Identifier
- St001123: Integer partitions ⟶ ℤ
Values
=>
Cc0002;cc-rep
[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
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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 $\lambda\vdash n$.
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
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))
Created
Mar 17, 2018 at 14:34 by Martin Rubey
Updated
Mar 17, 2018 at 14:34 by Martin Rubey
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