***************************************************************************** * 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: St000579 ----------------------------------------------------------------------------- Collection: Set partitions ----------------------------------------------------------------------------- Description: The number of occurrences of the pattern {{1},{2}} such that 2 is a maximal element. This is the number of pairs \$i\lt j\$ in different blocks such that \$j\$ is the maximal element of a block. ----------------------------------------------------------------------------- References: [1] Chern, B., Diaconis, P., Kane, D. M., Rhoades, R. C. Closed expressions for averages of set partition statistics [[MathSciNet:3338726]] [[arXiv:1304.4309]] ----------------------------------------------------------------------------- Code: def statistic(pi): return len(pattern_occurrences(pi, [[1],[2]], [], [2], [], [])) def pattern_occurrences(pi, P, First, Last, Arcs, Consecutives): """We assume that pi is a SetPartition of {1,2,...,n} and P is a SetPartition of {1,2,...,k}. """ occurrences = [] pi = SetPartition(pi) P = SetPartition(P) openers = [min(B) for B in pi] closers = [max(B) for B in pi] pi_sorted = sorted([sorted(b) for b in pi]) edges = [(a,b) for B in pi_sorted for (a,b) in zip(B, B[1:])] for s in Subsets(pi.base_set(), P.size()): s = sorted(s) pi_r = pi.restriction(s) if pi_r.standardization() != P: continue X = pi_r.base_set() if any(X[i-1] not in openers for i in First): continue if any(X[i-1] not in closers for i in Last): continue if any((X[i-1], X[j-1]) not in edges for (i,j) in Arcs): continue if any(abs(X[i-1]-X[j-1]) != 1 for (i,j) in Consecutives): continue occurrences += [s] return occurrences ----------------------------------------------------------------------------- Statistic values: {{1,2}} => 0 {{1},{2}} => 1 {{1,2,3}} => 0 {{1},{2,3}} => 1 {{1,3},{2}} => 2 {{1,2},{3}} => 2 {{1},{2},{3}} => 3 {{1,2,3,4}} => 0 {{1},{2,3,4}} => 1 {{1,3,4},{2}} => 2 {{1,2,4},{3}} => 3 {{1,2,3},{4}} => 3 {{1,2},{3,4}} => 2 {{1,3},{2,4}} => 3 {{1,4},{2,3}} => 3 {{1},{2},{3,4}} => 3 {{1},{2,4},{3}} => 4 {{1},{2,3},{4}} => 4 {{1,4},{2},{3}} => 5 {{1,3},{2},{4}} => 5 {{1,2},{3},{4}} => 5 {{1},{2},{3},{4}} => 6 {{1,2,3,4,5}} => 0 {{1},{2,3,4,5}} => 1 {{1,3,4,5},{2}} => 2 {{1,2,4,5},{3}} => 3 {{1,2,3,5},{4}} => 4 {{1,2,3,4},{5}} => 4 {{1,2},{3,4,5}} => 2 {{1,3},{2,4,5}} => 3 {{1,4},{2,3,5}} => 4 {{1,5},{2,3,4}} => 4 {{1,4,5},{2,3}} => 3 {{1,3,5},{2,4}} => 4 {{1,3,4},{2,5}} => 4 {{1,2,5},{3,4}} => 4 {{1,2,4},{3,5}} => 4 {{1,2,3},{4,5}} => 3 {{1},{2},{3,4,5}} => 3 {{1},{2,4,5},{3}} => 4 {{1},{2,3,5},{4}} => 5 {{1},{2,3,4},{5}} => 5 {{1,4,5},{2},{3}} => 5 {{1,3,5},{2},{4}} => 6 {{1,3,4},{2},{5}} => 6 {{1,2,5},{3},{4}} => 7 {{1,2,4},{3},{5}} => 7 {{1,2,3},{4},{5}} => 7 {{1},{2,3},{4,5}} => 4 {{1},{2,4},{3,5}} => 5 {{1},{2,5},{3,4}} => 5 {{1,3},{2},{4,5}} => 5 {{1,4},{2},{3,5}} => 6 {{1,5},{2},{3,4}} => 6 {{1,2},{3},{4,5}} => 5 {{1,4},{2,5},{3}} => 7 {{1,5},{2,4},{3}} => 7 {{1,2},{3,5},{4}} => 6 {{1,3},{2,5},{4}} => 7 {{1,5},{2,3},{4}} => 7 {{1,2},{3,4},{5}} => 6 {{1,3},{2,4},{5}} => 7 {{1,4},{2,3},{5}} => 7 {{1},{2},{3},{4,5}} => 6 {{1},{2},{3,5},{4}} => 7 {{1},{2},{3,4},{5}} => 7 {{1},{2,5},{3},{4}} => 8 {{1},{2,4},{3},{5}} => 8 {{1},{2,3},{4},{5}} => 8 {{1,5},{2},{3},{4}} => 9 {{1,4},{2},{3},{5}} => 9 {{1,3},{2},{4},{5}} => 9 {{1,2},{3},{4},{5}} => 9 {{1},{2},{3},{4},{5}} => 10 {{1,2,3,4,5,6}} => 0 {{1},{2,3,4,5,6}} => 1 {{1,3,4,5,6},{2}} => 2 {{1,2,4,5,6},{3}} => 3 {{1,2,3,5,6},{4}} => 4 {{1,2,3,4,6},{5}} => 5 {{1,2,3,4,5},{6}} => 5 {{1,2},{3,4,5,6}} => 2 {{1,3},{2,4,5,6}} => 3 {{1,4},{2,3,5,6}} => 4 {{1,5},{2,3,4,6}} => 5 {{1,6},{2,3,4,5}} => 5 {{1,4,5,6},{2,3}} => 3 {{1,3,5,6},{2,4}} => 4 {{1,3,4,6},{2,5}} => 5 {{1,3,4,5},{2,6}} => 5 {{1,2,5,6},{3,4}} => 4 {{1,2,4,6},{3,5}} => 5 {{1,2,4,5},{3,6}} => 5 {{1,2,3,6},{4,5}} => 5 {{1,2,3,5},{4,6}} => 5 {{1,2,3,4},{5,6}} => 4 {{1},{2},{3,4,5,6}} => 3 {{1},{2,4,5,6},{3}} => 4 {{1},{2,3,5,6},{4}} => 5 {{1},{2,3,4,6},{5}} => 6 {{1},{2,3,4,5},{6}} => 6 {{1,4,5,6},{2},{3}} => 5 {{1,3,5,6},{2},{4}} => 6 {{1,3,4,6},{2},{5}} => 7 {{1,3,4,5},{2},{6}} => 7 {{1,2,5,6},{3},{4}} => 7 {{1,2,4,6},{3},{5}} => 8 {{1,2,4,5},{3},{6}} => 8 {{1,2,3,6},{4},{5}} => 9 {{1,2,3,5},{4},{6}} => 9 {{1,2,3,4},{5},{6}} => 9 {{1,2,3},{4,5,6}} => 3 {{1,2,4},{3,5,6}} => 4 {{1,2,5},{3,4,6}} => 5 {{1,2,6},{3,4,5}} => 5 {{1,3,4},{2,5,6}} => 4 {{1,3,5},{2,4,6}} => 