***************************************************************************** * 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: St001818 ----------------------------------------------------------------------------- Collection: Decorated permutations ----------------------------------------------------------------------------- Description: The number of alignments of a decorated permutation. An alignment in a decorated permutation $w$ is a pair of directed edges $(i\mapsto w(i), j\mapsto w(j))$ which can be drawn as distinct noncrossing arcs oriented in the same direction in the circular chord diagram. According to [1], the number of alignments of a decorated permutation on $n$ letters with $k$ anit-exceedances equals and associated Grassmann interval $[u, v]$ equals $k(n - k) - [\ell(v) − \ell(u)]$. ----------------------------------------------------------------------------- References: [1] A. Postnikov, ''Total positivity, Grassmannians, and networks.'' 27 Sep 2006. Postnikov, A. Total positivity, Grassmannians, and networks [[arXiv:math/0609764]] [2] Billey, S. C., Weaver, J. E. Criteria for smoothness of Positroid varieties via pattern avoidance, Johnson graphs, and spirographs [[arXiv:2207.06508]] ----------------------------------------------------------------------------- Code: # please simplify def cyclic_interval(a, b, n): if a <= b: return list(range(a, b+1)) return list(range(a, n+1)) + list(range(1, b+1)) def alignments(x): pi = x.to_signed_permutation().permutation() n = len(pi) al = [] for i, e in enumerate(pi, 1): for j, f in enumerate(pi, 1): if i == j: continue a, b = sorted([i, e]) c, d = sorted([j, f]) if a <= c <= b <= d or c <= a <= d <= b: continue if (e not in cyclic_interval(i, f-1, n) or f not in cyclic_interval(e+1, j, n)): continue if j == f and x[j-1] < 0: continue if i == e and x[i-1] > 0: continue al.append((i, e, j, f)) return al def statistic(x): return len(alignments(x)) ----------------------------------------------------------------------------- Statistic values: [+] => 0 [-] => 0 [+,+] => 0 [-,+] => 1 [+,-] => 1 [-,-] => 0 [2,1] => 0 [+,+,+] => 0 [-,+,+] => 2 [+,-,+] => 2 [+,+,-] => 2 [-,-,+] => 2 [-,+,-] => 2 [+,-,-] => 2 [-,-,-] => 0 [+,3,2] => 1 [-,3,2] => 1 [2,1,+] => 1 [2,1,-] => 1 [2,3,1] => 0 [3,1,2] => 0 [3,+,1] => 1 [3,-,1] => 1 [+,+,+,+] => 0 [-,+,+,+] => 3 [+,-,+,+] => 3 [+,+,-,+] => 3 [+,+,+,-] => 3 [-,-,+,+] => 4 [-,+,-,+] => 4 [-,+,+,-] => 4 [+,-,-,+] => 4 [+,-,+,-] => 4 [+,+,-,-] => 4 [-,-,-,+] => 3 [-,-,+,-] => 3 [-,+,-,-] => 3 [+,-,-,-] => 3 [-,-,-,-] => 0 [+,+,4,3] => 2 [-,+,4,3] => 3 [+,-,4,3] => 3 [-,-,4,3] => 2 [+,3,2,+] => 2 [-,3,2,+] => 3 [+,3,2,-] => 3 [-,3,2,-] => 2 [+,3,4,2] => 2 [-,3,4,2] => 1 [+,4,2,3] => 1 [-,4,2,3] => 2 [+,4,+,2] => 2 [-,4,+,2] => 3 [+,4,-,2] => 3 [-,4,-,2] => 2 [2,1,+,+] => 2 [2,1,-,+] => 3 [2,1,+,-] => 3 [2,1,-,-] => 2 [2,1,4,3] => 2 [2,3,1,+] => 2 [2,3,1,-] => 1 [2,3,4,1] => 0 [2,4,1,3] => 1 [2,4,+,1] => 2 [2,4,-,1] => 1 [3,1,2,+] => 1 [3,1,2,-] => 2 [3,1,4,2] => 