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* Copyright (C) 2019 The FindStatCrew <info@findstat.org> *
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* 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. *
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-----------------------------------------------------------------------------
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