***************************************************************************** * 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: St000036 ----------------------------------------------------------------------------- Collection: Permutations ----------------------------------------------------------------------------- Description: The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. These are multiplicities of Verma modules. ----------------------------------------------------------------------------- References: [1] Kazhdan, D., Lusztig, G. Representations of Coxeter groups and Hecke algebras [[MathSciNet:0560412]] [2] Björner, A., Brenti, F. Combinatorics of Coxeter groups [[MathSciNet:2133266]] [3] Humphreys, J. E. Reflection groups and Coxeter groups [[MathSciNet:1066460]] [4] Number of permutations w in S_n whose Kazhdan-Lusztig polynomial evaluated at q=1 is less than or equal to 2; P_id,w(1) <= 2. [[OEIS:A224255]] [5] Number of permutations w in S_n whose Kazhdan-Lusztig polynomial P_id,w(1) <= 3. [[OEIS:A224316]] [6] Woo, A. Permutations with Kazhdan-Lusztig polynomial $P_{id,w}(q)=1+q^h$ [[MathSciNet:2515773]] ----------------------------------------------------------------------------- Code: # takes a long time! def statistic(x): if x.size() == 1: return 1 G = WeylGroup(['A',x.size()-1]) R.=LaurentPolynomialRing(QQ) KL=KazhdanLusztigPolynomial(G,q) w = G.from_reduced_word(x.reduced_word()) return KL.P(1,w).substitute({q:1}) # alternative, with coxeter3 installed: def statistic_alt(x): if x.size() == 1: return 1 G = CoxeterGroup(['A',x.size()-1], implementation="coxeter3") R. = PolynomialRing(QQ) P = G.kazhdan_lusztig_polynomial([],x.reduced_word()) return P.substitute({q:1}) ----------------------------------------------------------------------------- Statistic values: [1] => 1 [1,2] => 1 [2,1] => 1 [1,2,3] => 1 [1,3,2] => 1 [2,1,3] => 1 [2,3,1] => 1 [3,1,2] => 1 [3,2,1] => 1 [1,2,3,4] => 1 [1,2,4,3] => 1 [1,3,2,4] => 1 [1,3,4,2] => 1 [1,4,2,3] => 1 [1,4,3,2] => 1 [2,1,3,4] => 1 [2,1,4,3] => 1 [2,3,1,4] => 1 [2,3,4,1] => 1 [2,4,1,3] => 1 [2,4,3,1] => 1 [3,1,2,4] => 1 [3,1,4,2] => 1 [3,2,1,4] => 1 [3,2,4,1] => 1 [3,4,1,2] => 2 [3,4,2,1] => 1 [4,1,2,3] => 1 [4,1,3,2] => 1 [4,2,1,3] => 1 [4,2,3,1] => 2 [4,3,1,2] => 1 [4,3,2,1] => 1 [1,2,3,4,5] => 1 [1,2,3,5,4] => 1 [1,2,4,3,5] => 1 [1,2,4,5,3] => 1 [1,2,5,3,4] => 1 [1,2,5,4,3] => 1 [1,3,2,4,5] => 1 [1,3,2,5,4] => 1 [1,3,4,2,5] => 1 [1,3,4,5,2] => 1 [1,3,5,2,4] => 1 [1,3,5,4,2] => 1 [1,4,2,3,5] => 1 [1,4,2,5,3] => 1 [1,4,3,2,5] => 1 [1,4,3,5,2] => 1 [1,4,5,2,3] => 2 [1,4,5,3,2] => 1 [1,5,2,3,4] => 1 [1,5,2,4,3] => 1 [1,5,3,2,4] => 1 [1,5,3,4,2] => 2 [1,5,4,2,3] => 1 [1,5,4,3,2] => 1 [2,1,3,4,5] => 1 [2,1,3,5,4] => 1 [2,1,4,3,5] => 1 [2,1,4,5,3] => 1 [2,1,5,3,4] => 1 [2,1,5,4,3] => 1 [2,3,1,4,5] => 1 [2,3,1,5,4] => 1 [2,3,4,1,5] => 1 [2,3,4,5,1] => 1 [2,3,5,1,4] => 1 [2,3,5,4,1] => 1 [2,4,1,3,5] => 1 [2,4,1,5,3] => 1 [2,4,3,1,5] => 1 [2,4,3,5,1] => 1 [2,4,5,1,3] => 2 [2,4,5,3,1] => 1 [2,5,1,3,4] => 1 [2,5,1,4,3] => 1 [2,5,3,1,4] => 1 [2,5,3,4,1] => 2 [2,5,4,1,3] => 1 [2,5,4,3,1] => 1 [3,1,2,4,5] => 1 [3,1,2,5,4] => 1 [3,1,4,2,5] => 1 [3,1,4,5,2] => 1 [3,1,5,2,4] => 1 [3,1,5,4,2] => 1 [3,2,1,4,5] => 1 [3,2,1,5,4] => 1 [3,2,4,1,5] => 1 [3,2,4,5,1] => 1 [3,2,5,1,4] => 1 [3,2,5,4,1] => 1 [3,4,1,2,5] => 2 [3,4,1,5,2] => 2 [3,4,2,1,5] => 1 [3,4,2,5,1] => 1 [3,4,5,1,2] => 3 [3,4,5,2,1] => 1 [3,5,1,2,4] => 2 [3,5,1,4,2] => 2 [3,5,2,1,4] => 1 [3,5,2,4,1] => 2 [3,5,4,1,2] => 2 [3,5,4,2,1] => 1 [4,1,2,3,5] => 1 [4,1,2,5,3] => 1 [4,1,3,2,5] => 1 [4,1,3,5,2] => 1 [4,1,5,2,3] => 2 [4,1,5,3,2] => 1 [4,2,1,3,5] => 1 [4,2,1,5,3] => 1 [4,2,3,1,5] => 2 [4,2,3,5,1] => 2 [4,2,5,1,3] => 2 [4,2,5,3,1] => 2 [4,3,1,2,5] => 1 [4,3,1,5,2] => 1 [4,3,2,1,5] => 1 [4,3,2,5,1] => 1 [4,3,5,1,2] => 2 [4,3,5,2,1] => 1 [4,5,1,2,3] => 3 [4,5,1,3,2] => 2 [4,5,2,1,3] => 2 [4,5,2,3,1] => 3 [4,5,3,1,2] => 2 [4,5,3,2,1] => 1 [5,1,2,3,4] => 1 [5,1,2,4,3] => 1 [5,1,3,2,4] => 1 [5,1,3,4,2] => 2 [5,1,4,2,3] => 1 [5,1,4,3,2] => 1 [5,2,1,3,4] => 1 [5,2,1,4,3] => 1 [5,2,3,1,4] => 2 [5,2,3,4,1] => 4 [5,2,4,1,3] => 2 [5,2,4,3,1] => 2 [5,3,1,2,4] => 1 [5,3,1,4,2] => 2 [5,3,2,1,4] => 1 [5,3,2,4,1] => 2 [5,3,4,1,2] => 3 [5,3,4,2,1] => 2 [5,4,1,2,3] => 1 [5,4,1,3,2] => 1 [5,4,2,1,3] => 1 [5,4,2,3,1] => 2 [5,4,3,1,2] => 1 [5,4,3,2,1] => 1 [1,2,3,4,5,6] => 1 [1,2,3,4,6,5] => 1 [1,2,3,5,4,6] => 