***************************************************************************** * 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: St000029 ----------------------------------------------------------------------------- Collection: Permutations ----------------------------------------------------------------------------- Description: The depth of a permutation. This is given by $$\operatorname{dp}(\sigma) = \sum_{\sigma_i>i} (\sigma_i-i) = |\{ i \leq j : \sigma_i > j\}|.$$ The depth is half of the total displacement [4], Problem 5.1.1.28, or Spearmanâ€™s disarray [3] $\sum_i |\sigma_i-i|$. Permutations with depth at most $1$ are called ''almost-increasing'' in [5]. ----------------------------------------------------------------------------- References: [1] Petersen, T. K., Tenner, B. E. The depth of a permutation [[arXiv:1202.4765]] [2] Bagno, E., Biagioli, R., Novick, M., Woo, A. Depth in classical Coexter groups [[arXiv:1507.01180]] [3] Diaconis, P., Graham, R. L. Spearman's footrule as a measure of disarray [[MathSciNet:0652736]] [4] Knuth, D. E. The art of computer programming. Vol. 3 [[MathSciNet:3077154]] [5] Elizalde, S. The $X$-class and almost-increasing permutations [[arXiv:0710.5168]] ----------------------------------------------------------------------------- Code: def statistic(w): return sum(w[i]-i-1 for i in range(len(w)) if w[i]>i+1) ----------------------------------------------------------------------------- Statistic values: [1] => 0 [1,2] => 0 [2,1] => 1 [1,2,3] => 0 [1,3,2] => 1 [2,1,3] => 1 [2,3,1] => 2 [3,1,2] => 2 [3,2,1] => 2 [1,2,3,4] => 0 [1,2,4,3] => 1 [1,3,2,4] => 1 [1,3,4,2] => 2 [1,4,2,3] => 2 [1,4,3,2] => 2 [2,1,3,4] => 1 [2,1,4,3] => 2 [2,3,1,4] => 2 [2,3,4,1] => 3 [2,4,1,3] => 3 [2,4,3,1] => 3 [3,1,2,4] => 2 [3,1,4,2] => 3 [3,2,1,4] => 2 [3,2,4,1] => 3 [3,4,1,2] => 4 [3,4,2,1] => 4 [4,1,2,3] => 3 [4,1,3,2] => 3 [4,2,1,3] => 3 [4,2,3,1] => 3 [4,3,1,2] => 4 [4,3,2,1] => 4 [1,2,3,4,5] => 0 [1,2,3,5,4] => 1 [1,2,4,3,5] => 1 [1,2,4,5,3] => 2 [1,2,5,3,4] => 2 [1,2,5,4,3] => 2 [1,3,2,4,5] => 1 [1,3,2,5,4] => 2 [1,3,4,2,5] => 2 [1,3,4,5,2] => 3 [1,3,5,2,4] => 3 [1,3,5,4,2] => 3 [1,4,2,3,5] => 2 [1,4,2,5,3] => 3 [1,4,3,2,5] => 2 [1,4,3,5,2] => 3 [1,4,5,2,3] => 4 [1,4,5,3,2] => 4 [1,5,2,3,4] => 3 [1,5,2,4,3] => 3 [1,5,3,2,4] => 3 [1,5,3,4,2] => 3 [1,5,4,2,3] => 4 [1,5,4,3,2] => 4 [2,1,3,4,5] => 1 [2,1,3,5,4] => 2 [2,1,4,3,5] => 2 [2,1,4,5,3] => 3 [2,1,5,3,4] => 3 [2,1,5,4,3] => 3 [2,3,1,4,5] => 2 [2,3,1,5,4] => 3 [2,3,4,1,5] => 3 [2,3,4,5,1] => 4 [2,3,5,1,4] => 4 [2,3,5,4,1] => 4 [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] => 5 [2,4,5,3,1] => 5 [2,5,1,3,4] => 4 [2,5,1,4,3] => 4 [2,5,3,1,4] => 4 [2,5,3,4,1] => 4 [2,5,4,1,3] => 5 [2,5,4,3,1] => 5 [3,1,2,4,5] => 2 [3,1,2,5,4] => 3 [3,1,4,2,5] => 3 [3,1,4,5,2] => 4 [3,1,5,2,4] => 4 [3,1,5,4,2] => 4 [3,2,1,4,5] => 2 [3,2,1,5,4] => 3 [3,2,4,1,5] => 3 [3,2,4,5,1] => 4 [3,2,5,1,4] => 4 [3,2,5,4,1] => 4 [3,4,1,2,5] => 4 [3,4,1,5,2] => 5 [3,4,2,1,5] => 4 [3,4,2,5,1] => 5 [3,4,5,1,2] => 6 [3,4,5,2,1] => 6 [3,5,1,2,4] => 5 [3,5,1,4,2] => 5 [3,5,2,1,4] => 5 [3,5,2,4,1] => 5 [3,5,4,1,2] => 6 [3,5,4,2,1] => 6 [4,1,2,3,5] => 3 [4,1,2,5,3] => 4 [4,1,3,2,5] => 3 [4,1,3,5,2] => 4 [4,1,5,2,3] => 5 [4,1,5,3,2] => 5 [4,2,1,3,5] => 3 [4,2,1,5,3] => 4 [4,2,3,1,5] => 3 [4,2,3,5,1] => 4 [4,2,5,1,3] => 5 [4,2,5,3,1] => 5 [4,3,1,2,5] => 4 [4,3,1,5,2] => 5 [4,3,2,1,5] => 4 [4,3,2,5,1] => 5 [4,3,5,1,2] => 6 [4,3,5,2,1] => 6 [4,5,1,2,3] => 6 [4,5,1,3,2] => 6 [4,5,2,1,3] => 6 [4,5,2,3,1] => 6 [4,5,3,1,2] => 6 [4,5,3,2,1] => 6 [5,1,2,3,4] => 4 [5,1,2,4,3] => 4 [5,1,3,2,4] => 4 [5,1,3,4,2] => 4 [5,1,4,2,3] => 5 [5,1,4,3,2] => 5 [5,2,1,3,4] => 4 [5,2,1,4,3] => 4 [5,2,3,1,4] => 4 [5,2,3,4,1] => 4 [5,2,4,1,3] => 5 [5,2,4,3,1] => 5 [5,3,1,2,4] => 5 [5,3,1,4,2] => 5 [5,3,2,1,4] => 5 [5,3,2,4,1] => 5 [5,3,4,1,2] => 6 [5,3,4,2,1] => 6 [5,4,1,2,3] => 6 [5,4,1,3,2] => 6 [5,4,2,1,3] => 6 [5,4,2,3,1] => 6 [5,4,3,1,2] => 6 [5,4,3,2,1] => 6 [1,2,3,4,5,6] => 0 [1,2,3,4,6,5] => 1 [1,2,3,5,4,6] => 1 [1,2,3,5,6,4] => 2 [1,2,3,6,4,5] => 2 [1,2,3,6,5,4] => 2 [1,2,4,3,5,6] => 1 [1,2,4,3,6,5] => 2 [1,2,4,5,3,6] => 2 [1,2,4,5,6,3] => 3 [1,2,4,6,3,5] => 3 [1,2,4,6,5,3] => 3 [1,2,5,3,4,6] => 2 [1,2,5,3,6,4] => 3 [1,2,5,4,3,6] => 2 [1,2,5,4,6,3] => 3 [1,2,5,6,3,4] => 4 [1,2,5,6,4,3] => 4 [1,2,6,3,4,5] => 3 [1,2,6,3,5,4] => 3 [1,2,6,4,3,5] => 3 [1,2,6,4,5,3] => 3 [1,2,6,5,3,4] => 4 [1,2,6,5,4,3] => 4 [1,3,2,4,5,6] => 