5 {{1,3,6},{2,4,5}} => 5 {{1,4,5},{2,3,6}} => 5 {{1,4,6},{2,3,5}} => 5 {{1,5,6},{2,3,4}} => 4 {{1},{2,3},{4,5,6}} => 4 {{1},{2,4},{3,5,6}} => 5 {{1},{2,5},{3,4,6}} => 6 {{1},{2,6},{3,4,5}} => 6 {{1},{2,5,6},{3,4}} => 5 {{1},{2,4,6},{3,5}} => 6 {{1},{2,4,5},{3,6}} => 6 {{1},{2,3,6},{4,5}} => 6 {{1},{2,3,5},{4,6}} => 6 {{1},{2,3,4},{5,6}} => 5 {{1,3},{2},{4,5,6}} => 5 {{1,4},{2},{3,5,6}} => 6 {{1,5},{2},{3,4,6}} => 7 {{1,6},{2},{3,4,5}} => 7 {{1,2},{3},{4,5,6}} => 5 {{1,2},{3,5,6},{4}} => 6 {{1,2},{3,4,6},{5}} => 7 {{1,2},{3,4,5},{6}} => 7 {{1,4},{2,5,6},{3}} => 7 {{1,5},{2,4,6},{3}} => 8 {{1,6},{2,4,5},{3}} => 8 {{1,3},{2,5,6},{4}} => 7 {{1,3},{2,4,6},{5}} => 8 {{1,3},{2,4,5},{6}} => 8 {{1,5},{2,3,6},{4}} => 9 {{1,6},{2,3,5},{4}} => 9 {{1,4},{2,3,6},{5}} => 9 {{1,4},{2,3,5},{6}} => 9 {{1,6},{2,3,4},{5}} => 9 {{1,5},{2,3,4},{6}} => 9 {{1,5,6},{2},{3,4}} => 6 {{1,4,6},{2},{3,5}} => 7 {{1,4,5},{2},{3,6}} => 7 {{1,3,6},{2},{4,5}} => 7 {{1,3,5},{2},{4,6}} => 7 {{1,3,4},{2},{5,6}} => 6 {{1,5,6},{2,4},{3}} => 7 {{1,4,6},{2,5},{3}} => 8 {{1,4,5},{2,6},{3}} => 8 {{1,5,6},{2,3},{4}} => 7 {{1,4,6},{2,3},{5}} => 8 {{1,4,5},{2,3},{6}} => 8 {{1,3,6},{2,5},{4}} => 9 {{1,3,5},{2,6},{4}} => 9 {{1,3,6},{2,4},{5}} => 9 {{1,3,5},{2,4},{6}} => 9 {{1,3,4},{2,6},{5}} => 9 {{1,3,4},{2,5},{6}} => 9 {{1,2,6},{3},{4,5}} => 8 {{1,2,5},{3},{4,6}} => 8 {{1,2,4},{3},{5,6}} => 7 {{1,2,6},{3,5},{4}} => 9 {{1,2,5},{3,6},{4}} => 9 {{1,2,6},{3,4},{5}} => 9 {{1,2,5},{3,4},{6}} => 9 {{1,2,4},{3,6},{5}} => 9 {{1,2,4},{3,5},{6}} => 9 {{1,2,3},{4},{5,6}} => 7 {{1,2,3},{4,6},{5}} => 8 {{1,2,3},{4,5},{6}} => 8 {{1},{2},{3},{4,5,6}} => 6 {{1},{2},{3,5,6},{4}} => 7 {{1},{2},{3,4,6},{5}} => 8 {{1},{2},{3,4,5},{6}} => 8 {{1},{2,5,6},{3},{4}} => 8 {{1},{2,4,6},{3},{5}} => 9 {{1},{2,4,5},{3},{6}} => 9 {{1},{2,3,6},{4},{5}} => 10 {{1},{2,3,5},{4},{6}} => 10 {{1},{2,3,4},{5},{6}} => 10 {{1,5,6},{2},{3},{4}} => 9 {{1,4,6},{2},{3},{5}} => 10 {{1,4,5},{2},{3},{6}} => 10 {{1,3,6},{2},{4},{5}} => 11 {{1,3,5},{2},{4},{6}} => 11 {{1,3,4},{2},{5},{6}} => 11 {{1,2,6},{3},{4},{5}} => 12 {{1,2,5},{3},{4},{6}} => 12 {{1,2,4},{3},{5},{6}} => 12 {{1,2,3},{4},{5},{6}} => 12 {{1,2},{3,4},{5,6}} => 6 {{1,2},{3,5},{4,6}} => 7 {{1,2},{3,6},{4,5}} => 7 {{1,3},{2,4},{5,6}} => 7 {{1,3},{2,5},{4,6}} => 8 {{1,3},{2,6},{4,5}} => 8 {{1,4},{2,3},{5,6}} => 7 {{1,4},{2,5},{3,6}} => 9 {{1,4},{2,6},{3,5}} => 9 {{1,5},{2,3},{4,6}} => 8 {{1,5},{2,4},{3,6}} => 9 {{1,5},{2,6},{3,4}} => 9 {{1,6},{2,3},{4,5}} => 8 {{1,6},{2,4},{3,5}} => 9 {{1,6},{2,5},{3,4}} => 9 {{1},{2},{3,4},{5,6}} => 7 {{1},{2},{3,5},{4,6}} => 8 {{1},{2},{3,6},{4,5}} => 8 {{1},{2,4},{3},{5,6}} => 8 {{1},{2,5},{3},{4,6}} => 9 {{1},{2,6},{3},{4,5}} => 9 {{1},{2,3},{4},{5,6}} => 8 {{1},{2,5},{3,6},{4}} => 10 {{1},{2,6},{3,5},{4}} => 10 {{1},{2,3},{4,6},{5}} => 9 {{1},{2,4},{3,6},{5}} => 10 {{1},{2,6},{3,4},{5}} => 10 {{1},{2,3},{4,5},{6}} => 9 {{1},{2,4},{3,5},{6}} => 10 {{1},{2,5},{3,4},{6}} => 10 {{1,4},{2},{3},{5,6}} => 9 {{1,5},{2},{3},{4,6}} => 10 {{1,6},{2},{3},{4,5}} => 10 {{1,3},{2},{4},{5,6}} => 9 {{1,5},{2},{3,6},{4}} => 11 {{1,6},{2},{3,5},{4}} => 11 {{1,3},{2},{4,6},{5}} => 10 {{1,4},{2},{3,6},{5}} => 11 {{1,6},{2},{3,4},{5}} => 11 {{1,3},{2},{4,5},{6}} => 10 {{1,4},{2},{3,5},{6}} => 11 {{1,5},{2},{3,4},{6}} => 11 {{1,2},{3},{4},{5,6}} => 9 {{1,5},{2,6},{3},{4}} => 12 {{1,6},{2,5},{3},{4}} => 12 {{1,2},{3},{4,6},{5}} => 10 {{1,4},{2,6},{3},{5}} => 12 {{1,6},{2,4},{3},{5}} => 12 {{1,2},{3},{4,5},{6}} => 10 {{1,4},{2,5},{3},{6}} => 12 {{1,5},{2,4},{3},{6}} => 12 {{1,2},{3,6},{4},{5}} => 11 {{1,3},{2,6},{4},{5}} => 12 {{1,6},{2,3},{4},{5}} => 12 {{1,2},{3,5},{4},{6}} => 11 {{1,3},{2,5},{4},{6}} => 12 {{1,5},{2,3},{4},{6}} => 12 {{1,2},{3,4},{5},{6}} => 11 {{1,3},{2,4},{5},{6}} => 12 {{1,4},{2,3},{5},{6}} => 12 {{1},{2},{3},{4},{5,6}} => 10 {{1},{2},{3},{4,6},{5}} => 11 {{1},{2},{3},{4,5},{6}} => 11 {{1},{2},{3,6},{4},{5}} => 12 {{1},{2},{3,5},{4},{6}} => 12 {{1},{2},{3,4},{5},{6}} => 12 {{1},{2,6},{3},{4},{5}} => 13 {{1},{2,5},{3},{4},{6}} => 13 {{1},{2,4},{3},{5},{6}} => 13 {{1},{2,3},{4},{5},{6}} => 13 {{1,6},{2},{3},{4},{5}} => 14 {{1,5},{2},{3},{4},{6}} => 14 {{1,4},{2},{3},{5},{6}} => 14 {{1,3},{2},{4},{5},{6}} => 14 {{1,2},{3},{4},{5},{6}} => 14 {{1},{2},{3},{4},{5},{6}} => 15 {{1,2,3,4,5,6,7}} => 0 {{1},{2,3,4,5,6,7}} => 1 {{1,3,4,5,6,7},{2}} => 2 {{1,2,4,5,6,7},{3}} => 3 {{1,2,3,5,6,7},{4}} => 4 {{1,2,3,4,6,7},{5}} => 5 {{1,2,3,4,5,7},{6}} => 6 {{1,2,3,4,5,6},{7}} => 6 {{1,2},{3,4,5,6,7}} => 2 {{1,3},{2,4,5,6,7}} => 3 {{1,4},{2,3,5,6,7}} => 4 {{1,5},{2,3,4,6,7}} => 5 {{1,6},{2,3,4,5,7}} => 6 {{1,7},{2,3,4,5,6}} => 6 {{1,4,5,6,7},{2,3}} => 3 {{1,3,5,6,7},{2,4}} => 4 {{1,3,4,6,7},{2,5}} => 5 {{1,3,4,5,7},{2,6}} => 6 {{1,3,4,5,6},{2,7}} => 6 {{1,2,5,6,7},{3,4}} => 4 {{1,2,4,6,7},{3,5}} => 5 {{1,2,4,5,7},{3,6}} => 6 {{1,2,4,5,6},{3,7}} => 6 {{1,2,3,6,7},{4,5}} => 5 {{1,2,3,5,7},{4,6}} => 6 {{1,2,3,5,6},{4,7}} => 6 {{1,2,3,4,7},{5,6}} => 6 {{1,2,3,4,6},{5,7}} => 6 {{1,2,3,4,5},{6,7}} => 5 {{1},{2},{3,4,5,6,7}} => 3 {{1},{2,4,5,6,7},{3}} => 4 {{1},{2,3,5,6,7},{4}} => 5 {{1},{2,3,4,6,7},{5}} => 6 {{1},{2,3,4,5,7},{6}} => 7 {{1},{2,3,4,5,6},{7}} => 7 {{1,4,5,6,7},{2},{3}} => 5 {{1,3,5,6,7},{2},{4}} => 6 {{1,3,4,6,7},{2},{5}} => 7 {{1,3,4,5,7},{2},{6}} => 8 {{1,3,4,5,6},{2},{7}} => 8 {{1,2,5,6,7},{3},{4}} => 7 {{1,2,4,6,7},{3},{5}} => 8 {{1,2,4,5,7},{3},{6}} => 9 {{1,2,4,5,6},{3},{7}} => 9 {{1,2,3,6,7},{4},{5}} => 9 {{1,2,3,5,7},{4},{6}} => 10 {{1,2,3,5,6},{4},{7}} => 10 {{1,2,3,4,7},{5},{6}} => 11 {{1,2,3,4,6},{5},{7}} => 11 {{1,2,3,4,5},{6},{7}} => 11 {{1,2,3},{4,5,6,7}} => 3 {{1,2,4},{3,5,6,7}} => 4 {{1,2,5},{3,4,6,7}} => 5 {{1,2,6},{3,4,5,7}} => 6 {{1,2,7},{3,4,5,6}} => 6 {{1,3,4},{2,5,6,7}} => 4 {{1,3,5},{2,4,6,7}} => 5 {{1,3,6},{2,4,5,7}} => 6 {{1,3,7},{2,4,5,6}} => 6 {{1,4,5},{2,3,6,7}} => 5 {{1,4,6},{2,3,5,7}} => 6 {{1,4,7},{2,3,5,6}} => 6 {{1,5,6},{2,3,4,7}} => 6 {{1,5,7},{2,3,4,6}} => 6 {{1,6,7},{2,3,4,5}} => 5 {{1,5,6,7},{2,3,4}} => 4 {{1,4,6,7},{2,3,5}} => 5 {{1,4,5,7},{2,3,6}} => 6 {{1,4,5,6},{2,3,7}} => 6 {{1,3,6,7},{2,4,5}} => 5 {{1,3,5,7},{2,4,6}} => 6 {{1,3,5,6},{2,4,7}} => 6 {{1,3,4,7},{2,5,6}} => 6 {{1,3,4,6},{2,5,7}} => 6 {{1,3,4,5},{2,6,7}} => 5 {{1,2,6,7},{3,4,5}} => 5 {{1,2,5,7},{3,4,6}} => 6 {{1,2,5,6},{3,4,7}} => 6 {{1,2,4,7},{3,5,6}} => 6 {{1,2,4,6},{3,5,7}} => 6 {{1,2,4,5},{3,6,7}} => 5 {{1,2,3,7},{4,5,6}} => 6 {{1,2,3,6},{4,5,7}} => 6 {{1,2,3,5},{4,6,7}} => 5 {{1,2,3,4},{5,6,7}} => 4 {{1},{2,3},{4,5,6,7}} => 4 {{1},{2,4},{3,5,6,7}} => 5 {{1},{2,5},{3,4,6,7}} => 6 {{1},{2,6},{3,4,5,7}} => 7 {{1},{2,7},{3,4,5,6}} => 7 {{1},{2,5,6,7},{3,4}} => 5 {{1},{2,4,6,7},{3,5}} => 6 {{1},{2,4,5,7},{3,6}} => 7 {{1},{2,4,5,6},{3,7}} => 7 {{1},{2,3,6,7},{4,5}} => 6 {{1},{2,3,5,7},{4,6}} => 7 {{1},{2,3,5,6},{4,7}} => 7 {{1},{2,3,4,7},{5,6}} => 7 {{1},{2,3,4,6},{5,7}} => 7 {{1},{2,3,4,5},{6,7}} => 6 {{1,3},{2},{4,5,6,7}} => 5 {{1,4},{2},{3,5,6,7}} => 6 {{1,5},{2},{3,4,6,7}} => 7 {{1,6},{2},{3,4,5,7}} => 8 {{1,7},{2},{3,4,5,6}} => 8 {{1,2},{3},{4,5,6,7}} => 5 {{1,2},{3,5,6,7},{4}} => 6 {{1,2},{3,4,6,7},{5}} => 7 {{1,2},{3,4,5,7},{6}} => 8 {{1,2},{3,4,5,6},{7}} => 8 {{1,4},{2,5,6,7},{3}} => 7 {{1,5},{2,4,6,7},{3}} => 8 {{1,6},{2,4,5,7},{3}} => 9 {{1,7},{2,4,5,6},{3}} => 9 {{1,3},{2,5,6,7},{4}} => 7 {{1,3},{2,4,6,7},{5}} => 8 {{1,3},{2,4,5,7},{6}} => 9 {{1,3},{2,4,5,6},{7}} => 9 {{1,5},{2,3,6,7},{4}} => 9 {{1,6},{2,3,5,7},{4}} => 10 {{1,7},{2,3,5,6},{4}} => 10 {{1,4},{2,3,6,7},{5}} => 9 {{1,4},{2,3,5,7},{6}} => 10 {{1,4},{2,3,5,6},{7}} => 10 {{1,6},{2,3,4,7},{5}} => 11 {{1,7},{2,3,4,6},{5}} => 11 {{1,5},{2,3,4,7},{6}} => 11 {{1,5},{2,3,4,6},{7}} => 11 {{1,7},{2,3,4,5},{6}} => 11 {{1,6},{2,3,4,5},{7}} => 11 {{1,5,6,7},{2},{3,4}} => 6 {{1,4,6,7},{2},{3,5}} => 7 {{1,4,5,7},{2},{3,6}} => 8 {{1,4,5,6},{2},{3,7}} => 8 {{1,3,6,7},{2},{4,5}} => 7 {{1,3,5,7},{2},{4,6}} => 8 {{1,3,5,6},{2},{4,7}} => 8 {{1,3,4,7},{2},{5,6}} => 8 {{1,3,4,6},{2},{5,7}} => 8 {{1,3,4,5},{2},{6,7}} => 7 {{1,5,6,7},{2,4},{3}} => 7 {{1,4,6,7},{2,5},{3}} => 8 {{1,4,5,7},{2,6},{3}} => 9 {{1,4,5,6},{2,7},{3}} => 9 {{1,5,6,7},{2,3},{4}} => 7 {{1,4,6,7},{2,3},{5}} => 8 {{1,4,5,7},{2,3},{6}} => 9 {{1,4,5,6},{2,3},{7}} => 9 {{1,3,6,7},{2,5},{4}} => 9 {{1,3,5,7},{2,6},{4}} => 10 {{1,3,5,6},{2,7},{4}} => 10 {{1,3,6,7},{2,4},{5}} => 9 {{1,3,5,7},{2,4},{6}} => 10 {{1,3,5,6},{2,4},{7}} => 10 {{1,3,4,7},{2,6},{5}} => 11 {{1,3,4,6},{2,7},{5}} => 11 {{1,3,4,7},{2,5},{6}} => 11 {{1,3,4,6},{2,5},{7}} => 11 {{1,3,4,5},{2,7},{6}} => 11 {{1,3,4,5},{2,6},{7}} => 11 {{1,2,6,7},{3},{4,5}} => 8 {{1,2,5,7},{3},{4,6}} => 9 {{1,2,5,6},{3},{4,7}} => 9 {{1,2,4,7},{3},{5,6}} => 9 {{1,2,4,6},{3},{5,7}} => 9 {{1,2,4,5},{3},{6,7}} => 8 {{1,2,6,7},{3,5},{4}} => 9 {{1,2,5,7},{3,6},{4}} => 10 {{1,2,5,6},{3,7},{4}} => 10 {{1,2,6,7},{3,4},{5}} => 9 {{1,2,5,7},{3,4},{6}} => 10 {{1,2,5,6},{3,4},{7}} => 10 {{1,2,4,7},{3,6},{5}} => 11 {{1,2,4,6},{3,7},{5}} => 11 {{1,2,4,7},{3,5},{6}} => 11 {{1,2,4,6},{3,5},{7}} => 11 {{1,2,4,5},{3,7},{6}} => 11 {{1,2,4,5},{3,6},{7}} => 