***************************************************************************** * 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: St001824 ----------------------------------------------------------------------------- Collection: Decorated permutations ----------------------------------------------------------------------------- Description: The number of crossed alignments in a decorated permutation. Let $w$ be a decorated permutation. An alignment $(i \mapsto w(i), j \mapsto w(j))$ is crossed if there exists a third arc $(h \mapsto w(h))$ which forms a crossing with both arcs in the alignment. ----------------------------------------------------------------------------- References: [1] Billey, S. C., Weaver, J. E. Criteria for smoothness of Positroid varieties via pattern avoidance, Johnson graphs, and spirographs [[arXiv:2207.06508]] ----------------------------------------------------------------------------- Code: 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): """ sage: x = DecoratedPermutation([8,9,5,-4,7,6,1,3,2]) sage: (3, 5, 6, 6) in alignments(x) True sage: len(alignments(x)) 13 sage: x = DecoratedPermutation([8,9,5,4,7,-6,1,3,2]) sage: (3, 5, 6, 6) in alignments(x) False """ 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): A = alignments(x) pi = x.to_signed_permutation().permutation() n = len(pi) cal = [] for i, e, j, f in A: a, b = sorted([i, e]) c, d = sorted([j, f]) for k, g in enumerate(pi, 1): r, s = sorted([k, g]) if ((a <= r <= b <= s or r <= a <= s <= b) and (c <= r <= d <= s or r <= c <= s <= d)): cal.append((i, e, j, f, r, k)) break return len(cal) ----------------------------------------------------------------------------- Statistic values: [+] => 0 [-] => 0 [+,+] => 0 [-,+] => 0 [+,-] => 0 [-,-] => 0 [2,1] => 0 [+,+,+] => 0 [-,+,+] => 0 [+,-,+] => 0 [+,+,-] => 0 [-,-,+] => 0 [-,+,-] => 0 [+,-,-] => 0 [-,-,-] => 0 [+,3,2] => 0 [-,3,2] => 0 [2,1,+] => 0 [2,1,-] => 0 [2,3,1] => 0 [3,1,2] => 0 [3,+,1] => 0 [3,-,1] => 0 [+,+,+,+] => 0 [-,+,+,+] => 0 [+,-,+,+] => 0 [+,+,-,+] => 0 [+,+,+,-] => 0 [-,-,+,+] => 0 [-,+,-,+] => 0 [-,+,+,-] => 0 [+,-,-,+] => 0 [+,-,+,-] => 0 [+,+,-,-] => 0 [-,-,-,+] => 0 [-,-,+,-] => 0 [-,+,-,-] => 0 [+,-,-,-] => 0 [-,-,-,-] => 0 [+,+,4,3] => 0 [-,+,4,3] => 0 [+,-,4,3] => 0 [-,-,4,3] => 0 [+,3,2,+] => 0 [-,3,2,+] => 0 [+,3,2,-] => 0 [-,3,2,-] => 0 [+,3,4,2] => 0 [-,3,4,2] => 0 [+,4,2,3] => 0 [-,4,2,3] => 0 [+,4,+,2] => 0 [-,4,+,2] => 0 [+,4,-,2] => 0 [-,4,-,2] => 0 [2,1,+,+] => 0 [2,1,-,+] => 0 [2,1,+,-] => 0 [2,1,-,-] => 0 [2,1,4,3] => 0 [2,3,1,+] => 0 [2,3,1,-] => 0 [2,3,4,1] => 0 [2,4,1,3] => 1 [2,4,+,1] => 0 [2,4,-,1] => 0 [3,1,2,+] => 0 [3,1,2,-] => 0 [3,1,4,2] => 1 [3,+,1,+] => 0 [3,-,1,+] => 0 [3,+,1,-] => 0 [3,-,1,-] => 0 [3,+,4,1] => 0 [3,-,4,1] => 0 [3,4,1,2] => 0 [3,4,2,1] => 1 [4,1,2,3] => 0 [4,1,+,2] => 0 [4,1,-,2] => 0 [4,+,1,3] => 0 [4,-,1,3] => 0 [4,+,+,1] => 0 [4,-,+,1] => 0 [4,+,-,1] => 0 [4,-,-,1] => 0 [4,3,1,2] => 