***************************************************************************** * 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: St001989 ----------------------------------------------------------------------------- Collection: Finite Cartan types ----------------------------------------------------------------------------- Description: The number of reduced decompositions of the Weyl Group of the given Cartan type. Equivalently, the number of chains based at the identity in the weak order. ----------------------------------------------------------------------------- References: [1] Total number of reduced decompositions for all permutations in S_n. [[OEIS:A246865]] [2] The number of reduced decompositions of the longest element of the Weyl group of the given Cartan type. [[St001585]] [3] The number of reduced words for a permutation. [[St000001]] [4] The number of reduced words of a signed permutation. [[St001768]] ----------------------------------------------------------------------------- Code: def statistic(c): return sum(1 for e in WeylGroup(c) for _ in e.reduced_words_iter()) ----------------------------------------------------------------------------- Statistic values: ['A',1] => 2 ['A',2] => 7 ['B',2] => 9 ['G',2] => 13 ['A',3] => 66 ['B',3] => 209 ['C',3] => 209 ['A',4] => 3061 ['B',4] => 103484 ['C',4] => 103484 ['D',4] => 9719 ['F',4] => 9593133 ['A',5] => 1095266 ----------------------------------------------------------------------------- Created: Nov 18, 2025 at 08:57 by Martin Rubey ----------------------------------------------------------------------------- Last Updated: Nov 18, 2025 at 13:03 by Martin Rubey