***************************************************************************** * 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: St000940 ----------------------------------------------------------------------------- Collection: Integer partitions ----------------------------------------------------------------------------- Description: The number of characters of the symmetric group whose value on the partition is zero. The maximal value for any given size is recorded in [2]. ----------------------------------------------------------------------------- References: [1] Miller, A. R. Note on parity and the irreducible characters of the symmetric group [[arXiv:1708.03267]] [2] Maximal number of zeros in a column of the character table of the symmetric group S_n. [[OEIS:A086642]] ----------------------------------------------------------------------------- Code: def statistic(la): s = SymmetricFunctions(ZZ).s() p = SymmetricFunctions(ZZ).p() P = Partitions(la.size()) r = P.cardinality() res = [0]*r for i, (mu, v) in enumerate(s(p(la))): res[i] = v return len([1 for e in res if is_even(e)]) ----------------------------------------------------------------------------- Statistic values: [2] => 0 [1,1] => 0 [3] => 0 [2,1] => 1 [1,1,1] => 0 [4] => 1 [3,1] => 2 [2,2] => 0 [2,1,1] => 1 [1,1,1,1] => 0 [5] => 2 [4,1] => 3 [3,2] => 1 [3,1,1] => 1 [2,2,1] => 2 [2,1,1,1] => 1 [1,1,1,1,1] => 0 [6] => 5 [5,1] => 6 [4,2] => 3 [4,1,1] => 3 [3,3] => 2 [3,2,1] => 5 [3,1,1,1] => 2 [2,2,2] => 1 [2,2,1,1] => 1 [2,1,1,1,1] => 1 [1,1,1,1,1,1] => 0 [7] => 8 [6,1] => 9 [5,2] => 5 [5,1,1] => 5 [4,3] => 3 [4,2,1] => 7 [4,1,1,1] => 3 [3,3,1] => 6 [3,2,2] => 0 [3,2,1,1] => 3 [3,1,1,1,1] => 0 [2,2,2,1] => 5 [2,2,1,1,1] => 0 [2,1,1,1,1,1] => 1 [1,1,1,1,1,1,1] => 0 [8] => 14 [7,1] => 15 [6,2] => 10 [6,1,1] => 10 [5,3] => 7 [5,2,1] => 12 [5,1,1,1] => 7 [4,4] => 8 [4,3,1] => 10 [4,2,2] => 8 [4,2,1,1] => 12 [4,1,1,1,1] => 6 [3,3,2] => 4 [3,3,1,1] => 4 [3,2,2,1] => 7 [3,2,1,1,1] => 4 [3,1,1,1,1,1] => 1 [2,2,2,2] => 2 [2,2,2,1,1] => 6 [2,2,1,1,1,1] => 4 [2,1,1,1,1,1,1] => 2 [1,1,1,1,1,1,1,1] => 0 [9] => 21 [8,1] => 22 [7,2] => 16 [7,1,1] => 16 [6,3] => 12 [6,2,1] => 18 [6,1,1,1] => 12 [5,4] => 10 [5,3,1] => 15 [5,2,2] => 5 [5,2,1,1] => 10 [5,1,1,1,1] => 5 [4,4,1] => 16 [4,3,2] => 12 [4,3,1,1] => 12 [4,2,2,1] => 16 [4,2,1,1,1] => 12 [4,1,1,1,1,1] => 6 [3,3,3] => 8 [3,3,2,1] => 12 [3,3,1,1,1] => 13 [3,2,2,2] => 4 [3,2,2,1,1] => 6 [3,2,1,1,1,1] => 4 [3,1,1,1,1,1,1] => 6 [2,2,2,2,1] => 10 [2,2,2,1,1,1] => 4 [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] => 0 [10] => 32 [9,1] => 33 [8,2] => 26 [8,1,1] => 26 [7,3] => 21 [7,2,1] => 28 [7,1,1,1] => 21 [6,4] => 18 [6,3,1] => 24 [6,2,2] => 12 [6,2,1,1] => 18 [6,1,1,1,1] => 12 [5,5] => 22 [5,4,1] => 22 [5,3,2] => 18 [5,3,1,1] => 18 [5,2,2,1] => 17 [5,2,1,1,1] => 16 [5,1,1,1,1,1] => 12 [4,4,2] => 14 [4,4,1,1] => 14 [4,3,3] => 6 [4,3,2,1] => 18 [4,3,1,1,1] => 6 [4,2,2,2] => 10 [4,2,2,1,1] => 14 [4,2,1,1,1,1] => 10 [4,1,1,1,1,1,1] => 2 [3,3,3,1] => 20 [3,3,2,2] => 7 [3,3,2,1,1] => 6 [3,3,1,1,1,1] => 7 [3,2,2,2,1] => 12 [3,2,2,1,1,1] => 7 [3,2,1,1,1,1,1] => 4 [3,1,1,1,1,1,1,1] => 6 [2,2,2,2,2] => 6 [2,2,2,2,1,1] => 6 [2,2,2,1,1,1,1] => 10 [2,2,1,1,1,1,1,1] => 5 [2,1,1,1,1,1,1,1,1] => 2 [1,1,1,1,1,1,1,1,1,1] => 0 [11] => 45 [10,1] => 46 [9,2] => 38 [9,1,1] => 38 [8,3] => 32 [8,2,1] => 40 [8,1,1,1] => 32 [7,4] => 28 [7,3,1] => 35 [7,2,2] => 21 [7,2,1,1] => 28 [7,1,1,1,1] => 21 [6,5] => 26 [6,4,1] => 32 [6,3,2] => 20 [6,3,1,1] => 20 [6,2,2,1] => 26 [6,2,1,1,1] => 20 [6,1,1,1,1,1] => 14 [5,5,1] => 36 [5,4,2] => 22 [5,4,1,1] => 22 [5,3,3] => 11 [5,3,2,1] => 26 [5,3,1,1,1] => 11 [5,2,2,2] => 10 [5,2,2,1,1] => 12 [5,2,1,1,1,1] => 10 [5,1,1,1,1,1,1] => 6 [4,4,3] => 14 [4,4,2,1] => 28 [4,4,1,1,1] => 14 [4,3,3,1] => 20 [4,3,2,2] => 14 [4,3,2,1,1] => 26 [4,3,1,1,1,1] => 8 [4,2,2,2,1] => 24 [4,2,2,1,1,1] => 14 [4,2,1,1,1,1,1] => 14 [4,1,1,1,1,1,1,1] => 4 [3,3,3,2] => 12 [3,3,3,1,1] => 12 [3,3,2,2,1] => 11 [3,3,2,1,1,1] => 22 [3,3,1,1,1,1,1] => 5 [3,2,2,2,2] => 0 [3,2,2,2,1,1] => 8 [3,2,2,1,1,1,1] => 2 [3,2,1,1,1,1,1,1] => 10 [3,1,1,1,1,1,1,1,1] => 0 [2,2,2,2,2,1] => 20 [2,2,2,2,1,1,1] => 0 [2,2,2,1,1,1,1,1] => 4 [2,2,1,1,1,1,1,1,1] => 2 [2,1,1,1,1,1,1,1,1,1] => 2 [1,1,1,1,1,1,1,1,1,1,1] => 0 [12] => 65 [11,1] => 66 [10,2] => 57 [10,1,1] => 57 [9,3] => 50 [9,2,1] => 59 [9,1,1,1] => 50 [8,4] => 45 [8,3,1] => 53 [8,2,2] => 37 [8,2,1,1] => 45 [8,1,1,1,1] => 37 [7,5] => 42 [7,4,1] => 49 [7,3,2] => 35 [7,3,1,1] => 35 [7,2,2,1] => 42 [7,2,1,1,1] => 35 [7,1,1,1,1,1] => 28 [6,6] => 50 [6,5,1] => 47 [6,4,2] => 35 [6,4,1,1] => 35 [6,3,3] => 41 [6,3,2,1] => 47 [6,3,1,1,1] => 37 [6,2,2,2] => 26 [6,2,2,1,1] => 31 [6,2,1,1,1,1] => 26 [6,1,1,1,1,1,1] => 23 [5,5,2] => 37 [5,5,1,1] => 37 [5,4,3] => 29 [5,4,2,1] => 37 [5,4,1,1,1] => 25 [5,3,3,1] => 32 [5,3,2,2] => 18 [5,3,2,1,1] => 29 [5,3,1,1,1,1] => 14 [5,2,2,2,1] => 27 [5,2,2,1,1,1] => 16 [5,2,1,1,1,1,1] => 17 [5,1,1,1,1,1,1,1] => 12 [4,4,4] => 37 [4,4,3,1] => 35 [4,4,2,2] => 23 [4,4,2,1,1] => 37 [4,4,1,1,1,1] => 23 [4,3,3,2] => 19 [4,3,3,1,1] => 19 [4,3,2,2,1] => 39 [4,3,2,1,1,1] => 23 [4,3,1,1,1,1,1] => 15 [4,2,2,2,2] => 19 [4,2,2,2,1,1] => 19 [4,2,2,1,1,1,1] => 27 [4,2,1,1,1,1,1,1] => 11 [4,1,1,1,1,1,1,1,1] => 9 [3,3,3,3] => 26 [3,3,3,2,1] => 37 [3,3,3,1,1,1] => 26 [3,3,2,2,2] => 13 [3,3,2,2,1,1] => 12 [3,3,2,1,1,1,1] => 17 [3,3,1,1,1,1,1,1] => 8 [3,2,2,2,2,1] => 22 [3,2,2,2,1,1,1] => 17 [3,2,2,1,1,1,1,1] => 8 [3,2,1,1,1,1,1,1,1] => 15 [3,1,1,1,1,1,1,1,1,1] => 2 [2,2,2,2,2,2] => 12 [2,2,2,2,2,1,1] => 21 [2,2,2,2,1,1,1,1] => 12 [2,2,2,1,1,1,1,1,1] => 7 [2,2,1,1,1,1,1,1,1,1] => 2 [2,1,1,1,1,1,1,1,1,1,1] => 3 [1,1,1,1,1,1,1,1,1,1,1,1] => 0 ----------------------------------------------------------------------------- Created: Aug 12, 2017 at 11:37 by Martin Rubey ----------------------------------------------------------------------------- Last Updated: Aug 12, 2017 at 13:44 by Martin Rubey