***************************************************************************** * www.FindStat.org - The Combinatorial Statistic Finder * * * * Copyright (C) 2019 The FindStatCrew * * * * This information is distributed in the hope that it will be useful, * * but WITHOUT ANY WARRANTY; without even the implied warranty of * * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. * ***************************************************************************** ----------------------------------------------------------------------------- Statistic identifier: St001124 ----------------------------------------------------------------------------- Collection: Integer partitions ----------------------------------------------------------------------------- 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 [[St000159]], 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]) ----------------------------------------------------------------------------- Statistic 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 ----------------------------------------------------------------------------- Created: Mar 18, 2018 at 07:46 by Martin Rubey ----------------------------------------------------------------------------- Last Updated: Jun 25, 2021 at 10:01 by Martin Rubey