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-----------------------------------------------------------------------------
Statistic identifier: St001593
-----------------------------------------------------------------------------
Collection: Integer partitions
-----------------------------------------------------------------------------
Description: This is the number of standard Young tableaux of the given shifted shape.
For an integer partition $\lambda = (\lambda_1,\dots,\lambda_k)$, the shifted diagram is obtained by moving the $i$-th row in the diagram $i-1$ boxes to the right, i.e.,
$$\lambda^∗ = \{(i, j) | 1 \leq i \leq k, i \leq j \leq \lambda_i + i − 1 \}.$$
In particular, this statistic is zero if and only if $\lambda_{i+1} = \lambda_i$ for some $i$.
-----------------------------------------------------------------------------
References: [1] Cho, Y., Kim, J. S., Lee, E. Enumeration of Gelfand-Cetlin type reduced words [[arXiv:2009.06906]]
-----------------------------------------------------------------------------
Code:
def statistic(L):
return multinomial(list(L)) * prod( ZZ(L[i] - L[j]) / ZZ(L[i] + L[j]) for j in range(len(L)) for i in range(j))
-----------------------------------------------------------------------------
Statistic values:
[1] => 1
[2] => 1
[1,1] => 0
[3] => 1
[2,1] => 1
[1,1,1] => 0
[4] => 1
[3,1] => 2
[2,2] => 0
[2,1,1] => 0
[1,1,1,1] => 0
[5] => 1
[4,1] => 3
[3,2] => 2
[3,1,1] => 0
[2,2,1] => 0
[2,1,1,1] => 0
[1,1,1,1,1] => 0
[6] => 1
[5,1] => 4
[4,2] => 5
[4,1,1] => 0
[3,3] => 0
[3,2,1] => 2
[3,1,1,1] => 0
[2,2,2] => 0
[2,2,1,1] => 0
[2,1,1,1,1] => 0
[1,1,1,1,1,1] => 0
[7] => 1
[6,1] => 5
[5,2] => 9
[5,1,1] => 0
[4,3] => 5
[4,2,1] => 7
[4,1,1,1] => 0
[3,3,1] => 0
[3,2,2] => 0
[3,2,1,1] => 0
[3,1,1,1,1] => 0
[2,2,2,1] => 0
[2,2,1,1,1] => 0
[2,1,1,1,1,1] => 0
[1,1,1,1,1,1,1] => 0
[8] => 1
[7,1] => 6
[6,2] => 14
[6,1,1] => 0
[5,3] => 14
[5,2,1] => 16
[5,1,1,1] => 0
[4,4] => 0
[4,3,1] => 12
[4,2,2] => 0
[4,2,1,1] => 0
[4,1,1,1,1] => 0
[3,3,2] => 0
[3,3,1,1] => 0
[3,2,2,1] => 0
[3,2,1,1,1] => 0
[3,1,1,1,1,1] => 0
[2,2,2,2] => 0
[2,2,2,1,1] => 0
[2,2,1,1,1,1] => 0
[2,1,1,1,1,1,1] => 0
[1,1,1,1,1,1,1,1] => 0
[9] => 1
[8,1] => 7
[7,2] => 20
[7,1,1] => 0
[6,3] => 28
[6,2,1] => 30
[6,1,1,1] => 0
[5,4] => 14
[5,3,1] => 42
[5,2,2] => 0
[5,2,1,1] => 0
[5,1,1,1,1] => 0
[4,4,1] => 0
[4,3,2] => 12
[4,3,1,1] => 0
[4,2,2,1] => 0
[4,2,1,1,1] => 0
[4,1,1,1,1,1] => 0
[3,3,3] => 0
[3,3,2,1] => 0
[3,3,1,1,1] => 0
[3,2,2,2] => 0
[3,2,2,1,1] => 0
[3,2,1,1,1,1] => 0
[3,1,1,1,1,1,1] => 0
[2,2,2,2,1] => 0
[2,2,2,1,1,1] => 0
[2,2,1,1,1,1,1] => 0
[2,1,1,1,1,1,1,1] => 0
[1,1,1,1,1,1,1,1,1] => 0
[10] => 1
[9,1] => 8
[8,2] => 27
[8,1,1] => 0
[7,3] => 48
[7,2,1] => 50
[7,1,1,1] => 0
[6,4] => 42
[6,3,1] => 100
[6,2,2] => 0
[6,2,1,1] => 0
[6,1,1,1,1] => 0
[5,5] => 0
[5,4,1] => 56
[5,3,2] => 54
[5,3,1,1] => 0
[5,2,2,1] => 0
[5,2,1,1,1] => 0
[5,1,1,1,1,1] => 0
[4,4,2] => 0
[4,4,1,1] => 0
[4,3,3] => 0
[4,3,2,1] => 12
[4,3,1,1,1] => 0
[4,2,2,2] => 0
[4,2,2,1,1] => 0
[4,2,1,1,1,1] => 0
