***************************************************************************** * 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: St000545 ----------------------------------------------------------------------------- Collection: Permutations ----------------------------------------------------------------------------- Description: The number of parabolic double cosets with minimal element being the given permutation. For $w \in S_n$, this is $$\big| W_I \tau W_J\ :\ \tau \in S_n,\ I,J \subseteq S,\ w = \min\{W_I \tau W_J\}\big|$$ where $S$ is the set of simple transpositions, $W_K$ is the parabolic subgroup generated by $K \subseteq S$, and $\min\{W_I \tau W_J\}$ is the unique minimal element in weak order in the double coset $W_I \tau W_J$. [1] contains a combinatorial description of these parabolic double cosets which can be used to compute this statistic. ----------------------------------------------------------------------------- References: [1] S. Billey, M. Konvalinka, T.K. Petersen, W. Slofstra, B. Tenner, "Parabolic double cosets in Coxeter groups", to appear in Discrete Mathematics and Theoretical Computer Science, preprint 2016 [2] Number of distinct parabolic double cosets of the symmetric group S_n. [[OEIS:A260700]] ----------------------------------------------------------------------------- Code: ----------------------------------------------------------------------------- Statistic values: [1] => 1 [1,2] => 2 [2,1] => 1 [1,2,3] => 6 [1,3,2] => 4 [2,1,3] => 4 [2,3,1] => 2 [3,1,2] => 2 [3,2,1] => 1 [1,2,3,4] => 20 [1,2,4,3] => 12 [1,3,2,4] => 16 [1,3,4,2] => 10 [1,4,2,3] => 10 [1,4,3,2] => 4 [2,1,3,4] => 12 [2,1,4,3] => 4 [2,3,1,4] => 10 [2,3,4,1] => 6 [2,4,1,3] => 8 [2,4,3,1] => 4 [3,1,2,4] => 10 [3,1,4,2] => 8 [3,2,1,4] => 4 [3,2,4,1] => 4 [3,4,1,2] => 4 [3,4,2,1] => 2 [4,1,2,3] => 6 [4,1,3,2] => 4 [4,2,1,3] => 4 [4,2,3,1] => 2 [4,3,1,2] => 2 [4,3,2,1] => 1 [1,2,3,4,5] => 66 [1,2,3,5,4] => 36 [1,2,4,3,5] => 48 [1,2,4,5,3] => 31 [1,2,5,3,4] => 31 [1,2,5,4,3] => 12 [1,3,2,4,5] => 48 [1,3,2,5,4] => 16 [1,3,4,2,5] => 48 [1,3,4,5,2] => 30 [1,3,5,2,4] => 32 [1,3,5,4,2] => 16 [1,4,2,3,5] => 48 [1,4,2,5,3] => 32 [1,4,3,2,5] => 16 [1,4,3,5,2] => 16 [1,4,5,2,3] => 25 [1,4,5,3,2] => 10 [1,5,2,3,4] => 30 [1,5,2,4,3] => 16 [1,5,3,2,4] => 16 [1,5,3,4,2] => 8 [1,5,4,2,3] => 10 [1,5,4,3,2] => 4 [2,1,3,4,5] => 36 [2,1,3,5,4] => 16 [2,1,4,3,5] => 16 [2,1,4,5,3] => 10 [2,1,5,3,4] => 10 [2,1,5,4,3] => 4 [2,3,1,4,5] => 31 [2,3,1,5,4] => 10 [2,3,4,1,5] => 30 [2,3,4,5,1] => 20 [2,3,5,1,4] => 24 [2,3,5,4,1] => 12 [2,4,1,3,5] => 32 [2,4,1,5,3] => 16 [2,4,3,1,5] => 16 [2,4,3,5,1] => 16 [2,4,5,1,3] => 20 [2,4,5,3,1] => 10 [2,5,1,3,4] => 24 [2,5,1,4,3] => 8 [2,5,3,1,4] => 16 [2,5,3,4,1] => 10 [2,5,4,1,3] => 8 [2,5,4,3,1] => 4 [3,1,2,4,5] => 31 [3,1,2,5,4] => 10 [3,1,4,2,5] => 32 [3,1,4,5,2] => 24 [3,1,5,2,4] => 16 [3,1,5,4,2] => 8 [3,2,1,4,5] => 12 [3,2,1,5,4] => 4 [3,2,4,1,5] => 16 [3,2,4,5,1] => 12 [3,2,5,1,4] => 8 [3,2,5,4,1] => 4 [3,4,1,2,5] => 25 [3,4,1,5,2] => 20 [3,4,2,1,5] => 10 [3,4,2,5,1] => 10 [3,4,5,1,2] => 12 [3,4,5,2,1] => 6 [3,5,1,2,4] => 20 [3,5,1,4,2] => 16 [3,5,2,1,4] => 8 [3,5,2,4,1] => 8 [3,5,4,1,2] => 8 [3,5,4,2,1] => 4 [4,1,2,3,5] => 30 [4,1,2,5,3] => 24 [4,1,3,2,5] => 16 [4,1,3,5,2] => 16 [4,1,5,2,3] => 20 [4,1,5,3,2] => 8 [4,2,1,3,5] => 16 [4,2,1,5,3] => 8 [4,2,3,1,5] => 8 [4,2,3,5,1] => 10 [4,2,5,1,3] => 16 [4,2,5,3,1] => 8 [4,3,1,2,5] => 10 [4,3,1,5,2] => 8 [4,3,2,1,5] => 4 [4,3,2,5,1] => 4 [4,3,5,1,2] => 8 [4,3,5,2,1] => 4 [4,5,1,2,3] => 12 [4,5,1,3,2] => 8 [4,5,2,1,3] => 8 [4,5,2,3,1] => 4 [4,5,3,1,2] => 4 [4,5,3,2,1] => 2 [5,1,2,3,4] => 20 [5,1,2,4,3] => 12 [5,1,3,2,4] => 16 [5,1,3,4,2] => 10 [5,1,4,2,3] => 10 [5,1,4,3,2] => 4 [5,2,1,3,4] => 12 [5,2,1,4,3] => 4 [5,2,3,1,4] => 10 [5,2,3,4,1] => 6 [5,2,4,1,3] => 8 [5,2,4,3,1] => 4 [5,3,1,2,4] => 10 [5,3,1,4,2] => 8 [5,3,2,1,4] => 4 [5,3,2,4,1] => 4 [5,3,4,1,2] => 4 [5,3,4,2,1] => 2 [5,4,1,2,3] => 6 [5,4,1,3,2] => 4 [5,4,2,1,3] => 4 [5,4,2,3,1] => 2 [5,4,3,1,2] => 2 [5,4,3,2,1] => 1 ----------------------------------------------------------------------------- Created: Jul 12, 2016 at 13:51 by Sara Billey ----------------------------------------------------------------------------- Last Updated: Dec 30, 2016 at 10:32 by Christian Stump