1 [4,3,2,1] => 0 [+,+,+,+,+] => 0 [-,+,+,+,+] => 0 [+,-,+,+,+] => 0 [+,+,-,+,+] => 0 [+,+,+,-,+] => 0 [+,+,+,+,-] => 0 [-,-,+,+,+] => 0 [-,+,-,+,+] => 0 [-,+,+,-,+] => 0 [-,+,+,+,-] => 0 [+,-,-,+,+] => 0 [+,-,+,-,+] => 0 [+,-,+,+,-] => 0 [+,+,-,-,+] => 0 [+,+,-,+,-] => 0 [+,+,+,-,-] => 0 [-,-,-,+,+] => 0 [-,-,+,-,+] => 0 [-,-,+,+,-] => 0 [-,+,-,-,+] => 0 [-,+,-,+,-] => 0 [-,+,+,-,-] => 0 [+,-,-,-,+] => 0 [+,-,-,+,-] => 0 [+,-,+,-,-] => 0 [+,+,-,-,-] => 0 [-,-,-,-,+] => 0 [-,-,-,+,-] => 0 [-,-,+,-,-] => 0 [-,+,-,-,-] => 0 [+,-,-,-,-] => 0 [-,-,-,-,-] => 0 [+,+,+,5,4] => 0 [-,+,+,5,4] => 0 [+,-,+,5,4] => 0 [+,+,-,5,4] => 0 [-,-,+,5,4] => 0 [-,+,-,5,4] => 0 [+,-,-,5,4] => 0 [-,-,-,5,4] => 0 [+,+,4,3,+] => 0 [-,+,4,3,+] => 0 [+,-,4,3,+] => 0 [+,+,4,3,-] => 0 [-,-,4,3,+] => 0 [-,+,4,3,-] => 0 [+,-,4,3,-] => 0 [-,-,4,3,-] => 0 [+,+,4,5,3] => 0 [-,+,4,5,3] => 0 [+,-,4,5,3] => 0 [-,-,4,5,3] => 0 [+,+,5,3,4] => 0 [-,+,5,3,4] => 0 [+,-,5,3,4] => 0 [-,-,5,3,4] => 0 [+,+,5,+,3] => 0 [-,+,5,+,3] => 0 [+,-,5,+,3] => 0 [+,+,5,-,3] => 0 [-,-,5,+,3] => 0 [-,+,5,-,3] => 0 [+,-,5,-,3] => 0 [-,-,5,-,3] => 0 [+,3,2,+,+] => 0 [-,3,2,+,+] => 0 [+,3,2,-,+] => 0 [+,3,2,+,-] => 0 [-,3,2,-,+] => 0 [-,3,2,+,-] => 0 [+,3,2,-,-] => 0 [-,3,2,-,-] => 0 [+,3,2,5,4] => 0 [-,3,2,5,4] => 0 [+,3,4,2,+] => 0 [-,3,4,2,+] => 0 [+,3,4,2,-] => 0 [-,3,4,2,-] => 0 [+,3,4,5,2] => 0 [-,3,4,5,2] => 0 [+,3,5,2,4] => 1 [-,3,5,2,4] => 1 [+,3,5,+,2] => 0 [-,3,5,+,2] => 0 [+,3,5,-,2] => 0 [-,3,5,-,2] => 0 [+,4,2,3,+] => 0 [-,4,2,3,+] => 0 [+,4,2,3,-] => 0 [-,4,2,3,-] => 0 [+,4,2,5,3] => 1 [-,4,2,5,3] => 1 [+,4,+,2,+] => 0 [-,4,+,2,+] => 0 [+,4,-,2,+] => 0 [+,4,+,2,-] => 0 [-,4,-,2,+] => 0 [-,4,+,2,-] => 0 [+,4,-,2,-] => 0 [-,4,-,2,-] => 0 [+,4,+,5,2] => 0 [-,4,+,5,2] => 0 [+,4,-,5,2] => 0 [-,4,-,5,2] => 0 [+,4,5,2,3] => 0 [-,4,5,2,3] => 0 [+,4,5,3,2] => 1 [-,4,5,3,2] => 1 [+,5,2,3,4] => 0 [-,5,2,3,4] => 0 [+,5,2,+,3] => 0 [-,5,2,+,3] => 0 [+,5,2,-,3] => 0 [-,5,2,-,3] => 0 [+,5,+,2,4] => 0 [-,5,+,2,4] => 0 [+,5,-,2,4] => 0 [-,5,-,2,4] => 0 [+,5,+,+,2] => 0 [-,5,+,+,2] => 0 [+,5,-,+,2] => 0 [+,5,+,-,2] => 0 [-,5,-,+,2] => 0 [-,5,+,-,2] => 0 [+,5,-,-,2] => 0 [-,5,-,-,2] => 0 [+,5,4,2,3] => 1 [-,5,4,2,3] => 1 [+,5,4,3,2] => 0 [-,5,4,3,2] => 0 [2,1,+,+,+] => 0 [2,1,-,+,+] => 0 [2,1,+,-,+] => 0 [2,1,+,+,-] => 0 [2,1,-,-,+] => 0 [2,1,-,+,-] => 0 [2,1,+,-,-] => 0 [2,1,-,-,-] => 0 [2,1,+,5,4] => 0 [2,1,-,5,4] => 0 [2,1,4,3,+] => 0 [2,1,4,3,-] => 0 [2,1,4,5,3] => 0 [2,1,5,3,4] => 0 [2,1,5,+,3] => 0 [2,1,5,-,3] => 0 [2,3,1,+,+] => 0 [2,3,1,-,+] => 0 [2,3,1,+,-] => 0 [2,3,1,-,-] => 0 [2,3,1,5,4] => 0 [2,3,4,1,+] => 0 [2,3,4,1,-] => 0 [2,3,4,5,1] => 0 [2,3,5,1,4] => 2 [2,3,5,+,1] => 0 [2,3,5,-,1] => 0 [2,4,1,3,+] => 1 [2,4,1,3,-] => 1 [2,4,1,5,3] => 2 [2,4,+,1,+] => 0 [2,4,-,1,+] => 0 [2,4,+,1,-] => 0 [2,4,-,1,-] => 0 [2,4,+,5,1] => 0 [2,4,-,5,1] => 0 [2,4,5,1,3] => 1 [2,4,5,3,1] => 2 [2,5,1,3,4] => 2 [2,5,1,+,3] => 1 [2,5,1,-,3] => 1 [2,5,+,1,4] => 1 [2,5,-,1,4] => 