[4,1,1,1,1,1,1] => 0
[3,3,3,1] => 0
[3,3,2,2] => 0
[3,3,2,1,1] => 0
[3,3,1,1,1,1] => 0
[3,2,2,2,1] => 0
[3,2,2,1,1,1] => 0
[3,2,1,1,1,1,1] => 0
[3,1,1,1,1,1,1,1] => 0
[2,2,2,2,2] => 0
[2,2,2,2,1,1] => 0
[2,2,2,1,1,1,1] => 0
[2,2,1,1,1,1,1,1] => 0
[2,1,1,1,1,1,1,1,1] => 0
[1,1,1,1,1,1,1,1,1,1] => 0
[11] => 1
[10,1] => 9
[9,2] => 35
[9,1,1] => 0
[8,3] => 75
[8,2,1] => 77
[8,1,1,1] => 0
[7,4] => 90
[7,3,1] => 198
[7,2,2] => 0
[7,2,1,1] => 0
[7,1,1,1,1] => 0
[6,5] => 42
[6,4,1] => 198
[6,3,2] => 154
[6,3,1,1] => 0
[6,2,2,1] => 0
[6,2,1,1,1] => 0
[6,1,1,1,1,1] => 0
[5,5,1] => 0
[5,4,2] => 110
[5,4,1,1] => 0
[5,3,3] => 0
[5,3,2,1] => 66
[5,3,1,1,1] => 0
[5,2,2,2] => 0
[5,2,2,1,1] => 0
[5,2,1,1,1,1] => 0
[5,1,1,1,1,1,1] => 0
[4,4,3] => 0
[4,4,2,1] => 0
[4,4,1,1,1] => 0
[4,3,3,1] => 0
[4,3,2,2] => 0
[4,3,2,1,1] => 0
[4,3,1,1,1,1] => 0
[4,2,2,2,1] => 0
[4,2,2,1,1,1] => 0
[4,2,1,1,1,1,1] => 0
[4,1,1,1,1,1,1,1] => 0
[3,3,3,2] => 0
[3,3,3,1,1] => 0
[3,3,2,2,1] => 0
[3,3,2,1,1,1] => 0
[3,3,1,1,1,1,1] => 0
[3,2,2,2,2] => 0
[3,2,2,2,1,1] => 0
[3,2,2,1,1,1,1] => 0
[3,2,1,1,1,1,1,1] => 0
[3,1,1,1,1,1,1,1,1] => 0
[2,2,2,2,2,1] => 0
[2,2,2,2,1,1,1] => 0
[2,2,2,1,1,1,1,1] => 0
[2,2,1,1,1,1,1,1,1] => 0
[2,1,1,1,1,1,1,1,1,1] => 0
[1,1,1,1,1,1,1,1,1,1,1] => 0
[12] => 1
[11,1] => 10
[10,2] => 44
[10,1,1] => 0
[9,3] => 110
[9,2,1] => 112
[9,1,1,1] => 0
[8,4] => 165
[8,3,1] => 350
[8,2,2] => 0
[8,2,1,1] => 0
[8,1,1,1,1] => 0
[7,5] => 132
[7,4,1] => 486
[7,3,2] => 352
[7,3,1,1] => 0
[7,2,2,1] => 0
[7,2,1,1,1] => 0
[7,1,1,1,1,1] => 0
[6,6] => 0
[6,5,1] => 240
[6,4,2] => 462
[6,4,1,1] => 0
[6,3,3] => 0
[6,3,2,1] => 220
[6,3,1,1,1] => 0
[6,2,2,2] => 0
[6,2,2,1,1] => 0
[6,2,1,1,1,1] => 0
[6,1,1,1,1,1,1] => 0
[5,5,2] => 0
[5,5,1,1] => 0
[5,4,3] => 110
[5,4,2,1] => 176
[5,4,1,1,1] => 0
[5,3,3,1] => 0
[5,3,2,2] => 0
[5,3,2,1,1] => 0
[5,3,1,1,1,1] => 0
[5,2,2,2,1] => 0
[5,2,2,1,1,1] => 0
[5,2,1,1,1,1,1] => 0
[5,1,1,1,1,1,1,1] => 0
[4,4,4] => 0
[4,4,3,1] => 0
[4,4,2,2] => 0
[4,4,2,1,1] => 0
[4,4,1,1,1,1] => 0
[4,3,3,2] => 0
[4,3,3,1,1] => 0
[4,3,2,2,1] => 0
[4,3,2,1,1,1] => 0
[4,3,1,1,1,1,1] => 0
[4,2,2,2,2] => 0
[4,2,2,2,1,1] => 0
[4,2,2,1,1,1,1] => 0
[4,2,1,1,1,1,1,1] => 0
[4,1,1,1,1,1,1,1,1] => 0
[3,3,3,3] => 0
[3,3,3,2,1] => 0
[3,3,3,1,1,1] => 0
[3,3,2,2,2] => 0
[3,3,2,2,1,1] => 0
[3,3,2,1,1,1,1] => 0
[3,3,1,1,1,1,1,1] => 0
[3,2,2,2,2,1] => 0
[3,2,2,2,1,1,1] => 0
[3,2,2,1,1,1,1,1] => 0
[3,2,1,1,1,1,1,1,1] => 0
[3,1,1,1,1,1,1,1,1,1] => 0
[2,2,2,2,2,2] => 0
[2,2,2,2,2,1,1] => 0
[2,2,2,2,1,1,1,1] => 0
[2,2,2,1,1,1,1,1,1] => 0
[2,2,1,1,1,1,1,1,1,1] => 0
[2,1,1,1,1,1,1,1,1,1,1] => 0
[1,1,1,1,1,1,1,1,1,1,1,1] => 0
[13] => 1
[12,1] => 11
[11,2] => 54
[11,1,1] => 0
[10,3] => 154
[10,2,1] => 156
[10,1,1,1] => 0
[9,4] => 275
[9,3,1] => 572
[9,2,2] => 0
[9,2,1,1] => 0
[9,1,1,1,1] => 0
[8,5] => 297
[8,4,1] => 1001
[8,3,2] => 702
[8,3,1,1] => 0
[8,2,2,1] => 0
[8,2,1,1,1] => 0
[8,1,1,1,1,1] => 0
[7,6] => 132
[7,5,1] => 858
[7,4,2] => 1300
[7,4,1,1] => 0
[7,3,3] => 0
[7,3,2,1] => 572
[7,3,1,1,1] => 0
[7,2,2,2] => 0
[7,2,2,1,1] => 0
