***************************************************************************** * 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: St001121 ----------------------------------------------------------------------------- Collection: Integer partitions ----------------------------------------------------------------------------- Description: The multiplicity of the irreducible representation indexed by the partition in the Kronecker square corresponding to the 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 $g_{\lambda,\lambda}^\lambda$. ----------------------------------------------------------------------------- References: [1] [[wikipedia:Kronecker coefficient]] ----------------------------------------------------------------------------- Code: def statistic(la): s = SymmetricFunctions(ZZ).schur() return s[la].internal_product(s[la]).coefficient(la) ----------------------------------------------------------------------------- Statistic values: [] => 1 [1] => 1 [2] => 1 [1,1] => 0 [3] => 1 [2,1] => 1 [1,1,1] => 0 [4] => 1 [3,1] => 1 [2,2] => 1 [2,1,1] => 1 [1,1,1,1] => 0 [5] => 1 [4,1] => 1 [3,2] => 1 [3,1,1] => 1 [2,2,1] => 1 [2,1,1,1] => 0 [1,1,1,1,1] => 0 [6] => 1 [5,1] => 1 [4,2] => 2 [4,1,1] => 1 [3,3] => 0 [3,2,1] => 5 [3,1,1,1] => 1 [2,2,2] => 1 [2,2,1,1] => 0 [2,1,1,1,1] => 0 [1,1,1,1,1,1] => 0 [7] => 1 [6,1] => 1 [5,2] => 2 [5,1,1] => 1 [4,3] => 1 [4,2,1] => 9 [4,1,1,1] => 1 [3,3,1] => 1 [3,2,2] => 2 [3,2,1,1] => 8 [3,1,1,1,1] => 1 [2,2,2,1] => 1 [2,2,1,1,1] => 0 [2,1,1,1,1,1] => 0 [1,1,1,1,1,1,1] => 0 [8] => 1 [7,1] => 1 [6,2] => 2 [6,1,1] => 1 [5,3] => 1 [5,2,1] => 9 [5,1,1,1] => 1 [4,4] => 1 [4,3,1] => 8 [4,2,2] => 6 [4,2,1,1] => 17 [4,1,1,1,1] => 1 [3,3,2] => 1 [3,3,1,1] => 5 [3,2,2,1] => 8 [3,2,1,1,1] => 4 [3,1,1,1,1,1] => 0 [2,2,2,2] => 1 [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] => 1 [8,1] => 1 [7,2] => 2 [7,1,1] => 1 [6,3] => 2 [6,2,1] => 9 [6,1,1,1] => 1 [5,4] => 1 [5,3,1] => 15 [5,2,2] => 7 [5,2,1,1] => 18 [5,1,1,1,1] => 1 [4,4,1] => 2 [4,3,2] => 12 [4,3,1,1] => 27 [4,2,2,1] => 28 [4,2,1,1,1] => 17 [4,1,1,1,1,1] => 1 [3,3,3] => 1 [3,3,2,1] => 11 [3,3,1,1,1] => 5 [3,2,2,2] => 2 [3,2,2,1,1] => 7 [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] => 1 [9,1] => 1 [8,2] => 2 [8,1,1] => 1 [7,3] => 2 [7,2,1] => 9 [7,1,1,1] => 1 [6,4] => 2 [6,3,1] => 19 [6,2,2] => 7 [6,2,1,1] => 18 [6,1,1,1,1] => 1 [5,5] => 0 [5,4,1] => 9 [5,3,2] => 29 [5,3,1,1] => 53 [5,2,2,1] => 39 [5,2,1,1,1] => 21 [5,1,1,1,1,1] => 1 [4,4,2] => 6 [4,4,1,1] => 5 [4,3,3] => 2 [4,3,2,1] => 117 [4,3,1,1,1] => 40 [4,2,2,2] => 10 [4,2,2,1,1] => 46 [4,2,1,1,1,1] => 11 [4,1,1,1,1,1,1] => 