***************************************************************************** * 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: St001157 ----------------------------------------------------------------------------- Collection: Finite Cartan types ----------------------------------------------------------------------------- Description: The exponent of the Weyl group of given type. This is the least common multiple of the orders of the elements of the group. In a comment to [4], see also [5], it is asked whether this is the same as the least common multiple of the degrees of the Weyl group. ----------------------------------------------------------------------------- References: [1] [[groupprops:Exponent_of_a_group]] [2] Least common multiple (or LCM) of 1, 2, ..., n for n >= 1, a(0) = 1. [[OEIS:A003418]] [3] Least common multiple of 2, 4, 6, ..., 2n. [[OEIS:A051426]] [4] user66288 The maximal order of an element in a Coxeter group [[MathOverflow:343118]] [5] user66288 The least common multiple of all degrees of a finite Coxeter group and indecomposable elements in the generalized cycle decomposition [[MathOverflow:345908]] ----------------------------------------------------------------------------- Code: def statistic(C): return WeylGroup(C).as_permutation_group().exponent() ----------------------------------------------------------------------------- Statistic values: ['A',1] => 2 ['A',2] => 6 ['B',2] => 4 ['G',2] => 6 ['A',3] => 12 ['B',3] => 12 ['C',3] => 12 ['A',4] => 60 ['B',4] => 24 ['C',4] => 24 ['D',4] => 12 ['F',4] => 24 ['A',5] => 60 ['B',5] => 120 ['C',5] => 120 ['D',5] => 120 ['A',6] => 420 ['B',6] => 120 ['C',6] => 120 ['D',6] => 120 ['E',6] => 360 ['A',7] => 840 ['B',7] => 840 ['C',7] => 840 ['D',7] => 840 ['E',7] => 2520 ['A',8] => 2520 ['B',8] => 1680 ['C',8] => 1680 ['D',8] => 840 ['E',8] => 2520 ['D',9] => 5040 ['D',10] => 5040 ['D',11] => 55440 ----------------------------------------------------------------------------- Created: Apr 22, 2018 at 15:19 by Martin Rubey ----------------------------------------------------------------------------- Last Updated: Nov 13, 2019 at 10:37 by Martin Rubey