^11*X[1,3,1,2,3]^3+x^13-8*x^12*X[1,3,1,2,3]-5*x^11*X[1,3, 1,2,3]^2+3*x^12+7*x^11*X[1,3,1,2,3]-3*x^10*X[1,3,1,2,3]^2-3*x^11+6*x^10*X[1,3,1 ,2,3]+4*x^9*X[1,3,1,2,3]^2-3*x^10-9*x^9*X[1,3,1,2,3]-x^8*X[1,3,1,2,3]^2+5*x^9+x ^8+5*x^7*X[1,3,1,2,3]-5*x^7-5*x^5*X[1,3,1,2,3]+5*x^5+4*x^4*X[1,3,1,2,3]-2*x^4-x ^3*X[1,3,1,2,3]-6*x^3+9*x^2-5*x+1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [1, 3, 1, 2, 3], equals , - 3/16 + ---- 64 247 73 n The variance equals , - ---- + ---- 1024 4096 3111 759 n The , 3, -th moment about the mean is , - ---- + ----- 8192 32768 627493 15987 2 95531 The , 4, -th moment about the mean is , - ------- + -------- n + ------- n 1048576 16777216 8388608 The compositions of, 10, that yield the, 36, -th largest growth, that is, 1.9709013528640663101, are , [3, 1, 1, 2, 3], [3, 2, 1, 1, 3] Theorem Number, 36, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [3, 1, 1, 2, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 38 37 36 35 34 33 32 ) a(n) x = - (x + 3 x + 3 x + 2 x + 3 x + 3 x + 3 x / ----- n = 0 31 30 29 28 27 25 24 23 22 21 + 2 x - 2 x + x + 3 x - 3 x + 3 x - x + 2 x - x - 5 x 20 19 18 17 16 15 14 13 12 + 6 x + 4 x - 9 x + 3 x + 5 x - 11 x + 12 x + x - 22 x 11 9 8 7 6 5 4 3 2 + 25 x - 31 x + 48 x - 44 x + 13 x + 34 x - 63 x + 55 x - 28 x / 40 39 38 37 36 35 34 + 8 x - 1) / (x + 4 x + 5 x + 2 x + 2 x + 4 x + 3 x / 33 32 31 30 29 28 27 26 25 + 2 x - 2 x - 2 x + 5 x - x - 5 x + 6 x + 3 x - 4 x 24 23 21 20 19 18 17 16 15 + x - 4 x + 13 x - 8 x - 10 x + 14 x - 4 x - 8 x + 19 x 14 13 12 11 10 9 8 7 6 - 19 x - 7 x + 41 x - 37 x - 8 x + 53 x - 74 x + 62 x - 8 x 5 4 3 2 - 63 x + 97 x - 76 x + 35 x - 9 x + 1) and in Maple format -(x^38+3*x^37+3*x^36+2*x^35+3*x^34+3*x^33+3*x^32+2*x^31-2*x^30+x^29+3*x^28-3*x^ 27+3*x^25-x^24+2*x^23-x^22-5*x^21+6*x^20+4*x^19-9*x^18+3*x^17+5*x^16-11*x^15+12 *x^14+x^13-22*x^12+25*x^11-31*x^9+48*x^8-44*x^7+13*x^6+34*x^5-63*x^4+55*x^3-28* x^2+8*x-1)/(x^40+4*x^39+5*x^38+2*x^37+2*x^36+4*x^35+3*x^34+2*x^33-2*x^32-2*x^31 +5*x^30-x^29-5*x^28+6*x^27+3*x^26-4*x^25+x^24-4*x^23+13*x^21-8*x^20-10*x^19+14* x^18-4*x^17-8*x^16+19*x^15-19*x^14-7*x^13+41*x^12-37*x^11-8*x^10+53*x^9-74*x^8+ 62*x^7-8*x^6-63*x^5+97*x^4-76*x^3+35*x^2-9*x+1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1017, 2018, 3994, 7889, 15560, 30660, 60377, 118861, 233980, 460640, 907046, 1786474, 3519332, 6934352, 13665181, 26932105, 53082840, 104629347, 206233088, 406501207] The limit of a(n+1)/a(n) as n goes to infinity is 1.97090135286 a(n) is asymptotic to .587757686511*1.97090135286^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [3, 1, 1, 2, 3], denoted by the variable, X[3, 1, 1, 2, 3], is 38 38 37 38 37 36 38 - (x %7 - 8 x %6 + 3 x %7 + 28 x %5 - 24 x %6 + 3 x %7 - 56 x %4 37 36 35 38 37 36 + 84 x %5 - 24 x %6 + x %7 + 70 x %3 - 168 x %4 + 84 x %5 35 38 37 36 35 34 - 9 x %6 - 56 x %2 + 210 x %3 - 168 x %4 + 35 x %5 - 3 x %6 38 37 36 35 34 33 + 28 x %1 - 168 x %2 + 210 x %3 - 77 x %4 + 21 x %5 - 3 x %6 38 37 36 35 34 - 8 x X[3, 1, 1, 2, 3] + 84 x %1 - 168 x %2 + 105 x %3 - 63 x %4 33 32 38 37 36 + 21 x %5 - 3 x %6 + x - 24 x X[3, 1, 1, 2, 3] + 84 x %1 35 34 33 32 31 37 - 91 x %2 + 105 x %3 - 63 x %4 + 21 x %5 - 3 x %6 + 3 x 36 35 34 33 - 24 x X[3, 1, 1, 2, 3] + 49 x %1 - 105 x %2 + 105 x %3 32 31 36 35 34 - 63 x %4 + 20 x %5 + 3 x - 15 x X[3, 1, 1, 2, 3] + 63 x %1 33 32 31 30 35 - 105 x %2 + 105 x %3 - 57 x %4 - 2 x %5 + 2 x 34 33 32 31 30 - 21 x X[3, 1, 1, 2, 3] + 63 x %1 - 105 x %2 + 90 x %3 + 12 x %4 29 28 34 33 32 + x %5 - x %6 + 3 x - 21 x X[3, 1, 1, 2, 3] + 63 x %1 31 30 29 28 33 - 85 x %2 - 30 x %3 - 6 x %4 + 10 x %5 + 3 x 32 31 30 29 28 - 21 x X[3, 1, 1, 2, 3] + 48 x %1 + 40 x %2 + 15 x %3 - 38 x %4 27 32 31 30 29 - x %5 + 3 x - 15 x X[3, 1, 1, 2, 3] - 30 x %1 - 20 x %2 28 27 31 30 29 + 75 x %3 + 8 x %4 + 2 x + 12 x X[3, 1, 1, 2, 3] + 15 x %1 28 27 30 29 28 - 85 x %2 - 25 x %3 - 2 x - 6 x X[3, 1, 1, 2, 3] + 56 x %1 27 25 24 29 28 + 40 x %2 - 3 x %4 - x %5 + x - 20 x X[3, 1, 1, 2, 3] 27 25 24 23 28 - 35 x %1 + 15 x %3 + 6 x %4 + x %5 + 3 x 27 25 24 23 27 + 16 x X[3, 1, 1, 2, 3] - 30 x %2 - 15 x %3 - 9 x %4 - 3 x 25 24 23 25 24 + 30 x %1 + 20 x %2 + 28 x %3 - 15 x X[3, 1, 1, 2, 3] - 15 x %1 23 22 21 25 24 - 42 x %2 - x %3 + 6 x %4 + 3 x + 6 x X[3, 1, 1, 2, 3] 23 22 21 20 24 + 33 x %1 + 4 x %2 - 29 x %3 - 4 x %4 - x 23 22 21 20 19 - 13 x X[3, 1, 1, 2, 3] - 6 x %1 + 56 x %2 + 22 x %3 + x %4 23 22 21 20 19 22 + 2 x + 4 x X[3, 1, 1, 2, 3] - 54 x %1 - 48 x %2 - 2 x %3 - x 21 20 19 18 21 + 26 x X[3, 1, 1, 2, 3] + 52 x %1 - 4 x %2 - 7 x %3 - 5 x 20 19 18 17 20 - 28 x X[3, 1, 1, 2, 3] + 14 x %1 + 30 x %2 + 8 x %3 + 6 x 19 18 17 16 19 - 13 x X[3, 1, 1, 2, 3] - 48 x %1 - 28 x %2 - 7 x %3 + 4 x 18 17 16 15 18 + 34 x X[3, 1, 1, 2, 3] + 35 x %1 + 20 x %2 + 4 x %3 - 9 x 17 16 15 14 17 - 18 x X[3, 1, 1, 2, 3] - 14 x %1 - 6 x %2 - x %3 + 3 x 16 15 14 16 - 4 x X[3, 1, 1, 2, 3] - 11 x %1 - 10 x %2 + 5 x 15 14 13 15 + 24 x X[3, 1, 1, 2, 3] + 35 x %1 + 13 x %2 - 11 x 14 13 12 14 - 36 x X[3, 1, 1, 2, 3] - 25 x %1 - 6 x %2 + 12 x 13 12 11 13 + 11 x X[3, 1, 1, 2, 3] - 11 x %1 + x %2 + x 12 11 12 11 + 39 x X[3, 1, 1, 2, 3] + 30 x %1 - 22 x - 56 x X[3, 1, 1, 2, 3] 10 11 10 9 - 21 x %1 + 25 x + 21 x X[3, 1, 1, 2, 3] + 7 x %1 9 8 9 8 8 + 24 x X[3, 1, 1, 2, 3] - x %1 - 31 x - 48 x X[3, 1, 1, 2, 3] + 48 x 7 7 6 6 + 52 x X[3, 1, 1, 2, 3] - 44 x - 41 x X[3, 1, 1, 2, 3] + 13 x 5 5 4 4 + 22 x X[3, 1, 1, 2, 3] + 34 x - 7 x X[3, 1, 1, 2, 3] - 63 x 3 3 2 / 40 40 + x X[3, 1, 1, 2, 3] + 55 x - 28 x + 8 x - 1) / (x %7 - 8 x %6 / 39 40 39 38 40 39 + 4 x %7 + 28 x %5 - 32 x %6 + 5 x %7 - 56 x %4 + 112 x %5 38 37 40 39 38 37 - 40 x %6 + x %7 + 70 x %3 - 224 x %4 + 140 x %5 - 9 x %6 36 40 39 38 37 36 - 2 x %7 - 56 x %2 + 280 x %3 - 280 x %4 + 35 x %5 + 12 x %6 35 40 39 38 37 36 - x %7 + 28 x %1 - 224 x %2 + 350 x %3 - 77 x %4 - 28 x %5 35 40 39 38 37 + 3 x %6 - 8 x X[3, 1, 1, 2, 3] + 112 x %1 - 280 x %2 + 105 x %3 36 35 34 40 39 + 28 x %4 + 7 x %5 - 3 x %6 + x - 32 x X[3, 1, 1, 2, 3] 38 37 35 34 33 39 + 140 x %1 - 91 x %2 - 49 x %4 + 21 x %5 - 3 x %6 + 4 x 38 37 36 35 34 - 40 x X[3, 1, 1, 2, 3] + 49 x %1 - 28 x %2 + 105 x %3 - 63 x %4 33 32 38 37 36 + 20 x %5 - x %6 + 5 x - 15 x X[3, 1, 1, 2, 3] + 28 x %1 35 34 33 32 31 37 - 119 x %2 + 105 x %3 - 57 x %4 + 4 x %5 + 2 x %6 + 2 x 36 35 34 33 32 - 12 x X[3, 1, 1, 2, 3] + 77 x %1 - 105 x %2 + 90 x %3 - 3 x %4 31 30 36 35 34 - 14 x %5 - x %6 + 2 x - 27 x X[3, 1, 1, 2, 3] + 63 x %1 33 32 31 30 29 35 - 85 x %2 - 10 x %3 + 42 x %4 + 12 x %5 - x %6 + 4 x 34 33 32 31 30 - 21 x X[3, 1, 1, 2, 3] + 48 x %1 + 25 x %2 - 70 x %3 - 50 x %4 29 28 34 33 32 + 8 x %5 + x %6 + 3 x - 15 x X[3, 1, 1, 2, 3] - 24 x %1 31 30 29 28 33 + 70 x %2 + 105 x %3 - 24 x %4 - 11 x %5 + 2 x 32 31 30 29 28 + 11 x X[3, 1, 1, 2, 3] - 42 x %1 - 125 x %2 + 35 x %3 + 45 x %4 27 32 31 30 29 + x %5 - 2 x + 14 x X[3, 1, 1, 2, 3] + 86 x %1 - 25 x %2 28 27 31 30 29 - 95 x %3 - 11 x %4 - 2 x - 32 x X[3, 1, 1, 2, 3] + 6 x %1 28 27 26 30 29 + 115 x %2 + 40 x %3 - 4 x %4 + 5 x + 2 x X[3, 1, 1, 2, 3] 28 27 26 25 24 29 - 81 x %1 - 70 x %2 + 19 x %3 + 2 x %4 + 2 x %5 - x 28 27 26 25 24 + 31 x X[3, 1, 1, 2, 3] + 65 x %1 - 36 x %2 - 12 x %3 - 13 x %4 23 28 27 26 25 - x %5 - 5 x - 31 x X[3, 1, 1, 2, 3] + 34 x %1 + 28 x %2 24 23 27 26 25 + 33 x %3 + 12 x %4 + 6 x - 16 x X[3, 1, 1, 2, 3] - 32 x %1 24 23 22 26 25 - 42 x %2 - 42 x %3 + 3 x %4 + 3 x + 18 x X[3, 1, 1, 2, 3] 24 23 22 21 25 + 28 x %1 + 68 x %2 - 12 x %3 - 9 x %4 - 4 x 24 23 22 21 20 - 9 x X[3, 1, 1, 2, 3] - 57 x %1 + 18 x %2 + 47 x %3 + 5 x %4 24 23 22 21 20 + x + 24 x X[3, 1, 1, 2, 3] - 12 x %1 - 100 x %2 - 27 x %3 19 23 22 21 20 - x %4 - 4 x + 3 x X[3, 1, 1, 2, 3] + 108 x %1 + 59 x %2 19 21 20 19 18 - x %3 - 59 x X[3, 1, 1, 2, 3] - 65 x %1 + 19 x %2 + 11 x %3 21 20 19 18 17 + 13 x + 36 x X[3, 1, 1, 2, 3] - 41 x %1 - 46 x %2 - 12 x %3 20 19 18 17 16 - 8 x + 34 x X[3, 1, 1, 2, 3] + 73 x %1 + 40 x %2 + 10 x %3 19 18 17 16 15 - 10 x - 52 x X[3, 1, 1, 2, 3] - 48 x %1 - 28 x %2 - 5 x %3 18 17 16 15 14 17 + 14 x + 24 x X[3, 1, 1, 2, 3] + 18 x %1 + 4 x %2 + x %3 - 4 x 16 15 14 16 + 8 x X[3, 1, 1, 2, 3] + 26 x %1 + 18 x %2 - 8 x 15 14 13 15 - 44 x X[3, 1, 1, 2, 3] - 58 x %1 - 18 x %2 + 19 x 14 13 12 14 + 58 x X[3, 1, 1, 2, 3] + 30 x %1 + 7 x %2 - 19 x 13 12 11 13 - 5 x X[3, 1, 1, 2, 3] + 26 x %1 - x %2 - 7 x 12 11 12 11 - 74 x X[3, 1, 1, 2, 3] - 45 x %1 + 41 x + 83 x X[3, 1, 1, 2, 3] 10 11 10 9 10 + 27 x %1 - 37 x - 19 x X[3, 1, 1, 2, 3] - 8 x %1 - 8 x 9 8 9 8 8 - 45 x X[3, 1, 1, 2, 3] + x %1 + 53 x + 75 x X[3, 1, 1, 2, 3] - 74 x 7 7 6 6 - 77 x X[3, 1, 1, 2, 3] + 62 x + 57 x X[3, 1, 1, 2, 3] - 8 x 5 5 4 4 - 28 x X[3, 1, 1, 2, 3] - 63 x + 8 x X[3, 1, 1, 2, 3] + 97 x 3 3 2 - x X[3, 1, 1, 2, 3] - 76 x + 35 x - 9 x + 1) 2 %1 := X[3, 1, 1, 2, 3] 3 %2 := X[3, 1, 1, 2, 3] 4 %3 := X[3, 1, 1, 2, 3] 5 %4 := X[3, 1, 1, 2, 3] 6 %5 := X[3, 1, 1, 2, 3] 7 %6 := X[3, 1, 1, 2, 3] 8 %7 := X[3, 1, 1, 2, 3] and in Maple format -(x^38*X[3,1,1,2,3]^8-8*x^38*X[3,1,1,2,3]^7+3*x^37*X[3,1,1,2,3]^8+28*x^38*X[3,1 ,1,2,3]^6-24*x^37*X[3,1,1,2,3]^7+3*x^36*X[3,1,1,2,3]^8-56*x^38*X[3,1,1,2,3]^5+ 84*x^37*X[3,1,1,2,3]^6-24*x^36*X[3,1,1,2,3]^7+x^35*X[3,1,1,2,3]^8+70*x^38*X[3,1 ,1,2,3]^4-168*x^37*X[3,1,1,2,3]^5+84*x^36*X[3,1,1,2,3]^6-9*x^35*X[3,1,1,2,3]^7-\ 56*x^38*X[3,1,1,2,3]^3+210*x^37*X[3,1,1,2,3]^4-168*x^36*X[3,1,1,2,3]^5+35*x^35* X[3,1,1,2,3]^6-3*x^34*X[3,1,1,2,3]^7+28*x^38*X[3,1,1,2,3]^2-168*x^37*X[3,1,1,2, 3]^3+210*x^36*X[3,1,1,2,3]^4-77*x^35*X[3,1,1,2,3]^5+21*x^34*X[3,1,1,2,3]^6-3*x^ 33*X[3,1,1,2,3]^7-8*x^38*X[3,1,1,2,3]+84*x^37*X[3,1,1,2,3]^2-168*x^36*X[3,1,1,2 ,3]^3+105*x^35*X[3,1,1,2,3]^4-63*x^34*X[3,1,1,2,3]^5+21*x^33*X[3,1,1,2,3]^6-3*x ^32*X[3,1,1,2,3]^7+x^38-24*x^37*X[3,1,1,2,3]+84*x^36*X[3,1,1,2,3]^2-91*x^35*X[3 ,1,1,2,3]^3+105*x^34*X[3,1,1,2,3]^4-63*x^33*X[3,1,1,2,3]^5+21*x^32*X[3,1,1,2,3] ^6-3*x^31*X[3,1,1,2,3]^7+3*x^37-24*x^36*X[3,1,1,2,3]+49*x^35*X[3,1,1,2,3]^2-105 *x^34*X[3,1,1,2,3]^3+105*x^33*X[3,1,1,2,3]^4-63*x^32*X[3,1,1,2,3]^5+20*x^31*X[3 ,1,1,2,3]^6+3*x^36-15*x^35*X[3,1,1,2,3]+63*x^34*X[3,1,1,2,3]^2-105*x^33*X[3,1,1 ,2,3]^3+105*x^32*X[3,1,1,2,3]^4-57*x^31*X[3,1,1,2,3]^5-2*x^30*X[3,1,1,2,3]^6+2* x^35-21*x^34*X[3,1,1,2,3]+63*x^33*X[3,1,1,2,3]^2-105*x^32*X[3,1,1,2,3]^3+90*x^ 31*X[3,1,1,2,3]^4+12*x^30*X[3,1,1,2,3]^5+x^29*X[3,1,1,2,3]^6-x^28*X[3,1,1,2,3]^ 7+3*x^34-21*x^33*X[3,1,1,2,3]+63*x^32*X[3,1,1,2,3]^2-85*x^31*X[3,1,1,2,3]^3-30* x^30*X[3,1,1,2,3]^4-6*x^29*X[3,1,1,2,3]^5+10*x^28*X[3,1,1,2,3]^6+3*x^33-21*x^32 *X[3,1,1,2,3]+48*x^31*X[3,1,1,2,3]^2+40*x^30*X[3,1,1,2,3]^3+15*x^29*X[3,1,1,2,3 ]^4-38*x^28*X[3,1,1,2,3]^5-x^27*X[3,1,1,2,3]^6+3*x^32-15*x^31*X[3,1,1,2,3]-30*x ^30*X[3,1,1,2,3]^2-20*x^29*X[3,1,1,2,3]^3+75*x^28*X[3,1,1,2,3]^4+8*x^27*X[3,1,1 ,2,3]^5+2*x^31+12*x^30*X[3,1,1,2,3]+15*x^29*X[3,1,1,2,3]^2-85*x^28*X[3,1,1,2,3] ^3-25*x^27*X[3,1,1,2,3]^4-2*x^30-6*x^29*X[3,1,1,2,3]+56*x^28*X[3,1,1,2,3]^2+40* x^27*X[3,1,1,2,3]^3-3*x^25*X[3,1,1,2,3]^5-x^24*X[3,1,1,2,3]^6+x^29-20*x^28*X[3, 1,1,2,3]-35*x^27*X[3,1,1,2,3]^2+15*x^25*X[3,1,1,2,3]^4+6*x^24*X[3,1,1,2,3]^5+x^ 23*X[3,1,1,2,3]^6+3*x^28+16*x^27*X[3,1,1,2,3]-30*x^25*X[3,1,1,2,3]^3-15*x^24*X[ 3,1,1,2,3]^4-9*x^23*X[3,1,1,2,3]^5-3*x^27+30*x^25*X[3,1,1,2,3]^2+20*x^24*X[3,1, 1,2,3]^3+28*x^23*X[3,1,1,2,3]^4-15*x^25*X[3,1,1,2,3]-15*x^24*X[3,1,1,2,3]^2-42* x^23*X[3,1,1,2,3]^3-x^22*X[3,1,1,2,3]^4+6*x^21*X[3,1,1,2,3]^5+3*x^25+6*x^24*X[3 ,1,1,2,3]+33*x^23*X[3,1,1,2,3]^2+4*x^22*X[3,1,1,2,3]^3-29*x^21*X[3,1,1,2,3]^4-4 *x^20*X[3,1,1,2,3]^5-x^24-13*x^23*X[3,1,1,2,3]-6*x^22*X[3,1,1,2,3]^2+56*x^21*X[ 3,1,1,2,3]^3+22*x^20*X[3,1,1,2,3]^4+x^19*X[3,1,1,2,3]^5+2*x^23+4*x^22*X[3,1,1,2 ,3]-54*x^21*X[3,1,1,2,3]^2-48*x^20*X[3,1,1,2,3]^3-2*x^19*X[3,1,1,2,3]^4-x^22+26 *x^21*X[3,1,1,2,3]+52*x^20*X[3,1,1,2,3]^2-4*x^19*X[3,1,1,2,3]^3-7*x^18*X[3,1,1, 2,3]^4-5*x^21-28*x^20*X[3,1,1,2,3]+14*x^19*X[3,1,1,2,3]^2+30*x^18*X[3,1,1,2,3]^ 3+8*x^17*X[3,1,1,2,3]^4+6*x^20-13*x^19*X[3,1,1,2,3]-48*x^18*X[3,1,1,2,3]^2-28*x ^17*X[3,1,1,2,3]^3-7*x^16*X[3,1,1,2,3]^4+4*x^19+34*x^18*X[3,1,1,2,3]+35*x^17*X[ 3,1,1,2,3]^2+20*x^16*X[3,1,1,2,3]^3+4*x^15*X[3,1,1,2,3]^4-9*x^18-18*x^17*X[3,1, 1,2,3]-14*x^16*X[3,1,1,2,3]^2-6*x^15*X[3,1,1,2,3]^3-x^14*X[3,1,1,2,3]^4+3*x^17-\ 4*x^16*X[3,1,1,2,3]-11*x^15*X[3,1,1,2,3]^2-10*x^14*X[3,1,1,2,3]^3+5*x^16+24*x^ 15*X[3,1,1,2,3]+35*x^14*X[3,1,1,2,3]^2+13*x^13*X[3,1,1,2,3]^3-11*x^15-36*x^14*X [3,1,1,2,3]-25*x^13*X[3,1,1,2,3]^2-6*x^12*X[3,1,1,2,3]^3+12*x^14+11*x^13*X[3,1, 1,2,3]-11*x^12*X[3,1,1,2,3]^2+x^11*X[3,1,1,2,3]^3+x^13+39*x^12*X[3,1,1,2,3]+30* x^11*X[3,1,1,2,3]^2-22*x^12-56*x^11*X[3,1,1,2,3]-21*x^10*X[3,1,1,2,3]^2+25*x^11 +21*x^10*X[3,1,1,2,3]+7*x^9*X[3,1,1,2,3]^2+24*x^9*X[3,1,1,2,3]-x^8*X[3,1,1,2,3] ^2-31*x^9-48*x^8*X[3,1,1,2,3]+48*x^8+52*x^7*X[3,1,1,2,3]-44*x^7-41*x^6*X[3,1,1, 2,3]+13*x^6+22*x^5*X[3,1,1,2,3]+34*x^5-7*x^4*X[3,1,1,2,3]-63*x^4+x^3*X[3,1,1,2, 3]+55*x^3-28*x^2+8*x-1)/(x^40*X[3,1,1,2,3]^8-8*x^40*X[3,1,1,2,3]^7+4*x^39*X[3,1 ,1,2,3]^8+28*x^40*X[3,1,1,2,3]^6-32*x^39*X[3,1,1,2,3]^7+5*x^38*X[3,1,1,2,3]^8-\ 56*x^40*X[3,1,1,2,3]^5+112*x^39*X[3,1,1,2,3]^6-40*x^38*X[3,1,1,2,3]^7+x^37*X[3, 1,1,2,3]^8+70*x^40*X[3,1,1,2,3]^4-224*x^39*X[3,1,1,2,3]^5+140*x^38*X[3,1,1,2,3] ^6-9*x^37*X[3,1,1,2,3]^7-2*x^36*X[3,1,1,2,3]^8-56*x^40*X[3,1,1,2,3]^3+280*x^39* X[3,1,1,2,3]^4-280*x^38*X[3,1,1,2,3]^5+35*x^37*X[3,1,1,2,3]^6+12*x^36*X[3,1,1,2 ,3]^7-x^35*X[3,1,1,2,3]^8+28*x^40*X[3,1,1,2,3]^2-224*x^39*X[3,1,1,2,3]^3+350*x^ 38*X[3,1,1,2,3]^4-77*x^37*X[3,1,1,2,3]^5-28*x^36*X[3,1,1,2,3]^6+3*x^35*X[3,1,1, 2,3]^7-8*x^40*X[3,1,1,2,3]+112*x^39*X[3,1,1,2,3]^2-280*x^38*X[3,1,1,2,3]^3+105* x^37*X[3,1,1,2,3]^4+28*x^36*X[3,1,1,2,3]^5+7*x^35*X[3,1,1,2,3]^6-3*x^34*X[3,1,1 ,2,3]^7+x^40-32*x^39*X[3,1,1,2,3]+140*x^38*X[3,1,1,2,3]^2-91*x^37*X[3,1,1,2,3]^ 3-49*x^35*X[3,1,1,2,3]^5+21*x^34*X[3,1,1,2,3]^6-3*x^33*X[3,1,1,2,3]^7+4*x^39-40 *x^38*X[3,1,1,2,3]+49*x^37*X[3,1,1,2,3]^2-28*x^36*X[3,1,1,2,3]^3+105*x^35*X[3,1 ,1,2,3]^4-63*x^34*X[3,1,1,2,3]^5+20*x^33*X[3,1,1,2,3]^6-x^32*X[3,1,1,2,3]^7+5*x ^38-15*x^37*X[3,1,1,2,3]+28*x^36*X[3,1,1,2,3]^2-119*x^35*X[3,1,1,2,3]^3+105*x^ 34*X[3,1,1,2,3]^4-57*x^33*X[3,1,1,2,3]^5+4*x^32*X[3,1,1,2,3]^6+2*x^31*X[3,1,1,2 ,3]^7+2*x^37-12*x^36*X[3,1,1,2,3]+77*x^35*X[3,1,1,2,3]^2-105*x^34*X[3,1,1,2,3]^ 3+90*x^33*X[3,1,1,2,3]^4-3*x^32*X[3,1,1,2,3]^5-14*x^31*X[3,1,1,2,3]^6-x^30*X[3, 1,1,2,3]^7+2*x^36-27*x^35*X[3,1,1,2,3]+63*x^34*X[3,1,1,2,3]^2-85*x^33*X[3,1,1,2 ,3]^3-10*x^32*X[3,1,1,2,3]^4+42*x^31*X[3,1,1,2,3]^5+12*x^30*X[3,1,1,2,3]^6-x^29 *X[3,1,1,2,3]^7+4*x^35-21*x^34*X[3,1,1,2,3]+48*x^33*X[3,1,1,2,3]^2+25*x^32*X[3, 1,1,2,3]^3-70*x^31*X[3,1,1,2,3]^4-50*x^30*X[3,1,1,2,3]^5+8*x^29*X[3,1,1,2,3]^6+ x^28*X[3,1,1,2,3]^7+3*x^34-15*x^33*X[3,1,1,2,3]-24*x^32*X[3,1,1,2,3]^2+70*x^31* X[3,1,1,2,3]^3+105*x^30*X[3,1,1,2,3]^4-24*x^29*X[3,1,1,2,3]^5-11*x^28*X[3,1,1,2 ,3]^6+2*x^33+11*x^32*X[3,1,1,2,3]-42*x^31*X[3,1,1,2,3]^2-125*x^30*X[3,1,1,2,3]^ 3+35*x^29*X[3,1,1,2,3]^4+45*x^28*X[3,1,1,2,3]^5+x^27*X[3,1,1,2,3]^6-2*x^32+14*x ^31*X[3,1,1,2,3]+86*x^30*X[3,1,1,2,3]^2-25*x^29*X[3,1,1,2,3]^3-95*x^28*X[3,1,1, 2,3]^4-11*x^27*X[3,1,1,2,3]^5-2*x^31-32*x^30*X[3,1,1,2,3]+6*x^29*X[3,1,1,2,3]^2 +115*x^28*X[3,1,1,2,3]^3+40*x^27*X[3,1,1,2,3]^4-4*x^26*X[3,1,1,2,3]^5+5*x^30+2* x^29*X[3,1,1,2,3]-81*x^28*X[3,1,1,2,3]^2-70*x^27*X[3,1,1,2,3]^3+19*x^26*X[3,1,1 ,2,3]^4+2*x^25*X[3,1,1,2,3]^5+2*x^24*X[3,1,1,2,3]^6-x^29+31*x^28*X[3,1,1,2,3]+ 65*x^27*X[3,1,1,2,3]^2-36*x^26*X[3,1,1,2,3]^3-12*x^25*X[3,1,1,2,3]^4-13*x^24*X[ 3,1,1,2,3]^5-x^23*X[3,1,1,2,3]^6-5*x^28-31*x^27*X[3,1,1,2,3]+34*x^26*X[3,1,1,2, 3]^2+28*x^25*X[3,1,1,2,3]^3+33*x^24*X[3,1,1,2,3]^4+12*x^23*X[3,1,1,2,3]^5+6*x^ 27-16*x^26*X[3,1,1,2,3]-32*x^25*X[3,1,1,2,3]^2-42*x^24*X[3,1,1,2,3]^3-42*x^23*X [3,1,1,2,3]^4+3*x^22*X[3,1,1,2,3]^5+3*x^26+18*x^25*X[3,1,1,2,3]+28*x^24*X[3,1,1 ,2,3]^2+68*x^23*X[3,1,1,2,3]^3-12*x^22*X[3,1,1,2,3]^4-9*x^21*X[3,1,1,2,3]^5-4*x ^25-9*x^24*X[3,1,1,2,3]-57*x^23*X[3,1,1,2,3]^2+18*x^22*X[3,1,1,2,3]^3+47*x^21*X [3,1,1,2,3]^4+5*x^20*X[3,1,1,2,3]^5+x^24+24*x^23*X[3,1,1,2,3]-12*x^22*X[3,1,1,2 ,3]^2-100*x^21*X[3,1,1,2,3]^3-27*x^20*X[3,1,1,2,3]^4-x^19*X[3,1,1,2,3]^5-4*x^23 +3*x^22*X[3,1,1,2,3]+108*x^21*X[3,1,1,2,3]^2+59*x^20*X[3,1,1,2,3]^3-x^19*X[3,1, 1,2,3]^4-59*x^21*X[3,1,1,2,3]-65*x^20*X[3,1,1,2,3]^2+19*x^19*X[3,1,1,2,3]^3+11* x^18*X[3,1,1,2,3]^4+13*x^21+36*x^20*X[3,1,1,2,3]-41*x^19*X[3,1,1,2,3]^2-46*x^18 *X[3,1,1,2,3]^3-12*x^17*X[3,1,1,2,3]^4-8*x^20+34*x^19*X[3,1,1,2,3]+73*x^18*X[3, 1,1,2,3]^2+40*x^17*X[3,1,1,2,3]^3+10*x^16*X[3,1,1,2,3]^4-10*x^19-52*x^18*X[3,1, 