***************************************************************************** * 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: St001122 ----------------------------------------------------------------------------- Collection: Integer partitions ----------------------------------------------------------------------------- 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. ----------------------------------------------------------------------------- References: [1] [[wikipedia:Kronecker coefficient]] ----------------------------------------------------------------------------- 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()) ----------------------------------------------------------------------------- Statistic values: [1] => 1 [2] => 0 [1,1] => 0 [3] => 0 [2,1] => 1 [1,1,1] => 0 [4] => 0 [3,1] => 0 [2,2] => 1 [2,1,1] => 0 [1,1,1,1] => 0 [5] => 0 [4,1] => 0 [3,2] => 0 [3,1,1] => 1 [2,2,1] => 0 [2,1,1,1] => 0 [1,1,1,1,1] => 0 [6] => 0 [5,1] => 0 [4,2] => 0 [4,1,1] => 0 [3,3] => 0 [3,2,1] => 1 [3,1,1,1] => 0 [2,2,2] => 0 [2,2,1,1] => 0 [2,1,1,1,1] => 0 [1,1,1,1,1,1] => 0 [7] => 0 [6,1] => 0 [5,2] => 0 [5,1,1] => 0 [4,3] => 0 [4,2,1] => 0 [4,1,1,1] => 1 [3,3,1] => 0 [3,2,2] => 0 [3,2,1,1] => 0 [3,1,1,1,1] => 0 [2,2,2,1] => 0 [2,2,1,1,1] => 0 [2,1,1,1,1,1] => 0 [1,1,1,1,1,1,1] => 0 [8] => 0 [7,1] => 0 [6,2] => 0 [6,1,1] => 0 [5,3] => 0 [5,2,1] => 0 [5,1,1,1] => 0 [4,4] => 0 [4,3,1] => 0 [4,2,2] => 0 [4,2,1,1] => 1 [4,1,1,1,1] => 0 [3,3,2] => 1 [3,3,1,1] => 0 [3,2,2,1] => 0 [3,2,1,1,1] => 0 [3,1,1,1,1,1] => 0 [2,2,2,2] => 0 [2,2,2,1,1] => 0 [2,2,1,1,1,1] => 0 [2,1,1,1,1,1,1] => 0 [1,1,1,1,1,1,1,1] => 0 [9] => 0 [8,1] => 0 [7,2] => 0 [7,1,1] => 0 [6,3] => 0 [6,2,1] => 0 [6,1,1,1] => 0 [5,4] => 0 [5,3,1] => 0 [5,2,2] => 0 [5,2,1,1] => 0 [5,1,1,1,1] => 1 [4,4,1] => 0 [4,3,2] => 0 [4,3,1,1] => 0 [4,2,2,1] => 0 [4,2,1,1,1] => 0 [4,1,1,1,1,1] => 0 [3,3,3] => 1 [3,3,2,1] => 0 [3,3,1,1,1] => 0 [3,2,2,2] => 0 [3,2,2,1,1] => 0 [3,2,1,1,1,1] => 0 [3,1,1,1,1,1,1] => 0 [2,2,2,2,1] => 0 [2,2,2,1,1,1] => 0 [2,2,1,1,1,1,1] => 0 [2,1,1,1,1,1,1,1] => 0 [1,1,1,1,1,1,1,1,1] => 0 [10] => 0 [9,1] => 0 [8,2] => 0 [8,1,1] => 0 [7,3] => 0 [7,2,1] => 0 [7,1,1,1] => 0 [6,4] => 0 [6,3,1] => 0 [6,2,2] => 0 [6,2,1,1] => 0 [6,1,1,1,1] => 0 [5,5] => 0 [5,4,1] => 0 [5,3,2] => 0 [5,3,1,1] => 0 [5,2,2,1] => 0 [5,2,1,1,1] => 1 [5,1,1,1,1,1] => 0 [4,4,2] => 0 [4,4,1,1] => 0 [4,3,3] => 0 [4,3,2,1] => 1 [4,3,1,1,1] => 0 [4,2,2,2] => 0 [4,2,2,1,1] => 0 [4,2,1,1,1,1] => 0 [4,1,1,1,1,1,1] => 0 [3,3,3,1] => 0 [3,3,2,2] => 0 [3,3,2,1,1] => 0 [3,3,1,1,1,1] => 0 [3,2,2,2,1] => 0 [3,2,2,1,1,1] => 0 [3,2,1,1,1,1,1] => 0 [3,1,1,1,1,1,1,1] => 0 [2,2,2,2,2] => 0 [2,2,2,2,1,1] => 0 [2,2,2,1,1,1,1] => 0 [2,2,1,1,1,1,1,1] => 0 [2,1,1,1,1,1,1,1,1] => 0 [1,1,1,1,1,1,1,1,1,1] => 0 [11] => 