[7,2,1,1,1,1] => 0
[7,1,1,1,1,1,1] => 0
[6,6,1] => 0
[6,5,2] => 702
[6,5,1,1] => 0
[6,4,3] => 572
[6,4,2,1] => 858
[6,4,1,1,1] => 0
[6,3,3,1] => 0
[6,3,2,2] => 0
[6,3,2,1,1] => 0
[6,3,1,1,1,1] => 0
[6,2,2,2,1] => 0
[6,2,2,1,1,1] => 0
[6,2,1,1,1,1,1] => 0
[6,1,1,1,1,1,1,1] => 0
[5,5,3] => 0
[5,5,2,1] => 0
[5,5,1,1,1] => 0
[5,4,4] => 0
[5,4,3,1] => 286
[5,4,2,2] => 0
[5,4,2,1,1] => 0
[5,4,1,1,1,1] => 0
[5,3,3,2] => 0
[5,3,3,1,1] => 0
[5,3,2,2,1] => 0
[5,3,2,1,1,1] => 0
[5,3,1,1,1,1,1] => 0
[5,2,2,2,2] => 0
[5,2,2,2,1,1] => 0
[5,2,2,1,1,1,1] => 0
[5,2,1,1,1,1,1,1] => 0
[5,1,1,1,1,1,1,1,1] => 0
[4,4,4,1] => 0
[4,4,3,2] => 0
[4,4,3,1,1] => 0
[4,4,2,2,1] => 0
[4,4,2,1,1,1] => 0
[4,4,1,1,1,1,1] => 0
[4,3,3,3] => 0
[4,3,3,2,1] => 0
[4,3,3,1,1,1] => 0
[4,3,2,2,2] => 0
[4,3,2,2,1,1] => 0
[4,3,2,1,1,1,1] => 0
[4,3,1,1,1,1,1,1] => 0
[4,2,2,2,2,1] => 0
[4,2,2,2,1,1,1] => 0
[4,2,2,1,1,1,1,1] => 0
[4,2,1,1,1,1,1,1,1] => 0
[4,1,1,1,1,1,1,1,1,1] => 0
[3,3,3,3,1] => 0
[3,3,3,2,2] => 0
[3,3,3,2,1,1] => 0
[3,3,3,1,1,1,1] => 0
[3,3,2,2,2,1] => 0
[3,3,2,2,1,1,1] => 0
[3,3,2,1,1,1,1,1] => 0
[3,3,1,1,1,1,1,1,1] => 0
[3,2,2,2,2,2] => 0
[3,2,2,2,2,1,1] => 0
[3,2,2,2,1,1,1,1] => 0
[3,2,2,1,1,1,1,1,1] => 0
[3,2,1,1,1,1,1,1,1,1] => 0
[3,1,1,1,1,1,1,1,1,1,1] => 0
[2,2,2,2,2,2,1] => 0
[2,2,2,2,2,1,1,1] => 0
[2,2,2,2,1,1,1,1,1] => 0
[2,2,2,1,1,1,1,1,1,1] => 0
[2,2,1,1,1,1,1,1,1,1,1] => 0
[2,1,1,1,1,1,1,1,1,1,1,1] => 0
[1,1,1,1,1,1,1,1,1,1,1,1,1] => 0
[14] => 1
[13,1] => 12
[12,2] => 65
[12,1,1] => 0
[11,3] => 208
[11,2,1] => 210
[11,1,1,1] => 0
[10,4] => 429
[10,3,1] => 882
[10,2,2] => 0
[10,2,1,1] => 0
[10,1,1,1,1] => 0
[9,5] => 572
[9,4,1] => 1848
[9,3,2] => 1274
[9,3,1,1] => 0
[9,2,2,1] => 0
[9,2,1,1,1] => 0
[9,1,1,1,1,1] => 0
[8,6] => 429
[8,5,1] => 2156
[8,4,2] => 3003
[8,4,1,1] => 0
[8,3,3] => 0
[8,3,2,1] => 1274
[8,3,1,1,1] => 0
[8,2,2,2] => 0
[8,2,2,1,1] => 0
[8,2,1,1,1,1] => 0
[8,1,1,1,1,1,1] => 0
[7,7] => 0
[7,6,1] => 990
[7,5,2] => 2860
[7,5,1,1] => 0
[7,4,3] => 1872
[7,4,2,1] => 2730
[7,4,1,1,1] => 0
[7,3,3,1] => 0
[7,3,2,2] => 0
[7,3,2,1,1] => 0
[7,3,1,1,1,1] => 0
[7,2,2,2,1] => 0
[7,2,2,1,1,1] => 0
[7,2,1,1,1,1,1] => 0
[7,1,1,1,1,1,1,1] => 0
[6,6,2] => 0
[6,6,1,1] => 0
[6,5,3] => 1274
[6,5,2,1] => 1560
[6,5,1,1,1] => 0
[6,4,4] => 0
[6,4,3,1] => 1716
[6,4,2,2] => 0
[6,4,2,1,1] => 0
[6,4,1,1,1,1] => 0
[6,3,3,2] => 0
[6,3,3,1,1] => 0
[6,3,2,2,1] => 0
[6,3,2,1,1,1] => 0
[6,3,1,1,1,1,1] => 0
[6,2,2,2,2] => 0
[6,2,2,2,1,1] => 0
[6,2,2,1,1,1,1] => 0
[6,2,1,1,1,1,1,1] => 0
[6,1,1,1,1,1,1,1,1] => 0
[5,5,4] => 0
[5,5,3,1] => 0
[5,5,2,2] => 0
[5,5,2,1,1] => 0
[5,5,1,1,1,1] => 0
[5,4,4,1] => 0
[5,4,3,2] => 286
[5,4,3,1,1] => 0
[5,4,2,2,1] => 0
[5,4,2,1,1,1] => 0
[5,4,1,1,1,1,1] => 0
[5,3,3,3] => 0
[5,3,3,2,1] => 0
[5,3,3,1,1,1] => 0
[5,3,2,2,2] => 0
[5,3,2,2,1,1] => 0
[5,3,2,1,1,1,1] => 0
[5,3,1,1,1,1,1,1] => 0
[5,2,2,2,2,1] => 0
[5,2,2,2,1,1,1] => 0
[5,2,2,1,1,1,1,1] => 0
[5,2,1,1,1,1,1,1,1] => 0
[5,1,1,1,1,1,1,1,1,1] => 0
[4,4,4,2] => 0
[4,4,4,1,1] => 0
[4,4,3,3] => 0
[4,4,3,2,1] => 0
[4,4,3,1,1,1] => 0
[4,4,2,2,2] => 0
