Identifier
Identifier
Values
[1] generating graphics... => 1
[2] generating graphics... => 0
[1,1] generating graphics... => 0
[3] generating graphics... => 0
[2,1] generating graphics... => 1
[1,1,1] generating graphics... => 0
[4] generating graphics... => 0
[3,1] generating graphics... => 0
[2,2] generating graphics... => 1
[2,1,1] generating graphics... => 0
[1,1,1,1] generating graphics... => 0
[5] generating graphics... => 0
[4,1] generating graphics... => 0
[3,2] generating graphics... => 0
[3,1,1] generating graphics... => 1
[2,2,1] generating graphics... => 0
[2,1,1,1] generating graphics... => 0
[1,1,1,1,1] generating graphics... => 0
[6] generating graphics... => 0
[5,1] generating graphics... => 0
[4,2] generating graphics... => 0
[4,1,1] generating graphics... => 0
[3,3] generating graphics... => 0
[3,2,1] generating graphics... => 1
[3,1,1,1] generating graphics... => 0
[2,2,2] generating graphics... => 0
[2,2,1,1] generating graphics... => 0
[2,1,1,1,1] generating graphics... => 0
[1,1,1,1,1,1] generating graphics... => 0
[7] generating graphics... => 0
[6,1] generating graphics... => 0
[5,2] generating graphics... => 0
[5,1,1] generating graphics... => 0
[4,3] generating graphics... => 0
[4,2,1] generating graphics... => 0
[4,1,1,1] generating graphics... => 1
[3,3,1] generating graphics... => 0
[3,2,2] generating graphics... => 0
[3,2,1,1] generating graphics... => 0
[3,1,1,1,1] generating graphics... => 0
[2,2,2,1] generating graphics... => 0
[2,2,1,1,1] generating graphics... => 0
[2,1,1,1,1,1] generating graphics... => 0
[1,1,1,1,1,1,1] generating graphics... => 0
[8] generating graphics... => 0
[7,1] generating graphics... => 0
[6,2] generating graphics... => 0
[6,1,1] generating graphics... => 0
[5,3] generating graphics... => 0
[5,2,1] generating graphics... => 0
[5,1,1,1] generating graphics... => 0
[4,4] generating graphics... => 0
[4,3,1] generating graphics... => 0
[4,2,2] generating graphics... => 0
[4,2,1,1] generating graphics... => 1
[4,1,1,1,1] generating graphics... => 0
[3,3,2] generating graphics... => 1
[3,3,1,1] generating graphics... => 0
[3,2,2,1] generating graphics... => 0
[3,2,1,1,1] generating graphics... => 0
[3,1,1,1,1,1] generating graphics... => 0
[2,2,2,2] generating graphics... => 0
[2,2,2,1,1] generating graphics... => 0
[2,2,1,1,1,1] generating graphics... => 0
[2,1,1,1,1,1,1] generating graphics... => 0
[1,1,1,1,1,1,1,1] generating graphics... => 0
[9] generating graphics... => 0
[8,1] generating graphics... => 0
[7,2] generating graphics... => 0
[7,1,1] generating graphics... => 0
[6,3] generating graphics... => 0
[6,2,1] generating graphics... => 0
[6,1,1,1] generating graphics... => 0
[5,4] generating graphics... => 0
[5,3,1] generating graphics... => 0
[5,2,2] generating graphics... => 0
[5,2,1,1] generating graphics... => 0
[5,1,1,1,1] generating graphics... => 1
[4,4,1] generating graphics... => 0
[4,3,2] generating graphics... => 0
[4,3,1,1] generating graphics... => 0
[4,2,2,1] generating graphics... => 0
[4,2,1,1,1] generating graphics... => 0
[4,1,1,1,1,1] generating graphics... => 0
[3,3,3] generating graphics... => 1
[3,3,2,1] generating graphics... => 0
[3,3,1,1,1] generating graphics... => 0
[3,2,2,2] generating graphics... => 0
[3,2,2,1,1] generating graphics... => 0
[3,2,1,1,1,1] generating graphics... => 0
[3,1,1,1,1,1,1] generating graphics... => 0
[2,2,2,2,1] generating graphics... => 0
[2,2,2,1,1,1] generating graphics... => 0
[2,2,1,1,1,1,1] generating graphics... => 0
[2,1,1,1,1,1,1,1] generating graphics... => 0
[1,1,1,1,1,1,1,1,1] generating graphics... => 0
[10] generating graphics... => 0
[9,1] generating graphics... => 0
[8,2] generating graphics... => 0
[8,1,1] generating graphics... => 0
[7,3] generating graphics... => 0
[7,2,1] generating graphics... => 0
[7,1,1,1] generating graphics... => 0
[6,4] generating graphics... => 0