1 [1,3,2,4,6,5] => 2 [1,3,2,5,4,6] => 2 [1,3,2,5,6,4] => 3 [1,3,2,6,4,5] => 3 [1,3,2,6,5,4] => 3 [1,3,4,2,5,6] => 2 [1,3,4,2,6,5] => 3 [1,3,4,5,2,6] => 3 [1,3,4,5,6,2] => 4 [1,3,4,6,2,5] => 4 [1,3,4,6,5,2] => 4 [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] => 5 [1,3,5,6,4,2] => 5 [1,3,6,2,4,5] => 4 [1,3,6,2,5,4] => 4 [1,3,6,4,2,5] => 4 [1,3,6,4,5,2] => 4 [1,3,6,5,2,4] => 5 [1,3,6,5,4,2] => 5 [1,4,2,3,5,6] => 2 [1,4,2,3,6,5] => 3 [1,4,2,5,3,6] => 3 [1,4,2,5,6,3] => 4 [1,4,2,6,3,5] => 4 [1,4,2,6,5,3] => 4 [1,4,3,2,5,6] => 2 [1,4,3,2,6,5] => 3 [1,4,3,5,2,6] => 3 [1,4,3,5,6,2] => 4 [1,4,3,6,2,5] => 4 [1,4,3,6,5,2] => 4 [1,4,5,2,3,6] => 4 [1,4,5,2,6,3] => 5 [1,4,5,3,2,6] => 4 [1,4,5,3,6,2] => 5 [1,4,5,6,2,3] => 6 [1,4,5,6,3,2] => 6 [1,4,6,2,3,5] => 5 [1,4,6,2,5,3] => 5 [1,4,6,3,2,5] => 5 [1,4,6,3,5,2] => 5 [1,4,6,5,2,3] => 6 [1,4,6,5,3,2] => 6 [1,5,2,3,4,6] => 3 [1,5,2,3,6,4] => 4 [1,5,2,4,3,6] => 3 [1,5,2,4,6,3] => 4 [1,5,2,6,3,4] => 5 [1,5,2,6,4,3] => 5 [1,5,3,2,4,6] => 3 [1,5,3,2,6,4] => 4 [1,5,3,4,2,6] => 3 [1,5,3,4,6,2] => 4 [1,5,3,6,2,4] => 5 [1,5,3,6,4,2] => 5 [1,5,4,2,3,6] => 4 [1,5,4,2,6,3] => 5 [1,5,4,3,2,6] => 4 [1,5,4,3,6,2] => 5 [1,5,4,6,2,3] => 6 [1,5,4,6,3,2] => 6 [1,5,6,2,3,4] => 6 [1,5,6,2,4,3] => 6 [1,5,6,3,2,4] => 6 [1,5,6,3,4,2] => 6 [1,5,6,4,2,3] => 6 [1,5,6,4,3,2] => 6 [1,6,2,3,4,5] => 4 [1,6,2,3,5,4] => 4 [1,6,2,4,3,5] => 4 [1,6,2,4,5,3] => 4 [1,6,2,5,3,4] => 5 [1,6,2,5,4,3] => 5 [1,6,3,2,4,5] => 4 [1,6,3,2,5,4] => 4 [1,6,3,4,2,5] => 4 [1,6,3,4,5,2] => 4 [1,6,3,5,2,4] => 5 [1,6,3,5,4,2] => 5 [1,6,4,2,3,5] => 5 [1,6,4,2,5,3] => 5 [1,6,4,3,2,5] => 5 [1,6,4,3,5,2] => 5 [1,6,4,5,2,3] => 6 [1,6,4,5,3,2] => 6 [1,6,5,2,3,4] => 6 [1,6,5,2,4,3] => 6 [1,6,5,3,2,4] => 6 [1,6,5,3,4,2] => 6 [1,6,5,4,2,3] => 6 [1,6,5,4,3,2] => 6 [2,1,3,4,5,6] => 1 [2,1,3,4,6,5] => 2 [2,1,3,5,4,6] => 2 [2,1,3,5,6,4] => 3 [2,1,3,6,4,5] => 3 [2,1,3,6,5,4] => 3 [2,1,4,3,5,6] => 2 [2,1,4,3,6,5] => 3 [2,1,4,5,3,6] => 3 [2,1,4,5,6,3] => 4 [2,1,4,6,3,5] => 4 [2,1,4,6,5,3] => 4 [2,1,5,3,4,6] => 3 [2,1,5,3,6,4] => 4 [2,1,5,4,3,6] => 3 [2,1,5,4,6,3] => 4 [2,1,5,6,3,4] => 5 [2,1,5,6,4,3] => 5 [2,1,6,3,4,5] => 4 [2,1,6,3,5,4] => 4 [2,1,6,4,3,5] => 4 [2,1,6,4,5,3] => 4 [2,1,6,5,3,4] => 5 [2,1,6,5,4,3] => 5 [2,3,1,4,5,6] => 2 [2,3,1,4,6,5] => 3 [2,3,1,5,4,6] => 3 [2,3,1,5,6,4] => 4 [2,3,1,6,4,5] => 4 [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] => 4 [2,3,4,5,6,1] => 5 [2,3,4,6,1,5] => 5 [2,3,4,6,5,1] => 5 [2,3,5,1,4,6] => 4 [2,3,5,1,6,4] => 5 [2,3,5,4,1,6] => 4 [2,3,5,4,6,1] => 5 [2,3,5,6,1,4] => 6 [2,3,5,6,4,1] => 6 [2,3,6,1,4,5] => 5 [2,3,6,1,5,4] => 5 [2,3,6,4,1,5] => 5 [2,3,6,4,5,1] => 5 [2,3,6,5,1,4] => 6 [2,3,6,5,4,1] => 6 [2,4,1,3,5,6] => 3 [2,4,1,3,6,5] => 4 [2,4,1,5,3,6] => 4 [2,4,1,5,6,3] => 5 [2,4,1,6,3,5] => 5 [2,4,1,6,5,3] => 5 [2,4,3,1,5,6] => 3 [2,4,3,1,6,5] => 4 [2,4,3,5,1,6] => 4 [2,4,3,5,6,1] => 5 [2,4,3,6,1,5] => 5 [2,4,3,6,5,1] => 5 [2,4,5,1,3,6] => 5 [2,4,5,1,6,3] => 6 [2,4,5,3,1,6] => 5 [2,4,5,3,6,1] => 6 [2,4,5,6,1,3] => 7 [2,4,5,6,3,1] => 7 [2,4,6,1,3,5] => 6 [2,4,6,1,5,3] => 6 [2,4,6,3,1,5] => 6 [2,4,6,3,5,1] => 6 [2,4,6,5,1,3] => 7 [2,4,6,5,3,1] => 7 [2,5,1,3,4,6] => 4 [2,5,1,3,6,4] => 5 [2,5,1,4,3,6] => 4 [2,5,1,4,6,3] => 5 [2,5,1,6,3,4] => 6 [2,5,1,6,4,3] => 6 [2,5,3,1,4,6] => 4 [2,5,3,1,6,4] => 5 [2,5,3,4,1,6] => 4 [2,5,3,4,6,1] => 5 [2,5,3,6,1,4] => 6 [2,5,3,6,4,1] => 6 [2,5,4,1,3,6] => 5 [2,5,4,1,6,3] => 6 [2,5,4,3,1,6] => 5 [2,5,4,3,6,1] => 6 [2,5,4,6,1,3] => 7 [2,5,4,6,3,1] => 7 [2,5,6,1,3,4] => 7 [2,5,6,1,4,3] => 7 [2,5,6,3,1,4] => 7 [2,5,6,3,4,1] => 7 [2,5,6,4,1,3] => 7 [2,5,6,4,3,1] => 7 [2,6,1,3,4,5] => 5 [2,6,1,3,5,4] => 5 [2,6,1,4,3,5] => 5 [2,6,1,4,5,3] => 5 [2,6,1,5,3,4] => 6 [2,6,1,5,4,3] => 6 [2,6,3,1,4,5] => 5 [2,6,3,1,5,4] => 5 [2,6,3,4,1,5] => 5 [2,6,3,4,5,1] => 5 [2,6,3,5,1,4] => 6 [2,6,3,5,4,1] => 6 [2,6,4,1,3,5] => 6 [2,6,4,1,5,3] => 6 [2,6,4,3,1,5] => 6 [2,6,4,3,5,1] => 6 [2,6,4,5,1,3] => 7 [2,6,4,5,3,1] => 7 [2,6,5,1,3,4] => 7 [2,6,5,1,4,3] => 7 [2,6,5,3,1,4] => 7 [2,6,5,3,4,1] => 7 [2,6,5,4,1,3] => 7 [2,6,5,4,3,1] => 7 [3,1,2,4,5,6] => 2 [3,1,2,4,6,5] => 3 [3,1,2,5,4,6] => 3 [3,1,2,5,6,4] => 4 [3,1,2,6,4,5] => 4 [3,1,2,6,5,4] => 4 [3,1,4,2,5,6] => 3 [3,1,4,2,6,5] => 4 [3,1,4,5,2,6] => 4 [3,1,4,5,6,2] => 5 [3,1,4,6,2,5] => 5 [3,1,4,6,5,2] => 5 [3,1,5,2,4,6] => 4 [3,1,5,2,6,4] => 5 [3,1,5,4,2,6] => 4 [3,1,5,4,6,2] => 5 [3,1,5,6,2,4] => 6 [3,1,5,6,4,2] => 6 [3,1,6,2,4,5] => 5 [3,1,6,2,5,4] => 5 [3,1,6,4,2,5] => 5 [3,1,6,4,5,2] => 5 [3,1,6,5,2,4] => 6 [3,1,6,5,4,2] => 6 [3,2,1,4,5,6] => 2 [3,2,1,4,6,5] => 3 [3,2,1,5,4,6] => 3 [3,2,1,5,6,4] => 4 [3,2,1,6,4,5] => 4 [3,2,1,6,5,4] => 4 [3,2,4,1,5,6] => 3 [3,2,4,1,6,5] => 4 [3,2,4,5,1,6] => 4 [3,2,4,5,6,1] => 5 [3,2,4,6,1,5] => 5 [3,2,4,6,5,1] => 5 [3,2,5,1,4,6] => 4 [3,2,5,1,6,4] => 5 [3,2,5,4,1,6] => 4 [3,2,5,4,6,1] => 5 [3,2,5,6,1,4] => 6 [3,2,5,6,4,1] => 6 [3,2,6,1,4,5] => 5 [3,2,6,1,5,4] => 5 [3,2,6,4,1,5] => 5 [3,2,6,4,5,1] => 5 [3,2,6,5,1,4] => 6 [3,2,6,5,4,1] => 6 [3,4,1,2,5,6] => 4 [3,4,1,2,6,5] => 5 [3,4,1,5,2,6] => 5 [3,4,1,5,6,2] => 6 [3,4,1,6,2,5] => 6 [3,4,1,6,5,2] => 6 [3,4,2,1,5,6] => 4 [3,4,2,1,6,5] => 5 [3,4,2,5,1,6] => 5 [3,4,2,5,6,1] => 6 [3,4,2,6,1,5] => 6 [3,4,2,6,5,1] => 6 [3,4,5,1,2,6] => 6 [3,4,5,1,6,2] => 7 [3,4,5,2,1,6] => 6 [3,4,5,2,6,1] => 7 [3,4,5,6,1,2] => 8 [3,4,5,6,2,1] => 8 [3,4,6,1,2,5] => 7 [3,4,6,1,5,2] => 7 [3,4,6,2,1,5] => 7 [3,4,6,2,5,1] => 7 [3,4,6,5,1,2] => 8 [3,4,6,5,2,1] => 8 [3,5,1,2,4,6] => 5 [3,5,1,2,6,4] => 6 [3,5,1,4,2,6] => 5 [3,5,1,4,6,2] => 6 [3,5,1,6,2,4] => 7 [3,5,1,6,4,2] => 7 [3,5,2,1,4,6] => 5 [3,5,2,1,6,4] => 6 [3,5,2,4,1,6] => 5 [3,5,2,4,6,1] => 6 [3,5,2,6,1,4] => 7 [3,5,2,6,4,1] => 7 [3,5,4,1,2,6] => 6 [3,5,4,1,6,2] => 7 [3,5,4,2,1,6] => 6 [3,5,4,2,6,1] => 7 [3,5,4,6,1,2] => 8 [3,5,4,6,2,1] => 8 [3,5,6,1,2,4] => 8 [3,5,6,1,4,2] => 8 [3,5,6,2,1,4] => 8 [3,5,6,2,4,1] => 8 [3,5,6,4,1,2] => 8 [3,5,6,4,2,1] => 8 [3,6,1,2,4,5] => 6 [3,6,1,2,5,4] => 6 [3,6,1,4,2,5] => 6 [3,6,1,4,5,2] => 6 [3,6,1,5,2,4] => 7 [3,6,1,5,4,2] => 7 [3,6,2,1,4,5] => 6 [3,6,2,1,5,4] => 6 [3,6,2,4,1,5] => 6 [3,6,2,4,5,1] => 6 [3,6,2,5,1,4] => 7 [3,6,2,5,4,1] => 7 [3,6,4,1,2,5] => 7 [3,6,4,1,5,2] => 7 [3,6,4,2,1,5] => 7 [3,6,4,2,5,1] => 7 [3,6,4,5,1,2] => 8 [3,6,4,5,2,1] => 8 [3,6,5,1,2,4] => 8 [3,6,5,1,4,2] => 8 [3,6,5,2,1,4] => 8 [3,6,5,2,4,1] => 8 [3,6,5,4,1,2] => 8 [3,6,5,4,2,1] => 8 [4,1,2,3,5,6] => 3 [4,1,2,3,6,5] => 4 [4,1,2,5,3,6] => 4 [4,1,2,5,6,3] => 5 [4,1,2,6,3,5] => 5 [4,1,2,6,5,3] => 5 [4,1,3,2,5,6] => 3 [4,1,3,2,6,5] => 4 [4,1,3,5,2,6] => 4 [4,1,3,5,6,2] => 5 [4,1,3,6,2,5] => 5 [4,1,3,6,5,2] => 5 [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] => 7 [4,1,5,6,3,2] => 7 [4,1,6,2,3,5] => 