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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, *
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-----------------------------------------------------------------------------
Statistic identifier: St001785
-----------------------------------------------------------------------------
Collection: Integer partitions
-----------------------------------------------------------------------------
Description: The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition.
Given a partition $\lambda\vdash n$, let $\alpha(\lambda)$ be the partition given by the lengths of the antidiagonals of the Ferrers diagram of $\lambda$. Then, the value of the statistic on $\mu$ is the number of times $\mu$ appears in the multiset $\{\{\alpha(\lambda)\mid \lambda\vdash n\}\}$.
-----------------------------------------------------------------------------
References: [1] Amdeberhan, T. Agreement between two sets of data on partitions [[MathOverflow:420033]]
-----------------------------------------------------------------------------
Code:
def alpha(la):
la = Partition(la)
if not la:
return la
result = [0]*(2*max(len(la), la[0])-1)
for i, j in la.cells():
result[i+j] += 1
return Partition(sorted(result, reverse=True))
@cached_function
def frequencies(l):
return {d: l.count(d) for d in set(l)}
def statistic(la):
F = frequencies(tuple(alpha(mu) for mu in Partitions(la.size())))
return F.get(la, 0)
-----------------------------------------------------------------------------
Statistic values:
[1] => 1
[2] => 0
[1,1] => 2
[3] => 0
[2,1] => 1
[1,1,1] => 2
[4] => 0
[3,1] => 0
[2,2] => 0
[2,1,1] => 3
[1,1,1,1] => 2
[5] => 0
[4,1] => 0
[3,2] => 0
[3,1,1] => 0
[2,2,1] => 3
[2,1,1,1] => 2
[1,1,1,1,1] => 2
[6] => 0
[5,1] => 0
[4,2] => 0
[4,1,1] => 0
[3,3] => 0
[3,2,1] => 1
[3,1,1,1] => 0
[2,2,2] => 0
[2,2,1,1] => 6
[2,1,1,1,1] => 2
[1,1,1,1,1,1] => 2
[7] => 0
[6,1] => 0
[5,2] => 0
[5,1,1] => 0
[4,3] => 0
[4,2,1] => 0
[4,1,1,1] => 0
[3,3,1] => 0
[3,2,2] => 0
[3,2,1,1] => 4
[3,1,1,1,1] => 0
[2,2,2,1] => 3
[2,2,1,1,1] => 4
[2,1,1,1,1,1] => 2
[1,1,1,1,1,1,1] => 2
[8] => 0
[7,1] => 0
[6,2] => 0
[6,1,1] => 0
[5,3] => 0
[5,2,1] => 0
[5,1,1,1] => 0
[4,4] => 0
[4,3,1] => 0
[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] => 6
[3,2,1,1,1] => 2
[3,1,1,1,1,1] => 0
[2,2,2,2] => 0
[2,2,2,1,1] => 6
[2,2,1,1,1,1] => 4
[2,1,1,1,1,1,1] => 2
[1,1,1,1,1,1,1,1] => 2
[9] => 0
[8,1] => 0
[7,2] => 0
[7,1,1] => 0
[6,3] => 0
[6,2,1] => 0
[6,1,1,1] => 0
[5,4] => 0
[5,3,1] => 0
[5,2,2] => 0
[5,2,1,1] => 0
[5,1,1,1,1] => 0
[4,4,1] => 0
[4,3,2] => 0
[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] => 4
[3,3,1,1,1] => 0
[3,2,2,2] => 0
[3,2,2,1,1] => 9
[3,2,1,1,1,1] => 2
[3,1,1,1,1,1,1] => 0
[2,2,2,2,1] => 3
[2,2,2,1,1,1] => 4
[2,2,1,1,1,1,1] => 4