1 [2,5,+,+,1] => 0 [2,5,-,+,1] => 0 [2,5,+,-,1] => 0 [2,5,-,-,1] => 0 [2,5,4,1,3] => 2 [2,5,4,3,1] => 0 [3,1,2,+,+] => 0 [3,1,2,-,+] => 0 [3,1,2,+,-] => 0 [3,1,2,-,-] => 0 [3,1,2,5,4] => 0 [3,1,4,2,+] => 1 [3,1,4,2,-] => 1 [3,1,4,5,2] => 2 [3,1,5,2,4] => 2 [3,1,5,+,2] => 1 [3,1,5,-,2] => 1 [3,+,1,+,+] => 0 [3,-,1,+,+] => 0 [3,+,1,-,+] => 0 [3,+,1,+,-] => 0 [3,-,1,-,+] => 0 [3,-,1,+,-] => 0 [3,+,1,-,-] => 0 [3,-,1,-,-] => 0 [3,+,1,5,4] => 0 [3,-,1,5,4] => 0 [3,+,4,1,+] => 0 [3,-,4,1,+] => 0 [3,+,4,1,-] => 0 [3,-,4,1,-] => 0 [3,+,4,5,1] => 0 [3,-,4,5,1] => 0 [3,+,5,1,4] => 1 [3,-,5,1,4] => 1 [3,+,5,+,1] => 0 [3,-,5,+,1] => 0 [3,+,5,-,1] => 0 [3,-,5,-,1] => 0 [3,4,1,2,+] => 0 [3,4,1,2,-] => 0 [3,4,1,5,2] => 1 [3,4,2,1,+] => 1 [3,4,2,1,-] => 1 [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] => 0 [3,5,1,-,2] => 0 [3,5,2,1,4] => 2 [3,5,2,+,1] => 1 [3,5,2,-,1] => 1 [3,5,4,1,2] => 1 [3,5,4,2,1] => 2 [4,1,2,3,+] => 0 [4,1,2,3,-] => 0 [4,1,2,5,3] => 2 [4,1,+,2,+] => 0 [4,1,-,2,+] => 0 [4,1,+,2,-] => 0 [4,1,-,2,-] => 0 [4,1,+,5,2] => 1 [4,1,-,5,2] => 1 [4,1,5,2,3] => 1 [4,1,5,3,2] => 2 [4,+,1,3,+] => 0 [4,-,1,3,+] => 0 [4,+,1,3,-] => 0 [4,-,1,3,-] => 0 [4,+,1,5,3] => 1 [4,-,1,5,3] => 1 [4,+,+,1,+] => 0 [4,-,+,1,+] => 0 [4,+,-,1,+] => 0 [4,+,+,1,-] => 0 [4,-,-,1,+] => 0 [4,-,+,1,-] => 0 [4,+,-,1,-] => 0 [4,-,-,1,-] => 0 [4,+,+,5,1] => 0 [4,-,+,5,1] => 0 [4,+,-,5,1] => 0 [4,-,-,5,1] => 0 [4,+,5,1,3] => 0 [4,-,5,1,3] => 0 [4,+,5,3,1] => 1 [4,-,5,3,1] => 1 [4,3,1,2,+] => 1 [4,3,1,2,-] => 1 [4,3,1,5,2] => 2 [4,3,2,1,+] => 0 [4,3,2,1,-] => 0 [4,3,2,5,1] => 0 [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] => 0 [4,5,-,1,2] => 0 [4,5,+,2,1] => 1 [4,5,-,2,1] => 1 [5,1,2,3,4] => 0 [5,1,2,+,3] => 0 [5,1,2,-,3] => 0 [5,1,+,2,4] => 0 [5,1,-,2,4] => 0 [5,1,+,+,2] => 0 [5,1,-,+,2] => 0 [5,1,+,-,2] => 0 [5,1,-,-,2] => 0 [5,1,4,2,3] => 2 [5,1,4,3,2] => 0 [5,+,1,3,4] => 0 [5,-,1,3,4] => 0 [5,+,1,+,3] => 0 [5,-,1,+,3] => 0 [5,+,1,-,3] => 0 [5,-,1,-,3] => 0 [5,+,+,1,4] => 0 [5,-,+,1,4] => 0 [5,+,-,1,4] => 0 [5,-,-,1,4] => 0 [5,+,+,+,1] => 0 [5,-,+,+,1] => 0 [5,+,-,+,1] => 0 [5,+,+,-,1] => 0 [5,-,-,+,1] => 0 [5,-,+,-,1] => 0 [5,+,-,-,1] => 0 [5,-,-,-,1] => 0 [5,+,4,1,3] => 1 [5,-,4,1,3] => 1 [5,+,4,3,1] => 0 [5,-,4,3,1] => 0 [5,3,1,2,4] => 2 [5,3,1,+,2] => 1 [5,3,1,-,2] => 1 [5,3,2,1,4] => 0 [5,3,2,+,1] => 0 [5,3,2,-,1] => 0 [5,3,4,1,2] => 2 [5,3,4,2,1] => 0 [5,4,1,2,3] => 1 [5,4,1,3,2] => 2 [5,4,2,1,3] => 2 [5,4,2,3,1] => 0 [5,4,+,1,2] => 1 [5,4,-,1,2] => 1 [5,4,+,2,1] => 0 [5,4,-,2,1] => 0 [+,+,+,+,+,+] => 0 [-,+,+,+,+,+] => 0 [+,-,+,+,+,+] => 0 [+,+,-,+,+,+] => 0 [+,+,+,-,+,+] => 0 [+,+,+,+,-,+] => 0 [+,+,+,+,+,-] => 0 [-,-,+,+,+,+] => 0 [-,+,-,+,+,+] => 0 [-,+,+,-,+,+] => 0 [-,+,+,+,-,+] => 0 [-,+,+,+,+,-] => 0 [+,-,-,+,+,+] => 0 [+,-,+,-,+,+] => 0 [+,-,+,+,-,+] => 0 [+,-,+,+,+,-] => 0 [+,+,-,-,+,+] => 0 [+,+,-,+,-,+] => 0 [+,+,-,+,+,-] => 0 [+,+,+,-,-,+] => 0 [+,+,+,-,+,-] => 0 [+,+,+,+,-,-] => 0 [-,-,-,+,+,+] => 