1 [1,2,3,5,6,4] => 1 [1,2,3,6,4,5] => 1 [1,2,3,6,5,4] => 1 [1,2,4,3,5,6] => 1 [1,2,4,3,6,5] => 1 [1,2,4,5,3,6] => 1 [1,2,4,5,6,3] => 1 [1,2,4,6,3,5] => 1 [1,2,4,6,5,3] => 1 [1,2,5,3,4,6] => 1 [1,2,5,3,6,4] => 1 [1,2,5,4,3,6] => 1 [1,2,5,4,6,3] => 1 [1,2,5,6,3,4] => 2 [1,2,5,6,4,3] => 1 [1,2,6,3,4,5] => 1 [1,2,6,3,5,4] => 1 [1,2,6,4,3,5] => 1 [1,2,6,4,5,3] => 2 [1,2,6,5,3,4] => 1 [1,2,6,5,4,3] => 1 [1,3,2,4,5,6] => 1 [1,3,2,4,6,5] => 1 [1,3,2,5,4,6] => 1 [1,3,2,5,6,4] => 1 [1,3,2,6,4,5] => 1 [1,3,2,6,5,4] => 1 [1,3,4,2,5,6] => 1 [1,3,4,2,6,5] => 1 [1,3,4,5,2,6] => 1 [1,3,4,5,6,2] => 1 [1,3,4,6,2,5] => 1 [1,3,4,6,5,2] => 1 [1,3,5,2,4,6] => 1 [1,3,5,2,6,4] => 1 [1,3,5,4,2,6] => 1 [1,3,5,4,6,2] => 1 [1,3,5,6,2,4] => 2 [1,3,5,6,4,2] => 1 [1,3,6,2,4,5] => 1 [1,3,6,2,5,4] => 1 [1,3,6,4,2,5] => 1 [1,3,6,4,5,2] => 2 [1,3,6,5,2,4] => 1 [1,3,6,5,4,2] => 1 [1,4,2,3,5,6] => 1 [1,4,2,3,6,5] => 1 [1,4,2,5,3,6] => 1 [1,4,2,5,6,3] => 1 [1,4,2,6,3,5] => 1 [1,4,2,6,5,3] => 1 [1,4,3,2,5,6] => 1 [1,4,3,2,6,5] => 1 [1,4,3,5,2,6] => 1 [1,4,3,5,6,2] => 1 [1,4,3,6,2,5] => 1 [1,4,3,6,5,2] => 1 [1,4,5,2,3,6] => 2 [1,4,5,2,6,3] => 2 [1,4,5,3,2,6] => 1 [1,4,5,3,6,2] => 1 [1,4,5,6,2,3] => 3 [1,4,5,6,3,2] => 1 [1,4,6,2,3,5] => 2 [1,4,6,2,5,3] => 2 [1,4,6,3,2,5] => 1 [1,4,6,3,5,2] => 2 [1,4,6,5,2,3] => 2 [1,4,6,5,3,2] => 1 [1,5,2,3,4,6] => 1 [1,5,2,3,6,4] => 1 [1,5,2,4,3,6] => 1 [1,5,2,4,6,3] => 1 [1,5,2,6,3,4] => 2 [1,5,2,6,4,3] => 1 [1,5,3,2,4,6] => 1 [1,5,3,2,6,4] => 1 [1,5,3,4,2,6] => 2 [1,5,3,4,6,2] => 2 [1,5,3,6,2,4] => 2 [1,5,3,6,4,2] => 2 [1,5,4,2,3,6] => 1 [1,5,4,2,6,3] => 1 [1,5,4,3,2,6] => 1 [1,5,4,3,6,2] => 1 [1,5,4,6,2,3] => 2 [1,5,4,6,3,2] => 1 [1,5,6,2,3,4] => 3 [1,5,6,2,4,3] => 2 [1,5,6,3,2,4] => 2 [1,5,6,3,4,2] => 3 [1,5,6,4,2,3] => 2 [1,5,6,4,3,2] => 1 [1,6,2,3,4,5] => 1 [1,6,2,3,5,4] => 1 [1,6,2,4,3,5] => 1 [1,6,2,4,5,3] => 2 [1,6,2,5,3,4] => 1 [1,6,2,5,4,3] => 1 [1,6,3,2,4,5] => 1 [1,6,3,2,5,4] => 1 [1,6,3,4,2,5] => 2 [1,6,3,4,5,2] => 4 [1,6,3,5,2,4] => 2 [1,6,3,5,4,2] => 2 [1,6,4,2,3,5] => 1 [1,6,4,2,5,3] => 2 [1,6,4,3,2,5] => 1 [1,6,4,3,5,2] => 2 [1,6,4,5,2,3] => 3 [1,6,4,5,3,2] => 2 [1,6,5,2,3,4] => 1 [1,6,5,2,4,3] => 1 [1,6,5,3,2,4] => 1 [1,6,5,3,4,2] => 2 [1,6,5,4,2,3] => 1 [1,6,5,4,3,2] => 1 [2,1,3,4,5,6] => 1 [2,1,3,4,6,5] => 1 [2,1,3,5,4,6] => 1 [2,1,3,5,6,4] => 1 [2,1,3,6,4,5] => 1 [2,1,3,6,5,4] => 1 [2,1,4,3,5,6] => 1 [2,1,4,3,6,5] => 1 [2,1,4,5,3,6] => 1 [2,1,4,5,6,3] => 1 [2,1,4,6,3,5] => 1 [2,1,4,6,5,3] => 1 [2,1,5,3,4,6] => 1 [2,1,5,3,6,4] => 1 [2,1,5,4,3,6] => 1 [2,1,5,4,6,3] => 1 [2,1,5,6,3,4] => 2 [2,1,5,6,4,3] => 1 [2,1,6,3,4,5] => 1 [2,1,6,3,5,4] => 1 [2,1,6,4,3,5] => 1 [2,1,6,4,5,3] => 2 [2,1,6,5,3,4] => 1 [2,1,6,5,4,3] => 1 [2,3,1,4,5,6] => 1 [2,3,1,4,6,5] => 1 [2,3,1,5,4,6] => 1 [2,3,1,5,6,4] => 1 [2,3,1,6,4,5] => 1 [2,3,1,6,5,4] => 1 [2,3,4,1,5,6] => 1 [2,3,4,1,6,5] => 1 [2,3,4,5,1,6] => 1 [2,3,4,5,6,1] => 1 [2,3,4,6,1,5] => 1 [2,3,4,6,5,1] => 1 [2,3,5,1,4,6] => 1 [2,3,5,1,6,4] => 1 [2,3,5,4,1,6] => 1 [2,3,5,4,6,1] => 1 [2,3,5,6,1,4] => 2 [2,3,5,6,4,1] => 1 [2,3,6,1,4,5] => 1 [2,3,6,1,5,4] => 1 [2,3,6,4,1,5] => 1 [2,3,6,4,5,1] => 2 [2,3,6,5,1,4] => 1 [2,3,6,5,4,1] => 1 [2,4,1,3,5,6] => 1 [2,4,1,3,6,5] => 1 [2,4,1,5,3,6] => 1 [2,4,1,5,6,3] => 1 [2,4,1,6,3,5] => 1 [2,4,1,6,5,3] => 1 [2,4,3,1,5,6] => 1 [2,4,3,1,6,5] => 1 [2,4,3,5,1,6] => 1 [2,4,3,5,6,1] => 1 [2,4,3,6,1,5] => 1 [2,4,3,6,5,1] => 1 [2,4,5,1,3,6] => 2 [2,4,5,1,6,3] => 2 [2,4,5,3,1,6] => 1 [2,4,5,3,6,1] => 1 [2,4,5,6,1,3] => 3 [2,4,5,6,3,1] => 1 [2,4,6,1,3,5] => 2 [2,4,6,1,5,3] => 2 [2,4,6,3,1,5] => 1 [2,4,6,3,5,1] => 2 [2,4,6,5,1,3] => 2 [2,4,6,5,3,1] => 1 [2,5,1,3,4,6] => 1 [2,5,1,3,6,4] => 1 [2,5,1,4,3,6] => 1 [2,5,1,4,6,3] => 1 [2,5,1,6,3,4] => 2 [2,5,1,6,4,3] => 1 [2,5,3,1,4,6] => 1 [2,5,3,1,6,4] => 1 [2,5,3,4,1,6] => 2 [2,5,3,4,6,1] => 2 [2,5,3,6,1,4] => 2 [2,5,3,6,4,1] => 2 [2,5,4,1,3,6] => 1 [2,5,4,1,6,3] => 1 [2,5,4,3,1,6] => 1 [2,5,4,3,6,1] => 1 [2,5,4,6,1,3] => 2 [2,5,4,6,3,1] => 1 [2,5,6,1,3,4] => 3 [2,5,6,1,4,3] => 2 [2,5,6,3,1,4] => 2 [2,5,6,3,4,1] => 3 [2,5,6,4,1,3] => 2 [2,5,6,4,3,1] => 1 [2,6,1,3,4,5] => 1 [2,6,1,3,5,4] => 1 [2,6,1,4,3,5] => 1 [2,6,1,4,5,3] => 2 [2,6,1,5,3,4] => 1 [2,6,1,5,4,3] => 1 [2,6,3,1,4,5] => 1 [2,6,3,1,5,4] => 1 [2,6,3,4,1,5] => 2 [2,6,3,4,5,1] => 4 [2,6,3,5,1,4] => 2 [2,6,3,5,4,1] => 2 [2,6,4,1,3,5] => 1 [2,6,4,1,5,3] => 2 [2,6,4,3,1,5] => 1 [2,6,4,3,5,1] => 2 [2,6,4,5,1,3] => 3 [2,6,4,5,3,1] => 