***************************************************************************** * 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: St000511 ----------------------------------------------------------------------------- Collection: Integer partitions ----------------------------------------------------------------------------- Description: The number of invariant subsets 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): c = species.SubsetSpecies().cycle_index_series() return c.count(la) ----------------------------------------------------------------------------- Statistic values: [] => 1 [1] => 2 [2] => 2 [1,1] => 4 [3] => 2 [2,1] => 4 [1,1,1] => 8 [4] => 2 [3,1] => 4 [2,2] => 4 [2,1,1] => 8 [1,1,1,1] => 16 [5] => 2 [4,1] => 4 [3,2] => 4 [3,1,1] => 8 [2,2,1] => 8 [2,1,1,1] => 16 [1,1,1,1,1] => 32 [6] => 2 [5,1] => 4 [4,2] => 4 [4,1,1] => 8 [3,3] => 4 [3,2,1] => 8 [3,1,1,1] => 16 [2,2,2] => 8 [2,2,1,1] => 16 [2,1,1,1,1] => 32 [1,1,1,1,1,1] => 64 [7] => 2 [6,1] => 4 [5,2] => 4 [5,1,1] => 8 [4,3] => 4 [4,2,1] => 8 [4,1,1,1] => 16 [3,3,1] => 8 [3,2,2] => 8 [3,2,1,1] => 16 [3,1,1,1,1] => 32 [2,2,2,1] => 16 [2,2,1,1,1] => 32 [2,1,1,1,1,1] => 64 [1,1,1,1,1,1,1] => 128 [8] => 2 [7,1] => 4 [6,2] => 4 [6,1,1] => 8 [5,3] => 4 [5,2,1] => 8 [5,1,1,1] => 16 [4,4] => 4 [4,3,1] => 8 [4,2,2] => 8 [4,2,1,1] => 16 [4,1,1,1,1] => 32 [3,3,2] => 8 [3,3,1,1] => 16 [3,2,2,1] => 16 [3,2,1,1,1] => 32 [3,1,1,1,1,1] => 64 [2,2,2,2] => 16 [2,2,2,1,1] => 32 [2,2,1,1,1,1] => 64 [2,1,1,1,1,1,1] => 128 [1,1,1,1,1,1,1,1] => 256 [9] => 2 [8,1] => 4 [7,2] => 4 [7,1,1] => 8 [6,3] => 4 [6,2,1] => 8 [6,1,1,1] => 16 [5,4] => 4 [5,3,1] => 8 [5,2,2] => 8 [5,2,1,1] => 16 [5,1,1,1,1] => 32 [4,4,1] => 8 [4,3,2] => 8 [4,3,1,1] => 16 [4,2,2,1] => 16 [4,2,1,1,1] => 32 [4,1,1,1,1,1] => 64 [3,3,3] => 8 [3,3,2,1] => 16 [3,3,1,1,1] => 32 [3,2,2,2] => 16 [3,2,2,1,1] => 32 [3,2,1,1,1,1] => 64 [3,1,1,1,1,1,1] => 128 [2,2,2,2,1] => 32 [2,2,2,1,1,1] => 64 [2,2,1,1,1,1,1] => 128 [2,1,1,1,1,1,1,1] => 256 [1,1,1,1,1,1,1,1,1] => 512 [10] => 2 [9,1] => 4 [8,2] => 4 [8,1,1] => 8 [7,3] => 4 [7,2,1] => 8 [7,1,1,1] => 16 [6,4] => 4 [6,3,1] => 8 [6,2,2] => 8 [6,2,1,1] => 16 [6,1,1,1,1] => 32 [5,5] => 4 [5,4,1] => 8 [5,3,2] => 8 [5,3,1,1] => 16 [5,2,2,1] => 16 [5,2,1,1,1] => 32 [5,1,1,1,1,1] => 64 [4,4,2] => 8 [4,4,1,1] => 16 [4,3,3] => 8 [4,3,2,1] => 16 [4,3,1,1,1] => 32 [4,2,2,2] => 16 [4,2,2,1,1] => 32 [4,2,1,1,1,1] => 64 [4,1,1,1,1,1,1] => 128 [3,3,3,1] => 16 [3,3,2,2] => 16 [3,3,2,1,1] => 32 [3,3,1,1,1,1] => 64 [3,2,2,2,1] => 32 [3,2,2,1,1,1] => 64 [3,2,1,1,1,1,1] => 128 [3,1,1,1,1,1,1,1] => 256 [2,2,2,2,2] => 32 [2,2,2,2,1,1] => 64 [2,2,2,1,1,1,1] => 128 [2,2,1,1,1,1,1,1] => 256 [2,1,1,1,1,1,1,1,1] => 512 [1,1,1,1,1,1,1,1,1,1] => 1024 [11] => 2 [10,1] => 4 [9,2] => 4 [9,1,1] => 8 [8,3] => 4 [8,2,1] => 8 [8,1,1,1] => 16 [7,4] => 4 [7,3,1] => 8 [7,2,2] => 8 [7,2,1,1] => 16 [7,1,1,1,1] => 32 [6,5] => 4 [6,4,1] => 8 [6,3,2] => 8 [6,3,1,1] => 16 [6,2,2,1] => 16 [6,2,1,1,1] => 32 [6,1,1,1,1,1] => 64 [5,5,1] => 8 [5,4,2] => 8 [5,4,1,1] => 16 [5,3,3] => 8 [5,3,2,1] => 16 [5,3,1,1,1] => 32 [5,2,2,2] => 16 [5,2,2,1,1] => 32 [5,2,1,1,1,1] => 64 [5,1,1,1,1,1,1] => 128 [4,4,3] => 8 [4,4,2,1] => 16 [4,4,1,1,1] => 32 [4,3,3,1] => 16 [4,3,2,2] => 16 [4,3,2,1,1] => 32 [4,3,1,1,1,1] => 64 [4,2,2,2,1] => 32 [4,2,2,1,1,1] => 64 [4,2,1,1,1,1,1] => 128 [4,1,1,1,1,1,1,1] => 256 [3,3,3,2] => 16 [3,3,3,1,1] => 32 [3,3,2,2,1] => 32 [3,3,2,1,1,1] => 64 [3,3,1,1,1,1,1] => 128 [3,2,2,2,2] => 32 [3,2,2,2,1,1] => 64 [3,2,2,1,1,1,1] => 128 [3,2,1,1,1,1,1,1] => 256 [3,1,1,1,1,1,1,1,1] => 512 [2,2,2,2,2,1] => 64 [2,2,2,2,1,1,1] => 128 [2,2,2,1,1,1,1,1] => 256 [2,2,1,1,1,1,1,1,1] => 512 [2,1,1,1,1,1,1,1,1,1] => 1024 [1,1,1,1,1,1,1,1,1,1,1] => 2048 [12] => 2 [11,1] => 4 [10,2] => 4 [10,1,1] => 8 [9,3] => 4 [9,2,1] => 8 [9,1,1,1] => 16 [8,4] => 4 [8,3,1] => 8 [8,2,2] => 8 [8,2,1,1] => 16 [8,1,1,1,1] => 32 [7,5] => 4 [7,4,1] => 8 [7,3,2] => 8 [7,3,1,1] => 16 [7,2,2,1] => 16 [7,2,1,1,1] => 32 [7,1,1,1,1,1] => 64 [6,6] => 4 [6,5,1] => 8 [6,4,2] => 8 [6,4,1,1] => 16 [6,3,3] => 8 [6,3,2,1] => 16 [6,3,1,1,1] => 32 [6,2,2,2] => 16 [6,2,2,1,1] => 32 [6,2,1,1,1,1] => 64 [6,1,1,1,1,1,1] => 128 [5,5,2] => 8 [5,5,1,1] => 16 [5,4,3] => 8 [5,4,2,1] => 16 [5,4,1,1,1] => 32 [5,3,3,1] => 16 [5,3,2,2] => 16 [5,3,2,1,1] => 32 [5,3,1,1,1,1] => 64 [5,2,2,2,1] => 32 [5,2,2,1,1,1] => 64 [5,2,1,1,1,1,1] => 128 [5,1,1,1,1,1,1,1] => 256 [4,4,4] => 8 [4,4,3,1] => 16 [4,4,2,2] => 16 [4,4,2,1,1] => 32 [4,4,1,1,1,1] => 64 [4,3,3,2] => 16 [4,3,3,1,1] => 32 [4,3,2,2,1] => 32 [4,3,2,1,1,1] => 64 [4,3,1,1,1,1,1] => 128 [4,2,2,2,2] => 32 [4,2,2,2,1,1] => 64 [4,2,2,1,1,1,1] => 128 [4,2,1,1,1,1,1,1] => 256 [4,1,1,1,1,1,1,1,1] => 512 [3,3,3,3] => 16 [3,3,3,2,1] => 32 [3,3,3,1,1,1] => 64 [3,3,2,2,2] => 32 [3,3,2,2,1,1] => 64 [3,3,2,1,1,1,1] => 128 [3,3,1,1,1,1,1,1] => 256 [3,2,2,2,2,1] => 64 [3,2,2,2,1,1,1] => 128 [3,2,2,1,1,1,1,1] => 256 [3,2,1,1,1,1,1,1,1] => 512 [3,1,1,1,1,1,1,1,1,1] => 1024 [2,2,2,2,2,2] => 64 [2,2,2,2,2,1,1] => 128 [2,2,2,2,1,1,1,1] => 256 [2,2,2,1,1,1,1,1,1] => 512 [2,2,1,1,1,1,1,1,1,1] => 1024 [2,1,1,1,1,1,1,1,1,1,1] => 2048 [1,1,1,1,1,1,1,1,1,1,1,1] => 4096 ----------------------------------------------------------------------------- Created: May 26, 2016 at 21:00 by Martin Rubey ----------------------------------------------------------------------------- Last Updated: Oct 29, 2017 at 21:35 by Martin Rubey