***************************************************************************** * 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: St000858 ----------------------------------------------------------------------------- Collection: Finite Cartan types ----------------------------------------------------------------------------- Description: The number of factorizations of any Coxeter element into reflections of a finite Cartan type. The number of such factorizations is given by $n!h^n / |W|$ where $n$ is the rank, $h$ is the Coxeter number and $W$ is the Weyl group of the given Cartan type. This was originally proven in a letter from Deligne to Looijenga in the 1970s, and then recovered in [2, Theorem 3.6]. As an example, consider the three ($=2!3^2/6$) factorizations of the Coxeter element $$(1,2,3) = (1,2)(2,3) = (1,3)(1,2) = (2,3)(1,3)$$ in type $A_2$. ----------------------------------------------------------------------------- References: [1] Letter from Deligne to Looijenga [[http://homepage.univie.ac.at/christian.stump/Deligne_Looijenga_Letter_09-03-1974.pdf]] [2] Reading, N. Chains in the noncrossing partition lattice [[MathSciNet:2424827]] ----------------------------------------------------------------------------- Code: ----------------------------------------------------------------------------- Statistic values: ['A',1] => 1 ['A',2] => 3 ['B',2] => 4 ['G',2] => 6 ['A',3] => 16 ['B',3] => 27 ['C',3] => 27 ['A',4] => 125 ['B',4] => 256 ['C',4] => 256 ['D',4] => 162 ['F',4] => 432 ['A',5] => 1296 ['B',5] => 3125 ['C',5] => 3125 ['D',5] => 2048 ['A',6] => 16807 ['B',6] => 46656 ['C',6] => 46656 ['D',6] => 31250 ['E',6] => 41472 ['A',7] => 262144 ['B',7] => 823543 ['C',7] => 823543 ['D',7] => 559872 ['E',7] => 1062882 ['A',8] => 4782969 ['B',8] => 16777216 ['C',8] => 16777216 ['D',8] => 11529602 ['E',8] => 37968750 ----------------------------------------------------------------------------- Created: Jun 25, 2017 at 10:00 by Christian Stump ----------------------------------------------------------------------------- Last Updated: Jun 25, 2017 at 10:00 by Christian Stump