[4,4,2,2,1,1] => 0
[4,4,2,1,1,1,1] => 0
[4,4,1,1,1,1,1,1] => 0
[4,3,3,3,1] => 0
[4,3,3,2,2] => 0
[4,3,3,2,1,1] => 0
[4,3,3,1,1,1,1] => 0
[4,3,2,2,2,1] => 0
[4,3,2,2,1,1,1] => 0
[4,3,2,1,1,1,1,1] => 0
[4,3,1,1,1,1,1,1,1] => 0
[4,2,2,2,2,2] => 0
[4,2,2,2,2,1,1] => 0
[4,2,2,2,1,1,1,1] => 0
[4,2,2,1,1,1,1,1,1] => 0
[4,2,1,1,1,1,1,1,1,1] => 0
[4,1,1,1,1,1,1,1,1,1,1] => 0
[3,3,3,3,2] => 0
[3,3,3,3,1,1] => 0
[3,3,3,2,2,1] => 0
[3,3,3,2,1,1,1] => 0
[3,3,3,1,1,1,1,1] => 0
[3,3,2,2,2,2] => 0
[3,3,2,2,2,1,1] => 0
[3,3,2,2,1,1,1,1] => 0
[3,3,2,1,1,1,1,1,1] => 0
[3,3,1,1,1,1,1,1,1,1] => 0
[3,2,2,2,2,2,1] => 0
[3,2,2,2,2,1,1,1] => 0
[3,2,2,2,1,1,1,1,1] => 0
[3,2,2,1,1,1,1,1,1,1] => 0
[3,2,1,1,1,1,1,1,1,1,1] => 0
[3,1,1,1,1,1,1,1,1,1,1,1] => 0
[2,2,2,2,2,2,2] => 0
[2,2,2,2,2,2,1,1] => 0
[2,2,2,2,2,1,1,1,1] => 0
[2,2,2,2,1,1,1,1,1,1] => 0
[2,2,2,1,1,1,1,1,1,1,1] => 0
[2,2,1,1,1,1,1,1,1,1,1,1] => 0
[2,1,1,1,1,1,1,1,1,1,1,1,1] => 0
[1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 0
[15] => 1
[14,1] => 13
[13,2] => 77
[13,1,1] => 0
[12,3] => 273
[12,2,1] => 275
[12,1,1,1] => 0
[11,4] => 637
[11,3,1] => 1300
[11,2,2] => 0
[11,2,1,1] => 0
[11,1,1,1,1] => 0
[10,5] => 1001
[10,4,1] => 3159
[10,3,2] => 2156
[10,3,1,1] => 0
[10,2,2,1] => 0
[10,2,1,1,1] => 0
[10,1,1,1,1,1] => 0
[9,6] => 1001
[9,5,1] => 4576
[9,4,2] => 6125
[9,4,1,1] => 0
[9,3,3] => 0
[9,3,2,1] => 2548
[9,3,1,1,1] => 0
[9,2,2,2] => 0
[9,2,2,1,1] => 0
[9,2,1,1,1,1] => 0
[9,1,1,1,1,1,1] => 0
[8,7] => 429
[8,6,1] => 3575
[8,5,2] => 8019
[8,5,1,1] => 0
[8,4,3] => 4875
[8,4,2,1] => 7007
[8,4,1,1,1] => 0
[8,3,3,1] => 0
[8,3,2,2] => 0
[8,3,2,1,1] => 0
[8,3,1,1,1,1] => 0
[8,2,2,2,1] => 0
[8,2,2,1,1,1] => 0
[8,2,1,1,1,1,1] => 0
[8,1,1,1,1,1,1,1] => 0
[7,7,1] => 0
[7,6,2] => 3850
[7,6,1,1] => 0
[7,5,3] => 6006
[7,5,2,1] => 7150
[7,5,1,1,1] => 0
[7,4,4] => 0
[7,4,3,1] => 6318
[7,4,2,2] => 0
[7,4,2,1,1] => 0
[7,4,1,1,1,1] => 0
[7,3,3,2] => 0
[7,3,3,1,1] => 0
[7,3,2,2,1] => 0
[7,3,2,1,1,1] => 0
[7,3,1,1,1,1,1] => 0
[7,2,2,2,2] => 0
[7,2,2,2,1,1] => 0
[7,2,2,1,1,1,1] => 0
[7,2,1,1,1,1,1,1] => 0
[7,1,1,1,1,1,1,1,1] => 0
[6,6,3] => 0
[6,6,2,1] => 0
[6,6,1,1,1] => 0
[6,5,4] => 1274
[6,5,3,1] => 4550
[6,5,2,2] => 0
[6,5,2,1,1] => 0
[6,5,1,1,1,1] => 0
[6,4,4,1] => 0
[6,4,3,2] => 2002
[6,4,3,1,1] => 0
[6,4,2,2,1] => 0
[6,4,2,1,1,1] => 0
[6,4,1,1,1,1,1] => 0
[6,3,3,3] => 0
[6,3,3,2,1] => 0
[6,3,3,1,1,1] => 0
[6,3,2,2,2] => 0
[6,3,2,2,1,1] => 0
[6,3,2,1,1,1,1] => 0
[6,3,1,1,1,1,1,1] => 0
[6,2,2,2,2,1] => 0
[6,2,2,2,1,1,1] => 0
[6,2,2,1,1,1,1,1] => 0
[6,2,1,1,1,1,1,1,1] => 0
[6,1,1,1,1,1,1,1,1,1] => 0
[5,5,5] => 0
[5,5,4,1] => 0
[5,5,3,2] => 0
[5,5,3,1,1] => 0
[5,5,2,2,1] => 0
[5,5,2,1,1,1] => 0
[5,5,1,1,1,1,1] => 0
[5,4,4,2] => 0
[5,4,4,1,1] => 0
[5,4,3,3] => 0
[5,4,3,2,1] => 286
[5,4,3,1,1,1] => 0
[5,4,2,2,2] => 0
[5,4,2,2,1,1] => 0
[5,4,2,1,1,1,1] => 0
[5,4,1,1,1,1,1,1] => 0
[5,3,3,3,1] => 0
[5,3,3,2,2] => 0
[5,3,3,2,1,1] => 0
[5,3,3,1,1,1,1] => 0
[5,3,2,2,2,1] => 0
[5,3,2,2,1,1,1] => 0
[5,3,2,1,1,1,1,1] => 0
[5,3,1,1,1,1,1,1,1] => 0
[5,2,2,2,2,2] => 0
[5,2,2,2,2,1,1] => 0
