***************************************************************************** * 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: St001585 ----------------------------------------------------------------------------- Collection: Finite Cartan types ----------------------------------------------------------------------------- Description: The number of reduced decompositions of the longest element of the Weyl group of the given Cartan type. Equivalently, this is the number of chains in the weak order from the identity to the longest element. In type $A_n$, this is $$ \binom{n+1}{2}!/(1^n 3^{n-1} \dots (2n-1)^1). $$ In type $B_n$ and $C_n$ this is $$ (n^2)!\prod_{k=1}^{n-1} k! / \prod_{k=n}^{2n-1} k!. $$ ----------------------------------------------------------------------------- References: [1] Stanley, R. P. On the number of reduced decompositions of elements of Coxeter groups [[MathSciNet:0782057]] [2] Kraƛkiewicz, W. Reduced decompositions in Weyl groups [[MathSciNet:1330543]] [3] Winkelman, B. Number of reduced decompositions of the longest element of the Weyl group [[MathOverflow:370333]] ----------------------------------------------------------------------------- Code: def statistic(c): if c.type() == "A": n = c.rank() return binomial(n+1, 2).factorial()/prod((2*i-1)^(n+1-i) for i in range(1,n+1)) if c.type() in ["B", "C"]: n = c.rank() return ZZ(n^2).factorial()*prod(ZZ(k).factorial() for k in range(1, n))/ prod(ZZ(k).factorial() for k in range(n, 2*n)) P = WeylGroup(c).weak_poset() p = P.chain_polynomial() return p.leading_coefficient() ----------------------------------------------------------------------------- Statistic values: ['A',1] => 1 ['A',2] => 2 ['B',2] => 2 ['G',2] => 2 ['A',3] => 16 ['B',3] => 42 ['C',3] => 42 ['A',4] => 768 ['B',4] => 24024 ['C',4] => 24024 ['D',4] => 2316 ['F',4] => 2144892 ['A',5] => 292864 ['B',5] => 701149020 ['C',5] => 701149020 ['D',5] => 12985968 ['A',6] => 1100742656 ----------------------------------------------------------------------------- Created: Aug 28, 2020 at 19:40 by Martin Rubey ----------------------------------------------------------------------------- Last Updated: Aug 28, 2020 at 22:18 by Martin Rubey