[2,1,1,1,1,1,1,1] => 2
[1,1,1,1,1,1,1,1,1] => 2
[10] => 0
[9,1] => 0
[8,2] => 0
[8,1,1] => 0
[7,3] => 0
[7,2,1] => 0
[7,1,1,1] => 0
[6,4] => 0
[6,3,1] => 0
[6,2,2] => 0
[6,2,1,1] => 0
[6,1,1,1,1] => 0
[5,5] => 0
[5,4,1] => 0
[5,3,2] => 0
[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] => 1
[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] => 12
[3,3,1,1,1,1] => 0
[3,2,2,2,1] => 3
[3,2,2,1,1,1] => 6
[3,2,1,1,1,1,1] => 2
[3,1,1,1,1,1,1,1] => 0
[2,2,2,2,2] => 0
[2,2,2,2,1,1] => 6
[2,2,2,1,1,1,1] => 4
[2,2,1,1,1,1,1,1] => 4
[2,1,1,1,1,1,1,1,1] => 2
[1,1,1,1,1,1,1,1,1,1] => 2
[11] => 0
[10,1] => 0
[9,2] => 0
[9,1,1] => 0
[8,3] => 0
[8,2,1] => 0
[8,1,1,1] => 0
[7,4] => 0
[7,3,1] => 0
[7,2,2] => 0
[7,2,1,1] => 0
[7,1,1,1,1] => 0
[6,5] => 0
[6,4,1] => 0
[6,3,2] => 0
[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] => 0
[5,4,1,1] => 0
[5,3,3] => 0
[5,3,2,1] => 0
[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] => 5
[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] => 12
[3,3,2,1,1,1] => 6
[3,3,1,1,1,1,1] => 0
[3,2,2,2,2] => 0
[3,2,2,2,1,1] => 6
[3,2,2,1,1,1,1] => 6
[3,2,1,1,1,1,1,1] => 2
[3,1,1,1,1,1,1,1,1] => 0
[2,2,2,2,2,1] => 3
[2,2,2,2,1,1,1] => 4
[2,2,2,1,1,1,1,1] => 4
[2,2,1,1,1,1,1,1,1] => 4
[2,1,1,1,1,1,1,1,1,1] => 2
[1,1,1,1,1,1,1,1,1,1,1] => 2
[12] => 0
[11,1] => 0
[10,2] => 0
[10,1,1] => 0
[9,3] => 0
[9,2,1] => 0
[9,1,1,1] => 0
[8,4] => 0
[8,3,1] => 0
[8,2,2] => 0
[8,2,1,1] => 0
[8,1,1,1,1] => 0
[7,5] => 0
[7,4,1] => 0
[7,3,2] => 0
[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] => 0
[6,4,2] => 0
[6,4,1,1] => 0
[6,3,3] => 0
[6,3,2,1] => 0
[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] => 0
[5,4,2,1] => 0
[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] => 10
[4,3,2,1,1,1] => 2
[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] => 4
[3,3,3,1,1,1] => 0
[3,3,2,2,2] => 0
[3,3,2,2,1,1] => 18
[3,3,2,1,1,1,1] => 6
[3,3,1,1,1,1,1,1] => 0
[3,2,2,2,2,1] => 3
[3,2,2,2,1,1,1] => 4
[3,2,2,1,1,1,1,1] => 6
[3,2,1,1,1,1,1,1,1] => 2
[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] => 6
[2,2,2,2,1,1,1,1] => 4
[2,2,2,1,1,1,1,1,1] => 4
[2,2,1,1,1,1,1,1,1,1] => 4
[2,1,1,1,1,1,1,1,1,1,1] => 2
[1,1,1,1,1,1,1,1,1,1,1,1] => 2
[13] => 0
[12,1] => 0
[11,2] => 0
[11,1,1] => 0
[10,3] => 0
[10,2,1] => 0
[10,1,1,1] => 0
[9,4] => 0
[9,3,1] => 0
[9,2,2] => 0
[9,2,1,1] => 0
[9,1,1,1,1] => 0
[8,5] => 0
[8,4,1] => 0
[8,3,2] => 0
[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] => 0
[7,5,1] => 0
[7,4,2] => 0
[7,4,1,1] => 0
[7,3,3] => 0
[7,3,2,1] => 0
[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] => 0
[6,5,1,1] => 0
[6,4,3] => 0
[6,4,2,1] => 0
[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] => 0
[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] => 10
[4,3,3,1,1,1] => 0
[4,3,2,2,2] => 0
[4,3,2,2,1,1] => 12
[4,3,2,1,1,1,1] => 2
