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Identifier
Values
=>
Cc0002;cc-rep
[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 [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
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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