***************************************************************************** * 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: St000638 ----------------------------------------------------------------------------- Collection: Permutations ----------------------------------------------------------------------------- Description: The number of up-down runs of a permutation. An '''up-down run''' of a permutation $\pi=\pi_{1}\pi_{2}\cdots\pi_{n}$ is either a maximal monotone consecutive subsequence or $\pi_{1}$ if 1 is a descent of $\pi$. For example, the up-down runs of $\pi=85712643$ are $8$, $85$, $57$, $71$, $126$, and $643$. ----------------------------------------------------------------------------- References: [1] Zhuang, Y. Counting permutations by runs [[MathSciNet:3499494]] [2] Zhuang, Y. Eulerian polynomials and descent statistics [[arXiv:1610.07218]] ----------------------------------------------------------------------------- Code: def statistic(x): if len(x)>1 and x[0]>x[1]: return x.number_of_peaks() + x.complement().number_of_peaks() + 2 elif len(x)>0: return x.number_of_peaks() + x.complement().number_of_peaks() + 1 else: return 0 ----------------------------------------------------------------------------- Statistic values: [1] => 1 [1,2] => 1 [2,1] => 2 [1,2,3] => 1 [1,3,2] => 2 [2,1,3] => 3 [2,3,1] => 2 [3,1,2] => 3 [3,2,1] => 2 [1,2,3,4] => 1 [1,2,4,3] => 2 [1,3,2,4] => 3 [1,3,4,2] => 2 [1,4,2,3] => 3 [1,4,3,2] => 2 [2,1,3,4] => 3 [2,1,4,3] => 4 [2,3,1,4] => 3 [2,3,4,1] => 2 [2,4,1,3] => 3 [2,4,3,1] => 2 [3,1,2,4] => 3 [3,1,4,2] => 4 [3,2,1,4] => 3 [3,2,4,1] => 4 [3,4,1,2] => 3 [3,4,2,1] => 2 [4,1,2,3] => 3 [4,1,3,2] => 4 [4,2,1,3] => 3 [4,2,3,1] => 4 [4,3,1,2] => 3 [4,3,2,1] => 2 [1,2,3,4,5] => 1 [1,2,3,5,4] => 2 [1,2,4,3,5] => 3 [1,2,4,5,3] => 2 [1,2,5,3,4] => 3 [1,2,5,4,3] => 2 [1,3,2,4,5] => 3 [1,3,2,5,4] => 4 [1,3,4,2,5] => 3 [1,3,4,5,2] => 2 [1,3,5,2,4] => 3 [1,3,5,4,2] => 2 [1,4,2,3,5] => 3 [1,4,2,5,3] => 4 [1,4,3,2,5] => 3 [1,4,3,5,2] => 4 [1,4,5,2,3] => 3 [1,4,5,3,2] => 2 [1,5,2,3,4] => 3 [1,5,2,4,3] => 4 [1,5,3,2,4] => 3 [1,5,3,4,2] => 4 [1,5,4,2,3] => 3 [1,5,4,3,2] => 2 [2,1,3,4,5] => 3 [2,1,3,5,4] => 4 [2,1,4,3,5] => 5 [2,1,4,5,3] => 4 [2,1,5,3,4] => 5 [2,1,5,4,3] => 4 [2,3,1,4,5] => 3 [2,3,1,5,4] => 4 [2,3,4,1,5] => 3 [2,3,4,5,1] => 2 [2,3,5,1,4] => 3 [2,3,5,4,1] => 2 [2,4,1,3,5] => 3 [2,4,1,5,3] => 4 [2,4,3,1,5] => 3 [2,4,3,5,1] => 4 [2,4,5,1,3] => 3 [2,4,5,3,1] => 2 [2,5,1,3,4] => 3 [2,5,1,4,3] => 4 [2,5,3,1,4] => 3 [2,5,3,4,1] => 4 [2,5,4,1,3] => 3 [2,5,4,3,1] => 2 [3,1,2,4,5] => 3 [3,1,2,5,4] => 4 [3,1,4,2,5] => 5 [3,1,4,5,2] => 4 [3,1,5,2,4] => 5 [3,1,5,4,2] => 4 [3,2,1,4,5] => 3 [3,2,1,5,4] => 4 [3,2,4,1,5] => 5 [3,2,4,5,1] => 4 [3,2,5,1,4] => 5 [3,2,5,4,1] => 4 [3,4,1,2,5] => 3 [3,4,1,5,2] => 4 [3,4,2,1,5] => 3 [3,4,2,5,1] => 4 [3,4,5,1,2] => 3 [3,4,5,2,1] => 2 [3,5,1,2,4] => 3 [3,5,1,4,2] => 4 [3,5,2,1,4] => 3 [3,5,2,4,1] => 4 [3,5,4,1,2] => 3 [3,5,4,2,1] => 2 [4,1,2,3,5] => 3 [4,1,2,5,3] => 4 [4,1,3,2,5] => 5 [4,1,3,5,2] => 4 [4,1,5,2,3] => 5 [4,1,5,3,2] => 4 [4,2,1,3,5] => 3 [4,2,1,5,3] => 4 [4,2,3,1,5] => 5 [4,2,3,5,1] => 4 [4,2,5,1,3] => 5 [4,2,5,3,1] => 4 [4,3,1,2,5] => 3 [4,3,1,5,2] => 4 [4,3,2,1,5] => 3 [4,3,2,5,1] => 4 [4,3,5,1,2] => 5 [4,3,5,2,1] => 4 [4,5,1,2,3] => 3 [4,5,1,3,2] => 4 [4,5,2,1,3] => 3 [4,5,2,3,1] => 4 [4,5,3,1,2] => 3 [4,5,3,2,1] => 2 [5,1,2,3,4] => 3 [5,1,2,4,3] => 4 [5,1,3,2,4] => 5 [5,1,3,4,2] => 4 [5,1,4,2,3] => 5 [5,1,4,3,2] => 4 [5,2,1,3,4] => 3 [5,2,1,4,3] => 4 [5,2,3,1,4] => 5 [5,2,3,4,1] => 4 [5,2,4,1,3] => 5 [5,2,4,3,1] => 4 [5,3,1,2,4] => 3 [5,3,1,4,2] => 4 [5,3,2,1,4] => 3 [5,3,2,4,1] => 4 [5,3,4,1,2] => 5 [5,3,4,2,1] => 4 [5,4,1,2,3] => 3 [5,4,1,3,2] => 4 [5,4,2,1,3] => 3 [5,4,2,3,1] => 4 [5,4,3,1,2] => 3 [5,4,3,2,1] => 2 [1,2,3,4,5,6] => 1 [1,2,3,4,6,5] => 2 [1,2,3,5,4,6] => 3 [1,2,3,5,6,4] => 2 [1,2,3,6,4,5] => 3 [1,2,3,6,5,4] => 2 [1,2,4,3,5,6] => 3 [1,2,4,3,6,5] => 4 [1,2,4,5,3,6] => 3 [1,2,4,5,6,3] => 2 [1,2,4,6,3,5] => 3 [1,2,4,6,5,3] => 2 [1,2,5,3,4,6] => 3 [1,2,5,3,6,4] => 4 [1,2,5,4,3,6] => 3 [1,2,5,4,6,3] => 4 [1,2,5,6,3,4] => 3 [1,2,5,6,4,3] => 2 [1,2,6,3,4,5] => 3 [1,2,6,3,5,4] => 4 [1,2,6,4,3,5] => 3 [1,2,6,4,5,3] => 4 [1,2,6,5,3,4] => 3 [1,2,6,5,4,3] => 2 [1,3,2,4,5,6] => 3 [1,3,2,4,6,5] => 4 [1,3,2,5,4,6] => 5 [1,3,2,5,6,4] => 4 [1,3,2,6,4,5] => 5 [1,3,2,6,5,4] => 4 [1,3,4,2,5,6] => 3 [1,3,4,2,6,5] => 4 [1,3,4,5,2,6] => 3 [1,3,4,5,6,2] => 2 [1,3,4,6,2,5] => 3 [1,3,4,6,5,2] => 2 [1,3,5,2,4,6] => 3 [1,3,5,2,6,4] => 4 [1,3,5,4,2,6] => 3 [1,3,5,4,6,2] => 4 [1,3,5,6,2,4] => 3 [1,3,5,6,4,2] => 2 [1,3,6,2,4,5] => 3 [1,3,6,2,5,4] => 4 [1,3,6,4,2,5] => 3 [1,3,6,4,5,2] => 4 [1,3,6,5,2,4] => 3 [1,3,6,5,4,2] => 2 [1,4,2,3,5,6] => 3 [1,4,2,3,6,5] => 4 [1,4,2,5,3,6] => 5 [1,4,2,5,6,3] => 4 [1,4,2,6,3,5] => 5 [1,4,2,6,5,3] => 4 [1,4,3,2,5,6] => 3 [1,4,3,2,6,5] => 4 [1,4,3,5,2,6] => 5 [1,4,3,5,6,2] => 4 [1,4,3,6,2,5] => 5 [1,4,3,6,5,2] => 4 [1,4,5,2,3,6] => 3 [1,4,5,2,6,3] => 4 [1,4,5,3,2,6] => 3 [1,4,5,3,6,2] => 4 [1,4,5,6,2,3] => 3 [1,4,5,6,3,2] => 2 [1,4,6,2,3,5] => 3 [1,4,6,2,5,3] => 4 [1,4,6,3,2,5] => 3 [1,4,6,3,5,2] => 4 [1,4,6,5,2,3] => 3 [1,4,6,5,3,2] => 2 [1,5,2,3,4,6] => 3 [1,5,2,3,6,4] => 4 [1,5,2,4,3,6] => 5 [1,5,2,4,6,3] => 4 [1,5,2,6,3,4] => 5 [1,5,2,6,4,3] => 4 [1,5,3,2,4,6] => 3 [1,5,3,2,6,4] => 4 [1,5,3,4,2,6] => 5 [1,5,3,4,6,2] => 4 [1,5,3,6,2,4] => 5 [1,5,3,6,4,2] => 4 [1,5,4,2,3,6] => 3 [1,5,4,2,6,3] => 4 [1,5,4,3,2,6] => 3 [1,5,4,3,6,2] => 4 [1,5,4,6,2,3] => 5 [1,5,4,6,3,2] => 4 [1,5,6,2,3,4] => 3 [1,5,6,2,4,3] => 4 [1,5,6,3,2,4] => 3 [1,5,6,3,4,2] => 4 [1,5,6,4,2,3] => 3 [1,5,6,4,3,2] => 2 [1,6,2,3,4,5] => 3 [1,6,2,3,5,4] => 4 [1,6,2,4,3,5] => 5 [1,6,2,4,5,3] => 4 [1,6,2,5,3,4] => 5 [1,6,2,5,4,3] => 4 [1,6,3,2,4,5] => 3 [1,6,3,2,5,4] => 4 [1,6,3,4,2,5] => 5 [1,6,3,4,5,2] => 4 [1,6,3,5,2,4] => 5 [1,6,3,5,4,2] => 4 [1,6,4,2,3,5] => 3 [1,6,4,2,5,3] => 4 [1,6,4,3,2,5] => 3 [1,6,4,3,5,2] => 4 [1,6,4,5,2,3] => 5 [1,6,4,5,3,2] => 4 [1,6,5,2,3,4] => 3 [1,6,5,2,4,3] => 4 [1,6,5,3,2,4] => 3 [1,6,5,3,4,2] => 4 [1,6,5,4,2,3] => 3 [1,6,5,4,3,2] => 2 [2,1,3,4,5,6] => 3 [2,1,3,4,6,5] => 4 [2,1,3,5,4,6] => 5 [2,1,3,5,6,4] => 4 [2,1,3,6,4,5] => 5 [2,1,3,6,5,4] => 4 [2,1,4,3,5,6] => 5 [2,1,4,3,6,5] => 6 [2,1,4,5,3,6] => 5 [2,1,4,5,6,3] => 4 [2,1,4,6,3,5] => 5 [2,1,4,6,5,3] => 4 [2,1,5,3,4,6] => 5 [2,1,5,3,6,4] => 6 [2,1,5,4,3,6] => 5 [2,1,5,4,6,3] => 6 [2,1,5,6,3,4] => 5 [2,1,5,6,4,3] => 4 [2,1,6,3,4,5] => 5 [2,1,6,3,5,4] => 6 [2,1,6,4,3,5] => 5 [2,1,6,4,5,3] => 6 [2,1,6,5,3,4] => 5 [2,1,6,5,4,3] => 4 [2,3,1,4,5,6] => 3 [2,3,1,4,6,5] => 4 [2,3,1,5,4,6] => 5 [2,3,1,5,6,4] => 4 [2,3,1,6,4,5] => 5 [2,3,1,6,5,4] => 4 [2,3,4,1,5,6] => 