1,2,3]-48*x^17*X[3,1,1,2,3]^2-28*x^16*X[3,1,1,2,3]^3-5*x^15*X[3,1,1,2,3]^4+14*x ^18+24*x^17*X[3,1,1,2,3]+18*x^16*X[3,1,1,2,3]^2+4*x^15*X[3,1,1,2,3]^3+x^14*X[3, 1,1,2,3]^4-4*x^17+8*x^16*X[3,1,1,2,3]+26*x^15*X[3,1,1,2,3]^2+18*x^14*X[3,1,1,2, 3]^3-8*x^16-44*x^15*X[3,1,1,2,3]-58*x^14*X[3,1,1,2,3]^2-18*x^13*X[3,1,1,2,3]^3+ 19*x^15+58*x^14*X[3,1,1,2,3]+30*x^13*X[3,1,1,2,3]^2+7*x^12*X[3,1,1,2,3]^3-19*x^ 14-5*x^13*X[3,1,1,2,3]+26*x^12*X[3,1,1,2,3]^2-x^11*X[3,1,1,2,3]^3-7*x^13-74*x^ 12*X[3,1,1,2,3]-45*x^11*X[3,1,1,2,3]^2+41*x^12+83*x^11*X[3,1,1,2,3]+27*x^10*X[3 ,1,1,2,3]^2-37*x^11-19*x^10*X[3,1,1,2,3]-8*x^9*X[3,1,1,2,3]^2-8*x^10-45*x^9*X[3 ,1,1,2,3]+x^8*X[3,1,1,2,3]^2+53*x^9+75*x^8*X[3,1,1,2,3]-74*x^8-77*x^7*X[3,1,1,2 ,3]+62*x^7+57*x^6*X[3,1,1,2,3]-8*x^6-28*x^5*X[3,1,1,2,3]-63*x^5+8*x^4*X[3,1,1,2 ,3]+97*x^4-x^3*X[3,1,1,2,3]-76*x^3+35*x^2-9*x+1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [3, 1, 1, 2, 3], equals , - 3/16 + ---- 64 255 73 n The variance equals , - ---- + ---- 1024 4096 6663 189 n The , 3, -th moment about the mean is , - ----- + ----- 16384 8192 694941 15987 2 82715 The , 4, -th moment about the mean is , - ------- + -------- n + ------- n 1048576 16777216 8388608 The compositions of, 10, that yield the, 37, -th largest growth, that is, 1.9709021410209757284, are , [2, 3, 1, 1, 3], [3, 1, 1, 3, 2] Theorem Number, 37, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 3, 1, 1, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 21 20 17 16 15 13 12 ) a(n) x = - (x + 2 x + 3 x + 2 x - 2 x + 4 x - 3 x / ----- n = 0 11 10 9 8 7 6 5 4 3 2 - 2 x + 5 x - 2 x - 4 x + 4 x + 2 x - 5 x - x + 9 x - 10 x / 4 2 16 15 12 10 + 5 x - 1) / ((x - x + 1) (x - x + 1) (x + 2 x + 3 x - 2 x / 9 7 6 5 4 3 2 + 2 x - 4 x + 2 x + 2 x - 5 x + 2 x + 4 x - 4 x + 1)) and in Maple format -(x^21+2*x^20+3*x^17+2*x^16-2*x^15+4*x^13-3*x^12-2*x^11+5*x^10-2*x^9-4*x^8+4*x^ 7+2*x^6-5*x^5-x^4+9*x^3-10*x^2+5*x-1)/(x^4-x+1)/(x^2-x+1)/(x^16+2*x^15+3*x^12-2 *x^10+2*x^9-4*x^7+2*x^6+2*x^5-5*x^4+2*x^3+4*x^2-4*x+1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1017, 2018, 3994, 7889, 15560, 30661, 60385, 118898, 234110, 461024, 908052, 1788892, 3524789, 6946103, 13689603, 26981513, 53180821, 104820955, 206604525, 407218156] The limit of a(n+1)/a(n) as n goes to infinity is 1.97090214102 a(n) is asymptotic to .588714283570*1.97090214102^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [2, 3, 1, 1, 3], denoted by the variable, X[2, 3, 1, 1, 3], is 21 21 20 21 20 21 - (x %3 - 4 x %2 + 2 x %3 + 6 x %1 - 8 x %2 - 4 x X[2, 3, 1, 1, 3] 20 21 20 17 20 17 + 12 x %1 + x - 8 x X[2, 3, 1, 1, 3] + x %3 + 2 x - 6 x %2 17 16 17 16 15 + 12 x %1 - 2 x %2 - 10 x X[2, 3, 1, 1, 3] + 6 x %1 + 2 x %2 17 16 15 14 16 + 3 x - 6 x X[2, 3, 1, 1, 3] - 6 x %1 - x %2 + 2 x 15 14 13 15 + 6 x X[2, 3, 1, 1, 3] + 2 x %1 - 2 x %2 - 2 x 14 13 12 13 - x X[2, 3, 1, 1, 3] + 8 x %1 + 2 x %2 - 10 x X[2, 3, 1, 1, 3] 12 11 13 12 12 - 7 x %1 - x %2 + 4 x + 8 x X[2, 3, 1, 1, 3] - 3 x 11 10 11 10 + 3 x X[2, 3, 1, 1, 3] + 5 x %1 - 2 x - 10 x X[2, 3, 1, 1, 3] 9 10 9 8 9 - 4 x %1 + 5 x + 6 x X[2, 3, 1, 1, 3] + x %1 - 2 x 8 8 7 7 + 3 x X[2, 3, 1, 1, 3] - 4 x - 4 x X[2, 3, 1, 1, 3] + 4 x 6 6 5 5 - 2 x X[2, 3, 1, 1, 3] + 2 x + 6 x X[2, 3, 1, 1, 3] - 5 x 4 4 3 3 2 - 4 x X[2, 3, 1, 1, 3] - x + x X[2, 3, 1, 1, 3] + 9 x - 10 x + 5 x / 22 22 21 22 21 20 - 1) / (x %3 - 4 x %2 + x %3 + 6 x %1 - 4 x %2 - x %3 / 22 21 20 19 22 - 4 x X[2, 3, 1, 1, 3] + 6 x %1 + 4 x %2 + x %3 + x 21 20 19 18 21 - 4 x X[2, 3, 1, 1, 3] - 6 x %1 - 4 x %2 + x %3 + x 20 19 18 17 20 + 4 x X[2, 3, 1, 1, 3] + 6 x %1 - 6 x %2 - x %3 - x 19 18 17 19 - 4 x X[2, 3, 1, 1, 3] + 12 x %1 + 4 x %2 + x 18 17 16 18 - 10 x X[2, 3, 1, 1, 3] - 6 x %1 + 2 x %2 + 3 x 17 16 15 17 + 4 x X[2, 3, 1, 1, 3] - 6 x %1 - 4 x %2 - x 16 15 16 15 + 6 x X[2, 3, 1, 1, 3] + 11 x %1 - 2 x - 10 x X[2, 3, 1, 1, 3] 14 13 15 14 13 + 2 x %1 + 3 x %2 + 3 x - 4 x X[2, 3, 1, 1, 3] - 12 x %1 12 14 13 12 11 13 - 3 x %2 + 2 x + 15 x X[2, 3, 1, 1, 3] + 9 x %1 + x %2 - 6 x 12 11 12 11 - 9 x X[2, 3, 1, 1, 3] + 2 x %1 + 3 x - 7 x X[2, 3, 1, 1, 3] 10 11 10 9 10 - 8 x %1 + 4 x + 15 x X[2, 3, 1, 1, 3] + 5 x %1 - 7 x 9 8 9 8 8 - 6 x X[2, 3, 1, 1, 3] - x %1 + x - 6 x X[2, 3, 1, 1, 3] + 7 x 7 7 6 6 + 5 x X[2, 3, 1, 1, 3] - 5 x + 5 x X[2, 3, 1, 1, 3] - 5 x 5 5 4 4 - 9 x X[2, 3, 1, 1, 3] + 7 x + 5 x X[2, 3, 1, 1, 3] + 4 x 3 3 2 - x X[2, 3, 1, 1, 3] - 15 x + 14 x - 6 x + 1) 2 %1 := X[2, 3, 1, 1, 3] 3 %2 := X[2, 3, 1, 1, 3] 4 %3 := X[2, 3, 1, 1, 3] and in Maple format -(x^21*X[2,3,1,1,3]^4-4*x^21*X[2,3,1,1,3]^3+2*x^20*X[2,3,1,1,3]^4+6*x^21*X[2,3, 1,1,3]^2-8*x^20*X[2,3,1,1,3]^3-4*x^21*X[2,3,1,1,3]+12*x^20*X[2,3,1,1,3]^2+x^21-\ 8*x^20*X[2,3,1,1,3]+x^17*X[2,3,1,1,3]^4+2*x^20-6*x^17*X[2,3,1,1,3]^3+12*x^17*X[ 2,3,1,1,3]^2-2*x^16*X[2,3,1,1,3]^3-10*x^17*X[2,3,1,1,3]+6*x^16*X[2,3,1,1,3]^2+2 *x^15*X[2,3,1,1,3]^3+3*x^17-6*x^16*X[2,3,1,1,3]-6*x^15*X[2,3,1,1,3]^2-x^14*X[2, 3,1,1,3]^3+2*x^16+6*x^15*X[2,3,1,1,3]+2*x^14*X[2,3,1,1,3]^2-2*x^13*X[2,3,1,1,3] ^3-2*x^15-x^14*X[2,3,1,1,3]+8*x^13*X[2,3,1,1,3]^2+2*x^12*X[2,3,1,1,3]^3-10*x^13 *X[2,3,1,1,3]-7*x^12*X[2,3,1,1,3]^2-x^11*X[2,3,1,1,3]^3+4*x^13+8*x^12*X[2,3,1,1 ,3]-3*x^12+3*x^11*X[2,3,1,1,3]+5*x^10*X[2,3,1,1,3]^2-2*x^11-10*x^10*X[2,3,1,1,3 ]-4*x^9*X[2,3,1,1,3]^2+5*x^10+6*x^9*X[2,3,1,1,3]+x^8*X[2,3,1,1,3]^2-2*x^9+3*x^8 *X[2,3,1,1,3]-4*x^8-4*x^7*X[2,3,1,1,3]+4*x^7-2*x^6*X[2,3,1,1,3]+2*x^6+6*x^5*X[2 ,3,1,1,3]-5*x^5-4*x^4*X[2,3,1,1,3]-x^4+x^3*X[2,3,1,1,3]+9*x^3-10*x^2+5*x-1)/(x^ 22*X[2,3,1,1,3]^4-4*x^22*X[2,3,1,1,3]^3+x^21*X[2,3,1,1,3]^4+6*x^22*X[2,3,1,1,3] ^2-4*x^21*X[2,3,1,1,3]^3-x^20*X[2,3,1,1,3]^4-4*x^22*X[2,3,1,1,3]+6*x^21*X[2,3,1 ,1,3]^2+4*x^20*X[2,3,1,1,3]^3+x^19*X[2,3,1,1,3]^4+x^22-4*x^21*X[2,3,1,1,3]-6*x^ 20*X[2,3,1,1,3]^2-4*x^19*X[2,3,1,1,3]^3+x^18*X[2,3,1,1,3]^4+x^21+4*x^20*X[2,3,1 ,1,3]+6*x^19*X[2,3,1,1,3]^2-6*x^18*X[2,3,1,1,3]^3-x^17*X[2,3,1,1,3]^4-x^20-4*x^ 19*X[2,3,1,1,3]+12*x^18*X[2,3,1,1,3]^2+4*x^17*X[2,3,1,1,3]^3+x^19-10*x^18*X[2,3 ,1,1,3]-6*x^17*X[2,3,1,1,3]^2+2*x^16*X[2,3,1,1,3]^3+3*x^18+4*x^17*X[2,3,1,1,3]-\ 6*x^16*X[2,3,1,1,3]^2-4*x^15*X[2,3,1,1,3]^3-x^17+6*x^16*X[2,3,1,1,3]+11*x^15*X[ 2,3,1,1,3]^2-2*x^16-10*x^15*X[2,3,1,1,3]+2*x^14*X[2,3,1,1,3]^2+3*x^13*X[2,3,1,1 ,3]^3+3*x^15-4*x^14*X[2,3,1,1,3]-12*x^13*X[2,3,1,1,3]^2-3*x^12*X[2,3,1,1,3]^3+2 *x^14+15*x^13*X[2,3,1,1,3]+9*x^12*X[2,3,1,1,3]^2+x^11*X[2,3,1,1,3]^3-6*x^13-9*x ^12*X[2,3,1,1,3]+2*x^11*X[2,3,1,1,3]^2+3*x^12-7*x^11*X[2,3,1,1,3]-8*x^10*X[2,3, 1,1,3]^2+4*x^11+15*x^10*X[2,3,1,1,3]+5*x^9*X[2,3,1,1,3]^2-7*x^10-6*x^9*X[2,3,1, 1,3]-x^8*X[2,3,1,1,3]^2+x^9-6*x^8*X[2,3,1,1,3]+7*x^8+5*x^7*X[2,3,1,1,3]-5*x^7+5 *x^6*X[2,3,1,1,3]-5*x^6-9*x^5*X[2,3,1,1,3]+7*x^5+5*x^4*X[2,3,1,1,3]+4*x^4-x^3*X [2,3,1,1,3]-15*x^3+14*x^2-6*x+1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [2, 3, 1, 1, 3], equals , - 3/16 + ---- 64 251 73 n The variance equals , - ---- + ---- 1024 4096 201 189 n The , 3, -th moment about the mean is , - --- + ----- 512 8192 655685 15987 2 86411 The , 4, -th moment about the mean is , - ------- + -------- n + ------- n 1048576 16777216 8388608 The compositions of, 10, that yield the, 38, -th largest growth, that is, 1.9709232598962379131, are , [2, 1, 1, 3, 3], [3, 3, 1, 1, 2] Theorem Number, 38, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 1, 3, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 14 11 8 7 5 4 3 2 ) a(n) x = - (x + x + x - x + 3 x - 2 x - 3 x + 6 x - 4 x + 1 / ----- n = 0 / 15 14 12 11 9 8 7 6 5 4 3 ) / (x - x + x - x + x - x + 2 x + x - 5 x + 2 x + 6 x / 2 - 9 x + 5 x - 1) and in Maple format -(x^14+x^11+x^8-x^7+3*x^5-2*x^4-3*x^3+6*x^2-4*x+1)/(x^15-x^14+x^12-x^11+x^9-x^8 +2*x^7+x^6-5*x^5+2*x^4+6*x^3-9*x^2+5*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1017, 2018, 3994, 7890, 15567, 30690, 60478, 119153, 234739, 462464, 911177, 1795417, 3538041, 6972502, 13741532, 27082926, 53378271, 105205374, 207354351, 408684842] The limit of a(n+1)/a(n) as n goes to infinity is 1.97092325990 a(n) is asymptotic to .590624128719*1.97092325990^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [2, 1, 1, 3, 3], denoted by the variable, X[2, 1, 1, 3, 3], is 15 15 14 15 14 13 - (x %2 - 3 x %1 - x %2 + 3 x X[2, 1, 1, 3, 3] + 3 x %1 + x %2 15 14 13 14 13 - x - 3 x X[2, 1, 1, 3, 3] - 2 x %1 + x + x X[2, 1, 1, 3, 3] 12 12 11 12 11 - x %1 + 2 x X[2, 1, 1, 3, 3] + x %1 - x - 2 x X[2, 1, 1, 3, 3] 11 9 9 8 8 + x + x X[2, 1, 1, 3, 3] - x - 2 x X[2, 1, 1, 3, 3] + 2 x 7 7 6 6 + x X[2, 1, 1, 3, 3] - x + 3 x X[2, 1, 1, 3, 3] - 3 x 5 5 4 4 - 6 x X[2, 1, 1, 3, 3] + 5 x + 4 x X[2, 1, 1, 3, 3] + x 3 3 2 / 16 16 - x X[2, 1, 1, 3, 3] - 9 x + 10 x - 5 x + 1) / (x %2 - 3 x %1 / 15 16 15 14 16 - x %2 + 3 x X[2, 1, 1, 3, 3] + 4 x %1 + 2 x %2 - x 15 14 13 15 - 5 x X[2, 1, 1, 3, 3] - 5 x %1 - x %2 + 2 x 14 13 14 13 12 + 4 x X[2, 1, 1, 3, 3] + x %1 - x + x X[2, 1, 1, 3, 3] + 2 x %1 13 12 11 12 11 - x - 4 x X[2, 1, 1, 3, 3] - x %1 + 2 x + 2 x X[2, 1, 1, 3, 3] 11 10 10 9 9 - x + x X[2, 1, 1, 3, 3] - x - 2 x X[2, 1, 1, 3, 3] + 2 x 8 8 7 7 + 3 x X[2, 1, 1, 3, 3] - 3 x - x X[2, 1, 1, 3, 3] + x 6 6 5 5 - 6 x X[2, 1, 1, 3, 3] + 6 x + 9 x X[2, 1, 1, 3, 3] - 7 x 4 4 3 3 2 - 5 x X[2, 1, 1, 3, 3] - 4 x + x X[2, 1, 1, 3, 3] + 15 x - 14 x + 6 x - 1) 2 %1 := X[2, 1, 1, 3, 3] 3 %2 := X[2, 1, 1, 3, 3] and in Maple format -(x^15*X[2,1,1,3,3]^3-3*x^15*X[2,1,1,3,3]^2-x^14*X[2,1,1,3,3]^3+3*x^15*X[2,1,1, 3,3]+3*x^14*X[2,1,1,3,3]^2+x^13*X[2,1,1,3,3]^3-x^15-3*x^14*X[2,1,1,3,3]-2*x^13* X[2,1,1,3,3]^2+x^14+x^13*X[2,1,1,3,3]-x^12*X[2,1,1,3,3]^2+2*x^12*X[2,1,1,3,3]+x ^11*X[2,1,1,3,3]^2-x^12-2*x^11*X[2,1,1,3,3]+x^11+x^9*X[2,1,1,3,3]-x^9-2*x^8*X[2 ,1,1,3,3]+2*x^8+x^7*X[2,1,1,3,3]-x^7+3*x^6*X[2,1,1,3,3]-3*x^6-6*x^5*X[2,1,1,3,3 ]+5*x^5+4*x^4*X[2,1,1,3,3]+x^4-x^3*X[2,1,1,3,3]-9*x^3+10*x^2-5*x+1)/(x^16*X[2,1 ,1,3,3]^3-3*x^16*X[2,1,1,3,3]^2-x^15*X[2,1,1,3,3]^3+3*x^16*X[2,1,1,3,3]+4*x^15* X[2,1,1,3,3]^2+2*x^14*X[2,1,1,3,3]^3-x^16-5*x^15*X[2,1,1,3,3]-5*x^14*X[2,1,1,3, 3]^2-x^13*X[2,1,1,3,3]^3+2*x^15+4*x^14*X[2,1,1,3,3]+x^13*X[2,1,1,3,3]^2-x^14+x^ 13*X[2,1,1,3,3]+2*x^12*X[2,1,1,3,3]^2-x^13-4*x^12*X[2,1,1,3,3]-x^11*X[2,1,1,3,3 ]^2+2*x^12+2*x^11*X[2,1,1,3,3]-x^11+x^10*X[2,1,1,3,3]-x^10-2*x^9*X[2,1,1,3,3]+2 *x^9+3*x^8*X[2,1,1,3,3]-3*x^8-x^7*X[2,1,1,3,3]+x^7-6*x^6*X[2,1,1,3,3]+6*x^6+9*x ^5*X[2,1,1,3,3]-7*x^5-5*x^4*X[2,1,1,3,3]-4*x^4+x^3*X[2,1,1,3,3]+15*x^3-14*x^2+6 *x-1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [2, 1, 1, 3, 3], equals , - 3/16 + ---- 64 243 73 n The variance equals , - ---- + ---- 1024 4096 2997 189 n The , 3, -th moment about the mean is , - ---- + ----- 8192 8192 592773 15987 2 95723 The , 4, -th moment about the mean is , - ------- + -------- n + ------- n 1048576 16777216 8388608 The compositions of, 10, that yield the, 39, -th largest growth, that is, 1.9717270001741243154, are , [3, 1, 2, 1, 3] Theorem Number, 39, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [3, 1, 2, 1, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 38 37 36 35 34 33 32 ) a(n) x = - (x + 3 x + 3 x + 2 x + 3 x + 4 x + 4 x / ----- n = 0 31 30 29 28 26 25 24 23 22 20 + 2 x - x + x + 2 x - 2 x - 2 x + 4 x + x - 6 x + 4 x 19 17 16 15 14 13 11 10 9 + x - 6 x - x + 11 x - 2 x - 9 x + 10 x - x - 11 x 8 7 6 5 4 3 2 / + 13 x - 9 x - 8 x + 41 x - 64 x + 55 x - 28 x + 8 x - 1) / ( / 10 9 8 7 6 5 4 3 2 30 29 (x + x - x + x + x - x + 2 x - x - 2 x + 3 x - 1) (x + 3 x 28 27 25 24 23 22 21 20 19 18 16 + 3 x + x + x + x - x - x + x + x - 2 x - 2 x + 4 x 15 14 12 10 9 8 7 6 5 4 - x - 3 x + 4 x - 4 x + 2 x - 3 x + 9 x - 11 x + 11 x - 16 x 3 2 + 20 x - 15 x + 6 x - 1)) and in Maple format -(x^38+3*x^37+3*x^36+2*x^35+3*x^34+4*x^33+4*x^32+2*x^31-x^30+x^29+2*x^28-2*x^26 -2*x^25+4*x^24+x^23-6*x^22+4*x^20+x^19-6*x^17-x^16+11*x^15-2*x^14-9*x^13+10*x^ 11-x^10-11*x^9+13*x^8-9*x^7-8*x^6+41*x^5-64*x^4+55*x^3-28*x^2+8*x-1)/(x^10+x^9- x^8+x^7+x^6-x^5+2*x^4-x^3-2*x^2+3*x-1)/(x^30+3*x^29+3*x^28+x^27+x^25+x^24-x^23- x^22+x^21+x^20-2*x^19-2*x^18+4*x^16-x^15-3*x^14+4*x^12-4*x^10+2*x^9-3*x^8+9*x^7 -11*x^6+11*x^5-16*x^4+20*x^3-15*x^2+6*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1017, 2018, 3994, 7889, 15560, 30662, 60392, 118928, 234212, 461329, 908896, 1791110, 3530396, 6959833, 13722323, 27057686, 53354627, 105210778, 207466154, 409099187] The limit of a(n+1)/a(n) as n goes to infinity is 1.97172700017 a(n) is asymptotic to .584082883820*1.97172700017^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [3, 1, 2, 1, 3], denoted by the variable, X[3, 1, 2, 1, 3], is 38 38 37 38 37 36 38 - (x %7 - 8 x %6 + 3 x %7 + 28 x %5 - 24 x %6 + 3 x %7 - 56 x %4 37 36 35 38 37 36 + 84 x %5 - 24 x %6 + x %7 + 70 x %3 - 168 x %4 + 84 x %5 35 38 37 36 35 34 - 9 x %6 - 56 x %2 + 210 x %3 - 168 x %4 + 35 x %5 - 3 x %6 38 37 36 35 34 33 + 28 x %1 - 168 x %2 + 210 x %3 - 77 x %4 + 21 x %5 - 4 x %6 38 37 36 35 34 - 8 x X[3, 1, 2, 1, 3] + 84 x %1 - 168 x %2 + 105 x %3 - 63 x %4 33 32 38 37 36 + 28 x %5 - 4 x %6 + x - 24 x X[3, 1, 2, 1, 3] + 84 x %1 35 34 33 32 31 37 - 91 x %2 + 105 x %3 - 84 x %4 + 28 x %5 - 2 x %6 + 3 x 36 35 34 33 - 24 x X[3, 1, 2, 1, 3] + 49 x %1 - 105 x %2 + 140 x %3 32 31 30 36 35 - 84 x %4 + 14 x %5 + x %6 + 3 x - 15 x X[3, 1, 2, 1, 3] 34 33 32 31 30 35 + 63 x %1 - 140 x %2 + 140 x %3 - 42 x %4 - 7 x %5 + 2 x 34 33 32 31 30 - 21 x X[3, 1, 2, 1, 3] + 84 x %1 - 140 x %2 + 70 x %3 + 21 x %4 29 28 34 33 32 + x %5 - x %6 + 3 x - 28 x X[3, 1, 2, 1, 3] + 84 x %1 31 30 29 28 33 - 70 x %2 - 35 x %3 - 6 x %4 + 8 x %5 + 4 x 32 31 30 29 28 - 28 x X[3, 1, 2, 1, 3] + 42 x %1 + 35 x %2 + 15 x %3 - 27 x %4 32 31 30 29 28 + 4 x - 14 x X[3, 1, 2, 1, 3] - 21 x %1 - 20 x %2 + 50 x %3 26 31 30 29 28 - x %5 + 2 x + 7 x X[3, 1, 2, 1, 3] + 15 x %1 - 55 x %2 26 25 30 29 28 + 7 x %4 - 2 x %5 - x - 6 x X[3, 1, 2, 1, 3] + 36 x %1 26 25 24 29 28 - 20 x %3 + 12 x %4 + x %5 + x - 13 x X[3, 1, 2, 1, 3] 26 25 24 28 26 25 + 30 x %2 - 30 x %3 - 8 x %4 + 2 x - 25 x %1 + 40 x %2 24 23 26 25 24 + 26 x %3 - 3 x %4 + 11 x X[3, 1, 2, 1, 3] - 30 x %1 - 44 x %2 23 22 26 25 24 + 13 x %3 + 2 x %4 - 2 x + 12 x X[3, 1, 2, 1, 3] + 41 x %1 23 22 21 25 24 - 22 x %2 - 14 x %3 + 3 x %4 - 2 x - 20 x X[3, 1, 2, 1, 3] 23 22 21 24 23 + 18 x %1 + 36 x %2 - 13 x %3 + 4 x - 7 x X[3, 1, 2, 1, 3] 22 21 20 19 23 - 44 x %1 + 21 x %2 + 4 x %3 - 2 x %4 + x 22 21 20 19 18 + 26 x X[3, 1, 2, 1, 3] - 15 x %1 - 16 x %2 + 11 x %3 + x %4 22 21 20 19 18 - 6 x + 4 x X[3, 1, 2, 1, 3] + 24 x %1 - 22 x %2 - 8 x %3 20 19 18 17 20 - 16 x X[3, 1, 2, 1, 3] + 20 x %1 + 18 x %2 + 2 x %3 + 4 x 19 18 16 19 - 8 x X[3, 1, 2, 1, 3] - 16 x %1 - 3 x %3 + x 18 17 16 15 + 5 x X[3, 1, 2, 1, 3] - 12 x %1 + 8 x %2 + 3 x %3 17 16 15 14 17 + 16 x X[3, 1, 2, 1, 3] - 8 x %1 - 14 x %2 - x %3 - 6 x 16 15 14 16 + 4 x X[3, 1, 2, 1, 3] + 30 x %1 + x %2 - x 15 14 13 15 - 30 x X[3, 1, 2, 1, 3] - x %1 + 8 x %2 + 11 x 14 13 12 14 + 3 x X[3, 1, 2, 1, 3] - 26 x %1 - 5 x %2 - 2 x 13 12 11 13 + 27 x X[3, 1, 2, 1, 3] + 15 x %1 + x %2 - 9 x 12 11 11 10 - 10 x X[3, 1, 2, 1, 3] - x %1 - 10 x X[3, 1, 2, 1, 3] + 5 x %1 11 10 9 10 9 + 10 x - 4 x X[3, 1, 2, 1, 3] - 9 x %1 - x + 20 x X[3, 1, 2, 1, 3] 8 9 8 7 8 + 5 x %1 - 11 x - 19 x X[3, 1, 2, 1, 3] - x %1 + 13 x 7 7 6 6 + 18 x X[3, 1, 2, 1, 3] - 9 x - 20 x X[3, 1, 2, 1, 3] - 8 x 5 5 4 4 + 15 x X[3, 1, 2, 1, 3] + 41 x - 6 x X[3, 1, 2, 1, 3] - 64 x 3 3 2 / 10 + x X[3, 1, 2, 1, 3] + 55 x - 28 x + 8 x - 1) / ((x %1 / 10 9 10 9 8 - 2 x X[3, 1, 2, 1, 3] + x %1 + x - 2 x X[3, 1, 2, 1, 3] - x %1 9 8 8 7 7 + x + 2 x X[3, 1, 2, 1, 3] - x - x X[3, 1, 2, 1, 3] + x 6 6 5 5 - x X[3, 1, 2, 1, 3] + x + x X[3, 1, 2, 1, 3] - x 4 4 3 3 2 - 2 x X[3, 1, 2, 1, 3] + 2 x + x X[3, 1, 2, 1, 3] - x - 2 x + 3 x - 1 30 30 29 30 29 28 ) (x %5 - 6 x %4 + 3 x %5 + 15 x %3 - 18 x %4 + 3 x %5 30 29 28 27 30 29 - 20 x %2 + 45 x %3 - 18 x %4 + x %5 + 15 x %1 - 60 x %2 28 27 30 29 28 + 45 x %3 - 6 x %4 - 6 x X[3, 1, 2, 1, 3] + 45 x %1 - 60 x %2 27 30 29 28 27 + 15 x %3 + x - 18 x X[3, 1, 2, 1, 3] + 45 x %1 - 20 x %2 25 29 28 27 25 24 - x %4 + 3 x - 18 x X[3, 1, 2, 1, 3] + 15 x %1 + 5 x %3 - x %4 28 27 25 24 23 27 + 3 x - 6 x X[3, 1, 2, 1, 3] - 10 x %2 + 5 x %3 + x %4 + x 25 24 23 22 25 + 10 x %1 - 10 x %2 - 5 x %3 + 2 x %4 - 5 x X[3, 1, 2, 1, 3] 24 23 22 25 24 + 10 x %1 + 10 x %2 - 9 x %3 + x - 5 x X[3, 1, 2, 1, 3] 23 22 21 20 24 23 - 10 x %1 + 16 x %2 + x %3 - x %4 + x + 5 x X[3, 1, 2, 1, 3] 22 21 20 23 22 - 14 x %1 - 4 x %2 + 5 x %3 - x + 6 x X[3, 1, 2, 1, 3] 21 20 19 22 21 20 + 6 x %1 - 10 x %2 - x %3 - x - 4 x X[3, 1, 2, 1, 3] + 10 x %1 19 18 21 20 19 18 + 5 x %2 - x %3 + x - 5 x X[3, 1, 2, 1, 3] - 9 x %1 + 5 x %2 17 20 19 18 17 16 - x %3 + x + 7 x X[3, 1, 2, 1, 3] - 9 x %1 + 3 x %2 + 2 x %3 19 18 17 16 15 18 - 2 x + 7 x X[3, 1, 2, 1, 3] - 3 x %1 - 9 x %2 - x %3 - 2 x 17 16 15 16 + x X[3, 1, 2, 1, 3] + 16 x %1 + 4 x %2 - 13 x X[3, 1, 2, 1, 3] 15 14 16 15 14 13 - 6 x %1 - x %2 + 4 x + 4 x X[3, 1, 2, 1, 3] - x %1 + 3 x %2 15 14 13 12 14 - x + 5 x X[3, 1, 2, 1, 3] - 6 x %1 - 3 x %2 - 3 x 13 12 11 12 + 3 x X[3, 1, 2, 1, 3] + 10 x %1 + x %2 - 11 x X[3, 1, 2, 1, 3] 11 12 11 10 - 3 x %1 + 4 x + 2 x X[3, 1, 2, 1, 3] - 2 x %1 10 9 10 9 9 + 6 x X[3, 1, 2, 1, 3] + x %1 - 4 x - 3 x X[3, 1, 2, 1, 3] + 2 x 8 8 7 7 + 3 x X[3, 1, 2, 1, 3] - 3 x - 9 x X[3, 1, 2, 1, 3] + 9 x 6 6 5 5 + 10 x X[3, 1, 2, 1, 3] - 11 x - 5 x X[3, 1, 2, 1, 3] + 11 x 4 4 3 2 + x X[3, 1, 2, 1, 3] - 16 x + 20 x - 15 x + 6 x - 1)) 2 %1 := X[3, 1, 2, 1, 3] 3 %2 := X[3, 1, 2, 1, 3] 4 %3 := X[3, 1, 2, 1, 3] 5 %4 := X[3, 1, 2, 1, 3] 6 %5 := X[3, 1, 2, 1, 3] 7 %6 := X[3, 1, 2, 1, 3] 8 %7 := X[3, 1, 2, 1, 3] and in Maple format -(x^38*X[3,1,2,1,3]^8-8*x^38*X[3,1,2,1,3]^7+3*x^37*X[3,1,2,1,3]^8+28*x^38*X[3,1 ,2,1,3]^6-24*x^37*X[3,1,2,1,3]^7+3*x^36*X[3,1,2,1,3]^8-56*x^38*X[3,1,2,1,3]^5+ 84*x^37*X[3,1,2,1,3]^6-24*x^36*X[3,1,2,1,3]^7+x^35*X[3,1,2,1,3]^8+70*x^38*X[3,1 ,2,1,3]^4-168*x^37*X[3,1,2,1,3]^5+84*x^36*X[3,1,2,1,3]^6-9*x^35*X[3,1,2,1,3]^7-\ 56*x^38*X[3,1,2,1,3]^3+210*x^37*X[3,1,2,1,3]^4-168*x^36*X[3,1,2,1,3]^5+35*x^35* X[3,1,2,1,3]^6-3*x^34*X[3,1,2,1,3]^7+28*x^38*X[3,1,2,1,3]^2-168*x^37*X[3,1,2,1, 3]^3+210*x^36*X[3,1,2,1,3]^4-77*x^35*X[3,1,2,1,3]^5+21*x^34*X[3,1,2,1,3]^6-4*x^ 33*X[3,1,2,1,3]^7-8*x^38*X[3,1,2,1,3]+84*x^37*X[3,1,2,1,3]^2-168*x^36*X[3,1,2,1 ,3]^3+105*x^35*X[3,1,2,1,3]^4-63*x^34*X[3,1,2,1,3]^5+28*x^33*X[3,1,2,1,3]^6-4*x ^32*X[3,1,2,1,3]^7+x^38-24*x^37*X[3,1,2,1,3]+84*x^36*X[3,1,2,1,3]^2-91*x^35*X[3 ,1,2,1,3]^3+105*x^34*X[3,1,2,1,3]^4-84*x^33*X[3,1,2,1,3]^5+28*x^32*X[3,1,2,1,3] ^6-2*x^31*X[3,1,2,1,3]^7+3*x^37-24*x^36*X[3,1,2,1,3]+49*x^35*X[3,1,2,1,3]^2-105 *x^34*X[3,1,2,1,3]^3+140*x^33*X[3,1,2,1,3]^4-84*x^32*X[3,1,2,1,3]^5+14*x^31*X[3 ,1,2,1,3]^6+x^30*X[3,1,2,1,3]^7+3*x^36-15*x^35*X[3,1,2,1,3]+63*x^34*X[3,1,2,1,3 ]^2-140*x^33*X[3,1,2,1,3]^3+140*x^32*X[3,1,2,1,3]^4-42*x^31*X[3,1,2,1,3]^5-7*x^ 30*X[3,1,2,1,3]^6+2*x^35-21*x^34*X[3,1,2,1,3]+84*x^33*X[3,1,2,1,3]^2-140*x^32*X [3,1,2,1,3]^3+70*x^31*X[3,1,2,1,3]^4+21*x^30*X[3,1,2,1,3]^5+x^29*X[3,1,2,1,3]^6 -x^28*X[3,1,2,1,3]^7+3*x^34-28*x^33*X[3,1,2,1,3]+84*x^32*X[3,1,2,1,3]^2-70*x^31 *X[3,1,2,1,3]^3-35*x^30*X[3,1,2,1,3]^4-6*x^29*X[3,1,2,1,3]^5+8*x^28*X[3,1,2,1,3 ]^6+4*x^33-28*x^32*X[3,1,2,1,3]+42*x^31*X[3,1,2,1,3]^2+35*x^30*X[3,1,2,1,3]^3+ 15*x^29*X[3,1,2,1,3]^4-27*x^28*X[3,1,2,1,3]^5+4*x^32-14*x^31*X[3,1,2,1,3]-21*x^ 30*X[3,1,2,1,3]^2-20*x^29*X[3,1,2,1,3]^3+50*x^28*X[3,1,2,1,3]^4-x^26*X[3,1,2,1, 3]^6+2*x^31+7*x^30*X[3,1,2,1,3]+15*x^29*X[3,1,2,1,3]^2-55*x^28*X[3,1,2,1,3]^3+7 *x^26*X[3,1,2,1,3]^5-2*x^25*X[3,1,2,1,3]^6-x^30-6*x^29*X[3,1,2,1,3]+36*x^28*X[3 ,1,2,1,3]^2-20*x^26*X[3,1,2,1,3]^4+12*x^25*X[3,1,2,1,3]^5+x^24*X[3,1,2,1,3]^6+x ^29-13*x^28*X[3,1,2,1,3]+30*x^26*X[3,1,2,1,3]^3-30*x^25*X[3,1,2,1,3]^4-8*x^24*X [3,1,2,1,3]^5+2*x^28-25*x^26*X[3,1,2,1,3]^2+40*x^25*X[3,1,2,1,3]^3+26*x^24*X[3, 1,2,1,3]^4-3*x^23*X[3,1,2,1,3]^5+11*x^26*X[3,1,2,1,3]-30*x^25*X[3,1,2,1,3]^2-44 *x^24*X[3,1,2,1,3]^3+13*x^23*X[3,1,2,1,3]^4+2*x^22*X[3,1,2,1,3]^5-2*x^26+12*x^ 25*X[3,1,2,1,3]+41*x^24*X[3,1,2,1,3]^2-22*x^23*X[3,1,2,1,3]^3-14*x^22*X[3,1,2,1 ,3]^4+3*x^21*X[3,1,2,1,3]^5-2*x^25-20*x^24*X[3,1,2,1,3]+18*x^23*X[3,1,2,1,3]^2+ 36*x^22*X[3,1,2,1,3]^3-13*x^21*X[3,1,2,1,3]^4+4*x^24-7*x^23*X[3,1,2,1,3]-44*x^ 22*X[3,1,2,1,3]^2+21*x^21*X[3,1,2,1,3]^3+4*x^20*X[3,1,2,1,3]^4-2*x^19*X[3,1,2,1 ,3]^5+x^23+26*x^22*X[3,1,2,1,3]-15*x^21*X[3,1,2,1,3]^2-16*x^20*X[3,1,2,1,3]^3+ 11*x^19*X[3,1,2,1,3]^4+x^18*X[3,1,2,1,3]^5-6*x^22+4*x^21*X[3,1,2,1,3]+24*x^20*X [3,1,2,1,3]^2-22*x^19*X[3,1,2,1,3]^3-8*x^18*X[3,1,2,1,3]^4-16*x^20*X[3,1,2,1,3] +20*x^19*X[3,1,2,1,3]^2+18*x^18*X[3,1,2,1,3]^3+2*x^17*X[3,1,2,1,3]^4+4*x^20-8*x ^19*X[3,1,2,1,3]-16*x^18*X[3,1,2,1,3]^2-3*x^16*X[3,1,2,1,3]^4+x^19+5*x^18*X[3,1 ,2,1,3]-12*x^17*X[3,1,2,1,3]^2+8*x^16*X[3,1,2,1,3]^3+3*x^15*X[3,1,2,1,3]^4+16*x ^17*X[3,1,2,1,3]-8*x^16*X[3,1,2,1,3]^2-14*x^15*X[3,1,2,1,3]^3-x^14*X[3,1,2,1,3] ^4-6*x^17+4*x^16*X[3,1,2,1,3]+30*x^15*X[3,1,2,1,3]^2+x^14*X[3,1,2,1,3]^3-x^16-\ 30*x^15*X[3,1,2,1,3]-x^14*X[3,1,2,1,3]^2+8*x^13*X[3,1,2,1,3]^3+11*x^15+3*x^14*X [3,1,2,1,3]-26*x^13*X[3,1,2,1,3]^2-5*x^12*X[3,1,2,1,3]^3-2*x^14+27*x^13*X[3,1,2 ,1,3]+15*x^12*X[3,1,2,1,3]^2+x^11*X[3,1,2,1,3]^3-9*x^13-10*x^12*X[3,1,2,1,3]-x^ 11*X[3,1,2,1,3]^2-10*x^11*X[3,1,2,1,3]+5*x^10*X[3,1,2,1,3]^2+10*x^11-4*x^10*X[3 ,1,2,1,3]-9*x^9*X[3,1,2,1,3]^2-x^10+20*x^9*X[3,1,2,1,3]+5*x^8*X[3,1,2,1,3]^2-11 *x^9-19*x^8*X[3,1,2,1,3]-x^7*X[3,1,2,1,3]^2+13*x^8+18*x^7*X[3,1,2,1,3]-9*x^7-20 *x^6*X[3,1,2,1,3]-8*x^6+15*x^5*X[3,1,2,1,3]+41*x^5-6*x^4*X[3,1,2,1,3]-64*x^4+x^ 3*X[3,1,2,1,3]+55*x^3-28*x^2+8*x-1)/(x^10*X[3,1,2,1,3]^2-2*x^10*X[3,1,2,1,3]+x^ 9*X[3,1,2,1,3]^2+x^10-2*x^9*X[3,1,2,1,3]-x^8*X[3,1,2,1,3]^2+x^9+2*x^8*X[3,1,2,1 ,3]-x^8-x^7*X[3,1,2,1,3]+x^7-x^6*X[3,1,2,1,3]+x^6+x^5*X[3,1,2,1,3]-x^5-2*x^4*X[ 3,1,2,1,3]+2*x^4+x^3*X[3,1,2,1,3]-x^3-2*x^2+3*x-1)/(x^30*X[3,1,2,1,3]^6-6*x^30* X[3,1,2,1,3]^5+3*x^29*X[3,1,2,1,3]^6+15*x^30*X[3,1,2,1,3]^4-18*x^29*X[3,1,2,1,3 ]^5+3*x^28*X[3,1,2,1,3]^6-20*x^30*X[3,1,2,1,3]^3+45*x^29*X[3,1,2,1,3]^4-18*x^28 *X[3,1,2,1,3]^5+x^27*X[3,1,2,1,3]^6+15*x^30*X[3,1,2,1,3]^2-60*x^29*X[3,1,2,1,3] ^3+45*x^28*X[3,1,2,1,3]^4-6*x^27*X[3,1,2,1,3]^5-6*x^30*X[3,1,2,1,3]+45*x^29*X[3 ,1,2,1,3]^2-60*x^28*X[3,1,2,1,3]^3+15*x^27*X[3,1,2,1,3]^4+x^30-18*x^29*X[3,1,2, 1,3]+45*x^28*X[3,1,2,1,3]^2-20*x^27*X[3,1,2,1,3]^3-x^25*X[3,1,2,1,3]^5+3*x^29-\ 18*x^28*X[3,1,2,1,3]+15*x^27*X[3,1,2,1,3]^2+5*x^25*X[3,1,2,1,3]^4-x^24*X[3,1,2, 1,3]^5+3*x^28-6*x^27*X[3,1,2,1,3]-10*x^25*X[3,1,2,1,3]^3+5*x^24*X[3,1,2,1,3]^4+ x^23*X[3,1,2,1,3]^5+x^27+10*x^25*X[3,1,2,1,3]^2-10*x^24*X[3,1,2,1,3]^3-5*x^23*X [3,1,2,1,3]^4+2*x^22*X[3,1,2,1,3]^5-5*x^25*X[3,1,2,1,3]+10*x^24*X[3,1,2,1,3]^2+ 10*x^23*X[3,1,2,1,3]^3-9*x^22*X[3,1,2,1,3]^4+x^25-5*x^24*X[3,1,2,1,3]-10*x^23*X [3,1,2,1,3]^2+16*x^22*X[3,1,2,1,3]^3+x^21*X[3,1,2,1,3]^4-x^20*X[3,1,2,1,3]^5+x^ 24+5*x^23*X[3,1,2,1,3]-14*x^22*X[3,1,2,1,3]^2-4*x^21*X[3,1,2,1,3]^3+5*x^20*X[3, 1,2,1,3]^4-x^23+6*x^22*X[3,1,2,1,3]+6*x^21*X[3,1,2,1,3]^2-10*x^20*X[3,1,2,1,3]^ 3-x^19*X[3,1,2,1,3]^4-x^22-4*x^21*X[3,1,2,1,3]+10*x^20*X[3,1,2,1,3]^2+5*x^19*X[ 3,1,2,1,3]^3-x^18*X[3,1,2,1,3]^4+x^21-5*x^20*X[3,1,2,1,3]-9*x^19*X[3,1,2,1,3]^2 +5*x^18*X[3,1,2,1,3]^3-x^17*X[3,1,2,1,3]^4+x^20+7*x^19*X[3,1,2,1,3]-9*x^18*X[3, 1,2,1,3]^2+3*x^17*X[3,1,2,1,3]^3+2*x^16*X[3,1,2,1,3]^4-2*x^19+7*x^18*X[3,1,2,1, 3]-3*x^17*X[3,1,2,1,3]^2-9*x^16*X[3,1,2,1,3]^3-x^15*X[3,1,2,1,3]^4-2*x^18+x^17* X[3,1,2,1,3]+16*x^16*X[3,1,2,1,3]^2+4*x^15*X[3,1,2,1,3]^3-13*x^16*X[3,1,2,1,3]-\ 6*x^15*X[3,1,2,1,3]^2-x^14*X[3,1,2,1,3]^3+4*x^16+4*x^15*X[3,1,2,1,3]-x^14*X[3,1 ,2,1,3]^2+3*x^13*X[3,1,2,1,3]^3-x^15+5*x^14*X[3,1,2,1,3]-6*x^13*X[3,1,2,1,3]^2-\ 3*x^12*X[3,1,2,1,3]^3-3*x^14+3*x^13*X[3,1,2,1,3]+10*x^12*X[3,1,2,1,3]^2+x^11*X[ 3,1,2,1,3]^3-11*x^12*X[3,1,2,1,3]-3*x^11*X[3,1,2,1,3]^2+4*x^12+2*x^11*X[3,1,2,1 ,3]-2*x^10*X[3,1,2,1,3]^2+6*x^10*X[3,1,2,1,3]+x^9*X[3,1,2,1,3]^2-4*x^10-3*x^9*X [3,1,2,1,3]+2*x^9+3*x^8*X[3,1,2,1,3]-3*x^8-9*x^7*X[3,1,2,1,3]+9*x^7+10*x^6*X[3, 1,2,1,3]-11*x^6-5*x^5*X[3,1,2,1,3]+11*x^5+x^4*X[3,1,2,1,3]-16*x^4+20*x^3-15*x^2 +6*x-1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [3, 1, 2, 1, 3], equals , - 3/16 + ---- 64 17 77 n The variance equals , - -- + ---- 64 4096 7719 867 n The , 3, -th moment about the mean is , - ----- + ----- 16384 32768 222797 17787 2 135871 The , 4, -th moment about the mean is , - ------ + -------- n + ------- n 262144 16777216 8388608 The compositions of, 10, that yield the, 40, -th largest growth, that is, 1.9728837077631717755, are , [2, 1, 2, 2, 3], [2, 1, 2, 3, 2], [2, 1, 3, 2, 2], [2, 2, 2, 1, 3], [2, 2, 3, 1, 2], [2, 3, 2, 1, 2], [3, 1, 2, 2, 2], [3, 2, 2, 1, 2] Theorem Number, 40, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 2, 2, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n ) a(n) x = / ----- n = 0 11 7 6 5 4 3 2 x + 2 x - 3 x + 4 x - 2 x - 3 x + 6 x - 4 x + 1 - ----------------------------------------------------------------------- 12 11 9 8 7 6 5 4 3 2 x - x + x + x - 3 x + 5 x - 6 x + 2 x + 6 x - 9 x + 5 x - 1 and in Maple format -(x^11+2*x^7-3*x^6+4*x^5-2*x^4-3*x^3+6*x^2-4*x+1)/(x^12-x^11+x^9+x^8-3*x^7+5*x^ 6-6*x^5+2*x^4+6*x^3-9*x^2+5*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1017, 2018, 3994, 7890, 15568, 30699, 60523, 119322, 235273, 463970, 915104, 1805103, 3560988, 7025271, 13860194, 27345239, 53950507, 106440784, 209999459, 414310242] The limit of a(n+1)/a(n) as n goes to infinity is 1.97288370776 a(n) is asymptotic to .581145958125*1.97288370776^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [2, 1, 2, 2, 3], denoted by the variable, X[2, 1, 2, 2, 3], is 11 2 11 11 7 - (x X[2, 1, 2, 2, 3] - 2 x X[2, 1, 2, 2, 3] + x - 2 x X[2, 1, 2, 2, 3] 7 6 6 5 5 + 2 x + 3 x X[2, 1, 2, 2, 3] - 3 x - 4 x X[2, 1, 2, 2, 3] + 4 x 4 4 3 3 2 + 3 x X[2, 1, 2, 2, 3] - 2 x - x X[2, 1, 2, 2, 3] - 3 x + 6 x - 4 x / 12 2 12 + 1) / (x X[2, 1, 2, 2, 3] - 2 x X[2, 1, 2, 2, 3] / 11 2 12 11 11 - x X[2, 1, 2, 2, 3] + x + 2 x X[2, 1, 2, 2, 3] - x 9 9 8 8 - x X[2, 1, 2, 2, 3] + x - x X[2, 1, 2, 2, 3] + x 7 7 6 6 + 3 x X[2, 1, 2, 2, 3] - 3 x - 5 x X[2, 1, 2, 2, 3] + 5 x 5 5 4 4 + 6 x X[2, 1, 2, 2, 3] - 6 x - 4 x X[2, 1, 2, 2, 3] + 2 x 3 3 2 + x X[2, 1, 2, 2, 3] + 6 x - 9 x + 5 x - 1) and in Maple format -(x^11*X[2,1,2,2,3]^2-2*x^11*X[2,1,2,2,3]+x^11-2*x^7*X[2,1,2,2,3]+2*x^7+3*x^6*X [2,1,2,2,3]-3*x^6-4*x^5*X[2,1,2,2,3]+4*x^5+3*x^4*X[2,1,2,2,3]-2*x^4-x^3*X[2,1,2 ,2,3]-3*x^3+6*x^2-4*x+1)/(x^12*X[2,1,2,2,3]^2-2*x^12*X[2,1,2,2,3]-x^11*X[2,1,2, 2,3]^2+x^12+2*x^11*X[2,1,2,2,3]-x^11-x^9*X[2,1,2,2,3]+x^9-x^8*X[2,1,2,2,3]+x^8+ 3*x^7*X[2,1,2,2,3]-3*x^7-5*x^6*X[2,1,2,2,3]+5*x^6+6*x^5*X[2,1,2,2,3]-6*x^5-4*x^ 4*X[2,1,2,2,3]+2*x^4+x^3*X[2,1,2,2,3]+6*x^3-9*x^2+5*x-1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [2, 1, 2, 2, 3], equals , - 3/16 + ---- 64 149 85 n The variance equals , - --- + ---- 512 4096 4983 9 n The , 3, -th moment about the mean is , - ---- + --- 8192 256 745331 21675 2 358631 The , 4, -th moment about the mean is , - ------ + -------- n + ------- n 524288 16777216 8388608 The compositions of, 10, that yield the, 41, -th largest growth, that is, 1.9730623001088685615, are , [2, 2, 1, 2, 3], [2, 2, 1, 3, 2], [2, 3, 1, 2, 2], [3, 2, 1, 2, 2] Theorem Number, 41, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 2, 1, 2, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 13 12 11 10 8 7 6 5 ) a(n) x = - (x - 2 x + 3 x - 2 x + x + 2 x - 6 x + 6 x / ----- n = 0 4 3 2 / 14 13 12 11 10 + x - 9 x + 10 x - 5 x + 1) / (x - 2 x + 4 x - 4 x + 2 x / 9 8 7 6 5 4 3 2 + x - x - 4 x + 10 x - 8 x - 4 x + 15 x - 14 x + 6 x - 1) and in Maple format -(x^13-2*x^12+3*x^11-2*x^10+x^8+2*x^7-6*x^6+6*x^5+x^4-9*x^3+10*x^2-5*x+1)/(x^14 -2*x^13+4*x^12-4*x^11+2*x^10+x^9-x^8-4*x^7+10*x^6-8*x^5-4*x^4+15*x^3-14*x^2+6*x -1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1017, 2018, 3994, 7890, 15567, 30692, 60494, 119230, 235026, 463379, 913804, 1802420, 3555718, 7015316, 13841965, 27312680, 53893508, 106342634, 209832870, 414031474] The limit of a(n+1)/a(n) as n goes to infinity is 1.97306230011 a(n) is asymptotic to .579193495541*1.97306230011^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [2, 2, 1, 2, 3], denoted by the variable, X[2, 2, 1, 2, 3], is 13 13 12 13 12 - (x %1 - 2 x X[2, 2, 1, 2, 3] - 2 x %1 + x + 4 x X[2, 2, 1, 2, 3] 11 12 11 10 11 + 3 x %1 - 2 x - 6 x X[2, 2, 1, 2, 3] - 2 x %1 + 3 x 10 9 10 9 + 4 x X[2, 2, 1, 2, 3] + x %1 - 2 x - x X[2, 2, 1, 2, 3] 8 8 7 7 - x X[2, 2, 1, 2, 3] + x - 2 x X[2, 2, 1, 2, 3] + 2 x 6 6 5 5 + 6 x X[2, 2, 1, 2, 3] - 6 x - 7 x X[2, 2, 1, 2, 3] + 6 x 4 4 3 3 2 + 4 x X[2, 2, 1, 2, 3] + x - x X[2, 2, 1, 2, 3] - 9 x + 10 x - 5 x / 14 14 13 14 + 1) / (x %1 - 2 x X[2, 2, 1, 2, 3] - 2 x %1 + x / 13 12 13 12 + 4 x X[2, 2, 1, 2, 3] + 4 x %1 - 2 x - 8 x X[2, 2, 1, 2, 3] 11 12 11 10 11 - 4 x %1 + 4 x + 8 x X[2, 2, 1, 2, 3] + 3 x %1 - 4 x 10 9 10 9 8 8 - 5 x X[2, 2, 1, 2, 3] - x %1 + 2 x + x + x X[2, 2, 1, 2, 3] - x 7 7 6 6 + 4 x X[2, 2, 1, 2, 3] - 4 x - 10 x X[2, 2, 1, 2, 3] + 10 x 5 5 4 4 + 10 x X[2, 2, 1, 2, 3] - 8 x - 5 x X[2, 2, 1, 2, 3] - 4 x 3 3 2 + x X[2, 2, 1, 2, 3] + 15 x - 14 x + 6 x - 1) 2 %1 := X[2, 2, 1, 2, 3] and in Maple format -(x^13*X[2,2,1,2,3]^2-2*x^13*X[2,2,1,2,3]-2*x^12*X[2,2,1,2,3]^2+x^13+4*x^12*X[2 ,2,1,2,3]+3*x^11*X[2,2,1,2,3]^2-2*x^12-6*x^11*X[2,2,1,2,3]-2*x^10*X[2,2,1,2,3]^ 2+3*x^11+4*x^10*X[2,2,1,2,3]+x^9*X[2,2,1,2,3]^2-2*x^10-x^9*X[2,2,1,2,3]-x^8*X[2 ,2,1,2,3]+x^8-2*x^7*X[2,2,1,2,3]+2*x^7+6*x^6*X[2,2,1,2,3]-6*x^6-7*x^5*X[2,2,1,2 ,3]+6*x^5+4*x^4*X[2,2,1,2,3]+x^4-x^3*X[2,2,1,2,3]-9*x^3+10*x^2-5*x+1)/(x^14*X[2 ,2,1,2,3]^2-2*x^14*X[2,2,1,2,3]-2*x^13*X[2,2,1,2,3]^2+x^14+4*x^13*X[2,2,1,2,3]+ 4*x^12*X[2,2,1,2,3]^2-2*x^13-8*x^12*X[2,2,1,2,3]-4*x^11*X[2,2,1,2,3]^2+4*x^12+8 *x^11*X[2,2,1,2,3]+3*x^10*X[2,2,1,2,3]^2-4*x^11-5*x^10*X[2,2,1,2,3]-x^9*X[2,2,1 ,2,3]^2+2*x^10+x^9+x^8*X[2,2,1,2,3]-x^8+4*x^7*X[2,2,1,2,3]-4*x^7-10*x^6*X[2,2,1 ,2,3]+10*x^6+10*x^5*X[2,2,1,2,3]-8*x^5-5*x^4*X[2,2,1,2,3]-4*x^4+x^3*X[2,2,1,2,3 ]+15*x^3-14*x^2+6*x-1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [2, 2, 1, 2, 3], equals , - 3/16 + ---- 64 151 85 n The variance equals , - --- + ---- 512 4096 2505 567 n The , 3, -th moment about the mean is , - ---- + ----- 4096 16384 731215 21675 2 327479 The , 4, -th moment about the mean is , - ------ + -------- n + ------- n 524288 16777216 8388608 The compositions of, 10, that yield the, 42, -th largest growth, that is, 1.9735704833094816886, are , [2, 1, 3, 1, 3], [3, 1, 3, 1, 2] Theorem Number, 42, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 3, 1, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 20 16 13 12 11 10 9 8 ) a(n) x = - (x + x - 2 x + 3 x + 2 x - 6 x + 3 x + x / ----- n = 0 7 6 4 3 2 / - 2 x + 3 x - 11 x + 19 x - 15 x + 6 x - 1) / ( / 7 6 4 3 2 (x - x + 2 x - x - 2 x + 3 x - 1) 14 11 10 9 7 5 4 3 2 (x - x + x + x - 2 x + 2 x - 2 x + 4 x - 6 x + 4 x - 1)) and in Maple format -(x^20+x^16-2*x^13+3*x^12+2*x^11-6*x^10+3*x^9+x^8-2*x^7+3*x^6-11*x^4+19*x^3-15* x^2+6*x-1)/(x^7-x^6+2*x^4-x^3-2*x^2+3*x-1)/(x^14-x^11+x^10+x^9-2*x^7+2*x^5-2*x^ 4+4*x^3-6*x^2+4*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1017, 2018, 3994, 7890, 15568, 30700, 60531, 119360, 235412, 464402, 916306, 1808197, 3568521, 7042891, 13900230, 27434310, 54145622, 106863271, 210906206, 416242784] The limit of a(n+1)/a(n) as n goes to infinity is 1.97357048331 a(n) is asymptotic to .577784303828*1.97357048331^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [2, 1, 3, 1, 3], denoted by the variable, X[2, 1, 3, 1, 3], is 20 20 20 20 16 16 - (x %2 - 3 x %1 + 3 x X[2, 1, 3, 1, 3] - x + x %2 - 3 x %1 15 16 15 14 16 + x %2 + 3 x X[2, 1, 3, 1, 3] - 2 x %1 - 2 x %2 - x 15 14 13 14 + x X[2, 1, 3, 1, 3] + 4 x %1 + x %2 - 2 x X[2, 1, 3, 1, 3] 13 12 13 12 - 3 x X[2, 1, 3, 1, 3] - 3 x %1 + 2 x + 6 x X[2, 1, 3, 1, 3] 11 12 11 10 11 - x %1 - 3 x + 3 x X[2, 1, 3, 1, 3] + 2 x %1 - 2 x 10 9 10 9 8 - 8 x X[2, 1, 3, 1, 3] + 2 x %1 + 6 x + x X[2, 1, 3, 1, 3] - 3 x %1 9 8 7 8 7 7 - 3 x + 4 x X[2, 1, 3, 1, 3] + x %1 - x - 3 x X[2, 1, 3, 1, 3] + 2 x 6 6 5 + 4 x X[2, 1, 3, 1, 3] - 3 x - 6 x X[2, 1, 3, 1, 3] 4 4 3 3 2 + 4 x X[2, 1, 3, 1, 3] + 11 x - x X[2, 1, 3, 1, 3] - 19 x + 15 x / 7 7 6 6 - 6 x + 1) / ((x X[2, 1, 3, 1, 3] - x - x X[2, 1, 3, 1, 3] + x / 4 4 3 3 2 + 2 x X[2, 1, 3, 1, 3] - 2 x - x X[2, 1, 3, 1, 3] + x + 2 x - 3 x + 1 14 14 14 11 11 ) (x %1 - 2 x X[2, 1, 3, 1, 3] + x - x %1 + 2 x X[2, 1, 3, 1, 3] 10 11 10 10 9 9 + x %1 - x - 2 x X[2, 1, 3, 1, 3] + x - x X[2, 1, 3, 1, 3] + x 7 7 5 5 + 2 x X[2, 1, 3, 1, 3] - 2 x - 2 x X[2, 1, 3, 1, 3] + 2 x 4 4 3 2 + x X[2, 1, 3, 1, 3] - 2 x + 4 x - 6 x + 4 x - 1)) 2 %1 := X[2, 1, 3, 1, 3] 3 %2 := X[2, 1, 3, 1, 3] and in Maple format -(x^20*X[2,1,3,1,3]^3-3*x^20*X[2,1,3,1,3]^2+3*x^20*X[2,1,3,1,3]-x^20+x^16*X[2,1 ,3,1,3]^3-3*x^16*X[2,1,3,1,3]^2+x^15*X[2,1,3,1,3]^3+3*x^16*X[2,1,3,1,3]-2*x^15* X[2,1,3,1,3]^2-2*x^14*X[2,1,3,1,3]^3-x^16+x^15*X[2,1,3,1,3]+4*x^14*X[2,1,3,1,3] ^2+x^13*X[2,1,3,1,3]^3-2*x^14*X[2,1,3,1,3]-3*x^13*X[2,1,3,1,3]-3*x^12*X[2,1,3,1 ,3]^2+2*x^13+6*x^12*X[2,1,3,1,3]-x^11*X[2,1,3,1,3]^2-3*x^12+3*x^11*X[2,1,3,1,3] +2*x^10*X[2,1,3,1,3]^2-2*x^11-8*x^10*X[2,1,3,1,3]+2*x^9*X[2,1,3,1,3]^2+6*x^10+x ^9*X[2,1,3,1,3]-3*x^8*X[2,1,3,1,3]^2-3*x^9+4*x^8*X[2,1,3,1,3]+x^7*X[2,1,3,1,3]^ 2-x^8-3*x^7*X[2,1,3,1,3]+2*x^7+4*x^6*X[2,1,3,1,3]-3*x^6-6*x^5*X[2,1,3,1,3]+4*x^ 4*X[2,1,3,1,3]+11*x^4-x^3*X[2,1,3,1,3]-19*x^3+15*x^2-6*x+1)/(x^7*X[2,1,3,1,3]-x ^7-x^6*X[2,1,3,1,3]+x^6+2*x^4*X[2,1,3,1,3]-2*x^4-x^3*X[2,1,3,1,3]+x^3+2*x^2-3*x +1)/(x^14*X[2,1,3,1,3]^2-2*x^14*X[2,1,3,1,3]+x^14-x^11*X[2,1,3,1,3]^2+2*x^11*X[ 2,1,3,1,3]+x^10*X[2,1,3,1,3]^2-x^11-2*x^10*X[2,1,3,1,3]+x^10-x^9*X[2,1,3,1,3]+x ^9+2*x^7*X[2,1,3,1,3]-2*x^7-2*x^5*X[2,1,3,1,3]+2*x^5+x^4*X[2,1,3,1,3]-2*x^4+4*x ^3-6*x^2+4*x-1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [2, 1, 3, 1, 3], equals , - 3/16 + ---- 64 37 85 n The variance equals , - --- + ---- 128 4096 8895 531 n The , 3, -th moment about the mean is , - ----- + ----- 16384 16384 252269 21675 2 197663 The , 4, -th moment about the mean is , - ------ + -------- n + ------- n 262144 16777216 8388608 The compositions of, 10, that yield the, 43, -th largest growth, that is, 1.9756564557792322769, are , [1, 2, 2, 2, 3], [1, 2, 2, 3, 2], [1, 2, 3, 2, 2], [1, 3, 2, 2, 2], [2, 2, 2, 3, 1], [2, 2, 3, 2, 1], [2, 3, 2, 2, 1], [3, 2, 2, 2, 1] Theorem Number, 43, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 2, 2, 2, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n ) a(n) x = / ----- n = 0 9 8 7 6 5 4 3 2 x - x + 2 x - 3 x + 4 x - 2 x - 3 x + 6 x - 4 x + 1 ------------------------------------------------------------- 8 7 6 5 4 3 2 (-1 + x) (x - x + 2 x - 3 x + 3 x + x - 5 x + 4 x - 1) and in Maple format (x^9-x^8+2*x^7-3*x^6+4*x^5-2*x^4-3*x^3+6*x^2-4*x+1)/(-1+x)/(x^8-x^7+2*x^6-3*x^5 +3*x^4+x^3-5*x^2+4*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1017, 2018, 3995, 7897, 15598, 30800, 60818, 120107, 237231, 468631, 925825, 1829143, 3613885, 7140047, 14106662, 27870391, 55062778, 108785398, 214922391, 424611892] The limit of a(n+1)/a(n) as n goes to infinity is 1.97565645578 a(n) is asymptotic to .571014297866*1.97565645578^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [1, 2, 2, 2, 3], denoted by the variable, X[1, 2, 2, 2, 3], is 9 9 8 8 7 (x X[1, 2, 2, 2, 3] - x - x X[1, 2, 2, 2, 3] + x + 2 x X[1, 2, 2, 2, 3] 7 6 6 5 5 - 2 x - 3 x X[1, 2, 2, 2, 3] + 3 x + 4 x X[1, 2, 2, 2, 3] - 4 x 4 4 3 3 2 - 3 x X[1, 2, 2, 2, 3] + 2 x + x X[1, 2, 2, 2, 3] + 3 x - 6 x + 4 x / 8 8 7 7 - 1) / ((-1 + x) (x X[1, 2, 2, 2, 3] - x - x X[1, 2, 2, 2, 3] + x / 6 6 5 5 + 2 x X[1, 2, 2, 2, 3] - 2 x - 3 x X[1, 2, 2, 2, 3] + 3 x 4 4 3 3 2 + 3 x X[1, 2, 2, 2, 3] - 3 x - x X[1, 2, 2, 2, 3] - x + 5 x - 4 x + 1 )) and in Maple format (x^9*X[1,2,2,2,3]-x^9-x^8*X[1,2,2,2,3]+x^8+2*x^7*X[1,2,2,2,3]-2*x^7-3*x^6*X[1,2 ,2,2,3]+3*x^6+4*x^5*X[1,2,2,2,3]-4*x^5-3*x^4*X[1,2,2,2,3]+2*x^4+x^3*X[1,2,2,2,3 ]+3*x^3-6*x^2+4*x-1)/(-1+x)/(x^8*X[1,2,2,2,3]-x^8-x^7*X[1,2,2,2,3]+x^7+2*x^6*X[ 1,2,2,2,3]-2*x^6-3*x^5*X[1,2,2,2,3]+3*x^5+3*x^4*X[1,2,2,2,3]-3*x^4-x^3*X[1,2,2, 2,3]-x^3+5*x^2-4*x+1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [1, 2, 2, 2, 3], equals , - 3/16 + ---- 64 335 97 n The variance equals , - ---- + ---- 1024 4096 5817 45 n The , 3, -th moment about the mean is , - ---- + ---- 8192 1024 1581461 28227 2 426347 The , 4, -th moment about the mean is , - ------- + -------- n + ------- n 1048576 16777216 8388608 The compositions of, 10, that yield the, 44, -th largest growth, that is, 1.9822984210734063491, are , [2, 2, 2, 2, 2] Theorem Number, 44, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 2, 2, 2, 2] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n ) a(n) x = / ----- n = 0 8 7 6 5 4 3 2 x - x + 2 x - 3 x + 5 x - 7 x + 7 x - 4 x + 1 - ------------------------------------------------------------- 9 8 7 6 5 4 3 2 x - x + 2 x - 3 x + 5 x - 8 x + 11 x - 10 x + 5 x - 1 and in Maple format -(x^8-x^7+2*x^6-3*x^5+5*x^4-7*x^3+7*x^2-4*x+1)/(x^9-x^8+2*x^7-3*x^6+5*x^5-8*x^4 +11*x^3-10*x^2+5*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1017, 2019, 4002, 7927, 15699, 31095, 61604, 122075, 241949, 479595, 950727, 1884732, 3736324, 7406869, 14683110, 29106845, 57698887, 114376475, 226727625, 449439958] The limit of a(n+1)/a(n) as n goes to infinity is 1.98229842107 a(n) is asymptotic to .546508435333*1.98229842107^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [2, 2, 2, 2, 2], denoted by the variable, X[2, 2, 2, 2, 2], is 8 8 7 7 6 - (x X[2, 2, 2, 2, 2] - x - x X[2, 2, 2, 2, 2] + x + 2 x X[2, 2, 2, 2, 2] 6 5 5 4 4 - 2 x - 3 x X[2, 2, 2, 2, 2] + 3 x + 4 x X[2, 2, 2, 2, 2] - 5 x 3 3 2 2 - 3 x X[2, 2, 2, 2, 2] + 7 x + x X[2, 2, 2, 2, 2] - 7 x + 4 x - 1) / 9 9 8 8 / (x X[2, 2, 2, 2, 2] - x - x X[2, 2, 2, 2, 2] + x / 7 7 6 6 + 2 x X[2, 2, 2, 2, 2] - 2 x - 3 x X[2, 2, 2, 2, 2] + 3 x 5 5 4 4 + 5 x X[2, 2, 2, 2, 2] - 5 x - 6 x X[2, 2, 2, 2, 2] + 8 x 3 3 2 2 + 4 x X[2, 2, 2, 2, 2] - 11 x - x X[2, 2, 2, 2, 2] + 10 x - 5 x + 1) and in Maple format -(x^8*X[2,2,2,2,2]-x^8-x^7*X[2,2,2,2,2]+x^7+2*x^6*X[2,2,2,2,2]-2*x^6-3*x^5*X[2, 2,2,2,2]+3*x^5+4*x^4*X[2,2,2,2,2]-5*x^4-3*x^3*X[2,2,2,2,2]+7*x^3+x^2*X[2,2,2,2, 2]-7*x^2+4*x-1)/(x^9*X[2,2,2,2,2]-x^9-x^8*X[2,2,2,2,2]+x^8+2*x^7*X[2,2,2,2,2]-2 *x^7-3*x^6*X[2,2,2,2,2]+3*x^6+5*x^5*X[2,2,2,2,2]-5*x^5-6*x^4*X[2,2,2,2,2]+8*x^4 +4*x^3*X[2,2,2,2,2]-11*x^3-x^2*X[2,2,2,2,2]+10*x^2-5*x+1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [2, 2, 2, 2, 2], equals , - 3/16 + ---- 64 289 157 n The variance equals , - --- + ----- 512 4096 9327 4203 n The , 3, -th moment about the mean is , - ---- + ------ 4096 32768 5451115 73947 2 3467975 The , 4, -th moment about the mean is , - ------- + -------- n + ------- n 524288 16777216 8388608 The compositions of, 10, that yield the, 45, -th largest growth, that is, 1.9835828434243263304, are , [1, 1, 1, 7], [1, 1, 7, 1], [1, 7, 1, 1], [7, 1, 1, 1] Theorem Number, 45, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 1, 1, 7] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 9 8 7 3 2 \ n x - x + x - x + 3 x - 3 x + 1 ) a(n) x = ------------------------------------------ / 6 5 4 3 2 3 ----- (x + x + x + x + x + x - 1) (-1 + x) n = 0 and in Maple format (x^9-x^8+x^7-x^3+3*x^2-3*x+1)/(x^6+x^5+x^4+x^3+x^2+x-1)/(-1+x)^3 The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1018, 2025, 4024, 7991, 15862, 31477, 62453, 123899, 245785, 487559, 967141, 1918435, 3805409, 7548382, 14972883, 29700000, 58912461, 116857802, 231797191, 459788996] The limit of a(n+1)/a(n) as n goes to infinity is 1.98358284342 a(n) is asymptotic to .548338004018*1.98358284342^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [1, 1, 1, 7], denoted by the variable, X[1, 1, 1, 7], is 9 9 8 8 7 7 3 (x X[1, 1, 1, 7] - x - x X[1, 1, 1, 7] + x + x X[1, 1, 1, 7] - x + x 2 / 2 7 7 - 3 x + 3 x - 1) / ((-1 + x) (x X[1, 1, 1, 7] - x + 2 x - 1)) / and in Maple format (x^9*X[1,1,1,7]-x^9-x^8*X[1,1,1,7]+x^8+x^7*X[1,1,1,7]-x^7+x^3-3*x^2+3*x-1)/(-1+ x)^2/(x^7*X[1,1,1,7]-x^7+2*x-1) Furthermore, the expectation of the random variable number of occurences (by\ 11 n containment) of , [1, 1, 1, 7], equals , - --- + --- 128 128 1219 115 n The variance equals , - ----- + ----- 16384 16384 28545 2943 n The , 3, -th moment about the mean is , - ------ + ------ 524288 524288 1610839 39675 2 9847 The , 4, -th moment about the mean is , - --------- + --------- n + -------- n 268435456 268435456 67108864 The compositions of, 10, that yield the, 46, -th largest growth, that is, 1.9838613961621262283, are , [2, 1, 1, 6], [6, 1, 1, 2] Theorem Number, 46, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 1, 6] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 27 21 20 19 15 14 13 10 ) a(n) x = - (x + 2 x - x + x + 2 x - 3 x + x - x / ----- n = 0 9 8 7 6 4 3 2 / + 4 x - 5 x + 3 x - x - x + 4 x - 6 x + 4 x - 1) / ( / 21 14 13 9 8 3 2 7 6 (x - x + x + x - x + x - 3 x + 3 x - 1) (x - x + 2 x - 1)) and in Maple format -(x^27+2*x^21-x^20+x^19+2*x^15-3*x^14+x^13-x^10+4*x^9-5*x^8+3*x^7-x^6-x^4+4*x^3 -6*x^2+4*x-1)/(x^21-x^14+x^13+x^9-x^8+x^3-3*x^2+3*x-1)/(x^7-x^6+2*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1018, 2025, 4024, 7991, 15862, 31477, 62453, 123900, 245793, 487597, 967281, 1918878, 3806677, 7551764, 14981449, 29720868, 58961799, 116971749, 232055497, 460365831] The limit of a(n+1)/a(n) as n goes to infinity is 1.98386139616 a(n) is asymptotic to .546718776972*1.98386139616^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [2, 1, 1, 6], denoted by the variable, X[2, 1, 1, 6], is 27 4 27 3 27 2 - (x X[2, 1, 1, 6] - 4 x X[2, 1, 1, 6] + 6 x X[2, 1, 1, 6] 27 27 21 3 21 2 - 4 x X[2, 1, 1, 6] + x - 2 x X[2, 1, 1, 6] + 6 x X[2, 1, 1, 6] 20 3 21 20 2 + x X[2, 1, 1, 6] - 6 x X[2, 1, 1, 6] - 3 x X[2, 1, 1, 6] 19 3 21 20 19 2 - x X[2, 1, 1, 6] + 2 x + 3 x X[2, 1, 1, 6] + 3 x X[2, 1, 1, 6] 20 19 19 15 2 - x - 3 x X[2, 1, 1, 6] + x + 2 x X[2, 1, 1, 6] 15 14 2 15 14 - 4 x X[2, 1, 1, 6] - 3 x X[2, 1, 1, 6] + 2 x + 6 x X[2, 1, 1, 6] 13 2 14 13 13 + x X[2, 1, 1, 6] - 3 x - 2 x X[2, 1, 1, 6] + x 10 10 9 9 8 + x X[2, 1, 1, 6] - x - 4 x X[2, 1, 1, 6] + 4 x + 5 x X[2, 1, 1, 6] 8 7 7 6 6 4 3 - 5 x - 3 x X[2, 1, 1, 6] + 3 x + x X[2, 1, 1, 6] - x - x + 4 x 2 / - 6 x + 4 x - 1) / ( / 7 7 6 6 (x X[2, 1, 1, 6] - x - x X[2, 1, 1, 6] + x - 2 x + 1) ( 21 3 21 2 21 21 x X[2, 1, 1, 6] - 3 x X[2, 1, 1, 6] + 3 x X[2, 1, 1, 6] - x 14 2 14 13 2 14 + x X[2, 1, 1, 6] - 2 x X[2, 1, 1, 6] - x X[2, 1, 1, 6] + x 13 13 9 9 8 + 2 x X[2, 1, 1, 6] - x + x X[2, 1, 1, 6] - x - x X[2, 1, 1, 6] 8 3 2 + x - x + 3 x - 3 x + 1)) and in Maple format -(x^27*X[2,1,1,6]^4-4*x^27*X[2,1,1,6]^3+6*x^27*X[2,1,1,6]^2-4*x^27*X[2,1,1,6]+x ^27-2*x^21*X[2,1,1,6]^3+6*x^21*X[2,1,1,6]^2+x^20*X[2,1,1,6]^3-6*x^21*X[2,1,1,6] -3*x^20*X[2,1,1,6]^2-x^19*X[2,1,1,6]^3+2*x^21+3*x^20*X[2,1,1,6]+3*x^19*X[2,1,1, 6]^2-x^20-3*x^19*X[2,1,1,6]+x^19+2*x^15*X[2,1,1,6]^2-4*x^15*X[2,1,1,6]-3*x^14*X [2,1,1,6]^2+2*x^15+6*x^14*X[2,1,1,6]+x^13*X[2,1,1,6]^2-3*x^14-2*x^13*X[2,1,1,6] +x^13+x^10*X[2,1,1,6]-x^10-4*x^9*X[2,1,1,6]+4*x^9+5*x^8*X[2,1,1,6]-5*x^8-3*x^7* X[2,1,1,6]+3*x^7+x^6*X[2,1,1,6]-x^6-x^4+4*x^3-6*x^2+4*x-1)/(x^7*X[2,1,1,6]-x^7- x^6*X[2,1,1,6]+x^6-2*x+1)/(x^21*X[2,1,1,6]^3-3*x^21*X[2,1,1,6]^2+3*x^21*X[2,1,1 ,6]-x^21+x^14*X[2,1,1,6]^2-2*x^14*X[2,1,1,6]-x^13*X[2,1,1,6]^2+x^14+2*x^13*X[2, 1,1,6]-x^13+x^9*X[2,1,1,6]-x^9-x^8*X[2,1,1,6]+x^8-x^3+3*x^2-3*x+1) Furthermore, the expectation of the random variable number of occurences (by\ 11 n containment) of , [2, 1, 1, 6], equals , - --- + --- 128 128 1303 119 n The variance equals , - ----- + ----- 16384 16384 555 819 n The , 3, -th moment about the mean is , - ---- + ------ 8192 131072 7534111 42483 2 66107 The , 4, -th moment about the mean is , - --------- + --------- n + -------- n 268435456 268435456 67108864 The compositions of, 10, that yield the, 47, -th largest growth, that is, 1.9843693628442243022, are , [2, 1, 2, 5], [2, 1, 5, 2], [2, 2, 1, 5], [2, 5, 1, 2], [5, 1, 2, 2], [5, 2, 1, 2] Theorem Number, 47, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 2, 5] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n ) a(n) x = / ----- n = 0 13 7 6 5 3 2 x + 2 x - 2 x + x - x + 3 x - 3 x + 1 - -------------------------------------------------------------- 14 13 9 8 7 6 5 3 2 x - x + x + x - 3 x + 3 x - x + 2 x - 5 x + 4 x - 1 and in Maple format -(x^13+2*x^7-2*x^6+x^5-x^3+3*x^2-3*x+1)/(x^14-x^13+x^9+x^8-3*x^7+3*x^6-x^5+2*x^ 3-5*x^2+4*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1018, 2025, 4024, 7991, 15862, 31478, 62460, 123930, 245895, 487900, 968105, 1920985, 3811829, 7563949, 15009547, 29784402, 59103259, 117282855, 232732901, 461828768] The limit of a(n+1)/a(n) as n goes to infinity is 1.98436936284 a(n) is asymptotic to .544260748124*1.98436936284^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [2, 1, 2, 5], denoted by the variable, X[2, 1, 2, 5], is 13 2 13 13 7 7 - (x X[2, 1, 2, 5] - 2 x X[2, 1, 2, 5] + x - 2 x X[2, 1, 2, 5] + 2 x 6 6 5 5 3 2 + 2 x X[2, 1, 2, 5] - 2 x - x X[2, 1, 2, 5] + x - x + 3 x - 3 x + 1) / 14 2 14 13 2 14 / (x X[2, 1, 2, 5] - 2 x X[2, 1, 2, 5] - x X[2, 1, 2, 5] + x / 13 13 9 9 8 + 2 x X[2, 1, 2, 5] - x - x X[2, 1, 2, 5] + x - x X[2, 1, 2, 5] 8 7 7 6 6 + x + 3 x X[2, 1, 2, 5] - 3 x - 3 x X[2, 1, 2, 5] + 3 x 5 5 3 2 + x X[2, 1, 2, 5] - x + 2 x - 5 x + 4 x - 1) and in Maple format -(x^13*X[2,1,2,5]^2-2*x^13*X[2,1,2,5]+x^13-2*x^7*X[2,1,2,5]+2*x^7+2*x^6*X[2,1,2 ,5]-2*x^6-x^5*X[2,1,2,5]+x^5-x^3+3*x^2-3*x+1)/(x^14*X[2,1,2,5]^2-2*x^14*X[2,1,2 ,5]-x^13*X[2,1,2,5]^2+x^14+2*x^13*X[2,1,2,5]-x^13-x^9*X[2,1,2,5]+x^9-x^8*X[2,1, 2,5]+x^8+3*x^7*X[2,1,2,5]-3*x^7-3*x^6*X[2,1,2,5]+3*x^6+x^5*X[2,1,2,5]-x^5+2*x^3 -5*x^2+4*x-1) Furthermore, the expectation of the random variable number of occurences (by\ 11 n containment) of , [2, 1, 2, 5], equals , - --- + --- 128 128 1447 127 n The variance equals , - ----- + ----- 16384 16384 12249 501 n The , 3, -th moment about the mean is , - ------ + ----- 131072 65536 21793087 48387 2 225391 The , 4, -th moment about the mean is , - --------- + --------- n + -------- n 268435456 268435456 67108864 The compositions of, 10, that yield the, 48, -th largest growth, that is, 1.9843858253440954550, are , [1, 1, 3, 5], [1, 1, 5, 3], [1, 3, 5, 1], [1, 5, 3, 1], [3, 5, 1, 1], [5, 3, 1, 1] Theorem Number, 48, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 1, 3, 5] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 3 5 4 3 2 \ n (x - x + 1) (x + x - x + x - 2 x + 1) ) a(n) x = - ------------------------------------------ / 7 6 5 2 ----- (x + x - x + 2 x - 1) (-1 + x) n = 0 and in Maple format -(x^3-x+1)*(x^5+x^4-x^3+x^2-2*x+1)/(x^7+x^6-x^5+2*x-1)/(-1+x)^2 The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1018, 2025, 4024, 7991, 15863, 31484, 62482, 123994, 246058, 488281, 968946, 1922772, 3815532, 7571499, 15024787, 29814986, 59164447, 117405101, 232977029, 462316325] The limit of a(n+1)/a(n) as n goes to infinity is 1.98438582534 a(n) is asymptotic to .544697969901*1.98438582534^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [1, 1, 3, 5], denoted by the variable, X[1, 1, 3, 5], is 8 8 7 7 6 6 - (x X[1, 1, 3, 5] - x + x X[1, 1, 3, 5] - x - 2 x X[1, 1, 3, 5] + 2 x 5 5 3 2 / 2 + x X[1, 1, 3, 5] - x + x - 3 x + 3 x - 1) / ((-1 + x) ( / 7 7 6 6 5 5 x X[1, 1, 3, 5] - x + x X[1, 1, 3, 5] - x - x X[1, 1, 3, 5] + x - 2 x + 1)) and in Maple format -(x^8*X[1,1,3,5]-x^8+x^7*X[1,1,3,5]-x^7-2*x^6*X[1,1,3,5]+2*x^6+x^5*X[1,1,3,5]-x ^5+x^3-3*x^2+3*x-1)/(-1+x)^2/(x^7*X[1,1,3,5]-x^7+x^6*X[1,1,3,5]-x^6-x^5*X[1,1,3 ,5]+x^5-2*x+1) Furthermore, the expectation of the random variable number of occurences (by\ 11 n containment) of , [1, 1, 3, 5], equals , - --- + --- 128 128 1415 127 n The variance equals , - ----- + ----- 16384 16384 11355 249 n The , 3, -th moment about the mean is , - ------ + ----- 131072 32768 16936319 48387 2 214111 The , 4, -th moment about the mean is , - --------- + --------- n + -------- n 268435456 268435456 67108864 The compositions of, 10, that yield the, 49, -th largest growth, that is, 1.9846407398915826487, are , [1, 2, 2, 5], [1, 2, 5, 2], [1, 5, 2, 2], [2, 2, 5, 1], [2, 5, 2, 1], [5, 2, 2, 1] Theorem Number, 49, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 2, 2, 5] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 9 8 7 6 5 3 2 \ n x - x + 2 x - 2 x + x - x + 3 x - 3 x + 1 ) a(n) x = ------------------------------------------------ / 8 7 6 5 2 ----- (-1 + x) (x - x + 2 x - x - 2 x + 3 x - 1) n = 0 and in Maple format (x^9-x^8+2*x^7-2*x^6+x^5-x^3+3*x^2-3*x+1)/(-1+x)/(x^8-x^7+2*x^6-x^5-2*x^2+3*x-1 ) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1018, 2025, 4024, 7991, 15863, 31484, 62483, 124001, 246088, 488383, 969249, 1923595, 3817632, 7576622, 15036878, 29842821, 59227316, 117544999, 233284662, 462986318] The limit of a(n+1)/a(n) as n goes to infinity is 1.98464073989 a(n) is asymptotic to .543389350197*1.98464073989^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [1, 2, 2, 5], denoted by the variable, X[1, 2, 2, 5], is 9 9 8 8 7 7 (x X[1, 2, 2, 5] - x - x X[1, 2, 2, 5] + x + 2 x X[1, 2, 2, 5] - 2 x 6 6 5 5 3 2 - 2 x X[1, 2, 2, 5] + 2 x + x X[1, 2, 2, 5] - x + x - 3 x + 3 x - 1) / 8 8 7 7 / ((-1 + x) (x X[1, 2, 2, 5] - x - x X[1, 2, 2, 5] + x / 6 6 5 5 2 + 2 x X[1, 2, 2, 5] - 2 x - x X[1, 2, 2, 5] + x + 2 x - 3 x + 1)) and in Maple format (x^9*X[1,2,2,5]-x^9-x^8*X[1,2,2,5]+x^8+2*x^7*X[1,2,2,5]-2*x^7-2*x^6*X[1,2,2,5]+ 2*x^6+x^5*X[1,2,2,5]-x^5+x^3-3*x^2+3*x-1)/(-1+x)/(x^8*X[1,2,2,5]-x^8-x^7*X[1,2, 2,5]+x^7+2*x^6*X[1,2,2,5]-2*x^6-x^5*X[1,2,2,5]+x^5+2*x^2-3*x+1) Furthermore, the