1 [3,+,1,+] => 2 [3,-,1,+] => 3 [3,+,1,-] => 3 [3,-,1,-] => 2 [3,+,4,1] => 2 [3,-,4,1] => 1 [3,4,1,2] => 0 [3,4,2,1] => 1 [4,1,2,3] => 0 [4,1,+,2] => 1 [4,1,-,2] => 2 [4,+,1,3] => 1 [4,-,1,3] => 2 [4,+,+,1] => 2 [4,-,+,1] => 3 [4,+,-,1] => 3 [4,-,-,1] => 2 [4,3,1,2] => 1 [4,3,2,1] => 2 [+,+,+,+,+] => 0 [-,+,+,+,+] => 4 [+,-,+,+,+] => 4 [+,+,-,+,+] => 4 [+,+,+,-,+] => 4 [+,+,+,+,-] => 4 [-,-,+,+,+] => 6 [-,+,-,+,+] => 6 [-,+,+,-,+] => 6 [-,+,+,+,-] => 6 [+,-,-,+,+] => 6 [+,-,+,-,+] => 6 [+,-,+,+,-] => 6 [+,+,-,-,+] => 6 [+,+,-,+,-] => 6 [+,+,+,-,-] => 6 [-,-,-,+,+] => 6 [-,-,+,-,+] => 6 [-,-,+,+,-] => 6 [-,+,-,-,+] => 6 [-,+,-,+,-] => 6 [-,+,+,-,-] => 6 [+,-,-,-,+] => 6 [+,-,-,+,-] => 6 [+,-,+,-,-] => 6 [+,+,-,-,-] => 6 [-,-,-,-,+] => 4 [-,-,-,+,-] => 4 [-,-,+,-,-] => 4 [-,+,-,-,-] => 4 [+,-,-,-,-] => 4 [-,-,-,-,-] => 0 [+,+,+,5,4] => 3 [-,+,+,5,4] => 5 [+,-,+,5,4] => 5 [+,+,-,5,4] => 5 [-,-,+,5,4] => 5 [-,+,-,5,4] => 5 [+,-,-,5,4] => 5 [-,-,-,5,4] => 3 [+,+,4,3,+] => 3 [-,+,4,3,+] => 5 [+,-,4,3,+] => 5 [+,+,4,3,-] => 5 [-,-,4,3,+] => 5 [-,+,4,3,-] => 5 [+,-,4,3,-] => 5 [-,-,4,3,-] => 3 [+,+,4,5,3] => 4 [-,+,4,5,3] => 4 [+,-,4,5,3] => 4 [-,-,4,5,3] => 2 [+,+,5,3,4] => 2 [-,+,5,3,4] => 4 [+,-,5,3,4] => 4 [-,-,5,3,4] => 4 [+,+,5,+,3] => 3 [-,+,5,+,3] => 5 [+,-,5,+,3] => 5 [+,+,5,-,3] => 5 [-,-,5,+,3] => 5 [-,+,5,-,3] => 5 [+,-,5,-,3] => 5 [-,-,5,-,3] => 3 [+,3,2,+,+] => 3 [-,3,2,+,+] => 5 [+,3,2,-,+] => 5 [+,3,2,+,-] => 5 [-,3,2,-,+] => 5 [-,3,2,+,-] => 5 [+,3,2,-,-] => 5 [-,3,2,-,-] => 3 [+,3,2,5,4] => 4 [-,3,2,5,4] => 4 [+,3,4,2,+] => 4 [-,3,4,2,+] => 4 [+,3,4,2,-] => 4 [-,3,4,2,-] => 2 [+,3,4,5,2] => 3 [-,3,4,5,2] => 1 [+,3,5,2,4] => 3 [-,3,5,2,4] => 3 [+,3,5,+,2] => 4 [-,3,5,+,2] => 4 [+,3,5,-,2] => 4 [-,3,5,-,2] => 2 [+,4,2,3,+] => 2 [-,4,2,3,+] => 4 [+,4,2,3,-] => 4 [-,4,2,3,-] => 4 [+,4,2,5,3] => 3 [-,4,2,5,3] => 3 [+,4,+,2,+] => 3 [-,4,+,2,+] => 5 [+,4,-,2,+] => 5 [+,4,+,2,-] => 5 [-,4,-,2,+] => 5 [-,4,+,2,-] => 5 [+,4,-,2,-] => 5 [-,4,-,2,-] => 3 [+,4,+,5,2] => 4 [-,4,+,5,2] => 4 [+,4,-,5,2] => 4 [-,4,-,5,2] => 2 [+,4,5,2,3] => 2 [-,4,5,2,3] => 2 [+,4,5,3,2] => 3 [-,4,5,3,2] => 3 [+,5,2,3,4] => 1 [-,5,2,3,4] => 3 [+,5,2,+,3] => 2 [-,5,2,+,3] => 4 [+,5,2,-,3] => 4 [-,5,2,-,3] => 4 [+,5,+,2,4] => 2 [-,5,+,2,4] => 4 [+,5,-,2,4] => 4 [-,5,-,2,4] => 4 [+,5,+,+,2] => 3 [-,5,+,+,2] => 5 [+,5,-,+,2] => 5 [+,5,+,-,2] => 5 [-,5,-,+,2] => 5 [-,5,+,-,2] => 5 [+,5,-,-,2] => 5 [-,5,-,-,2] => 3 [+,5,4,2,3] => 3 [-,5,4,2,3] => 3 [+,5,4,3,2] => 4 [-,5,4,3,2] => 4 [2,1,+,+,+] => 3 [2,1,-,+,+] => 5 [2,1,+,-,+] => 5 [2,1,+,+,-] => 5 [2,1,-,-,+] => 5 [2,1,-,+,-] => 5 [2,1,+,-,-] => 5 [2,1,-,-,-] => 3 [2,1,+,5,4] => 4 [2,1,-,5,4] => 4 [2,1,4,3,+] => 4 [2,1,4,3,-] => 4 [2,1,4,5,3] => 3 [2,1,5,3,4] => 3 [2,1,5,+,3] => 4 [2,1,5,-,3] => 4 [2,3,1,+,+] => 4 [2,3,1,-,+] => 4 [2,3,1,+,-] => 4 [2,3,1,-,-] => 2 [2,3,1,5,4] => 3 [2,3,4,1,+] => 3 [2,3,4,1,-] => 1 [2,3,4,5,1] => 0 [2,3,5,1,4] => 2 [2,3,5,+,1] => 3 [2,3,5,-,1] => 1 [2,4,1,3,+] => 3 [2,4,1,3,-] => 3 [2,4,1,5,3] => 2 [2,4,+,1,+] => 4 [2,4,-,1,+] => 4 [2,4,+,1,-] => 4 [2,4,-,1,-] => 2 [2,4,+,5,1] => 3 [2,4,-,5,1] => 1 [2,4,5,1,3] => 1 [2,4,5,3,1] => 2 [2,5,1,3,4] => 2 [2,5,1,+,3] => 3 [2,5,1,-,3] => 3 [2,5,+,1,4] => 3 [2,5,-,1,4] => 3 [2,5,+,+,1] => 4 [2,5,-,+,1] => 4 [2,5,+,-,1] => 4 [2,5,-,-,1] => 2 [2,5,4,1,3] => 2 [2,5,4,3,1] => 3 [3,1,2,+,+] => 2 [3,1,2,-,+] => 4 [3,1,2,+,-] => 4 [3,1,2,-,-] => 4 [3,1,2,5,4] => 3 [3,1,4,2,+] => 3 [3,1,4,2,-] => 3 [3,1,4,5,2] => 2 [3,1,5,2,4] => 2 [3,1,5,+,2] => 3 [3,1,5,-,2] => 3 [3,+,1,+,+] => 3 [3,-,1,+,+] => 5 [3,+,1,-,+] => 5 [3,+,1,+,-] => 5 [3,-,1,-,+] => 5 [3,-,1,+,-] => 5 [3,+,1,-,-] => 5 [3,-,1,-,-] => 3 [3,+,1,5,4] => 4 [3,-,1,5,4] => 4 [3,+,4,1,+] => 4 [3,-,4,1,+] => 4 [3,+,4,1,-] => 4 [3,-,4,1,-] => 2 [3,+,4,5,1] => 3 [3,-,4,5,1] => 1 [3,+,5,1,4] => 3 [3,-,5,1,4] => 3 [3,+,5,+,1] => 4 [3,-,5,+,1] => 4 [3,+,5,-,1] => 4 [3,-,5,-,1] => 2 [3,4,1,2,+] => 2 [3,4,1,2,-] => 2 [3,4,1,5,2] => 1 [3,4,2,1,+] => 3 [3,4,2,1,-] => 3 [3,4,2,5,1] => 2 [3,4,5,1,2] => 0 [3,4,5,2,1] => 1 [3,5,1,2,4] => 1 [3,5,1,+,2] => 2 [3,5,1,-,2] => 2 [3,5,2,1,4] => 2 [3,5,2,+,1] => 3 [3,5,2,-,1] => 3 [3,5,4,1,2] => 1 [3,5,4,2,1] => 2 [4,1,2,3,+] => 1 [4,1,2,3,-] => 3 [4,1,2,5,3] => 2 [4,1,+,2,+] => 2 [4,1,-,2,+] => 4 [4,1,+,2,-] => 4 [4,1,-,2,-] => 4 [4,1,+,5,2] => 3 [4,1,-,5,2] => 3 [4,1,5,2,3] => 1 [4,1,5,3,2] => 2 [4,+,1,3,+] => 2 [4,-,1,3,+] => 4 [4,+,1,3,-] => 4 [4,-,1,3,-] => 4 [4,+,1,5,3] => 3 [4,-,1,5,3] => 3 [4,+,+,1,+] => 3 [4,-,+,1,+] => 5 [4,+,-,1,+] => 5 [4,+,+,1,-] => 5 [4,-,-,1,+] => 5 [4,-,+,1,-] => 5 [4,+,-,1,-] => 5 [4,-,-,1,-] => 3 [4,+,+,5,1] => 4 [4,-,+,5,1] => 4 [4,+,-,5,1] => 4 [4,-,-,5,1] => 2 [4,+,5,1,3] => 2 [4,-,5,1,3] => 2 [4,+,5,3,1] => 3 [4,-,5,3,1] => 3 [4,3,1,2,+] => 3 [4,3,1,2,-] => 3 [4,3,1,5,2] => 2 [4,3,2,1,+] => 4 [4,3,2,1,-] => 4 [4,3,2,5,1] => 3 [4,3,5,1,2] => 1 [4,3,5,2,1] => 2 [4,5,1,2,3] => 0 [4,5,1,3,2] => 1 [4,5,2,1,3] => 1 [4,5,2,3,1] => 2 [4,5,+,1,2] => 2 [4,5,-,1,2] => 2 [4,5,+,2,1] => 3 [4,5,-,2,1] => 3 [5,1,2,3,4] => 0 [5,1,2,+,3] => 1 [5,1,2,-,3] => 3 [5,1,+,2,4] => 1 [5,1,-,2,4] => 3 [5,1,+,+,2] => 2 [5,1,-,+,2] => 4 [5,1,+,-,2] => 4 [5,1,-,-,2] => 4 [5,1,4,2,3] => 2 [5,1,4,3,2] => 3 [5,+,1,3,4] => 1 [5,-,1,3,4] => 3 [5,+,1,+,3] => 2 [5,-,1,+,3] => 4 [5,+,1,-,3] => 4 [5,-,1,-,3] => 4 [5,+,+,1,4] => 2 [5,-,+,1,4] => 4 [5,+,-,1,4] => 4 [5,-,-,1,4] => 4 [5,+,+,+,1] => 3 [5,-,+,+,1] => 5 [5,+,-,+,1] => 5 [5,+,+,-,1] => 5 [5,-,-,+,1] => 5 [5,-,+,-,1] => 5 [5,+,-,-,1] => 5 [5,-,-,-,1] => 3 [5,+,4,1,3] => 3 [5,-,4,1,3] => 3 [5,+,4,3,1] => 4 [5,-,4,3,1] => 4 [5,3,1,2,4] => 2 [5,3,1,+,2] => 3 [5,3,1,-,2] => 3 [5,3,2,1,4] => 3 [5,3,2,+,1] => 4 [5,3,2,-,1] => 4 [5,3,4,1,2] => 2 [5,3,4,2,1] => 3 [5,4,1,2,3] => 1 [5,4,1,3,2] => 2 [5,4,2,1,3] => 2 [5,4,2,3,1] => 3 [5,4,+,1,2] => 3 [5,4,-,1,2] => 3 [5,4,+,2,1] => 4 [5,4,-,2,1] => 4 [+,+,+,+,+,+] => 0 [-,+,+,+,+,+] => 5 [+,-,+,+,+,+] => 5 [+,+,-,+,+,+] => 5 [+,+,+,-,+,+] => 5 [+,+,+,+,-,+] => 5 [+,+,+,+,+,-] => 5 [-,-,+,+,+,+] => 8 [-,+,-,+,+,+] => 8 [-,+,+,-,+,+] => 8 [-,+,+,+,-,+] => 8 [-,+,+,+,+,-] => 8 [+,-,-,+,+,+] => 8 [+,-,+,-,+,+] => 8 [+,-,+,+,-,+] => 8 [+,-,+,+,+,-] => 8 [+,+,-,-,+,+] => 8 [+,+,-,+,-,+] => 8 [+,+,-,+,+,-] => 8 [+,+,+,-,-,+] => 8 [+,+,+,-,+,-] => 8 [+,+,+,+,-,-] => 8 [-,-,-,+,+,+] => 9 [-,-,+,-,+,+] => 9 [-,-,+,+,-,+] => 9 [-,-,+,+,+,-] => 9 [-,+,-,-,+,+] => 9 [-,+,-,+,-,+] => 9 [-,+,-,+,+,-] => 9 [-,+,+,-,-,+] => 9 [-,+,+,-,+,-] => 9 [-,+,+,+,-,-] => 9 [+,-,-,-,+,+] => 9 [+,-,-,+,-,+] => 9 [+,-,-,+,+,-] => 9 [+,-,+,-,-,+] => 9 [+,-,+,-,+,-] => 9 [+,-,+,+,-,-] => 9 [+,+,-,-,-,+] => 9 [+,+,-,-,+,-] => 9 [+,+,-,+,-,-] => 9 [+,+,+,-,-,-] => 9 [-,-,-,-,+,+] => 8 [-,-,-,+,-,+] => 8 [-,-,-,+,+,-] => 8 [-,-,+,-,-,+] => 8 [-,-,+,-,+,-] => 8 [-,-,+,+,-,-] => 8 [-,+,-,-,-,+] => 8 [-,+,-,-,+,-] => 8 [-,+,-,+,-,-] => 8 [-,+,+,-,-,-] => 8 [+,-,-,-,-,+] => 8 [+,-,-,-,+,-] => 8 [+,-,-,+,-,-] => 8 [+,-,+,-,-,-] => 8 [+,+,-,-,-,-] => 8 [-,-,-,-,-,+] => 5 [-,-,-,-,+,-] => 5 [-,-,-,+,-,-] => 5 [-,-,+,-,-,-] => 5 [-,+,-,-,-,-] => 5 [+,-,-,-,-,-] => 5 [-,-,-,-,-,-] => 0 [+,+,+,+,6,5] => 4 [-,+,+,+,6,5] => 7 [+,-,+,+,6,5] => 7 [+,+,-,+,6,5] => 7 [+,+,+,-,6,5] => 7 [-,-,+,+,6,5] => 8 [-,+,-,+,6,5] => 8 [-,+,+,-,6,5] => 8 [+,-,-,+,6,5] => 8 [+,-,+,-,6,5] => 8 [+,+,-,-,6,5] => 8 [-,-,-,+,6,5] => 7 [-,-,+,-,6,5] => 7 [-,+,-,-,6,5] => 7 [+,-,-,-,6,5] => 7 [-,-,-,-,6,5] => 4 [+,+,+,5,4,+] => 4 [-,+,+,5,4,+] => 7 [+,-,+,5,4,+] => 7 [+,+,-,5,4,+] => 7 [+,+,+,5,4,-] => 7 [-,-,+,5,4,+] => 8 [-,+,-,5,4,+] => 8 [-,+,+,5,4,-] => 8 [+,-,-,5,4,+] => 8 [+,-,+,5,4,-] => 8 [+,+,-,5,4,-] => 8 [-,-,-,5,4,+] => 7 [-,-,+,5,4,-] => 7 [-,+,-,5,4,-] => 7 [+,-,-,5,4,-] => 7 [-,-,-,5,4,-] => 4 [+,+,+,5,6,4] => 6 [-,+,+,5,6,4] => 7 [+,-,+,5,6,4] => 7 [+,+,-,5,6,4] => 7 [-,-,+,5,6,4] => 6 [-,+,-,5,6,4] => 6 [+,-,-,5,6,4] => 6 [-,-,-,5,6,4] => 3 [+,+,+,6,4,5] => 3 [-,+,+,6,4,5] => 6 [+,-,+,6,4,5] => 6 [+,+,-,6,4,5] => 6 [-,-,+,6,4,5] => 7 [-,+,-,6,4,5] => 7 [+,-,-,6,4,5] => 7 [-,-,-,6,4,5] => 6 [+,+,+,6,+,4] => 4 [-,+,+,6,+,4] => 7 [+,-,+,6,+,4] => 7 [+,+,-,6,+,4] => 7 [+,+,+,6,-,4] => 7 [-,-,+,6,+,4] => 8 [-,+,-,6,+,4] => 8 [-,+,+,6,-,4] => 8 [+,-,-,6,+,4] => 8 [+,-,+,6,-,4] => 8 [+,+,-,6,-,4] => 8 [-,-,-,6,+,4] => 7 [-,-,+,6,-,4] => 7 [-,+,-,6,-,4] => 7 [+,-,-,6,-,4] => 7 [-,-,-,6,-,4] => 4 [+,+,4,3,+,+] => 4 [-,+,4,3,+,+] => 7 [+,-,4,3,+,+] => 7 [+,+,4,3,-,+] => 7 [+,+,4,3,+,-] => 7 [-,-,4,3,+,+] => 8 [-,+,4,3,-,+] => 8 [-,+,4,3,+,-] => 8 [+,-,4,3,-,+] => 8 [+,-,4,3,+,-] => 8 [+,+,4,3,-,-] => 8 [-,-,4,3,-,+] => 7 [-,-,4,3,+,-] => 7 [-,+,4,3,-,-] => 7 [+,-,4,3,-,-] => 7 [-,-,4,3,-,-] => 4 [+,+,4,3,6,5] => 6 [-,+,4,3,6,5] => 7 [+,-,4,3,6,5] => 7 [-,-,4,3,6,5] => 6 [+,+,4,5,3,+] => 6 [-,+,4,5,3,+] => 7 [+,-,4,5,3,+] => 7 [+,+,4,5,3,-] => 7 [-,-,4,5,3,+] => 6 [-,+,4,5,3,-] => 6 [+,-,4,5,3,-] => 6 [-,-,4,5,3,-] => 3 [+,+,4,5,6,3] => 6 [-,+,4,5,6,3] => 5 [+,-,4,5,6,3] => 5 [-,-,4,5,6,3] => 2 [+,+,4,6,3,5] => 5 [-,+,4,6,3,5] => 6 [+,-,4,6,3,5] => 6 [-,-,4,6,3,5] => 5 [+,+,4,6,+,3] => 6 [-,+,4,6,+,3] => 7 [+,-,4,6,+,3] => 7 [+,+,4,6,-,3] => 7 [-,-,4,6,+,3] => 6 [-,+,4,6,-,3] => 6 [+,-,4,6,-,3] => 6 [-,-,4,6,-,3] => 3 [+,+,5,3,4,+] => 3 [-,+,5,3,4,+] => 6 [+,-,5,3,4,+] => 6 [+,+,5,3,4,-] => 6 [-,-,5,3,4,+] => 7 [-,+,5,3,4,-] => 7 [+,-,5,3,4,-] => 7 [-,-,5,3,4,-] => 6 [+,+,5,3,6,4] => 5 [-,+,5,3,6,4] => 6 [+,-,5,3,6,4] => 6 [-,-,5,3,6,4] => 5 [+,+,5,+,3,+] => 4 [-,+,5,+,3,+] => 7 [+,-,5,+,3,+] => 7 [+,+,5,-,3,+] => 7 [+,+,5,+,3,-] => 7 [-,-,5,+,3,+] => 8 [-,+,5,-,3,+] => 8 [-,+,5,+,3,-] => 8 [+,-,5,-,3,+] => 8 [+,-,5,+,3,-] => 8 [+,+,5,-,3,-] => 8 [-,-,5,-,3,+] => 7 [-,-,5,+,3,-] => 7 [-,+,5,-,3,-] => 7 [+,-,5,-,3,-] => 7 [-,-,5,-,3,-] => 4 [+,+,5,+,6,3] => 6 [-,+,5,+,6,3] => 7 [+,-,5,+,6,3] => 7 [+,+,5,-,6,3] => 7 [-,-,5,+,6,3] => 6 [-,+,5,-,6,3] => 6 [+,-,5,-,6,3] => 6 [-,-,5,-,6,3] => 3 [+,+,5,6,3,4] => 4 [-,+,5,6,3,4] => 5 [+,-,5,6,3,4] => 5 [-,-,5,6,3,4] => 4 [+,+,5,6,4,3] => 5 [-,+,5,6,4,3] => 6 [+,-,5,6,4,3] => 6 [-,-,5,6,4,3] => 5 [+,+,6,3,4,5] => 2 [-,+,6,3,4,5] => 5 [+,-,6,3,4,5] => 5 [-,-,6,3,4,5] => 6 [+,+,6,3,+,4] => 3 [-,+,6,3,+,4] => 6 [+,-,6,3,+,4] => 6 [+,+,6,3,-,4] => 6 [-,-,6,3,+,4] => 7 [-,+,6,3,-,4] => 7 [+,-,6,3,-,4] => 7 [-,-,6,3,-,4] => 6 [+,+,6,+,3,5] => 3 [-,+,6,+,3,5] => 6 [+,-,6,+,3,5] => 6 [+,+,6,-,3,5] => 6 [-,-,6,+,3,5] => 7 [-,+,6,-,3,5] => 7 [+,-,6,-,3,5] => 7 [-,-,6,-,3,5] => 6 [+,+,6,+,+,3] => 4 [-,+,6,+,+,3] => 7 [+,-,6,+,+,3] => 7 [+,+,6,-,+,3] => 7 [+,+,6,+,-,3] => 7 [-,-,6,+,+,3] => 8 [-,+,6,-,+,3] => 8 [-,+,6,+,-,3] => 8 [+,-,6,-,+,3] => 8 [+,-,6,+,-,3] => 8 [+,+,6,-,-,3] => 8 [-,-,6,-,+,3] => 7 [-,-,6,+,-,3] => 7 [-,+,6,-,-,3] => 7 [+,-,6,-,-,3] => 7 [-,-,6,-,-,3] => 4 [+,+,6,5,3,4] => 5 [-,+,6,5,3,4] => 6 [+,-,6,5,3,4] => 6 [-,-,6,5,3,4] => 5 [+,+,6,5,4,3] => 6 [-,+,6,5,4,3] => 7 [+,-,6,5,4,3] => 7 [-,-,6,5,4,3] => 6 [+,3,2,+,+,+] => 4 [-,3,2,+,+,+] => 7 [+,3,2,-,+,+] => 7 [+,3,2,+,-,+] => 7 [+,3,2,+,+,-] => 7 [-,3,2,-,+,+] => 8 [-,3,2,+,-,+] => 8 [-,3,2,+,+,-] => 8 [+,3,2,-,-,+] => 8 [+,3,2,-,+,-] => 8 [+,3,2,+,-,-] => 8 [-,3,2,-,-,+] => 7 [-,3,2,-,+,-] => 7 [-,3,2,+,-,-] => 7 [+,3,2,-,-,-] => 7 [-,3,2,-,-,-] => 4 [+,3,2,+,6,5] => 6 [-,3,2,+,6,5] => 7 [+,3,2,-,6,5] => 7 [-,3,2,-,6,5] => 6 [+,3,2,5,4,+] => 6 [-,3,2,5,4,+] => 7 [+,3,2,5,4,-] => 7 [-,3,2,5,4,-] => 6 [+,3,2,5,6,4] => 6 [-,3,2,5,6,4] => 5 [+,3,2,6,4,5] => 5 [-,3,2,6,4,5] => 6 [+,3,2,6,+,4] => 6 [-,3,2,6,+,4] => 7 [+,3,2,6,-,4] => 7 [-,3,2,6,-,4] => 6 [+,3,4,2,+,+] => 6 [-,3,4,2,+,+] => 7 [+,3,4,2,-,+] => 7 [+,3,4,2,+,-] => 7 [-,3,4,2,-,+] => 6 [-,3,4,2,+,-] => 6 [+,3,4,2,-,-] => 6 [-,3,4,2,-,-] => 3 [+,3,4,2,6,5] => 6 [-,3,4,2,6,5] => 5 [+,3,4,5,2,+] => 6 [-,3,4,5,2,+] => 5 [+,3,4,5,2,-] => 5 [-,3,4,5,2,-] => 2 [+,3,4,5,6,2] => 4 [-,3,4,5,6,2] => 1 [+,3,4,6,2,5] => 5 [-,3,4,6,2,5] => 4 [+,3,4,6,+,2] => 6 [-,3,4,6,+,2] => 5 [+,3,4,6,-,2] => 5 [-,3,4,6,-,2] => 2 [+,3,5,2,4,+] => 5 [-,3,5,2,4,+] => 6 [+,3,5,2,4,-] => 6 [-,3,5,2,4,-] => 5 [+,3,5,2,6,4] => 5 [-,3,5,2,6,4] => 4 [+,3,5,+,2,+] => 6 [-,3,5,+,2,+] => 7 [+,3,5,-,2,+] => 7 [+,3,5,+,2,-] => 7 [-,3,5,-,2,+] => 6 [-,3,5,+,2,-] => 6 [+,3,5,-,2,-] => 6 [-,3,5,-,2,-] => 3 [+,3,5,+,6,2] => 6 [-,3,5,+,6,2] => 5 [+,3,5,-,6,2] => 5 [-,3,5,-,6,2] => 2 [+,3,5,6,2,4] => 4 [-,3,5,6,2,4] => 3 [+,3,5,6,4,2] => 5 [-,3,5,6,4,2] => 4 [+,3,6,2,4,5] => 4 [-,3,6,2,4,5] => 5 [+,3,6,2,+,4] => 5 [-,3,6,2,+,4] => 6 [+,3,6,2,-,4] => 6 [-,3,6,2,-,4] => 5 [+,3,6,+,2,5] => 5 [-,3,6,+,2,5] => 6 [+,3,6,-,2,5] => 6 [-,3,6,-,2,5] => 5 [+,3,6,+,+,2] => 6 [-,3,6,+,+,2] => 7 [+,3,6,-,+,2] => 7 [+,3,6,+,-,2] => 7 [-,3,6,-,+,2] => 6 [-,3,6,+,-,2] => 6 [+,3,6,-,-,2] => 6 [-,3,6,-,-,2] => 3 [+,3,6,5,2,4] => 5 [-,3,6,5,2,4] => 4 [+,3,6,5,4,2] => 6 [-,3,6,5,4,2] => 5 [+,4,2,3,+,+] => 3 [-,4,2,3,+,+] => 6 [+,4,2,3,-,+] => 6 [+,4,2,3,+,-] => 6 [-,4,2,3,-,+] => 7 [-,4,2,3,+,-] => 7 [+,4,2,3,-,-] => 7 [-,4,2,3,-,-] => 6 [+,4,2,3,6,5] => 5 [-,4,2,3,6,5] => 6 [+,4,2,5,3,+] => 5 [-,4,2,5,3,+] => 6 [+,4,2,5,3,-] => 6 [-,4,2,5,3,-] => 5 [+,4,2,5,6,3] => 5 [-,4,2,5,6,3] => 4 [+,4,2,6,3,5] => 4 [-,4,2,6,3,5] => 5 [+,4,2,6,+,3] => 5 [-,4,2,6,+,3] => 6 [+,4,2,6,-,3] => 6 [-,4,2,6,-,3] => 5 [+,4,+,2,+,+] => 4 [-,4,+,2,+,+] => 7 [+,4,-,2,+,+] => 7 [+,4,+,2,-,+] => 7 [+,4,+,2,+,-] => 7 [-,4,-,2,+,+] => 8 [-,4,+,2,-,+] => 8 [-,4,+,2,+,-] => 8 [+,4,-,2,-,+] => 8 [+,4,-,2,+,-] => 8 [+,4,+,2,-,-] => 8 [-,4,-,2,-,+] => 7 [-,4,-,2,+,-] => 7 [-,4,+,2,-,-] => 7 [+,4,-,2,-,-] => 7 [-,4,-,2,-,-] => 4 [+,4,+,2,6,5] => 6 [-,4,+,2,6,5] => 7 [+,4,-,2,6,5] => 7 [-,4,-,2,6,5] => 6 [+,4,+,5,2,+] => 6 [-,4,+,5,2,+] => 7 [+,4,-,5,2,+] => 7 [+,4,+,5,2,-] => 7 [-,4,-,5,2,+] => 6 [-,4,+,5,2,-] => 6 [+,4,-,5,2,-] => 6 [-,4,-,5,2,-] => 3 [+,4,+,5,6,2] => 6 [-,4,+,5,6,2] => 5 [+,4,-,5,6,2] => 5 [-,4,-,5,6,2] => 2 [+,4,+,6,2,5] => 5 [-,4,+,6,2,5] => 6 [+,4,-,6,2,5] => 6 [-,4,-,6,2,5] => 5 [+,4,+,6,+,2] => 6 [-,4,+,6,+,2] => 7 [+,4,-,6,+,2] => 7 [+,4,+,6,-,2] => 7 [-,4,-,6,+,2] => 6 [-,4,+,6,-,2] => 6 [+,4,-,6,-,2] => 6 [-,4,-,6,-,2] => 3 [+,4,5,2,3,+] => 4 [-,4,5,2,3,+] => 5 [+,4,5,2,3,-] => 5 [-,4,5,2,3,-] => 4 [+,4,5,2,6,3] => 4 [-,4,5,2,6,3] => 3 [+,4,5,3,2,+] => 5 [-,4,5,3,2,+] => 6 [+,4,5,3,2,-] => 6 [-,4,5,3,2,-] => 5 [+,4,5,3,6,2] => 5 [-,4,5,3,6,2] => 4 [+,4,5,6,2,3] => 3 [-,4,5,6,2,3] => 2 [+,4,5,6,3,2] => 4 [-,4,5,6,3,2] => 3 [+,4,6,2,3,5] => 3 [-,4,6,2,3,5] => 4 [+,4,6,2,+,3] => 4 [-,4,6,2,+,3] => 5 [+,4,6,2,-,3] => 5 [-,4,6,2,-,3] => 4 [+,4,6,3,2,5] => 4 [-,4,6,3,2,5] => 5 [+,4,6,3,+,2] => 5 [-,4,6,3,+,2] => 6 [+,4,6,3,-,2] => 6 [-,4,6,3,-,2] => 5 [+,4,6,5,2,3] => 4 [-,4,6,5,2,3] => 3 [+,4,6,5,3,2] => 5 [-,4,6,5,3,2] => 4 [+,5,2,3,4,+] => 2 [-,5,2,3,4,+] => 5 [+,5,2,3,4,-] => 5 [-,5,2,3,4,-] => 6 [+,5,2,3,6,4] => 4 [-,5,2,3,6,4] => 5 [+,5,2,+,3,+] => 3 [-,5,2,+,3,+] => 6 [+,5,2,-,3,+] => 6 [+,5,2,+,3,-] => 6 [-,5,2,-,3,+] => 7 [-,5,2,+,3,-] => 7 [+,5,2,-,3,-] => 7 [-,5,2,-,3,-] => 6 [+,5,2,+,6,3] => 5 [-,5,2,+,6,3] => 6 [+,5,2,-,6,3] => 6 [-,5,2,-,6,3] => 5 [+,5,2,6,3,4] => 3 [-,5,2,6,3,4] => 4 [+,5,2,6,4,3] => 4 [-,5,2,6,4,3] => 5 [+,5,+,2,4,+] => 3 [-,5,+,2,4,+] => 6 [+,5,-,2,4,+] => 6 [+,5,+,2,4,-] => 6 [-,5,-,2,4,+] => 7 [-,5,+,2,4,-] => 7 [+,5,-,2,4,-] => 7 [-,5,-,2,4,-] => 6 [+,5,+,2,6,4] => 5 [-,5,+,2,6,4] => 6 [+,5,-,2,6,4] => 6 [-,5,-,2,6,4] => 5 [+,5,+,+,2,+] => 4 [-,5,+,+,2,+] => 7 [+,5,-,+,2,+] => 7 [+,5,+,-,2,+] => 7 [+,5,+,+,2,-] => 7 [-,5,-,+,2,+] => 8 [-,5,+,-,2,+] => 8 [-,5,+,+,2,-] => 8 [+,5,-,-,2,+] => 8 [+,5,-,+,2,-] => 8 [+,5,+,-,2,-] => 8 [-,5,-,-,2,+] => 7 [-,5,-,+,2,-] => 7 [-,5,+,-,2,-] => 7 [+,5,-,-,2,-] => 7 [-,5,-,-,2,-] => 4 [+,5,+,+,6,2] => 6 [-,5,+,+,6,2] => 7 [+,5,-,+,6,2] => 7 [+,5,+,-,6,2] => 7 [-,5,-,+,6,2] => 6 [-,5,+,-,6,2] => 6 [+,5,-,-,6,2] => 6 [-,5,-,-,6,2] => 3 [+,5,+,6,2,4] => 4 [-,5,+,6,2,4] => 5 [+,5,-,6,2,4] => 5 [-,5,-,6,2,4] => 4 [+,5,+,6,4,2] => 5 [-,5,+,6,4,2] => 6 [+,5,-,6,4,2] => 6 [-,5,-,6,4,2] => 5 [+,5,4,2,3,+] => 5 [-,5,4,2,3,+] => 6 [+,5,4,2,3,-] => 6 [-,5,4,2,3,-] => 5 [+,5,4,2,6,3] => 5 [-,5,4,2,6,3] => 4 [+,5,4,3,2,+] => 6 [-,5,4,3,2,+] => 7 [+,5,4,3,2,-] => 7 [-,5,4,3,2,-] => 6 [+,5,4,3,6,2] => 6 [-,5,4,3,6,2] => 5 [+,5,4,6,2,3] => 4 [-,5,4,6,2,3] => 3 [+,5,4,6,3,2] => 5 [-,5,4,6,3,2] => 4 [+,5,6,2,3,4] => 2 [-,5,6,2,3,4] => 3 [+,5,6,2,4,3] => 3 [-,5,6,2,4,3] => 4 [+,5,6,3,2,4] => 3 [-,5,6,3,2,4] => 4 [+,5,6,3,4,2] => 4 [-,5,6,3,4,2] => 5 [+,5,6,+,2,3] => 4 [-,5,6,+,2,3] => 5 [+,5,6,-,2,3] => 5 [-,5,6,-,2,3] => 4 [+,5,6,+,3,2] => 5 [-,5,6,+,3,2] => 6 [+,5,6,-,3,2] => 6 [-,5,6,-,3,2] => 5 [+,6,2,3,4,5] => 1 [-,6,2,3,4,5] => 4 [+,6,2,3,+,4] => 2 [-,6,2,3,+,4] => 5 [+,6,2,3,-,4] => 5 [-,6,2,3,-,4] => 6 [+,6,2,+,3,5] => 2 [-,6,2,+,3,5] => 5 [+,6,2,-,3,5] => 5 [-,6,2,-,3,5] => 6 [+,6,2,+,+,3] => 3 [-,6,2,+,+,3] => 6 [+,6,2,-,+,3] => 6 [+,6,2,+,-,3] => 6 [-,6,2,-,+,3] => 7 [-,6,2,+,-,3] => 7 [+,6,2,-,-,3] => 7 [-,6,2,-,-,3] => 6 [+,6,2,5,3,4] => 4 [-,6,2,5,3,4] => 5 [+,6,2,5,4,3] => 5 [-,6,2,5,4,3] => 6 [+,6,+,2,4,5] => 2 [-,6,+,2,4,5] => 5 [+,6,-,2,4,5] => 5 [-,6,-,2,4,5] => 6 [+,6,+,2,+,4] => 3 [-,6,+,2,+,4] => 6 [+,6,-,2,+,4] => 6 [+,6,+,2,-,4] => 6 [-,6,-,2,+,4] => 7 [-,6,+,2,-,4] => 7 [+,6,-,2,-,4] => 7 [-,6,-,2,-,4] => 6 [+,6,+,+,2,5] => 3 [-,6,+,+,2,5] => 6 [+,6,-,+,2,5] => 6 [+,6,+,-,2,5] => 6 [-,6,-,+,2,5] => 7 [-,6,+,-,2,5] => 7 [+,6,-,-,2,5] => 7 [-,6,-,-,2,5] => 6 [+,6,+,+,+,2] => 4 [-,6,+,+,+,2] => 7 [+,6,-,+,+,2] => 7 [+,6,+,-,+,2] => 7 [+,6,+,+,-,2] => 7 [-,6,-,+,+,2] => 8 [-,6,+,-,+,2] => 8 [-,6,+,+,-,2] => 8 [+,6,-,-,+,2] => 8 [+,6,-,+,-,2] => 8 [+,6,+,-,-,2] => 8 [-,6,-,-,+,2] => 7 [-,6,-,+,-,2] => 7 [-,6,+,-,-,2] => 7 [+,6,-,-,-,2] => 7 [-,6,-,-,-,2] => 4 [+,6,+,5,2,4] => 5 [-,6,+,5,2,4] => 6 [+,6,-,5,2,4] => 6 [-,6,-,5,2,4] => 5 [+,6,+,5,4,2] => 6 [-,6,+,5,4,2] => 7 [+,6,-,5,4,2] => 7 [-,6,-,5,4,2] => 6 [+,6,4,2,3,5] => 4 [-,6,4,2,3,5] => 5 [+,6,4,2,+,3] => 5 [-,6,4,2,+,3] => 6 [+,6,4,2,-,3] => 6 [-,6,4,2,-,3] => 5 [+,6,4,3,2,5] => 5 [-,6,4,3,2,5] => 6 [+,6,4,3,+,2] => 6 [-,6,4,3,+,2] => 7 [+,6,4,3,-,2] => 7 [-,6,4,3,-,2] => 6 [+,6,4,5,2,3] => 5 [-,6,4,5,2,3] => 4 [+,6,4,5,3,2] => 6 [-,6,4,5,3,2] => 5 [+,6,5,2,3,4] => 3 [-,6,5,2,3,4] => 4 [+,6,5,2,4,3] => 4 [-,6,5,2,4,3] => 5 [+,6,5,3,2,4] => 4 [-,6,5,3,2,4] => 5 [+,6,5,3,4,2] => 5 [-,6,5,3,4,2] => 6 [+,6,5,+,2,3] => 5 [-,6,5,+,2,3] => 6 [+,6,5,-,2,3] => 6 [-,6,5,-,2,3] => 5 [+,6,5,+,3,2] => 6 [-,6,5,+,3,2] => 7 [+,6,5,-,3,2] => 7 [-,6,5,-,3,2] => 6 [2,1,+,+,+,+] => 4 [2,1,-,+,+,+] => 7 [2,1,+,-,+,+] => 7 [2,1,+,+,-,+] => 7 [2,1,+,+,+,-] => 7 [2,1,-,-,+,+] => 8 [2,1,-,+,-,+] => 8 [2,1,-,+,+,-] => 8 [2,1,+,-,-,+] => 8 [2,1,+,-,+,-] => 8 [2,1,+,+,-,-] => 8 [2,1,-,-,-,+] => 7 [2,1,-,-,+,-] => 7 [2,1,-,+,-,-] => 7 [2,1,+,-,-,-] => 7 [2,1,-,-,-,-] => 4 [2,1,+,+,6,5] => 6 [2,1,-,+,6,5] => 7 [2,1,+,-,6,5] => 7 [2,1,-,-,6,5] => 6 [2,1,+,5,4,+] => 6 [2,1,-,5,4,+] => 7 [2,1,+,5,4,-] => 7 [2,1,-,5,4,-] => 6 [2,1,+,5,6,4] => 6 [2,1,-,5,6,4] => 5 [2,1,+,6,4,5] => 5 [2,1,-,6,4,5] => 6 [2,1,+,6,+,4] => 6 [2,1,-,6,+,4] => 7 [2,1,+,6,-,4] => 7 [2,1,-,6,-,4] => 6 [2,1,4,3,+,+] => 6 [2,1,4,3,-,+] => 7 [2,1,4,3,+,-] => 7 [2,1,4,3,-,-] => 6 [2,1,4,3,6,5] => 6 [2,1,4,5,3,+] => 6 [2,1,4,5,3,-] => 5 [2,1,4,5,6,3] => 4 [2,1,4,6,3,5] => 5 [2,1,4,6,+,3] => 6 [2,1,4,6,-,3] => 5 [2,1,5,3,4,+] => 5 [2,1,5,3,4,-] => 6 [2,1,5,3,6,4] => 5 [2,1,5,+,3,+] => 6 [2,1,5,-,3,+] => 7 [2,1,5,+,3,-] => 7 [2,1,5,-,3,-] => 6 [2,1,5,+,6,3] => 6 [2,1,5,-,6,3] => 5 [2,1,5,6,3,4] => 4 [2,1,5,6,4,3] => 5 [2,1,6,3,4,5] => 4 [2,1,6,3,+,4] => 5 [2,1,6,3,-,4] => 6 [2,1,6,+,3,5] => 5 [2,1,6,-,3,5] => 6 [2,1,6,+,+,3] => 6 [2,1,6,-,+,3] => 7 [2,1,6,+,-,3] => 7 [2,1,6,-,-,3] => 6 [2,1,6,5,3,4] => 5 [2,1,6,5,4,3] => 6 [2,3,1,+,+,+] => 6 [2,3,1,-,+,+] => 7 [2,3,1,+,-,+] => 7 [2,3,1,+,+,-] => 7 [2,3,1,-,-,+] => 6 [2,3,1,-,+,-] => 6 [2,3,1,+,-,-] => 6 [2,3,1,-,-,-] => 3 [2,3,1,+,6,5] => 6 [2,3,1,-,6,5] => 5 [2,3,1,5,4,+] => 6 [2,3,1,5,4,-] => 5 [2,3,1,5,6,4] => 4 [2,3,1,6,4,5] => 5 [2,3,1,6,+,4] => 6 [2,3,1,6,-,4] => 5 [2,3,4,1,+,+] => 6 [2,3,4,1,-,+] => 5 [2,3,4,1,+,-] => 5 [2,3,4,1,-,-] => 2 [2,3,4,1,6,5] => 4 [2,3,4,5,1,+] => 4 [2,3,4,5,1,-] => 1 [2,3,4,5,6,1] => 0 [2,3,4,6,1,5] => 3 [2,3,4,6,+,1] => 4 [2,3,4,6,-,1] => 1 [2,3,5,1,4,+] => 5 [2,3,5,1,4,-] => 4 [2,3,5,1,6,4] => 3 [2,3,5,+,1,+] => 6 [2,3,5,-,1,+] => 5 [2,3,5,+,1,-] => 5 [2,3,5,-,1,-] => 2 [2,3,5,+,6,1] => 4 [2,3,5,-,6,1] => 1 [2,3,5,6,1,4] => 2 [2,3,5,6,4,1] => 3 [2,3,6,1,4,5] => 4 [2,3,6,1,+,4] => 5 [2,3,6,1,-,4] => 4 [2,3,6,+,1,5] => 5 [2,3,6,-,1,5] => 4 [2,3,6,+,+,1] => 6 [2,3,6,-,+,1] => 5 [2,3,6,+,-,1] => 5 [2,3,6,-,-,1] => 2 [2,3,6,5,1,4] => 3 [2,3,6,5,4,1] => 4 [2,4,1,3,+,+] => 5 [2,4,1,3,-,+] => 6 [2,4,1,3,+,-] => 6 [2,4,1,3,-,-] => 5 [2,4,1,3,6,5] => 5 [2,4,1,5,3,+] => 5 [2,4,1,5,3,-] => 4 [2,4,1,5,6,3] => 3 [2,4,1,6,3,5] => 4 [2,4,1,6,+,3] => 5 [2,4,1,6,-,3] => 4 [2,4,+,1,+,+] => 6 [2,4,-,1,+,+] => 7 [2,4,+,1,-,+] => 7 [2,4,+,1,+,-] => 7 [2,4,-,1,-,+] => 6 [2,4,-,1,+,-] => 6 [2,4,+,1,-,-] => 6 [2,4,-,1,-,-] => 3 [2,4,+,1,6,5] => 6 ----------------------------------------------------------------------------- Created: Jul 15, 2022 at 18:19 by Martin Rubey ----------------------------------------------------------------------------- Last Updated: Jul 15, 2022 at 18:19 by Martin Rubey