[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] => 12
[3,3,3,1,1,1,1] => 0
[3,3,2,2,2,1] => 6
[3,3,2,2,1,1,1] => 12
[3,3,2,1,1,1,1,1] => 6
[3,3,1,1,1,1,1,1,1] => 0
[3,2,2,2,2,2] => 0
[3,2,2,2,2,1,1] => 6
[3,2,2,2,1,1,1,1] => 4
[3,2,2,1,1,1,1,1,1] => 6
[3,2,1,1,1,1,1,1,1,1] => 2
[3,1,1,1,1,1,1,1,1,1,1] => 0
[2,2,2,2,2,2,1] => 3
[2,2,2,2,2,1,1,1] => 4
[2,2,2,2,1,1,1,1,1] => 4
[2,2,2,1,1,1,1,1,1,1] => 4
[2,2,1,1,1,1,1,1,1,1,1] => 4
[2,1,1,1,1,1,1,1,1,1,1,1] => 2
[1,1,1,1,1,1,1,1,1,1,1,1,1] => 2
[14] => 0
[13,1] => 0
[12,2] => 0
[12,1,1] => 0
[11,3] => 0
[11,2,1] => 0
[11,1,1,1] => 0
[10,4] => 0
[10,3,1] => 0
[10,2,2] => 0
[10,2,1,1] => 0
[10,1,1,1,1] => 0
[9,5] => 0
[9,4,1] => 0
[9,3,2] => 0
[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] => 0
[8,5,1] => 0
[8,4,2] => 0
[8,4,1,1] => 0
[8,3,3] => 0
[8,3,2,1] => 0
[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] => 0
[7,5,2] => 0
[7,5,1,1] => 0
[7,4,3] => 0
[7,4,2,1] => 0
[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] => 0
[6,5,2,1] => 0
[6,5,1,1,1] => 0
[6,4,4] => 0
[6,4,3,1] => 0
[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] => 0
[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] => 5
[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] => 24
[4,3,3,1,1,1,1] => 0
[4,3,2,2,2,1] => 3
[4,3,2,2,1,1,1] => 8
[4,3,2,1,1,1,1,1] => 2
[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] => 12
[3,3,3,2,1,1,1] => 6
[3,3,3,1,1,1,1,1] => 0
[3,3,2,2,2,2] => 0
[3,3,2,2,2,1,1] => 12
[3,3,2,2,1,1,1,1] => 12
[3,3,2,1,1,1,1,1,1] => 6
[3,3,1,1,1,1,1,1,1,1] => 0
[3,2,2,2,2,2,1] => 3
[3,2,2,2,2,1,1,1] => 4
[3,2,2,2,1,1,1,1,1] => 4
[3,2,2,1,1,1,1,1,1,1] => 6
[3,2,1,1,1,1,1,1,1,1,1] => 2
[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] => 6
[2,2,2,2,2,1,1,1,1] => 4
[2,2,2,2,1,1,1,1,1,1] => 4
[2,2,2,1,1,1,1,1,1,1,1] => 4
[2,2,1,1,1,1,1,1,1,1,1,1] => 4
[2,1,1,1,1,1,1,1,1,1,1,1,1] => 2
[1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 2
[15] => 0
[14,1] => 0
[13,2] => 0
[13,1,1] => 0
[12,3] => 0
[12,2,1] => 0
[12,1,1,1] => 0
[11,4] => 0
[11,3,1] => 0
[11,2,2] => 0
[11,2,1,1] => 0
[11,1,1,1,1] => 0
[10,5] => 0
[10,4,1] => 0
[10,3,2] => 0
[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] => 0
[9,5,1] => 0
[9,4,2] => 0
[9,4,1,1] => 0
[9,3,3] => 0
[9,3,2,1] => 0
[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] => 0
[8,6,1] => 0
[8,5,2] => 0
[8,5,1,1] => 0
[8,4,3] => 0
[8,4,2,1] => 0
[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] => 0
[7,6,1,1] => 0
[7,5,3] => 0
[7,5,2,1] => 0
[7,5,1,1,1] => 0
[7,4,4] => 0
[7,4,3,1] => 0
[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] => 0
[6,5,3,1] => 0
[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] => 0
[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] => 1
[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] => 20