2 [2,6,5,1,3,4] => 1 [2,6,5,1,4,3] => 1 [2,6,5,3,1,4] => 1 [2,6,5,3,4,1] => 2 [2,6,5,4,1,3] => 1 [2,6,5,4,3,1] => 1 [3,1,2,4,5,6] => 1 [3,1,2,4,6,5] => 1 [3,1,2,5,4,6] => 1 [3,1,2,5,6,4] => 1 [3,1,2,6,4,5] => 1 [3,1,2,6,5,4] => 1 [3,1,4,2,5,6] => 1 [3,1,4,2,6,5] => 1 [3,1,4,5,2,6] => 1 [3,1,4,5,6,2] => 1 [3,1,4,6,2,5] => 1 [3,1,4,6,5,2] => 1 [3,1,5,2,4,6] => 1 [3,1,5,2,6,4] => 1 [3,1,5,4,2,6] => 1 [3,1,5,4,6,2] => 1 [3,1,5,6,2,4] => 2 [3,1,5,6,4,2] => 1 [3,1,6,2,4,5] => 1 [3,1,6,2,5,4] => 1 [3,1,6,4,2,5] => 1 [3,1,6,4,5,2] => 2 [3,1,6,5,2,4] => 1 [3,1,6,5,4,2] => 1 [3,2,1,4,5,6] => 1 [3,2,1,4,6,5] => 1 [3,2,1,5,4,6] => 1 [3,2,1,5,6,4] => 1 [3,2,1,6,4,5] => 1 [3,2,1,6,5,4] => 1 [3,2,4,1,5,6] => 1 [3,2,4,1,6,5] => 1 [3,2,4,5,1,6] => 1 [3,2,4,5,6,1] => 1 [3,2,4,6,1,5] => 1 [3,2,4,6,5,1] => 1 [3,2,5,1,4,6] => 1 [3,2,5,1,6,4] => 1 [3,2,5,4,1,6] => 1 [3,2,5,4,6,1] => 1 [3,2,5,6,1,4] => 2 [3,2,5,6,4,1] => 1 [3,2,6,1,4,5] => 1 [3,2,6,1,5,4] => 1 [3,2,6,4,1,5] => 1 [3,2,6,4,5,1] => 2 [3,2,6,5,1,4] => 1 [3,2,6,5,4,1] => 1 [3,4,1,2,5,6] => 2 [3,4,1,2,6,5] => 2 [3,4,1,5,2,6] => 2 [3,4,1,5,6,2] => 2 [3,4,1,6,2,5] => 2 [3,4,1,6,5,2] => 2 [3,4,2,1,5,6] => 1 [3,4,2,1,6,5] => 1 [3,4,2,5,1,6] => 1 [3,4,2,5,6,1] => 1 [3,4,2,6,1,5] => 1 [3,4,2,6,5,1] => 1 [3,4,5,1,2,6] => 3 [3,4,5,1,6,2] => 3 [3,4,5,2,1,6] => 1 [3,4,5,2,6,1] => 1 [3,4,5,6,1,2] => 5 [3,4,5,6,2,1] => 1 [3,4,6,1,2,5] => 3 [3,4,6,1,5,2] => 3 [3,4,6,2,1,5] => 1 [3,4,6,2,5,1] => 2 [3,4,6,5,1,2] => 3 [3,4,6,5,2,1] => 1 [3,5,1,2,4,6] => 2 [3,5,1,2,6,4] => 2 [3,5,1,4,2,6] => 2 [3,5,1,4,6,2] => 2 [3,5,1,6,2,4] => 4 [3,5,1,6,4,2] => 2 [3,5,2,1,4,6] => 1 [3,5,2,1,6,4] => 1 [3,5,2,4,1,6] => 2 [3,5,2,4,6,1] => 2 [3,5,2,6,1,4] => 2 [3,5,2,6,4,1] => 2 [3,5,4,1,2,6] => 2 [3,5,4,1,6,2] => 2 [3,5,4,2,1,6] => 1 [3,5,4,2,6,1] => 1 [3,5,4,6,1,2] => 4 [3,5,4,6,2,1] => 1 [3,5,6,1,2,4] => 5 [3,5,6,1,4,2] => 3 [3,5,6,2,1,4] => 2 [3,5,6,2,4,1] => 3 [3,5,6,4,1,2] => 3 [3,5,6,4,2,1] => 1 [3,6,1,2,4,5] => 2 [3,6,1,2,5,4] => 2 [3,6,1,4,2,5] => 2 [3,6,1,4,5,2] => 4 [3,6,1,5,2,4] => 2 [3,6,1,5,4,2] => 2 [3,6,2,1,4,5] => 1 [3,6,2,1,5,4] => 1 [3,6,2,4,1,5] => 2 [3,6,2,4,5,1] => 4 [3,6,2,5,1,4] => 2 [3,6,2,5,4,1] => 2 [3,6,4,1,2,5] => 2 [3,6,4,1,5,2] => 4 [3,6,4,2,1,5] => 1 [3,6,4,2,5,1] => 2 [3,6,4,5,1,2] => 6 [3,6,4,5,2,1] => 2 [3,6,5,1,2,4] => 2 [3,6,5,1,4,2] => 2 [3,6,5,2,1,4] => 1 [3,6,5,2,4,1] => 2 [3,6,5,4,1,2] => 2 [3,6,5,4,2,1] => 1 [4,1,2,3,5,6] => 1 [4,1,2,3,6,5] => 1 [4,1,2,5,3,6] => 1 [4,1,2,5,6,3] => 1 [4,1,2,6,3,5] => 1 [4,1,2,6,5,3] => 1 [4,1,3,2,5,6] => 1 [4,1,3,2,6,5] => 1 [4,1,3,5,2,6] => 1 [4,1,3,5,6,2] => 1 [4,1,3,6,2,5] => 1 [4,1,3,6,5,2] => 1 [4,1,5,2,3,6] => 2 [4,1,5,2,6,3] => 2 [4,1,5,3,2,6] => 1 [4,1,5,3,6,2] => 1 [4,1,5,6,2,3] => 3 [4,1,5,6,3,2] => 1 [4,1,6,2,3,5] => 2 [4,1,6,2,5,3] => 2 [4,1,6,3,2,5] => 1 [4,1,6,3,5,2] => 2 [4,1,6,5,2,3] => 2 [4,1,6,5,3,2] => 1 [4,2,1,3,5,6] => 1 [4,2,1,3,6,5] => 1 [4,2,1,5,3,6] => 1 [4,2,1,5,6,3] => 1 [4,2,1,6,3,5] => 1 [4,2,1,6,5,3] => 1 [4,2,3,1,5,6] => 2 [4,2,3,1,6,5] => 2 [4,2,3,5,1,6] => 2 [4,2,3,5,6,1] => 2 [4,2,3,6,1,5] => 2 [4,2,3,6,5,1] => 2 [4,2,5,1,3,6] => 2 [4,2,5,1,6,3] => 2 [4,2,5,3,1,6] => 2 [4,2,5,3,6,1] => 2 [4,2,5,6,1,3] => 3 [4,2,5,6,3,1] => 2 [4,2,6,1,3,5] => 2 [4,2,6,1,5,3] => 2 [4,2,6,3,1,5] => 2 [4,2,6,3,5,1] => 4 [4,2,6,5,1,3] => 2 [4,2,6,5,3,1] => 2 [4,3,1,2,5,6] => 1 [4,3,1,2,6,5] => 1 [4,3,1,5,2,6] => 1 [4,3,1,5,6,2] => 1 [4,3,1,6,2,5] => 1 [4,3,1,6,5,2] => 1 [4,3,2,1,5,6] => 1 [4,3,2,1,6,5] => 1 [4,3,2,5,1,6] => 1 [4,3,2,5,6,1] => 1 [4,3,2,6,1,5] => 1 [4,3,2,6,5,1] => 1 [4,3,5,1,2,6] => 2 [4,3,5,1,6,2] => 2 [4,3,5,2,1,6] => 1 [4,3,5,2,6,1] => 1 [4,3,5,6,1,2] => 3 [4,3,5,6,2,1] => 1 [4,3,6,1,2,5] => 2 [4,3,6,1,5,2] => 2 [4,3,6,2,1,5] => 1 [4,3,6,2,5,1] => 2 [4,3,6,5,1,2] => 2 [4,3,6,5,2,1] => 1 [4,5,1,2,3,6] => 3 [4,5,1,2,6,3] => 3 [4,5,1,3,2,6] => 2 [4,5,1,3,6,2] => 2 [4,5,1,6,2,3] => 5 [4,5,1,6,3,2] => 2 [4,5,2,1,3,6] => 2 [4,5,2,1,6,3] => 2 [4,5,2,3,1,6] => 3 [4,5,2,3,6,1] => 3 [4,5,2,6,1,3] => 3 [4,5,2,6,3,1] => 3 [4,5,3,1,2,6] => 2 [4,5,3,1,6,2] => 2 [4,5,3,2,1,6] => 1 [4,5,3,2,6,1] => 1 [4,5,3,6,1,2] => 3 [4,5,3,6,2,1] => 1 [4,5,6,1,2,3] => 10 [4,5,6,1,3,2] => 4 [4,5,6,2,1,3] => 4 [4,5,6,2,3,1] => 6 [4,5,6,3,1,2] => 3 [4,5,6,3,2,1] => 1 [4,6,1,2,3,5] => 3 [4,6,1,2,5,3] => 3 [4,6,1,3,2,5] => 2 [4,6,1,3,5,2] => 4 [4,6,1,5,2,3] => 3 [4,6,1,5,3,2] => 2 [4,6,2,1,3,5] => 2 [4,6,2,1,5,3] => 2 [4,6,2,3,1,5] => 3 [4,6,2,3,5,1] => 6 [4,6,2,5,1,3] => 3 [4,6,2,5,3,1] => 3 [4,6,3,1,2,5] => 2 [4,6,3,1,5,2] => 3 [4,6,3,2,1,5] => 1 [4,6,3,2,5,1] => 2 [4,6,3,5,1,2] => 4 [4,6,3,5,2,1] => 2 [4,6,5,1,2,3] => 4 [4,6,5,1,3,2] => 3 [4,6,5,2,1,3] => 2 [4,6,5,2,3,1] => 4 [4,6,5,3,1,2] => 2 [4,6,5,3,2,1] => 1 [5,1,2,3,4,6] => 1 [5,1,2,3,6,4] => 1 [5,1,2,4,3,6] => 1 [5,1,2,4,6,3] => 1 [5,1,2,6,3,4] => 2 [5,1,2,6,4,3] => 1 [5,1,3,2,4,6] => 1 [5,1,3,2,6,4] => 1 [5,1,3,4,2,6] => 2 [5,1,3,4,6,2] => 2 [5,1,3,6,2,4] => 2 [5,1,3,6,4,2] => 2 [5,1,4,2,3,6] => 1 [5,1,4,2,6,3] => 1 [5,1,4,3,2,6] => 1 [5,1,4,3,6,2] => 1 [5,1,4,6,2,3] => 2 [5,1,4,6,3,2] => 1 [5,1,6,2,3,4] => 3 [5,1,6,2,4,3] => 2 [5,1,6,3,2,4] => 2 [5,1,6,3,4,2] => 3 [5,1,6,4,2,3] => 2 [5,1,6,4,3,2] => 1 [5,2,1,3,4,6] => 1 [5,2,1,3,6,4] => 1 [5,2,1,4,3,6] => 1 [5,2,1,4,6,3] => 1 [5,2,1,6,3,4] => 2 [5,2,1,6,4,3] => 1 [5,2,3,1,4,6] => 2 [5,2,3,1,6,4] => 2 [5,2,3,4,1,6] => 4 [5,2,3,4,6,1] => 4 [5,2,3,6,1,4] => 4 [5,2,3,6,4,1] => 4 [5,2,4,1,3,6] => 2 [5,2,4,1,6,3] => 2 [5,2,4,3,1,6] => 2 [5,2,4,3,6,1] => 2 [5,2,4,6,1,3] => 4 [5,2,4,6,3,1] => 2 [5,2,6,1,3,4] => 3 [5,2,6,1,4,3] => 2 [5,2,6,3,1,4] => 4 [5,2,6,3,4,1] => 6 [5,2,6,4,1,3] => 3 [5,2,6,4,3,1] => 2 [5,3,1,2,4,6] => 1 [5,3,1,2,6,4] => 1 [5,3,1,4,2,6] => 2 [5,3,1,4,6,2] => 2 [5,3,1,6,2,4] => 2 [5,3,1,6,4,2] => 2 [5,3,2,1,4,6] => 1 [5,3,2,1,6,4] => 1 [5,3,2,4,1,6] => 2 [5,3,2,4,6,1] => 2 [5,3,2,6,1,4] => 2 [5,3,2,6,4,1] => 2 [5,3,4,1,2,6] => 3 [5,3,4,1,6,2] => 3 [5,3,4,2,1,6] => 2 [5,3,4,2,6,1] => 2 [5,3,4,6,1,2] => 6 [5,3,4,6,2,1] => 2 [5,3,6,1,2,4] => 3 [5,3,6,1,4,2] => 3 [5,3,6,2,1,4] => 2 [5,3,6,2,4,1] => 3 [5,3,6,4,1,2] => 4 [5,3,6,4,2,1] => 2 [5,4,1,2,3,6] => 1 [5,4,1,2,6,3] => 1 [5,4,1,3,2,6] => 1 [5,4,1,3,6,2] => 1 [5,4,1,6,2,3] => 2 [5,4,1,6,3,2] => 1 [5,4,2,1,3,6] => 1 [5,4,2,1,6,3] => 1 [5,4,2,3,1,6] => 2 [5,4,2,3,6,1] => 2 [5,4,2,6,1,3] => 2 [5,4,2,6,3,1] => 2 [5,4,3,1,2,6] => 1 [5,4,3,1,6,2] => 1 [5,4,3,2,1,6] => 1 [5,4,3,2,6,1] => 1 [5,4,3,6,1,2] => 2 [5,4,3,6,2,1] => 1 [5,4,6,1,2,3] => 4 [5,4,6,1,3,2] => 2 [5,4,6,2,1,3] => 3 [5,4,6,2,3,1] => 4 [5,4,6,3,1,2] => 2 [5,4,6,3,2,1] => 1 [5,6,1,2,3,4] => 5 [5,6,1,2,4,3] => 3 [5,6,1,3,2,4] => 4 [5,6,1,3,4,2] => 6 [5,6,1,4,2,3] => 3 [5,6,1,4,3,2] => 2 [5,6,2,1,3,4] => 3 [5,6,2,1,4,3] => 2 [5,6,2,3,1,4] => 6 [5,6,2,3,4,1] => 9 [5,6,2,4,1,3] => 4 [5,6,2,4,3,1] => 3 [5,6,3,1,2,4] => 3 [5,6,3,1,4,2] => 4 [5,6,3,2,1,4] => 2 [5,6,3,2,4,1] => 3 [5,6,3,4,1,2] => 8 [5,6,3,4,2,1] => 4 [5,6,4,1,2,3] => 3 [5,6,4,1,3,2] => 2 [5,6,4,2,1,3] => 2 [5,6,4,2,3,1] => 3 [5,6,4,3,1,2] => 2 [5,6,4,3,2,1] => 1 [6,1,2,3,4,5] => 1 [6,1,2,3,5,4] => 1 [6,1,2,4,3,5] => 1 [6,1,2,4,5,3] => 2 [6,1,2,5,3,4] => 1 [6,1,2,5,4,3] => 1 [6,1,3,2,4,5] => 1 [6,1,3,2,5,4] => 1 [6,1,3,4,2,5] => 2 [6,1,3,4,5,2] => 4 [6,1,3,5,2,4] => 2 [6,1,3,5,4,2] => 2 [6,1,4,2,3,5] => 1 [6,1,4,2,5,3] => 2 [6,1,4,3,2,5] => 1 [6,1,4,3,5,2] => 2 [6,1,4,5,2,3] => 3 [6,1,4,5,3,2] => 2 [6,1,5,2,3,4] => 1 [6,1,5,2,4,3] => 1 [6,1,5,3,2,4] => 1 [6,1,5,3,4,2] => 2 [6,1,5,4,2,3] => 1 [6,1,5,4,3,2] => 1 [6,2,1,3,4,5] => 1 [6,2,1,3,5,4] => 1 [6,2,1,4,3,5] => 1 [6,2,1,4,5,3] => 2 [6,2,1,5,3,4] => 1 [6,2,1,5,4,3] => 1 [6,2,3,1,4,5] => 2 [6,2,3,1,5,4] => 2 [6,2,3,4,1,5] => 4 [6,2,3,4,5,1] => 8 [6,2,3,5,1,4] => 4 [6,2,3,5,4,1] => 4 [6,2,4,1,3,5] => 2 [6,2,4,1,5,3] => 4 [6,2,4,3,1,5] => 2 [6,2,4,3,5,1] => 4 [6,2,4,5,1,3] => 6 [6,2,4,5,3,1] => 4 [6,2,5,1,3,4] => 2 [6,2,5,1,4,3] => 2 [6,2,5,3,1,4] => 2 [6,2,5,3,4,1] => 4 [6,2,5,4,1,3] => 2 [6,2,5,4,3,1] => 2 [6,3,1,2,4,5] => 1 [6,3,1,2,5,4] => 1 [6,3,1,4,2,5] => 2 [6,3,1,4,5,2] => 4 [6,3,1,5,2,4] => 2 [6,3,1,5,4,2] => 2 [6,3,2,1,4,5] => 1 [6,3,2,1,5,4] => 1 [6,3,2,4,1,5] => 2 [6,3,2,4,5,1] => 4 [6,3,2,5,1,4] => 2 [6,3,2,5,4,1] => 3 [6,3,4,1,2,5] => 3 [6,3,4,1,5,2] => 6 [6,3,4,2,1,5] => 2 [6,3,4,2,5,1] => 4 [6,3,4,5,1,2] => 9 [6,3,4,5,2,1] => 4 [6,3,5,1,2,4] => 3 [6,3,5,1,4,2] => 3 [6,3,5,2,1,4] => 2 [6,3,5,2,4,1] => 4 [6,3,5,4,1,2] => 3 [6,3,5,4,2,1] => 2 [6,4,1,2,3,5] => 1 [6,4,1,2,5,3] => 2 [6,4,1,3,2,5] => 1 [6,4,1,3,5,2] => 2 [6,4,1,5,2,3] => 3 [6,4,1,5,3,2] => 2 [6,4,2,1,3,5] => 1 [6,4,2,1,5,3] => 2 [6,4,2,3,1,5] => 2 [6,4,2,3,5,1] => 4 [6,4,2,5,1,3] => 3 [6,4,2,5,3,1] => 4 [6,4,3,1,2,5] => 1 [6,4,3,1,5,2] => 2 [6,4,3,2,1,5] => 1 [6,4,3,2,5,1] => 2 [6,4,3,5,1,2] => 3 [6,4,3,5,2,1] => 2 [6,4,5,1,2,3] => 6 [6,4,5,1,3,2] => 4 [6,4,5,2,1,3] => 4 [6,4,5,2,3,1] => 8 [6,4,5,3,1,2] => 3 [6,4,5,3,2,1] => 2 [6,5,1,2,3,4] => 1 [6,5,1,2,4,3] => 1 [6,5,1,3,2,4] => 1 [6,5,1,3,4,2] => 2 [6,5,1,4,2,3] => 1 [6,5,1,4,3,2] => 1 [6,5,2,1,3,4] => 1 [6,5,2,1,4,3] => 1 [6,5,2,3,1,4] => 2 [6,5,2,3,4,1] => 4 [6,5,2,4,1,3] => 2 [6,5,2,4,3,1] => 2 [6,5,3,1,2,4] => 1 [6,5,3,1,4,2] => 2 [6,5,3,2,1,4] => 1 [6,5,3,2,4,1] => 2 [6,5,3,4,1,2] => 4 [6,5,3,4,2,1] => 3 [6,5,4,1,2,3] => 1 [6,5,4,1,3,2] => 1 [6,5,4,2,1,3] => 1 [6,5,4,2,3,1] => 2 [6,5,4,3,1,2] => 1 [6,5,4,3,2,1] => 1 [1,2,3,4,5,6,7] => 1 [1,2,3,4,5,7,6] => 1 [1,2,3,4,6,5,7] => 1 [1,2,3,6,4,7,5] => 1 [1,2,6,3,4,7,5] => 1 [1,3,2,4,5,6,7] => 1 [1,3,7,2,4,5,6] => 1 [1,3,7,6,5,4,2] => 1 [1,4,6,7,2,3,5] => 5 [1,4,6,7,5,3,2] => 1 [1,4,7,2,3,5,6] => 2 [1,4,7,6,5,3,2] => 1 [1,5,2,3,4,7,6] => 1 [1,5,2,7,3,4,6] => 2 [1,5,4,7,6,3,2] => 1 [1,5,6,2,7,3,4] => 5 [1,5,6,4,7,3,2] => 1 [1,5,6,7,2,3,4] => 10 [1,5,6,7,4,3,2] => 1 [1,5,7,2,3,4,6] => 3 [1,5,7,6,4,3,2] => 1 [1,6,2,3,4,7,5] => 1 [1,6,2,3,7,4,5] => 2 [1,6,2,7,3,4,5] => 3 [1,6,5,4,3,7,2] => 1 [1,6,5,4,7,3,2] => 1 [1,6,5,7,4,3,2] => 1 [1,6,7,2,3,4,5] => 5 [1,6,7,5,4,3,2] => 1 [1,7,2,3,4,5,6] => 1 [1,7,3,5,6,4,2] => 4 [1,7,3,6,5,4,2] => 2 [1,7,4,3,6,5,2] => 3 [1,7,4,5,3,6,2] => 4 [1,7,4,5,6,3,2] => 4 [1,7,4,6,5,3,2] => 2 [1,7,5,4,3,6,2] => 2 [1,7,5,4,6,3,2] => 2 [1,7,6,4,5,3,2] => 3 [1,7,6,5,4,3,2] => 1 [2,1,3,4,5,6,7] => 1 [2,1,3,4,7,5,6] => 1 [2,1,3,7,4,5,6] => 1 [2,1,6,3,4,5,7] => 1 [2,1,7,3,4,5,6] => 1 [2,1,7,6,5,4,3] => 1 [2,3,1,4,5,6,7] => 1 [2,3,4,1,5,6,7] => 1 [2,3,4,1,5,7,6] => 1 [2,3,4,5,1,6,7] => 1 [2,3,4,5,1,7,6] => 1 [2,3,4,5,6,1,7] => 1 [2,3,4,5,6,7,1] => 1 [2,3,4,5,7,1,6] => 1 [2,3,4,6,1,7,5] => 1 [2,3,4,6,7,1,5] => 2 [2,3,4,7,1,5,6] => 1 [2,3,5,6,7,1,4] => 3 [2,4,1,3,5,6,7] => 1 [2,5,1,3,4,6,7] => 1 [2,5,7,1,3,4,6] => 3 [2,6,1,3,4,5,7] => 1 [2,6,1,7,3,4,5] => 3 [2,6,5,4,3,1,7] => 1 [2,6,7,1,3,4,5] => 5 [2,7,1,3,4,5,6] => 1 [3,1,2,4,5,6,7] => 1 [3,1,2,4,5,7,6] => 1 [3,2,1,4,5,6,7] => 1 [3,2,4,5,6,7,1] => 1 [3,4,1,2,5,6,7] => 2 [3,4,2,5,6,7,1] => 1 [3,4,5,2,6,7,1] => 1 [3,4,5,6,2,7,1] => 1 [3,4,5,6,7,1,2] => 8 [3,5,6,1,2,4,7] => 5 [3,5,6,4,2,1,7] => 1 [3,6,1,2,4,5,7] => 2 [3,6,5,4,2,1,7] => 1 [4,1,2,3,5,6,7] => 1 [4,1,2,3,5,7,6] => 1 [4,1,6,2,3,5,7] => 2 [4,2,3,5,6,7,1] => 2 [4,2,5,1,3,6,7] => 2 [4,3,2,1,5,6,7] => 1 [4,3,2,5,6,7,1] => 1 [4,3,5,2,6,7,1] => 1 [4,3,6,5,2,1,7] => 1 [4,5,1,6,2,3,7] => 5 [4,5,2,3,6,7,1] => 3 [4,5,3,6,2,1,7] => 1 [4,5,6,1,2,3,7] => 10 [4,5,6,3,2,1,7] => 1 [4,6,1,2,3,5,7] => 3 [4,6,1,2,3,7,5] => 3 [4,6,5,3,2,1,7] => 1 [5,1,2,3,4,6,7] => 1 [5,1,2,3,4,7,6] => 1 [5,1,2,3,6,4,7] => 1 [5,1,2,3,6,7,4] => 1 [5,1,2,3,7,4,6] => 1 [5,1,2,6,3,4,7] => 2 [5,1,6,2,3,4,7] => 3 [5,1,6,2,3,7,4] => 3 [5,2,3,4,6,7,1] => 4 [5,2,6,7,1,3,4] => 10 [5,3,2,4,6,7,1] => 2 [5,3,2,6,1,4,7] => 2 [5,3,6,7,1,2,4] => 10 [5,4,3,2,1,6,7] => 1 [5,4,3,2,1,7,6] => 1 [5,4,3,2,6,1,7] => 1 [5,4,3,6,2,1,7] => 1 [5,4,6,3,2,1,7] => 1 [5,4,6,7,1,2,3] => 12 [5,6,1,2,3,4,7] => 5 [5,6,1,2,3,7,4] => 5 [5,6,3,4,1,2,7] => 8 [5,6,4,3,2,1,7] => 1 [5,6,7,1,2,3,4] => 26 [6,1,2,3,4,5,7] => 1 [6,1,2,3,4,7,5] => 1 [6,1,2,3,7,4,5] => 2 [6,1,2,7,3,4,5] => 3 [6,2,3,4,5,7,1] => 8 [6,2,4,5,3,1,7] => 4 [6,2,5,4,3,1,7] => 2 [6,3,2,5,4,1,7] => 3 [6,3,4,2,5,1,7] => 4 [6,3,4,5,2,1,7] => 4 [6,3,4,7,1,2,5] => 9 [6,3,5,4,2,1,7] => 2 [6,4,3,2,5,1,7] => 2 [6,4,3,2,7,1,5] => 2 [6,4,3,5,2,1,7] => 2 [6,4,5,7,1,2,3] => 20 [6,4,7,2,5,1,3] => 8 [6,4,7,3,5,1,2] => 8 [6,5,3,4,2,1,7] => 3 [6,5,4,3,2,1,7] => 1 [6,5,7,3,4,1,2] => 10 [6,7,1,2,3,4,5] => 8 [6,7,3,4,1,2,5] => 14 [6,7,4,5,1,2,3] => 17 [7,1,2,3,4,5,6] => 1 [7,2,1,3,4,5,6] => 1 [7,2,3,1,4,5,6] => 2 [7,2,3,4,1,5,6] => 4 [7,2,3,4,5,1,6] => 8 [7,3,1,2,4,5,6] => 1 [7,3,2,1,4,5,6] => 1 [7,3,2,4,1,5,6] => 2 [7,3,4,1,2,5,6] => 3 [7,4,1,2,3,5,6] => 1 [7,4,2,1,3,5,6] => 1 [7,4,5,6,1,2,3] => 31 [7,5,1,2,3,4,6] => 1 [7,5,6,1,2,3,4] => 11 [7,5,6,3,4,1,2] => 17 [7,6,1,2,3,4,5] => 1 [7,6,4,5,1,2,3] => 7 [7,6,5,1,2,3,4] => 1 [7,6,5,3,4,1,2] => 4 [7,6,5,4,1,2,3] => 1 [7,6,5,4,3,1,2] => 1 [7,6,5,4,3,2,1] => 1 [8,7,6,5,4,3,2,1] => 1 [7,6,8,5,4,3,2,1] => 1 [7,8,5,6,4,3,2,1] => 4 [7,8,6,4,5,3,2,1] => 5 [8,6,7,4,5,3,2,1] => 12 [7,6,5,4,8,3,2,1] => 1 [6,5,7,4,8,3,2,1] => 1 [7,8,6,5,3,4,2,1] => 4 [8,6,7,5,3,4,2,1] => 10 [8,7,5,6,3,4,2,1] => 18 [6,7,8,3,4,5,2,1] => 49 [7,8,6,5,4,2,3,1] => 3 [8,6,7,5,4,2,3,1] => 8 [8,7,5,6,4,2,3,1] => 10 [8,7,6,4,5,2,3,1] => 12 [8,6,5,7,3,2,4,1] => 10 [6,7,8,5,2,3,4,1] => 31 [8,5,6,7,2,3,4,1] => 198 [7,8,3,2,4,5,6,1] => 27 [7,6,5,4,3,2,8,1] => 1 [6,5,7,4,3,2,8,1] => 1 [6,5,4,3,7,2,8,1] => 1 [5,4,6,3,7,2,8,1] => 1 [3,4,5,2,6,7,8,1] => 1 [5,2,3,4,6,7,8,1] => 4 [4,3,2,5,6,7,8,1] => 1 [3,4,2,5,6,7,8,1] => 1 [4,2,3,5,6,7,8,1] => 2 [3,2,4,5,6,7,8,1] => 1 [2,3,4,5,6,7,8,1] => 1 [8,7,6,5,4,3,1,2] => 1 [7,8,6,5,4,3,1,2] => 2 [8,6,7,5,4,3,1,2] => 3 [8,7,5,6,4,3,1,2] => 4 [8,7,6,4,5,3,1,2] => 5 [8,7,6,5,3,4,1,2] => 4 [8,7,5,6,3,4,1,2] => 28 [7,8,5,6,3,4,1,2] => 56 [5,6,7,8,3,4,1,2] => 42 [7,8,3,4,5,6,1,2] => 239 [3,4,5,6,7,8,1,2] => 13 [8,7,6,5,4,2,1,3] => 1 [7,6,8,5,4,2,1,3] => 3 [8,7,6,5,4,1,2,3] => 1 [6,7,8,5,4,1,2,3] => 10 [8,5,6,7,4,1,2,3] => 31 [8,7,6,4,5,1,2,3] => 8 [8,6,7,4,5,1,2,3] => 48 [8,7,4,5,6,1,2,3] => 49 [7,8,4,5,6,1,2,3] => 166 [6,7,4,5,8,1,2,3] => 62 [7,6,5,8,3,2,1,4] => 4 [7,5,6,8,2,3,1,4] => 72 [6,7,5,8,3,1,2,4] => 13 [6,5,7,8,2,1,3,4] => 24 [8,7,6,5,1,2,3,4] => 1 [8,7,5,6,1,2,3,4] => 14 [7,8,5,6,1,2,3,4] => 42 [8,5,6,7,1,2,3,4] => 114 [5,6,7,8,1,2,3,4] => 145 [8,7,6,4,3,2,1,5] => 1 [8,7,6,4,2,1,3,5] => 1 [8,7,6,1,2,3,4,5] => 1 [8,6,7,1,2,3,4,5] => 20 [6,7,8,1,2,3,4,5] => 73 [7,8,5,4,2,1,3,6] => 3 [7,8,5,4,1,2,3,6] => 5 [8,7,1,2,3,4,5,6] => 1 [7,8,1,2,3,4,5,6] => 13 [8,6,5,4,3,2,1,7] => 1 [8,6,5,4,3,1,2,7] => 1 [8,5,6,3,4,1,2,7] => 17 [8,6,5,4,2,1,3,7] => 1 [8,6,4,3,2,1,5,7] => 1 [8,6,4,2,1,3,5,7] => 1 [8,2,3,4,1,5,6,7] => 4 [8,4,1,2,3,5,6,7] => 1 [8,3,2,1,4,5,6,7] => 1 [8,2,3,1,4,5,6,7] => 2 [8,3,1,2,4,5,6,7] => 1 [8,2,1,3,4,5,6,7] => 1 [8,1,2,3,4,5,6,7] => 1 [7,6,5,4,3,2,1,8] => 1 [6,7,5,4,3,2,1,8] => 1 [6,5,7,4,3,2,1,8] => 1 [5,6,7,4,3,2,1,8] => 1 [7,6,4,5,3,2,1,8] => 3 [6,5,4,7,3,2,1,8] => 1 [7,5,6,3,4,2,1,8] => 10 [7,6,4,3,5,2,1,8] => 3 [7,6,3,4,5,2,1,8] => 9 [7,5,4,3,6,2,1,8] => 2 [6,7,5,4,2,3,1,8] => 3 [6,5,7,3,2,4,1,8] => 4 [5,6,7,2,3,4,1,8] => 31 [2,3,4,5,6,7,1,8] => 1 [5,6,7,1,2,3,4,8] => 26 [6,7,1,2,3,4,5,8] => 8 [7,1,2,3,4,5,6,8] => 1 [6,5,4,3,2,1,7,8] => 1 [2,3,4,5,6,1,7,8] => 1 [5,6,3,4,1,2,7,8] => 8 [4,5,6,1,2,3,7,8] => 10 [6,1,2,3,4,5,7,8] => 1 [5,4,3,2,1,6,7,8] => 1 [2,3,4,5,1,6,7,8] => 1 [5,1,2,3,4,6,7,8] => 1 [4,3,2,1,5,6,7,8] => 1 [3,4,1,2,5,6,7,8] => 2 [3,2,1,4,5,6,7,8] => 1 [2,1,3,4,5,6,7,8] => 1 [1,2,3,4,5,6,7,8] => 1 [6,5,8,7,4,3,2,1] => 1 [3,5,8,7,6,4,2,1] => 1 [4,3,6,5,8,7,2,1] => 1 [2,6,8,7,5,4,3,1] => 1 [2,3,5,6,8,7,4,1] => 1 [3,2,6,5,4,8,7,1] => 1 [4,3,6,5,7,2,8,1] => 1 [3,2,6,5,7,4,8,1] => 1 [4,3,5,2,7,6,8,1] => 1 [3,2,5,4,7,6,8,1] => 1 [1,8,7,6,5,4,3,2] => 1 [1,7,8,6,5,4,3,2] => 1 [1,6,8,7,5,4,3,2] => 1 [1,7,6,8,5,4,3,2] => 1 [1,6,7,8,5,4,3,2] => 1 [1,5,8,7,6,4,3,2] => 1 [1,7,6,5,8,4,3,2] => 1 [1,3,5,7,8,6,4,2] => 1 [1,4,3,8,7,6,5,2] => 1 [1,6,5,4,3,8,7,2] => 1 [1,4,3,6,5,8,7,2] => 1 [2,1,8,7,6,5,4,3] => 1 [2,1,6,5,7,4,8,3] => 1 [2,1,5,4,7,6,8,3] => 1 [1,2,6,7,8,5,4,3] => 1 [3,2,1,8,7,6,5,4] => 1 [3,2,1,6,5,8,7,4] => 1 [1,2,3,5,6,7,8,4] => 1 [4,3,2,1,8,7,6,5] => 1 [3,4,2,1,7,8,6,5] => 1 [2,4,3,1,6,8,7,5] => 1 [2,4,3,1,7,6,8,5] => 1 [3,2,4,1,6,8,7,5] => 1 [3,2,4,1,7,6,8,5] => 1 [2,3,4,1,6,7,8,5] => 1 [2,1,4,3,8,7,6,5] => 1 [2,1,4,3,6,8,7,5] => 1 [2,1,4,3,7,6,8,5] => 1 [5,4,3,2,1,8,7,6] => 1 [3,2,5,4,1,8,7,6] => 1 [3,2,1,4,5,8,7,6] => 1 [2,3,1,4,5,7,8,6] => 1 [6,5,4,3,2,1,8,7] => 1 [4,3,5,2,6,1,8,7] => 1 [3,2,5,4,6,1,8,7] => 1 [2,3,4,5,6,1,8,7] => 1 [2,1,6,5,4,3,8,7] => 1 [2,1,4,6,5,3,8,7] => 1 [2,1,5,4,6,3,8,7] => 1 [4,3,2,1,6,5,8,7] => 1 [2,4,3,1,6,5,8,7] => 1 [3,2,4,1,6,5,8,7] => 1 [2,1,4,3,6,5,8,7] => 1 [2,1,3,4,5,6,8,7] => 1 [1,2,3,4,5,6,8,7] => 1 [5,7,6,4,3,2,1,8] => 1 [4,7,6,5,3,2,1,8] => 1 [3,2,7,6,5,4,1,8] => 1 [5,4,3,2,7,6,1,8] => 1 [3,2,5,4,7,6,1,8] => 1 [1,4,3,2,7,6,5,8] => 1 [1,3,4,2,6,7,5,8] => 1 [1,3,2,5,4,7,6,8] => 1 [1,2,3,5,4,7,6,8] => 1 [1,3,2,4,5,7,6,8] => 1 [1,2,3,4,5,7,6,8] => 1 [1,2,4,3,6,5,7,8] => 1 [1,3,2,5,4,6,7,8] => 1 [1,2,3,5,4,6,7,8] => 1 [1,3,2,4,5,6,7,8] => 1 [1,8,4,7,6,5,3,2] => 2 [1,8,5,4,7,6,3,2] => 3 [1,8,7,4,5,6,3,2] => 9 [1,8,7,4,6,5,3,2] => 3 [1,8,6,5,4,7,3,2] => 2 [1,8,7,5,4,6,3,2] => 3 [1,8,7,5,6,4,3,2] => 3 [2,1,4,5,8,6,7,3] => 2 [2,1,8,5,4,7,6,3] => 3 [2,3,4,8,5,6,7,1] => 4 [2,3,6,4,5,1,8,7] => 2 [2,3,7,4,5,6,8,1] => 4 [2,8,4,3,5,7,6,1] => 5 [4,2,3,1,8,6,7,5] => 4 [8,2,7,4,6,5,3,1] => 6 [4,3,2,7,5,6,8,1] => 2 [4,3,2,8,5,7,6,1] => 2 [6,3,2,5,4,1,8,7] => 3 [8,3,2,5,4,7,6,1] => 9 [8,3,2,7,6,5,4,1] => 3 [8,3,7,4,6,5,2,1] => 6 [7,3,6,5,4,2,1,8] => 2 [7,4,3,6,5,2,1,8] => 3 [7,6,3,5,4,2,1,8] => 3 [8,5,4,3,2,7,6,1] => 