11 {{1,2,3,7},{4},{5,6}} => 10 {{1,2,3,6},{4},{5,7}} => 10 {{1,2,3,5},{4},{6,7}} => 9 {{1,2,3,7},{4,6},{5}} => 11 {{1,2,3,6},{4,7},{5}} => 11 {{1,2,3,7},{4,5},{6}} => 11 {{1,2,3,6},{4,5},{7}} => 11 {{1,2,3,5},{4,7},{6}} => 11 {{1,2,3,5},{4,6},{7}} => 11 {{1,2,3,4},{5},{6,7}} => 9 {{1,2,3,4},{5,7},{6}} => 10 {{1,2,3,4},{5,6},{7}} => 10 {{1},{2},{3},{4,5,6,7}} => 6 {{1},{2},{3,5,6,7},{4}} => 7 {{1},{2},{3,4,6,7},{5}} => 8 {{1},{2},{3,4,5,7},{6}} => 9 {{1},{2},{3,4,5,6},{7}} => 9 {{1},{2,5,6,7},{3},{4}} => 8 {{1},{2,4,6,7},{3},{5}} => 9 {{1},{2,4,5,7},{3},{6}} => 10 {{1},{2,4,5,6},{3},{7}} => 10 {{1},{2,3,6,7},{4},{5}} => 10 {{1},{2,3,5,7},{4},{6}} => 11 {{1},{2,3,5,6},{4},{7}} => 11 {{1},{2,3,4,7},{5},{6}} => 12 {{1},{2,3,4,6},{5},{7}} => 12 {{1},{2,3,4,5},{6},{7}} => 12 {{1,5,6,7},{2},{3},{4}} => 9 {{1,4,6,7},{2},{3},{5}} => 10 {{1,4,5,7},{2},{3},{6}} => 11 {{1,4,5,6},{2},{3},{7}} => 11 {{1,3,6,7},{2},{4},{5}} => 11 {{1,3,5,7},{2},{4},{6}} => 12 {{1,3,5,6},{2},{4},{7}} => 12 {{1,3,4,7},{2},{5},{6}} => 13 {{1,3,4,6},{2},{5},{7}} => 13 {{1,3,4,5},{2},{6},{7}} => 13 {{1,2,6,7},{3},{4},{5}} => 12 {{1,2,5,7},{3},{4},{6}} => 13 {{1,2,5,6},{3},{4},{7}} => 13 {{1,2,4,7},{3},{5},{6}} => 14 {{1,2,4,6},{3},{5},{7}} => 14 {{1,2,4,5},{3},{6},{7}} => 14 {{1,2,3,7},{4},{5},{6}} => 15 {{1,2,3,6},{4},{5},{7}} => 15 {{1,2,3,5},{4},{6},{7}} => 15 {{1,2,3,4},{5},{6},{7}} => 15 {{1},{2,3,4},{5,6,7}} => 5 {{1},{2,3,5},{4,6,7}} => 6 {{1},{2,3,6},{4,5,7}} => 7 {{1},{2,3,7},{4,5,6}} => 7 {{1},{2,4,5},{3,6,7}} => 6 {{1},{2,4,6},{3,5,7}} => 7 {{1},{2,4,7},{3,5,6}} => 7 {{1},{2,5,6},{3,4,7}} => 7 {{1},{2,5,7},{3,4,6}} => 7 {{1},{2,6,7},{3,4,5}} => 6 {{1,3,4},{2},{5,6,7}} => 6 {{1,3,5},{2},{4,6,7}} => 7 {{1,3,6},{2},{4,5,7}} => 8 {{1,3,7},{2},{4,5,6}} => 8 {{1,4,5},{2},{3,6,7}} => 7 {{1,4,6},{2},{3,5,7}} => 8 {{1,4,7},{2},{3,5,6}} => 8 {{1,5,6},{2},{3,4,7}} => 8 {{1,5,7},{2},{3,4,6}} => 8 {{1,6,7},{2},{3,4,5}} => 7 {{1,2,4},{3},{5,6,7}} => 7 {{1,2,5},{3},{4,6,7}} => 8 {{1,2,6},{3},{4,5,7}} => 9 {{1,2,7},{3},{4,5,6}} => 9 {{1,4,5},{2,6,7},{3}} => 8 {{1,4,6},{2,5,7},{3}} => 9 {{1,4,7},{2,5,6},{3}} => 9 {{1,5,6},{2,4,7},{3}} => 9 {{1,5,7},{2,4,6},{3}} => 9 {{1,6,7},{2,4,5},{3}} => 8 {{1,2,3},{4},{5,6,7}} => 7 {{1,2,5},{3,6,7},{4}} => 9 {{1,2,6},{3,5,7},{4}} => 10 {{1,2,7},{3,5,6},{4}} => 10 {{1,3,5},{2,6,7},{4}} => 9 {{1,3,6},{2,5,7},{4}} => 10 {{1,3,7},{2,5,6},{4}} => 10 {{1,5,6},{2,3,7},{4}} => 10 {{1,5,7},{2,3,6},{4}} => 10 {{1,6,7},{2,3,5},{4}} => 9 {{1,2,3},{4,6,7},{5}} => 8 {{1,2,4},{3,6,7},{5}} => 9 {{1,2,6},{3,4,7},{5}} => 11 {{1,2,7},{3,4,6},{5}} => 11 {{1,3,4},{2,6,7},{5}} => 9 {{1,3,6},{2,4,7},{5}} => 11 {{1,3,7},{2,4,6},{5}} => 11 {{1,4,6},{2,3,7},{5}} => 11 {{1,4,7},{2,3,6},{5}} => 11 {{1,6,7},{2,3,4},{5}} => 9 {{1,2,3},{4,5,7},{6}} => 9 {{1,2,4},{3,5,7},{6}} => 10 {{1,2,5},{3,4,7},{6}} => 11 {{1,2,7},{3,4,5},{6}} => 11 {{1,3,4},{2,5,7},{6}} => 10 {{1,3,5},{2,4,7},{6}} => 11 {{1,3,7},{2,4,5},{6}} => 11 {{1,4,5},{2,3,7},{6}} => 11 {{1,4,7},{2,3,5},{6}} => 11 {{1,5,7},{2,3,4},{6}} => 10 {{1,2,3},{4,5,6},{7}} => 9 {{1,2,4},{3,5,6},{7}} => 10 {{1,2,5},{3,4,6},{7}} => 11 {{1,2,6},{3,4,5},{7}} => 11 {{1,3,4},{2,5,6},{7}} => 10 {{1,3,5},{2,4,6},{7}} => 11 {{1,3,6},{2,4,5},{7}} => 11 {{1,4,5},{2,3,6},{7}} => 11 {{1,4,6},{2,3,5},{7}} => 11 {{1,5,6},{2,3,4},{7}} => 10 {{1,2},{3,4},{5,6,7}} => 6 {{1,3},{2,4},{5,6,7}} => 7 {{1,4},{2,3},{5,6,7}} => 7 {{1,2},{3,5},{4,6,7}} => 7 {{1,3},{2,5},{4,6,7}} => 8 {{1,5},{2,3},{4,6,7}} => 8 {{1,2},{3,6},{4,5,7}} => 8 {{1,3},{2,6},{4,5,7}} => 9 {{1,6},{2,3},{4,5,7}} => 9 {{1,2},{3,7},{4,5,6}} => 8 {{1,3},{2,7},{4,5,6}} => 9 {{1,7},{2,3},{4,5,6}} => 9 {{1,2},{3,6,7},{4,5}} => 7 {{1,4},{2,5},{3,6,7}} => 9 {{1,5},{2,4},{3,6,7}} => 9 {{1,2},{3,5,7},{4,6}} => 8 {{1,4},{2,6},{3,5,7}} => 10 {{1,6},{2,4},{3,5,7}} => 10 {{1,2},{3,5,6},{4,7}} => 8 {{1,4},{2,7},{3,5,6}} => 10 {{1,7},{2,4},{3,5,6}} => 10 {{1,2},{3,4,7},{5,6}} => 8 {{1,5},{2,6},{3,4,7}} => 11 {{1,6},{2,5},{3,4,7}} => 11 {{1,2},{3,4,6},{5,7}} => 8 {{1,5},{2,7},{3,4,6}} => 11 {{1,7},{2,5},{3,4,6}} => 11 {{1,2},{3,4,5},{6,7}} => 7 {{1,6},{2,7},{3,4,5}} => 11 {{1,7},{2,6},{3,4,5}} => 11 {{1,3},{2,6,7},{4,5}} => 8 {{1,4},{2,6,7},{3,5}} => 9 {{1,5},{2,6,7},{3,4}} => 9 {{1,3},{2,5,7},{4,6}} => 9 {{1,4},{2,5,7},{3,6}} => 10 {{1,6},{2,5,7},{3,4}} => 10 {{1,3},{2,5,6},{4,7}} => 9 {{1,4},{2,5,6},{3,7}} => 10 {{1,7},{2,5,6},{3,4}} => 10 {{1,3},{2,4,7},{5,6}} => 9 {{1,5},{2,4,7},{3,6}} => 11 {{1,6},{2,4,7},{3,5}} => 11 {{1,3},{2,4,6},{5,7}} => 