[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] => 18
[4,3,3,2,1,1,1] => 12
[4,3,3,1,1,1,1,1] => 0
[4,3,2,2,2,2] => 0
[4,3,2,2,2,1,1] => 6
[4,3,2,2,1,1,1,1] => 8
[4,3,2,1,1,1,1,1,1] => 2
[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] => 4
[3,3,3,3,1,1,1] => 0
[3,3,3,2,2,2] => 0
[3,3,3,2,2,1,1] => 18
[3,3,3,2,1,1,1,1] => 6
[3,3,3,1,1,1,1,1,1] => 0
[3,3,2,2,2,2,1] => 6
[3,3,2,2,2,1,1,1] => 8
[3,3,2,2,1,1,1,1,1] => 12
[3,3,2,1,1,1,1,1,1,1] => 6
[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] => 6
[3,2,2,2,2,1,1,1,1] => 4
[3,2,2,2,1,1,1,1,1,1] => 4
[3,2,2,1,1,1,1,1,1,1,1] => 6
[3,2,1,1,1,1,1,1,1,1,1,1] => 2
[3,1,1,1,1,1,1,1,1,1,1,1,1] => 0
[2,2,2,2,2,2,2,1] => 3
[2,2,2,2,2,2,1,1,1] => 4
[2,2,2,2,2,1,1,1,1,1] => 4
[2,2,2,2,1,1,1,1,1,1,1] => 4
[2,2,2,1,1,1,1,1,1,1,1,1] => 4
[2,2,1,1,1,1,1,1,1,1,1,1,1] => 4
[2,1,1,1,1,1,1,1,1,1,1,1,1,1] => 2
[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 2
[16] => 0
[15,1] => 0
[14,2] => 0
[14,1,1] => 0
[13,3] => 0
[13,2,1] => 0
[13,1,1,1] => 0
[12,4] => 0
[12,3,1] => 0
[12,2,2] => 0
[12,2,1,1] => 0
[12,1,1,1,1] => 0
[11,5] => 0
[11,4,1] => 0
[11,3,2] => 0
[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] => 0
[10,5,1] => 0
[10,4,2] => 0
[10,4,1,1] => 0
[10,3,3] => 0
[10,3,2,1] => 0
[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] => 0
[9,6,1] => 0
[9,5,2] => 0
[9,5,1,1] => 0
[9,4,3] => 0
[9,4,2,1] => 0
[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] => 0
[8,6,2] => 0
[8,6,1,1] => 0
[8,5,3] => 0
[8,5,2,1] => 0
[8,5,1,1,1] => 0
[8,4,4] => 0
[8,4,3,1] => 0
[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] => 0
[7,6,2,1] => 0
[7,6,1,1,1] => 0
[7,5,4] => 0
[7,5,3,1] => 0
[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] => 0
[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] => 0
[6,5,3,2] => 0
[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] => 0
[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] => 6
[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] => 30
[4,4,3,2,1,1,1] => 8
[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] => 4
[4,3,3,3,1,1,1] => 0
[4,3,3,2,2,2] => 0
[4,3,3,2,2,1,1] => 27
[4,3,3,2,1,1,1,1] => 12
[4,3,3,1,1,1,1,1,1] => 0
[4,3,2,2,2,2,1] => 3
[4,3,2,2,2,1,1,1] => 4
[4,3,2,2,1,1,1,1,1] => 8
[4,3,2,1,1,1,1,1,1,1] => 2
[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] => 12
[3,3,3,3,1,1,1,1] => 0
[3,3,3,2,2,2,1] => 6
[3,3,3,2,2,1,1,1] => 12
[3,3,3,2,1,1,1,1,1] => 6
[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] => 12
[3,3,2,2,2,1,1,1,1] => 8
[3,3,2,2,1,1,1,1,1,1] => 12
[3,3,2,1,1,1,1,1,1,1,1] => 6
[3,3,1,1,1,1,1,1,1,1,1,1] => 0
[3,2,2,2,2,2,2,1] => 3
[3,2,2,2,2,2,1,1,1] => 4
[3,2,2,2,2,1,1,1,1,1] => 4
[3,2,2,2,1,1,1,1,1,1,1] => 4
[3,2,2,1,1,1,1,1,1,1,1,1] => 6
[3,2,1,1,1,1,1,1,1,1,1,1,1] => 2