0 [-,-,+,-,+,+] => 0 [-,-,+,+,-,+] => 0 [-,-,+,+,+,-] => 0 [-,+,-,-,+,+] => 0 [-,+,-,+,-,+] => 0 [-,+,-,+,+,-] => 0 [-,+,+,-,-,+] => 0 [-,+,+,-,+,-] => 0 [-,+,+,+,-,-] => 0 [+,-,-,-,+,+] => 0 [+,-,-,+,-,+] => 0 [+,-,-,+,+,-] => 0 [+,-,+,-,-,+] => 0 [+,-,+,-,+,-] => 0 [+,-,+,+,-,-] => 0 [+,+,-,-,-,+] => 0 [+,+,-,-,+,-] => 0 [+,+,-,+,-,-] => 0 [+,+,+,-,-,-] => 0 [-,-,-,-,+,+] => 0 [-,-,-,+,-,+] => 0 [-,-,-,+,+,-] => 0 [-,-,+,-,-,+] => 0 [-,-,+,-,+,-] => 0 [-,-,+,+,-,-] => 0 [-,+,-,-,-,+] => 0 [-,+,-,-,+,-] => 0 [-,+,-,+,-,-] => 0 [-,+,+,-,-,-] => 0 [+,-,-,-,-,+] => 0 [+,-,-,-,+,-] => 0 [+,-,-,+,-,-] => 0 [+,-,+,-,-,-] => 0 [+,+,-,-,-,-] => 0 [-,-,-,-,-,+] => 0 [-,-,-,-,+,-] => 0 [-,-,-,+,-,-] => 0 [-,-,+,-,-,-] => 0 [-,+,-,-,-,-] => 0 [+,-,-,-,-,-] => 0 [-,-,-,-,-,-] => 0 [+,+,+,+,6,5] => 0 [-,+,+,+,6,5] => 0 [+,-,+,+,6,5] => 0 [+,+,-,+,6,5] => 0 [+,+,+,-,6,5] => 0 [-,-,+,+,6,5] => 0 [-,+,-,+,6,5] => 0 [-,+,+,-,6,5] => 0 [+,-,-,+,6,5] => 0 [+,-,+,-,6,5] => 0 [+,+,-,-,6,5] => 0 [-,-,-,+,6,5] => 0 [-,-,+,-,6,5] => 0 [-,+,-,-,6,5] => 0 [+,-,-,-,6,5] => 0 [-,-,-,-,6,5] => 0 [+,+,+,5,4,+] => 0 [-,+,+,5,4,+] => 0 [+,-,+,5,4,+] => 0 [+,+,-,5,4,+] => 0 [+,+,+,5,4,-] => 0 [-,-,+,5,4,+] => 0 [-,+,-,5,4,+] => 0 [-,+,+,5,4,-] => 0 [+,-,-,5,4,+] => 0 [+,-,+,5,4,-] => 0 [+,+,-,5,4,-] => 0 [-,-,-,5,4,+] => 0 [-,-,+,5,4,-] => 0 [-,+,-,5,4,-] => 0 [+,-,-,5,4,-] => 0 [-,-,-,5,4,-] => 0 [+,+,+,5,6,4] => 0 [-,+,+,5,6,4] => 0 [+,-,+,5,6,4] => 0 [+,+,-,5,6,4] => 0 [-,-,+,5,6,4] => 0 [-,+,-,5,6,4] => 0 [+,-,-,5,6,4] => 0 [-,-,-,5,6,4] => 0 [+,+,+,6,4,5] => 0 [-,+,+,6,4,5] => 0 [+,-,+,6,4,5] => 0 [+,+,-,6,4,5] => 0 [-,-,+,6,4,5] => 0 [-,+,-,6,4,5] => 0 [+,-,-,6,4,5] => 0 [-,-,-,6,4,5] => 0 [+,+,+,6,+,4] => 0 [-,+,+,6,+,4] => 0 [+,-,+,6,+,4] => 0 [+,+,-,6,+,4] => 0 [+,+,+,6,-,4] => 0 [-,-,+,6,+,4] => 0 [-,+,-,6,+,4] => 0 [-,+,+,6,-,4] => 0 [+,-,-,6,+,4] => 0 [+,-,+,6,-,4] => 0 [+,+,-,6,-,4] => 0 [-,-,-,6,+,4] => 0 [-,-,+,6,-,4] => 0 [-,+,-,6,-,4] => 0 [+,-,-,6,-,4] => 0 [-,-,-,6,-,4] => 0 [+,+,4,3,+,+] => 0 [-,+,4,3,+,+] => 0 [+,-,4,3,+,+] => 0 [+,+,4,3,-,+] => 0 [+,+,4,3,+,-] => 0 [-,-,4,3,+,+] => 0 [-,+,4,3,-,+] => 0 [-,+,4,3,+,-] => 0 [+,-,4,3,-,+] => 0 [+,-,4,3,+,-] => 0 [+,+,4,3,-,-] => 0 [-,-,4,3,-,+] => 0 [-,-,4,3,+,-] => 0 [-,+,4,3,-,-] => 0 [+,-,4,3,-,-] => 0 [-,-,4,3,-,-] => 0 [+,+,4,3,6,5] => 0 [-,+,4,3,6,5] => 0 [+,-,4,3,6,5] => 0 [-,-,4,3,6,5] => 0 [+,+,4,5,3,+] => 0 [-,+,4,5,3,+] => 0 [+,-,4,5,3,+] => 0 [+,+,4,5,3,-] => 0 [-,-,4,5,3,+] => 0 [-,+,4,5,3,-] => 0 [+,-,4,5,3,-] => 0 [-,-,4,5,3,-] => 0 [+,+,4,5,6,3] => 0 [-,+,4,5,6,3] => 0 [+,-,4,5,6,3] => 0 [-,-,4,5,6,3] => 0 [+,+,4,6,3,5] => 1 [-,+,4,6,3,5] => 1 [+,-,4,6,3,5] => 1 [-,-,4,6,3,5] => 1 [+,+,4,6,+,3] => 0 [-,+,4,6,+,3] => 0 [+,-,4,6,+,3] => 0 [+,+,4,6,-,3] => 0 [-,-,4,6,+,3] => 0 [-,+,4,6,-,3] => 0 [+,-,4,6,-,3] => 0 [-,-,4,6,-,3] => 0 [+,+,5,3,4,+] => 0 [-,+,5,3,4,+] => 0 [+,-,5,3,4,+] => 0 [+,+,5,3,4,-] => 0 [-,-,5,3,4,+] => 0 [-,+,5,3,4,-] => 