***************************************************************************** * 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: St001144 ----------------------------------------------------------------------------- Collection: Finite Cartan types ----------------------------------------------------------------------------- Description: The largest mu-coefficient of the Kazhdan Lusztig polynomial occurring in the Weyl group of given type. The $\mu$-coefficient of the Kazhdan-Lusztig polynomial $P_{u,w}(q)$ is the coefficient of $q^{\frac{l(w)-l(u)-1}{2}}$ in $P_{u,w}(q)$. ----------------------------------------------------------------------------- References: [1] Warrington, G. S. Equivalence classes for the ยต-coefficient of Kazhdan-Lusztig polynomials in $S_n$ [[MathSciNet:2859901]] ----------------------------------------------------------------------------- Code: def statistic(C): W = CoxeterGroup(C, implementation='coxeter3') r = [] for u in W: U = (W(v) for v in W.bruhat_interval(u, W.long_element())) next(U) for v in U: ldiff = v.length()-u.length()-1 if is_even(ldiff): p = W.kazhdan_lusztig_polynomial(u, v) r.append(p[ldiff//2]) return max(r) ----------------------------------------------------------------------------- Statistic values: ['A',1] => 1 ['A',2] => 1 ['B',2] => 1 ['G',2] => 1 ['A',3] => 1 ['B',3] => 1 ['C',3] => 1 ['A',4] => 1 ['B',4] => 1 ['C',4] => 1 ['D',4] => 1 ['F',4] => 1 ['A',5] => 1 ['B',5] => 2 ['C',5] => 2 ['D',5] => 1 ['A',6] => 1 ['A',7] => 1 ['A',8] => 1 ['A',9] => 5 ['A',10] => 28 ----------------------------------------------------------------------------- Created: Apr 18, 2018 at 22:49 by Martin Rubey ----------------------------------------------------------------------------- Last Updated: Apr 18, 2018 at 22:49 by Martin Rubey