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-----------------------------------------------------------------------------
Statistic identifier: St001121
-----------------------------------------------------------------------------
Collection: Integer partitions
-----------------------------------------------------------------------------
Description: The multiplicity of the irreducible representation indexed by the partition in the Kronecker square corresponding to the partition.
The Kronecker coefficient is the multiplicity $g_{\mu,\nu}^\lambda$ of the Specht module $S^\lambda$ in $S^\mu\otimes S^\nu$:
$$ S^\mu\otimes S^\nu = \bigoplus_\lambda g_{\mu,\nu}^\lambda S^\lambda $$
This statistic records the Kronecker coefficient $g_{\lambda,\lambda}^\lambda$.
-----------------------------------------------------------------------------
References: [1] [[wikipedia:Kronecker coefficient]]
-----------------------------------------------------------------------------
Code:
def statistic(la):
s = SymmetricFunctions(ZZ).schur()
return s[la].internal_product(s[la]).coefficient(la)
-----------------------------------------------------------------------------
Statistic values:
[] => 1
[1] => 1
[2] => 1
[1,1] => 0
[3] => 1
[2,1] => 1
[1,1,1] => 0
[4] => 1
[3,1] => 1
[2,2] => 1
[2,1,1] => 1
[1,1,1,1] => 0
[5] => 1
[4,1] => 1
[3,2] => 1
[3,1,1] => 1
[2,2,1] => 1
[2,1,1,1] => 0
[1,1,1,1,1] => 0
[6] => 1
[5,1] => 1
[4,2] => 2
[4,1,1] => 1
[3,3] => 0
[3,2,1] => 5
[3,1,1,1] => 1
[2,2,2] => 1
[2,2,1,1] => 0
[2,1,1,1,1] => 0
[1,1,1,1,1,1] => 0
[7] => 1
[6,1] => 1
[5,2] => 2
[5,1,1] => 1
[4,3] => 1
[4,2,1] => 9
[4,1,1,1] => 1
[3,3,1] => 1
[3,2,2] => 2
[3,2,1,1] => 8
[3,1,1,1,1] => 1
[2,2,2,1] => 1
[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] => 1
[6,2] => 2
[6,1,1] => 1
[5,3] => 1
[5,2,1] => 9
[5,1,1,1] => 1
[4,4] => 1
[4,3,1] => 8
[4,2,2] => 6
[4,2,1,1] => 17
[4,1,1,1,1] => 1
[3,3,2] => 1
[3,3,1,1] => 5
[3,2,2,1] => 8
[3,2,1,1,1] => 4
[3,1,1,1,1,1] => 0
[2,2,2,2] => 1
[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] => 1
[7,2] => 2
[7,1,1] => 1
[6,3] => 2
[6,2,1] => 9
[6,1,1,1] => 1
[5,4] => 1
[5,3,1] => 15
[5,2,2] => 7
[5,2,1,1] => 18
[5,1,1,1,1] => 1
[4,4,1] => 2
[4,3,2] => 12
[4,3,1,1] => 27
[4,2,2,1] => 28
[4,2,1,1,1] => 17
[4,1,1,1,1,1] => 1
[3,3,3] => 1
[3,3,2,1] => 11
[3,3,1,1,1] => 5
[3,2,2,2] => 2
[3,2,2,1,1] => 7
[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] => 1
[8,2] => 2
[8,1,1] => 1
[7,3] => 2
[7,2,1] => 9
[7,1,1,1] => 1
[6,4] => 2
[6,3,1] => 19
[6,2,2] => 7
[6,2,1,1] => 18
[6,1,1,1,1] => 1
[5,5] => 0
[5,4,1] => 9
[5,3,2] => 29
[5,3,1,1] => 53
[5,2,2,1] => 39
[5,2,1,1,1] => 21
[5,1,1,1,1,1] => 1
[4,4,2] => 6
[4,4,1,1] => 5