9 {{1,5},{2,4,6},{3,7}} => 11 {{1,7},{2,4,6},{3,5}} => 11 {{1,3},{2,4,5},{6,7}} => 8 {{1,6},{2,4,5},{3,7}} => 11 {{1,7},{2,4,5},{3,6}} => 11 {{1,4},{2,3,7},{5,6}} => 10 {{1,5},{2,3,7},{4,6}} => 11 {{1,6},{2,3,7},{4,5}} => 11 {{1,4},{2,3,6},{5,7}} => 10 {{1,5},{2,3,6},{4,7}} => 11 {{1,7},{2,3,6},{4,5}} => 11 {{1,4},{2,3,5},{6,7}} => 9 {{1,6},{2,3,5},{4,7}} => 11 {{1,7},{2,3,5},{4,6}} => 11 {{1,5},{2,3,4},{6,7}} => 9 {{1,6},{2,3,4},{5,7}} => 10 {{1,7},{2,3,4},{5,6}} => 10 {{1,6,7},{2,3},{4,5}} => 8 {{1,6,7},{2,4},{3,5}} => 9 {{1,6,7},{2,5},{3,4}} => 9 {{1,5,7},{2,3},{4,6}} => 9 {{1,5,7},{2,4},{3,6}} => 10 {{1,5,7},{2,6},{3,4}} => 10 {{1,5,6},{2,3},{4,7}} => 9 {{1,5,6},{2,4},{3,7}} => 10 {{1,5,6},{2,7},{3,4}} => 10 {{1,4,7},{2,3},{5,6}} => 9 {{1,4,7},{2,5},{3,6}} => 11 {{1,4,7},{2,6},{3,5}} => 11 {{1,4,6},{2,3},{5,7}} => 9 {{1,4,6},{2,5},{3,7}} => 11 {{1,4,6},{2,7},{3,5}} => 11 {{1,4,5},{2,3},{6,7}} => 8 {{1,4,5},{2,6},{3,7}} => 11 {{1,4,5},{2,7},{3,6}} => 11 {{1,3,7},{2,4},{5,6}} => 10 {{1,3,7},{2,5},{4,6}} => 11 {{1,3,7},{2,6},{4,5}} => 11 {{1,3,6},{2,4},{5,7}} => 10 {{1,3,6},{2,5},{4,7}} => 11 {{1,3,6},{2,7},{4,5}} => 11 {{1,3,5},{2,4},{6,7}} => 9 {{1,3,5},{2,6},{4,7}} => 11 {{1,3,5},{2,7},{4,6}} => 11 {{1,3,4},{2,5},{6,7}} => 9 {{1,3,4},{2,6},{5,7}} => 10 {{1,3,4},{2,7},{5,6}} => 10 {{1,2,7},{3,4},{5,6}} => 10 {{1,2,7},{3,5},{4,6}} => 11 {{1,2,7},{3,6},{4,5}} => 11 {{1,2,6},{3,4},{5,7}} => 10 {{1,2,6},{3,5},{4,7}} => 11 {{1,2,6},{3,7},{4,5}} => 11 {{1,2,5},{3,4},{6,7}} => 9 {{1,2,5},{3,6},{4,7}} => 11 {{1,2,5},{3,7},{4,6}} => 11 {{1,2,4},{3,5},{6,7}} => 9 {{1,2,4},{3,6},{5,7}} => 10 {{1,2,4},{3,7},{5,6}} => 10 {{1,2,3},{4,5},{6,7}} => 8 {{1,2,3},{4,6},{5,7}} => 9 {{1,2,3},{4,7},{5,6}} => 9 {{1},{2},{3,4},{5,6,7}} => 7 {{1},{2},{3,5},{4,6,7}} => 8 {{1},{2},{3,6},{4,5,7}} => 9 {{1},{2},{3,7},{4,5,6}} => 9 {{1},{2},{3,6,7},{4,5}} => 8 {{1},{2},{3,5,7},{4,6}} => 9 {{1},{2},{3,5,6},{4,7}} => 9 {{1},{2},{3,4,7},{5,6}} => 9 {{1},{2},{3,4,6},{5,7}} => 9 {{1},{2},{3,4,5},{6,7}} => 8 {{1},{2,4},{3},{5,6,7}} => 8 {{1},{2,5},{3},{4,6,7}} => 9 {{1},{2,6},{3},{4,5,7}} => 10 {{1},{2,7},{3},{4,5,6}} => 10 {{1},{2,3},{4},{5,6,7}} => 8 {{1},{2,3},{4,6,7},{5}} => 9 {{1},{2,3},{4,5,7},{6}} => 10 {{1},{2,3},{4,5,6},{7}} => 10 {{1},{2,5},{3,6,7},{4}} => 10 {{1},{2,6},{3,5,7},{4}} => 11 {{1},{2,7},{3,5,6},{4}} => 11 {{1},{2,4},{3,6,7},{5}} => 10 {{1},{2,4},{3,5,7},{6}} => 11 {{1},{2,4},{3,5,6},{7}} => 11 {{1},{2,6},{3,4,7},{5}} => 12 {{1},{2,7},{3,4,6},{5}} => 12 {{1},{2,5},{3,4,7},{6}} => 12 {{1},{2,5},{3,4,6},{7}} => 12 {{1},{2,7},{3,4,5},{6}} => 12 {{1},{2,6},{3,4,5},{7}} => 12 {{1},{2,6,7},{3},{4,5}} => 9 {{1},{2,5,7},{3},{4,6}} => 10 {{1},{2,5,6},{3},{4,7}} => 10 {{1},{2,4,7},{3},{5,6}} => 10 {{1},{2,4,6},{3},{5,7}} => 10 {{1},{2,4,5},{3},{6,7}} => 9 {{1},{2,6,7},{3,5},{4}} => 10 {{1},{2,5,7},{3,6},{4}} => 11 {{1},{2,5,6},{3,7},{4}} => 11 {{1},{2,6,7},{3,4},{5}} => 10 {{1},{2,5,7},{3,4},{6}} => 11 {{1},{2,5,6},{3,4},{7}} => 11 {{1},{2,4,7},{3,6},{5}} => 12 {{1},{2,4,6},{3,7},{5}} => 12 {{1},{2,4,7},{3,5},{6}} => 12 {{1},{2,4,6},{3,5},{7}} => 12 {{1},{2,4,5},{3,7},{6}} => 12 {{1},{2,4,5},{3,6},{7}} => 12 {{1},{2,3,7},{4},{5,6}} => 11 {{1},{2,3,6},{4},{5,7}} => 11 {{1},{2,3,5},{4},{6,7}} => 10 {{1},{2,3,7},{4,6},{5}} => 12 {{1},{2,3,6},{4,7},{5}} => 12 {{1},{2,3,7},{4,5},{6}} => 12 {{1},{2,3,6},{4,5},{7}} => 12 {{1},{2,3,5},{4,7},{6}} => 12 {{1},{2,3,5},{4,6},{7}} => 12 {{1},{2,3,4},{5},{6,7}} => 10 {{1},{2,3,4},{5,7},{6}} => 11 {{1},{2,3,4},{5,6},{7}} => 11 {{1,4},{2},{3},{5,6,7}} => 9 {{1,5},{2},{3},{4,6,7}} => 10 {{1,6},{2},{3},{4,5,7}} => 11 {{1,7},{2},{3},{4,5,6}} => 11 {{1,3},{2},{4},{5,6,7}} => 9 {{1,3},{2},{4,6,7},{5}} => 10 {{1,3},{2},{4,5,7},{6}} => 11 {{1,3},{2},{4,5,6},{7}} => 11 {{1,5},{2},{3,6,7},{4}} => 11 {{1,6},{2},{3,5,7},{4}} => 12 {{1,7},{2},{3,5,6},{4}} => 12 {{1,4},{2},{3,6,7},{5}} => 11 {{1,4},{2},{3,5,7},{6}} => 12 {{1,4},{2},{3,5,6},{7}} => 12 {{1,6},{2},{3,4,7},{5}} => 13 {{1,7},{2},{3,4,6},{5}} => 13 {{1,5},{2},{3,4,7},{6}} => 13 {{1,5},{2},{3,4,6},{7}} => 13 {{1,7},{2},{3,4,5},{6}} => 13 {{1,6},{2},{3,4,5},{7}} => 13 {{1,2},{3},{4},{5,6,7}} => 9 {{1,2},{3},{4,6,7},{5}} => 10 {{1,2},{3},{4,5,7},{6}} => 11 {{1,2},{3},{4,5,6},{7}} => 11 {{1,2},{3,6,7},{4},{5}} => 11 {{1,2},{3,5,7},{4},{6}} => 12 {{1,2},{3,5,6},{4},{7}} => 12 {{1,2},{3,4,7},{5},{6}} => 13 {{1,2},{3,4,6},{5},{7}} => 13 {{1,2},{3,4,5},{6},{7}} => 13 {{1,5},{2,6,7},{3},{4}} => 12 {{1,6},{2,5,7},{3},{4}} => 13 {{1,7},{2,5,6},{3},{4}} => 13 {{1,4},{2,6,7},{3},{5}} => 12 {{1,4},{2,5,7},{3},{6}} => 13 {{1,4},{2,5,6},{3},{7}} => 13 {{1,6},{2,4,7},{3},{5}} => 14 {{1,7},{2,4,6},{3},{5}} => 14 {{1,5},{2,4,7},{3},{6}} => 14 {{1,5},{2,4,6},{3},{7}} => 14 {{1,7},{2,4,5},{3},{6}} => 14 {{1,6},{2,4,5},{3},{7}} => 14 {{1,3},{2,6,7},{4},{5}} => 12 {{1,3},{2,5,7},{4},{6}} => 13 {{1,3},{2,5,6},{4},{7}} => 13 {{1,3},{2,4,7},{5},{6}} => 14 {{1,3},{2,4,6},{5},{7}} => 14 {{1,3},{2,4,5},{6},{7}} => 14 {{1,6},{2,3,7},{4},{5}} => 15 {{1,7},{2,3,6},{4},{5}} => 15 {{1,5},{2,3,7},{4},{6}} => 15 {{1,5},{2,3,6},{4},{7}} => 15 {{1,7},{2,3,5},{4},{6}} => 15 {{1,6},{2,3,5},{4},{7}} => 15 {{1,4},{2,3,7},{5},{6}} => 15 {{1,4},{2,3,6},{5},{7}} => 15 {{1,4},{2,3,5},{6},{7}} => 15 {{1,7},{2,3,4},{5},{6}} => 15 {{1,6},{2,3,4},{5},{7}} => 15 {{1,5},{2,3,4},{6},{7}} => 15 {{1,6,7},{2},{3},{4,5}} => 10 {{1,5,7},{2},{3},{4,6}} => 11 {{1,5,6},{2},{3},{4,7}} => 11 {{1,4,7},{2},{3},{5,6}} => 11 {{1,4,6},{2},{3},{5,7}} => 11 {{1,4,5},{2},{3},{6,7}} => 10 {{1,6,7},{2},{3,5},{4}} => 11 {{1,5,7},{2},{3,6},{4}} => 12 {{1,5,6},{2},{3,7},{4}} => 12 {{1,6,7},{2},{3,4},{5}} => 11 {{1,5,7},{2},{3,4},{6}} => 12 {{1,5,6},{2},{3,4},{7}} => 12 {{1,4,7},{2},{3,6},{5}} => 13 {{1,4,6},{2},{3,7},{5}} => 13 {{1,4,7},{2},{3,5},{6}} => 13 {{1,4,6},{2},{3,5},{7}} => 13 {{1,4,5},{2},{3,7},{6}} => 13 {{1,4,5},{2},{3,6},{7}} => 13 {{1,3,7},{2},{4},{5,6}} => 12 {{1,3,6},{2},{4},{5,7}} => 12 {{1,3,5},{2},{4},{6,7}} => 11 {{1,3,7},{2},{4,6},{5}} => 13 {{1,3,6},{2},{4,7},{5}} => 13 {{1,3,7},{2},{4,5},{6}} => 13 {{1,3,6},{2},{4,5},{7}} => 13 {{1,3,5},{2},{4,7},{6}} => 13 {{1,3,5},{2},{4,6},{7}} => 13 {{1,3,4},{2},{5},{6,7}} => 11 {{1,3,4},{2},{5,7},{6}} => 12 {{1,3,4},{2},{5,6},{7}} => 12 {{1,6,7},{2,5},{3},{4}} => 12 {{1,5,7},{2,6},{3},{4}} => 13 {{1,5,6},{2,7},{3},{4}} => 13 {{1,6,7},{2,4},{3},{5}} => 12 {{1,5,7},{2,4},{3},{6}} => 13 {{1,5,6},{2,4},{3},{7}} => 13 {{1,4,7},{2,6},{3},{5}} => 14 {{1,4,6},{2,7},{3},{5}} => 14 {{1,4,7},{2,5},{3},{6}} => 14 {{1,4,6},{2,5},{3},{7}} => 14 {{1,4,5},{2,7},{3},{6}} => 14 {{1,4,5},{2,6},{3},{7}} => 14 {{1,6,7},{2,3},{4},{5}} => 12 {{1,5,7},{2,3},{4},{6}} => 13 {{1,5,6},{2,3},{4},{7}} => 13 {{1,4,7},{2,3},{5},{6}} => 14 {{1,4,6},{2,3},{5},{7}} => 14 {{1,4,5},{2,3},{6},{7}} => 14 {{1,3,7},{2,6},{4},{5}} => 15 {{1,3,6},{2,7},{4},{5}} => 15 {{1,3,7},{2,5},{4},{6}} => 15 {{1,3,6},{2,5},{4},{7}} => 15 {{1,3,5},{2,7},{4},{6}} => 15 {{1,3,5},{2,6},{4},{7}} => 15 {{1,3,7},{2,4},{5},{6}} => 15 {{1,3,6},{2,4},{5},{7}} => 15 {{1,3,5},{2,4},{6},{7}} => 15 {{1,3,4},{2,7},{5},{6}} => 15 {{1,3,4},{2,6},{5},{7}} => 15 {{1,3,4},{2,5},{6},{7}} => 15 {{1,2,7},{3},{4},{5,6}} => 13 {{1,2,6},{3},{4},{5,7}} => 13 {{1,2,5},{3},{4},{6,7}} => 12 {{1,2,7},{3},{4,6},{5}} => 14 {{1,2,6},{3},{4,7},{5}} => 14 {{1,2,7},{3},{4,5},{6}} => 14 {{1,2,6},{3},{4,5},{7}} => 14 {{1,2,5},{3},{4,7},{6}} => 14 {{1,2,5},{3},{4,6},{7}} => 14 {{1,2,4},{3},{5},{6,7}} => 12 {{1,2,4},{3},{5,7},{6}} => 13 {{1,2,4},{3},{5,6},{7}} => 13 {{1,2,7},{3,6},{4},{5}} => 15 {{1,2,6},{3,7},{4},{5}} => 15 {{1,2,7},{3,5},{4},{6}} => 15 {{1,2,6},{3,5},{4},{7}} => 15 {{1,2,5},{3,7},{4},{6}} => 15 {{1,2,5},{3,6},{4},{7}} => 15 {{1,2,7},{3,4},{5},{6}} => 15 {{1,2,6},{3,4},{5},{7}} => 15 {{1,2,5},{3,4},{6},{7}} => 15 {{1,2,4},{3,7},{5},{6}} => 15 {{1,2,4},{3,6},{5},{7}} => 15 {{1,2,4},{3,5},{6},{7}} => 15 {{1,2,3},{4},{5},{6,7}} => 12 {{1,2,3},{4},{5,7},{6}} => 13 {{1,2,3},{4},{5,6},{7}} => 13 {{1,2,3},{4,7},{5},{6}} => 14 {{1,2,3},{4,6},{5},{7}} => 14 {{1,2,3},{4,5},{6},{7}} => 14 {{1},{2},{3},{4},{5,6,7}} => 10 {{1},{2},{3},{4,6,7},{5}} => 11 {{1},{2},{3},{4,5,7},{6}} => 12 {{1},{2},{3},{4,5,6},{7}} => 12 {{1},{2},{3,6,7},{4},{5}} => 12 {{1},{2},{3,5,7},{4},{6}} => 13 {{1},{2},{3,5,6},{4},{7}} => 13 {{1},{2},{3,4,7},{5},{6}} => 14 {{1},{2},{3,4,6},{5},{7}} => 14 {{1},{2},{3,4,5},{6},{7}} => 14 {{1},{2,6,7},{3},{4},{5}} => 13 {{1},{2,5,7},{3},{4},{6}} => 14 {{1},{2,5,6},{3},{4},{7}} => 14 {{1},{2,4,7},{3},{5},{6}} => 15 {{1},{2,4,6},{3},{5},{7}} => 15 {{1},{2,4,5},{3},{6},{7}} => 15 {{1},{2,3,7},{4},{5},{6}} => 16 {{1},{2,3,6},{4},{5},{7}} => 16 {{1},{2,3,5},{4},{6},{7}} => 16 {{1},{2,3,4},{5},{6},{7}} => 16 {{1,6,7},{2},{3},{4},{5}} => 14 {{1,5,7},{2},{3},{4},{6}} => 15 {{1,5,6},{2},{3},{4},{7}} => 15 {{1,4,7},{2},{3},{5},{6}} => 16 {{1,4,6},{2},{3},{5},{7}} => 16 {{1,4,5},{2},{3},{6},{7}} => 16 {{1,3,7},{2},{4},{5},{6}} => 17 {{1,3,6},{2},{4},{5},{7}} => 17 {{1,3,5},{2},{4},{6},{7}} => 17 {{1,3,4},{2},{5},{6},{7}} => 17 {{1,2,7},{3},{4},{5},{6}} => 18 {{1,2,6},{3},{4},{5},{7}} => 18 {{1,2,5},{3},{4},{6},{7}} => 18 {{1,2,4},{3},{5},{6},{7}} => 18 {{1,2,3},{4},{5},{6},{7}} => 18 {{1},{2,3},{4,5},{6,7}} => 9 {{1},{2,3},{4,6},{5,7}} => 10 {{1},{2,3},{4,7},{5,6}} => 10 {{1},{2,4},{3,5},{6,7}} => 10 {{1},{2,4},{3,6},{5,7}} => 11 {{1},{2,4},{3,7},{5,6}} => 11 {{1},{2,5},{3,4},{6,7}} => 10 {{1},{2,5},{3,6},{4,7}} => 12 {{1},{2,5},{3,7},{4,6}} => 12 {{1},{2,6},{3,4},{5,7}} => 11 {{1},{2,6},{3,5},{4,7}} => 12 {{1},{2,6},{3,7},{4,5}} => 12 {{1},{2,7},{3,4},{5,6}} => 11 {{1},{2,7},{3,5},{4,6}} => 12 {{1},{2,7},{3,6},{4,5}} => 12 {{1,3},{2},{4,5},{6,7}} => 10 {{1,3},{2},{4,6},{5,7}} => 11 {{1,3},{2},{4,7},{5,6}} => 11 {{1,4},{2},{3,5},{6,7}} => 11 {{1,4},{2},{3,6},{5,7}} => 12 {{1,4},{2},{3,7},{5,6}} => 12 {{1,5},{2},{3,4},{6,7}} => 11 {{1,5},{2},{3,6},{4,7}} => 13 {{1,5},{2},{3,7},{4,6}} => 13 {{1,6},{2},{3,4},{5,7}} => 12 {{1,6},{2},{3,5},{4,7}} => 13 {{1,6},{2},{3,7},{4,5}} => 13 {{1,7},{2},{3,4},{5,6}} => 12 {{1,7},{2},{3,5},{4,6}} => 13 {{1,7},{2},{3,6},{4,5}} => 13 {{1,2},{3},{4,5},{6,7}} => 10 {{1,2},{3},{4,6},{5,7}} => 11 {{1,2},{3},{4,7},{5,6}} => 11 {{1,4},{2,5},{3},{6,7}} => 12 {{1,4},{2,6},{3},{5,7}} => 13 {{1,4},{2,7},{3},{5,6}} => 13 {{1,5},{2,4},{3},{6,7}} => 12 {{1,5},{2,6},{3},{4,7}} => 14 {{1,5},{2,7},{3},{4,6}} => 14 {{1,6},{2,4},{3},{5,7}} => 13 {{1,6},{2,5},{3},{4,7}} => 14 {{1,6},{2,7},{3},{4,5}} => 14 {{1,7},{2,4},{3},{5,6}} => 13 {{1,7},{2,5},{3},{4,6}} => 14 {{1,7},{2,6},{3},{4,5}} => 14 {{1,2},{3,5},{4},{6,7}} => 11 {{1,2},{3,6},{4},{5,7}} => 12 {{1,2},{3,7},{4},{5,6}} => 12 {{1,3},{2,5},{4},{6,7}} => 12 {{1,3},{2,6},{4},{5,7}} => 13 {{1,3},{2,7},{4},{5,6}} => 13 {{1,5},{2,3},{4},{6,7}} => 12 {{1,5},{2,6},{3,7},{4}} => 15 {{1,5},{2,7},{3,6},{4}} => 15 {{1,6},{2,3},{4},{5,7}} => 13 {{1,6},{2,5},{3,7},{4}} => 15 {{1,6},{2,7},{3,5},{4}} => 15 {{1,7},{2,3},{4},{5,6}} => 13 {{1,7},{2,5},{3,6},{4}} => 15 {{1,7},{2,6},{3,5},{4}} => 15 {{1,2},{3,4},{5},{6,7}} => 11 {{1,2},{3,6},{4,7},{5}} => 13 {{1,2},{3,7},{4,6},{5}} => 13 {{1,3},{2,4},{5},{6,7}} => 12 {{1,3},{2,6},{4,7},{5}} => 14 {{1,3},{2,7},{4,6},{5}} => 14 {{1,4},{2,3},{5},{6,7}} => 12 {{1,4},{2,6},{3,7},{5}} => 15 {{1,4},{2,7},{3,6},{5}} => 15 {{1,6},{2,3},{4,7},{5}} => 14 {{1,6},{2,4},{3,7},{5}} => 15 {{1,6},{2,7},{3,4},{5}} => 15 {{1,7},{2,3},{4,6},{5}} => 14 {{1,7},{2,4},{3,6},{5}} => 15 {{1,7},{2,6},{3,4},{5}} => 15 {{1,2},{3,4},{5,7},{6}} => 12 {{1,2},{3,5},{4,7},{6}} => 13 {{1,2},{3,7},{4,5},{6}} => 13 {{1,3},{2,4},{5,7},{6}} => 13 {{1,3},{2,5},{4,7},{6}} => 14 {{1,3},{2,7},{4,5},{6}} => 14 {{1,4},{2,3},{5,7},{6}} => 13 {{1,4},{2,5},{3,7},{6}} => 15 {{1,4},{2,7},{3,5},{6}} => 15 {{1,5},{2,3},{4,7},{6}} => 14 {{1,5},{2,4},{3,7},{6}} => 15 {{1,5},{2,7},{3,4},{6}} => 15 {{1,7},{2,3},{4,5},{6}} => 14 {{1,7},{2,4},{3,5},{6}} => 15 {{1,7},{2,5},{3,4},{6}} => 15 {{1,2},{3,4},{5,6},{7}} => 12 {{1,2},{3,5},{4,6},{7}} => 13 {{1,2},{3,6},{4,5},{7}} => 13 {{1,3},{2,4},{5,6},{7}} => 13 {{1,3},{2,5},{4,6},{7}} => 14 {{1,3},{2,6},{4,5},{7}} => 14 {{1,4},{2,3},{5,6},{7}} => 13 {{1,4},{2,5},{3,6},{7}} => 15 {{1,4},{2,6},{3,5},{7}} => 15 {{1,5},{2,3},{4,6},{7}} => 14 {{1,5},{2,4},{3,6},{7}} => 15 {{1,5},{2,6},{3,4},{7}} => 15 {{1,6},{2,3},{4,5},{7}} => 14 {{1,6},{2,4},{3,5},{7}} => 15 {{1,6},{2,5},{3,4},{7}} => 15 {{1},{2},{3},{4,5},{6,7}} => 11 {{1},{2},{3},{4,6},{5,7}} => 12 {{1},{2},{3},{4,7},{5,6}} => 12 {{1},{2},{3,5},{4},{6,7}} => 12 {{1},{2},{3,6},{4},{5,7}} => 13 {{1},{2},{3,7},{4},{5,6}} => 13 {{1},{2},{3,4},{5},{6,7}} => 12 {{1},{2},{3,6},{4,7},{5}} => 14 {{1},{2},{3,7},{4,6},{5}} => 14 {{1},{2},{3,4},{5,7},{6}} => 13 {{1},{2},{3,5},{4,7},{6}} => 14 {{1},{2},{3,7},{4,5},{6}} => 14 {{1},{2},{3,4},{5,6},{7}} => 13 {{1},{2},{3,5},{4,6},{7}} => 14 {{1},{2},{3,6},{4,5},{7}} => 14 {{1},{2,5},{3},{4},{6,7}} => 13 {{1},{2,6},{3},{4},{5,7}} => 14 {{1},{2,7},{3},{4},{5,6}} => 14 {{1},{2,4},{3},{5},{6,7}} => 13 {{1},{2,6},{3},{4,7},{5}} => 15 {{1},{2,7},{3},{4,6},{5}} => 15 {{1},{2,4},{3},{5,7},{6}} => 14 {{1},{2,5},{3},{4,7},{6}} => 15 {{1},{2,7},{3},{4,5},{6}} => 15 {{1},{2,4},{3},{5,6},{7}} => 14 {{1},{2,5},{3},{4,6},{7}} => 15 {{1},{2,6},{3},{4,5},{7}} => 15 {{1},{2,3},{4},{5},{6,7}} => 13 {{1},{2,6},{3,7},{4},{5}} => 16 {{1},{2,7},{3,6},{4},{5}} => 16 {{1},{2,3},{4},{5,7},{6}} => 14 {{1},{2,5},{3,7},{4},{6}} => 16 {{1},{2,7},{3,5},{4},{6}} => 16 {{1},{2,3},{4},{5,6},{7}} => 14 {{1},{2,5},{3,6},{4},{7}} => 16 {{1},{2,6},{3,5},{4},{7}} => 