[6,3,1] generating graphics... => 0
[6,2,2] generating graphics... => 0
[6,2,1,1] generating graphics... => 0
[6,1,1,1,1] generating graphics... => 0
[5,5] generating graphics... => 0
[5,4,1] generating graphics... => 0
[5,3,2] generating graphics... => 0
[5,3,1,1] generating graphics... => 0
[5,2,2,1] generating graphics... => 0
[5,2,1,1,1] generating graphics... => 1
[5,1,1,1,1,1] generating graphics... => 0
[4,4,2] generating graphics... => 0
[4,4,1,1] generating graphics... => 0
[4,3,3] generating graphics... => 0
[4,3,2,1] generating graphics... => 1
[4,3,1,1,1] generating graphics... => 0
[4,2,2,2] generating graphics... => 0
[4,2,2,1,1] generating graphics... => 0
[4,2,1,1,1,1] generating graphics... => 0
[4,1,1,1,1,1,1] generating graphics... => 0
[3,3,3,1] generating graphics... => 0
[3,3,2,2] generating graphics... => 0
[3,3,2,1,1] generating graphics... => 0
[3,3,1,1,1,1] generating graphics... => 0
[3,2,2,2,1] generating graphics... => 0
[3,2,2,1,1,1] generating graphics... => 0
[3,2,1,1,1,1,1] generating graphics... => 0
[3,1,1,1,1,1,1,1] generating graphics... => 0
[2,2,2,2,2] generating graphics... => 0
[2,2,2,2,1,1] generating graphics... => 0
[2,2,2,1,1,1,1] generating graphics... => 0
[2,2,1,1,1,1,1,1] generating graphics... => 0
[2,1,1,1,1,1,1,1,1] generating graphics... => 0
[1,1,1,1,1,1,1,1,1,1] generating graphics... => 0
[11] generating graphics... => 0
[10,1] generating graphics... => 0
[9,2] generating graphics... => 0
[9,1,1] generating graphics... => 0
[8,3] generating graphics... => 0
[8,2,1] generating graphics... => 0
[8,1,1,1] generating graphics... => 0
[7,4] generating graphics... => 0
[7,3,1] generating graphics... => 0
[7,2,2] generating graphics... => 0
[7,2,1,1] generating graphics... => 0
[7,1,1,1,1] generating graphics... => 0
[6,5] generating graphics... => 0
[6,4,1] generating graphics... => 0
[6,3,2] generating graphics... => 0
[6,3,1,1] generating graphics... => 0
[6,2,2,1] generating graphics... => 0
[6,2,1,1,1] generating graphics... => 0
[6,1,1,1,1,1] generating graphics... => 1
[5,5,1] generating graphics... => 0
[5,4,2] generating graphics... => 0
[5,4,1,1] generating graphics... => 0
[5,3,3] generating graphics... => 0
[5,3,2,1] generating graphics... => 0
[5,3,1,1,1] generating graphics... => 0
[5,2,2,2] generating graphics... => 0
[5,2,2,1,1] generating graphics... => 0
[5,2,1,1,1,1] generating graphics... => 0
[5,1,1,1,1,1,1] generating graphics... => 0
[4,4,3] generating graphics... => 0
[4,4,2,1] generating graphics... => 0
[4,4,1,1,1] generating graphics... => 0
[4,3,3,1] generating graphics... => 1
[4,3,2,2] generating graphics... => 0
[4,3,2,1,1] generating graphics... => 0
[4,3,1,1,1,1] generating graphics... => 0
[4,2,2,2,1] generating graphics... => 0
[4,2,2,1,1,1] generating graphics... => 0
[4,2,1,1,1,1,1] generating graphics... => 0
[4,1,1,1,1,1,1,1] generating graphics... => 0
[3,3,3,2] generating graphics... => 0
[3,3,3,1,1] generating graphics... => 0
[3,3,2,2,1] generating graphics... => 0
[3,3,2,1,1,1] generating graphics... => 0
[3,3,1,1,1,1,1] generating graphics... => 0
[3,2,2,2,2] generating graphics... => 0
[3,2,2,2,1,1] generating graphics... => 0
[3,2,2,1,1,1,1] generating graphics... => 0
[3,2,1,1,1,1,1,1] generating graphics... => 0
[3,1,1,1,1,1,1,1,1] generating graphics... => 0
[2,2,2,2,2,1] generating graphics... => 0
[2,2,2,2,1,1,1] generating graphics... => 0
[2,2,2,1,1,1,1,1] generating graphics... => 0
[2,2,1,1,1,1,1,1,1] generating graphics... => 0
[2,1,1,1,1,1,1,1,1,1] generating graphics... => 0
[1,1,1,1,1,1,1,1,1,1,1] generating graphics... => 0
[12] generating graphics... => 0
[11,1] generating graphics... => 0
[10,2] generating graphics... => 0
[10,1,1] generating graphics... => 0
[9,3] generating graphics... => 0
[9,2,1] generating graphics... => 0
[9,1,1,1] generating graphics... => 0
[8,4] generating graphics... => 0