[5,2,2,2,1,1,1,1] => 0
[5,2,2,1,1,1,1,1,1] => 0
[5,2,1,1,1,1,1,1,1,1] => 0
[5,1,1,1,1,1,1,1,1,1,1] => 0
[4,4,4,3] => 0
[4,4,4,2,1] => 0
[4,4,4,1,1,1] => 0
[4,4,3,3,1] => 0
[4,4,3,2,2] => 0
[4,4,3,2,1,1] => 0
[4,4,3,1,1,1,1] => 0
[4,4,2,2,2,1] => 0
[4,4,2,2,1,1,1] => 0
[4,4,2,1,1,1,1,1] => 0
[4,4,1,1,1,1,1,1,1] => 0
[4,3,3,3,2] => 0
[4,3,3,3,1,1] => 0
[4,3,3,2,2,1] => 0
[4,3,3,2,1,1,1] => 0
[4,3,3,1,1,1,1,1] => 0
[4,3,2,2,2,2] => 0
[4,3,2,2,2,1,1] => 0
[4,3,2,2,1,1,1,1] => 0
[4,3,2,1,1,1,1,1,1] => 0
[4,3,1,1,1,1,1,1,1,1] => 0
[4,2,2,2,2,2,1] => 0
[4,2,2,2,2,1,1,1] => 0
[4,2,2,2,1,1,1,1,1] => 0
[4,2,2,1,1,1,1,1,1,1] => 0
[4,2,1,1,1,1,1,1,1,1,1] => 0
[4,1,1,1,1,1,1,1,1,1,1,1] => 0
[3,3,3,3,3] => 0
[3,3,3,3,2,1] => 0
[3,3,3,3,1,1,1] => 0
[3,3,3,2,2,2] => 0
[3,3,3,2,2,1,1] => 0
[3,3,3,2,1,1,1,1] => 0
[3,3,3,1,1,1,1,1,1] => 0
[3,3,2,2,2,2,1] => 0
[3,3,2,2,2,1,1,1] => 0
[3,3,2,2,1,1,1,1,1] => 0
[3,3,2,1,1,1,1,1,1,1] => 0
[3,3,1,1,1,1,1,1,1,1,1] => 0
[3,2,2,2,2,2,2] => 0
[3,2,2,2,2,2,1,1] => 0
[3,2,2,2,2,1,1,1,1] => 0
[3,2,2,2,1,1,1,1,1,1] => 0
[3,2,2,1,1,1,1,1,1,1,1] => 0
[3,2,1,1,1,1,1,1,1,1,1,1] => 0
[3,1,1,1,1,1,1,1,1,1,1,1,1] => 0
[2,2,2,2,2,2,2,1] => 0
[2,2,2,2,2,2,1,1,1] => 0
[2,2,2,2,2,1,1,1,1,1] => 0
[2,2,2,2,1,1,1,1,1,1,1] => 0
[2,2,2,1,1,1,1,1,1,1,1,1] => 0
[2,2,1,1,1,1,1,1,1,1,1,1,1] => 0
[2,1,1,1,1,1,1,1,1,1,1,1,1,1] => 0
[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 0
[16] => 1
[15,1] => 14
[14,2] => 90
[14,1,1] => 0
[13,3] => 350
[13,2,1] => 352
[13,1,1,1] => 0
[12,4] => 910
[12,3,1] => 1848
[12,2,2] => 0
[12,2,1,1] => 0
[12,1,1,1,1] => 0
[11,5] => 1638
[11,4,1] => 5096
[11,3,2] => 3456
[11,3,1,1] => 0
[11,2,2,1] => 0
[11,2,1,1,1] => 0
[11,1,1,1,1,1] => 0
[10,6] => 2002
[10,5,1] => 8736
[10,4,2] => 11440
[10,4,1,1] => 0
[10,3,3] => 0
[10,3,2,1] => 4704
[10,3,1,1,1] => 0
[10,2,2,2] => 0
[10,2,2,1,1] => 0
[10,2,1,1,1,1] => 0
[10,1,1,1,1,1,1] => 0
[9,7] => 1430
[9,6,1] => 9152
[9,5,2] => 18720
[9,5,1,1] => 0
[9,4,3] => 11000
[9,4,2,1] => 15680
[9,4,1,1,1] => 0
[9,3,3,1] => 0
[9,3,2,2] => 0
[9,3,2,1,1] => 0
[9,3,1,1,1,1] => 0
[9,2,2,2,1] => 0
[9,2,2,1,1,1] => 0
[9,2,1,1,1,1,1] => 0
[9,1,1,1,1,1,1,1] => 0
[8,8] => 0
[8,7,1] => 4004
[8,6,2] => 15444
[8,6,1,1] => 0
[8,5,3] => 18900
[8,5,2,1] => 22176
[8,5,1,1,1] => 0
[8,4,4] => 0
[8,4,3,1] => 18200
[8,4,2,2] => 0
[8,4,2,1,1] => 0
[8,4,1,1,1,1] => 0
[8,3,3,2] => 0
[8,3,3,1,1] => 0
[8,3,2,2,1] => 0
[8,3,2,1,1,1] => 0
[8,3,1,1,1,1,1] => 0
[8,2,2,2,2] => 0
[8,2,2,2,1,1] => 0
[8,2,2,1,1,1,1] => 0
[8,2,1,1,1,1,1,1] => 0
[8,1,1,1,1,1,1,1,1] => 0
[7,7,2] => 0
[7,7,1,1] => 0
[7,6,3] => 9856
[7,6,2,1] => 11000
[7,6,1,1,1] => 0
[7,5,4] => 7280
[7,5,3,1] => 24024
[7,5,2,2] => 0
[7,5,2,1,1] => 0
[7,5,1,1,1,1] => 0
[7,4,4,1] => 0
[7,4,3,2] => 8320
[7,4,3,1,1] => 0
[7,4,2,2,1] => 0
[7,4,2,1,1,1] => 0
[7,4,1,1,1,1,1] => 0
[7,3,3,3] => 0
[7,3,3,2,1] => 0
[7,3,3,1,1,1] => 0
[7,3,2,2,2] => 0
[7,3,2,2,1,1] => 0
[7,3,2,1,1,1,1] => 0
[7,3,1,1,1,1,1,1] => 0
[7,2,2,2,2,1] => 0
[7,2,2,2,1,1,1] => 0
[7,2,2,1,1,1,1,1] => 0
[7,2,1,1,1,1,1,1,1] => 0