3 [8,7,4,3,6,5,2,1] => 6 [2,4,6,8,1,3,5,7] => 5 [2,4,6,1,7,3,8,5] => 4 [2,1,4,3,7,5,8,6] => 1 [2,4,1,3,7,5,8,6] => 1 [2,1,4,3,7,8,5,6] => 2 [2,4,7,8,1,3,5,6] => 8 [2,1,4,3,8,5,6,7] => 1 [2,4,5,6,7,1,8,3] => 5 [2,1,5,3,6,4,8,7] => 1 [2,1,5,6,3,4,8,7] => 2 [2,5,6,8,1,3,4,7] => 10 [2,1,5,3,7,4,8,6] => 1 [2,5,1,7,3,4,8,6] => 2 [2,1,5,7,3,8,4,6] => 4 [2,5,7,8,1,3,4,6] => 16 [2,3,5,6,1,7,8,4] => 2 [2,1,6,3,7,4,8,5] => 2 [2,6,7,1,3,4,8,5] => 5 [2,1,6,7,8,3,4,5] => 10 [2,6,7,8,1,3,4,5] => 26 [2,1,6,3,4,7,8,5] => 1 [2,1,6,3,4,5,8,7] => 1 [2,6,1,3,4,5,7,8] => 1 [2,7,8,1,3,4,5,6] => 8 [2,7,1,3,4,5,6,8] => 1 [2,3,4,5,7,8,1,6] => 2 [2,1,8,3,4,5,6,7] => 1 [2,8,1,3,4,5,6,7] => 1 [2,1,3,8,4,5,6,7] => 1 [2,3,4,5,6,8,1,7] => 1 [3,1,4,2,6,5,8,7] => 1 [3,4,1,2,6,5,8,7] => 2 [3,1,4,2,6,8,5,7] => 1 [3,4,6,8,1,2,5,7] => 8 [3,1,4,2,7,5,8,6] => 1 [3,4,1,2,7,8,5,6] => 4 [3,4,7,8,1,2,5,6] => 13 [3,1,5,2,6,4,8,7] => 1 [3,5,1,6,2,4,8,7] => 4 [3,1,5,6,2,8,4,7] => 2 [3,5,6,8,1,2,4,7] => 16 [3,1,5,2,7,4,8,6] => 1 [3,5,1,7,2,8,4,6] => 8 [3,5,7,8,1,2,4,6] => 26 [1,3,2,5,4,8,6,7] => 1 [1,3,5,8,2,4,6,7] => 2 [1,3,5,6,2,7,4,8] => 2 [3,1,6,2,7,4,8,5] => 2 [3,6,7,1,2,8,4,5] => 13 [3,6,1,7,8,2,4,5] => 20 [3,6,7,8,1,2,4,5] => 43 [1,3,2,6,4,7,5,8] => 1 [3,1,2,6,7,4,8,5] => 2 [1,3,2,6,4,8,5,7] => 1 [1,3,2,6,4,5,7,8] => 1 [1,3,2,7,4,8,5,6] => 2 [1,3,2,7,4,5,6,8] => 1 [1,3,2,8,4,5,6,7] => 1 [3,1,2,4,5,8,6,7] => 1 [1,3,2,4,5,8,6,7] => 1 [4,1,5,2,6,3,8,7] => 2 [4,5,6,1,2,3,8,7] => 10 [4,1,5,6,8,2,3,7] => 5 [4,5,6,8,1,2,3,7] => 26 [4,1,5,2,7,3,8,6] => 2 [4,1,5,2,7,8,3,6] => 4 [4,5,7,1,2,8,3,6] => 20 [4,5,1,7,8,2,3,6] => 13 [4,5,7,8,1,2,3,6] => 43 [1,4,2,5,3,7,6,8] => 1 [1,4,2,5,3,8,6,7] => 1 [1,4,2,5,3,6,7,8] => 1 [1,4,5,6,7,2,3,8] => 5 [4,1,6,2,7,3,8,5] => 4 [4,6,7,1,8,2,3,5] => 51 [4,6,7,8,1,2,3,5] => 73 [1,4,2,6,3,7,5,8] => 1 [1,4,2,6,7,8,3,5] => 3 [1,4,2,6,3,8,5,7] => 1 [4,1,2,3,6,5,8,7] => 1 [1,4,2,6,3,5,7,8] => 1 [1,4,2,7,3,8,5,6] => 2 [4,7,1,2,3,5,6,8] => 3 [1,4,2,7,3,5,6,8] => 1 [1,4,2,3,7,5,6,8] => 1 [1,4,2,3,5,7,6,8] => 1 [1,4,2,8,3,5,6,7] => 1 [4,1,2,3,8,5,6,7] => 1 [1,4,2,3,5,8,6,7] => 1 [1,4,2,3,5,6,7,8] => 1 [5,1,6,2,7,3,8,4] => 5 [1,5,2,6,3,7,4,8] => 2 [1,5,2,6,3,8,4,7] => 2 [1,5,2,6,3,4,7,8] => 2 [1,2,5,6,3,4,7,8] => 2 [1,5,2,7,3,8,4,6] => 4 [5,7,1,2,3,4,6,8] => 5 [1,5,2,7,3,4,6,8] => 2 [1,5,2,3,4,7,6,8] => 1 [5,1,2,8,3,4,6,7] => 2 [1,5,2,8,3,4,6,7] => 2 [1,5,8,2,3,4,6,7] => 3 [1,5,2,3,4,8,6,7] => 1 [1,2,3,5,4,8,6,7] => 1 [5,1,2,3,4,6,8,7] => 1 [1,5,2,3,4,6,7,8] => 1 [1,6,2,7,3,8,4,5] => 5 [1,6,7,8,2,3,4,5] => 26 [6,7,1,2,3,4,8,5] => 8 [6,1,7,2,3,4,5,8] => 5 [6,1,2,7,3,4,5,8] => 3 [1,6,2,7,3,4,5,8] => 3 [1,6,7,2,3,4,5,8] => 5 [1,6,2,3,4,7,5,8] => 1 [1,2,3,6,4,7,5,8] => 1 [6,1,8,2,3,4,5,7] => 5 [1,6,2,8,3,4,5,7] => 3 [1,6,8,2,3,4,5,7] => 5 [1,6,2,3,4,8,5,7] => 1 [1,2,3,6,4,8,5,7] => 1 [6,1,2,3,4,5,8,7] => 1 [1,6,2,3,4,5,7,8] => 1 [1,2,3,6,4,5,7,8] => 1 [1,7,2,8,3,4,5,6] => 5 [1,7,8,2,3,4,5,6] => 8 [1,7,2,3,8,4,5,6] => 3 [7,1,2,3,4,8,5,6] => 2 [1,7,2,3,4,8,5,6] => 2 [1,2,3,7,4,8,5,6] => 2 [7,1,2,3,4,5,8,6] => 1 [1,7,2,3,4,5,8,6] => 1 [1,2,7,3,4,5,8,6] => 1 [1,7,2,3,4,5,6,8] => 1 [1,2,3,7,4,5,6,8] => 1 [1,8,2,3,4,5,6,7] => 1 [1,2,3,8,4,5,6,7] => 1 [1,2,3,4,5,8,6,7] => 1 [5,3,2,6,1,4,8,7] => 2 [7,3,2,5,4,8,1,6] => 6 [8,4,5,2,3,7,6,1] => 24 [7,4,5,2,3,8,1,6] => 16 [6,4,5,2,3,1,8,7] => 8 [5,4,6,2,1,3,8,7] => 3 [3,6,1,5,4,2,8,7] => 2 [4,6,5,1,3,2,8,7] => 3 [5,6,4,3,1,2,8,7] => 2 [7,5,4,3,2,8,1,6] => 2 [8,6,4,3,7,2,5,1] => 8 [7,6,4,3,8,2,1,5] => 3 [6,7,4,3,8,1,2,5] => 10 [5,7,4,3,1,8,2,6] => 4 [4,7,5,1,3,8,2,6] => 6 [3,7,1,5,4,8,2,6] => 4 [2,1,7,5,4,8,3,6] => 2 [3,8,1,5,4,7,6,2] => 6 [4,8,5,1,3,7,6,2] => 9 [5,8,4,3,1,7,6,2] => 4 [6,8,4,3,7,1,5,2] => 10 [7,8,4,3,6,5,1,2] => 18 [6,8,5,7,3,1,4,2] => 18 [5,8,6,7,1,3,4,2] => 72 [4,8,6,1,7,3,5,2] => 24 [3,8,1,6,7,4,5,2] => 16 [2,1,8,6,7,4,5,3] => 8 [2,1,7,6,8,4,3,5] => 3 [3,7,1,6,8,4,2,5] => 6 [4,7,6,1,8,3,2,5] => 9 [5,7,6,8,1,3,2,4] => 26 [5,4,7,2,1,8,3,6] => 6 [6,4,7,2,8,1,3,5] => 14 [7,4,6,2,8,3,1,5] => 24 [8,4,6,2,7,3,5,1] => 63 [8,3,2,6,7,4,5,1] => 24 [7,3,2,6,8,4,1,5] => 9 [6,3,2,7,8,1,4,5] => 10 [5,3,2,7,1,8,4,6] => 4 [4,3,2,1,7,8,5,6] => 2 [3,4,1,2,8,7,6,5] => 2 [5,3,2,8,1,7,6,4] => 2 [6,3,2,8,7,1,5,4] => 3 [7,3,2,8,6,5,1,4] => 4 [8,4,7,2,6,5,3,1] => 8 [7,4,8,2,6,5,1,3] => 10 [6,4,8,2,7,1,5,3] => 8 [5,4,8,2,1,7,6,3] => 3 [4,5,8,1,2,7,6,3] => 10 [3,5,1,8,2,7,6,4] => 4 [2,1,5,8,3,7,6,4] => 