expectation of the random variable number of occurences (by\ 11 n containment) of , [1, 2, 2, 5], equals , - --- + --- 128 128 1491 131 n The variance equals , - ----- + ----- 16384 16384 52413 4347 n The , 3, -th moment about the mean is , - ------ + ------ 524288 524288 23998263 51483 2 288839 The , 4, -th moment about the mean is , - --------- + --------- n + -------- n 268435456 268435456 67108864 The compositions of, 10, that yield the, 50, -th largest growth, that is, 1.9848029699614000332, are , [2, 1, 3, 4], [2, 1, 4, 3], [3, 4, 1, 2], [4, 3, 1, 2] Theorem Number, 50, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 3, 4] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n ) a(n) x = / ----- n = 0 12 7 6 5 4 3 2 x + x + x - 2 x + x - x + 3 x - 3 x + 1 - ------------------------------------------------------------------- 13 12 9 8 7 6 5 4 3 2 x - x + x + x - x - 2 x + 3 x - x + 2 x - 5 x + 4 x - 1 and in Maple format -(x^12+x^7+x^6-2*x^5+x^4-x^3+3*x^2-3*x+1)/(x^13-x^12+x^9+x^8-x^7-2*x^6+3*x^5-x^ 4+2*x^3-5*x^2+4*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1018, 2025, 4024, 7991, 15863, 31484, 62483, 124002, 246095, 488414, 969358, 1923928, 3818558, 7579032, 15042854, 29857113, 59260567, 117620727, 233454270, 463361173] The limit of a(n+1)/a(n) as n goes to infinity is 1.98480296996 a(n) is asymptotic to .542498019435*1.98480296996^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [2, 1, 3, 4], denoted by the variable, X[2, 1, 3, 4], is 12 2 12 12 7 7 - (x X[2, 1, 3, 4] - 2 x X[2, 1, 3, 4] + x - x X[2, 1, 3, 4] + x 6 6 5 5 4 - x X[2, 1, 3, 4] + x + 2 x X[2, 1, 3, 4] - 2 x - x X[2, 1, 3, 4] 4 3 2 / 13 2 13 + x - x + 3 x - 3 x + 1) / (x X[2, 1, 3, 4] - 2 x X[2, 1, 3, 4] / 12 2 13 12 12 9 - x X[2, 1, 3, 4] + x + 2 x X[2, 1, 3, 4] - x - x X[2, 1, 3, 4] 9 8 8 7 7 6 + x - x X[2, 1, 3, 4] + x + x X[2, 1, 3, 4] - x + 2 x X[2, 1, 3, 4] 6 5 5 4 4 3 2 - 2 x - 3 x X[2, 1, 3, 4] + 3 x + x X[2, 1, 3, 4] - x + 2 x - 5 x + 4 x - 1) and in Maple format -(x^12*X[2,1,3,4]^2-2*x^12*X[2,1,3,4]+x^12-x^7*X[2,1,3,4]+x^7-x^6*X[2,1,3,4]+x^ 6+2*x^5*X[2,1,3,4]-2*x^5-x^4*X[2,1,3,4]+x^4-x^3+3*x^2-3*x+1)/(x^13*X[2,1,3,4]^2 -2*x^13*X[2,1,3,4]-x^12*X[2,1,3,4]^2+x^13+2*x^12*X[2,1,3,4]-x^12-x^9*X[2,1,3,4] +x^9-x^8*X[2,1,3,4]+x^8+x^7*X[2,1,3,4]-x^7+2*x^6*X[2,1,3,4]-2*x^6-3*x^5*X[2,1,3 ,4]+3*x^5+x^4*X[2,1,3,4]-x^4+2*x^3-5*x^2+4*x-1) Furthermore, the expectation of the random variable number of occurences (by\ 11 n containment) of , [2, 1, 3, 4], equals , - --- + --- 128 128 1575 135 n The variance equals , - ----- + ----- 16384 16384 7941 1215 n The , 3, -th moment about the mean is , - ----- + ------ 65536 131072 42538239 54675 2 477579 The , 4, -th moment about the mean is , - --------- + --------- n + -------- n 268435456 268435456 67108864 The compositions of, 10, that yield the, 51, -th largest growth, that is, 1.9848266528671481993, are , [2, 3, 1, 4], [2, 4, 1, 3], [3, 1, 4, 2], [4, 1, 3, 2] Theorem Number, 51, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 3, 1, 4] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n ) a(n) x = - ( / ----- n = 0 13 12 11 10 8 7 6 5 4 3 2 x + x - x + x + x - x + 2 x - 2 x + x - x + 3 x - 3 x + 1) / 14 12 11 10 9 8 7 6 5 4 3 / (x - x + 2 x - x + x - x + 2 x - 3 x + 3 x - x + 2 x / 2 - 5 x + 4 x - 1) and in Maple format -(x^13+x^12-x^11+x^10+x^8-x^7+2*x^6-2*x^5+x^4-x^3+3*x^2-3*x+1)/(x^14-x^12+2*x^ 11-x^10+x^9-x^8+2*x^7-3*x^6+3*x^5-x^4+2*x^3-5*x^2+4*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1018, 2025, 4024, 7991, 15862, 31479, 62467, 123960, 245997, 488202, 968922, 1923061, 3816871, 7575792, 15036674, 29845357, 59238208, 117578158, 233373115, 463206289] The limit of a(n+1)/a(n) as n goes to infinity is 1.98482665287 a(n) is asymptotic to .542122764500*1.98482665287^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [2, 3, 1, 4], denoted by the variable, X[2, 3, 1, 4], is 13 2 13 12 2 13 - (x X[2, 3, 1, 4] - 2 x X[2, 3, 1, 4] + x X[2, 3, 1, 4] + x 12 11 2 12 11 - 2 x X[2, 3, 1, 4] - x X[2, 3, 1, 4] + x + 2 x X[2, 3, 1, 4] 10 2 11 10 10 8 + x X[2, 3, 1, 4] - x - 2 x X[2, 3, 1, 4] + x - x X[2, 3, 1, 4] 8 7 7 6 6 + x + x X[2, 3, 1, 4] - x - 2 x X[2, 3, 1, 4] + 2 x 5 5 4 4 3 2 + 2 x X[2, 3, 1, 4] - 2 x - x X[2, 3, 1, 4] + x - x + 3 x - 3 x + 1) / 14 2 14 14 12 2 / (x X[2, 3, 1, 4] - 2 x X[2, 3, 1, 4] + x - x X[2, 3, 1, 4] / 12 11 2 12 11 + 2 x X[2, 3, 1, 4] + 2 x X[2, 3, 1, 4] - x - 4 x X[2, 3, 1, 4] 10 2 11 10 10 - x X[2, 3, 1, 4] + 2 x + 2 x X[2, 3, 1, 4] - x 9 9 8 8 7 - x X[2, 3, 1, 4] + x + x X[2, 3, 1, 4] - x - 2 x X[2, 3, 1, 4] 7 6 6 5 5 + 2 x + 3 x X[2, 3, 1, 4] - 3 x - 3 x X[2, 3, 1, 4] + 3 x 4 4 3 2 + x X[2, 3, 1, 4] - x + 2 x - 5 x + 4 x - 1) and in Maple format -(x^13*X[2,3,1,4]^2-2*x^13*X[2,3,1,4]+x^12*X[2,3,1,4]^2+x^13-2*x^12*X[2,3,1,4]- x^11*X[2,3,1,4]^2+x^12+2*x^11*X[2,3,1,4]+x^10*X[2,3,1,4]^2-x^11-2*x^10*X[2,3,1, 4]+x^10-x^8*X[2,3,1,4]+x^8+x^7*X[2,3,1,4]-x^7-2*x^6*X[2,3,1,4]+2*x^6+2*x^5*X[2, 3,1,4]-2*x^5-x^4*X[2,3,1,4]+x^4-x^3+3*x^2-3*x+1)/(x^14*X[2,3,1,4]^2-2*x^14*X[2, 3,1,4]+x^14-x^12*X[2,3,1,4]^2+2*x^12*X[2,3,1,4]+2*x^11*X[2,3,1,4]^2-x^12-4*x^11 *X[2,3,1,4]-x^10*X[2,3,1,4]^2+2*x^11+2*x^10*X[2,3,1,4]-x^10-x^9*X[2,3,1,4]+x^9+ x^8*X[2,3,1,4]-x^8-2*x^7*X[2,3,1,4]+2*x^7+3*x^6*X[2,3,1,4]-3*x^6-3*x^5*X[2,3,1, 4]+3*x^5+x^4*X[2,3,1,4]-x^4+2*x^3-5*x^2+4*x-1) Furthermore, the expectation of the random variable number of occurences (by\ 11 n containment) of , [2, 3, 1, 4], equals , - --- + --- 128 128 1591 135 n The variance equals , - ----- + ----- 16384 16384 16113 603 n The , 3, -th moment about the mean is , - ------ + ----- 131072 65536 42410719 54675 2 447555 The , 4, -th moment about the mean is , - --------- + --------- n + -------- n 268435456 268435456 67108864 The compositions of, 10, that yield the, 52, -th largest growth, that is, 1.9848333098768844858, are , [3, 1, 2, 4], [3, 2, 1, 4], [4, 1, 2, 3], [4, 2, 1, 3] Theorem Number, 52, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [3, 1, 2, 4] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 22 21 18 17 15 14 13 12 11 ) a(n) x = - (x + x + x + 2 x - x + x + x + x - 3 x / ----- n = 0 10 9 8 7 6 5 4 3 2 / + x - x + 3 x - x - 3 x + 3 x - 2 x + 4 x - 6 x + 4 x - 1) / ( / 24 23 21 20 19 18 17 15 13 12 x + 2 x - x + x + 3 x + x - 2 x + 3 x - 2 x - 3 x 11 9 8 6 5 4 3 2 + 4 x + 2 x - 5 x + 5 x - 4 x + 3 x - 7 x + 9 x - 5 x + 1) and in Maple format -(x^22+x^21+x^18+2*x^17-x^15+x^14+x^13+x^12-3*x^11+x^10-x^9+3*x^8-x^7-3*x^6+3*x ^5-2*x^4+4*x^3-6*x^2+4*x-1)/(x^24+2*x^23-x^21+x^20+3*x^19+x^18-2*x^17+3*x^15-2* x^13-3*x^12+4*x^11+2*x^9-5*x^8+5*x^6-4*x^5+3*x^4-7*x^3+9*x^2-5*x+1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1018, 2025, 4024, 7991, 15862, 31478, 62461, 123938, 245933, 488040, 968547, 1922244, 3815165, 7572334, 15029804, 29831878, 59211938, 117527083, 233273778, 463012719] The limit of a(n+1)/a(n) as n goes to infinity is 1.98483330988 a(n) is asymptotic to .541844937295*1.98483330988^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [3, 1, 2, 4], denoted by the variable, X[3, 1, 2, 4], is 22 4 22 3 21 4 - (x X[3, 1, 2, 4] - 4 x X[3, 1, 2, 4] + x X[3, 1, 2, 4] 22 2 21 3 22 + 6 x X[3, 1, 2, 4] - 4 x X[3, 1, 2, 4] - 4 x X[3, 1, 2, 4] 21 2 22 21 21 + 6 x X[3, 1, 2, 4] + x - 4 x X[3, 1, 2, 4] + x 18 3 18 2 17 3 - x X[3, 1, 2, 4] + 3 x X[3, 1, 2, 4] - 2 x X[3, 1, 2, 4] 18 17 2 18 17 - 3 x X[3, 1, 2, 4] + 6 x X[3, 1, 2, 4] + x - 6 x X[3, 1, 2, 4] 15 3 17 15 2 14 3 + x X[3, 1, 2, 4] + 2 x - 3 x X[3, 1, 2, 4] - x X[3, 1, 2, 4] 15 14 2 15 14 + 3 x X[3, 1, 2, 4] + 3 x X[3, 1, 2, 4] - x - 3 x X[3, 1, 2, 4] 13 2 14 13 12 2 + x X[3, 1, 2, 4] + x - 2 x X[3, 1, 2, 4] + x X[3, 1, 2, 4] 13 12 11 2 12 + x - 2 x X[3, 1, 2, 4] - 3 x X[3, 1, 2, 4] + x 11 10 2 11 10 + 6 x X[3, 1, 2, 4] + x X[3, 1, 2, 4] - 3 x - 2 x X[3, 1, 2, 4] 10 9 9 8 8 + x + x X[3, 1, 2, 4] - x - 3 x X[3, 1, 2, 4] + 3 x 7 7 6 6 5 + x X[3, 1, 2, 4] - x + 3 x X[3, 1, 2, 4] - 3 x - 3 x X[3, 1, 2, 4] 5 4 4 3 2 / + 3 x + x X[3, 1, 2, 4] - 2 x + 4 x - 6 x + 4 x - 1) / ( / 24 4 24 3 23 4 x X[3, 1, 2, 4] - 4 x X[3, 1, 2, 4] + 2 x X[3, 1, 2, 4] 24 2 23 3 24 + 6 x X[3, 1, 2, 4] - 8 x X[3, 1, 2, 4] - 4 x X[3, 1, 2, 4] 23 2 21 4 24 23 + 12 x X[3, 1, 2, 4] - x X[3, 1, 2, 4] + x - 8 x X[3, 1, 2, 4] 21 3 23 21 2 20 3 + 4 x X[3, 1, 2, 4] + 2 x - 6 x X[3, 1, 2, 4] - x X[3, 1, 2, 4] 21 20 2 19 3 21 + 4 x X[3, 1, 2, 4] + 3 x X[3, 1, 2, 4] - 3 x X[3, 1, 2, 4] - x 20 19 2 18 3 20 - 3 x X[3, 1, 2, 4] + 9 x X[3, 1, 2, 4] - x X[3, 1, 2, 4] + x 19 18 2 17 3 - 9 x X[3, 1, 2, 4] + 3 x X[3, 1, 2, 4] + 2 x X[3, 1, 2, 4] 19 18 17 2 18 + 3 x - 3 x X[3, 1, 2, 4] - 6 x X[3, 1, 2, 4] + x 17 15 3 17 + 6 x X[3, 1, 2, 4] - 2 x X[3, 1, 2, 4] - 2 x 15 2 14 3 15 + 7 x X[3, 1, 2, 4] + x X[3, 1, 2, 4] - 8 x X[3, 1, 2, 4] 14 2 15 14 13 2 - 2 x X[3, 1, 2, 4] + 3 x + x X[3, 1, 2, 4] - 2 x X[3, 1, 2, 4] 13 12 2 13 12 + 4 x X[3, 1, 2, 4] - 3 x X[3, 1, 2, 4] - 2 x + 6 x X[3, 1, 2, 4] 11 2 12 11 10 2 + 4 x X[3, 1, 2, 4] - 3 x - 8 x X[3, 1, 2, 4] - x X[3, 1, 2, 4] 11 10 9 9 + 4 x + x X[3, 1, 2, 4] - 2 x X[3, 1, 2, 4] + 2 x 8 8 6 6 + 5 x X[3, 1, 2, 4] - 5 x - 5 x X[3, 1, 2, 4] + 5 x 5 5 4 4 3 2 + 4 x X[3, 1, 2, 4] - 4 x - x X[3, 1, 2, 4] + 3 x - 7 x + 9 x - 5 x + 1) and in Maple format -(x^22*X[3,1,2,4]^4-4*x^22*X[3,1,2,4]^3+x^21*X[3,1,2,4]^4+6*x^22*X[3,1,2,4]^2-4 *x^21*X[3,1,2,4]^3-4*x^22*X[3,1,2,4]+6*x^21*X[3,1,2,4]^2+x^22-4*x^21*X[3,1,2,4] +x^21-x^18*X[3,1,2,4]^3+3*x^18*X[3,1,2,4]^2-2*x^17*X[3,1,2,4]^3-3*x^18*X[3,1,2, 4]+6*x^17*X[3,1,2,4]^2+x^18-6*x^17*X[3,1,2,4]+x^15*X[3,1,2,4]^3+2*x^17-3*x^15*X [3,1,2,4]^2-x^14*X[3,1,2,4]^3+3*x^15*X[3,1,2,4]+3*x^14*X[3,1,2,4]^2-x^15-3*x^14 *X[3,1,2,4]+x^13*X[3,1,2,4]^2+x^14-2*x^13*X[3,1,2,4]+x^12*X[3,1,2,4]^2+x^13-2*x ^12*X[3,1,2,4]-3*x^11*X[3,1,2,4]^2+x^12+6*x^11*X[3,1,2,4]+x^10*X[3,1,2,4]^2-3*x ^11-2*x^10*X[3,1,2,4]+x^10+x^9*X[3,1,2,4]-x^9-3*x^8*X[3,1,2,4]+3*x^8+x^7*X[3,1, 2,4]-x^7+3*x^6*X[3,1,2,4]-3*x^6-3*x^5*X[3,1,2,4]+3*x^5+x^4*X[3,1,2,4]-2*x^4+4*x ^3-6*x^2+4*x-1)/(x^24*X[3,1,2,4]^4-4*x^24*X[3,1,2,4]^3+2*x^23*X[3,1,2,4]^4+6*x^ 24*X[3,1,2,4]^2-8*x^23*X[3,1,2,4]^3-4*x^24*X[3,1,2,4]+12*x^23*X[3,1,2,4]^2-x^21 *X[3,1,2,4]^4+x^24-8*x^23*X[3,1,2,4]+4*x^21*X[3,1,2,4]^3+2*x^23-6*x^21*X[3,1,2, 4]^2-x^20*X[3,1,2,4]^3+4*x^21*X[3,1,2,4]+3*x^20*X[3,1,2,4]^2-3*x^19*X[3,1,2,4]^ 3-x^21-3*x^20*X[3,1,2,4]+9*x^19*X[3,1,2,4]^2-x^18*X[3,1,2,4]^3+x^20-9*x^19*X[3, 1,2,4]+3*x^18*X[3,1,2,4]^2+2*x^17*X[3,1,2,4]^3+3*x^19-3*x^18*X[3,1,2,4]-6*x^17* X[3,1,2,4]^2+x^18+6*x^17*X[3,1,2,4]-2*x^15*X[3,1,2,4]^3-2*x^17+7*x^15*X[3,1,2,4 ]^2+x^14*X[3,1,2,4]^3-8*x^15*X[3,1,2,4]-2*x^14*X[3,1,2,4]^2+3*x^15+x^14*X[3,1,2 ,4]-2*x^13*X[3,1,2,4]^2+4*x^13*X[3,1,2,4]-3*x^12*X[3,1,2,4]^2-2*x^13+6*x^12*X[3 ,1,2,4]+4*x^11*X[3,1,2,4]^2-3*x^12-8*x^11*X[3,1,2,4]-x^10*X[3,1,2,4]^2+4*x^11+x ^10*X[3,1,2,4]-2*x^9*X[3,1,2,4]+2*x^9+5*x^8*X[3,1,2,4]-5*x^8-5*x^6*X[3,1,2,4]+5 *x^6+4*x^5*X[3,1,2,4]-4*x^5-x^4*X[3,1,2,4]+3*x^4-7*x^3+9*x^2-5*x+1) Furthermore, the expectation of the random variable number of occurences (by\ 11 n containment) of , [3, 1, 2, 4], equals , - --- + --- 128 128 1607 135 n The variance equals , - ----- + ----- 16384 16384 8235 1203 n The , 3, -th moment about the mean is , - ----- + ------ 65536 131072 43703615 54675 2 434427 The , 4, -th moment about the mean is , - --------- + --------- n + -------- n 268435456 268435456 67108864 The compositions of, 10, that yield the, 53, -th largest growth, that is, 1.9850654703526320630, are , [1, 3, 2, 4], [1, 4, 2, 3], [3, 2, 4, 1], [4, 2, 3, 1] Theorem Number, 53, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 3, 2, 4] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n ) a(n) x = / ----- n = 0 13 11 10 9 7 6 5 4 3 2 x - x + x + x - x + 2 x - 2 x + x - x + 3 x - 3 x + 1 ---------------------------------------------------------------------- 12 11 10 8 7 6 5 4 2 (-1 + x) (x + x - x + x + x - x + 2 x - x - 2 x + 3 x - 1) and in Maple format (x^13-x^11+x^10+x^9-x^7+2*x^6-2*x^5+x^4-x^3+3*x^2-3*x+1)/(-1+x)/(x^12+x^11-x^10 +x^8+x^7-x^6+2*x^5-x^4-2*x^2+3*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1018, 2025, 4024, 7991, 15863, 31485, 62490, 124031, 246189, 488679, 970042, 1925592, 3822444, 7587851, 15062467, 29900103, 59353801, 117821309, 233883084, 464273193] The limit of a(n+1)/a(n) as n goes to infinity is 1.98506547035 a(n) is asymptotic to .541412415763*1.98506547035^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [1, 3, 2, 4], denoted by the variable, X[1, 3, 2, 4], is 13 2 13 13 11 2 (x X[1, 3, 2, 4] - 2 x X[1, 3, 2, 4] + x - x X[1, 3, 2, 4] 11 10 2 11 10 + 2 x X[1, 3, 2, 4] + x X[1, 3, 2, 4] - x - 2 x X[1, 3, 2, 4] 10 9 9 7 7 6 + x - x X[1, 3, 2, 4] + x + x X[1, 3, 2, 4] - x - 2 x X[1, 3, 2, 4] 6 5 5 4 4 3 2 + 2 x + 2 x X[1, 3, 2, 4] - 2 x - x X[1, 3, 2, 4] + x - x + 3 x / 12 2 12 - 3 x + 1) / ((-1 + x) (x X[1, 3, 2, 4] - 2 x X[1, 3, 2, 4] / 11 2 12 11 10 2 + x X[1, 3, 2, 4] + x - 2 x X[1, 3, 2, 4] - x X[1, 3, 2, 4] 11 10 10 8 8 + x + 2 x X[1, 3, 2, 4] - x - x X[1, 3, 2, 4] + x 7 7 6 6 5 - x X[1, 3, 2, 4] + x + x X[1, 3, 2, 4] - x - 2 x X[1, 3, 2, 4] 5 4 4 2 + 2 x + x X[1, 3, 2, 4] - x - 2 x + 3 x - 1)) and in Maple format (x^13*X[1,3,2,4]^2-2*x^13*X[1,3,2,4]+x^13-x^11*X[1,3,2,4]^2+2*x^11*X[1,3,2,4]+x ^10*X[1,3,2,4]^2-x^11-2*x^10*X[1,3,2,4]+x^10-x^9*X[1,3,2,4]+x^9+x^7*X[1,3,2,4]- x^7-2*x^6*X[1,3,2,4]+2*x^6+2*x^5*X[1,3,2,4]-2*x^5-x^4*X[1,3,2,4]+x^4-x^3+3*x^2-\ 3*x+1)/(-1+x)/(x^12*X[1,3,2,4]^2-2*x^12*X[1,3,2,4]+x^11*X[1,3,2,4]^2+x^12-2*x^ 11*X[1,3,2,4]-x^10*X[1,3,2,4]^2+x^11+2*x^10*X[1,3,2,4]-x^10-x^8*X[1,3,2,4]+x^8- x^7*X[1,3,2,4]+x^7+x^6*X[1,3,2,4]-x^6-2*x^5*X[1,3,2,4]+2*x^5+x^4*X[1,3,2,4]-x^4 -2*x^2+3*x-1) Furthermore, the expectation of the random variable number of occurences (by\ 11 n containment) of , [1, 3, 2, 4], equals , - --- + --- 128 128 1635 139 n The variance equals , - ----- + ----- 16384 16384 68709 5211 n The , 3, -th moment about the mean is , - ------ + ------ 524288 524288 46972503 57963 2 544363 The , 4, -th moment about the mean is , - --------- + --------- n + -------- n 268435456 268435456 67108864 The compositions of, 10, that yield the, 54, -th largest growth, that is, 1.9853288885629234253, are , [4, 1, 1, 4] Theorem Number, 54, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [4, 1, 1, 4] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 25 24 23 22 21 20 19 ) a(n) x = - (x + 3 x + 6 x + 7 x + 6 x + 3 x + 3 x / ----- n = 0 18 17 16 15 13 12 10 9 8 7 + 3 x + 4 x + x + x + 3 x - x - 3 x + 3 x - x + 2 x 6 5 4 3 2 / - 4 x + 3 x - 2 x + 4 x - 6 x + 4 x - 1) / ( / 7 6 5 4 21 20 19 18 17 16 (x + x + x - x + 2 x - 1) (x + 3 x + 6 x + 7 x + 6 x + 3 x 15 12 9 6 3 2 + x - x + 2 x - x + x - 3 x + 3 x - 1)) and in Maple format -(x^25+3*x^24+6*x^23+7*x^22+6*x^21+3*x^20+3*x^19+3*x^18+4*x^17+x^16+x^15+3*x^13 -x^12-3*x^10+3*x^9-x^8+2*x^7-4*x^6+3*x^5-2*x^4+4*x^3-6*x^2+4*x-1)/(x^7+x^6+x^5- x^4+2*x-1)/(x^21+3*x^20+6*x^19+7*x^18+6*x^17+3*x^16+x^15-x^12+2*x^9-x^6+x^3-3*x ^2+3*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1018, 2025, 4024, 7991, 15862, 31478, 62462, 123946, 245971, 488180, 968992, 1923527, 3818613, 7581119, 15051276, 29882668, 59328998, 117791372, 233860753, 464299424] The limit of a(n+1)/a(n) as n goes to infinity is 1.98532888856 a(n) is asymptotic to .539298326604*1.98532888856^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [4, 1, 1, 4], denoted by the variable, X[4, 1, 1, 4], is 25 4 25 3 24 4 - (x X[4, 1, 1, 4] - 4 x X[4, 1, 1, 4] + 3 x X[4, 1, 1, 4] 25 2 24 3 23 4 + 6 x X[4, 1, 1, 4] - 12 x X[4, 1, 1, 4] + 6 x X[4, 1, 1, 4] 25 24 2 23 3 - 4 x X[4, 1, 1, 4] + 18 x X[4, 1, 1, 4] - 24 x X[4, 1, 1, 4] 22 4 25 24 + 7 x X[4, 1, 1, 4] + x - 12 x X[4, 1, 1, 4] 23 2 22 3 21 4 + 36 x X[4, 1, 1, 4] - 28 x X[4, 1, 1, 4] + 6 x X[4, 1, 1, 4] 24 23 22 2 + 3 x - 24 x X[4, 1, 1, 4] + 42 x X[4, 1, 1, 4] 21 3 20 4 23 - 24 x X[4, 1, 1, 4] + 3 x X[4, 1, 1, 4] + 6 x 22 21 2 20 3 - 28 x X[4, 1, 1, 4] + 36 x X[4, 1, 1, 4] - 12 x X[4, 1, 1, 4] 19 4 22 21 + x X[4, 1, 1, 4] + 7 x - 24 x X[4, 1, 1, 4] 20 2 19 3 21 + 18 x X[4, 1, 1, 4] - 6 x X[4, 1, 1, 4] + 6 x 20 19 2 18 3 - 12 x X[4, 1, 1, 4] + 12 x X[4, 1, 1, 4] - 3 x X[4, 1, 1, 4] 20 19 18 2 + 3 x - 10 x X[4, 1, 1, 4] + 9 x X[4, 1, 1, 4] 17 3 19 18 - 4 x X[4, 1, 1, 4] + 3 x - 9 x X[4, 1, 1, 4] 17 2 16 3 18 + 12 x X[4, 1, 1, 4] - x X[4, 1, 1, 4] + 3 x 17 16 2 15 3 17 - 12 x X[4, 1, 1, 4] + 3 x X[4, 1, 1, 4] - x X[4, 1, 1, 4] + 4 x 16 15 2 16 15 - 3 x X[4, 1, 1, 4] + 