***************************************************************************** * 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: St000936 ----------------------------------------------------------------------------- Collection: Integer partitions ----------------------------------------------------------------------------- Description: The number of even values of the symmetric group character corresponding to the partition. For example, the character values of the irreducible representation $S^{(2,2)}$ are $2$ on the conjugacy classes $(4)$ and $(2,2)$, $0$ on the conjugacy classes $(3,1)$ and $(1,1,1,1)$, and $-1$ on the conjugace class $(2,1,1)$. Therefore, the statistic on the partition $(2,2)$ is $4$. It is shown in [1] that the sum of the values of the statistic over all partitions of a given size is even. ----------------------------------------------------------------------------- References: [1] Miller, A. R. Note on parity and the irreducible characters of the symmetric group [[arXiv:1708.03267]] ----------------------------------------------------------------------------- Code: def table(n): s = SymmetricFunctions(ZZ).s() p = SymmetricFunctions(ZZ).p() res = dict() P = Partitions(n) r = P.cardinality() for mu in P: res[mu] = [0]*r for i, la in enumerate(P): for mu, v in s(p(la)): res[mu][i] = v return res def statistic(la): t = table(la.size()) return len([1 for e in t[la] if is_even(e)]) ----------------------------------------------------------------------------- Statistic values: [2] => 0 [1,1] => 0 [3] => 0 [2,1] => 2 [1,1,1] => 0 [4] => 0 [3,1] => 1 [2,2] => 4 [2,1,1] => 1 [1,1,1,1] => 0 [5] => 0 [4,1] => 4 [3,2] => 1 [3,1,1] => 6 [2,2,1] => 1 [2,1,1,1] => 4 [1,1,1,1,1] => 0 [6] => 0 [5,1] => 3 [4,2] => 4 [4,1,1] => 7 [3,3] => 3 [3,2,1] => 10 [3,1,1,1] => 7 [2,2,2] => 3 [2,2,1,1] => 4 [2,1,1,1,1] => 3 [1,1,1,1,1,1] => 0 [7] => 0 [6,1] => 8 [5,2] => 11 [5,1,1] => 4 [4,3] => 9 [4,2,1] => 3 [4,1,1,1] => 14 [3,3,1] => 3 [3,2,2] => 3 [3,2,1,1] => 3 [3,1,1,1,1] => 4 [2,2,2,1] => 9 [2,2,1,1,1] => 11 [2,1,1,1,1,1] => 8 [1,1,1,1,1,1,1] => 0 [8] => 0 [7,1] => 7 [6,2] => 13 [6,1,1] => 9 [5,3] => 13 [5,2,1] => 18 [5,1,1,1] => 8 [4,4] => 15 [4,3,1] => 14 [4,2,2] => 15 [4,2,1,1] => 21 [4,1,1,1,1] => 8 [3,3,2] => 21 [3,3,1,1] => 15 [3,2,2,1] => 14 [3,2,1,1,1] => 18 [3,1,1,1,1,1] => 9 [2,2,2,2] => 15 [2,2,2,1,1] => 13 [2,2,1,1,1,1] => 13 [2,1,1,1,1,1,1] => 7 [1,1,1,1,1,1,1,1] => 0 [9] => 0 [8,1] => 15 [7,2] => 11 [7,1,1] => 19 [6,3] => 25 [6,2,1] => 8 [6,1,1,1] => 18 [5,4] => 20 [5,3,1] => 24 [5,2,2] => 22 [5,2,1,1] => 9 [5,1,1,1,1] => 29 [4,4,1] => 19 [4,3,2] => 18 [4,3,1,1] => 17 [4,2,2,1] => 17 [4,2,1,1,1] => 9 [4,1,1,1,1,1] => 18 [3,3,3] => 29 [3,3,2,1] => 18 [3,3,1,1,1] => 22 [3,2,2,2] => 19 [3,2,2,1,1] => 24 [3,2,1,1,1,1] => 8 [3,1,1,1,1,1,1] => 19 [2,2,2,2,1] => 20 [2,2,2,1,1,1] => 25 [2,2,1,1,1,1,1] => 11 [2,1,1,1,1,1,1,1] => 15 [1,1,1,1,1,1,1,1,1] => 0 [10] => 0 [9,1] => 15 [8,2] => 19 [8,1,1] => 23 [7,3] => 19 [7,2,1] => 32 [7,1,1,1] => 22 [6,4] => 25 [6,3,1] => 8 [6,2,2] => 19 [6,2,1,1] => 25 [6,1,1,1,1] => 28 [5,5] => 28 [5,4,1] => 33 [5,3,2] => 27 [5,3,1,1] => 24 [5,2,2,1] => 