edit this statistic or download as text // json
Identifier
Values
[2] => 0
[1,1] => 0
[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] => 1
[3,2] => 1
[3,1,1] => 1
[2,2,1] => 1
[2,1,1,1] => 1
[1,1,1,1,1] => 0
[6] => 0
[5,1] => 1
[4,2] => 1
[4,1,1] => 1
[3,3] => 0
[3,2,1] => 2
[3,1,1,1] => 1
[2,2,2] => 0
[2,2,1,1] => 1
[2,1,1,1,1] => 1
[1,1,1,1,1,1] => 0
[7] => 0
[6,1] => 1
[5,2] => 1
[5,1,1] => 1
[4,3] => 1
[4,2,1] => 2
[4,1,1,1] => 1
[3,3,1] => 1
[3,2,2] => 1
[3,2,1,1] => 2
[3,1,1,1,1] => 1
[2,2,2,1] => 1
[2,2,1,1,1] => 1
[2,1,1,1,1,1] => 1
[1,1,1,1,1,1,1] => 0
[8] => 0
[7,1] => 1
[6,2] => 1
[6,1,1] => 1
[5,3] => 1
[5,2,1] => 2
[5,1,1,1] => 1
[4,4] => 0
[4,3,1] => 2
[4,2,2] => 1
[4,2,1,1] => 2
[4,1,1,1,1] => 1
[3,3,2] => 1
[3,3,1,1] => 1
[3,2,2,1] => 2
[3,2,1,1,1] => 2
[3,1,1,1,1,1] => 1
[2,2,2,2] => 0
[2,2,2,1,1] => 1
[2,2,1,1,1,1] => 1
[2,1,1,1,1,1,1] => 1
[1,1,1,1,1,1,1,1] => 0
[9] => 0
[8,1] => 1
[7,2] => 1
[7,1,1] => 1
[6,3] => 1
[6,2,1] => 2
[6,1,1,1] => 1
[5,4] => 1
[5,3,1] => 2
[5,2,2] => 1
[5,2,1,1] => 2
[5,1,1,1,1] => 1
[4,4,1] => 1
[4,3,2] => 2
[4,3,1,1] => 2
[4,2,2,1] => 2
[4,2,1,1,1] => 2
[4,1,1,1,1,1] => 1
[3,3,3] => 0
[3,3,2,1] => 2
[3,3,1,1,1] => 1
[3,2,2,2] => 1
[3,2,2,1,1] => 2
[3,2,1,1,1,1] => 2
[3,1,1,1,1,1,1] => 1
[2,2,2,2,1] => 1
[2,2,2,1,1,1] => 1
[2,2,1,1,1,1,1] => 1
[2,1,1,1,1,1,1,1] => 1
[1,1,1,1,1,1,1,1,1] => 0
[10] => 0
[9,1] => 1
[8,2] => 1
[8,1,1] => 1
[7,3] => 1
[7,2,1] => 2
>>> Load all 186 entries. <<<
[7,1,1,1] => 1
[6,4] => 1
[6,3,1] => 2
[6,2,2] => 1
[6,2,1,1] => 2
[6,1,1,1,1] => 1
[5,5] => 0
[5,4,1] => 2
[5,3,2] => 2
[5,3,1,1] => 2
[5,2,2,1] => 2
[5,2,1,1,1] => 2
[5,1,1,1,1,1] => 1
[4,4,2] => 1
[4,4,1,1] => 1
[4,3,3] => 1
[4,3,2,1] => 3
[4,3,1,1,1] => 2
[4,2,2,2] => 1
[4,2,2,1,1] => 2
[4,2,1,1,1,1] => 2
[4,1,1,1,1,1,1] => 1
[3,3,3,1] => 1
[3,3,2,2] => 1
[3,3,2,1,1] => 2
[3,3,1,1,1,1] => 1
[3,2,2,2,1] => 2
[3,2,2,1,1,1] => 2
[3,2,1,1,1,1,1] => 2
[3,1,1,1,1,1,1,1] => 1
[2,2,2,2,2] => 0
[2,2,2,2,1,1] => 1
[2,2,2,1,1,1,1] => 1
[2,2,1,1,1,1,1,1] => 1
[2,1,1,1,1,1,1,1,1] => 1
[1,1,1,1,1,1,1,1,1,1] => 0
[5,4,2] => 2
[5,4,1,1] => 2
[5,3,3] => 1
[5,3,2,1] => 3
[5,3,1,1,1] => 2
[5,2,2,2] => 1
[5,2,2,1,1] => 2
[4,4,3] => 1
[4,4,2,1] => 2
[4,4,1,1,1] => 1
[4,3,3,1] => 2
[4,3,2,2] => 2
[4,3,2,1,1] => 3
[4,2,2,2,1] => 2
[3,3,3,2] => 1
[3,3,3,1,1] => 1
[3,3,2,2,1] => 2
[6,4,2] => 2
[5,4,3] => 2
[5,4,2,1] => 3
[5,4,1,1,1] => 2
[5,3,3,1] => 2
[5,3,2,2] => 2
[5,3,2,1,1] => 3
[5,2,2,2,1] => 2
[4,4,3,1] => 2
[4,4,2,2] => 1
[4,4,2,1,1] => 2
[4,3,3,2] => 2
[4,3,3,1,1] => 2
[4,3,2,2,1] => 3
[3,3,3,2,1] => 2
[3,3,2,2,1,1] => 2
[5,4,3,1] => 3
[5,4,2,2] => 2
[5,4,2,1,1] => 3
[5,3,3,2] => 2
[5,3,3,1,1] => 2
[5,3,2,2,1] => 3
[4,4,3,2] => 2
[4,4,3,1,1] => 2
[4,4,2,2,1] => 2
[4,3,3,2,1] => 3
[5,4,3,2] => 3
[5,4,3,1,1] => 3
[5,4,2,2,1] => 3
[5,3,3,2,1] => 3
[4,4,3,2,1] => 3
[5,4,3,2,1] => 4
search for individual values
searching the database for the individual values of this statistic
/ search for generating function
searching the database for statistics with the same generating function
click to show known generating functions       
Description
The multiplicity 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 $g_{\lambda,\lambda}^{(n-1)1}$, for $\lambda\vdash n > 1$. For $n\leq1$ the statistic is undefined.
It follows from [3, Prop.4.1] (or, slightly easier from [3, Thm.4.2]) that this is one less than St000159The number of distinct parts of the integer partition., the number of distinct parts of the partition.
References
[1] wikipedia:Kronecker coefficient
[2] https://groupprops.subwiki.org/wiki/Standard_representation
[3] Ini Liu, R. A simplified Kronecker rule for one hook shape arXiv:1412.2180
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):
    if not la:
        raise ValueError("partition must not be empty")
    return kronecker_coefficient(la,la,[la.size()-1,1])
Created
Mar 18, 2018 at 07:46 by Martin Rubey
Updated
Jun 25, 2021 at 10:01 by Martin Rubey