[7,1,1,1,1,1,1,1,1,1] => 0
[6,6,4] => 0
[6,6,3,1] => 0
[6,6,2,2] => 0
[6,6,2,1,1] => 0
[6,6,1,1,1,1] => 0
[6,5,5] => 0
[6,5,4,1] => 5824
[6,5,3,2] => 6552
[6,5,3,1,1] => 0
[6,5,2,2,1] => 0
[6,5,2,1,1,1] => 0
[6,5,1,1,1,1,1] => 0
[6,4,4,2] => 0
[6,4,4,1,1] => 0
[6,4,3,3] => 0
[6,4,3,2,1] => 2288
[6,4,3,1,1,1] => 0
[6,4,2,2,2] => 0
[6,4,2,2,1,1] => 0
[6,4,2,1,1,1,1] => 0
[6,4,1,1,1,1,1,1] => 0
[6,3,3,3,1] => 0
[6,3,3,2,2] => 0
[6,3,3,2,1,1] => 0
[6,3,3,1,1,1,1] => 0
[6,3,2,2,2,1] => 0
[6,3,2,2,1,1,1] => 0
[6,3,2,1,1,1,1,1] => 0
[6,3,1,1,1,1,1,1,1] => 0
[6,2,2,2,2,2] => 0
[6,2,2,2,2,1,1] => 0
[6,2,2,2,1,1,1,1] => 0
[6,2,2,1,1,1,1,1,1] => 0
[6,2,1,1,1,1,1,1,1,1] => 0
[6,1,1,1,1,1,1,1,1,1,1] => 0
[5,5,5,1] => 0
[5,5,4,2] => 0
[5,5,4,1,1] => 0
[5,5,3,3] => 0
[5,5,3,2,1] => 0
[5,5,3,1,1,1] => 0
[5,5,2,2,2] => 0
[5,5,2,2,1,1] => 0
[5,5,2,1,1,1,1] => 0
[5,5,1,1,1,1,1,1] => 0
[5,4,4,3] => 0
[5,4,4,2,1] => 0
[5,4,4,1,1,1] => 0
[5,4,3,3,1] => 0
[5,4,3,2,2] => 0
[5,4,3,2,1,1] => 0
[5,4,3,1,1,1,1] => 0
[5,4,2,2,2,1] => 0
[5,4,2,2,1,1,1] => 0
[5,4,2,1,1,1,1,1] => 0
[5,4,1,1,1,1,1,1,1] => 0
[5,3,3,3,2] => 0
[5,3,3,3,1,1] => 0
[5,3,3,2,2,1] => 0
[5,3,3,2,1,1,1] => 0
[5,3,3,1,1,1,1,1] => 0
[5,3,2,2,2,2] => 0
[5,3,2,2,2,1,1] => 0
[5,3,2,2,1,1,1,1] => 0
[5,3,2,1,1,1,1,1,1] => 0
[5,3,1,1,1,1,1,1,1,1] => 0
[5,2,2,2,2,2,1] => 0
[5,2,2,2,2,1,1,1] => 0
[5,2,2,2,1,1,1,1,1] => 0
[5,2,2,1,1,1,1,1,1,1] => 0
[5,2,1,1,1,1,1,1,1,1,1] => 0
[5,1,1,1,1,1,1,1,1,1,1,1] => 0
[4,4,4,4] => 0
[4,4,4,3,1] => 0
[4,4,4,2,2] => 0
[4,4,4,2,1,1] => 0
[4,4,4,1,1,1,1] => 0
[4,4,3,3,2] => 0
[4,4,3,3,1,1] => 0
[4,4,3,2,2,1] => 0
[4,4,3,2,1,1,1] => 0
[4,4,3,1,1,1,1,1] => 0
[4,4,2,2,2,2] => 0
[4,4,2,2,2,1,1] => 0
[4,4,2,2,1,1,1,1] => 0
[4,4,2,1,1,1,1,1,1] => 0
[4,4,1,1,1,1,1,1,1,1] => 0
[4,3,3,3,3] => 0
[4,3,3,3,2,1] => 0
[4,3,3,3,1,1,1] => 0
[4,3,3,2,2,2] => 0
[4,3,3,2,2,1,1] => 0
[4,3,3,2,1,1,1,1] => 0
[4,3,3,1,1,1,1,1,1] => 0
[4,3,2,2,2,2,1] => 0
[4,3,2,2,2,1,1,1] => 0
[4,3,2,2,1,1,1,1,1] => 0
[4,3,2,1,1,1,1,1,1,1] => 0
[4,3,1,1,1,1,1,1,1,1,1] => 0
[4,2,2,2,2,2,2] => 0
[4,2,2,2,2,2,1,1] => 0
[4,2,2,2,2,1,1,1,1] => 0
[4,2,2,2,1,1,1,1,1,1] => 0
[4,2,2,1,1,1,1,1,1,1,1] => 0
[4,2,1,1,1,1,1,1,1,1,1,1] => 0
[4,1,1,1,1,1,1,1,1,1,1,1,1] => 0
[3,3,3,3,3,1] => 0
[3,3,3,3,2,2] => 0
[3,3,3,3,2,1,1] => 0
[3,3,3,3,1,1,1,1] => 0
[3,3,3,2,2,2,1] => 0
[3,3,3,2,2,1,1,1] => 0
[3,3,3,2,1,1,1,1,1] => 0
[3,3,3,1,1,1,1,1,1,1] => 0
[3,3,2,2,2,2,2] => 0
[3,3,2,2,2,2,1,1] => 0
[3,3,2,2,2,1,1,1,1] => 0
[3,3,2,2,1,1,1,1,1,1] => 0
[3,3,2,1,1,1,1,1,1,1,1] => 0
[3,3,1,1,1,1,1,1,1,1,1,1] => 0
[3,2,2,2,2,2,2,1] => 0
[3,2,2,2,2,2,1,1,1] => 0
[3,2,2,2,2,1,1,1,1,1] => 0
[3,2,2,2,1,1,1,1,1,1,1] => 0
[3,2,2,1,1,1,1,1,1,1,1,1] => 0
[3,2,1,1,1,1,1,1,1,1,1,1,1] => 0
[3,1,1,1,1,1,1,1,1,1,1,1,1,1] => 0
[2,2,2,2,2,2,2,2] => 0
[2,2,2,2,2,2,2,1,1] => 0
[2,2,2,2,2,2,1,1,1,1] => 0
[2,2,2,2,2,1,1,1,1,1,1] => 0
[2,2,2,2,1,1,1,1,1,1,1,1] => 0
[2,2,2,1,1,1,1,1,1,1,1,1,1] => 0
[2,2,1,1,1,1,1,1,1,1,1,1,1,1] => 0
[2,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 0
[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 0