[4,3,3] => 2
[4,3,2,1] => 117
[4,3,1,1,1] => 40
[4,2,2,2] => 10
[4,2,2,1,1] => 46
[4,2,1,1,1,1] => 11
[4,1,1,1,1,1,1] => 1
[3,3,3,1] => 2
[3,3,2,2] => 2
[3,3,2,1,1] => 21
[3,3,1,1,1,1] => 1
[3,2,2,2,1] => 5
[3,2,2,1,1,1] => 1
[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] => 1
[9,2] => 2
[9,1,1] => 1
[8,3] => 2
[8,2,1] => 9
[8,1,1,1] => 1
[7,4] => 2
[7,3,1] => 19
[7,2,2] => 7
[7,2,1,1] => 18
[7,1,1,1,1] => 1
[6,5] => 1
[6,4,1] => 19
[6,3,2] => 39
[6,3,1,1] => 62
[6,2,2,1] => 39
[6,2,1,1,1] => 21
[6,1,1,1,1,1] => 1
[5,5,1] => 1
[5,4,2] => 29
[5,4,1,1] => 40
[5,3,3] => 6
[5,3,2,1] => 312
[5,3,1,1,1] => 89
[5,2,2,2] => 17
[5,2,2,1,1] => 86
[5,2,1,1,1,1] => 20
[5,1,1,1,1,1,1] => 1
[4,4,3] => 2
[4,4,2,1] => 53
[4,4,1,1,1] => 14
[4,3,3,1] => 37
[4,3,2,2] => 53
[4,3,2,1,1] => 301
[4,3,1,1,1,1] => 30
[4,2,2,2,1] => 37
[4,2,2,1,1,1] => 32
[4,2,1,1,1,1,1] => 4
[4,1,1,1,1,1,1,1] => 0
[3,3,3,2] => 2
[3,3,3,1,1] => 8
[3,3,2,2,1] => 21
[3,3,2,1,1,1] => 11
[3,3,1,1,1,1,1] => 0
[3,2,2,2,2] => 1
[3,2,2,2,1,1] => 1
[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] => 1
[10,2] => 2
[10,1,1] => 1
[9,3] => 2
[9,2,1] => 9
[9,1,1,1] => 1
[8,4] => 3
[8,3,1] => 19
[8,2,2] => 7
[8,2,1,1] => 18
[8,1,1,1,1] => 1
[7,5] => 1
[7,4,1] => 26
[7,3,2] => 40
[7,3,1,1] => 63
[7,2,2,1] => 39
[7,2,1,1,1] => 21
[7,1,1,1,1,1] => 1
[6,6] => 1
[6,5,1] => 9
[6,4,2] => 71
[6,4,1,1] => 80
[6,3,3] => 13
[6,3,2,1] => 429
[6,3,1,1,1] => 108
[6,2,2,2] => 19
[6,2,2,1,1] => 90
[6,2,1,1,1,1] => 21
[6,1,1,1,1,1,1] => 1
[5,5,2] => 5
[5,5,1,1] => 9
[5,4,3] => 20
[5,4,2,1] => 407
[5,4,1,1,1] => 89
[5,3,3,1] => 146
[5,3,2,2] => 179
[5,3,2,1,1] => 945
[5,3,1,1,1,1] => 91
[5,2,2,2,1] => 94
[5,2,2,1,1,1] => 103
[5,2,1,1,1,1,1] => 17
[5,1,1,1,1,1,1,1] => 1
[4,4,4] => 2
[4,4,3,1] => 46
[4,4,2,2] => 45
[4,4,2,1,1] => 180
[4,4,1,1,1,1] => 18
[4,3,3,2] => 47
[4,3,3,1,1] => 144
[4,3,2,2,1] => 380
[4,3,2,1,1,1] => 312
[4,3,1,1,1,1,1] => 11
[4,2,2,2,2] => 9
[4,2,2,2,1,1] => 35
[4,2,2,1,1,1,1] => 10
[4,2,1,1,1,1,1,1] => 0
[4,1,1,1,1,1,1,1,1] => 0
[3,3,3,3] => 1
[3,3,3,2,1] => 15
[3,3,3,1,1,1] => 7
[3,3,2,2,2] => 4
[3,3,2,2,1,1] => 21
[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
[5,4,3,1] => 523
[5,4,2,2] => 333
[5,4,2,1,1] => 1573
[5,3,3,2] => 232
[5,3,3,1,1] => 661
[5,3,2,2,1] => 1580
[4,4,3,2] => 85
[4,4,3,1,1] => 235
[4,4,2,2,1] => 325
[4,3,3,2,1] => 494
[5,4,3,2] => 1169
[5,4,3,1,1] => 2929
[5,4,2,2,1] => 3649
[5,3,3,2,1] => 2933
[4,4,3,2,1] => 1154
[5,4,3,2,1] => 18269
-----------------------------------------------------------------------------
Created: Mar 17, 2018 at 10:22 by Martin Rubey
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Last Updated: Mar 17, 2018 at 10:29 by Martin Rubey