***************************************************************************** * 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: St001629 ----------------------------------------------------------------------------- Collection: Integer compositions ----------------------------------------------------------------------------- Description: The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. ----------------------------------------------------------------------------- References: ----------------------------------------------------------------------------- Code: def statistic(a): F = QuasiSymmetricFunctions(QQ).F() S = species.CycleSpecies().cycle_index_series() return F(S.coefficient(a.size())).coefficient(a) ----------------------------------------------------------------------------- Statistic values: [1,1,1] => 1 [1,2] => 0 [2,1] => 0 [3] => 1 [1,1,1,1] => 0 [1,1,2] => 1 [1,2,1] => 2 [1,3] => 0 [2,1,1] => 1 [2,2] => 1 [3,1] => 0 [4] => 1 [1,1,1,1,1] => 1 [1,1,1,2] => 0 [1,1,2,1] => 1 [1,1,3] => 2 [1,2,1,1] => 1 [1,2,2] => 4 [1,3,1] => 3 [1,4] => 0 [2,1,1,1] => 0 [2,1,2] => 3 [2,2,1] => 4 [2,3] => 1 [3,1,1] => 2 [3,2] => 1 [4,1] => 0 [5] => 1 [1,1,1,1,1,1] => 0 [1,1,1,1,2] => 1 [1,1,1,2,1] => 2 [1,1,1,3] => 2 [1,1,2,1,1] => 4 [1,1,2,2] => 5 [1,1,3,1] => 4 [1,1,4] => 2 [1,2,1,1,1] => 2 [1,2,1,2] => 7 [1,2,2,1] => 10 [1,2,3] => 6 [1,3,1,1] => 4 [1,3,2] => 7 [1,4,1] => 4 [1,5] => 0 [2,1,1,1,1] => 1 [2,1,1,2] => 3 [2,1,2,1] => 7 [2,1,3] => 4 [2,2,1,1] => 5 [2,2,2] => 11 [2,3,1] => 7 [2,4] => 2 [3,1,1,1] => 2 [3,1,2] => 4 [3,2,1] => 6 [3,3] => 3 [4,1,1] => 2 [4,2] => 2 [5,1] => 0 [6] => 1 [1,1,1,1,1,1,1] => 1 [1,1,1,1,1,2] => 0 [1,1,1,1,2,1] => 2 [1,1,1,1,3] => 3 [1,1,1,2,1,1] => 4 [1,1,1,2,2] => 10 [1,1,1,3,1] => 8 [1,1,1,4] => 2 [1,1,2,1,1,1] => 4 [1,1,2,1,2] => 15 [1,1,2,2,1] => 23 [1,1,2,3] => 12 [1,1,3,1,1] => 11 [1,1,3,2] => 15 [1,1,4,1] => 7 [1,1,5] => 3 [1,2,1,1,1,1] => 2 [1,2,1,1,2] => 12 [1,2,1,2,1] => 25 [1,2,1,3] => 15 [1,2,2,1,1] => 23 [1,2,2,2] => 38 [1,2,3,1] => 25 [1,2,4] => 10 [1,3,1,1,1] => 8 [1,3,1,2] => 18 [1,3,2,1] => 25 [1,3,3] => 15 [1,4,1,1] => 7 [1,4,2] => 12 [1,5,1] => 5 [1,6] => 0 [2,1,1,1,1,1] => 0 [2,1,1,1,2] => 5 [2,1,1,2,1] => 12 [2,1,1,3] => 7 [2,1,2,1,1] => 15 [2,1,2,2] => 25 [2,1,3,1] => 18 [2,1,4] => 8 [2,2,1,1,1] => 10 [2,2,1,2] => 25 [2,2,2,1] => 38 [2,2,3] => 23 [2,3,1,1] => 15 [2,3,2] => 25 [2,4,1] => 12 [2,5] => 2 [3,1,1,1,1] => 3 [3,1,1,2] => 7 [3,1,2,1] => 15 [3,1,3] => 11 [3,2,1,1] => 12 [3,2,2] => 23 [3,3,1] => 15 [3,4] => 