3 x X[4, 1, 1, 4] + x - 3 x X[4, 1, 1, 4] 13 3 15 13 2 13 - x X[4, 1, 1, 4] + x + 5 x X[4, 1, 1, 4] - 7 x X[4, 1, 1, 4] 12 2 13 12 12 - x X[4, 1, 1, 4] + 3 x + 2 x X[4, 1, 1, 4] - x 10 2 10 9 2 10 - 2 x X[4, 1, 1, 4] + 5 x X[4, 1, 1, 4] + x X[4, 1, 1, 4] - 3 x 9 9 8 8 7 - 4 x X[4, 1, 1, 4] + 3 x + x X[4, 1, 1, 4] - x - 2 x X[4, 1, 1, 4] 7 6 6 5 5 + 2 x + 4 x X[4, 1, 1, 4] - 4 x - 3 x X[4, 1, 1, 4] + 3 x 4 4 3 2 / 7 + x X[4, 1, 1, 4] - 2 x + 4 x - 6 x + 4 x - 1) / ((x X[4, 1, 1, 4] / 7 6 6 5 5 4 - x + x X[4, 1, 1, 4] - x + x X[4, 1, 1, 4] - x - x X[4, 1, 1, 4] 4 21 3 21 2 + x - 2 x + 1) (x X[4, 1, 1, 4] - 3 x X[4, 1, 1, 4] 20 3 21 20 2 + 3 x X[4, 1, 1, 4] + 3 x X[4, 1, 1, 4] - 9 x X[4, 1, 1, 4] 19 3 21 20 19 2 + 6 x X[4, 1, 1, 4] - x + 9 x X[4, 1, 1, 4] - 18 x X[4, 1, 1, 4] 18 3 20 19 + 7 x X[4, 1, 1, 4] - 3 x + 18 x X[4, 1, 1, 4] 18 2 17 3 19 - 21 x X[4, 1, 1, 4] + 6 x X[4, 1, 1, 4] - 6 x 18 17 2 16 3 + 21 x X[4, 1, 1, 4] - 18 x X[4, 1, 1, 4] + 3 x X[4, 1, 1, 4] 18 17 16 2 15 3 - 7 x + 18 x X[4, 1, 1, 4] - 9 x X[4, 1, 1, 4] + x X[4, 1, 1, 4] 17 16 15 2 16 - 6 x + 9 x X[4, 1, 1, 4] - 3 x X[4, 1, 1, 4] - 3 x 15 15 12 2 12 + 3 x X[4, 1, 1, 4] - x + x X[4, 1, 1, 4] - 2 x X[4, 1, 1, 4] 12 9 2 9 9 6 + x - x X[4, 1, 1, 4] + 3 x X[4, 1, 1, 4] - 2 x - x X[4, 1, 1, 4] 6 3 2 + x - x + 3 x - 3 x + 1)) and in Maple format -(x^25*X[4,1,1,4]^4-4*x^25*X[4,1,1,4]^3+3*x^24*X[4,1,1,4]^4+6*x^25*X[4,1,1,4]^2 -12*x^24*X[4,1,1,4]^3+6*x^23*X[4,1,1,4]^4-4*x^25*X[4,1,1,4]+18*x^24*X[4,1,1,4]^ 2-24*x^23*X[4,1,1,4]^3+7*x^22*X[4,1,1,4]^4+x^25-12*x^24*X[4,1,1,4]+36*x^23*X[4, 1,1,4]^2-28*x^22*X[4,1,1,4]^3+6*x^21*X[4,1,1,4]^4+3*x^24-24*x^23*X[4,1,1,4]+42* x^22*X[4,1,1,4]^2-24*x^21*X[4,1,1,4]^3+3*x^20*X[4,1,1,4]^4+6*x^23-28*x^22*X[4,1 ,1,4]+36*x^21*X[4,1,1,4]^2-12*x^20*X[4,1,1,4]^3+x^19*X[4,1,1,4]^4+7*x^22-24*x^ 21*X[4,1,1,4]+18*x^20*X[4,1,1,4]^2-6*x^19*X[4,1,1,4]^3+6*x^21-12*x^20*X[4,1,1,4 ]+12*x^19*X[4,1,1,4]^2-3*x^18*X[4,1,1,4]^3+3*x^20-10*x^19*X[4,1,1,4]+9*x^18*X[4 ,1,1,4]^2-4*x^17*X[4,1,1,4]^3+3*x^19-9*x^18*X[4,1,1,4]+12*x^17*X[4,1,1,4]^2-x^ 16*X[4,1,1,4]^3+3*x^18-12*x^17*X[4,1,1,4]+3*x^16*X[4,1,1,4]^2-x^15*X[4,1,1,4]^3 +4*x^17-3*x^16*X[4,1,1,4]+3*x^15*X[4,1,1,4]^2+x^16-3*x^15*X[4,1,1,4]-x^13*X[4,1 ,1,4]^3+x^15+5*x^13*X[4,1,1,4]^2-7*x^13*X[4,1,1,4]-x^12*X[4,1,1,4]^2+3*x^13+2*x ^12*X[4,1,1,4]-x^12-2*x^10*X[4,1,1,4]^2+5*x^10*X[4,1,1,4]+x^9*X[4,1,1,4]^2-3*x^ 10-4*x^9*X[4,1,1,4]+3*x^9+x^8*X[4,1,1,4]-x^8-2*x^7*X[4,1,1,4]+2*x^7+4*x^6*X[4,1 ,1,4]-4*x^6-3*x^5*X[4,1,1,4]+3*x^5+x^4*X[4,1,1,4]-2*x^4+4*x^3-6*x^2+4*x-1)/(x^7 *X[4,1,1,4]-x^7+x^6*X[4,1,1,4]-x^6+x^5*X[4,1,1,4]-x^5-x^4*X[4,1,1,4]+x^4-2*x+1) /(x^21*X[4,1,1,4]^3-3*x^21*X[4,1,1,4]^2+3*x^20*X[4,1,1,4]^3+3*x^21*X[4,1,1,4]-9 *x^20*X[4,1,1,4]^2+6*x^19*X[4,1,1,4]^3-x^21+9*x^20*X[4,1,1,4]-18*x^19*X[4,1,1,4 ]^2+7*x^18*X[4,1,1,4]^3-3*x^20+18*x^19*X[4,1,1,4]-21*x^18*X[4,1,1,4]^2+6*x^17*X [4,1,1,4]^3-6*x^19+21*x^18*X[4,1,1,4]-18*x^17*X[4,1,1,4]^2+3*x^16*X[4,1,1,4]^3-\ 7*x^18+18*x^17*X[4,1,1,4]-9*x^16*X[4,1,1,4]^2+x^15*X[4,1,1,4]^3-6*x^17+9*x^16*X [4,1,1,4]-3*x^15*X[4,1,1,4]^2-3*x^16+3*x^15*X[4,1,1,4]-x^15+x^12*X[4,1,1,4]^2-2 *x^12*X[4,1,1,4]+x^12-x^9*X[4,1,1,4]^2+3*x^9*X[4,1,1,4]-2*x^9-x^6*X[4,1,1,4]+x^ 6-x^3+3*x^2-3*x+1) Furthermore, the expectation of the random variable number of occurences (by\ 11 n containment) of , [4, 1, 1, 4], equals , - --- + --- 128 128 1767 143 n The variance equals , - ----- + ----- 16384 16384 5103 1389 n The , 3, -th moment about the mean is , - ----- + ------ 32768 131072 61363263 61347 2 588071 The , 4, -th moment about the mean is , - --------- + --------- n + -------- n 268435456 268435456 67108864 The compositions of, 10, that yield the, 55, -th largest growth, that is, 1.9855197870868743418, are , [3, 1, 3, 3], [3, 3, 1, 3] Theorem Number, 55, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [3, 1, 3, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n ) a(n) x = / ----- n = 0 3 8 7 6 2 (x - x + 1) (x + x + x + x - 2 x + 1) - ------------------------------------------------------------------------- 13 12 10 9 8 7 6 5 4 3 2 x + 2 x - x + x + x + x - x - 2 x + 3 x + x - 5 x + 4 x - 1 and in Maple format -(x^3-x+1)*(x^8+x^7+x^6+x^2-2*x+1)/(x^13+2*x^12-x^10+x^9+x^8+x^7-x^6-2*x^5+3*x^ 4+x^3-5*x^2+4*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1018, 2025, 4024, 7991, 15863, 31485, 62491, 124039, 246227, 488818, 970478, 1926830, 3825722, 7596093, 15082388, 29946803, 59460703, 118061507, 234414938, 465437241] The limit of a(n+1)/a(n) as n goes to infinity is 1.98551978709 a(n) is asymptotic to .539056792066*1.98551978709^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [3, 1, 3, 3], denoted by the variable, X[3, 1, 3, 3], is 11 2 11 10 2 11 - (x X[3, 1, 3, 3] - 2 x X[3, 1, 3, 3] + x X[3, 1, 3, 3] + x 10 10 6 6 5 - 2 x X[3, 1, 3, 3] + x - x X[3, 1, 3, 3] + x - x X[3, 1, 3, 3] 5 4 4 3 2 / + x + 2 x X[3, 1, 3, 3] - 2 x - x X[3, 1, 3, 3] + 3 x - 3 x + 1) / / 13 2 13 12 2 13 (x X[3, 1, 3, 3] - 2 x X[3, 1, 3, 3] + 2 x X[3, 1, 3, 3] + x 12 12 10 2 10 - 4 x X[3, 1, 3, 3] + 2 x - x X[3, 1, 3, 3] + 2 x X[3, 1, 3, 3] 10 9 9 8 8 7 - x - x X[3, 1, 3, 3] + x - x X[3, 1, 3, 3] + x - x X[3, 1, 3, 3] 7 6 6 5 5 + x + x X[3, 1, 3, 3] - x + 2 x X[3, 1, 3, 3] - 2 x 4 4 3 3 2 - 3 x X[3, 1, 3, 3] + 3 x + x X[3, 1, 3, 3] + x - 5 x + 4 x - 1) and in Maple format -(x^11*X[3,1,3,3]^2-2*x^11*X[3,1,3,3]+x^10*X[3,1,3,3]^2+x^11-2*x^10*X[3,1,3,3]+ x^10-x^6*X[3,1,3,3]+x^6-x^5*X[3,1,3,3]+x^5+2*x^4*X[3,1,3,3]-2*x^4-x^3*X[3,1,3,3 ]+3*x^2-3*x+1)/(x^13*X[3,1,3,3]^2-2*x^13*X[3,1,3,3]+2*x^12*X[3,1,3,3]^2+x^13-4* x^12*X[3,1,3,3]+2*x^12-x^10*X[3,1,3,3]^2+2*x^10*X[3,1,3,3]-x^10-x^9*X[3,1,3,3]+ x^9-x^8*X[3,1,3,3]+x^8-x^7*X[3,1,3,3]+x^7+x^6*X[3,1,3,3]-x^6+2*x^5*X[3,1,3,3]-2 *x^5-3*x^4*X[3,1,3,3]+3*x^4+x^3*X[3,1,3,3]+x^3-5*x^2+4*x-1) Furthermore, the expectation of the random variable number of occurences (by\ 11 n containment) of , [3, 1, 3, 3], equals , - --- + --- 128 128 1879 151 n The variance equals , - ----- + ----- 16384 16384 26307 27 n The , 3, -th moment about the mean is , - ------ + ---- 131072 2048 121493407 68403 2 1298635 The , 4, -th moment about the mean is , - --------- + --------- n + -------- n 268435456 268435456 67108864 The compositions of, 10, that yield the, 56, -th largest growth, that is, 1.9855529777414181545, are , [1, 2, 3, 4], [1, 2, 4, 3], [1, 3, 4, 2], [1, 4, 3, 2], [2, 3, 4, 1], [2, 4, 3, 1], [3, 4, 2, 1], [4, 3, 2, 1] Theorem Number, 56, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 2, 3, 4] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 3 6 5 4 3 2 \ n (x - x + 1) (x - x + 2 x - x + x - 2 x + 1) ) a(n) x = ------------------------------------------------- / 8 7 5 4 2 ----- (-1 + x) (x - x + 2 x - x - 2 x + 3 x - 1) n = 0 and in Maple format (x^3-x+1)*(x^6-x^5+2*x^4-x^3+x^2-2*x+1)/(-1+x)/(x^8-x^7+2*x^5-x^4-2*x^2+3*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1018, 2025, 4024, 7992, 15869, 31507, 62555, 124201, 246602, 489637, 972199, 1930356, 3832833, 7610307, 15110685, 30003083, 59572724, 118284804, 234860737, 466328414] The limit of a(n+1)/a(n) as n goes to infinity is 1.98555297774 a(n) is asymptotic to .539818135967*1.98555297774^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [1, 2, 3, 4], denoted by the variable, X[1, 2, 3, 4], is 9 9 8 8 7 7 (x X[1, 2, 3, 4] - x - x X[1, 2, 3, 4] + x + x X[1, 2, 3, 4] - x 6 6 5 5 4 + x X[1, 2, 3, 4] - x - 2 x X[1, 2, 3, 4] + 2 x + x X[1, 2, 3, 4] 4 3 2 / 8 8 - x + x - 3 x + 3 x - 1) / ((-1 + x) (x X[1, 2, 3, 4] - x / 7 7 5 5 4 - x X[1, 2, 3, 4] + x + 2 x X[1, 2, 3, 4] - 2 x - x X[1, 2, 3, 4] 4 2 + x + 2 x - 3 x + 1)) and in Maple format (x^9*X[1,2,3,4]-x^9-x^8*X[1,2,3,4]+x^8+x^7*X[1,2,3,4]-x^7+x^6*X[1,2,3,4]-x^6-2* x^5*X[1,2,3,4]+2*x^5+x^4*X[1,2,3,4]-x^4+x^3-3*x^2+3*x-1)/(-1+x)/(x^8*X[1,2,3,4] -x^8-x^7*X[1,2,3,4]+x^7+2*x^5*X[1,2,3,4]-2*x^5-x^4*X[1,2,3,4]+x^4+2*x^2-3*x+1) Furthermore, the expectation of the random variable number of occurences (by\ 11 n containment) of , [1, 2, 3, 4], equals , - --- + --- 128 128 1731 147 n The variance equals , - ----- + ----- 16384 16384 77697 5967 n The , 3, -th moment about the mean is , - ------ + ------ 524288 524288 55929687 64827 2 719559 The , 4, -th moment about the mean is , - --------- + --------- n + -------- n 268435456 268435456 67108864 The compositions of, 10, that yield the, 57, -th largest growth, that is, 1.9861840427096552271, are , [2, 2, 2, 4], [2, 2, 4, 2], [2, 4, 2, 2], [4, 2, 2, 2] Theorem Number, 57, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 2, 2, 4] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 8 7 6 5 4 3 2 \ n x - x + 2 x - 2 x + x - x + 3 x - 3 x + 1 ) a(n) x = - --------------------------------------------------------- / 9 8 7 6 5 4 3 2 ----- x - x + 2 x - 3 x + 3 x - x + 2 x - 5 x + 4 x - 1 n = 0 and in Maple format -(x^8-x^7+2*x^6-2*x^5+x^4-x^3+3*x^2-3*x+1)/(x^9-x^8+2*x^7-3*x^6+3*x^5-x^4+2*x^3 -5*x^2+4*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1018, 2025, 4024, 7992, 15869, 31508, 62562, 124232, 246712, 489976, 973148, 1932836, 3838993, 7625037, 15144909, 30080871, 59746556, 118668153, 235697297, 468138578] The limit of a(n+1)/a(n) as n goes to infinity is 1.98618404271 a(n) is asymptotic to .536770920616*1.98618404271^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [2, 2, 2, 4], denoted by the variable, X[2, 2, 2, 4], is 8 8 7 7 6 6 - (x X[2, 2, 2, 4] - x - x X[2, 2, 2, 4] + x + 2 x X[2, 2, 2, 4] - 2 x 5 5 4 4 3 2 - 2 x X[2, 2, 2, 4] + 2 x + x X[2, 2, 2, 4] - x + x - 3 x + 3 x - 1) / 9 9 8 8 7 / (x X[2, 2, 2, 4] - x - x X[2, 2, 2, 4] + x + 2 x X[2, 2, 2, 4] / 7 6 6 5 5 - 2 x - 3 x X[2, 2, 2, 4] + 3 x + 3 x X[2, 2, 2, 4] - 3 x 4 4 3 2 - x X[2, 2, 2, 4] + x - 2 x + 5 x - 4 x + 1) and in Maple format -(x^8*X[2,2,2,4]-x^8-x^7*X[2,2,2,4]+x^7+2*x^6*X[2,2,2,4]-2*x^6-2*x^5*X[2,2,2,4] +2*x^5+x^4*X[2,2,2,4]-x^4+x^3-3*x^2+3*x-1)/(x^9*X[2,2,2,4]-x^9-x^8*X[2,2,2,4]+x ^8+2*x^7*X[2,2,2,4]-2*x^7-3*x^6*X[2,2,2,4]+3*x^6+3*x^5*X[2,2,2,4]-3*x^5-x^4*X[2 ,2,2,4]+x^4-2*x^3+5*x^2-4*x+1) Furthermore, the expectation of the random variable number of occurences (by\ 11 n containment) of , [2, 2, 2, 4], equals , - --- + --- 128 128 1959 159 n The variance equals , - ----- + ----- 16384 16384 25977 909 n The , 3, -th moment about the mean is , - ------ + ----- 131072 65536 93189183 75843 2 1094271 The , 4, -th moment about the mean is , - --------- + --------- n + -------- n 268435456 268435456 67108864 The compositions of, 10, that yield the, 58, -th largest growth, that is, 1.9864180586156033445, are , [3, 2, 2, 3] Theorem Number, 58, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [3, 2, 2, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 2 16 15 12 11 10 8 7 6 ) a(n) x = - (x - x + 1) (x + x + x + x - x + x - x - x / ----- n = 0 5 4 3 2 / 20 19 18 17 16 + x + 2 x - 2 x - 2 x + 3 x - 1) / (x + x - x + x + 2 x / 14 13 12 11 9 8 7 6 5 4 3 - x + x + 2 x - 3 x + 3 x - x - 2 x - x + 5 x - 2 x - 6 x 2 + 9 x - 5 x + 1) and in Maple format -(x^2-x+1)*(x^16+x^15+x^12+x^11-x^10+x^8-x^7-x^6+x^5+2*x^4-2*x^3-2*x^2+3*x-1)/( x^20+x^19-x^18+x^17+2*x^16-x^14+x^13+2*x^12-3*x^11+3*x^9-x^8-2*x^7-x^6+5*x^5-2* x^4-6*x^3+9*x^2-5*x+1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1018, 2025, 4024, 7992, 15869, 31508, 62563, 124240, 246748, 490100, 973514, 1933815, 3841445, 7630904, 15158497, 30111604, 59814863, 118817998, 236022743, 468839898] The limit of a(n+1)/a(n) as n goes to infinity is 1.98641805862 a(n) is asymptotic to .535676599493*1.98641805862^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [3, 2, 2, 3], denoted by the variable, X[3, 2, 2, 3], is 18 4 18 3 18 2 - (x X[3, 2, 2, 3] - 4 x X[3, 2, 2, 3] + 6 x X[3, 2, 2, 3] 18 18 15 3 15 2 - 4 x X[3, 2, 2, 3] + x - x X[3, 2, 2, 3] + 3 x X[3, 2, 2, 3] 14 3 15 14 2 15 - x X[3, 2, 2, 3] - 3 x X[3, 2, 2, 3] + 3 x X[3, 2, 2, 3] + x 14 12 3 14 12 2 - 3 x X[3, 2, 2, 3] + x X[3, 2, 2, 3] + x - 3 x X[3, 2, 2, 3] 11 3 12 11 2 12 - x X[3, 2, 2, 3] + 3 x X[3, 2, 2, 3] + 4 x X[3, 2, 2, 3] - x 11 11 9 2 9 - 5 x X[3, 2, 2, 3] + 2 x - 2 x X[3, 2, 2, 3] + 4 x X[3, 2, 2, 3] 8 2 9 8 8 7 + x X[3, 2, 2, 3] - 2 x - 2 x X[3, 2, 2, 3] + x - x X[3, 2, 2, 3] 7 5 5 4 4 + x + 3 x X[3, 2, 2, 3] - 3 x - 3 x X[3, 2, 2, 3] + 2 x 3 3 2 / 20 4 + x X[3, 2, 2, 3] + 3 x - 6 x + 4 x - 1) / (x X[3, 2, 2, 3] / 20 3 19 4 20 2 - 4 x X[3, 2, 2, 3] + x X[3, 2, 2, 3] + 6 x X[3, 2, 2, 3] 19 3 18 4 20 - 4 x X[3, 2, 2, 3] - x X[3, 2, 2, 3] - 4 x X[3, 2, 2, 3] 19 2 18 3 20 19 + 6 x X[3, 2, 2, 3] + 4 x X[3, 2, 2, 3] + x - 4 x X[3, 2, 2, 3] 18 2 17 3 19 18 - 6 x X[3, 2, 2, 3] - x X[3, 2, 2, 3] + x + 4 x X[3, 2, 2, 3] 17 2 16 3 18 17 + 3 x X[3, 2, 2, 3] - 2 x X[3, 2, 2, 3] - x - 3 x X[3, 2, 2, 3] 16 2 17 16 14 3 + 6 x X[3, 2, 2, 3] + x - 6 x X[3, 2, 2, 3] + x X[3, 2, 2, 3] 16 14 2 14 13 2 + 2 x - 3 x X[3, 2, 2, 3] + 3 x X[3, 2, 2, 3] + x X[3, 2, 2, 3] 12 3 14 13 12 2 - 2 x X[3, 2, 2, 3] - x - 2 x X[3, 2, 2, 3] + 6 x X[3, 2, 2, 3] 11 3 13 12 11 2 + x X[3, 2, 2, 3] + x - 6 x X[3, 2, 2, 3] - 5 x X[3, 2, 2, 3] 12 11 10 2 11 + 2 x + 7 x X[3, 2, 2, 3] - x X[3, 2, 2, 3] - 3 x 10 9 2 9 + x X[3, 2, 2, 3] + 3 x X[3, 2, 2, 3] - 6 x X[3, 2, 2, 3] 8 2 9 8 8 7 - x X[3, 2, 2, 3] + 3 x + 2 x X[3, 2, 2, 3] - x + 2 x X[3, 2, 2, 3] 7 6 6 5 5 - 2 x + x X[3, 2, 2, 3] - x - 5 x X[3, 2, 2, 3] + 5 x 4 4 3 3 2 + 4 x X[3, 2, 2, 3] - 2 x - x X[3, 2, 2, 3] - 6 x + 9 x - 5 x + 1) and in Maple format -(x^18*X[3,2,2,3]^4-4*x^18*X[3,2,2,3]^3+6*x^18*X[3,2,2,3]^2-4*x^18*X[3,2,2,3]+x ^18-x^15*X[3,2,2,3]^3+3*x^15*X[3,2,2,3]^2-x^14*X[3,2,2,3]^3-3*x^15*X[3,2,2,3]+3 *x^14*X[3,2,2,3]^2+x^15-3*x^14*X[3,2,2,3]+x^12*X[3,2,2,3]^3+x^14-3*x^12*X[3,2,2 ,3]^2-x^11*X[3,2,2,3]^3+3*x^12*X[3,2,2,3]+4*x^11*X[3,2,2,3]^2-x^12-5*x^11*X[3,2 ,2,3]+2*x^11-2*x^9*X[3,2,2,3]^2+4*x^9*X[3,2,2,3]+x^8*X[3,2,2,3]^2-2*x^9-2*x^8*X [3,2,2,3]+x^8-x^7*X[3,2,2,3]+x^7+3*x^5*X[3,2,2,3]-3*x^5-3*x^4*X[3,2,2,3]+2*x^4+ x^3*X[3,2,2,3]+3*x^3-6*x^2+4*x-1)/(x^20*X[3,2,2,3]^4-4*x^20*X[3,2,2,3]^3+x^19*X [3,2,2,3]^4+6*x^20*X[3,2,2,3]^2-4*x^19*X[3,2,2,3]^3-x^18*X[3,2,2,3]^4-4*x^20*X[ 3,2,2,3]+6*x^19*X[3,2,2,3]^2+4*x^18*X[3,2,2,3]^3+x^20-4*x^19*X[3,2,2,3]-6*x^18* X[3,2,2,3]^2-x^17*X[3,2,2,3]^3+x^19+4*x^18*X[3,2,2,3]+3*x^17*X[3,2,2,3]^2-2*x^ 16*X[3,2,2,3]^3-x^18-3*x^17*X[3,2,2,3]+6*x^16*X[3,2,2,3]^2+x^17-6*x^16*X[3,2,2, 3]+x^14*X[3,2,2,3]^3+2*x^16-3*x^14*X[3,2,2,3]^2+3*x^14*X[3,2,2,3]+x^13*X[3,2,2, 3]^2-2*x^12*X[3,2,2,3]^3-x^14-2*x^13*X[3,2,2,3]+6*x^12*X[3,2,2,3]^2+x^11*X[3,2, 2,3]^3+x^13-6*x^12*X[3,2,2,3]-5*x^11*X[3,2,2,3]^2+2*x^12+7*x^11*X[3,2,2,3]-x^10 *X[3,2,2,3]^2-3*x^11+x^10*X[3,2,2,3]+3*x^9*X[3,2,2,3]^2-6*x^9*X[3,2,2,3]-x^8*X[ 3,2,2,3]^2+3*x^9+2*x^8*X[3,2,2,3]-x^8+2*x^7*X[3,2,2,3]-2*x^7+x^6*X[3,2,2,3]-x^6 -5*x^5*X[3,2,2,3]+5*x^5+4*x^4*X[3,2,2,3]-2*x^4-x^3*X[3,2,2,3]-6*x^3+9*x^2-5*x+1 ) Furthermore, the expectation of the random variable number of occurences (by\ 11 n containment) of , [3, 2, 2, 3], equals , - --- + --- 128 128 2119 167 n The variance equals , - ----- + ----- 16384 16384 2067 2145 n The , 3, -th moment about the mean is , - ---- + ------ 8192 131072 160327615 83667 2 1763627 The , 4, -th moment about the mean is , - --------- + --------- n + -------- n 268435456 268435456 67108864 The compositions of, 10, that yield the, 59, -th largest growth, that is, 1.9867708871840811416, are , [2, 3, 2, 3], [3, 2, 3, 2] Theorem Number, 59, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 3, 2, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n ) a(n) x = / ----- n = 0 3 8 5 2 (x - x + 1) (x + x + x - 2 x + 1) - ------------------------------------------------------------------ 12 11 9 8 6 5 4 3 2 x - x + 3 x - 2 x + 2 x - 3 x + 3 x + x - 5 x + 4 x - 1 and in Maple format -(x^3-x+1)*(x^8+x^5+x^2-2*x+1)/(x^12-x^11+3*x^9-2*x^8+2*x^6-3*x^5+3*x^4+x^3-5*x ^2+4*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1018, 2025, 4024, 7992, 15870, 31515, 62591, 124327, 246983, 490683, 974884, 1936923, 3848334, 7645940, 15190977, 30181266, 59963490, 119133778, 236691252, 470250505] The limit of a(n+1)/a(n) as n goes to infinity is 1.98677088718 a(n) is asymptotic to .534434032726*1.98677088718^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [2, 3, 2, 3], denoted by the variable, X[2, 3, 2, 3], is 11 2 11 11 9 2 - (x X[2, 3, 2, 3] - 2 x X[2, 3, 2, 3] + x - x X[2, 3, 2, 3] 9 8 2 9 8 8 + 2 x X[2, 3, 2, 3] + x X[2, 3, 2, 3] - x - 3 x X[2, 3, 2, 3] + 2 x 6 6 5 5 4 + x X[2, 3, 2, 3] - x - 2 x X[2, 3, 2, 3] + 2 x + 2 x X[2, 3, 2, 3] 4 3 2 / 12 2 - 2 x - x