6 [4,1,6,2,5,3] => 6 [4,1,6,3,2,5] => 6 [4,1,6,3,5,2] => 6 [4,1,6,5,2,3] => 7 [4,1,6,5,3,2] => 7 [4,2,1,3,5,6] => 3 [4,2,1,3,6,5] => 4 [4,2,1,5,3,6] => 4 [4,2,1,5,6,3] => 5 [4,2,1,6,3,5] => 5 [4,2,1,6,5,3] => 5 [4,2,3,1,5,6] => 3 [4,2,3,1,6,5] => 4 [4,2,3,5,1,6] => 4 [4,2,3,5,6,1] => 5 [4,2,3,6,1,5] => 5 [4,2,3,6,5,1] => 5 [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] => 7 [4,2,5,6,3,1] => 7 [4,2,6,1,3,5] => 6 [4,2,6,1,5,3] => 6 [4,2,6,3,1,5] => 6 [4,2,6,3,5,1] => 6 [4,2,6,5,1,3] => 7 [4,2,6,5,3,1] => 7 [4,3,1,2,5,6] => 4 [4,3,1,2,6,5] => 5 [4,3,1,5,2,6] => 5 [4,3,1,5,6,2] => 6 [4,3,1,6,2,5] => 6 [4,3,1,6,5,2] => 6 [4,3,2,1,5,6] => 4 [4,3,2,1,6,5] => 5 [4,3,2,5,1,6] => 5 [4,3,2,5,6,1] => 6 [4,3,2,6,1,5] => 6 [4,3,2,6,5,1] => 6 [4,3,5,1,2,6] => 6 [4,3,5,1,6,2] => 7 [4,3,5,2,1,6] => 6 [4,3,5,2,6,1] => 7 [4,3,5,6,1,2] => 8 [4,3,5,6,2,1] => 8 [4,3,6,1,2,5] => 7 [4,3,6,1,5,2] => 7 [4,3,6,2,1,5] => 7 [4,3,6,2,5,1] => 7 [4,3,6,5,1,2] => 8 [4,3,6,5,2,1] => 8 [4,5,1,2,3,6] => 6 [4,5,1,2,6,3] => 7 [4,5,1,3,2,6] => 6 [4,5,1,3,6,2] => 7 [4,5,1,6,2,3] => 8 [4,5,1,6,3,2] => 8 [4,5,2,1,3,6] => 6 [4,5,2,1,6,3] => 7 [4,5,2,3,1,6] => 6 [4,5,2,3,6,1] => 7 [4,5,2,6,1,3] => 8 [4,5,2,6,3,1] => 8 [4,5,3,1,2,6] => 6 [4,5,3,1,6,2] => 7 [4,5,3,2,1,6] => 6 [4,5,3,2,6,1] => 7 [4,5,3,6,1,2] => 8 [4,5,3,6,2,1] => 8 [4,5,6,1,2,3] => 9 [4,5,6,1,3,2] => 9 [4,5,6,2,1,3] => 9 [4,5,6,2,3,1] => 9 [4,5,6,3,1,2] => 9 [4,5,6,3,2,1] => 9 [4,6,1,2,3,5] => 7 [4,6,1,2,5,3] => 7 [4,6,1,3,2,5] => 7 [4,6,1,3,5,2] => 7 [4,6,1,5,2,3] => 8 [4,6,1,5,3,2] => 8 [4,6,2,1,3,5] => 7 [4,6,2,1,5,3] => 7 [4,6,2,3,1,5] => 7 [4,6,2,3,5,1] => 7 [4,6,2,5,1,3] => 8 [4,6,2,5,3,1] => 8 [4,6,3,1,2,5] => 7 [4,6,3,1,5,2] => 7 [4,6,3,2,1,5] => 7 [4,6,3,2,5,1] => 7 [4,6,3,5,1,2] => 8 [4,6,3,5,2,1] => 8 [4,6,5,1,2,3] => 9 [4,6,5,1,3,2] => 9 [4,6,5,2,1,3] => 9 [4,6,5,2,3,1] => 9 [4,6,5,3,1,2] => 9 [4,6,5,3,2,1] => 9 [5,1,2,3,4,6] => 4 [5,1,2,3,6,4] => 5 [5,1,2,4,3,6] => 4 [5,1,2,4,6,3] => 5 [5,1,2,6,3,4] => 6 [5,1,2,6,4,3] => 6 [5,1,3,2,4,6] => 4 [5,1,3,2,6,4] => 5 [5,1,3,4,2,6] => 4 [5,1,3,4,6,2] => 5 [5,1,3,6,2,4] => 6 [5,1,3,6,4,2] => 6 [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] => 7 [5,1,4,6,3,2] => 7 [5,1,6,2,3,4] => 