18 [5,2,1,1,1] => 41 [5,1,1,1,1,1] => 28 [4,4,2] => 26 [4,4,1,1] => 23 [4,3,3] => 28 [4,3,2,1] => 41 [4,3,1,1,1] => 18 [4,2,2,2] => 23 [4,2,2,1,1] => 24 [4,2,1,1,1,1] => 25 [4,1,1,1,1,1,1] => 22 [3,3,3,1] => 28 [3,3,2,2] => 26 [3,3,2,1,1] => 27 [3,3,1,1,1,1] => 19 [3,2,2,2,1] => 33 [3,2,2,1,1,1] => 8 [3,2,1,1,1,1,1] => 32 [3,1,1,1,1,1,1,1] => 23 [2,2,2,2,2] => 28 [2,2,2,2,1,1] => 25 [2,2,2,1,1,1,1] => 19 [2,2,1,1,1,1,1,1] => 19 [2,1,1,1,1,1,1,1,1] => 15 [1,1,1,1,1,1,1,1,1,1] => 0 [11] => 0 [10,1] => 27 [9,2] => 33 [9,1,1] => 23 [8,3] => 35 [8,2,1] => 18 [8,1,1,1] => 42 [7,4] => 8 [7,3,1] => 38 [7,2,2] => 22 [7,2,1,1] => 43 [7,1,1,1,1] => 35 [6,5] => 43 [6,4,1] => 22 [6,3,2] => 34 [6,3,1,1] => 46 [6,2,2,1] => 49 [6,2,1,1,1] => 32 [6,1,1,1,1,1] => 55 [5,5,1] => 35 [5,4,2] => 44 [5,4,1,1] => 16 [5,3,3] => 35 [5,3,2,1] => 34 [5,3,1,1,1] => 34 [5,2,2,2] => 27 [5,2,2,1,1] => 34 [5,2,1,1,1,1] => 32 [5,1,1,1,1,1,1] => 35 [4,4,3] => 38 [4,4,2,1] => 44 [4,4,1,1,1] => 27 [4,3,3,1] => 55 [4,3,2,2] => 44 [4,3,2,1,1] => 34 [4,3,1,1,1,1] => 49 [4,2,2,2,1] => 16 [4,2,2,1,1,1] => 46 [4,2,1,1,1,1,1] => 43 [4,1,1,1,1,1,1,1] => 42 [3,3,3,2] => 38 [3,3,3,1,1] => 35 [3,3,2,2,1] => 44 [3,3,2,1,1,1] => 34 [3,3,1,1,1,1,1] => 22 [3,2,2,2,2] => 35 [3,2,2,2,1,1] => 22 [3,2,2,1,1,1,1] => 38 [3,2,1,1,1,1,1,1] => 18 [3,1,1,1,1,1,1,1,1] => 23 [2,2,2,2,2,1] => 43 [2,2,2,2,1,1,1] => 8 [2,2,2,1,1,1,1,1] => 35 [2,2,1,1,1,1,1,1,1] => 33 [2,1,1,1,1,1,1,1,1,1] => 27 [1,1,1,1,1,1,1,1,1,1,1] => 0 [12] => 0 [11,1] => 29 [10,2] => 42 [10,1,1] => 35 [9,3] => 42 [9,2,1] => 55 [9,1,1,1] => 36 [8,4] => 32 [8,3,1] => 37 [8,2,2] => 44 [8,2,1,1] => 22 [8,1,1,1,1] => 44 [7,5] => 24 [7,4,1] => 71 [7,3,2] => 40 [7,3,1,1] => 46 [7,2,2,1] => 39 [7,2,1,1,1] => 59 [7,1,1,1,1,1] => 49 [6,6] => 60 [6,5,1] => 23 [6,4,2] => 50 [6,4,1,1] => 50 [6,3,3] => 45 [6,3,2,1] => 73 [6,3,1,1,1] => 67 [6,2,2,2] => 44 [6,2,2,1,1] => 62 [6,2,1,1,1,1] => 76 [6,1,1,1,1,1,1] => 49 [5,5,2] => 48 [5,5,1,1] => 40 [5,4,3] => 59 [5,4,2,1] => 41 [5,4,1,1,1] => 57 [5,3,3,1] => 49 [5,3,2,2] => 44 [5,3,2,1,1] => 76 [5,3,1,1,1,1] => 62 [5,2,2,2,1] => 57 [5,2,2,1,1,1] => 67 [5,2,1,1,1,1,1] => 59 [5,1,1,1,1,1,1,1] => 44 [4,4,4] => 55 [4,4,3,1] => 47 [4,4,2,2] => 76 [4,4,2,1,1] => 44 [4,4,1,1,1,1] => 44 [4,3,3,2] => 47 [4,3,3,1,1] => 49 [4,3,2,2,1] => 41 [4,3,2,1,1,1] => 73 [4,3,1,1,1,1,1] => 39 [4,2,2,2,2] => 40 [4,2,2,2,1,1] => 50 [4,2,2,1,1,1,1] => 46 [4,2,1,1,1,1,1,1] => 22 [4,1,1,1,1,1,1,1,1] => 36 [3,3,3,3] => 55 [3,3,3,2,1] => 59 [3,3,3,1,1,1] => 45 [3,3,2,2,2] => 48 [3,3,2,2,1,1] => 50 [3,3,2,1,1,1,1] => 40 [3,3,1,1,1,1,1,1] => 44 [3,2,2,2,2,1] => 23 [3,2,2,2,1,1,1] => 71 [3,2,2,1,1,1,1,1] => 37 [3,2,1,1,1,1,1,1,1] => 55 [3,1,1,1,1,1,1,1,1,1] => 35 [2,2,2,2,2,2] => 60 [2,2,2,2,2,1,1] => 24 [2,2,2,2,1,1,1,1] => 32 [2,2,2,1,1,1,1,1,1] => 42 [2,2,1,1,1,1,1,1,1,1] => 42 [2,1,1,1,1,1,1,1,1,1,1] => 29 [1,1,1,1,1,1,1,1,1,1,1,1] => 0 ----------------------------------------------------------------------------- Created: Aug 11, 2017 at 22:54 by Martin Rubey ----------------------------------------------------------------------------- Last Updated: Aug 11, 2017 at 22:54 by Martin Rubey