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-----------------------------------------------------------------------------
Statistic identifier: St001123
-----------------------------------------------------------------------------
Collection: Integer partitions
-----------------------------------------------------------------------------
Description: The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a 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 $g_{\lambda,\lambda}^{21^{n-2}}$, for $\lambda\vdash n$.
-----------------------------------------------------------------------------
References: [1] [[wikipedia:Kronecker coefficient]]
-----------------------------------------------------------------------------
Code:
from sage.libs.symmetrica.symmetrica import charvalue_symmetrica as chv
def kronecker_coefficient(*partns):
if partns == ():
return 1
else:
return sum(mul(chv(la,mu) for la in partns)/mu.centralizer_size() for mu in Partitions(sum(partns[0])))
def statistic(la):
return kronecker_coefficient(la,la,[2]+[1]*(la.size()-2))
-----------------------------------------------------------------------------
Statistic values:
[2] => 1
[1,1] => 1
[3] => 0
[2,1] => 1
[1,1,1] => 0
[4] => 0
[3,1] => 1
[2,2] => 0
[2,1,1] => 1
[1,1,1,1] => 0
[5] => 0
[4,1] => 0
[3,2] => 1
[3,1,1] => 1
[2,2,1] => 1
[2,1,1,1] => 0
[1,1,1,1,1] => 0
[6] => 0
[5,1] => 0
[4,2] => 0
[4,1,1] => 1
[3,3] => 0
[3,2,1] => 2
[3,1,1,1] => 1
[2,2,2] => 0
[2,2,1,1] => 0
[2,1,1,1,1] => 0
[1,1,1,1,1,1] => 0
[7] => 0
[6,1] => 0
[5,2] => 0
[5,1,1] => 0
[4,3] => 0
[4,2,1] => 1
[4,1,1,1] => 1
[3,3,1] => 1
[3,2,2] => 1
[3,2,1,1] => 1
[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] => 0
[7,1] => 0
[6,2] => 0
[6,1,1] => 0
[5,3] => 0
[5,2,1] => 0
[5,1,1,1] => 1
[4,4] => 0
[4,3,1] => 0
[4,2,2] => 0
[4,2,1,1] => 2
[4,1,1,1,1] => 1
[3,3,2] => 1
[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] => 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] => 1
[5,1,1,1,1] => 1
[4,4,1] => 0
[4,3,2] => 1
[4,3,1,1] => 1
[4,2,2,1] => 1
[4,2,1,1,1] => 1
[4,1,1,1,1,1] => 0
[3,3,3] => 0
[3,3,2,1] => 1
[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] => 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] => 1
[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] => 2
[5,1,1,1,1,1] => 1
[4,4,2] => 0
[4,4,1,1] => 0
[4,3,3] => 1
[4,3,2,1] => 3
[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] => 1
[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] => 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] => 1
[6,1,1,1,1,1] => 1
[5,5,1] => 0
[5,4,2] => 0
[5,4,1,1] => 0
[5,3,3] => 0
[5,3,2,1] => 1
[5,3,1,1,1] => 1
[5,2,2,2] => 0
[5,2,2,1,1] => 1
[5,2,1,1,1,1] => 1
[5,1,1,1,1,1,1] => 0
[4,4,3] => 0
[4,4,2,1] => 1
[4,4,1,1,1] => 0
[4,3,3,1] => 2
[4,3,2,2] => 1
[4,3,2,1,1] => 1
[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] => 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] => 1
[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] => 2
[6,1,1,1,1,1,1] => 1
[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] => 1
[5,3,2,2] => 0
[5,3,2,1,1] => 3
[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] => 1
[4,4,2,2] => 1
[4,4,2,1,1] => 0
[4,4,1,1,1,1] => 0
[4,3,3,2] => 1
[4,3,3,1,1] => 1
[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
-----------------------------------------------------------------------------
Created: Mar 17, 2018 at 14:34 by Martin Rubey
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Last Updated: Mar 17, 2018 at 14:34 by Martin Rubey