0 [+,-,5,3,4,-] => 0 [-,-,5,3,4,-] => 0 [+,+,5,3,6,4] => 1 [-,+,5,3,6,4] => 1 [+,-,5,3,6,4] => 1 [-,-,5,3,6,4] => 1 [+,+,5,+,3,+] => 0 [-,+,5,+,3,+] => 0 [+,-,5,+,3,+] => 0 [+,+,5,-,3,+] => 0 [+,+,5,+,3,-] => 0 [-,-,5,+,3,+] => 0 [-,+,5,-,3,+] => 0 [-,+,5,+,3,-] => 0 [+,-,5,-,3,+] => 0 [+,-,5,+,3,-] => 0 [+,+,5,-,3,-] => 0 [-,-,5,-,3,+] => 0 [-,-,5,+,3,-] => 0 [-,+,5,-,3,-] => 0 [+,-,5,-,3,-] => 0 [-,-,5,-,3,-] => 0 [+,+,5,+,6,3] => 0 [-,+,5,+,6,3] => 0 [+,-,5,+,6,3] => 0 [+,+,5,-,6,3] => 0 [-,-,5,+,6,3] => 0 [-,+,5,-,6,3] => 0 [+,-,5,-,6,3] => 0 [-,-,5,-,6,3] => 0 [+,+,5,6,3,4] => 0 [-,+,5,6,3,4] => 0 [+,-,5,6,3,4] => 0 [-,-,5,6,3,4] => 0 [+,+,5,6,4,3] => 1 [-,+,5,6,4,3] => 1 [+,-,5,6,4,3] => 1 [-,-,5,6,4,3] => 1 [+,+,6,3,4,5] => 0 [-,+,6,3,4,5] => 0 [+,-,6,3,4,5] => 0 [-,-,6,3,4,5] => 0 [+,+,6,3,+,4] => 0 [-,+,6,3,+,4] => 0 [+,-,6,3,+,4] => 0 [+,+,6,3,-,4] => 0 [-,-,6,3,+,4] => 0 [-,+,6,3,-,4] => 0 [+,-,6,3,-,4] => 0 [-,-,6,3,-,4] => 0 [+,+,6,+,3,5] => 0 [-,+,6,+,3,5] => 0 [+,-,6,+,3,5] => 0 [+,+,6,-,3,5] => 0 [-,-,6,+,3,5] => 0 [-,+,6,-,3,5] => 0 [+,-,6,-,3,5] => 0 [-,-,6,-,3,5] => 0 [+,+,6,+,+,3] => 0 [-,+,6,+,+,3] => 0 [+,-,6,+,+,3] => 0 [+,+,6,-,+,3] => 0 [+,+,6,+,-,3] => 0 [-,-,6,+,+,3] => 0 [-,+,6,-,+,3] => 0 [-,+,6,+,-,3] => 0 [+,-,6,-,+,3] => 0 [+,-,6,+,-,3] => 0 [+,+,6,-,-,3] => 0 [-,-,6,-,+,3] => 0 [-,-,6,+,-,3] => 0 [-,+,6,-,-,3] => 0 [+,-,6,-,-,3] => 0 [-,-,6,-,-,3] => 0 [+,+,6,5,3,4] => 1 [-,+,6,5,3,4] => 1 [+,-,6,5,3,4] => 1 [-,-,6,5,3,4] => 1 [+,+,6,5,4,3] => 0 [-,+,6,5,4,3] => 0 [+,-,6,5,4,3] => 0 [-,-,6,5,4,3] => 0 [+,3,2,+,+,+] => 0 [-,3,2,+,+,+] => 0 [+,3,2,-,+,+] => 0 [+,3,2,+,-,+] => 0 [+,3,2,+,+,-] => 0 [-,3,2,-,+,+] => 0 [-,3,2,+,-,+] => 0 [-,3,2,+,+,-] => 0 [+,3,2,-,-,+] => 0 [+,3,2,-,+,-] => 0 [+,3,2,+,-,-] => 0 [-,3,2,-,-,+] => 0 [-,3,2,-,+,-] => 0 [-,3,2,+,-,-] => 0 [+,3,2,-,-,-] => 0 [-,3,2,-,-,-] => 0 [+,3,2,+,6,5] => 0 [-,3,2,+,6,5] => 0 [+,3,2,-,6,5] => 0 [-,3,2,-,6,5] => 0 [+,3,2,5,4,+] => 0 [-,3,2,5,4,+] => 0 [+,3,2,5,4,-] => 0 [-,3,2,5,4,-] => 0 [+,3,2,5,6,4] => 0 [-,3,2,5,6,4] => 0 [+,3,2,6,4,5] => 0 [-,3,2,6,4,5] => 0 [+,3,2,6,+,4] => 0 [-,3,2,6,+,4] => 0 [+,3,2,6,-,4] => 0 [-,3,2,6,-,4] => 0 [+,3,4,2,+,+] => 0 [-,3,4,2,+,+] => 0 [+,3,4,2,-,+] => 0 [+,3,4,2,+,-] => 0 [-,3,4,2,-,+] => 0 [-,3,4,2,+,-] => 0 [+,3,4,2,-,-] => 0 [-,3,4,2,-,-] => 0 [+,3,4,2,6,5] => 0 [-,3,4,2,6,5] => 0 [+,3,4,5,2,+] => 0 [-,3,4,5,2,+] => 0 [+,3,4,5,2,-] => 0 [-,3,4,5,2,-] => 0 [+,3,4,5,6,2] => 0 [-,3,4,5,6,2] => 0 [+,3,4,6,2,5] => 2 [-,3,4,6,2,5] => 2 [+,3,4,6,+,2] => 0 [-,3,4,6,+,2] => 0 [+,3,4,6,-,2] => 0 [-,3,4,6,-,2] => 0 [+,3,5,2,4,+] => 1 [-,3,5,2,4,+] => 1 [+,3,5,2,4,-] => 1 [-,3,5,2,4,-] => 1 [+,3,5,2,6,4] => 2 [-,3,5,2,6,4] => 2 [+,3,5,+,2,+] => 0 [-,3,5,+,2,+] => 0 [+,3,5,-,2,+] => 0 [+,3,5,+,2,-] => 0 [-,3,5,-,2,+] => 0 [-,3,5,+,2,-] => 0 [+,3,5,-,2,-] => 0 [-,3,5,-,2,-] => 0 [+,3,5,+,6,2] => 0 [-,3,5,+,6,2] => 0 [+,3,5,-,6,2] => 0 [-,3,5,-,6,2] => 0 [+,3,5,6,2,4] => 1 [-,3,5,6,2,4] => 1 [+,3,5,6,4,2] => 2 [-,3,5,6,4,2] => 2 [+,3,6,2,4,5] => 2 [-,3,6,2,4,5] => 2 [+,3,6,2,+,4] => 1 [-,3,6,2,+,4] => 1 [+,3,6,2,-,4] => 1 [-,3,6,2,-,4] => 1 [+,3,6,+,2,5] => 1 [-,3,6,+,2,5] => 1 [+,3,6,-,2,5] => 1 [-,3,6,-,2,5] => 1 [+,3,6,+,+,2] => 0 [-,3,6,+,+,2] => 0 [+,3,6,-,+,2] => 0 [+,3,6,+,-,2] => 0 [-,3,6,-,+,2] => 0 [-,3,6,+,-,2] => 0 [+,3,6,-,-,2] => 0 [-,3,6,-,-,2] => 0 [+,3,6,5,2,4] => 2 [-,3,6,5,2,4] => 2 [+,3,6,5,4,2] => 0 [-,3,6,5,4,2] => 0 [+,4,2,3,+,+] => 0 [-,4,2,3,+,+] => 0 [+,4,2,3,-,+] => 0 [+,4,2,3,+,-] => 0 [-,4,2,3,-,+] => 0 [-,4,2,3,+,-] => 0 [+,4,2,3,-,-] => 0 [-,4,2,3,-,-] => 0 [+,4,2,3,6,5] => 0 [-,4,2,3,6,5] => 0 [+,4,2,5,3,+] => 1 [-,4,2,5,3,+] => 1 [+,4,2,5,3,-] => 1 [-,4,2,5,3,-] => 1 [+,4,2,5,6,3] => 2 [-,4,2,5,6,3] => 2 [+,4,2,6,3,5] => 2 [-,4,2,6,3,5] => 2 [+,4,2,6,+,3] => 1 [-,4,2,6,+,3] => 1 [+,4,2,6,-,3] => 1 [-,4,2,6,-,3] => 1 [+,4,+,2,+,+] => 0 [-,4,+,2,+,+] => 0 [+,4,-,2,+,+] => 0 [+,4,+,2,-,+] => 0 [+,4,+,2,+,-] => 0 [-,4,-,2,+,+] => 0 [-,4,+,2,-,+] => 0 [-,4,+,2,+,-] => 0 [+,4,-,2,-,+] => 0 [+,4,-,2,+,-] => 0 [+,4,+,2,-,-] => 0 [-,4,-,2,-,+] => 0 [-,4,-,2,+,-] => 0 [-,4,+,2,-,-] => 0 [+,4,-,2,-,-] => 0 [-,4,-,2,-,-] => 0 [+,4,+,2,6,5] => 0 [-,4,+,2,6,5] => 0 [+,4,-,2,6,5] => 0 [-,4,-,2,6,5] => 0 [+,4,+,5,2,+] => 0 [-,4,+,5,2,+] => 0 [+,4,-,5,2,+] => 0 [+,4,+,5,2,-] => 0 [-,4,-,5,2,+] => 0 [-,4,+,5,2,-] => 0 [+,4,-,5,2,-] => 0 [-,4,-,5,2,-] => 0 [+,4,+,5,6,2] => 0 [-,4,+,5,6,2] => 0 [+,4,-,5,6,2] => 0 [-,4,-,5,6,2] => 0 [+,4,+,6,2,5] => 1 [-,4,+,6,2,5] => 1 [+,4,-,6,2,5] => 1 [-,4,-,6,2,5] => 1 [+,4,+,6,+,2] => 0 [-,4,+,6,+,2] => 0 [+,4,-,6,+,2] => 0 [+,4,+,6,-,2] => 0 [-,4,-,6,+,2] => 0 [-,4,+,6,-,2] => 0 [+,4,-,6,-,2] => 0 [-,4,-,6,-,2] => 0 [+,4,5,2,3,+] => 0 [-,4,5,2,3,+] => 0 [+,4,5,2,3,-] => 0 [-,4,5,2,3,-] => 0 [+,4,5,2,6,3] => 1 [-,4,5,2,6,3] => 1 [+,4,5,3,2,+] => 1 [-,4,5,3,2,+] => 1 [+,4,5,3,2,-] => 1 [-,4,5,3,2,-] => 1 [+,4,5,3,6,2] => 2 [-,4,5,3,6,2] => 2 [+,4,5,6,2,3] => 0 [-,4,5,6,2,3] => 0 [+,4,5,6,3,2] => 1 [-,4,5,6,3,2] => 1 [+,4,6,2,3,5] => 1 [-,4,6,2,3,5] => 1 [+,4,6,2,+,3] => 0 [-,4,6,2,+,3] => 0 [+,4,6,2,-,3] => 0 [-,4,6,2,-,3] => 0 [+,4,6,3,2,5] => 2 [-,4,6,3,2,5] => 2 [+,4,6,3,+,2] => 1 [-,4,6,3,+,2] => 1 [+,4,6,3,-,2] => 1 [-,4,6,3,-,2] => 1 [+,4,6,5,2,3] => 1 [-,4,6,5,2,3] => 1 [+,4,6,5,3,2] => 2 [-,4,6,5,3,2] => 2 [+,5,2,3,4,+] => 0 [-,5,2,3,4,+] => 0 [+,5,2,3,4,-] => 0 [-,5,2,3,4,-] => 0 [+,5,2,3,6,4] => 2 [-,5,2,3,6,4] => 2 [+,5,2,+,3,+] => 0 [-,5,2,+,3,+] => 0 [+,5,2,-,3,+] => 0 [+,5,2,+,3,-] => 0 [-,5,2,-,3,+] => 0 [-,5,2,+,3,-] => 0 [+,5,2,-,3,-] => 0 [-,5,2,-,3,-] => 0 [+,5,2,+,6,3] => 1 [-,5,2,+,6,3] => 1 [+,5,2,-,6,3] => 1 [-,5,2,-,6,3] => 1 [+,5,2,6,3,4] => 1 [-,5,2,6,3,4] => 1 [+,5,2,6,4,3] => 2 [-,5,2,6,4,3] => 2 [+,5,+,2,4,+] => 0 [-,5,+,2,4,+] => 0 [+,5,-,2,4,+] => 0 [+,5,+,2,4,-] => 0 [-,5,-,2,4,+] => 0 [-,5,+,2,4,-] => 0 [+,5,-,2,4,-] => 0 [-,5,-,2,4,-] => 0 [+,5,+,2,6,4] => 1 [-,5,+,2,6,4] => 1 [+,5,-,2,6,4] => 