***************************************************************************** * 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: St001467 ----------------------------------------------------------------------------- Collection: Finite Cartan types ----------------------------------------------------------------------------- Description: The number of involutions in the Weyl group of a given Cartan type. For type $A_n$, the generating function is $\exp(x+x^2/2)$, for type $BC_n$ it is $\exp(x^2+2x)$ and for type $D_n$ it is $\exp(x^2)(\exp(2x)+1)/2$. ----------------------------------------------------------------------------- References: ----------------------------------------------------------------------------- Code: def statistic(C): return sum(1 for x in WeylGroup(C) if x == x.inverse()) ----------------------------------------------------------------------------- Statistic values: ['A',1] => 2 ['A',2] => 4 ['B',2] => 6 ['G',2] => 8 ['A',3] => 10 ['B',3] => 20 ['C',3] => 20 ['A',4] => 26 ['B',4] => 76 ['C',4] => 76 ['D',4] => 44 ['F',4] => 140 ['A',5] => 76 ['B',5] => 312 ['C',5] => 312 ['D',5] => 156 ['A',6] => 232 ['B',6] => 1384 ['C',6] => 1384 ['D',6] => 752 ----------------------------------------------------------------------------- Created: Sep 02, 2019 at 14:20 by Martin Rubey ----------------------------------------------------------------------------- Last Updated: Sep 02, 2019 at 14:20 by Martin Rubey