***************************************************************************** * 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: St001888 ----------------------------------------------------------------------------- Collection: Finite Cartan types ----------------------------------------------------------------------------- Description: The number of connected elements in the Coxeter group corresponding to a finite Cartan type. Let $(W, S)$ be a Coxeter system. Then, according to [1], the connectivity set of $w\in W$ is the cardinality of $S \setminus S(w)$, where $S(w)$ is the set of generators appearing in any reduced word for $w$. For $A_n$, this is [2], for $B_n$ this is [3] and for $D_n$ this is [4]. ----------------------------------------------------------------------------- References: [1] Bergeron, N., Hohlweg, C., Zabrocki, M. Posets related to the connectivity set of Coxeter groups [[arXiv:math/0509271]] [2] Number of connected permutations of [1..n] (those not fixing [1..j] for 0 < j < n). Also called indecomposable permutations, or irreducible permutations. [[OEIS:A003319]] [3] Number of elements of the Weyl group of type B where a reduced word contains all of the simple reflections. [[OEIS:A109253]] [4] Number of elements of a Weyl group of order 2^n-1 n! of type D for which a reduced word contains all of the simple reflections. [[OEIS:A112225]] ----------------------------------------------------------------------------- Code: def connected(ct): W = CoxeterGroup(ct) I = set(W.index_set()) return sum(1 for w in W if not I.difference(w.reduced_word())) ----------------------------------------------------------------------------- Statistic values: ['A',1] => 1 ['A',2] => 3 ['B',2] => 5 ['G',2] => 9 ['A',3] => 13 ['B',3] => 35 ['C',3] => 35 ['A',4] => 71 ['B',4] => 309 ['C',4] => 309 ['D',4] => 135 ['F',4] => 1057 ['A',5] => 461 ['B',5] => 3287 ['C',5] => 3287 ['D',5] => 1537 ['A',6] => 3447 ['B',6] => 41005 ['C',6] => 41005 ['D',6] => 19811 ['E',6] => 47527 ['A',7] => 29093 ----------------------------------------------------------------------------- Created: Feb 08, 2023 at 18:00 by Martin Rubey ----------------------------------------------------------------------------- Last Updated: Feb 08, 2023 at 18:00 by Martin Rubey