16 {{1},{2,3},{4,7},{5},{6}} => 15 {{1},{2,4},{3,7},{5},{6}} => 16 {{1},{2,7},{3,4},{5},{6}} => 16 {{1},{2,3},{4,6},{5},{7}} => 15 {{1},{2,4},{3,6},{5},{7}} => 16 {{1},{2,6},{3,4},{5},{7}} => 16 {{1},{2,3},{4,5},{6},{7}} => 15 {{1},{2,4},{3,5},{6},{7}} => 16 {{1},{2,5},{3,4},{6},{7}} => 16 {{1,5},{2},{3},{4},{6,7}} => 14 {{1,6},{2},{3},{4},{5,7}} => 15 {{1,7},{2},{3},{4},{5,6}} => 15 {{1,4},{2},{3},{5},{6,7}} => 14 {{1,6},{2},{3},{4,7},{5}} => 16 {{1,7},{2},{3},{4,6},{5}} => 16 {{1,4},{2},{3},{5,7},{6}} => 15 {{1,5},{2},{3},{4,7},{6}} => 16 {{1,7},{2},{3},{4,5},{6}} => 16 {{1,4},{2},{3},{5,6},{7}} => 15 {{1,5},{2},{3},{4,6},{7}} => 16 {{1,6},{2},{3},{4,5},{7}} => 16 {{1,3},{2},{4},{5},{6,7}} => 14 {{1,6},{2},{3,7},{4},{5}} => 17 {{1,7},{2},{3,6},{4},{5}} => 17 {{1,3},{2},{4},{5,7},{6}} => 15 {{1,5},{2},{3,7},{4},{6}} => 17 {{1,7},{2},{3,5},{4},{6}} => 17 {{1,3},{2},{4},{5,6},{7}} => 15 {{1,5},{2},{3,6},{4},{7}} => 17 {{1,6},{2},{3,5},{4},{7}} => 17 {{1,3},{2},{4,7},{5},{6}} => 16 {{1,4},{2},{3,7},{5},{6}} => 17 {{1,7},{2},{3,4},{5},{6}} => 17 {{1,3},{2},{4,6},{5},{7}} => 16 {{1,4},{2},{3,6},{5},{7}} => 17 {{1,6},{2},{3,4},{5},{7}} => 17 {{1,3},{2},{4,5},{6},{7}} => 16 {{1,4},{2},{3,5},{6},{7}} => 17 {{1,5},{2},{3,4},{6},{7}} => 17 {{1,2},{3},{4},{5},{6,7}} => 14 {{1,6},{2,7},{3},{4},{5}} => 18 {{1,7},{2,6},{3},{4},{5}} => 18 {{1,2},{3},{4},{5,7},{6}} => 15 {{1,5},{2,7},{3},{4},{6}} => 18 {{1,7},{2,5},{3},{4},{6}} => 18 {{1,2},{3},{4},{5,6},{7}} => 15 {{1,5},{2,6},{3},{4},{7}} => 18 {{1,6},{2,5},{3},{4},{7}} => 18 {{1,2},{3},{4,7},{5},{6}} => 16 {{1,4},{2,7},{3},{5},{6}} => 18 {{1,7},{2,4},{3},{5},{6}} => 18 {{1,2},{3},{4,6},{5},{7}} => 16 {{1,4},{2,6},{3},{5},{7}} => 18 {{1,6},{2,4},{3},{5},{7}} => 18 {{1,2},{3},{4,5},{6},{7}} => 16 {{1,4},{2,5},{3},{6},{7}} => 18 {{1,5},{2,4},{3},{6},{7}} => 18 {{1,2},{3,7},{4},{5},{6}} => 17 {{1,3},{2,7},{4},{5},{6}} => 18 {{1,7},{2,3},{4},{5},{6}} => 18 {{1,2},{3,6},{4},{5},{7}} => 17 {{1,3},{2,6},{4},{5},{7}} => 18 {{1,6},{2,3},{4},{5},{7}} => 18 {{1,2},{3,5},{4},{6},{7}} => 17 {{1,3},{2,5},{4},{6},{7}} => 18 {{1,5},{2,3},{4},{6},{7}} => 18 {{1,2},{3,4},{5},{6},{7}} => 17 {{1,3},{2,4},{5},{6},{7}} => 18 {{1,4},{2,3},{5},{6},{7}} => 18 {{1},{2},{3},{4},{5},{6,7}} => 15 {{1},{2},{3},{4},{5,7},{6}} => 16 {{1},{2},{3},{4},{5,6},{7}} => 16 {{1},{2},{3},{4,7},{5},{6}} => 17 {{1},{2},{3},{4,6},{5},{7}} => 17 {{1},{2},{3},{4,5},{6},{7}} => 17 {{1},{2},{3,7},{4},{5},{6}} => 18 {{1},{2},{3,6},{4},{5},{7}} => 18 {{1},{2},{3,5},{4},{6},{7}} => 18 {{1},{2},{3,4},{5},{6},{7}} => 18 {{1},{2,7},{3},{4},{5},{6}} => 19 {{1},{2,6},{3},{4},{5},{7}} => 19 {{1},{2,5},{3},{4},{6},{7}} => 19 {{1},{2,4},{3},{5},{6},{7}} => 19 {{1},{2,3},{4},{5},{6},{7}} => 19 {{1,7},{2},{3},{4},{5},{6}} => 20 {{1,6},{2},{3},{4},{5},{7}} => 20 {{1,5},{2},{3},{4},{6},{7}} => 20 {{1,4},{2},{3},{5},{6},{7}} => 20 {{1,3},{2},{4},{5},{6},{7}} => 20 {{1,2},{3},{4},{5},{6},{7}} => 20 {{1},{2},{3},{4},{5},{6},{7}} => 21 {{1,2,3,4,5,6,7,8}} => 0 {{1},{2,3,4,5,6,7,8}} => 1 {{1,3,4,5,6,7,8},{2}} => 2 {{1,2,4,5,6,7,8},{3}} => 3 {{1,2,3,5,6,7,8},{4}} => 4 {{1,2,3,4,6,7,8},{5}} => 5 {{1,2,3,4,5,7,8},{6}} => 6 {{1,2,3,4,5,6,8},{7}} => 7 {{1,2,3,4,5,6,7},{8}} => 7 {{1,2},{3,4,5,6,7,8}} => 2 {{1,3},{2,4,5,6,7,8}} => 3 {{1,4},{2,3,5,6,7,8}} => 4 {{1,5},{2,3,4,6,7,8}} => 5 {{1,6},{2,3,4,5,7,8}} => 6 {{1,7},{2,3,4,5,6,8}} => 7 {{1,8},{2,3,4,5,6,7}} => 7 {{1,4,5,6,7,8},{2,3}} => 3 {{1,3,5,6,7,8},{2,4}} => 4 {{1,3,4,6,7,8},{2,5}} => 5 {{1,3,4,5,7,8},{2,6}} => 6 {{1,3,4,5,6,8},{2,7}} => 7 {{1,3,4,5,6,7},{2,8}} => 7 {{1,2,5,6,7,8},{3,4}} => 4 {{1,2,4,6,7,8},{3,5}} => 5 {{1,2,4,5,7,8},{3,6}} => 6 {{1,2,4,5,6,8},{3,7}} => 7 {{1,2,4,5,6,7},{3,8}} => 7 {{1,2,3,6,7,8},{4,5}} => 5 {{1,2,3,5,7,8},{4,6}} => 6 {{1,2,3,5,6,8},{4,7}} => 7 {{1,2,3,5,6,7},{4,8}} => 7 {{1,2,3,4,7,8},{5,6}} => 6 {{1,2,3,4,6,8},{5,7}} => 7 {{1,2,3,4,6,7},{5,8}} => 7 {{1,2,3,4,5,8},{6,7}} => 7 {{1,2,3,4,5,7},{6,8}} => 7 {{1,2,3,4,5,6},{7,8}} => 6 {{1},{2},{3,4,5,6,7,8}} => 3 {{1},{2,4,5,6,7,8},{3}} => 4 {{1},{2,3,5,6,7,8},{4}} => 5 {{1},{2,3,4,6,7,8},{5}} => 6 {{1},{2,3,4,5,7,8},{6}} => 7 {{1},{2,3,4,5,6,8},{7}} => 8 {{1},{2,3,4,5,6,7},{8}} => 8 {{1,4,5,6,7,8},{2},{3}} => 5 {{1,3,5,6,7,8},{2},{4}} => 6 ----------------------------------------------------------------------------- Created: Aug 10, 2016 at 23:16 by Martin Rubey ----------------------------------------------------------------------------- Last Updated: Aug 10, 2016 at 23:16 by Martin Rubey