Identifier
Identifier
• St001124: ⟶ ℤ (values match St000159The number of distinct parts of the integer partition., St000318The number of addable cells of the Ferrers diagram of an integer partition.)
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
[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
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):
return kronecker_coefficient(la,la,[la.size()-1,1])


Created
Mar 18, 2018 at 07:46 by Martin Rubey
Updated
Mar 19, 2018 at 07:05 by Martin Rubey