7 [5,1,6,2,4,3] => 7 [5,1,6,3,2,4] => 7 [5,1,6,3,4,2] => 7 [5,1,6,4,2,3] => 7 [5,1,6,4,3,2] => 7 [5,2,1,3,4,6] => 4 [5,2,1,3,6,4] => 5 [5,2,1,4,3,6] => 4 [5,2,1,4,6,3] => 5 [5,2,1,6,3,4] => 6 [5,2,1,6,4,3] => 6 [5,2,3,1,4,6] => 4 [5,2,3,1,6,4] => 5 [5,2,3,4,1,6] => 4 [5,2,3,4,6,1] => 5 [5,2,3,6,1,4] => 6 [5,2,3,6,4,1] => 6 [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] => 7 [5,2,4,6,3,1] => 7 [5,2,6,1,3,4] => 7 [5,2,6,1,4,3] => 7 [5,2,6,3,1,4] => 7 [5,2,6,3,4,1] => 7 [5,2,6,4,1,3] => 7 [5,2,6,4,3,1] => 7 [5,3,1,2,4,6] => 5 [5,3,1,2,6,4] => 6 [5,3,1,4,2,6] => 5 [5,3,1,4,6,2] => 6 [5,3,1,6,2,4] => 7 [5,3,1,6,4,2] => 7 [5,3,2,1,4,6] => 5 [5,3,2,1,6,4] => 6 [5,3,2,4,1,6] => 5 [5,3,2,4,6,1] => 6 [5,3,2,6,1,4] => 7 [5,3,2,6,4,1] => 7 [5,3,4,1,2,6] => 6 [5,3,4,1,6,2] => 7 [5,3,4,2,1,6] => 6 [5,3,4,2,6,1] => 7 [5,3,4,6,1,2] => 8 [5,3,4,6,2,1] => 8 [5,3,6,1,2,4] => 8 [5,3,6,1,4,2] => 8 [5,3,6,2,1,4] => 8 [5,3,6,2,4,1] => 8 [5,3,6,4,1,2] => 8 [5,3,6,4,2,1] => 8 [5,4,1,2,3,6] => 6 [5,4,1,2,6,3] => 7 [5,4,1,3,2,6] => 6 [5,4,1,3,6,2] => 7 [5,4,1,6,2,3] => 8 [5,4,1,6,3,2] => 8 [5,4,2,1,3,6] => 6 [5,4,2,1,6,3] => 7 [5,4,2,3,1,6] => 6 [5,4,2,3,6,1] => 7 [5,4,2,6,1,3] => 8 [5,4,2,6,3,1] => 8 [5,4,3,1,2,6] => 6 [5,4,3,1,6,2] => 7 [5,4,3,2,1,6] => 6 [5,4,3,2,6,1] => 7 [5,4,3,6,1,2] => 8 [5,4,3,6,2,1] => 8 [5,4,6,1,2,3] => 9 [5,4,6,1,3,2] => 9 [5,4,6,2,1,3] => 9 [5,4,6,2,3,1] => 9 [5,4,6,3,1,2] => 9 [5,4,6,3,2,1] => 9 [5,6,1,2,3,4] => 8 [5,6,1,2,4,3] => 8 [5,6,1,3,2,4] => 8 [5,6,1,3,4,2] => 8 [5,6,1,4,2,3] => 8 [5,6,1,4,3,2] => 8 [5,6,2,1,3,4] => 8 [5,6,2,1,4,3] => 8 [5,6,2,3,1,4] => 8 [5,6,2,3,4,1] => 8 [5,6,2,4,1,3] => 8 [5,6,2,4,3,1] => 8 [5,6,3,1,2,4] => 8 [5,6,3,1,4,2] => 8 [5,6,3,2,1,4] => 8 [5,6,3,2,4,1] => 8 [5,6,3,4,1,2] => 8 [5,6,3,4,2,1] => 8 [5,6,4,1,2,3] => 9 [5,6,4,1,3,2] => 9 [5,6,4,2,1,3] => 9 [5,6,4,2,3,1] => 9 [5,6,4,3,1,2] => 9 [5,6,4,3,2,1] => 9 [6,1,2,3,4,5] => 5 [6,1,2,3,5,4] => 5 [6,1,2,4,3,5] => 5 [6,1,2,4,5,3] => 5 [6,1,2,5,3,4] => 6 [6,1,2,5,4,3] => 6 [6,1,3,2,4,5] => 5 [6,1,3,2,5,4] => 5 [6,1,3,4,2,5] => 5 [6,1,3,4,5,2] => 5 [6,1,3,5,2,4] => 6 [6,1,3,5,4,2] => 6 [6,1,4,2,3,5] => 6 [6,1,4,2,5,3] => 6 [6,1,4,3,2,5] => 6 [6,1,4,3,5,2] => 6 [6,1,4,5,2,3] => 