[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] => 6
[2,2,2,2,2,2,1,1,1,1] => 4
[2,2,2,2,2,1,1,1,1,1,1] => 4
[2,2,2,2,1,1,1,1,1,1,1,1] => 4
[2,2,2,1,1,1,1,1,1,1,1,1,1] => 4
[2,2,1,1,1,1,1,1,1,1,1,1,1,1] => 4
[2,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 2
[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 2
[17] => 0
[16,1] => 0
[15,2] => 0
[15,1,1] => 0
[14,3] => 0
[14,2,1] => 0
[14,1,1,1] => 0
[13,4] => 0
[13,3,1] => 0
[13,2,2] => 0
[13,2,1,1] => 0
[13,1,1,1,1] => 0
[12,5] => 0
[12,4,1] => 0
[12,3,2] => 0
[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] => 0
[11,5,1] => 0
[11,4,2] => 0
[11,4,1,1] => 0
[11,3,3] => 0
[11,3,2,1] => 0
[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] => 0
[10,6,1] => 0
[10,5,2] => 0
[10,5,1,1] => 0
[10,4,3] => 0
[10,4,2,1] => 0
[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] => 0
[9,7,1] => 0
[9,6,2] => 0
[9,6,1,1] => 0
[9,5,3] => 0
[9,5,2,1] => 0
[9,5,1,1,1] => 0
[9,4,4] => 0
[9,4,3,1] => 0
[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] => 0
[8,7,1,1] => 0
[8,6,3] => 0
[8,6,2,1] => 0
[8,6,1,1,1] => 0
[8,5,4] => 0
[8,5,3,1] => 0
[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] => 0
[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] => 0
[7,6,3,1] => 0
[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] => 0
[7,5,3,2] => 0
[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] => 0
[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] => 0
[6,5,4,1,1] => 0
[6,5,3,3] => 0
[6,5,3,2,1] => 0
[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] => 15
[5,4,3,2,1,1,1] => 2
[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] => 20
[4,4,3,3,1,1,1] => 0
[4,4,3,2,2,2] => 0
[4,4,3,2,2,1,1] => 36
[4,4,3,2,1,1,1,1] => 8
[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] => 12
[4,3,3,3,1,1,1,1] => 0
[4,3,3,2,2,2,1] => 9
[4,3,3,2,2,1,1,1] => 18
[4,3,3,2,1,1,1,1,1] => 12
[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] => 6
[4,3,2,2,2,1,1,1,1] => 4
[4,3,2,2,1,1,1,1,1,1] => 8
[4,3,2,1,1,1,1,1,1,1,1] => 2
[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] => 12
[3,3,3,3,2,1,1,1] => 6
[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] => 12
[3,3,3,2,2,1,1,1,1] => 12
[3,3,3,2,1,1,1,1,1,1] => 6
[3,3,3,1,1,1,1,1,1,1,1] => 0
[3,3,2,2,2,2,2,1] => 6
[3,3,2,2,2,2,1,1,1] => 8
[3,3,2,2,2,1,1,1,1,1] => 8
[3,3,2,2,1,1,1,1,1,1,1] => 12
[3,3,2,1,1,1,1,1,1,1,1,1] => 6
[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] => 6
[3,2,2,2,2,2,1,1,1,1] => 4
[3,2,2,2,2,1,1,1,1,1,1] => 4
[3,2,2,2,1,1,1,1,1,1,1,1] => 4
[3,2,2,1,1,1,1,1,1,1,1,1,1] => 6
-----------------------------------------------------------------------------
Created: Apr 14, 2022 at 15:39 by Martin Rubey
-----------------------------------------------------------------------------
Last Updated: Apr 14, 2022 at 15:39 by Martin Rubey