3 [2,3,4,1,6,5] => 4 [2,3,4,5,1,6] => 3 [2,3,4,5,6,1] => 2 [2,3,4,6,1,5] => 3 [2,3,4,6,5,1] => 2 [2,3,5,1,4,6] => 3 [2,3,5,1,6,4] => 4 [2,3,5,4,1,6] => 3 [2,3,5,4,6,1] => 4 [2,3,5,6,1,4] => 3 [2,3,5,6,4,1] => 2 [2,3,6,1,4,5] => 3 [2,3,6,1,5,4] => 4 [2,3,6,4,1,5] => 3 [2,3,6,4,5,1] => 4 [2,3,6,5,1,4] => 3 [2,3,6,5,4,1] => 2 [2,4,1,3,5,6] => 3 [2,4,1,3,6,5] => 4 [2,4,1,5,3,6] => 5 [2,4,1,5,6,3] => 4 [2,4,1,6,3,5] => 5 [2,4,1,6,5,3] => 4 [2,4,3,1,5,6] => 3 [2,4,3,1,6,5] => 4 [2,4,3,5,1,6] => 5 [2,4,3,5,6,1] => 4 [2,4,3,6,1,5] => 5 [2,4,3,6,5,1] => 4 [2,4,5,1,3,6] => 3 [2,4,5,1,6,3] => 4 [2,4,5,3,1,6] => 3 [2,4,5,3,6,1] => 4 [2,4,5,6,1,3] => 3 [2,4,5,6,3,1] => 2 [2,4,6,1,3,5] => 3 [2,4,6,1,5,3] => 4 [2,4,6,3,1,5] => 3 [2,4,6,3,5,1] => 4 [2,4,6,5,1,3] => 3 [2,4,6,5,3,1] => 2 [2,5,1,3,4,6] => 3 [2,5,1,3,6,4] => 4 [2,5,1,4,3,6] => 5 [2,5,1,4,6,3] => 4 [2,5,1,6,3,4] => 5 [2,5,1,6,4,3] => 4 [2,5,3,1,4,6] => 3 [2,5,3,1,6,4] => 4 [2,5,3,4,1,6] => 5 [2,5,3,4,6,1] => 4 [2,5,3,6,1,4] => 5 [2,5,3,6,4,1] => 4 [2,5,4,1,3,6] => 3 [2,5,4,1,6,3] => 4 [2,5,4,3,1,6] => 3 [2,5,4,3,6,1] => 4 [2,5,4,6,1,3] => 5 [2,5,4,6,3,1] => 4 [2,5,6,1,3,4] => 3 [2,5,6,1,4,3] => 4 [2,5,6,3,1,4] => 3 [2,5,6,3,4,1] => 4 [2,5,6,4,1,3] => 3 [2,5,6,4,3,1] => 2 [2,6,1,3,4,5] => 3 [2,6,1,3,5,4] => 4 [2,6,1,4,3,5] => 5 [2,6,1,4,5,3] => 4 [2,6,1,5,3,4] => 5 [2,6,1,5,4,3] => 4 [2,6,3,1,4,5] => 3 [2,6,3,1,5,4] => 4 [2,6,3,4,1,5] => 5 [2,6,3,4,5,1] => 4 [2,6,3,5,1,4] => 5 [2,6,3,5,4,1] => 4 [2,6,4,1,3,5] => 3 [2,6,4,1,5,3] => 4 [2,6,4,3,1,5] => 3 [2,6,4,3,5,1] => 4 [2,6,4,5,1,3] => 5 [2,6,4,5,3,1] => 4 [2,6,5,1,3,4] => 3 [2,6,5,1,4,3] => 4 [2,6,5,3,1,4] => 3 [2,6,5,3,4,1] => 4 [2,6,5,4,1,3] => 3 [2,6,5,4,3,1] => 2 [3,1,2,4,5,6] => 3 [3,1,2,4,6,5] => 4 [3,1,2,5,4,6] => 5 [3,1,2,5,6,4] => 4 [3,1,2,6,4,5] => 5 [3,1,2,6,5,4] => 4 [3,1,4,2,5,6] => 5 [3,1,4,2,6,5] => 6 [3,1,4,5,2,6] => 5 [3,1,4,5,6,2] => 4 [3,1,4,6,2,5] => 5 [3,1,4,6,5,2] => 4 [3,1,5,2,4,6] => 5 [3,1,5,2,6,4] => 6 [3,1,5,4,2,6] => 5 [3,1,5,4,6,2] => 6 [3,1,5,6,2,4] => 5 [3,1,5,6,4,2] => 4 [3,1,6,2,4,5] => 5 [3,1,6,2,5,4] => 6 [3,1,6,4,2,5] => 5 [3,1,6,4,5,2] => 6 [3,1,6,5,2,4] => 5 [3,1,6,5,4,2] => 4 [3,2,1,4,5,6] => 3 [3,2,1,4,6,5] => 4 [3,2,1,5,4,6] => 5 [3,2,1,5,6,4] => 4 [3,2,1,6,4,5] => 5 [3,2,1,6,5,4] => 4 [3,2,4,1,5,6] => 5 [3,2,4,1,6,5] => 6 [3,2,4,5,1,6] => 5 [3,2,4,5,6,1] => 4 [3,2,4,6,1,5] => 5 [3,2,4,6,5,1] => 4 [3,2,5,1,4,6] => 5 [3,2,5,1,6,4] => 6 [3,2,5,4,1,6] => 5 [3,2,5,4,6,1] => 6 [3,2,5,6,1,4] => 5 [3,2,5,6,4,1] => 4 [3,2,6,1,4,5] => 5 [3,2,6,1,5,4] => 6 [3,2,6,4,1,5] => 5 [3,2,6,4,5,1] => 6 [3,2,6,5,1,4] => 5 [3,2,6,5,4,1] => 4 [3,4,1,2,5,6] => 3 [3,4,1,2,6,5] => 4 [3,4,1,5,2,6] => 5 [3,4,1,5,6,2] => 4 [3,4,1,6,2,5] => 5 [3,4,1,6,5,2] => 4 [3,4,2,1,5,6] => 3 [3,4,2,1,6,5] => 4 [3,4,2,5,1,6] => 5 [3,4,2,5,6,1] => 4 [3,4,2,6,1,5] => 5 [3,4,2,6,5,1] => 4 [3,4,5,1,2,6] => 3 [3,4,5,1,6,2] => 4 [3,4,5,2,1,6] => 3 [3,4,5,2,6,1] => 4 [3,4,5,6,1,2] => 3 [3,4,5,6,2,1] => 2 [3,4,6,1,2,5] => 3 [3,4,6,1,5,2] => 4 [3,4,6,2,1,5] => 3 [3,4,6,2,5,1] => 4 [3,4,6,5,1,2] => 3 [3,4,6,5,2,1] => 2 [3,5,1,2,4,6] => 3 [3,5,1,2,6,4] => 4 [3,5,1,4,2,6] => 5 [3,5,1,4,6,2] => 4 [3,5,1,6,2,4] => 5 [3,5,1,6,4,2] => 4 [3,5,2,1,4,6] => 3 [3,5,2,1,6,4] => 4 [3,5,2,4,1,6] => 5 [3,5,2,4,6,1] => 4 [3,5,2,6,1,4] => 5 [3,5,2,6,4,1] => 4 [3,5,4,1,2,6] => 3 [3,5,4,1,6,2] => 4 [3,5,4,2,1,6] => 3 [3,5,4,2,6,1] => 4 [3,5,4,6,1,2] => 5 [3,5,4,6,2,1] => 4 [3,5,6,1,2,4] => 3 [3,5,6,1,4,2] => 4 [3,5,6,2,1,4] => 3 [3,5,6,2,4,1] => 4 [3,5,6,4,1,2] => 3 [3,5,6,4,2,1] => 2 [3,6,1,2,4,5] => 3 [3,6,1,2,5,4] => 4 [3,6,1,4,2,5] => 5 [3,6,1,4,5,2] => 4 [3,6,1,5,2,4] => 5 [3,6,1,5,4,2] => 4 [3,6,2,1,4,5] => 3 [3,6,2,1,5,4] => 4 [3,6,2,4,1,5] => 5 [3,6,2,4,5,1] => 4 [3,6,2,5,1,4] => 5 [3,6,2,5,4,1] => 4 [3,6,4,1,2,5] => 3 [3,6,4,1,5,2] => 4 [3,6,4,2,1,5] => 3 [3,6,4,2,5,1] => 4 [3,6,4,5,1,2] => 5 [3,6,4,5,2,1] => 4 [3,6,5,1,2,4] => 3 [3,6,5,1,4,2] => 4 [3,6,5,2,1,4] => 3 [3,6,5,2,4,1] => 4 [3,6,5,4,1,2] => 3 [3,6,5,4,2,1] => 2 [4,1,2,3,5,6] => 3 [4,1,2,3,6,5] => 4 [4,1,2,5,3,6] => 5 [4,1,2,5,6,3] => 4 [4,1,2,6,3,5] => 5 [4,1,2,6,5,3] => 4 [4,1,3,2,5,6] => 5 [4,1,3,2,6,5] => 6 [4,1,3,5,2,6] => 5 [4,1,3,5,6,2] => 4 [4,1,3,6,2,5] => 5 [4,1,3,6,5,2] => 4 [4,1,5,2,3,6] => 5 [4,1,5,2,6,3] => 6 [4,1,5,3,2,6] => 5 [4,1,5,3,6,2] => 6 [4,1,5,6,2,3] => 5 [4,1,5,6,3,2] => 4 [4,1,6,2,3,5] => 5 [4,1,6,2,5,3] => 6 [4,1,6,3,2,5] => 5 [4,1,6,3,5,2] => 6 [4,1,6,5,2,3] => 5 [4,1,6,5,3,2] => 