***************************************************************************** * 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: St001790 ----------------------------------------------------------------------------- Collection: Finite Cartan types ----------------------------------------------------------------------------- Description: The number of reflection subgroups of the associated Weyl group. Let $\mathcal{R} \subseteq W$ be the set of reflections in the Weyl group $W$. A (possibly empty) subset $X \subseteq \mathcal{R}$ generates a subgroup of $W$ that is again a reflection group. This is the number of all pairwise different subgroups of $W$ obtained this way (including the trivial subgroup). If $\Phi^+$ is an associated set of positive roots, then this also is the number of subsets $Y \subseteq \Phi^+$ such that $Y$ is a simple system of some type (including the empty system for type $A_0$). Such a subset $Y$ is identified as simple system if for all $x \neq y \in Y$ we have $\langle x,y \rangle \leq 0$. ----------------------------------------------------------------------------- References: ----------------------------------------------------------------------------- Code: ----------------------------------------------------------------------------- Statistic values: ['A',1] => 2 ['A',2] => 5 ['B',2] => 8 ['G',2] => 13 ['A',3] => 15 ['B',3] => 38 ['C',3] => 38 ['A',4] => 52 ['B',4] => 218 ['C',4] => 218 ['D',4] => 75 ['F',4] => 637 ['A',5] => 203 ['B',5] => 1430 ['C',5] => 1430 ['D',5] => 428 ['A',6] => 877 ['B',6] => 10514 ['C',6] => 10514 ['D',6] => 2781 ['E',6] => 5079 ['A',7] => 4140 ['B',7] => 85202 ['C',7] => 85202 ['D',7] => 20093 ['E',7] => 107911 ['A',8] => 21147 ['B',8] => 751982 ['C',8] => 751982 ['D',8] => 159340 ['E',8] => 7591975 ['C',2] => 8 ----------------------------------------------------------------------------- Created: May 03, 2022 at 13:32 by Dennis Jahn ----------------------------------------------------------------------------- Last Updated: May 04, 2022 at 11:00 by Dennis Jahn