7 [6,1,4,5,3,2] => 7 [6,1,5,2,3,4] => 7 [6,1,5,2,4,3] => 7 [6,1,5,3,2,4] => 7 [6,1,5,3,4,2] => 7 [6,1,5,4,2,3] => 7 [6,1,5,4,3,2] => 7 [6,2,1,3,4,5] => 5 [6,2,1,3,5,4] => 5 [6,2,1,4,3,5] => 5 [6,2,1,4,5,3] => 5 [6,2,1,5,3,4] => 6 [6,2,1,5,4,3] => 6 [6,2,3,1,4,5] => 5 [6,2,3,1,5,4] => 5 [6,2,3,4,1,5] => 5 [6,2,3,4,5,1] => 5 [6,2,3,5,1,4] => 6 [6,2,3,5,4,1] => 6 [6,2,4,1,3,5] => 6 [6,2,4,1,5,3] => 6 [6,2,4,3,1,5] => 6 [6,2,4,3,5,1] => 6 [6,2,4,5,1,3] => 7 [6,2,4,5,3,1] => 7 [6,2,5,1,3,4] => 7 [6,2,5,1,4,3] => 7 [6,2,5,3,1,4] => 7 [6,2,5,3,4,1] => 7 [6,2,5,4,1,3] => 7 [6,2,5,4,3,1] => 7 [6,3,1,2,4,5] => 6 [6,3,1,2,5,4] => 6 [6,3,1,4,2,5] => 6 [6,3,1,4,5,2] => 6 [6,3,1,5,2,4] => 7 [6,3,1,5,4,2] => 7 [6,3,2,1,4,5] => 6 [6,3,2,1,5,4] => 6 [6,3,2,4,1,5] => 6 [6,3,2,4,5,1] => 6 [6,3,2,5,1,4] => 7 [6,3,2,5,4,1] => 7 [6,3,4,1,2,5] => 7 [6,3,4,1,5,2] => 7 [6,3,4,2,1,5] => 7 [6,3,4,2,5,1] => 7 [6,3,4,5,1,2] => 8 [6,3,4,5,2,1] => 8 [6,3,5,1,2,4] => 8 [6,3,5,1,4,2] => 8 [6,3,5,2,1,4] => 8 [6,3,5,2,4,1] => 8 [6,3,5,4,1,2] => 8 [6,3,5,4,2,1] => 8 [6,4,1,2,3,5] => 7 [6,4,1,2,5,3] => 7 [6,4,1,3,2,5] => 7 [6,4,1,3,5,2] => 7 [6,4,1,5,2,3] => 8 [6,4,1,5,3,2] => 8 [6,4,2,1,3,5] => 7 [6,4,2,1,5,3] => 7 [6,4,2,3,1,5] => 7 [6,4,2,3,5,1] => 7 [6,4,2,5,1,3] => 8 [6,4,2,5,3,1] => 8 [6,4,3,1,2,5] => 7 [6,4,3,1,5,2] => 7 [6,4,3,2,1,5] => 7 [6,4,3,2,5,1] => 7 [6,4,3,5,1,2] => 8 [6,4,3,5,2,1] => 8 [6,4,5,1,2,3] => 9 [6,4,5,1,3,2] => 9 [6,4,5,2,1,3] => 9 [6,4,5,2,3,1] => 9 [6,4,5,3,1,2] => 9 [6,4,5,3,2,1] => 9 [6,5,1,2,3,4] => 8 [6,5,1,2,4,3] => 8 [6,5,1,3,2,4] => 8 [6,5,1,3,4,2] => 8 [6,5,1,4,2,3] => 8 [6,5,1,4,3,2] => 8 [6,5,2,1,3,4] => 8 [6,5,2,1,4,3] => 8 [6,5,2,3,1,4] => 8 [6,5,2,3,4,1] => 8 [6,5,2,4,1,3] => 8 [6,5,2,4,3,1] => 8 [6,5,3,1,2,4] => 8 [6,5,3,1,4,2] => 8 [6,5,3,2,1,4] => 8 [6,5,3,2,4,1] => 8 [6,5,3,4,1,2] => 8 [6,5,3,4,2,1] => 8 [6,5,4,1,2,3] => 9 [6,5,4,1,3,2] => 9 [6,5,4,2,1,3] => 9 [6,5,4,2,3,1] => 9 [6,5,4,3,1,2] => 9 [6,5,4,3,2,1] => 9 ----------------------------------------------------------------------------- Created: Nov 30, 2012 at 21:09 by Chris Berg ----------------------------------------------------------------------------- Last Updated: Jan 15, 2017 at 10:37 by Christian Stump