4 [4,2,1,3,5,6] => 3 [4,2,1,3,6,5] => 4 [4,2,1,5,3,6] => 5 [4,2,1,5,6,3] => 4 [4,2,1,6,3,5] => 5 [4,2,1,6,5,3] => 4 [4,2,3,1,5,6] => 5 [4,2,3,1,6,5] => 6 [4,2,3,5,1,6] => 5 [4,2,3,5,6,1] => 4 [4,2,3,6,1,5] => 5 [4,2,3,6,5,1] => 4 [4,2,5,1,3,6] => 5 [4,2,5,1,6,3] => 6 [4,2,5,3,1,6] => 5 [4,2,5,3,6,1] => 6 [4,2,5,6,1,3] => 5 [4,2,5,6,3,1] => 4 [4,2,6,1,3,5] => 5 [4,2,6,1,5,3] => 6 [4,2,6,3,1,5] => 5 [4,2,6,3,5,1] => 6 [4,2,6,5,1,3] => 5 [4,2,6,5,3,1] => 4 [4,3,1,2,5,6] => 3 [4,3,1,2,6,5] => 4 [4,3,1,5,2,6] => 5 [4,3,1,5,6,2] => 4 [4,3,1,6,2,5] => 5 [4,3,1,6,5,2] => 4 [4,3,2,1,5,6] => 3 [4,3,2,1,6,5] => 4 [4,3,2,5,1,6] => 5 [4,3,2,5,6,1] => 4 [4,3,2,6,1,5] => 5 [4,3,2,6,5,1] => 4 [4,3,5,1,2,6] => 5 [4,3,5,1,6,2] => 6 [4,3,5,2,1,6] => 5 [4,3,5,2,6,1] => 6 [4,3,5,6,1,2] => 5 [4,3,5,6,2,1] => 4 [4,3,6,1,2,5] => 5 [4,3,6,1,5,2] => 6 [4,3,6,2,1,5] => 5 [4,3,6,2,5,1] => 6 [4,3,6,5,1,2] => 5 [4,3,6,5,2,1] => 4 [4,5,1,2,3,6] => 3 [4,5,1,2,6,3] => 4 [4,5,1,3,2,6] => 5 [4,5,1,3,6,2] => 4 [4,5,1,6,2,3] => 5 [4,5,1,6,3,2] => 4 [4,5,2,1,3,6] => 3 [4,5,2,1,6,3] => 4 [4,5,2,3,1,6] => 5 [4,5,2,3,6,1] => 4 [4,5,2,6,1,3] => 5 [4,5,2,6,3,1] => 4 [4,5,3,1,2,6] => 3 [4,5,3,1,6,2] => 4 [4,5,3,2,1,6] => 3 [4,5,3,2,6,1] => 4 [4,5,3,6,1,2] => 5 [4,5,3,6,2,1] => 4 [4,5,6,1,2,3] => 3 [4,5,6,1,3,2] => 4 [4,5,6,2,1,3] => 3 [4,5,6,2,3,1] => 4 [4,5,6,3,1,2] => 3 [4,5,6,3,2,1] => 2 [4,6,1,2,3,5] => 3 [4,6,1,2,5,3] => 4 [4,6,1,3,2,5] => 5 [4,6,1,3,5,2] => 4 [4,6,1,5,2,3] => 5 [4,6,1,5,3,2] => 4 [4,6,2,1,3,5] => 3 [4,6,2,1,5,3] => 4 [4,6,2,3,1,5] => 5 [4,6,2,3,5,1] => 4 [4,6,2,5,1,3] => 5 [4,6,2,5,3,1] => 4 [4,6,3,1,2,5] => 3 [4,6,3,1,5,2] => 4 [4,6,3,2,1,5] => 3 [4,6,3,2,5,1] => 4 [4,6,3,5,1,2] => 5 [4,6,3,5,2,1] => 4 [4,6,5,1,2,3] => 3 [4,6,5,1,3,2] => 4 [4,6,5,2,1,3] => 3 [4,6,5,2,3,1] => 4 [4,6,5,3,1,2] => 3 [4,6,5,3,2,1] => 2 [5,1,2,3,4,6] => 3 [5,1,2,3,6,4] => 4 [5,1,2,4,3,6] => 5 [5,1,2,4,6,3] => 4 [5,1,2,6,3,4] => 5 [5,1,2,6,4,3] => 4 [5,1,3,2,4,6] => 5 [5,1,3,2,6,4] => 6 [5,1,3,4,2,6] => 5 [5,1,3,4,6,2] => 4 [5,1,3,6,2,4] => 5 [5,1,3,6,4,2] => 4 [5,1,4,2,3,6] => 5 [5,1,4,2,6,3] => 6 [5,1,4,3,2,6] => 5 [5,1,4,3,6,2] => 6 [5,1,4,6,2,3] => 5 [5,1,4,6,3,2] => 4 [5,1,6,2,3,4] => 5 [5,1,6,2,4,3] => 6 [5,1,6,3,2,4] => 5 [5,1,6,3,4,2] => 6 [5,1,6,4,2,3] => 5 [5,1,6,4,3,2] => 4 [5,2,1,3,4,6] => 3 [5,2,1,3,6,4] => 4 [5,2,1,4,3,6] => 5 [5,2,1,4,6,3] => 4 [5,2,1,6,3,4] => 5 [5,2,1,6,4,3] => 4 [5,2,3,1,4,6] => 5 [5,2,3,1,6,4] => 6 [5,2,3,4,1,6] => 5 [5,2,3,4,6,1] => 4 [5,2,3,6,1,4] => 5 [5,2,3,6,4,1] => 4 [5,2,4,1,3,6] => 5 [5,2,4,1,6,3] => 6 [5,2,4,3,1,6] => 5 [5,2,4,3,6,1] => 6 [5,2,4,6,1,3] => 5 [5,2,4,6,3,1] => 4 [5,2,6,1,3,4] => 5 [5,2,6,1,4,3] => 6 [5,2,6,3,1,4] => 5 [5,2,6,3,4,1] => 6 [5,2,6,4,1,3] => 5 [5,2,6,4,3,1] => 4 [5,3,1,2,4,6] => 3 [5,3,1,2,6,4] => 4 [5,3,1,4,2,6] => 5 [5,3,1,4,6,2] => 4 [5,3,1,6,2,4] => 5 [5,3,1,6,4,2] => 4 [5,3,2,1,4,6] => 3 [5,3,2,1,6,4] => 4 [5,3,2,4,1,6] => 5 [5,3,2,4,6,1] => 4 [5,3,2,6,1,4] => 5 [5,3,2,6,4,1] => 4 [5,3,4,1,2,6] => 5 [5,3,4,1,6,2] => 6 [5,3,4,2,1,6] => 5 [5,3,4,2,6,1] => 6 [5,3,4,6,1,2] => 5 [5,3,4,6,2,1] => 4 [5,3,6,1,2,4] => 5 [5,3,6,1,4,2] => 6 [5,3,6,2,1,4] => 5 [5,3,6,2,4,1] => 6 [5,3,6,4,1,2] => 5 [5,3,6,4,2,1] => 4 [5,4,1,2,3,6] => 3 [5,4,1,2,6,3] => 4 [5,4,1,3,2,6] => 5 [5,4,1,3,6,2] => 4 [5,4,1,6,2,3] => 5 [5,4,1,6,3,2] => 4 [5,4,2,1,3,6] => 3 [5,4,2,1,6,3] => 4 [5,4,2,3,1,6] => 5 [5,4,2,3,6,1] => 4 [5,4,2,6,1,3] => 5 [5,4,2,6,3,1] => 4 [5,4,3,1,2,6] => 3 [5,4,3,1,6,2] => 4 [5,4,3,2,1,6] => 3 [5,4,3,2,6,1] => 4 [5,4,3,6,1,2] => 5 [5,4,3,6,2,1] => 4 [5,4,6,1,2,3] => 5 [5,4,6,1,3,2] => 6 [5,4,6,2,1,3] => 5 [5,4,6,2,3,1] => 6 [5,4,6,3,1,2] => 5 [5,4,6,3,2,1] => 4 [5,6,1,2,3,4] => 3 [5,6,1,2,4,3] => 4 [5,6,1,3,2,4] => 5 [5,6,1,3,4,2] => 4 [5,6,1,4,2,3] => 5 [5,6,1,4,3,2] => 4 [5,6,2,1,3,4] => 3 [5,6,2,1,4,3] => 4 [5,6,2,3,1,4] => 5 [5,6,2,3,4,1] => 4 [5,6,2,4,1,3] => 5 [5,6,2,4,3,1] => 4 [5,6,3,1,2,4] => 3 [5,6,3,1,4,2] => 4 [5,6,3,2,1,4] => 3 [5,6,3,2,4,1] => 4 [5,6,3,4,1,2] => 5 [5,6,3,4,2,1] => 4 [5,6,4,1,2,3] => 3 [5,6,4,1,3,2] => 4 [5,6,4,2,1,3] => 3 [5,6,4,2,3,1] => 4 [5,6,4,3,1,2] => 3 [5,6,4,3,2,1] => 2 [6,1,2,3,4,5] => 3 [6,1,2,3,5,4] => 4 [6,1,2,4,3,5] => 5 [6,1,2,4,5,3] => 4 [6,1,2,5,3,4] => 5 [6,1,2,5,4,3] => 4 [6,1,3,2,4,5] => 5 [6,1,3,2,5,4] => 6 [6,1,3,4,2,5] => 5 [6,1,3,4,5,2] => 4 [6,1,3,5,2,4] => 5 [6,1,3,5,4,2] => 4 [6,1,4,2,3,5] => 5 [6,1,4,2,5,3] => 6 [6,1,4,3,2,5] => 5 [6,1,4,3,5,2] => 6 [6,1,4,5,2,3] => 