0 [10,1] => 0 [9,2] => 0 [9,1,1] => 0 [8,3] => 0 [8,2,1] => 0 [8,1,1,1] => 0 [7,4] => 0 [7,3,1] => 0 [7,2,2] => 0 [7,2,1,1] => 0 [7,1,1,1,1] => 0 [6,5] => 0 [6,4,1] => 0 [6,3,2] => 0 [6,3,1,1] => 0 [6,2,2,1] => 0 [6,2,1,1,1] => 0 [6,1,1,1,1,1] => 1 [5,5,1] => 0 [5,4,2] => 0 [5,4,1,1] => 0 [5,3,3] => 0 [5,3,2,1] => 0 [5,3,1,1,1] => 0 [5,2,2,2] => 0 [5,2,2,1,1] => 0 [5,2,1,1,1,1] => 0 [5,1,1,1,1,1,1] => 0 [4,4,3] => 0 [4,4,2,1] => 0 [4,4,1,1,1] => 0 [4,3,3,1] => 1 [4,3,2,2] => 0 [4,3,2,1,1] => 0 [4,3,1,1,1,1] => 0 [4,2,2,2,1] => 0 [4,2,2,1,1,1] => 0 [4,2,1,1,1,1,1] => 0 [4,1,1,1,1,1,1,1] => 0 [3,3,3,2] => 0 [3,3,3,1,1] => 0 [3,3,2,2,1] => 0 [3,3,2,1,1,1] => 0 [3,3,1,1,1,1,1] => 0 [3,2,2,2,2] => 0 [3,2,2,2,1,1] => 0 [3,2,2,1,1,1,1] => 0 [3,2,1,1,1,1,1,1] => 0 [3,1,1,1,1,1,1,1,1] => 0 [2,2,2,2,2,1] => 0 [2,2,2,2,1,1,1] => 0 [2,2,2,1,1,1,1,1] => 0 [2,2,1,1,1,1,1,1,1] => 0 [2,1,1,1,1,1,1,1,1,1] => 0 [1,1,1,1,1,1,1,1,1,1,1] => 0 [12] => 0 [11,1] => 0 [10,2] => 0 [10,1,1] => 0 [9,3] => 0 [9,2,1] => 0 [9,1,1,1] => 0 [8,4] => 0 [8,3,1] => 0 [8,2,2] => 0 [8,2,1,1] => 0 [8,1,1,1,1] => 0 [7,5] => 0 [7,4,1] => 0 [7,3,2] => 0 [7,3,1,1] => 0 [7,2,2,1] => 0 [7,2,1,1,1] => 0 [7,1,1,1,1,1] => 0 [6,6] => 0 [6,5,1] => 0 [6,4,2] => 0 [6,4,1,1] => 0 [6,3,3] => 0 [6,3,2,1] => 0 [6,3,1,1,1] => 0 [6,2,2,2] => 0 [6,2,2,1,1] => 0 [6,2,1,1,1,1] => 1 [6,1,1,1,1,1,1] => 0 [5,5,2] => 0 [5,5,1,1] => 0 [5,4,3] => 0 [5,4,2,1] => 0 [5,4,1,1,1] => 0 [5,3,3,1] => 0 [5,3,2,2] => 0 [5,3,2,1,1] => 1 [5,3,1,1,1,1] => 0 [5,2,2,2,1] => 0 [5,2,2,1,1,1] => 0 [5,2,1,1,1,1,1] => 0 [5,1,1,1,1,1,1,1] => 0 [4,4,4] => 0 [4,4,3,1] => 0 [4,4,2,2] => 1 [4,4,2,1,1] => 0 [4,4,1,1,1,1] => 0 [4,3,3,2] => 0 [4,3,3,1,1] => 0 [4,3,2,2,1] => 0 [4,3,2,1,1,1] => 0 [4,3,1,1,1,1,1] => 0 [4,2,2,2,2] => 0 [4,2,2,2,1,1] => 0 [4,2,2,1,1,1,1] => 0 [4,2,1,1,1,1,1,1] => 0 [4,1,1,1,1,1,1,1,1] => 0 [3,3,3,3] => 0 [3,3,3,2,1] => 0 [3,3,3,1,1,1] => 0 [3,3,2,2,2] => 0 [3,3,2,2,1,1] => 0 [3,3,2,1,1,1,1] => 0 [3,3,1,1,1,1,1,1] => 0 [3,2,2,2,2,1] => 0 [3,2,2,2,1,1,1] => 0 [3,2,2,1,1,1,1,1] => 0 [3,2,1,1,1,1,1,1,1] => 0 [3,1,1,1,1,1,1,1,1,1] => 0 [2,2,2,2,2,2] => 0 [2,2,2,2,2,1,1] => 0 [2,2,2,2,1,1,1,1] => 0 [2,2,2,1,1,1,1,1,1] => 0 [2,2,1,1,1,1,1,1,1,1] => 0 [2,1,1,1,1,1,1,1,1,1,1] => 0 [1,1,1,1,1,1,1,1,1,1,1,1] => 0 [5,4,3,1] => 0 [5,4,2,2] => 0 [5,4,2,1,1] => 0 [5,3,3,2] => 0 [5,3,3,1,1] => 1 [5,3,2,2,1] => 0 [4,4,3,2] => 1 [4,4,3,1,1] => 0 [4,4,2,2,1] => 0 [4,3,3,2,1] => 0 [5,4,3,2] => 0 [5,4,3,1,1] => 0 [5,4,2,2,1] => 1 [5,3,3,2,1] => 0 [4,4,3,2,1] => 0 [5,4,3,2,1] => 1 ----------------------------------------------------------------------------- Created: Mar 17, 2018 at 11:57 by Martin Rubey ----------------------------------------------------------------------------- Last Updated: Sep 14, 2018 at 19:08 by Martin Rubey