[8,3,1] generating graphics... => 0
[8,2,2] generating graphics... => 0
[8,2,1,1] generating graphics... => 0
[8,1,1,1,1] generating graphics... => 0
[7,5] generating graphics... => 0
[7,4,1] generating graphics... => 0
[7,3,2] generating graphics... => 0
[7,3,1,1] generating graphics... => 0
[7,2,2,1] generating graphics... => 0
[7,2,1,1,1] generating graphics... => 0
[7,1,1,1,1,1] generating graphics... => 0
[6,6] generating graphics... => 0
[6,5,1] generating graphics... => 0
[6,4,2] generating graphics... => 0
[6,4,1,1] generating graphics... => 0
[6,3,3] generating graphics... => 0
[6,3,2,1] generating graphics... => 0
[6,3,1,1,1] generating graphics... => 0
[6,2,2,2] generating graphics... => 0
[6,2,2,1,1] generating graphics... => 0
[6,2,1,1,1,1] generating graphics... => 1
[6,1,1,1,1,1,1] generating graphics... => 0
[5,5,2] generating graphics... => 0
[5,5,1,1] generating graphics... => 0
[5,4,3] generating graphics... => 0
[5,4,2,1] generating graphics... => 0
[5,4,1,1,1] generating graphics... => 0
[5,3,3,1] generating graphics... => 0
[5,3,2,2] generating graphics... => 0
[5,3,2,1,1] generating graphics... => 1
[5,3,1,1,1,1] generating graphics... => 0
[5,2,2,2,1] generating graphics... => 0
[5,2,2,1,1,1] generating graphics... => 0
[5,2,1,1,1,1,1] generating graphics... => 0
[5,1,1,1,1,1,1,1] generating graphics... => 0
[4,4,4] generating graphics... => 0
[4,4,3,1] generating graphics... => 0
[4,4,2,2] generating graphics... => 1
[4,4,2,1,1] generating graphics... => 0
[4,4,1,1,1,1] generating graphics... => 0
[4,3,3,2] generating graphics... => 0
[4,3,3,1,1] generating graphics... => 0
[4,3,2,2,1] generating graphics... => 0
[4,3,2,1,1,1] generating graphics... => 0
[4,3,1,1,1,1,1] generating graphics... => 0
[4,2,2,2,2] generating graphics... => 0
[4,2,2,2,1,1] generating graphics... => 0
[4,2,2,1,1,1,1] generating graphics... => 0
[4,2,1,1,1,1,1,1] generating graphics... => 0
[4,1,1,1,1,1,1,1,1] generating graphics... => 0
[3,3,3,3] generating graphics... => 0
[3,3,3,2,1] generating graphics... => 0
[3,3,3,1,1,1] generating graphics... => 0
[3,3,2,2,2] generating graphics... => 0
[3,3,2,2,1,1] generating graphics... => 0
[3,3,2,1,1,1,1] generating graphics... => 0
[3,3,1,1,1,1,1,1] generating graphics... => 0
[3,2,2,2,2,1] generating graphics... => 0
[3,2,2,2,1,1,1] generating graphics... => 0
[3,2,2,1,1,1,1,1] generating graphics... => 0
[3,2,1,1,1,1,1,1,1] generating graphics... => 0
[3,1,1,1,1,1,1,1,1,1] generating graphics... => 0
[2,2,2,2,2,2] generating graphics... => 0
[2,2,2,2,2,1,1] generating graphics... => 0
[2,2,2,2,1,1,1,1] generating graphics... => 0
[2,2,2,1,1,1,1,1,1] generating graphics... => 0
[2,2,1,1,1,1,1,1,1,1] generating graphics... => 0
[2,1,1,1,1,1,1,1,1,1,1] generating graphics... => 0
[1,1,1,1,1,1,1,1,1,1,1,1] generating graphics... => 0
[5,4,3,1] generating graphics... => 0
[5,4,2,2] generating graphics... => 0
[5,4,2,1,1] generating graphics... => 0
[5,3,3,2] generating graphics... => 0
[5,3,3,1,1] generating graphics... => 1
[5,3,2,2,1] generating graphics... => 0
[4,4,3,2] generating graphics... => 1
[4,4,3,1,1] generating graphics... => 0
[4,4,2,2,1] generating graphics... => 0
[4,3,3,2,1] generating graphics... => 0
[5,4,3,2] generating graphics... => 0
[5,4,3,1,1] generating graphics... => 0
[5,4,2,2,1] generating graphics... => 1
[5,3,3,2,1] generating graphics... => 0
[4,4,3,2,1] generating graphics... => 0
[5,4,3,2,1] generating graphics... => 1
click to show generating function       
Description
The multiplicity of the sign 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 $g_{\lambda,\lambda}^{1^n}$, for $\lambda\vdash n$. It equals $1$ if and only if $\lambda$ is self-conjugate.
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,[1]*la.size())
Created
Mar 17, 2018 at 11:57 by Martin Rubey
Updated
Sep 14, 2018 at 19:08 by Martin Rubey