***************************************************************************** * 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: St001143 ----------------------------------------------------------------------------- Collection: Finite Cartan types ----------------------------------------------------------------------------- Description: The number of pairs in the Weyl group of given type with mu-coefficient of the Kazhdan Lusztig polynomial being non-zero. 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] Vogan, D. Number of pairs of permutation in $S_n$ whose µ-coefficient (of their Kazhdan Lusztig polynomial) is non-zero [[MathOverflow:298028]] [2] Warrington, G. S. Equivalence classes for the µ-coefficient of Kazhdan-Lusztig polynomials in $S_n$ [[MathSciNet:2859901]] ----------------------------------------------------------------------------- Code: def statistic(C): """ sage: statistic(CartanType(["A", 4])) 482 """ W = CoxeterGroup(C, implementation='coxeter3') r = 0 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) if p[ldiff//2] != 0: r += 1 return r ----------------------------------------------------------------------------- Statistic values: ['A',1] => 1 ['A',2] => 8 ['B',2] => 12 ['G',2] => 20 ['A',3] => 60 ['B',3] => 152 ['C',3] => 152 ['A',4] => 482 ['B',4] => 2148 ['C',4] => 2148 ['D',4] => 892 ['F',4] => 8920 ['A',5] => 4268 ['B',5] => 35070 ['C',5] => 35070 ['D',5] => 14874 ['A',6] => 41934 ['B',6] => 679152 ['C',6] => 679152 ['D',6] => 287438 ['E',6] => 846476 ['A',7] => 457782 ----------------------------------------------------------------------------- Created: Apr 18, 2018 at 22:32 by Martin Rubey ----------------------------------------------------------------------------- Last Updated: Apr 18, 2018 at 22:32 by Martin Rubey