X[2, 3, 2, 3] + 3 x - 3 x + 1) / (x X[2, 3, 2, 3] / 12 11 2 12 11 - 2 x X[2, 3, 2, 3] - x X[2, 3, 2, 3] + x + 2 x X[2, 3, 2, 3] 11 9 2 9 8 2 - x + 2 x X[2, 3, 2, 3] - 5 x X[2, 3, 2, 3] - x X[2, 3, 2, 3] 9 8 8 6 6 + 3 x + 3 x X[2, 3, 2, 3] - 2 x - 2 x X[2, 3, 2, 3] + 2 x 5 5 4 4 3 + 3 x X[2, 3, 2, 3] - 3 x - 3 x X[2, 3, 2, 3] + 3 x + x X[2, 3, 2, 3] 3 2 + x - 5 x + 4 x - 1) and in Maple format -(x^11*X[2,3,2,3]^2-2*x^11*X[2,3,2,3]+x^11-x^9*X[2,3,2,3]^2+2*x^9*X[2,3,2,3]+x^ 8*X[2,3,2,3]^2-x^9-3*x^8*X[2,3,2,3]+2*x^8+x^6*X[2,3,2,3]-x^6-2*x^5*X[2,3,2,3]+2 *x^5+2*x^4*X[2,3,2,3]-2*x^4-x^3*X[2,3,2,3]+3*x^2-3*x+1)/(x^12*X[2,3,2,3]^2-2*x^ 12*X[2,3,2,3]-x^11*X[2,3,2,3]^2+x^12+2*x^11*X[2,3,2,3]-x^11+2*x^9*X[2,3,2,3]^2-\ 5*x^9*X[2,3,2,3]-x^8*X[2,3,2,3]^2+3*x^9+3*x^8*X[2,3,2,3]-2*x^8-2*x^6*X[2,3,2,3] +2*x^6+3*x^5*X[2,3,2,3]-3*x^5-3*x^4*X[2,3,2,3]+3*x^4+x^3*X[2,3,2,3]+x^3-5*x^2+4 *x-1) Furthermore, the expectation of the random variable number of occurences (by\ 11 n containment) of , [2, 3, 2, 3], equals , - --- + --- 128 128 2231 175 n The variance equals , - ----- + ----- 16384 16384 9195 2397 n The , 3, -th moment about the mean is , - ----- + ------ 32768 131072 185587487 91875 2 2138863 The , 4, -th moment about the mean is , - --------- + --------- n + -------- n 268435456 268435456 67108864 The compositions of, 10, that yield the, 60, -th largest growth, that is, 1.9874108030247649893, are , [1, 3, 3, 3], [3, 3, 3, 1] Theorem Number, 60, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 3, 3, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 3 6 2 \ n (x - x + 1) (x + x - 2 x + 1) ) a(n) x = ---------------------------------------------------- / 8 7 6 4 3 2 ----- (-1 + x) (x + x - x + 2 x - x - 2 x + 3 x - 1) n = 0 and in Maple format (x^3-x+1)*(x^6+x^2-2*x+1)/(-1+x)/(x^8+x^7-x^6+2*x^4-x^3-2*x^2+3*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1018, 2025, 4025, 7998, 15892, 31579, 62755, 124716, 247862, 492610, 979032, 1945758, 3867041, 7685414, 15274076, 30355844, 60329489, 119899416, 238289328, 473578737] The limit of a(n+1)/a(n) as n goes to infinity is 1.98741080302 a(n) is asymptotic to .533043756330*1.98741080302^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [1, 3, 3, 3], denoted by the variable, X[1, 3, 3, 3], is 9 9 7 7 6 6 (x X[1, 3, 3, 3] - x - x X[1, 3, 3, 3] + x + x X[1, 3, 3, 3] - x 5 5 4 4 3 + x X[1, 3, 3, 3] - x - 2 x X[1, 3, 3, 3] + 2 x + x X[1, 3, 3, 3] 2 / 8 8 7 - 3 x + 3 x - 1) / ((-1 + x) (x X[1, 3, 3, 3] - x + x X[1, 3, 3, 3] / 7 6 6 4 4 - x - x X[1, 3, 3, 3] + x + 2 x X[1, 3, 3, 3] - 2 x 3 3 2 - x X[1, 3, 3, 3] + x + 2 x - 3 x + 1)) and in Maple format (x^9*X[1,3,3,3]-x^9-x^7*X[1,3,3,3]+x^7+x^6*X[1,3,3,3]-x^6+x^5*X[1,3,3,3]-x^5-2* x^4*X[1,3,3,3]+2*x^4+x^3*X[1,3,3,3]-3*x^2+3*x-1)/(-1+x)/(x^8*X[1,3,3,3]-x^8+x^7 *X[1,3,3,3]-x^7-x^6*X[1,3,3,3]+x^6+2*x^4*X[1,3,3,3]-2*x^4-x^3*X[1,3,3,3]+x^3+2* x^2-3*x+1) Furthermore, the expectation of the random variable number of occurences (by\ 11 n containment) of , [1, 3, 3, 3], equals , - --- + --- 128 128 2323 187 n The variance equals , - ----- + ----- 16384 16384 155769 10863 n The , 3, -th moment about the mean is , - ------ + ------- 524288 524288 191645431 104907 2 2495899 The , 4, -th moment about the mean is , - --------- + --------- n + -------- n 268435456 268435456 67108864 The compositions of, 10, that yield the, 61, -th largest growth, that is, 1.9875763672979512663, are , [2, 2, 3, 3], [2, 3, 3, 2], [3, 3, 2, 2] Theorem Number, 61, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 2, 3, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 8 7 6 5 4 2 \ n x - x + x + x - 2 x + 3 x - 3 x + 1 ) a(n) x = - ------------------------------------------------------- / 9 8 7 6 5 4 3 2 ----- x - x + 2 x - x - 2 x + 3 x + x - 5 x + 4 x - 1 n = 0 and in Maple format -(x^8-x^7+x^6+x^5-2*x^4+3*x^2-3*x+1)/(x^9-x^8+2*x^7-x^6-2*x^5+3*x^4+x^3-5*x^2+4 *x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1018, 2025, 4025, 7998, 15892, 31579, 62755, 124717, 247870, 492648, 979170, 1946183, 3868215, 7688425, 15281410, 30373069, 60368903, 119987901, 238485167, 474007445] The limit of a(n+1)/a(n) as n goes to infinity is 1.98757636730 a(n) is asymptotic to .532194205807*1.98757636730^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [2, 2, 3, 3], denoted by the variable, X[2, 2, 3, 3], is 8 8 7 7 6 6 - (x X[2, 2, 3, 3] - x - x X[2, 2, 3, 3] + x + x X[2, 2, 3, 3] - x 5 5 4 4 3 + x X[2, 2, 3, 3] - x - 2 x X[2, 2, 3, 3] + 2 x + x X[2, 2, 3, 3] 2 / 9 9 8 8 - 3 x + 3 x - 1) / (x X[2, 2, 3, 3] - x - x X[2, 2, 3, 3] + x / 7 7 6 6 5 + 2 x X[2, 2, 3, 3] - 2 x - x X[2, 2, 3, 3] + x - 2 x X[2, 2, 3, 3] 5 4 4 3 3 2 + 2 x + 3 x X[2, 2, 3, 3] - 3 x - x X[2, 2, 3, 3] - x + 5 x - 4 x + 1) and in Maple format -(x^8*X[2,2,3,3]-x^8-x^7*X[2,2,3,3]+x^7+x^6*X[2,2,3,3]-x^6+x^5*X[2,2,3,3]-x^5-2 *x^4*X[2,2,3,3]+2*x^4+x^3*X[2,2,3,3]-3*x^2+3*x-1)/(x^9*X[2,2,3,3]-x^9-x^8*X[2,2 ,3,3]+x^8+2*x^7*X[2,2,3,3]-2*x^7-x^6*X[2,2,3,3]+x^6-2*x^5*X[2,2,3,3]+2*x^5+3*x^ 4*X[2,2,3,3]-3*x^4-x^3*X[2,2,3,3]-x^3+5*x^2-4*x+1) Furthermore, the expectation of the random variable number of occurences (by\ 11 n containment) of , [2, 2, 3, 3], equals , - --- + --- 128 128 2407 191 n The variance equals , - ----- + ----- 16384 16384 41973 1425 n The , 3, -th moment about the mean is , - ------ + ------ 131072 65536 215052991 109443 2 2708015 The , 4, -th moment about the mean is , - --------- + --------- n + -------- n 268435456 268435456 67108864 The compositions of, 10, that yield the, 62, -th largest growth, that is, 1.9919641966050350211, are , [1, 1, 8], [1, 8, 1], [8, 1, 1] Theorem Number, 62, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 1, 8] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 8 2 \ n x + x - 2 x + 1 ) a(n) x = - ----------------------------------------------- / 7 6 5 4 3 2 2 ----- (x + x + x + x + x + x + x - 1) (-1 + x) n = 0 and in Maple format -(x^8+x^2-2*x+1)/(x^7+x^6+x^5+x^4+x^3+x^2+x-1)/(-1+x)^2 The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1019, 2031, 4047, 8063, 16063, 31999, 63743, 126976, 252934, 503838, 1003630, 1999198, 3982334, 7932670, 15801598, 31476221, 62699509, 124895181, 248786733, 495574269] The limit of a(n+1)/a(n) as n goes to infinity is 1.99196419661 a(n) is asymptotic to .520790324966*1.99196419661^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [1, 1, 8], denoted by the variable, X[1, 1, 8], is 8 8 2 x X[1, 1, 8] - x - x + 2 x - 1 - --------------------------------------- 8 8 (-1 + x) (x X[1, 1, 8] - x + 2 x - 1) and in Maple format -(x^8*X[1,1,8]-x^8-x^2+2*x-1)/(-1+x)/(x^8*X[1,1,8]-x^8+2*x-1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [1, 1, 8], equals , - 5/128 + --- 256 147 241 n The variance equals , - ---- + ----- 4096 65536 126033 27261 n The , 3, -th moment about the mean is , - ------- + ------- 4194304 8388608 The , 4, -th moment about the mean is , 1067427 174243 2 3622595 - -------- + ---------- n + ---------- n 67108864 4294967296 2147483648 The compositions of, 10, that yield the, 63, -th largest growth, that is, 1.9920300868462484222, are , [2, 1, 7], [7, 1, 2] Theorem Number, 63, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 7] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 15 7 2 \ n x + x + x - 2 x + 1 ) a(n) x = - -------------------------------------------------------- / 6 5 3 2 8 7 2 ----- (x + x - x - x + 1) (x - x + 2 x - 1) (x - x + 1) n = 0 and in Maple format -(x^15+x^7+x^2-2*x+1)/(x^6+x^5-x^3-x^2+1)/(x^8-x^7+2*x-1)/(x^2-x+1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1019, 2031, 4047, 8063, 16063, 31999, 63743, 126977, 252940, 503861, 1003702, 1999399, 3982856, 7933961, 15804681, 31483393, 62715861, 124931880, 248868066, 495752683] The limit of a(n+1)/a(n) as n goes to infinity is 1.99203008685 a(n) is asymptotic to .520461112883*1.99203008685^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [2, 1, 7], denoted by the variable, X[2, 1, 7], is 15 2 15 15 7 7 2 - (x X[2, 1, 7] - 2 x X[2, 1, 7] + x - x X[2, 1, 7] + x + x - 2 x + 1 / 8 8 ) / ((x X[2, 1, 7] - x + x - 1) / 8 8 7 7 (x X[2, 1, 7] - x - x X[2, 1, 7] + x - 2 x + 1)) and in Maple format -(x^15*X[2,1,7]^2-2*x^15*X[2,1,7]+x^15-x^7*X[2,1,7]+x^7+x^2-2*x+1)/(x^8*X[2,1,7 ]-x^8+x-1)/(x^8*X[2,1,7]-x^8-x^7*X[2,1,7]+x^7-2*x+1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [2, 1, 7], equals , - 5/128 + --- 256 607 245 n The variance equals , - ----- + ----- 16384 65536 139401 28677 n The , 3, -th moment about the mean is , - ------- + ------- 4194304 8388608 The , 4, -th moment about the mean is , 5933779 180075 2 4280143 - --------- + ---------- n + ---------- n 268435456 4294967296 2147483648 The compositions of, 10, that yield the, 64, -th largest growth, that is, 1.9921580953553798820, are , [3, 1, 6], [6, 1, 3] Theorem Number, 64, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [3, 1, 6] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 14 13 6 2 \ n x + x + x + x - 2 x + 1 ) a(n) x = - ------------------------------------------ / 8 7 6 8 7 ----- (x + x - x + 2 x - 1) (x + x - x + 1) n = 0 and in Maple format -(x^14+x^13+x^6+x^2-2*x+1)/(x^8+x^7-x^6+2*x-1)/(x^8+x^7-x+1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1019, 2031, 4047, 8063, 16063, 31999, 63744, 126983, 252963, 503933, 1003903, 1999921, 3984147, 7937043, 15811845, 31499708, 62752428, 125012808, 249045352, 496137813] The limit of a(n+1)/a(n) as n goes to infinity is 1.99215809536 a(n) is asymptotic to .519862388220*1.99215809536^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [3, 1, 6], denoted by the variable, X[3, 1, 6], is 14 2 14 13 2 14 - (x X[3, 1, 6] - 2 x X[3, 1, 6] + x X[3, 1, 6] + x 13 13 6 6 2 / - 2 x X[3, 1, 6] + x - x X[3, 1, 6] + x + x - 2 x + 1) / ( / 8 8 7 7 (x X[3, 1, 6] - x + x X[3, 1, 6] - x + x - 1) 8 8 7 7 6 6 (x X[3, 1, 6] - x + x X[3, 1, 6] - x - x X[3, 1, 6] + x - 2 x + 1)) and in Maple format -(x^14*X[3,1,6]^2-2*x^14*X[3,1,6]+x^13*X[3,1,6]^2+x^14-2*x^13*X[3,1,6]+x^13-x^6 *X[3,1,6]+x^6+x^2-2*x+1)/(x^8*X[3,1,6]-x^8+x^7*X[3,1,6]-x^7+x-1)/(x^8*X[3,1,6]- x^8+x^7*X[3,1,6]-x^7-x^6*X[3,1,6]+x^6-2*x+1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [3, 1, 6], equals , - 5/128 + --- 256 643 253 n The variance equals , - ----- + ----- 16384 65536 165777 31593 n The , 3, -th moment about the mean is , - ------- + ------- 4194304 8388608 The , 4, -th moment about the mean is , 9450643 192027 2 5714111 - --------- + ---------- n + ---------- n 268435456 4294967296 2147483648 The compositions of, 10, that yield the, 65, -th largest growth, that is, 1.9922212637540336675, are , [2, 2, 6], [2, 6, 2], [6, 2, 2] Theorem Number, 65, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 2, 6] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 8 7 6 2 \ n x - x + x + x - 2 x + 1 ) a(n) x = - ------------------------------------ / 9 8 7 6 2 ----- x - x + 2 x - x - 2 x + 3 x - 1 n = 0 and in Maple format -(x^8-x^7+x^6+x^2-2*x+1)/(x^9-x^8+2*x^7-x^6-2*x^2+3*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1019, 2031, 4047, 8063, 16063, 32000, 63749, 126999, 253006, 504039, 1004152, 2000488, 3985411, 7939819, 15817878, 31512719, 62780320, 125072305, 249171727, 496405238] The limit of a(n+1)/a(n) as n goes to infinity is 1.99222126375 a(n) is asymptotic to .519647972626*1.99222126375^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [2, 2, 6], denoted by the variable, X[2, 2, 6], is 8 8 7 7 6 6 2 - (x X[2, 2, 6] - x - x X[2, 2, 6] + x + x X[2, 2, 6] - x - x + 2 x - 1) / 9 9 8 8 7 7 / (x X[2, 2, 6] - x - x X[2, 2, 6] + x + 2 x X[2, 2, 6] - 2 x / 6 6 2 - x X[2, 2, 6] + x + 2 x - 3 x + 1) and in Maple format -(x^8*X[2,2,6]-x^8-x^7*X[2,2,6]+x^7+x^6*X[2,2,6]-x^6-x^2+2*x-1)/(x^9*X[2,2,6]-x ^9-x^8*X[2,2,6]+x^8+2*x^7*X[2,2,6]-2*x^7-x^6*X[2,2,6]+x^6+2*x^2-3*x+1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [2, 2, 6], equals , - 5/128 + --- 256 41 257 n The variance equals , - ---- + ----- 1024 65536 175353 33069 n The , 3, -th moment about the mean is , - ------- + ------- 4194304 8388608 The , 4, -th moment about the mean is , 2682971 198147 2 6468499 - -------- + ---------- n + ---------- n 67108864 4294967296 2147483648 The compositions of, 10, that yield the, 66, -th largest growth, that is, 1.9923365951207343993, are , [3, 2, 5], [5, 2, 3] Theorem Number, 66, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [3, 2, 5] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n ) a(n) x = / ----- n = 0 12 7 6 5 2 x + x - x + x + x - 2 x + 1 - ----------------------------------------------------------- 14 13 12 9 8 7 6 5 2 x + x - x + x + x - x + 2 x - x - 2 x + 3 x - 1 and in Maple format -(x^12+x^7-x^6+x^5+x^2-2*x+1)/(x^14+x^13-x^12+x^9+x^8-x^7+2*x^6-x^5-2*x^2+3*x-1 ) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1019, 2031, 4047, 8063, 16063, 32000, 63750, 127005, 253029, 504111, 1004352, 2001004, 3986677, 7942816, 15824789, 31528346, 62815130, 125148943, 249338879, 496767018] The limit of a(n+1)/a(n) as n goes to infinity is 1.99233659512 a(n) is asymptotic to .519124274617*1.99233659512^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [3, 2, 5], denoted by the variable, X[3, 2, 5], is 12 2 12 12 7 7 - (x X[3, 2, 5] - 2 x X[3, 2, 5] + x - x X[3, 2, 5] + x 6 6 5 5 2 / + x X[3, 2, 5] - x - x X[3, 2, 5] + x + x - 2 x + 1) / ( / 14 2 14 13 2 14 x X[3, 2, 5] - 2 x X[3, 2, 5] + x X[3, 2, 5] + x 13 12 2 13 12 12 - 2 x X[3, 2, 5] - x X[3, 2, 5] + x + 2 x X[3, 2, 5] - x 9 9 8 8 7 7 - x X[3, 2, 5] + x - x X[3, 2, 5] + x + x X[3, 2, 5] - x 6 6 5 5 2 - 2 x X[3, 2, 5] + 2 x + x X[3, 2, 5] - x - 2 x + 3 x - 1) and in Maple format -(x^12*X[3,2,5]^2-2*x^12*X[3,2,5]+x^12-x^7*X[3,2,5]+x^7+x^6*X[3,2,5]-x^6-x^5*X[ 3,2,5]+x^5+x^2-2*x+1)/(x^14*X[3,2,5]^2-2*x^14*X[3,2,5]+x^13*X[3,2,5]^2+x^14-2*x ^13*X[3,2,5]-x^12*X[3,2,5]^2+x^13+2*x^12*X[3,2,5]-x^12-x^9*X[3,2,5]+x^9-x^8*X[3 ,2,5]+x^8+x^7*X[3,2,5]-x^7-2*x^6*X[3,2,5]+2*x^6+x^5*X[3,2,5]-x^5-2*x^2+3*x-1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [3, 2, 5], equals , - 5/128 + --- 256 173 265 n The variance equals , - ---- + ----- 4096 65536 205293 36297 n The , 3, -th moment about the mean is , - ------- + ------- 4194304 8388608 The , 4, -th moment about the mean is , 3899753 210675 2 8362979 - -------- + ---------- n + ---------- n 67108864 4294967296 2147483648 The compositions of, 10, that yield the, 67, -th largest growth, that is, 1.9924010004614550874, are , [4, 1, 5], [5, 1, 4] Theorem Number, 67, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [4, 1, 5] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 13 12 11 5 2 \ n x + x + x + x + x - 2 x + 1 ) a(n) x = - ---------------------------------------------------- / 8 7 6 8 7 6 5 ----- (x + x + x - x + 1) (x + x + x - x + 2 x - 1) n = 0 and in Maple format -(x^13+x^12+x^11+x^5+x^2-2*x+1)/(x^8+x^7+x^6-x+1)/(x^8+x^7+x^6-x^5+2*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1019, 2031, 4047, 8063, 16063, 32000, 63750, 127006, 253035, 504134, 1004425, 2001211, 3987221, 7944170, 15828028, 31535870, 62832228, 125187155, 249423186, 496951151] The limit of a(n+1)/a(n) as n goes to infinity is 1.99240100046 a(n) is asymptotic to .518813178100*1.99240100046^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [4, 1, 5], denoted by the variable, X[4, 1, 5], is 13 2 13 12 2 13 - (x X[4, 1, 5] - 2 x X[4, 1, 5] + x X[4, 1, 5] + x 12 11 2 12 11 11 - 2 x X[4, 1, 5] + x X[4, 1, 5] + x - 2 x X[4, 1, 5] + x 5 5 2 / - x X[4, 1, 5] + x + x - 2 x + 1) / ( / 8 8 7 7 6 6 (x X[4, 1, 5] - x + x X[4, 1, 5] - x + x X[4, 1, 5] - x + x - 1) ( 8 8 7 7 6 6 x X[4, 1, 5] - x + x X[4, 1, 5] - x + x X[4, 1, 5] - x 5 5 - x X[4, 1, 5] + x - 2 x + 1)) and in Maple format -(x^13*X[4,1,5]^2-2*x^13*X[4,1,5]+x^12*X[4,1,5]^2+x^13-2*x^12*X[4,1,5]+x^11*X[4 ,1,5]^2+x^12-2*x^11*X[4,1,5]+x^11-x^5*X[4,1,5]+x^5+x^2-2*x+1)/(x^8*X[4,1,5]-x^8 +x^7*X[4,1,5]-x^7+x^6*X[4,1,5]-x^6+x-1)/(x^8*X[4,1,5]-x^8+x^7*X[4,1,5]-x^7+x^6* X[4,1,5]-x^6-x^5*X[4,1,5]+x^5-2*x+1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [4, 1, 5], equals , - 5/128 + --- 256 711 269 n The variance equals , - ----- + ----- 16384 65536 219321 37737 n The , 3, -th moment about the mean is , - ------- + ------- 4194304 8388608 The , 4, -th moment about the mean is , 17463363 217083 2 9033487 - --------- + ---------- n + ---------- n 268435456 4294967296 2147483648 The compositions of, 10, that yield the, 68, -th largest growth, that is, 1.9924602335260952454, are , [2, 3, 5], [2, 5, 3], [3, 5, 2], [5, 3, 2] Theorem Number, 68, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 3, 5] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 8 6 5 2 \ n x - x + x + x - 2 x + 1 ) a(n) x = - ------------------------------------ / 9 8 6 5 2 ----- x - x + 2 x - x - 2 x + 3 x - 1 n = 0 and in Maple format -(x^8-x^6+x^5+x^2-2*x+1)/(x^9-x^8+2*x^6-x^5-2*x^2+3*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1019, 2031, 4047, 8063, 16064, 32005, 63766, 127048, 253135, 504359, 1004914, 2002252, 3989411, 7948749, 15837575, 31555749, 62873586, 125273129, 249601733, 497321526] The limit of a(n+1)/a(n) as n goes to infinity is 1.99246023353 a(n) is asymptotic to .518737176056*1.99246023353^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [2, 3, 5], denoted by the variable, X[2, 3, 5], is 8 8 6 6 5 5 2 - (x X[2, 3, 5] - x - x X[2, 3, 5] + x + x X[2, 3, 5] - x - x + 2 x - 1) / 9 9 8 8 6 6 / (x X[2, 3, 5] - x - x X[2, 3, 5] + x + 2 x X[2, 3, 5] - 2 x / 5 5 2 - x X[2, 3, 5] + x + 2 x - 3 x + 1) and in Maple format -(x^8*X[2,3,5]-x^8-x^6*X[2,3,5]+x^6+x^5*X[2,3,5]-x^5-x^2+2*x-1)/(x^9*X[2,3,5]-x ^9-x^8*X[2,3,5]+x^8+2*x^6*X[2,3,5]-2*x^6-x^5*X[2,3,5]+x^5+2*x^2-3*x+1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [2, 3, 5], equals , - 5/128 + --- 256 179 273 n The variance equals , - ---- + ----- 4096 65536 223281 39309 n The , 3, -th moment about the mean is , - ------- + ------- 4194304 8388608 The , 4, -th moment about the mean is , 4513043 223587 2 9951747 - -------- + ---------- n + ---------- n 67108864 4294967296 2147483648 The compositions of, 10, that yield the, 69, -th largest growth, that is, 1.9925574403170594250, are , [4, 2, 4] Theorem Number, 69, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [4, 2, 4] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 11 10 6 5 4 2 / ) a(n) x = - (x + x + x - x + x + x - 2 x + 1) / ( / / ----- n = 0 14 13 12 10 9 8 7 6 5 4 2 x + 2 x + 2 x - x + x + x + x - x + 2 x - x - 2 x + 3 x - 1) and in Maple format -(x^11+x^10+x^6-x^5+x^4+x^2-2*x+1)/(x^14+2*x^13+2*x^12-x^10+x^9+x^8+x^7-x^6+2*x ^5-x^4-2*x^2+3*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1019, 2031, 4047, 8063, 16063, 32001, 63756, 127028, 253101, 504311, 1004868, 2002269, 3989666, 7949691, 15840291, 31562782, 62890751, 125313504, 249694359, 497530239] The limit of a(n+1)/a(n) as n goes to infinity is 1.99255744032 a(n) is asymptotic to .518195458821*1.99255744032^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [4, 2, 4], denoted by the variable, X[4, 2, 4], is 11 2 11 10 2 11 - (x X[4, 2, 4] - 2 x X[4, 2, 4] + x X[4, 2, 4] + x 10 10 6 6 5 5 - 2 x X[4, 2, 4] + x - x X[4, 2, 4] + x + x X[4, 2, 4] - x 4 4 2 / 14 2 - x X[4, 2, 4] + x + x - 2 x + 1) / (x X[4, 2, 4] / 14 13 2 14 13 - 2 x X[4, 2, 4] + 2 x X[4, 2, 4] + x - 4 x X[4, 2, 4] 12 2 13 12 12 10 2 + 2 x X[4, 2, 4] + 2 x - 4 x X[4, 2, 4] + 2 x - x X[4, 2, 4] 10 10 9 9 8 8 + 2 x X[4, 2, 4] - x - x X[4, 2, 4] + x - x X[4, 2, 4] + x 7 7 6 6 5 5 - x X[4, 2, 4] + x + x X[4, 2, 4] - x - 2 x X[4, 2, 4] + 2 x 4 4 2 + x X[4, 2, 4] - x - 2 x + 3 x - 1) and in Maple format -(x^11*X[4,2,4]^2-2*x^11*X[4,2,4]+x^10*X[4,2,4]^2+x^11-2*x^10*X[4,2,4]+x^10-x^6 *X[4,2,4]+x^6+x^5*X[4,2,4]-x^5-x^4*X[4,2,4]+x^4+x^2-2*x+1)/(x^14*X[4,2,4]^2-2*x ^14*X[4,2,4]+2*x^13*X[4,2,4]^2+x^14-4*x^13*X[4,2,4]+2*x^12*X[4,2,4]^2+2*x^13-4* x^12*X[4,2,4]+2*x^12-x^10*X[4,2,4]^2+2*x^10*X[4,2,4]-x^10-x^9*X[4,2,4]+x^9-x^8* X[4,2,4]+x^8-x^7*X[4,2,4]+x^7+x^6*X[4,2,4]-x^6-2*x^5*X[4,2,4]+2*x^5+x^4*X[4,2,4 ]-x^4-2*x^2+3*x-1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [4, 2, 4], equals , - 5/128 + --- 256 95 281 n The variance equals , - ---- + ----- 2048 65536 265653 43065 n The , 3, -th moment about the mean is , - ------- + ------- 4194304 8388608 The , 4, -th moment about the mean is , 6626041 236883 2 12694579 - -------- + ---------- n + ---------- n 67108864 4294967296 2147483648 The compositions of, 10, that yield the, 70, -th largest growth, that is, 1.9928945584015041909, are , [2, 4, 4], [4, 4, 2] Theorem Number, 70, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 4, 4] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 8 5 4 2 \ n x - x + x + x - 2 x + 1 ) a(n) x = - ------------------------------------ / 9 8 5 4 2 ----- x - x + 2 x - x - 2 x + 3 x - 1 n = 0 and in Maple format -(x^8-x^5+x^4+x^2-2*x+1)/(x^9-x^8+2*x^5-x^4-2*x^2+3*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1019, 2031, 4047, 8064, 16069, 32022, 63815, 127176, 253449, 505099, 1006612, 2006075, 3997899, 7967393, 15878174, 31643524, 63062202, 125676313, 250459634, 499139637] The limit of a(n+1)/a(n) as n goes to infinity is 1.99289455840 a(n) is asymptotic to .517240378759*1.99289455840^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [2, 4, 4], denoted by the variable, X[2, 4, 4], is 8 8 5 5 4 4 2 - (x X[2, 4, 4] - x - x X[2, 4, 4] + x + x X[2, 4, 4] - x - x + 2 x - 1) / 9 9 8 8 5 5 / (x X[2, 4, 4] - x - x X[2, 4, 4] + x + 2 x X[2, 4, 4] - 2 x / 4 4 2 - x X[2, 4, 4] + x + 2 x - 3 x + 1) and in Maple format -(x^8*X[2,4,4]-x^8-x^5*X[2,4,4]+x^5+x^4*X[2,4,4]-x^4-x^2+2*x-1)/(x^9*X[2,4,4]-x ^9-x^8*X[2,4,4]+x^8+2*x^5*X[2,4,4]-2*x^5-x^4*X[2,4,4]+x^4+2*x^2-3*x+1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [2, 4, 4], equals , - 5/128 + --- 256 207 305 n The variance equals , - ---- + ----- 4096 65536 323793 52989 n The , 3, -th moment about the mean is , - ------- + ------- 4194304 8388608 The , 4, -th moment about the mean is , 9065487 279075 2 18820099 - -------- + ---------- n + ---------- n 67108864 4294967296 2147483648 The compositions of, 10, that yield the, 71, -th largest growth, that is, 1.9929967428173323177, are , [3, 3, 4], [3, 4, 3], [4, 3, 3] Theorem Number, 71, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [3, 3, 4] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 7 5 4 2 \ n x - x + x + x - 2 x + 1 ) a(n) x = - ----------------------------------------- / 9 8 7 5 4 2 ----- x + x - x + 2 x - x - 2 x + 3 x - 1 n = 0 and in Maple format -(x^7-x^5+x^4+x^2-2*x+1)/(x^9+x^8-x^7+2*x^5-x^4-2*x^2+3*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1019, 2031, 4047, 8064, 16069, 32022, 63816, 127182, 253472, 505171, 1006811, 2006583, 3999130, 7970272, 15884743, 31658250, 63094783, 125747669, 250614641, 499474085] The limit of a(n+1)/a(n) as n goes to infinity is 1.99299674282 a(n) is asymptotic to .516791351539*1.99299674282^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [3, 3, 4], denoted by the variable, X[3, 3, 4], is 7 7 5 5 4 4 2 - (x X[3, 3, 4] - x - x X[3, 3, 4] + x + x X[3, 3, 4] - x - x + 2 x - 1) / 9 9 8 8 7 7 / (x X[3, 3, 4] - x + x X[3, 3, 4] - x - x X[3, 3, 4] + x / 5 5 4 4 2 + 2 x X[3, 3, 4] - 2 x - x X[3, 3, 4] + x + 2 x - 3 x + 1) and in Maple format -(x^7*X[3,3,4]-x^7-x^5*X[3,3,4]+x^5+x^4*X[3,3,4]-x^4-x^2+2*x-1)/(x^9*X[3,3,4]-x ^9+x^8*X[3,3,4]-x^8-x^7*X[3,3,4]+x^7+2*x^5*X[3,3,4]-2*x^5-x^4*X[3,3,4]+x^4+2*x^ 2-3*x+1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [3, 3, 4], equals , - 5/128 + --- 256 27 313 n The variance equals , - --- + ----- 512 65536 358053 56601 n The , 3, -th moment about the mean is , - ------- + ------- 4194304 8388608 The , 4, -th moment about the mean is , 10743045 293907 2 21320915 - -------- + ---------- n + ---------- n 67108864 4294967296 2147483648 The compositions of, 10, that yield the, 72, -th largest growth, that is, 1.9960311797354145898, are , [1, 9], [9, 1] Theorem Number, 72, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 9] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 9 \ n x - x + 1 ) a(n) x = --------------------------------------------------- / 8 7 6 5 4 3 2 ----- (-1 + x) (x + x + x + x + x + x + x + x - 1) n = 0 and in Maple format (x^9-x+1)/(-1+x)/(x^8+x^7+x^6+x^5+x^4+x^3+x^2+x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1020, 2036, 4064, 8112, 16192, 32320, 64512, 128768, 257025, 513030, 1024024, 2043984, 4079856, 8143520, 16254720, 32444928, 64761088, 129265151, 258017272, 515010520] The limit of a(n+1)/a(n) as n goes to infinity is 1.99603117974 a(n) is asymptotic to .509092230016*1.99603117974^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [1, 9], denoted by the variable, X[1, 9], is 9 9 x X[1, 9] - x + x - 1 ------------------------- 9 9 x X[1, 9] - x + 2 x - 1 and in Maple format (x^9*X[1,9]-x^9+x-1)/(x^9*X[1,9]-x^9+2*x-1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [1, 9], equals , - 9/512 + --- 512 4383 495 n The variance equals , - ------ + ------ 262144 262144 1012221 118341 n The , 3, -th moment about the mean is , - -------- + -------- 67108864 67108864 The , 4, -th moment about the mean is , 769703319 735075 2 22990653 - ----------- + ----------- n + ----------- n 68719476736 68719476736 17179869184 The compositions of, 10, that yield the, 73, -th largest growth, that is, 1.9960471205602957908, are , [2, 8], [8, 2] Theorem Number, 73, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 8] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 2 6 5 3 2 \ n (x - x + 1) (x + x - x - x + 1) ) a(n) x = - ------------------------------------ / 9 8 ----- x - x + 2 x - 1 n = 0 and in Maple format -(x^2-x+1)*(x^6+x^5-x^3-x^2+1)/(x^9-x^8+2*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1020, 2036, 4064, 8112, 16192, 32320, 64512, 128769, 257029, 513042, 1024056, 2044064, 4080048, 8143968, 16255744, 32447231, 64766202, 129276391, 258041768, 515063528] The limit of a(n+1)/a(n) as n goes to infinity is 1.99604712056 a(n) is asymptotic to .509022658976*1.99604712056^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [2, 8], denoted by the variable, X[2, 8], is 8 8 x X[2, 8] - x + x - 1 - ------------------------------------------- 9 9 8 8 x X[2, 8] - x - x X[2, 8] + x - 2 x + 1 and in Maple format -(x^8*X[2,8]-x^8+x-1)/(x^9*X[2,8]-x^9-x^8*X[2,8]+x^8-2*x+1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [2, 8], equals , - 9/512 + --- 512 4451 499 n The variance equals , - ------ + ------ 262144 262144 1061763 121275 n The , 3, -th moment about the mean is , - -------- + -------- 67108864 67108864 The , 4, -th moment about the mean is , 876686687 747003 2 24542089 - ----------- + ----------- n + ----------- n 68719476736 68719476736 17179869184 The compositions of, 10, that yield the, 74, -th largest growth, that is, 1.9960785538884684114, are , [3, 7], [7, 3] Theorem Number, 74, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [3, 7] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 7 \ n x - x + 1 ) a(n) x = - ---------------------- / 9 8 7 ----- x + x - x + 2 x - 1 n = 0 and in Maple format -(x^7-x+1)/(x^9+x^8-x^7+2*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1020, 2036, 4064, 8112, 16192, 32320, 64513, 128773, 257041, 513074, 1024136, 2044256, 4080496, 8144991, 16258042, 32452329, 64777398, 129300775, 258094504, 515176904] The limit of a(n+1)/a(n) as n goes to infinity is 1.99607855389 a(n) is asymptotic to .508894231203*1.99607855389^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [3, 7], denoted by the variable, X[3, 7], is 7 7 x X[3, 7] - x + x - 1 - ------------------------------------------------------------- 9 9 8 8 7 7 x X[3, 7] - x + x X[3, 7] - x - x X[3, 7] + x - 2 x + 1 and in Maple format -(x^7*X[3,7]-x^7+x-1)/(x^9*X[3,7]-x^9+x^8*X[3,7]-x^8-x^7*X[3,7]+x^7-2*x+1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [3, 7], equals , - 9/512 + --- 512 4579 507 n The variance equals , - ------ + ------ 262144 262144 1156851 127227 n The , 3, -th moment about the mean is , - -------- + -------- 67108864 67108864 The , 4, -th moment about the mean is , 1088674015 771147 2 27771285 - ----------- + ----------- n + ----------- n 68719476736 68719476736 17179869184 The compositions of, 10, that yield the, 75, -th largest growth, that is, 1.9961398080808123357, are , [4, 6], [6, 4] Theorem Number, 75, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [4, 6] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 6 \ n x - x + 1 ) a(n) x = - --------------------------- / 9 8 7 6 ----- x + x + x - x + 2 x - 1 n = 0 and in Maple format -(x^6-x+1)/(x^9+x^8+x^7-x^6+2*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1020, 2036, 4064, 8112, 16192, 32321, 64517, 128785, 257073, 513154, 1024328, 2044703, 4081514, 8147273, 16263096, 32463413, 64801510, 129352872, 258206415, 515416102] The limit of a(n+1)/a(n) as n goes to infinity is 1.99613980808 a(n) is asymptotic to .508662020641*1.99613980808^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [4, 6], denoted by the variable, X[4, 6], is 6 6 / 9 9 8 8 7 - (x X[4, 6] - x + x - 1) / (x X[4, 6] - x + x X[4, 6] - x + x X[4, 6] / 7 6 6 - x - x X[4, 6] + x - 2 x + 1) and in Maple format -(x^6*X[4,6]-x^6+x-1)/(x^9*X[4,6]-x^9+x^8*X[4,6]-x^8+x^7*X[4,6]-x^7-x^6*X[4,6]+ x^6-2*x+1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [4, 6], equals , - 9/512 + --- 512 4819 523 n The variance equals , - ------ + ------ 262144 262144 1341627 139443 n The , 3, -th moment about the mean is , - -------- + -------- 67108864 67108864 The , 4, -th moment about the mean is , 1524741247 820587 2 34698613 - ----------- + ----------- n + ----------- n 68719476736 68719476736 17179869184 The compositions of, 10, that yield the, 76, -th largest growth, that is, 1.9962564778474358596, are , [5, 5] Theorem Number, 76, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [5, 5] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 5 \ n x - x + 1 ) a(n) x = - -------------------------------- / 9 8 7 6 5 ----- x + x + x + x - x + 2 x - 1 n = 0 and in Maple format -(x^5-x+1)/(x^9+x^8+x^7+x^6-x^5+2*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1020, 2036, 4064, 8112, 16193, 32325, 64529, 128817, 257153, 513345, 1024770, 2045705, 4083752, 8152215, 16273909, 32486892, 64852164, 129461549, 258438455, 515909443] The limit of a(n+1)/a(n) as n goes to infinity is 1.99625647785 a(n) is asymptotic to .508256955954*1.99625647785^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [5, 5], denoted by the variable, X[5, 5], is 5 5 / 9 9 8 8 7 - (x X[5, 5] - x + x - 1) / (x X[5, 5] - x + x X[5, 5] - x + x X[5, 5] / 7 6 6 5 5 - x + x X[5, 5] - x - x X[5, 5] + x - 2 x + 1) and in Maple format -(x^5*X[5,5]-x^5+x-1)/(x^9*X[5,5]-x^9+x^8*X[5,5]-x^8+x^7*X[5,5]-x^7+x^6*X[5,5]- x^6-x^5*X[5,5]+x^5-2*x+1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [5, 5], equals , - 9/512 + --- 512 5267 555 n The variance equals , - ------ + ------ 262144 262144 1709691 165075 n The , 3, -th moment about the mean is , - -------- + -------- 67108864 67108864 The , 4, -th moment about the mean is , 2483602943 924075 2 50409189 - ----------- + ----------- n + ----------- n 68719476736 68719476736 17179869184 The compositions of, 10, that yield the, 77, -th largest growth, that is, 1.9980294702622866987, are , [10] Theorem Number, 77, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [10] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 1 ) a(n) x = - --------------------------------------------- / 9 8 7 6 5 4 3 2 ----- x + x + x + x + x + x + x + x + x - 1 n = 0 and in Maple format -1/(x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1021, 2040, 4076, 8144, 16272, 32512, 64960, 129792, 259328, 518145, 1035269, 2068498, 4132920, 8257696, 16499120, 32965728, 65866496, 131603200, 262947072, 525375999] The limit of a(n+1)/a(n) as n goes to infinity is 1.99802947026 a(n) is asymptotic to .503980275734*1.99802947026^n ---------------------------------------------------------------------------- The generating function keeping track of the number of occurrences of, [10], denoted by the variable, X[10], is -1 + x ------------------------- 10 10 x X[10] - x + 2 x - 1 and in Maple format (-1+x)/(x^10*X[10]-x^10+2*x-1) Furthermore, the expectation of the random variable number of occurences (by\ n containment) of , [10], equals , - 1/128 + ---- 1024 497 1005 n The variance equals , - ----- + ------- 65536 1048576 239517 247755 n The , 3, -th moment about the mean is , - -------- + --------- 33554432 268435456 The , 4, -th moment about the mean is , 3030075 2 104980631 445632375 ------------- n - ----------- + ------------ n 1099511627776 17179869184 549755813888 This ends this article, that took, 23303.212, seconds to generate. ---------------------------------------------------------- -------------------------------------------- This took, 23393.135, seconds.