***************************************************************************** * 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: St000865 ----------------------------------------------------------------------------- Collection: Finite Cartan types ----------------------------------------------------------------------------- Description: The number of Coxeter elements in the Weyl group of a finite Cartan type. This is, the elements that are conjugate to the product of the simple generators in any order, or, equivalently, the elements that admit a primitive \$h\$-th root of unity as an eigenvalue where \$h\$ is the Coxeter number. ----------------------------------------------------------------------------- References: [1] Reiner, V., Ripoll, V., Stump, C. On non-conjugate Coxeter elements in well-generated reflection groups [[MathSciNet:3623739]] ----------------------------------------------------------------------------- Code: def statistic(cartan_type): W = ReflectionGroup(cartan_type) return len(W.coxeter_elements()) ----------------------------------------------------------------------------- Statistic values: ['A',1] => 1 ['A',2] => 2 ['B',2] => 2 ['G',2] => 2 ['A',3] => 6 ['B',3] => 8 ['C',3] => 8 ['A',4] => 24 ['B',4] => 48 ['C',4] => 48 ['D',4] => 32 ['F',4] => 96 ['A',5] => 120 ['B',5] => 384 ['C',5] => 384 ['D',5] => 240 ['A',6] => 720 ['B',6] => 3840 ['C',6] => 3840 ['D',6] => 2304 ['E',6] => 4320 ['A',7] => 5040 ['B',7] => 46080 ['C',7] => 46080 ['D',7] => 26880 ['E',7] => 161280 ['A',8] => 40320 ['B',8] => 645120 ----------------------------------------------------------------------------- Created: Jun 25, 2017 at 20:14 by Christian Stump ----------------------------------------------------------------------------- Last Updated: Jun 26, 2017 at 08:34 by Christian Stump