2 [2,1,6,8,7,3,5,4] => 3 [3,6,1,8,7,2,5,4] => 6 [4,6,8,1,7,2,5,3] => 14 [5,6,8,7,1,2,4,3] => 24 [6,5,8,7,2,1,4,3] => 6 [7,5,8,6,2,4,1,3] => 18 [8,5,7,6,2,4,3,1] => 10 [5,7,8,6,1,4,2,3] => 13 [4,7,8,1,6,5,2,3] => 10 [3,7,1,8,6,5,2,4] => 4 [2,1,7,8,6,5,3,4] => 2 [3,8,1,7,6,5,4,2] => 2 [4,8,7,1,6,5,3,2] => 3 [5,8,7,6,1,4,3,2] => 4 [6,8,7,5,4,1,3,2] => 3 [4,2,5,1,3,6,7,8] => 2 [5,2,6,7,1,3,4,8] => 10 [3,2,7,8,1,4,5,6] => 5 [5,3,2,6,1,4,7,8] => 2 [7,4,3,8,1,2,5,6] => 5 [4,7,3,6,8,1,2,5] => 16 [6,7,3,4,8,1,2,5] => 44 [5,4,2,7,8,1,3,6] => 5 [6,4,7,2,5,1,3,8] => 8 [6,4,3,2,7,1,5,8] => 2 [5,4,3,2,8,1,6,7] => 1 [7,5,3,8,2,6,1,4] => 8 [8,5,4,7,2,6,1,3] => 10 [8,5,4,7,3,6,1,2] => 10 [2,1,4,3,6,5,8,7,10,9] => 1 [4,3,2,1,6,5,8,7,10,9] => 1 [6,3,2,5,4,1,8,7,10,9] => 3 [8,3,2,5,4,7,6,1,10,9] => 9 [10,3,2,5,4,7,6,9,8,1] => 27 [2,1,6,5,4,3,8,7,10,9] => 1 [6,5,4,3,2,1,8,7,10,9] => 1 [8,5,4,3,2,7,6,1,10,9] => 3 [10,5,4,3,2,7,6,9,8,1] => 9 [2,1,8,5,4,7,6,3,10,9] => 3 [8,7,4,3,6,5,2,1,10,9] => 6 [10,7,4,3,6,5,2,9,8,1] => 18 [2,1,10,5,4,7,6,9,8,3] => 9 [10,9,4,3,6,5,8,7,2,1] => 36 [8,9,5,6,3,4,10,1,2,7] => 320 [2,1,4,3,8,7,6,5,10,9] => 1 [4,3,2,1,8,7,6,5,10,9] => 1 [8,3,2,7,6,5,4,1,10,9] => 3 [10,3,2,7,6,5,4,9,8,1] => 9 [2,1,8,7,6,5,4,3,10,9] => 1 [8,7,6,5,4,3,2,1,10,9] => 1 [9,7,6,5,4,3,2,10,1,8] => 2 [10,7,6,5,4,3,2,9,8,1] => 3 [2,1,10,7,6,5,4,9,8,3] => 3 [10,9,6,5,4,3,8,7,2,1] => 6 [2,1,4,3,10,7,6,9,8,5] => 3 [4,3,2,1,10,7,6,9,8,5] => 3 [10,3,2,9,6,5,8,7,4,1] => 18 [2,1,10,9,6,5,8,7,4,3] => 6 [10,9,8,5,4,7,6,3,2,1] => 10 [6,7,8,9,10,1,2,3,4,5] => 6307 [7,3,2,8,9,10,1,4,5,6] => 147 [8,5,4,3,2,9,10,1,6,7] => 10 [2,1,4,3,6,5,10,9,8,7] => 1 [4,3,2,1,6,5,10,9,8,7] => 1 [6,3,2,5,4,1,10,9,8,7] => 3 [10,3,2,5,4,9,8,7,6,1] => 9 [2,1,6,5,4,3,10,9,8,7] => 1 [6,5,4,3,2,1,10,9,8,7] => 1 [10,5,4,3,2,9,8,7,6,1] => 5 [2,1,10,5,4,9,8,7,6,3] => 3 [10,9,4,3,8,7,6,5,2,1] => 6 [9,7,5,10,3,8,2,6,1,4] => 56 [2,1,4,3,10,9,8,7,6,5] => 1 [4,3,2,1,10,9,8,7,6,5] => 1 [10,3,2,9,8,7,6,5,4,1] => 3 [2,1,10,9,8,7,6,5,4,3] => 1 [10,9,8,7,6,5,4,3,2,1] => 1 [10,8,9,5,6,7,1,2,3,4] => 2902 [10,8,9,4,5,6,1,2,3,7] => 1166 [10,7,8,5,6,9,1,2,3,4] => 1334 [10,6,7,4,5,8,1,2,3,9] => 262 [10,7,8,3,4,9,1,2,5,6] => 662 [10,6,7,3,4,8,1,2,5,9] => 181 [9,8,10,5,6,7,1,2,3,4] => 1664 [9,8,10,4,5,6,1,2,3,7] => 806 [8,7,9,5,6,10,1,2,3,4] => 484 [7,6,8,4,5,9,1,2,3,10] => 96 [8,7,9,3,4,10,1,2,5,6] => 260 [7,6,8,3,4,9,1,2,5,10] => 68 [9,6,10,5,7,8,1,2,3,4] => 928 [9,5,10,4,6,7,1,2,3,8] => 348 [8,6,9,5,7,10,1,2,3,4] => 340 [7,5,8,4,6,9,1,2,3,10] => 72 [8,4,9,3,5,10,1,2,6,7] => 120 [7,4,8,3,5,9,1,2,6,10] => 44 [9,6,10,2,7,8,1,3,4,5] => 735 [9,5,10,2,6,7,1,3,4,8] => 301 [8,6,9,2,7,10,1,3,4,5] => 268 [7,5,8,2,6,9,1,3,4,10] => 62 [8,4,9,2,5,10,1,3,6,7] => 120 [7,4,8,2,5,9,1,3,6,10] => 44 [2,4,6,8,10,1,3,5,7,9] => 26 [2,4,6,9,10,1,3,5,7,8] => 42 [2,4,7,8,10,1,3,5,6,9] => 43 [2,4,7,9,10,1,3,5,6,8] => 71 [2,4,8,9,10,1,3,5,6,7] => 120 [2,5,6,8,10,1,3,4,7,9] => 43 [2,5,6,9,10,1,3,4,7,8] => 69 [2,5,7,8,10,1,3,4,6,9] => 73 [2,5,7,9,10,1,3,4,6,8] => 120 [2,5,8,9,10,1,3,4,6,7] => 202 [2,6,7,8,10,1,3,4,5,9] => 145 [2,6,7,9,10,1,3,4,5,8] => 238 [2,6,8,9,10,1,3,4,5,7] => 394 [2,7,8,9,10,1,3,4,5,6] => 673 [3,4,6,8,10,1,2,5,7,9] => 42 [3,4,6,9,10,1,2,5,7,8] => 68 [3,4,7,8,10,1,2,5,6,9] => 69 [3,4,7,9,10,1,2,5,6,8] => 114 [3,4,8,9,10,1,2,5,6,7] => 193 [3,5,6,8,10,1,2,4,7,9] => 71 [3,5,6,9,10,1,2,4,7,8] => 114 [3,5,7,8,10,1,2,4,6,9] => 120 [3,5,7,9,10,1,2,4,6,8] => 197 [3,5,8,9,10,1,2,4,6,7] => 334 [3,6,7,8,10,1,2,4,5,9] => 238 [3,6,7,9,10,1,2,4,5,8] => 390 [3,6,8,9,10,1,2,4,5,7] => 652 [3,7,8,9,10,1,2,4,5,6] => 1114 [4,5,6,8,10,1,2,3,7,9] => 120 [4,5,6,9,10,1,2,3,7,8] => 193 [4,5,7,8,10,1,2,3,6,9] => 202 [4,5,7,9,10,1,2,3,6,8] => 334 [4,5,8,9,10,1,2,3,6,7] => 566 [4,6,7,8,10,1,2,3,5,9] => 394 [4,6,7,9,10,1,2,3,5,8] => 652 [4,6,8,9,10,1,2,3,5,7] => 1084 [4,7,8,9,10,1,2,3,5,6] => 1872 [5,6,7,8,10,1,2,3,4,9] => 673 [5,6,7,9,10,1,2,3,4,8] => 1114 [5,6,8,9,10,1,2,3,4,7] => 1872 [5,7,8,9,10,1,2,3,4,6] => 3206 [2,1,12,11,10,9,8,7,6,5,4,3] => 1 [10,9,8,7,6,5,4,3,2,1,12,11] => 1 [2,8,9,10,11,12,1,3,4,5,6,7] => 51312 [6,7,8,9,10,12,1,2,3,4,5,11] => 51312 ----------------------------------------------------------------------------- Created: Feb 14, 2013 at 00:43 by Sara Billey ----------------------------------------------------------------------------- Last Updated: Dec 28, 2017 at 20:18 by Martin Rubey