1 [-,5,-,2,6,4] => 1 [+,5,+,+,2,+] => 0 [-,5,+,+,2,+] => 0 [+,5,-,+,2,+] => 0 [+,5,+,-,2,+] => 0 [+,5,+,+,2,-] => 0 [-,5,-,+,2,+] => 0 [-,5,+,-,2,+] => 0 [-,5,+,+,2,-] => 0 [+,5,-,-,2,+] => 0 [+,5,-,+,2,-] => 0 [+,5,+,-,2,-] => 0 [-,5,-,-,2,+] => 0 [-,5,-,+,2,-] => 0 [-,5,+,-,2,-] => 0 [+,5,-,-,2,-] => 0 [-,5,-,-,2,-] => 0 [+,5,+,+,6,2] => 0 [-,5,+,+,6,2] => 0 [+,5,-,+,6,2] => 0 [+,5,+,-,6,2] => 0 [-,5,-,+,6,2] => 0 [-,5,+,-,6,2] => 0 [+,5,-,-,6,2] => 0 [-,5,-,-,6,2] => 0 [+,5,+,6,2,4] => 0 [-,5,+,6,2,4] => 0 [+,5,-,6,2,4] => 0 [-,5,-,6,2,4] => 0 [+,5,+,6,4,2] => 1 [-,5,+,6,4,2] => 1 [+,5,-,6,4,2] => 1 [-,5,-,6,4,2] => 1 [+,5,4,2,3,+] => 1 [-,5,4,2,3,+] => 1 [+,5,4,2,3,-] => 1 [-,5,4,2,3,-] => 1 [+,5,4,2,6,3] => 2 [-,5,4,2,6,3] => 2 [+,5,4,3,2,+] => 0 [-,5,4,3,2,+] => 0 [+,5,4,3,2,-] => 0 [-,5,4,3,2,-] => 0 [+,5,4,3,6,2] => 0 [-,5,4,3,6,2] => 0 [+,5,4,6,2,3] => 1 [-,5,4,6,2,3] => 1 [+,5,4,6,3,2] => 2 [-,5,4,6,3,2] => 2 [+,5,6,2,3,4] => 0 [-,5,6,2,3,4] => 0 [+,5,6,2,4,3] => 1 [-,5,6,2,4,3] => 1 [+,5,6,3,2,4] => 1 [-,5,6,3,2,4] => 1 [+,5,6,3,4,2] => 2 [-,5,6,3,4,2] => 2 [+,5,6,+,2,3] => 0 [-,5,6,+,2,3] => 0 [+,5,6,-,2,3] => 0 [-,5,6,-,2,3] => 0 [+,5,6,+,3,2] => 1 [-,5,6,+,3,2] => 1 [+,5,6,-,3,2] => 1 [-,5,6,-,3,2] => 1 [+,6,2,3,4,5] => 0 [-,6,2,3,4,5] => 0 [+,6,2,3,+,4] => 0 [-,6,2,3,+,4] => 0 [+,6,2,3,-,4] => 0 [-,6,2,3,-,4] => 0 [+,6,2,+,3,5] => 0 [-,6,2,+,3,5] => 0 [+,6,2,-,3,5] => 0 [-,6,2,-,3,5] => 0 [+,6,2,+,+,3] => 0 [-,6,2,+,+,3] => 0 [+,6,2,-,+,3] => 0 [+,6,2,+,-,3] => 0 [-,6,2,-,+,3] => 0 [-,6,2,+,-,3] => 0 [+,6,2,-,-,3] => 0 [-,6,2,-,-,3] => 0 [+,6,2,5,3,4] => 2 [-,6,2,5,3,4] => 2 [+,6,2,5,4,3] => 0 [-,6,2,5,4,3] => 0 [+,6,+,2,4,5] => 0 [-,6,+,2,4,5] => 0 [+,6,-,2,4,5] => 0 [-,6,-,2,4,5] => 0 [+,6,+,2,+,4] => 0 [-,6,+,2,+,4] => 0 [+,6,-,2,+,4] => 0 [+,6,+,2,-,4] => 0 [-,6,-,2,+,4] => 0 [-,6,+,2,-,4] => 0 [+,6,-,2,-,4] => 0 [-,6,-,2,-,4] => 0 [+,6,+,+,2,5] => 0 [-,6,+,+,2,5] => 0 [+,6,-,+,2,5] => 0 [+,6,+,-,2,5] => 0 [-,6,-,+,2,5] => 0 [-,6,+,-,2,5] => 0 [+,6,-,-,2,5] => 0 [-,6,-,-,2,5] => 0 [+,6,+,+,+,2] => 0 [-,6,+,+,+,2] => 0 [+,6,-,+,+,2] => 0 [+,6,+,-,+,2] => 0 [+,6,+,+,-,2] => 0 [-,6,-,+,+,2] => 0 [-,6,+,-,+,2] => 0 [-,6,+,+,-,2] => 0 [+,6,-,-,+,2] => 0 [+,6,-,+,-,2] => 0 [+,6,+,-,-,2] => 0 [-,6,-,-,+,2] => 0 [-,6,-,+,-,2] => 0 [-,6,+,-,-,2] => 0 [+,6,-,-,-,2] => 0 [-,6,-,-,-,2] => 0 [+,6,+,5,2,4] => 1 [-,6,+,5,2,4] => 1 [+,6,-,5,2,4] => 1 [-,6,-,5,2,4] => 1 [+,6,+,5,4,2] => 0 [-,6,+,5,4,2] => 0 [+,6,-,5,4,2] => 0 [-,6,-,5,4,2] => 0 [+,6,4,2,3,5] => 2 [-,6,4,2,3,5] => 2 [+,6,4,2,+,3] => 1 [-,6,4,2,+,3] => 1 [+,6,4,2,-,3] => 1 [-,6,4,2,-,3] => 1 [+,6,4,3,2,5] => 0 [-,6,4,3,2,5] => 0 [+,6,4,3,+,2] => 0 [-,6,4,3,+,2] => 0 [+,6,4,3,-,2] => 0 [-,6,4,3,-,2] => 0 [+,6,4,5,2,3] => 2 [-,6,4,5,2,3] => 2 [+,6,4,5,3,2] => 0 [-,6,4,5,3,2] => 0 [+,6,5,2,3,4] => 1 [-,6,5,2,3,4] => 1 [+,6,5,2,4,3] => 2 [-,6,5,2,4,3] => 2 [+,6,5,3,2,4] => 2 [-,6,5,3,2,4] => 2 [+,6,5,3,4,2] => 0 [-,6,5,3,4,2] => 0 [+,6,5,+,2,3] => 1 [-,6,5,+,2,3] => 1 [+,6,5,-,2,3] => 1 [-,6,5,-,2,3] => 1 [+,6,5,+,3,2] => 0 [-,6,5,+,3,2] => 0 [+,6,5,-,3,2] => 0 [-,6,5,-,3,2] => 0 [2,1,+,+,+,+] => 0 [2,1,-,+,+,+] => 0 [2,1,+,-,+,+] => 0 [2,1,+,+,-,+] => 0 [2,1,+,+,+,-] => 0 [2,1,-,-,+,+] => 0 [2,1,-,+,-,+] => 0 [2,1,-,+,+,-] => 0 [2,1,+,-,-,+] => 0 [2,1,+,-,+,-] => 0 [2,1,+,+,-,-] => 0 [2,1,-,-,-,+] => 0 [2,1,-,-,+,-] => 0 [2,1,-,+,-,-] => 0 [2,1,+,-,-,-] => 0 [2,1,-,-,-,-] => 0 [2,1,+,+,6,5] => 0 [2,1,-,+,6,5] => 0 [2,1,+,-,6,5] => 0 [2,1,-,-,6,5] => 0 [2,1,+,5,4,+] => 0 [2,1,-,5,4,+] => 0 [2,1,+,5,4,-] => 0 [2,1,-,5,4,-] => 0 [2,1,+,5,6,4] => 0 [2,1,-,5,6,4] => 0 [2,1,+,6,4,5] => 0 [2,1,-,6,4,5] => 0 [2,1,+,6,+,4] => 0 [2,1,-,6,+,4] => 0 [2,1,+,6,-,4] => 0 [2,1,-,6,-,4] => 0 [2,1,4,3,+,+] => 0 [2,1,4,3,-,+] => 0 [2,1,4,3,+,-] => 0 [2,1,4,3,-,-] => 0 [2,1,4,3,6,5] => 0 [2,1,4,5,3,+] => 0 [2,1,4,5,3,-] => 0 [2,1,4,5,6,3] => 0 [2,1,4,6,3,5] => 1 [2,1,4,6,+,3] => 0 [2,1,4,6,-,3] => 0 [2,1,5,3,4,+] => 0 [2,1,5,3,4,-] => 0 [2,1,5,3,6,4] => 1 [2,1,5,+,3,+] => 0 [2,1,5,-,3,+] => 0 [2,1,5,+,3,-] => 0 [2,1,5,-,3,-] => 0 [2,1,5,+,6,3] => 0 [2,1,5,-,6,3] => 0 [2,1,5,6,3,4] => 0 [2,1,5,6,4,3] => 1 [2,1,6,3,4,5] => 0 [2,1,6,3,+,4] => 0 [2,1,6,3,-,4] => 0 [2,1,6,+,3,5] => 0 [2,1,6,-,3,5] => 0 [2,1,6,+,+,3] => 0 [2,1,6,-,+,3] => 0 [2,1,6,+,-,3] => 0 [2,1,6,-,-,3] => 0 [2,1,6,5,3,4] => 1 [2,1,6,5,4,3] => 0 [2,3,1,+,+,+] => 0 [2,3,1,-,+,+] => 0 [2,3,1,+,-,+] => 0 [2,3,1,+,+,-] => 0 [2,3,1,-,-,+] => 0 [2,3,1,-,+,-] => 0 [2,3,1,+,-,-] => 0 [2,3,1,-,-,-] => 0 [2,3,1,+,6,5] => 0 [2,3,1,-,6,5] => 0 [2,3,1,5,4,+] => 0 [2,3,1,5,4,-] => 0 [2,3,1,5,6,4] => 0 [2,3,1,6,4,5] => 0 [2,3,1,6,+,4] => 0 [2,3,1,6,-,4] => 0 [2,3,4,1,+,+] => 0 [2,3,4,1,-,+] => 0 [2,3,4,1,+,-] => 0 [2,3,4,1,-,-] => 0 [2,3,4,1,6,5] => 0 [2,3,4,5,1,+] => 0 [2,3,4,5,1,-] => 0 [2,3,4,5,6,1] => 0 [2,3,4,6,1,5] => 2 [2,3,4,6,+,1] => 0 [2,3,4,6,-,1] => 0 [2,3,5,1,4,+] => 2 [2,3,5,1,4,-] => 2 [2,3,5,1,6,4] => 3 [2,3,5,+,1,+] => 0 [2,3,5,-,1,+] => 0 [2,3,5,+,1,-] => 0 [2,3,5,-,1,-] => 0 [2,3,5,+,6,1] => 0 [2,3,5,-,6,1] => 0 [2,3,5,6,1,4] => 2 [2,3,5,6,4,1] => 2 [2,3,6,1,4,5] => 2 [2,3,6,1,+,4] => 2 [2,3,6,1,-,4] => 2 [2,3,6,+,1,5] => 2 [2,3,6,-,1,5] => 2 [2,3,6,+,+,1] => 0 [2,3,6,-,+,1] => 0 [2,3,6,+,-,1] => 0 [2,3,6,-,-,1] => 0 [2,3,6,5,1,4] => 3 [2,3,6,5,4,1] => 0 [2,4,1,3,+,+] => 1 [2,4,1,3,-,+] => 1 [2,4,1,3,+,-] => 1 [2,4,1,3,-,-] => 1 [2,4,1,3,6,5] => 1 [2,4,1,5,3,+] => 2 [2,4,1,5,3,-] => 2 [2,4,1,5,6,3] => 3 [2,4,1,6,3,5] => 3 [2,4,1,6,+,3] => 2 [2,4,1,6,-,3] => 2 [2,4,+,1,+,+] => 0 [2,4,-,1,+,+] => 0 [2,4,+,1,-,+] => 0 [2,4,+,1,+,-] => 0 [2,4,-,1,-,+] => 0 [2,4,-,1,+,-] => 0 [2,4,+,1,-,-] => 0 [2,4,-,1,-,-] => 0 [2,4,+,1,6,5] => 0 ----------------------------------------------------------------------------- Created: Jul 22, 2022 at 12:07 by Martin Rubey ----------------------------------------------------------------------------- Last Updated: Jul 22, 2022 at 12:07 by Martin Rubey