1 [3,3,3,1] => 2 [3,3,2,2] => 2 [3,3,2,1,1] => 21 [3,3,1,1,1,1] => 1 [3,2,2,2,1] => 5 [3,2,2,1,1,1] => 1 [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] => 1 [10,1] => 1 [9,2] => 2 [9,1,1] => 1 [8,3] => 2 [8,2,1] => 9 [8,1,1,1] => 1 [7,4] => 2 [7,3,1] => 19 [7,2,2] => 7 [7,2,1,1] => 18 [7,1,1,1,1] => 1 [6,5] => 1 [6,4,1] => 19 [6,3,2] => 39 [6,3,1,1] => 62 [6,2,2,1] => 39 [6,2,1,1,1] => 21 [6,1,1,1,1,1] => 1 [5,5,1] => 1 [5,4,2] => 29 [5,4,1,1] => 40 [5,3,3] => 6 [5,3,2,1] => 312 [5,3,1,1,1] => 89 [5,2,2,2] => 17 [5,2,2,1,1] => 86 [5,2,1,1,1,1] => 20 [5,1,1,1,1,1,1] => 1 [4,4,3] => 2 [4,4,2,1] => 53 [4,4,1,1,1] => 14 [4,3,3,1] => 37 [4,3,2,2] => 53 [4,3,2,1,1] => 301 [4,3,1,1,1,1] => 30 [4,2,2,2,1] => 37 [4,2,2,1,1,1] => 32 [4,2,1,1,1,1,1] => 4 [4,1,1,1,1,1,1,1] => 0 [3,3,3,2] => 2 [3,3,3,1,1] => 8 [3,3,2,2,1] => 21 [3,3,2,1,1,1] => 11 [3,3,1,1,1,1,1] => 0 [3,2,2,2,2] => 1 [3,2,2,2,1,1] => 1 [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] => 1 [11,1] => 1 [10,2] => 2 [10,1,1] => 1 [9,3] => 2 [9,2,1] => 9 [9,1,1,1] => 1 [8,4] => 3 [8,3,1] => 19 [8,2,2] => 7 [8,2,1,1] => 18 [8,1,1,1,1] => 1 [7,5] => 1 [7,4,1] => 26 [7,3,2] => 40 [7,3,1,1] => 63 [7,2,2,1] => 39 [7,2,1,1,1] => 21 [7,1,1,1,1,1] => 1 [6,6] => 1 [6,5,1] => 9 [6,4,2] => 71 [6,4,1,1] => 80 [6,3,3] => 13 [6,3,2,1] => 429 [6,3,1,1,1] => 108 [6,2,2,2] => 19 [6,2,2,1,1] => 90 [6,2,1,1,1,1] => 21 [6,1,1,1,1,1,1] => 1 [5,5,2] => 5 [5,5,1,1] => 9 [5,4,3] => 20 [5,4,2,1] => 407 [5,4,1,1,1] => 89 [5,3,3,1] => 146 [5,3,2,2] => 179 [5,3,2,1,1] => 945 [5,3,1,1,1,1] => 91 [5,2,2,2,1] => 94 [5,2,2,1,1,1] => 103 [5,2,1,1,1,1,1] => 17 [5,1,1,1,1,1,1,1] => 1 [4,4,4] => 2 [4,4,3,1] => 46 [4,4,2,2] => 45 [4,4,2,1,1] => 180 [4,4,1,1,1,1] => 18 [4,3,3,2] => 47 [4,3,3,1,1] => 144 [4,3,2,2,1] => 380 [4,3,2,1,1,1] => 312 [4,3,1,1,1,1,1] => 11 [4,2,2,2,2] => 9 [4,2,2,2,1,1] => 35 [4,2,2,1,1,1,1] => 10 [4,2,1,1,1,1,1,1] => 0 [4,1,1,1,1,1,1,1,1] => 0 [3,3,3,3] => 1 [3,3,3,2,1] => 15 [3,3,3,1,1,1] => 7 [3,3,2,2,2] => 4 [3,3,2,2,1,1] => 21 [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] => 523 [5,4,2,2] => 333 [5,4,2,1,1] => 1573 [5,3,3,2] => 232 [5,3,3,1,1] => 661 [5,3,2,2,1] => 1580 [4,4,3,2] => 85 [4,4,3,1,1] => 235 [4,4,2,2,1] => 325 [4,3,3,2,1] => 494 [5,4,3,2] => 1169 [5,4,3,1,1] => 2929 [5,4,2,2,1] => 3649 [5,3,3,2,1] => 2933 [4,4,3,2,1] => 1154 [5,4,3,2,1] => 18269 ----------------------------------------------------------------------------- Created: Mar 17, 2018 at 10:22 by Martin Rubey ----------------------------------------------------------------------------- Last Updated: Mar 17, 2018 at 10:29 by Martin Rubey