***************************************************************************** * www.FindStat.org - The Combinatorial Statistic Finder * * * * Copyright (C) 2019 The FindStatCrew * * * * This information is distributed in the hope that it will be useful, * * but WITHOUT ANY WARRANTY; without even the implied warranty of * * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. * ***************************************************************************** ----------------------------------------------------------------------------- Statistic identifier: St001177 ----------------------------------------------------------------------------- Collection: Integer partitions ----------------------------------------------------------------------------- Description: Twice the mean value of the major index among all standard Young tableaux of a partition. For a partition $\lambda$ of $n$, this mean value is given in [1, Proposition 3.1] by $$\frac{1}{2}\Big(\binom{n}{2} - \sum_i\binom{\lambda_i}{2} + \sum_i\binom{\lambda_i'}{2}\Big),$$ where $\lambda_i$ is the size of the $i$-th row of $\lambda$ and $\lambda_i'$ is the size of the $i$-th column. ----------------------------------------------------------------------------- References: [1] , Adin, R. M., Roichman, Y. Descent functions and random Young tableaux [[MathSciNet:1841639]] [[arXiv:math/9910165]] ----------------------------------------------------------------------------- Code: def statistic(L): n = sum(L) return binomial(n,2) - sum(binomial(i,2) for i in L) + sum(binomial(i,2) for i in L.conjugate()) ----------------------------------------------------------------------------- Statistic values: [1] => 0 [2] => 0 [1,1] => 2 [3] => 0 [2,1] => 3 [1,1,1] => 6 [4] => 0 [3,1] => 4 [2,2] => 6 [2,1,1] => 8 [1,1,1,1] => 12 [5] => 0 [4,1] => 5 [3,2] => 8 [3,1,1] => 10 [2,2,1] => 12 [2,1,1,1] => 15 [1,1,1,1,1] => 20 [6] => 0 [5,1] => 6 [4,2] => 10 [4,1,1] => 12 [3,3] => 12 [3,2,1] => 15 [3,1,1,1] => 18 [2,2,2] => 18 [2,2,1,1] => 20 [2,1,1,1,1] => 24 [1,1,1,1,1,1] => 30 [7] => 0 [6,1] => 7 [5,2] => 12 [5,1,1] => 14 [4,3] => 15 [4,2,1] => 18 [4,1,1,1] => 21 [3,3,1] => 20 [3,2,2] => 22 [3,2,1,1] => 24 [3,1,1,1,1] => 28 [2,2,2,1] => 27 [2,2,1,1,1] => 30 [2,1,1,1,1,1] => 35 [1,1,1,1,1,1,1] => 42 [8] => 0 [7,1] => 8 [6,2] => 14 [6,1,1] => 16 [5,3] => 18 [5,2,1] => 21 [5,1,1,1] => 24 [4,4] => 20 [4,3,1] => 24 [4,2,2] => 26 [4,2,1,1] => 28 [4,1,1,1,1] => 32 [3,3,2] => 28 [3,3,1,1] => 30 [3,2,2,1] => 32 [3,2,1,1,1] => 35 [3,1,1,1,1,1] => 40 [2,2,2,2] => 36 [2,2,2,1,1] => 38 [2,2,1,1,1,1] => 42 [2,1,1,1,1,1,1] => 48 [1,1,1,1,1,1,1,1] => 56 [9] => 0 [8,1] => 9 [7,2] => 16 [7,1,1] => 18 [6,3] => 21 [6,2,1] => 24 [6,1,1,1] => 27 [5,4] => 24 [5,3,1] => 28 [5,2,2] => 30 [5,2,1,1] => 32 [5,1,1,1,1] => 36 [4,4,1] => 30 [4,3,2] => 33 [4,3,1,1] => 35 [4,2,2,1] => 37 [4,2,1,1,1] => 40 [4,1,1,1,1,1] => 45 [3,3,3] => 36 [3,3,2,1] => 39 [3,3,1,1,1] => 42 [3,2,2,2] => 42 [3,2,2,1,1] => 44 [3,2,1,1,1,1] => 48 [3,1,1,1,1,1,1] => 54 [2,2,2,2,1] => 48 [2,2,2,1,1,1] => 51 [2,2,1,1,1,1,1] => 56 [2,1,1,1,1,1,1,1] => 63 [1,1,1,1,1,1,1,1,1] => 72 [10] => 0 [9,1] => 10 [8,2] => 18 [8,1,1] => 20 [7,3] => 24 [7,2,1] => 27 [7,1,1,1] => 30 [6,4] => 28 [6,3,1] => 32 [6,2,2] => 34 [6,2,1,1] => 36 [6,1,1,1,1] => 40 [5,5] => 30 [5,4,1] => 35 [5,3,2] => 38 [5,3,1,1] => 40 [5,2,2,1] => 42 [5,2,1,1,1] => 45 [5,1,1,1,1,1] => 50 [4,4,2] => 40 [4,4,1,1] => 42 [4,3,3] => 42 [4,3,2,1] => 45 [4,3,1,1,1] => 48 [4,2,2,2] => 48 [4,2,2,1,1] => 50 [4,2,1,1,1,1] => 54 [4,1,1,1,1,1,1] => 60 [3,3,3,1] => 48 [3,3,2,2] => 50 [3,3,2,1,1] => 52 [3,3,1,1,1,1] => 56 [3,2,2,2,1] => 55 [3,2,2,1,1,1] => 58 [3,2,1,1,1,1,1] => 63 [3,1,1,1,1,1,1,1] => 70 [2,2,2,2,2] => 60 [2,2,2,2,1,1] => 62 [2,2,2,1,1,1,1] => 66 [2,2,1,1,1,1,1,1] => 72 [2,1,1,1,1,1,1,1,1] => 80 [1,1,1,1,1,1,1,1,1,1] => 90 [11] => 0 [10,1] => 11 [9,2] => 20 [9,1,1] => 22 [8,3] => 27 [8,2,1] => 30 [8,1,1,1] => 33 [7,4] => 32 [7,3,1] => 36 [7,2,2] => 38 [7,2,1,1] => 40 [7,1,1,1,1] => 44 [6,5] => 35 [6,4,1] => 40 [6,3,2] => 43 [6,3,1,1] => 45 [6,2,2,1] => 47 [6,2,1,1,1] => 50 [6,1,1,1,1,1] => 55 [5,5,1] => 42 [5,4,2] => 46 [5,4,1,1] => 48 [5,3,3] => 48 [5,3,2,1] => 51 [5,3,1,1,1] => 54 [5,2,2,2] => 54 [5,2,2,1,1] => 56 [5,2,1,1,1,1] => 60 [5,1,1,1,1,1,1] => 66 [4,4,3] => 50 [4,4,2,1] => 53 [4,4,1,1,1] => 56 [4,3,3,1] => 55 [4,3,2,2] => 57 [4,3,2,1,1] => 59 [4,3,1,1,1,1] => 63 [4,2,2,2,1] => 62 [4,2,2,1,1,1] => 65 [4,2,1,1,1,1,1] => 70 [4,1,1,1,1,1,1,1] => 77 [3,3,3,2] => 60 [3,3,3,1,1] => 62 [3,3,2,2,1] => 64 [3,3,2,1,1,1] => 67 [3,3,1,1,1,1,1] => 72 [3,2,2,2,2] => 68 [3,2,2,2,1,1] => 70 [3,2,2,1,1,1,1] => 74 [3,2,1,1,1,1,1,1] => 80 [3,1,1,1,1,1,1,1,1] => 88 [2,2,2,2,2,1] => 75 [2,2,2,2,1,1,1] => 78 [2,2,2,1,1,1,1,1] => 83 [2,2,1,1,1,1,1,1,1] => 90 [2,1,1,1,1,1,1,1,1,1] => 99 [1,1,1,1,1,1,1,1,1,1,1] => 110 [12] => 0 [11,1] => 12 [10,2] => 22 [10,1,1] => 24 [9,3] => 30 [9,2,1] => 33 [9,1,1,1] => 36 [8,4] => 36 [8,3,1] => 40 [8,2,2] => 42 [8,2,1,1] => 44 [8,1,1,1,1] => 48 [7,5] => 40 [7,4,1] => 45 [7,3,2] => 48 [7,3,1,1] => 50 [7,2,2,1] => 52 [7,2,1,1,1] => 55 [7,1,1,1,1,1] => 60 [6,6] => 42 [6,5,1] => 48 [6,4,2] => 52 [6,4,1,1] => 54 [6,3,3] => 54 [6,3,2,1] => 57 [6,3,1,1,1] => 60 [6,2,2,2] => 60 [6,2,2,1,1] => 62 [6,2,1,1,1,1] => 66 [6,1,1,1,1,1,1] => 72 [5,5,2] => 54 [5,5,1,1] => 56 [5,4,3] => 57 [5,4,2,1] => 60 [5,4,1,1,1] => 63 [5,3,3,1] => 62 [5,3,2,2] => 64 [5,3,2,1,1] => 66 [5,3,1,1,1,1] => 70 [5,2,2,2,1] => 69 [5,2,2,1,1,1] => 72 [5,2,1,1,1,1,1] => 77 [5,1,1,1,1,1,1,1] => 84 [4,4,4] => 60 [4,4,3,1] => 64 [4,4,2,2] => 66 [4,4,2,1,1] => 68 [4,4,1,1,1,1] => 72 [4,3,3,2] => 68 [4,3,3,1,1] => 70 [4,3,2,2,1] => 72 [4,3,2,1,1,1] => 75 [4,3,1,1,1,1,1] => 80 [4,2,2,2,2] => 76 [4,2,2,2,1,1] => 78 [4,2,2,1,1,1,1] => 82 [4,2,1,1,1,1,1,1] => 88 [4,1,1,1,1,1,1,1,1] => 96 [3,3,3,3] => 72 [3,3,3,2,1] => 75 [3,3,3,1,1,1] => 78 [3,3,2,2,2] => 78 [3,3,2,2,1,1] => 80 [3,3,2,1,1,1,1] => 84 [3,3,1,1,1,1,1,1] => 90 [3,2,2,2,2,1] => 84 [3,2,2,2,1,1,1] => 87 [3,2,2,1,1,1,1,1] => 92 [3,2,1,1,1,1,1,1,1] => 99 [3,1,1,1,1,1,1,1,1,1] => 108 [2,2,2,2,2,2] => 90 [2,2,2,2,2,1,1] => 92 [2,2,2,2,1,1,1,1] => 96 [2,2,2,1,1,1,1,1,1] => 102 [2,2,1,1,1,1,1,1,1,1] => 110 [2,1,1,1,1,1,1,1,1,1,1] => 120 [1,1,1,1,1,1,1,1,1,1,1,1] => 132 [13] => 0 [12,1] => 13 [11,2] => 24 [11,1,1] => 26 [10,3] => 33 [10,2,1] => 36 [10,1,1,1] => 39 [9,4] => 40 [9,3,1] => 44 [9,2,2] => 46 [9,2,1,1] => 48 [9,1,1,1,1] => 52 [8,5] => 45 [8,4,1] => 50 [8,3,2] => 53 [8,3,1,1] => 55 [8,2,2,1] => 57 [8,2,1,1,1] => 60 [8,1,1,1,1,1] => 