5 [6,1,4,5,3,2] => 4 [6,1,5,2,3,4] => 5 [6,1,5,2,4,3] => 6 [6,1,5,3,2,4] => 5 [6,1,5,3,4,2] => 6 [6,1,5,4,2,3] => 5 [6,1,5,4,3,2] => 4 [6,2,1,3,4,5] => 3 [6,2,1,3,5,4] => 4 [6,2,1,4,3,5] => 5 [6,2,1,4,5,3] => 4 [6,2,1,5,3,4] => 5 [6,2,1,5,4,3] => 4 [6,2,3,1,4,5] => 5 [6,2,3,1,5,4] => 6 [6,2,3,4,1,5] => 5 [6,2,3,4,5,1] => 4 [6,2,3,5,1,4] => 5 [6,2,3,5,4,1] => 4 [6,2,4,1,3,5] => 5 [6,2,4,1,5,3] => 6 [6,2,4,3,1,5] => 5 [6,2,4,3,5,1] => 6 [6,2,4,5,1,3] => 5 [6,2,4,5,3,1] => 4 [6,2,5,1,3,4] => 5 [6,2,5,1,4,3] => 6 [6,2,5,3,1,4] => 5 [6,2,5,3,4,1] => 6 [6,2,5,4,1,3] => 5 [6,2,5,4,3,1] => 4 [6,3,1,2,4,5] => 3 [6,3,1,2,5,4] => 4 [6,3,1,4,2,5] => 5 [6,3,1,4,5,2] => 4 [6,3,1,5,2,4] => 5 [6,3,1,5,4,2] => 4 [6,3,2,1,4,5] => 3 [6,3,2,1,5,4] => 4 [6,3,2,4,1,5] => 5 [6,3,2,4,5,1] => 4 [6,3,2,5,1,4] => 5 [6,3,2,5,4,1] => 4 [6,3,4,1,2,5] => 5 [6,3,4,1,5,2] => 6 [6,3,4,2,1,5] => 5 [6,3,4,2,5,1] => 6 [6,3,4,5,1,2] => 5 [6,3,4,5,2,1] => 4 [6,3,5,1,2,4] => 5 [6,3,5,1,4,2] => 6 [6,3,5,2,1,4] => 5 [6,3,5,2,4,1] => 6 [6,3,5,4,1,2] => 5 [6,3,5,4,2,1] => 4 [6,4,1,2,3,5] => 3 [6,4,1,2,5,3] => 4 [6,4,1,3,2,5] => 5 [6,4,1,3,5,2] => 4 [6,4,1,5,2,3] => 5 [6,4,1,5,3,2] => 4 [6,4,2,1,3,5] => 3 [6,4,2,1,5,3] => 4 [6,4,2,3,1,5] => 5 [6,4,2,3,5,1] => 4 [6,4,2,5,1,3] => 5 [6,4,2,5,3,1] => 4 [6,4,3,1,2,5] => 3 [6,4,3,1,5,2] => 4 [6,4,3,2,1,5] => 3 [6,4,3,2,5,1] => 4 [6,4,3,5,1,2] => 5 [6,4,3,5,2,1] => 4 [6,4,5,1,2,3] => 5 [6,4,5,1,3,2] => 6 [6,4,5,2,1,3] => 5 [6,4,5,2,3,1] => 6 [6,4,5,3,1,2] => 5 [6,4,5,3,2,1] => 4 [6,5,1,2,3,4] => 3 [6,5,1,2,4,3] => 4 [6,5,1,3,2,4] => 5 [6,5,1,3,4,2] => 4 [6,5,1,4,2,3] => 5 [6,5,1,4,3,2] => 4 [6,5,2,1,3,4] => 3 [6,5,2,1,4,3] => 4 [6,5,2,3,1,4] => 5 [6,5,2,3,4,1] => 4 [6,5,2,4,1,3] => 5 [6,5,2,4,3,1] => 4 [6,5,3,1,2,4] => 3 [6,5,3,1,4,2] => 4 [6,5,3,2,1,4] => 3 [6,5,3,2,4,1] => 4 [6,5,3,4,1,2] => 5 [6,5,3,4,2,1] => 4 [6,5,4,1,2,3] => 3 [6,5,4,1,3,2] => 4 [6,5,4,2,1,3] => 3 [6,5,4,2,3,1] => 4 [6,5,4,3,1,2] => 3 [6,5,4,3,2,1] => 2 ----------------------------------------------------------------------------- Created: Oct 27, 2016 at 21:11 by Yan Zhuang ----------------------------------------------------------------------------- Last Updated: Oct 27, 2016 at 21:11 by Yan Zhuang