[17] => 1
[16,1] => 15
[15,2] => 104
[15,1,1] => 0
[14,3] => 440
[14,2,1] => 442
[14,1,1,1] => 0
[13,4] => 1260
[13,3,1] => 2550
[13,2,2] => 0
[13,2,1,1] => 0
[13,1,1,1,1] => 0
[12,5] => 2548
[12,4,1] => 7854
[12,3,2] => 5304
[12,3,1,1] => 0
[12,2,2,1] => 0
[12,2,1,1,1] => 0
[12,1,1,1,1,1] => 0
[11,6] => 3640
[11,5,1] => 15470
[11,4,2] => 19992
[11,4,1,1] => 0
[11,3,3] => 0
[11,3,2,1] => 8160
[11,3,1,1,1] => 0
[11,2,2,2] => 0
[11,2,2,1,1] => 0
[11,2,1,1,1,1] => 0
[11,1,1,1,1,1,1] => 0
[10,7] => 3432
[10,6,1] => 19890
[10,5,2] => 38896
[10,5,1,1] => 0
[10,4,3] => 22440
[10,4,2,1] => 31824
[10,4,1,1,1] => 0
[10,3,3,1] => 0
[10,3,2,2] => 0
[10,3,2,1,1] => 0
[10,3,1,1,1,1] => 0
[10,2,2,2,1] => 0
[10,2,2,1,1,1] => 0
[10,2,1,1,1,1,1] => 0
[10,1,1,1,1,1,1,1] => 0
[9,8] => 1430
[9,7,1] => 14586
[9,6,2] => 43316
[9,6,1,1] => 0
[9,5,3] => 48620
[9,5,2,1] => 56576
[9,5,1,1,1] => 0
[9,4,4] => 0
[9,4,3,1] => 44880
[9,4,2,2] => 0
[9,4,2,1,1] => 0
[9,4,1,1,1,1] => 0
[9,3,3,2] => 0
[9,3,3,1,1] => 0
[9,3,2,2,1] => 0
[9,3,2,1,1,1] => 0
[9,3,1,1,1,1,1] => 0
[9,2,2,2,2] => 0
[9,2,2,2,1,1] => 0
[9,2,2,1,1,1,1] => 0
[9,2,1,1,1,1,1,1] => 0
[9,1,1,1,1,1,1,1,1] => 0
[8,8,1] => 0
[8,7,2] => 19448
[8,7,1,1] => 0
[8,6,3] => 44200
[8,6,2,1] => 48620
[8,6,1,1,1] => 0
[8,5,4] => 26180
[8,5,3,1] => 83300
[8,5,2,2] => 0
[8,5,2,1,1] => 0
[8,5,1,1,1,1] => 0
[8,4,4,1] => 0
[8,4,3,2] => 26520
[8,4,3,1,1] => 0
[8,4,2,2,1] => 0
[8,4,2,1,1,1] => 0
[8,4,1,1,1,1,1] => 0
[8,3,3,3] => 0
[8,3,3,2,1] => 0
[8,3,3,1,1,1] => 0
[8,3,2,2,2] => 0
[8,3,2,2,1,1] => 0
[8,3,2,1,1,1,1] => 0
[8,3,1,1,1,1,1,1] => 0
[8,2,2,2,2,1] => 0
[8,2,2,2,1,1,1] => 0
[8,2,2,1,1,1,1,1] => 0
[8,2,1,1,1,1,1,1,1] => 0
[8,1,1,1,1,1,1,1,1,1] => 0
[7,7,3] => 0
[7,7,2,1] => 0
[7,7,1,1,1] => 0
[7,6,4] => 17136
[7,6,3,1] => 44880
[7,6,2,2] => 0
[7,6,2,1,1] => 0
[7,6,1,1,1,1] => 0
[7,5,5] => 0
[7,5,4,1] => 37128
[7,5,3,2] => 38896
[7,5,3,1,1] => 0
[7,5,2,2,1] => 0
[7,5,2,1,1,1] => 0
[7,5,1,1,1,1,1] => 0
[7,4,4,2] => 0
[7,4,4,1,1] => 0
[7,4,3,3] => 0
[7,4,3,2,1] => 10608
[7,4,3,1,1,1] => 0
[7,4,2,2,2] => 0
[7,4,2,2,1,1] => 0
[7,4,2,1,1,1,1] => 0
[7,4,1,1,1,1,1,1] => 0
[7,3,3,3,1] => 0
[7,3,3,2,2] => 0
[7,3,3,2,1,1] => 0
[7,3,3,1,1,1,1] => 0
[7,3,2,2,2,1] => 0
[7,3,2,2,1,1,1] => 0
[7,3,2,1,1,1,1,1] => 0
[7,3,1,1,1,1,1,1,1] => 0
[7,2,2,2,2,2] => 0
[7,2,2,2,2,1,1] => 0
[7,2,2,2,1,1,1,1] => 0
[7,2,2,1,1,1,1,1,1] => 0
[7,2,1,1,1,1,1,1,1,1] => 0
[7,1,1,1,1,1,1,1,1,1,1] => 0
[6,6,5] => 0
[6,6,4,1] => 0
[6,6,3,2] => 0
[6,6,3,1,1] => 0
[6,6,2,2,1] => 0
[6,6,2,1,1,1] => 0
[6,6,1,1,1,1,1] => 0
[6,5,5,1] => 0
[6,5,4,2] => 12376
[6,5,4,1,1] => 0
[6,5,3,3] => 0
[6,5,3,2,1] => 8840
[6,5,3,1,1,1] => 0
[6,5,2,2,2] => 0
[6,5,2,2,1,1] => 0
[6,5,2,1,1,1,1] => 0
[6,5,1,1,1,1,1,1] => 0
[6,4,4,3] => 0
[6,4,4,2,1] => 0
[6,4,4,1,1,1] => 0
[6,4,3,3,1] => 0
[6,4,3,2,2] => 0
[6,4,3,2,1,1] => 0
[6,4,3,1,1,1,1] => 0
[6,4,2,2,2,1] => 0
[6,4,2,2,1,1,1] => 0
[6,4,2,1,1,1,1,1] => 0
[6,4,1,1,1,1,1,1,1] => 0
[6,3,3,3,2] => 0
[6,3,3,3,1,1] => 0
[6,3,3,2,2,1] => 0
[6,3,3,2,1,1,1] => 0