65 [7,6] => 48 [7,5,1] => 54 [7,4,2] => 58 [7,4,1,1] => 60 [7,3,3] => 60 [7,3,2,1] => 63 [7,3,1,1,1] => 66 [7,2,2,2] => 66 [7,2,2,1,1] => 68 [7,2,1,1,1,1] => 72 [7,1,1,1,1,1,1] => 78 [6,6,1] => 56 [6,5,2] => 61 [6,5,1,1] => 63 [6,4,3] => 64 [6,4,2,1] => 67 [6,4,1,1,1] => 70 [6,3,3,1] => 69 [6,3,2,2] => 71 [6,3,2,1,1] => 73 [6,3,1,1,1,1] => 77 [6,2,2,2,1] => 76 [6,2,2,1,1,1] => 79 [6,2,1,1,1,1,1] => 84 [6,1,1,1,1,1,1,1] => 91 [5,5,3] => 66 [5,5,2,1] => 69 [5,5,1,1,1] => 72 [5,4,4] => 68 [5,4,3,1] => 72 [5,4,2,2] => 74 [5,4,2,1,1] => 76 [5,4,1,1,1,1] => 80 [5,3,3,2] => 76 [5,3,3,1,1] => 78 [5,3,2,2,1] => 80 [5,3,2,1,1,1] => 83 [5,3,1,1,1,1,1] => 88 [5,2,2,2,2] => 84 [5,2,2,2,1,1] => 86 [5,2,2,1,1,1,1] => 90 [5,2,1,1,1,1,1,1] => 96 [5,1,1,1,1,1,1,1,1] => 104 [4,4,4,1] => 75 [4,4,3,2] => 78 [4,4,3,1,1] => 80 [4,4,2,2,1] => 82 [4,4,2,1,1,1] => 85 [4,4,1,1,1,1,1] => 90 [4,3,3,3] => 81 [4,3,3,2,1] => 84 [4,3,3,1,1,1] => 87 [4,3,2,2,2] => 87 [4,3,2,2,1,1] => 89 [4,3,2,1,1,1,1] => 93 [4,3,1,1,1,1,1,1] => 99 [4,2,2,2,2,1] => 93 [4,2,2,2,1,1,1] => 96 [4,2,2,1,1,1,1,1] => 101 [4,2,1,1,1,1,1,1,1] => 108 [4,1,1,1,1,1,1,1,1,1] => 117 [3,3,3,3,1] => 88 [3,3,3,2,2] => 90 [3,3,3,2,1,1] => 92 [3,3,3,1,1,1,1] => 96 [3,3,2,2,2,1] => 95 [3,3,2,2,1,1,1] => 98 [3,3,2,1,1,1,1,1] => 103 [3,3,1,1,1,1,1,1,1] => 110 [3,2,2,2,2,2] => 100 [3,2,2,2,2,1,1] => 102 [3,2,2,2,1,1,1,1] => 106 [3,2,2,1,1,1,1,1,1] => 112 [3,2,1,1,1,1,1,1,1,1] => 120 [3,1,1,1,1,1,1,1,1,1,1] => 130 [2,2,2,2,2,2,1] => 108 [2,2,2,2,2,1,1,1] => 111 [2,2,2,2,1,1,1,1,1] => 116 [2,2,2,1,1,1,1,1,1,1] => 123 [2,2,1,1,1,1,1,1,1,1,1] => 132 [2,1,1,1,1,1,1,1,1,1,1,1] => 143 [1,1,1,1,1,1,1,1,1,1,1,1,1] => 156 [14] => 0 [13,1] => 14 [12,2] => 26 [12,1,1] => 28 [11,3] => 36 [11,2,1] => 39 [11,1,1,1] => 42 [10,4] => 44 [10,3,1] => 48 [10,2,2] => 50 [10,2,1,1] => 52 [10,1,1,1,1] => 56 [9,5] => 50 [9,4,1] => 55 [9,3,2] => 58 [9,3,1,1] => 60 [9,2,2,1] => 62 [9,2,1,1,1] => 65 [9,1,1,1,1,1] => 70 [8,6] => 54 [8,5,1] => 60 [8,4,2] => 64 [8,4,1,1] => 66 [8,3,3] => 66 [8,3,2,1] => 69 [8,3,1,1,1] => 72 [8,2,2,2] => 72 [8,2,2,1,1] => 74 [8,2,1,1,1,1] => 78 [8,1,1,1,1,1,1] => 84 [7,7] => 56 [7,6,1] => 63 [7,5,2] => 68 [7,5,1,1] => 70 [7,4,3] => 71 [7,4,2,1] => 74 [7,4,1,1,1] => 77 [7,3,3,1] => 76 [7,3,2,2] => 78 [7,3,2,1,1] => 80 [7,3,1,1,1,1] => 84 [7,2,2,2,1] => 83 [7,2,2,1,1,1] => 86 [7,2,1,1,1,1,1] => 91 [7,1,1,1,1,1,1,1] => 98 [6,6,2] => 70 [6,6,1,1] => 72 [6,5,3] => 74 [6,5,2,1] => 77 [6,5,1,1,1] => 80 [6,4,4] => 76 [6,4,3,1] => 80 [6,4,2,2] => 82 [6,4,2,1,1] => 84 [6,4,1,1,1,1] => 88 [6,3,3,2] => 84 [6,3,3,1,1] => 86 [6,3,2,2,1] => 88 [6,3,2,1,1,1] => 91 [6,3,1,1,1,1,1] => 96 [6,2,2,2,2] => 92 [6,2,2,2,1,1] => 94 [6,2,2,1,1,1,1] => 98 [6,2,1,1,1,1,1,1] => 104 [6,1,1,1,1,1,1,1,1] => 112 [5,5,4] => 78 [5,5,3,1] => 82 [5,5,2,2] => 84 [5,5,2,1,1] => 86 [5,5,1,1,1,1] => 90 [5,4,4,1] => 84 [5,4,3,2] => 87 [5,4,3,1,1] => 89 [5,4,2,2,1] => 91 [5,4,2,1,1,1] => 94 [5,4,1,1,1,1,1] => 99 [5,3,3,3] => 90 [5,3,3,2,1] => 93 [5,3,3,1,1,1] => 96 [5,3,2,2,2] => 96 [5,3,2,2,1,1] => 98 [5,3,2,1,1,1,1] => 102 [5,3,1,1,1,1,1,1] => 108 [5,2,2,2,2,1] => 102 [5,2,2,2,1,1,1] => 105 [5,2,2,1,1,1,1,1] => 110 [5,2,1,1,1,1,1,1,1] => 117 [5,1,1,1,1,1,1,1,1,1] => 126 [4,4,4,2] => 90 [4,4,4,1,1] => 92 [4,4,3,3] => 92 [4,4,3,2,1] => 95 [4,4,3,1,1,1] => 98 [4,4,2,2,2] => 98 [4,4,2,2,1,1] => 100 [4,4,2,1,1,1,1] => 104 [4,4,1,1,1,1,1,1] => 110 [4,3,3,3,1] => 98 [4,3,3,2,2] => 100 [4,3,3,2,1,1] => 102 [4,3,3,1,1,1,1] => 106 [4,3,2,2,2,1] => 105 [4,3,2,2,1,1,1] => 108 [4,3,2,1,1,1,1,1] => 113 [4,3,1,1,1,1,1,1,1] => 120 [4,2,2,2,2,2] => 110 [4,2,2,2,2,1,1] => 112 [4,2,2,2,1,1,1,1] => 116 [4,2,2,1,1,1,1,1,1] => 122 [4,2,1,1,1,1,1,1,1,1] => 130 [4,1,1,1,1,1,1,1,1,1,1] => 140 [3,3,3,3,2] => 104 [3,3,3,3,1,1] => 106 [3,3,3,2,2,1] => 108 [3,3,3,2,1,1,1] => 111 [3,3,3,1,1,1,1,1] => 116 [3,3,2,2,2,2] => 112 [3,3,2,2,2,1,1] => 114 [3,3,2,2,1,1,1,1] => 118 [3,3,2,1,1,1,1,1,1] => 124 [3,3,1,1,1,1,1,1,1,1] => 132 [3,2,2,2,2,2,1] => 119 [3,2,2,2,2,1,1,1] => 122 [3,2,2,2,1,1,1,1,1] => 127 [3,2,2,1,1,1,1,1,1,1] => 134 [3,2,1,1,1,1,1,1,1,1,1] => 143 [3,1,1,1,1,1,1,1,1,1,1,1] => 154 [2,2,2,2,2,2,2] => 126 [2,2,2,2,2,2,1,1] => 128 [2,2,2,2,2,1,1,1,1] => 132 [2,2,2,2,1,1,1,1,1,1] => 138 [2,2,2,1,1,1,1,1,1,1,1] => 146 [2,2,1,1,1,1,1,1,1,1,1,1] => 156 [2,1,1,1,1,1,1,1,1,1,1,1,1] => 168 [1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 182 [15] => 0 [14,1] => 15 [13,2] => 28 [13,1,1] => 30 [12,3] => 39 [12,2,1] => 42 [12,1,1,1] => 45 [11,4] => 48 [11,3,1] => 52 [11,2,2] => 54 [11,2,1,1] => 56 [11,1,1,1,1] => 60 [10,5] => 55 [10,4,1] => 60 [10,3,2] => 63 [10,3,1,1] => 65 [10,2,2,1] => 67 [10,2,1,1,1] => 70 [10,1,1,1,1,1] => 75 [9,6] => 60 [9,5,1] => 66 [9,4,2] => 70 [9,4,1,1] => 72 [9,3,3] => 72 [9,3,2,1] => 75 [9,3,1,1,1] => 78 [9,2,2,2] => 78 [9,2,2,1,1] => 80 [9,2,1,1,1,1] => 84 [9,1,1,1,1,1,1] => 90 [8,7] => 63 [8,6,1] => 70 [8,5,2] => 75 [8,5,1,1] => 77 [8,4,3] => 78 [8,4,2,1] => 81 [8,4,1,1,1] => 84 [8,3,3,1] => 83 [8,3,2,2] => 85 [8,3,2,1,1] => 87 [8,3,1,1,1,1] => 91 [8,2,2,2,1] => 90 [8,2,2,1,1,1] => 93 [8,2,1,1,1,1,1] => 98 [8,1,1,1,1,1,1,1] => 105 [7,7,1] => 72 [7,6,2] => 78 [7,6,1,1] => 80 [7,5,3] => 82 [7,5,2,1] => 85 [7,5,1,1,1] => 88 [7,4,4] => 84 [7,4,3,1] => 88 [7,4,2,2] => 90 [7,4,2,1,1] => 92 [7,4,1,1,1,1] => 96 [7,3,3,2] => 92 [7,3,3,1,1] => 94 [7,3,2,2,1] => 96 [7,3,2,1,1,1] => 99 [7,3,1,1,1,1,1] => 104 [7,2,2,2,2] => 100 [7,2,2,2,1,1] => 102 [7,2,2,1,1,1,1] => 106 [7,2,1,1,1,1,1,1] => 112 [7,1,1,1,1,1,1,1,1] => 120 [6,6,3] => 84 [6,6,2,1] => 87 [6,6,1,1,1] => 90 [6,5,4] => 87 [6,5,3,1] => 91 [6,5,2,2] => 93 [6,5,2,1,1] => 95 [6,5,1,1,1,1] => 99 [6,4,4,1] => 93 [6,4,3,2] => 96 [6,4,3,1,1] => 98 [6,4,2,2,1] => 100 [6,4,2,1,1,1] => 103 [6,4,1,1,1,1,1] => 108 [6,3,3,3] => 99 [6,3,3,2,1] => 102 [6,3,3,1,1,1] => 105 [6,3,2,2,2] => 105 [6,3,2,2,1,1] => 107 [6,3,2,1,1,1,1] => 111 [6,3,1,1,1,1,1,1] => 117 [6,2,2,2,2,1] => 111 [6,2,2,2,1,1,1] => 114 [6,2,2,1,1,1,1,1] => 119 [6,2,1,1,1,1,1,1,1] => 126 [6,1,1,1,1,1,1,1,1,1] => 135 [5,5,5] => 90 [5,5,4,1] => 95 [5,5,3,2] => 98 [5,5,3,1,1] => 100 [5,5,2,2,1] => 102 [5,5,2,1,1,1] => 105 [5,5,1,1,1,1,1] => 110 [5,4,4,2] => 100 [5,4,4,1,1] => 102 [5,4,3,3] => 102 [5,4,3,2,1] => 105 [5,4,3,1,1,1] => 108 [5,4,2,2,2] => 108 [5,4,2,2,1,1] => 110 [5,4,2,1,1,1,1] => 114 [5,4,1,1,1,1,1,1] => 120 [5,3,3,3,1] => 108 [5,3,3,2,2] => 110 [5,3,3,2,1,1] => 112 [5,3,3,1,1,1,1] => 116 [5,3,2,2,2,1] => 115 [5,3,2,2,1,1,1] => 118 [5,3,2,1,1,1,1,1] => 123 [5,3,1,1,1,1,1,1,1] => 130 [5,2,2,2,2,2] => 120 [5,2,2,2,2,1,1] => 122 [5,2,2,2,1,1,1,1] => 126 [5,2,2,1,1,1,1,1,1] => 132 [5,2,1,1,1,1,1,1,1,1] => 140 [5,1,1,1,1,1,1,1,1,1,1] => 150 [4,4,4,3] => 105 [4,4,4,2,1] => 108 [4,4,4,1,1,1] => 111 [4,4,3,3,1] => 110 [4,4,3,2,2] => 112 [4,4,3,2,1,1] => 114 [4,4,3,1,1,1,1] => 118 [4,4,2,2,2,1] => 117 [4,4,2,2,1,1,1] => 120 [4,4,2,1,1,1,1,1] => 125 [4,4,1,1,1,1,1,1,1] => 132 [4,3,3,3,2] => 115 [4,3,3,3,1,1] => 117 [4,3,3,2,2,1] => 119 [4,3,3,2,1,1,1] => 122 [4,3,3,1,1,1,1,1] => 127 [4,3,2,2,2,2] => 123 [4,3,2,2,2,1,1] => 125 [4,3,2,2,1,1,1,1] => 129 [4,3,2,1,1,1,1,1,1] => 135 [4,3,1,1,1,1,1,1,1,1] => 143 [4,2,2,2,2,2,1] => 130 [4,2,2,2,2,1,1,1] => 133 [4,2,2,2,1,1,1,1,1] => 138 [4,2,2,1,1,1,1,1,1,1] => 145 [4,2,1,1,1,1,1,1,1,1,1] => 154 [4,1,1,1,1,1,1,1,1,1,1,1] => 165 [3,3,3,3,3] => 120 [3,3,3,3,2,1] => 123 [3,3,3,3,1,1,1] => 126 [3,3,3,2,2,2] => 126 [3,3,3,2,2,1,1] => 128 [3,3,3,2,1,1,1,1] => 132 [3,3,3,1,1,1,1,1,1] => 138 [3,3,2,2,2,2,1] => 132 [3,3,2,2,2,1,1,1] => 135 [3,3,2,2,1,1,1,1,1] => 140 [3,3,2,1,1,1,1,1,1,1] => 147 [3,3,1,1,1,1,1,1,1,1,1] => 156 [3,2,2,2,2,2,2] => 138 [3,2,2,2,2,2,1,1] => 140 [3,2,2,2,2,1,1,1,1] => 144 [3,2,2,2,1,1,1,1,1,1] => 150 [3,2,2,1,1,1,1,1,1,1,1] => 158 [3,2,1,1,1,1,1,1,1,1,1,1] => 168 [3,1,1,1,1,1,1,1,1,1,1,1,1] => 180 [2,2,2,2,2,2,2,1] => 147 [2,2,2,2,2,2,1,1,1] => 150 [2,2,2,2,2,1,1,1,1,1] => 155 [2,2,2,2,1,1,1,1,1,1,1] => 162 [2,2,2,1,1,1,1,1,1,1,1,1] => 171 [2,2,1,1,1,1,1,1,1,1,1,1,1] => 182 [2,1,1,1,1,1,1,1,1,1,1,1,1,1] => 195 [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 210 [16] => 0 [15,1] => 16 [14,2] => 30 [14,1,1] => 32 [13,3] => 42 [13,2,1] => 45 [13,1,1,1] => 48 [12,4] => 52 [12,3,1] => 56 [12,2,2] => 58 [12,2,1,1] => 60 [12,1,1,1,1] => 64 [11,5] => 60 [11,4,1] => 65 [11,3,2] => 68 [11,3,1,1] => 70 [11,2,2,1] => 72 [11,2,1,1,1] => 75 [11,1,1,1,1,1] => 80 [10,6] => 66 [10,5,1] => 72 [10,4,2] => 76 [10,4,1,1] => 78 [10,3,3] => 78 [10,3,2,1] => 81 [10,3,1,1,1] => 84 [10,2,2,2] => 84 [10,2,2,1,1] => 86 [10,2,1,1,1,1] => 90 [10,1,1,1,1,1,1] => 96 [9,7] => 70 [9,6,1] => 77 [9,5,2] => 82 [9,5,1,1] => 84 [9,4,3] => 85 [9,4,2,1] => 88 [9,4,1,1,1] => 91 [9,3,3,1] => 90 [9,3,2,2] => 92 [9,3,2,1,1] => 94 [9,3,1,1,1,1] => 98 [9,2,2,2,1] => 97 [9,2,2,1,1,1] => 100 [9,2,1,1,1,1,1] => 105 [9,1,1,1,1,1,1,1] => 112 [8,8] => 72 [8,7,1] => 80 [8,6,2] => 86 [8,6,1,1] => 88 [8,5,3] => 90 [8,5,2,1] => 93 [8,5,1,1,1] => 96 [8,4,4] => 92 [8,4,3,1] => 96 [8,4,2,2] => 98 [8,4,2,1,1] => 100 [8,4,1,1,1,1] => 104 [8,3,3,2] => 100 [8,3,3,1,1] => 102 [8,3,2,2,1] => 104 [8,3,2,1,1,1] => 107 [8,3,1,1,1,1,1] => 112 [8,2,2,2,2] => 108 [8,2,2,2,1,1] => 110 [8,2,2,1,1,1,1] => 114 [8,2,1,1,1,1,1,1] => 120 [8,1,1,1,1,1,1,1,1] => 128 [7,7,2] => 88 [7,7,1,1] => 90 [7,6,3] => 93 [7,6,2,1] => 96 [7,6,1,1,1] => 99 [7,5,4] => 96 [7,5,3,1] => 100 [7,5,2,2] => 102 [7,5,2,1,1] => 104 [7,5,1,1,1,1] => 108 [7,4,4,1] => 102 [7,4,3,2] => 105 [7,4,3,1,1] => 107 [7,4,2,2,1] => 109 [7,4,2,1,1,1] => 112 [7,4,1,1,1,1,1] => 117 [7,3,3,3] => 108 [7,3,3,2,1] => 111 [7,3,3,1,1,1] => 114 [7,3,2,2,2] => 114 [7,3,2,2,1,1] => 116 [7,3,2,1,1,1,1] => 120 [7,3,1,1,1,1,1,1] => 126 [7,2,2,2,2,1] => 120 [7,2,2,2,1,1,1] => 123 [7,2,2,1,1,1,1,1] => 128 [7,2,1,1,1,1,1,1,1] => 135 [7,1,1,1,1,1,1,1,1,1] => 144 [6,6,4] => 98 [6,6,3,1] => 102 [6,6,2,2] => 104 [6,6,2,1,1] => 106 [6,6,1,1,1,1] => 110 [6,5,5] => 100 [6,5,4,1] => 105 [6,5,3,2] => 108 [6,5,3,1,1] => 110 [6,5,2,2,1] => 112 [6,5,2,1,1,1] => 115 [6,5,1,1,1,1,1] => 120 [6,4,4,2] => 110 [6,4,4,1,1] => 112 [6,4,3,3] => 112 [6,4,3,2,1] => 115 [6,4,3,1,1,1] => 118 [6,4,2,2,2] => 118 [6,4,2,2,1,1] => 120 [6,4,2,1,1,1,1] => 124 [6,4,1,1,1,1,1,1] => 130 [6,3,3,3,1] => 118 [6,3,3,2,2] => 120 [6,3,3,2,1,1] => 122 [6,3,3,1,1,1,1] => 126 [6,3,2,2,2,1] => 125 [6,3,2,2,1,1,1] => 128 [6,3,2,1,1,1,1,1] => 133 [6,3,1,1,1,1,1,1,1] => 140 [6,2,2,2,2,2] => 130 [6,2,2,2,2,1,1] => 132 [6,2,2,2,1,1,1,1] => 136 [6,2,2,1,1,1,1,1,1] => 142 [6,2,1,1,1,1,1,1,1,1] => 150 [6,1,1,1,1,1,1,1,1,1,1] => 160 [5,5,5,1] => 108 [5,5,4,2] => 112 [5,5,4,1,1] => 114 [5,5,3,3] => 114 [5,5,3,2,1] => 117 [5,5,3,1,1,1] => 120 [5,5,2,2,2] => 120 [5,5,2,2,1,1] => 122 [5,5,2,1,1,1,1] => 126 [5,5,1,1,1,1,1,1] => 132 [5,4,4,3] => 116 [5,4,4,2,1] => 119 [5,4,4,1,1,1] => 122 [5,4,3,3,1] => 121 [5,4,3,2,2] => 123 [5,4,3,2,1,1] => 125 [5,4,3,1,1,1,1] => 129 [5,4,2,2,2,1] => 128 [5,4,2,2,1,1,1] => 131 [5,4,2,1,1,1,1,1] => 136 [5,4,1,1,1,1,1,1,1] => 143 [5,3,3,3,2] => 126 [5,3,3,3,1,1] => 128 [5,3,3,2,2,1] => 130 [5,3,3,2,1,1,1] => 133 [5,3,3,1,1,1,1,1] => 138 [5,3,2,2,2,2] => 134 [5,3,2,2,2,1,1] => 136 [5,3,2,2,1,1,1,1] => 140 [5,3,2,1,1,1,1,1,1] => 146 [5,3,1,1,1,1,1,1,1,1] => 154 [5,2,2,2,2,2,1] => 141 [5,2,2,2,2,1,1,1] => 144 [5,2,2,2,1,1,1,1,1] => 149 [5,2,2,1,1,1,1,1,1,1] => 156 [5,2,1,1,1,1,1,1,1,1,1] => 165 [5,1,1,1,1,1,1,1,1,1,1,1] => 176 [4,4,4,4] => 120 [4,4,4,3,1] => 124 [4,4,4,2,2] => 126 [4,4,4,2,1,1] => 128 [4,4,4,1,1,1,1] => 132 [4,4,3,3,2] => 128 [4,4,3,3,1,1] => 130 [4,4,3,2,2,1] => 132 [4,4,3,2,1,1,1] => 135 [4,4,3,1,1,1,1,1] => 140 [4,4,2,2,2,2] => 136 [4,4,2,2,2,1,1] => 138 [4,4,2,2,1,1,1,1] => 142 [4,4,2,1,1,1,1,1,1] => 148 [4,4,1,1,1,1,1,1,1,1] => 156 [4,3,3,3,3] => 132 [4,3,3,3,2,1] => 135 [4,3,3,3,1,1,1] => 138 [4,3,3,2,2,2] => 138 [4,3,3,2,2,1,1] => 140 [4,3,3,2,1,1,1,1] => 144 [4,3,3,1,1,1,1,1,1] => 150 [4,3,2,2,2,2,1] => 144 [4,3,2,2,2,1,1,1] => 147 [4,3,2,2,1,1,1,1,1] => 152 [4,3,2,1,1,1,1,1,1,1] => 159 [4,3,1,1,1,1,1,1,1,1,1] => 168 [4,2,2,2,2,2,2] => 150 [4,2,2,2,2,2,1,1] => 152 [4,2,2,2,2,1,1,1,1] => 156 [4,2,2,2,1,1,1,1,1,1] => 162 [4,2,2,1,1,1,1,1,1,1,1] => 170 [4,2,1,1,1,1,1,1,1,1,1,1] => 180 [4,1,1,1,1,1,1,1,1,1,1,1,1] => 192 [3,3,3,3,3,1] => 140 [3,3,3,3,2,2] => 142 [3,3,3,3,2,1,1] => 144 [3,3,3,3,1,1,1,1] => 148 [3,3,3,2,2,2,1] => 147 [3,3,3,2,2,1,1,1] => 150 [3,3,3,2,1,1,1,1,1] => 155 [3,3,3,1,1,1,1,1,1,1] => 162 [3,3,2,2,2,2,2] => 152 [3,3,2,2,2,2,1,1] => 154 [3,3,2,2,2,1,1,1,1] => 158 [3,3,2,2,1,1,1,1,1,1] => 164 [3,3,2,1,1,1,1,1,1,1,1] => 172 [3,3,1,1,1,1,1,1,1,1,1,1] => 182 [3,2,2,2,2,2,2,1] => 160 [3,2,2,2,2,2,1,1,1] => 163 [3,2,2,2,2,1,1,1,1,1] => 168 [3,2,2,2,1,1,1,1,1,1,1] => 175 [3,2,2,1,1,1,1,1,1,1,1,1] => 184 [3,2,1,1,1,1,1,1,1,1,1,1,1] => 195 [3,1,1,1,1,1,1,1,1,1,1,1,1,1] => 208 [2,2,2,2,2,2,2,2] => 168 [2,2,2,2,2,2,2,1,1] => 170 [2,2,2,2,2,2,1,1,1,1] => 174 [2,2,2,2,2,1,1,1,1,1,1] => 180 [2,2,2,2,1,1,1,1,1,1,1,1] => 188 [2,2,2,1,1,1,1,1,1,1,1,1,1] => 198 [2,2,1,1,1,1,1,1,1,1,1,1,1,1] => 210 [2,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 224 [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 240 [17] => 0 [16,1] => 17 [15,2] => 32 [15,1,1] => 34 [14,3] => 45 [14,2,1] => 48 [14,1,1,1] => 51 [13,4] => 56 [13,3,1] => 60 [13,2,2] => 62 [13,2,1,1] => 64 [13,1,1,1,1] => 68 [12,5] => 65 [12,4,1] => 70 [12,3,2] => 73 [12,3,1,1] => 75 [12,2,2,1] => 77 [12,2,1,1,1] => 80 [12,1,1,1,1,1] => 85 [11,6] => 72 [11,5,1] => 78 [11,4,2] => 82 [11,4,1,1] => 84 [11,3,3] => 84 [11,3,2,1] => 87 [11,3,1,1,1] => 90 [11,2,2,2] => 90 [11,2,2,1,1] => 92 [11,2,1,1,1,1] => 96 [11,1,1,1,1,1,1] => 102 [10,7] => 77 [10,6,1] => 84 [10,5,2] => 89 [10,5,1,1] => 91 [10,4,3] => 92 [10,4,2,1] => 95 [10,4,1,1,1] => 98 [10,3,3,1] => 97 [10,3,2,2] => 99 [10,3,2,1,1] => 101 [10,3,1,1,1,1] => 105 [10,2,2,2,1] => 104 [10,2,2,1,1,1] => 107 [10,2,1,1,1,1,1] => 112 [10,1,1,1,1,1,1,1] => 119 [9,8] => 80 [9,7,1] => 88 [9,6,2] => 94 [9,6,1,1] => 96 [9,5,3] => 98 [9,5,2,1] => 101 [9,5,1,1,1] => 104 [9,4,4] => 100 [9,4,3,1] => 104 [9,4,2,2] => 106 [9,4,2,1,1] => 108 [9,4,1,1,1,1] => 112 [9,3,3,2] => 108 [9,3,3,1,1] => 110 [9,3,2,2,1] => 112 [9,3,2,1,1,1] => 115 [9,3,1,1,1,1,1] => 120 [9,2,2,2,2] => 116 [9,2,2,2,1,1] => 118 [9,2,2,1,1,1,1] => 122 [9,2,1,1,1,1,1,1] => 128 [9,1,1,1,1,1,1,1,1] => 136 [8,8,1] => 90 [8,7,2] => 97 [8,7,1,1] => 99 [8,6,3] => 102 [8,6,2,1] => 105 [8,6,1,1,1] => 108 [8,5,4] => 105 [8,5,3,1] => 109 [8,5,2,2] => 111 [8,5,2,1,1] => 113 [8,5,1,1,1,1] => 117 [8,4,4,1] => 111 [8,4,3,2] => 114 [8,4,3,1,1] => 116 [8,4,2,2,1] => 118 [8,4,2,1,1,1] => 121 [8,4,1,1,1,1,1] => 126 [8,3,3,3] => 117 [8,3,3,2,1] => 120 [8,3,3,1,1,1] => 123 [8,3,2,2,2] => 123 [8,3,2,2,1,1] => 125 [8,3,2,1,1,1,1] => 129 [8,3,1,1,1,1,1,1] => 135 [8,2,2,2,2,1] => 129 [8,2,2,2,1,1,1] => 132 [8,2,2,1,1,1,1,1] => 137 [8,2,1,1,1,1,1,1,1] => 144 [8,1,1,1,1,1,1,1,1,1] => 153 [7,7,3] => 104 [7,7,2,1] => 107 [7,7,1,1,1] => 110 [7,6,4] => 108 [7,6,3,1] => 112 [7,6,2,2] => 114 [7,6,2,1,1] => 116 [7,6,1,1,1,1] => 120 [7,5,5] => 110 [7,5,4,1] => 115 [7,5,3,2] => 118 [7,5,3,1,1] => 120 [7,5,2,2,1] => 122 [7,5,2,1,1,1] => 125 [7,5,1,1,1,1,1] => 130 [7,4,4,2] => 120 [7,4,4,1,1] => 122 [7,4,3,3] => 122 [7,4,3,2,1] => 125 [7,4,3,1,1,1] => 128 [7,4,2,2,2] => 128 [7,4,2,2,1,1] => 130 [7,4,2,1,1,1,1] => 134 [7,4,1,1,1,1,1,1] => 140 [7,3,3,3,1] => 128 [7,3,3,2,2] => 130 [7,3,3,2,1,1] => 132 [7,3,3,1,1,1,1] => 136 [7,3,2,2,2,1] => 135 [7,3,2,2,1,1,1] => 138 [7,3,2,1,1,1,1,1] => 143 [7,3,1,1,1,1,1,1,1] => 150 [7,2,2,2,2,2] => 140 [7,2,2,2,2,1,1] => 142 [7,2,2,2,1,1,1,1] => 146 [7,2,2,1,1,1,1,1,1] => 152 [7,2,1,1,1,1,1,1,1,1] => 160 [7,1,1,1,1,1,1,1,1,1,1] => 170 [6,6,5] => 112 [6,6,4,1] => 117 [6,6,3,2] => 120 [6,6,3,1,1] => 122 [6,6,2,2,1] => 124 [6,6,2,1,1,1] => 127 [6,6,1,1,1,1,1] => 132 [6,5,5,1] => 119 [6,5,4,2] => 123 [6,5,4,1,1] => 125 [6,5,3,3] => 125 [6,5,3,2,1] => 128 [6,5,3,1,1,1] => 131 [6,5,2,2,2] => 131 [6,5,2,2,1,1] => 133 [6,5,2,1,1,1,1] => 137 [6,5,1,1,1,1,1,1] => 143 [6,4,4,3] => 127 [6,4,4,2,1] => 130 [6,4,4,1,1,1] => 133 [6,4,3,3,1] => 132 [6,4,3,2,2] => 134 [6,4,3,2,1,1] => 136 [6,4,3,1,1,1,1] => 140 [6,4,2,2,2,1] => 139 [6,4,2,2,1,1,1] => 142 [6,4,2,1,1,1,1,1] => 147 [6,4,1,1,1,1,1,1,1] => 154 [6,3,3,3,2] => 137 [6,3,3,3,1,1] => 139 [6,3,3,2,2,1] => 141 [6,3,3,2,1,1,1] => 144 [6,3,3,1,1,1,1,1] => 149 [6,3,2,2,2,2] => 145 [6,3,2,2,2,1,1] => 147 [6,3,2,2,1,1,1,1] => 151 [6,3,2,1,1,1,1,1,1] => 157 [6,3,1,1,1,1,1,1,1,1] => 165 [6,2,2,2,2,2,1] => 152 [6,2,2,2,2,1,1,1] => 155 [6,2,2,2,1,1,1,1,1] => 160 [6,2,2,1,1,1,1,1,1,1] => 167 [6,2,1,1,1,1,1,1,1,1,1] => 176 [6,1,1,1,1,1,1,1,1,1,1,1] => 187 [5,5,5,2] => 126 [5,5,5,1,1] => 128 [5,5,4,3] => 129 [5,5,4,2,1] => 132 [5,5,4,1,1,1] => 135 [5,5,3,3,1] => 134 [5,5,3,2,2] => 136 [5,5,3,2,1,1] => 138 [5,5,3,1,1,1,1] => 142 [5,5,2,2,2,1] => 141 [5,5,2,2,1,1,1] => 144 [5,5,2,1,1,1,1,1] => 149 [5,5,1,1,1,1,1,1,1] => 156 [5,4,4,4] => 132 [5,4,4,3,1] => 136 [5,4,4,2,2] => 138 [5,4,4,2,1,1] => 140 [5,4,4,1,1,1,1] => 144 [5,4,3,3,2] => 140 [5,4,3,3,1,1] => 142 [5,4,3,2,2,1] => 144 [5,4,3,2,1,1,1] => 147 [5,4,3,1,1,1,1,1] => 152 [5,4,2,2,2,2] => 148 [5,4,2,2,2,1,1] => 150 [5,4,2,2,1,1,1,1] => 154 [5,4,2,1,1,1,1,1,1] => 160 [5,4,1,1,1,1,1,1,1,1] => 168 [5,3,3,3,3] => 144 [5,3,3,3,2,1] => 147 [5,3,3,3,1,1,1] => 150 [5,3,3,2,2,2] => 150 [5,3,3,2,2,1,1] => 152 [5,3,3,2,1,1,1,1] => 156 [5,3,3,1,1,1,1,1,1] => 162 [5,3,2,2,2,2,1] => 156 [5,3,2,2,2,1,1,1] => 159 [5,3,2,2,1,1,1,1,1] => 164 [5,3,2,1,1,1,1,1,1,1] => 171 [5,3,1,1,1,1,1,1,1,1,1] => 180 [5,2,2,2,2,2,2] => 162 [5,2,2,2,2,2,1,1] => 164 [5,2,2,2,2,1,1,1,1] => 168 [5,2,2,2,1,1,1,1,1,1] => 174 [5,2,2,1,1,1,1,1,1,1,1] => 182 [5,2,1,1,1,1,1,1,1,1,1,1] => 192 [5,1,1,1,1,1,1,1,1,1,1,1,1] => 204 [4,4,4,4,1] => 140 [4,4,4,3,2] => 143 [4,4,4,3,1,1] => 145 [4,4,4,2,2,1] => 147 [4,4,4,2,1,1,1] => 150 [4,4,4,1,1,1,1,1] => 155 [4,4,3,3,3] => 146 [4,4,3,3,2,1] => 149 [4,4,3,3,1,1,1] => 152 [4,4,3,2,2,2] => 152 [4,4,3,2,2,1,1] => 154 [4,4,3,2,1,1,1,1] => 158 [4,4,3,1,1,1,1,1,1] => 164 [4,4,2,2,2,2,1] => 158 [4,4,2,2,2,1,1,1] => 161 [4,4,2,2,1,1,1,1,1] => 166 [4,4,2,1,1,1,1,1,1,1] => 173 [4,4,1,1,1,1,1,1,1,1,1] => 182 [4,3,3,3,3,1] => 153 [4,3,3,3,2,2] => 155 [4,3,3,3,2,1,1] => 157 [4,3,3,3,1,1,1,1] => 161 [4,3,3,2,2,2,1] => 160 [4,3,3,2,2,1,1,1] => 163 [4,3,3,2,1,1,1,1,1] => 168 [4,3,3,1,1,1,1,1,1,1] => 175 [4,3,2,2,2,2,2] => 165 [4,3,2,2,2,2,1,1] => 167 [4,3,2,2,2,1,1,1,1] => 171 [4,3,2,2,1,1,1,1,1,1] => 177 [4,3,2,1,1,1,1,1,1,1,1] => 185 [4,3,1,1,1,1,1,1,1,1,1,1] => 195 [4,2,2,2,2,2,2,1] => 173 [4,2,2,2,2,2,1,1,1] => 176 [4,2,2,2,2,1,1,1,1,1] => 181 [4,2,2,2,1,1,1,1,1,1,1] => 188 [4,2,2,1,1,1,1,1,1,1,1,1] => 197 [4,2,1,1,1,1,1,1,1,1,1,1,1] => 208 [4,1,1,1,1,1,1,1,1,1,1,1,1,1] => 221 [3,3,3,3,3,2] => 160 [3,3,3,3,3,1,1] => 162 [3,3,3,3,2,2,1] => 164 [3,3,3,3,2,1,1,1] => 167 [3,3,3,3,1,1,1,1,1] => 172 [3,3,3,2,2,2,2] => 168 [3,3,3,2,2,2,1,1] => 170 [3,3,3,2,2,1,1,1,1] => 174 [3,3,3,2,1,1,1,1,1,1] => 180 [3,3,3,1,1,1,1,1,1,1,1] => 188 [3,3,2,2,2,2,2,1] => 175 [3,3,2,2,2,2,1,1,1] => 178 [3,3,2,2,2,1,1,1,1,1] => 183 [3,3,2,2,1,1,1,1,1,1,1] => 190 [3,3,2,1,1,1,1,1,1,1,1,1] => 199 [3,3,1,1,1,1,1,1,1,1,1,1,1] => 210 [3,2,2,2,2,2,2,2] => 182 [3,2,2,2,2,2,2,1,1] => 184 [3,2,2,2,2,2,1,1,1,1] => 188 [3,2,2,2,2,1,1,1,1,1,1] => 194 [3,2,2,2,1,1,1,1,1,1,1,1] => 202 [3,2,2,1,1,1,1,1,1,1,1,1,1] => 212 ----------------------------------------------------------------------------- Created: May 07, 2018 at 20:47 by Christian Stump ----------------------------------------------------------------------------- Last Updated: May 07, 2018 at 20:47 by Christian Stump