4 [4,1,1,1] => 2 [4,1,2] => 8 [4,2,1] => 10 [4,3] => 4 [5,1,1] => 3 [5,2] => 2 [6,1] => 0 [7] => 1 [1,1,1,1,1,1,1,1] => 0 [1,1,1,1,1,1,2] => 1 [1,1,1,1,1,2,1] => 4 [1,1,1,1,1,3] => 2 [1,1,1,1,2,1,1] => 7 [1,1,1,1,2,2] => 13 [1,1,1,1,3,1] => 10 [1,1,1,1,4] => 5 [1,1,1,2,1,1,1] => 10 [1,1,1,2,1,2] => 24 [1,1,1,2,2,1] => 40 [1,1,1,2,3] => 24 [1,1,1,3,1,1] => 18 [1,1,1,3,2] => 32 [1,1,1,4,1] => 16 [1,1,1,5] => 4 [1,1,2,1,1,1,1] => 7 [1,1,2,1,1,2] => 27 [1,1,2,1,2,1] => 59 [1,1,2,1,3] => 40 [1,1,2,2,1,1] => 56 [1,1,2,2,2] => 99 [1,1,2,3,1] => 67 [1,1,2,4] => 23 [1,1,3,1,1,1] => 18 [1,1,3,1,2] => 53 [1,1,3,2,1] => 72 [1,1,3,3] => 39 [1,1,4,1,1] => 24 [1,1,4,2] => 31 [1,1,5,1] => 12 [1,1,6] => 3 [1,2,1,1,1,1,1] => 4 [1,2,1,1,1,2] => 16 [1,2,1,1,2,1] => 46 [1,2,1,1,3] => 32 [1,2,1,2,1,1] => 59 [1,2,1,2,2] => 110 [1,2,1,3,1] => 80 [1,2,1,4] => 31 [1,2,2,1,1,1] => 40 [1,2,2,1,2] => 115 [1,2,2,2,1] => 174 [1,2,2,3] => 98 [1,2,3,1,1] => 72 [1,2,3,2] => 109 [1,2,4,1] => 50 [1,2,5] => 14 [1,3,1,1,1,1] => 10 [1,3,1,1,2] => 40 [1,3,1,2,1] => 80 [1,3,1,3] => 52 [1,3,2,1,1] => 67 [1,3,2,2] => 114 [1,3,3,1] => 74 [1,3,4] => 25 [1,4,1,1,1] => 16 [1,4,1,2] => 39 [1,4,2,1] => 50 [1,4,3] => 28 [1,5,1,1] => 12 [1,5,2] => 17 [1,6,1] => 6 [1,7] => 0 [2,1,1,1,1,1,1] => 1 [2,1,1,1,1,2] => 5 [2,1,1,1,2,1] => 16 [2,1,1,1,3] => 13 [2,1,1,2,1,1] => 27 [2,1,1,2,2] => 51 [2,1,1,3,1] => 40 [2,1,1,4] => 15 [2,1,2,1,1,1] => 24 [2,1,2,1,2] => 75 [2,1,2,2,1] => 115 [2,1,2,3] => 66 [2,1,3,1,1] => 53 [2,1,3,2] => 79 [2,1,4,1] => 39 [2,1,5] => 11 [2,2,1,1,1,1] => 13 [2,2,1,1,2] => 51 [2,2,1,2,1] => 110 [2,2,1,3] => 71 [2,2,2,1,1] => 99 [2,2,2,2] => 173 [2,2,3,1] => 114 [2,2,4] => 41 [2,3,1,1,1] => 32 [2,3,1,2] => 79 [2,3,2,1] => 109 [2,3,3] => 60 [2,4,1,1] => 31 [2,4,2] => 47 [2,5,1] => 17 [2,6] => 3 [3,1,1,1,1,1] => 2 [3,1,1,1,2] => 13 [3,1,1,2,1] => 32 [3,1,1,3] => 23 [3,1,2,1,1] => 40 [3,1,2,2] => 71 [3,1,3,1] => 52 [3,1,4] => 19 [3,2,1,1,1] => 24 [3,2,1,2] => 66 [3,2,2,1] => 98 [3,2,3] => 57 [3,3,1,1] => 39 [3,3,2] => 60 [3,4,1] => 28 [3,5] => 6 [4,1,1,1,1] => 5 [4,1,1,2] => 15 [4,1,2,1] => 31 [4,1,3] => 19 [4,2,1,1] => 23 [4,2,2] => 41 [4,3,1] => 25 [4,4] => 9 [5,1,1,1] => 4 [5,1,2] => 11 [5,2,1] => 14 [5,3] => 6 [6,1,1] => 3 [6,2] => 3 [7,1] => 0 [8] => 1 ----------------------------------------------------------------------------- Created: Oct 02, 2020 at 07:18 by Martin Rubey ----------------------------------------------------------------------------- Last Updated: Oct 02, 2020 at 07:18 by Martin Rubey