[6,3,3,1,1,1,1,1] => 0
[6,3,2,2,2,2] => 0
[6,3,2,2,2,1,1] => 0
[6,3,2,2,1,1,1,1] => 0
[6,3,2,1,1,1,1,1,1] => 0
[6,3,1,1,1,1,1,1,1,1] => 0
[6,2,2,2,2,2,1] => 0
[6,2,2,2,2,1,1,1] => 0
[6,2,2,2,1,1,1,1,1] => 0
[6,2,2,1,1,1,1,1,1,1] => 0
[6,2,1,1,1,1,1,1,1,1,1] => 0
[6,1,1,1,1,1,1,1,1,1,1,1] => 0
[5,5,5,2] => 0
[5,5,5,1,1] => 0
[5,5,4,3] => 0
[5,5,4,2,1] => 0
[5,5,4,1,1,1] => 0
[5,5,3,3,1] => 0
[5,5,3,2,2] => 0
[5,5,3,2,1,1] => 0
[5,5,3,1,1,1,1] => 0
[5,5,2,2,2,1] => 0
[5,5,2,2,1,1,1] => 0
[5,5,2,1,1,1,1,1] => 0
[5,5,1,1,1,1,1,1,1] => 0
[5,4,4,4] => 0
[5,4,4,3,1] => 0
[5,4,4,2,2] => 0
[5,4,4,2,1,1] => 0
[5,4,4,1,1,1,1] => 0
[5,4,3,3,2] => 0
[5,4,3,3,1,1] => 0
[5,4,3,2,2,1] => 0
[5,4,3,2,1,1,1] => 0
[5,4,3,1,1,1,1,1] => 0
[5,4,2,2,2,2] => 0
[5,4,2,2,2,1,1] => 0
[5,4,2,2,1,1,1,1] => 0
[5,4,2,1,1,1,1,1,1] => 0
[5,4,1,1,1,1,1,1,1,1] => 0
[5,3,3,3,3] => 0
[5,3,3,3,2,1] => 0
[5,3,3,3,1,1,1] => 0
[5,3,3,2,2,2] => 0
[5,3,3,2,2,1,1] => 0
[5,3,3,2,1,1,1,1] => 0
[5,3,3,1,1,1,1,1,1] => 0
[5,3,2,2,2,2,1] => 0
[5,3,2,2,2,1,1,1] => 0
[5,3,2,2,1,1,1,1,1] => 0
[5,3,2,1,1,1,1,1,1,1] => 0
[5,3,1,1,1,1,1,1,1,1,1] => 0
[5,2,2,2,2,2,2] => 0
[5,2,2,2,2,2,1,1] => 0
[5,2,2,2,2,1,1,1,1] => 0
[5,2,2,2,1,1,1,1,1,1] => 0
[5,2,2,1,1,1,1,1,1,1,1] => 0
[5,2,1,1,1,1,1,1,1,1,1,1] => 0
[5,1,1,1,1,1,1,1,1,1,1,1,1] => 0
[4,4,4,4,1] => 0
[4,4,4,3,2] => 0
[4,4,4,3,1,1] => 0
[4,4,4,2,2,1] => 0
[4,4,4,2,1,1,1] => 0
[4,4,4,1,1,1,1,1] => 0
[4,4,3,3,3] => 0
[4,4,3,3,2,1] => 0
[4,4,3,3,1,1,1] => 0
[4,4,3,2,2,2] => 0
[4,4,3,2,2,1,1] => 0
[4,4,3,2,1,1,1,1] => 0
[4,4,3,1,1,1,1,1,1] => 0
[4,4,2,2,2,2,1] => 0
[4,4,2,2,2,1,1,1] => 0
[4,4,2,2,1,1,1,1,1] => 0
[4,4,2,1,1,1,1,1,1,1] => 0
[4,4,1,1,1,1,1,1,1,1,1] => 0
[4,3,3,3,3,1] => 0
[4,3,3,3,2,2] => 0
[4,3,3,3,2,1,1] => 0
[4,3,3,3,1,1,1,1] => 0
[4,3,3,2,2,2,1] => 0
[4,3,3,2,2,1,1,1] => 0
[4,3,3,2,1,1,1,1,1] => 0
[4,3,3,1,1,1,1,1,1,1] => 0
[4,3,2,2,2,2,2] => 0
[4,3,2,2,2,2,1,1] => 0
[4,3,2,2,2,1,1,1,1] => 0
[4,3,2,2,1,1,1,1,1,1] => 0
[4,3,2,1,1,1,1,1,1,1,1] => 0
[4,3,1,1,1,1,1,1,1,1,1,1] => 0
[4,2,2,2,2,2,2,1] => 0
[4,2,2,2,2,2,1,1,1] => 0
[4,2,2,2,2,1,1,1,1,1] => 0
[4,2,2,2,1,1,1,1,1,1,1] => 0
[4,2,2,1,1,1,1,1,1,1,1,1] => 0
[4,2,1,1,1,1,1,1,1,1,1,1,1] => 0
[4,1,1,1,1,1,1,1,1,1,1,1,1,1] => 0
[3,3,3,3,3,2] => 0
[3,3,3,3,3,1,1] => 0
[3,3,3,3,2,2,1] => 0
[3,3,3,3,2,1,1,1] => 0
[3,3,3,3,1,1,1,1,1] => 0
[3,3,3,2,2,2,2] => 0
[3,3,3,2,2,2,1,1] => 0
[3,3,3,2,2,1,1,1,1] => 0
[3,3,3,2,1,1,1,1,1,1] => 0
[3,3,3,1,1,1,1,1,1,1,1] => 0
[3,3,2,2,2,2,2,1] => 0
[3,3,2,2,2,2,1,1,1] => 0
[3,3,2,2,2,1,1,1,1,1] => 0
[3,3,2,2,1,1,1,1,1,1,1] => 0
[3,3,2,1,1,1,1,1,1,1,1,1] => 0
[3,3,1,1,1,1,1,1,1,1,1,1,1] => 0
[3,2,2,2,2,2,2,2] => 0
[3,2,2,2,2,2,2,1,1] => 0
[3,2,2,2,2,2,1,1,1,1] => 0
[3,2,2,2,2,1,1,1,1,1,1] => 0
[3,2,2,2,1,1,1,1,1,1,1,1] => 0
[3,2,2,1,1,1,1,1,1,1,1,1,1] => 0
-----------------------------------------------------------------------------
Created: Sep 18, 2020 at 10:32 by Christian Stump
-----------------------------------------------------------------------------
Last Updated: Sep 18, 2020 at 10:32 by Christian Stump