***************************************************************************** * 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: St000512 ----------------------------------------------------------------------------- Collection: Integer partitions ----------------------------------------------------------------------------- Description: The number of invariant subsets of size 3 when acting with a permutation of given cycle type. ----------------------------------------------------------------------------- References: [1] Bergeron, F., Labelle, G., Leroux, P. Combinatorial species and tree-like structures [[MathSciNet:1629341]] ----------------------------------------------------------------------------- Code: def statistic(la): E = species.SetSpecies() E2 = species.CharacteristicSpecies(3) c = (E2*E).cycle_index_series() return c.count(la) ----------------------------------------------------------------------------- Statistic values: [2] => 0 [1,1] => 0 [3] => 1 [2,1] => 1 [1,1,1] => 1 [4] => 0 [3,1] => 1 [2,2] => 0 [2,1,1] => 2 [1,1,1,1] => 4 [5] => 0 [4,1] => 0 [3,2] => 1 [3,1,1] => 1 [2,2,1] => 2 [2,1,1,1] => 4 [1,1,1,1,1] => 10 [6] => 0 [5,1] => 0 [4,2] => 0 [4,1,1] => 0 [3,3] => 2 [3,2,1] => 2 [3,1,1,1] => 2 [2,2,2] => 0 [2,2,1,1] => 4 [2,1,1,1,1] => 8 [1,1,1,1,1,1] => 20 [7] => 0 [6,1] => 0 [5,2] => 0 [5,1,1] => 0 [4,3] => 1 [4,2,1] => 1 [4,1,1,1] => 1 [3,3,1] => 2 [3,2,2] => 1 [3,2,1,1] => 3 [3,1,1,1,1] => 5 [2,2,2,1] => 3 [2,2,1,1,1] => 7 [2,1,1,1,1,1] => 15 [1,1,1,1,1,1,1] => 35 [8] => 0 [7,1] => 0 [6,2] => 0 [6,1,1] => 0 [5,3] => 1 [5,2,1] => 1 [5,1,1,1] => 1 [4,4] => 0 [4,3,1] => 1 [4,2,2] => 0 [4,2,1,1] => 2 [4,1,1,1,1] => 4 [3,3,2] => 2 [3,3,1,1] => 2 [3,2,2,1] => 3 [3,2,1,1,1] => 5 [3,1,1,1,1,1] => 11 [2,2,2,2] => 0 [2,2,2,1,1] => 6 [2,2,1,1,1,1] => 12 [2,1,1,1,1,1,1] => 26 [1,1,1,1,1,1,1,1] => 56 [9] => 0 [8,1] => 0 [7,2] => 0 [7,1,1] => 0 [6,3] => 1 [6,2,1] => 1 [6,1,1,1] => 1 [5,4] => 0 [5,3,1] => 1 [5,2,2] => 0 [5,2,1,1] => 2 [5,1,1,1,1] => 4 [4,4,1] => 0 [4,3,2] => 1 [4,3,1,1] => 1 [4,2,2,1] => 2 [4,2,1,1,1] => 4 [4,1,1,1,1,1] => 10 [3,3,3] => 3 [3,3,2,1] => 3 [3,3,1,1,1] => 3 [3,2,2,2] => 1 [3,2,2,1,1] => 5 [3,2,1,1,1,1] => 9 [3,1,1,1,1,1,1] => 21 [2,2,2,2,1] => 4 [2,2,2,1,1,1] => 10 [2,2,1,1,1,1,1] => 20 [2,1,1,1,1,1,1,1] => 42 [1,1,1,1,1,1,1,1,1] => 84 [10] => 0 [9,1] => 0 [8,2] => 0 [8,1,1] => 0 [7,3] => 1 [7,2,1] => 1 [7,1,1,1] => 1 [6,4] => 0 [6,3,1] => 1 [6,2,2] => 0 [6,2,1,1] => 2 [6,1,1,1,1] => 4 [5,5] => 0 [5,4,1] => 0 [5,3,2] => 1 [5,3,1,1] => 1 [5,2,2,1] => 2 [5,2,1,1,1] => 4 [5,1,1,1,1,1] => 10 [4,4,2] => 0 [4,4,1,1] => 0 [4,3,3] => 2 [4,3,2,1] => 2 [4,3,1,1,1] => 2 [4,2,2,2] => 0 [4,2,2,1,1] => 4 [4,2,1,1,1,1] => 8 [4,1,1,1,1,1,1] => 20 [3,3,3,1] => 3 [3,3,2,2] => 2 [3,3,2,1,1] => 4 [3,3,1,1,1,1] => 6 [3,2,2,2,1] => 4 [3,2,2,1,1,1] => 8 [3,2,1,1,1,1,1] => 16 [3,1,1,1,1,1,1,1] => 36 [2,2,2,2,2] => 0 [2,2,2,2,1,1] => 8 [2,2,2,1,1,1,1] => 16 [2,2,1,1,1,1,1,1] => 32 [2,1,1,1,1,1,1,1,1] => 64 [1,1,1,1,1,1,1,1,1,1] => 120 [11] => 0 [10,1] => 0 [9,2] => 0 [9,1,1] => 0 [8,3] => 1 [8,2,1] => 1 [8,1,1,1] => 1 [7,4] => 0 [7,3,1] => 1 [7,2,2] => 0 [7,2,1,1] => 2 [7,1,1,1,1] => 4 [6,5] => 0 [6,4,1] => 0 [6,3,2] => 1 [6,3,1,1] => 1 [6,2,2,1] => 2 [6,2,1,1,1] => 4 [6,1,1,1,1,1] => 10 [5,5,1] => 0 [5,4,2] => 0 [5,4,1,1] => 0 [5,3,3] => 2 [5,3,2,1] => 2 [5,3,1,1,1] => 2 [5,2,2,2] => 0 [5,2,2,1,1] => 4 [5,2,1,1,1,1] => 8 [5,1,1,1,1,1,1] => 20 [4,4,3] => 1 [4,4,2,1] => 1 [4,4,1,1,1] => 1 [4,3,3,1] => 2 [4,3,2,2] => 1 [4,3,2,1,1] => 3 [4,3,1,1,1,1] => 5 [4,2,2,2,1] => 3 [4,2,2,1,1,1] => 7 [4,2,1,1,1,1,1] => 15 [4,1,1,1,1,1,1,1] => 35 [3,3,3,2] => 3 [3,3,3,1,1] => 3 [3,3,2,2,1] => 4 [3,3,2,1,1,1] => 6 [3,3,1,1,1,1,1] => 12 [3,2,2,2,2] => 1 [3,2,2,2,1,1] => 7 [3,2,2,1,1,1,1] => 13 [3,2,1,1,1,1,1,1] => 27 [3,1,1,1,1,1,1,1,1] => 57 [2,2,2,2,2,1] => 5 [2,2,2,2,1,1,1] => 13 [2,2,2,1,1,1,1,1] => 25 [2,2,1,1,1,1,1,1,1] => 49 [2,1,1,1,1,1,1,1,1,1] => 93 [1,1,1,1,1,1,1,1,1,1,1] => 165 [12] => 0 [11,1] => 0 [10,2] => 0 [10,1,1] => 0 [9,3] => 1 [9,2,1] => 1 [9,1,1,1] => 1 [8,4] => 0 [8,3,1] => 1 [8,2,2] => 0 [8,2,1,1] => 2 [8,1,1,1,1] => 4 [7,5] => 0 [7,4,1] => 0 [7,3,2] => 1 [7,3,1,1] => 1 [7,2,2,1] => 2 [7,2,1,1,1] => 4 [7,1,1,1,1,1] => 10 [6,6] => 0 [6,5,1] => 0 [6,4,2] => 0 [6,4,1,1] => 0 [6,3,3] => 2 [6,3,2,1] => 2 [6,3,1,1,1] => 2 [6,2,2,2] => 0 [6,2,2,1,1] => 4 [6,2,1,1,1,1] => 8 [6,1,1,1,1,1,1] => 20 [5,5,2] => 0 [5,5,1,1] => 0 [5,4,3] => 1 [5,4,2,1] => 1 [5,4,1,1,1] => 1 [5,3,3,1] => 2 [5,3,2,2] => 1 [5,3,2,1,1] => 3 [5,3,1,1,1,1] => 5 [5,2,2,2,1] => 3 [5,2,2,1,1,1] => 7 [5,2,1,1,1,1,1] => 15 [5,1,1,1,1,1,1,1] => 35 [4,4,4] => 0 [4,4,3,1] => 1 [4,4,2,2] => 0 [4,4,2,1,1] => 2 [4,4,1,1,1,1] => 4 [4,3,3,2] => 2 [4,3,3,1,1] => 2 [4,3,2,2,1] => 3 [4,3,2,1,1,1] => 5 [4,3,1,1,1,1,1] => 11 [4,2,2,2,2] => 0 [4,2,2,2,1,1] => 6 [4,2,2,1,1,1,1] => 12 [4,2,1,1,1,1,1,1] => 26 [4,1,1,1,1,1,1,1,1] => 56 [3,3,3,3] => 4 [3,3,3,2,1] => 4 [3,3,3,1,1,1] => 4 [3,3,2,2,2] => 2 [3,3,2,2,1,1] => 6 [3,3,2,1,1,1,1] => 10 [3,3,1,1,1,1,1,1] => 22 [3,2,2,2,2,1] => 5 [3,2,2,2,1,1,1] => 11 [3,2,2,1,1,1,1,1] => 21 [3,2,1,1,1,1,1,1,1] => 43 [3,1,1,1,1,1,1,1,1,1] => 85 [2,2,2,2,2,2] => 0 [2,2,2,2,2,1,1] => 10 [2,2,2,2,1,1,1,1] => 20 [2,2,2,1,1,1,1,1,1] => 38 [2,2,1,1,1,1,1,1,1,1] => 72 [2,1,1,1,1,1,1,1,1,1,1] => 130 [1,1,1,1,1,1,1,1,1,1,1,1] => 220 ----------------------------------------------------------------------------- Created: May 26, 2016 at 20:58 by Martin Rubey ----------------------------------------------------------------------------- Last Updated: May 26, 2016 at 20:58 by Martin Rubey