***************************************************************************** * 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: St000823 ----------------------------------------------------------------------------- Collection: Set partitions ----------------------------------------------------------------------------- Description: The number of unsplittable factors of the set partition. Let $\pi$ be a set partition of $m$ into $k$ parts and $\sigma$ a set partition of $n$ into $\ell$ parts. Then their split product is the set partition of $m+n$ with blocks $$ B_1 \cup (C_1+m),\dots,B_k \cup (C_k+m), C_{k+1}+m,\dots,C_\ell+m $$ if $k\leq \ell$ and $$ B_1 \cup (C_1+m),\dots,B_\ell \cup (C_\ell+m), B_{\ell+1},\dots,B_k $$ otherwise. ----------------------------------------------------------------------------- References: [1] Chen, W. Y. C., Li, T. X. S., Wang, D. G. L. A bijection between atomic partitions and unsplitable partitions [[MathSciNet:2770112]] [2] Triangle read by rows: T(n,k) = number of noncommutative symmetric polynomials of degree n that have exactly k different variables appearing in each monomial and which generate the algebra of all noncommutative symmetric polynomials (n >= 1, 1 <= k <= n). [[OEIS:A055105]] ----------------------------------------------------------------------------- Code: def statistic(pi): return len(split(pi)) def split_product(pi, sigma): """ An associative product. sage: split_product([[1,2]],[[1,3],[2]]) {{1, 2, 3, 5}, {4}} sage: split_product([[1,3],[2]],[[1,2]]) {{1, 3, 4, 5}, {2}} """ pi = sorted([sorted(B) for B in pi]) sigma = sorted([sorted(B) for B in sigma]) k = len(pi) l = len(sigma) m = SetPartition(pi).size() if k <= l: return SetPartition([B + [c+m for c in C] for B, C in zip(pi, sigma)] + [[c+m for c in C] for C in sigma[k:]]) else: return SetPartition([B + [c+m for c in C] for B, C in zip(pi, sigma)] + pi[l:]) @cached_function def unsplittable(n): """ sage: [len(unsplittable(n)) for n in range(1,8)] [1, 1, 2, 6, 22, 92, 426] sage: oeis(_) 0: A074664: Number of algebraically independent elements of degree n in the algebra of symmetric polynomials in noncommuting variables. """ return Set(SetPartitions(n)).difference(Set([split_product(pi, sigma) for k in range(1,n) for pi in SetPartitions(k) for sigma in SetPartitions(n-k)])) def split(pi): pi = SetPartition(pi) n = pi.size() if pi in unsplittable(n): return [pi] for l in range(1, n): for sigma in unsplittable(l): for kappa in SetPartitions(n-l): if pi == split_product(sigma, kappa): return [sigma] + split(kappa) ----------------------------------------------------------------------------- Statistic values: {{1,2}} => 2 {{1},{2}} => 1 {{1,2,3}} => 3 {{1,2},{3}} => 2 {{1,3},{2}} => 2 {{1},{2,3}} => 1 {{1},{2},{3}} => 1 {{1,2,3,4}} => 4 {{1,2,3},{4}} => 3 {{1,2,4},{3}} => 3 {{1,2},{3,4}} => 2 {{1,2},{3},{4}} => 2 {{1,3,4},{2}} => 3 {{1,3},{2,4}} => 2 {{1,3},{2},{4}} => 1 {{1,4},{2,3}} => 2 {{1},{2,3,4}} => 1 {{1},{2,3},{4}} => 1 {{1,4},{2},{3}} => 2 {{1},{2,4},{3}} => 1 {{1},{2},{3,4}} => 1 {{1},{2},{3},{4}} => 1 {{1,2,3,4,5}} => 5 {{1,2,3,4},{5}} => 4 {{1,2,3,5},{4}} => 4 {{1,2,3},{4,5}} => 3 {{1,2,3},{4},{5}} => 3 {{1,2,4,5},{3}} => 4 {{1,2,4},{3,5}} => 3 {{1,2,4},{3},{5}} => 2 {{1,2,5},{3,4}} => 3 {{1,2},{3,4,5}} => 2 {{1,2},{3,4},{5}} => 2 {{1,2,5},{3},{4}} => 3 {{1,2},{3,5},{4}} => 2 {{1,2},{3},{4,5}} => 2 {{1,2},{3},{4},{5}} => 2 {{1,3,4,5},{2}} => 4 {{1,3,4},{2,5}} => 3 {{1,3,4},{2},{5}} => 1 {{1,3,5},{2,4}} => 3 {{1,3},{2,4,5}} => 2 {{1,3},{2,4},{5}} => 2 {{1,3,5},{2},{4}} => 2 {{1,3},{2,5},{4}} => 1 {{1,3},{2},{4,5}} => 1 {{1,3},{2},{4},{5}} => 1 {{1,4,5},{2,3}} => 3 {{1,4},{2,3,5}} => 2 {{1,4},{2,3},{5}} => 1 {{1,5},{2,3,4}} => 2 {{1},{2,3,4,5}} => 1 {{1},{2,3,4},{5}} => 1 {{1,5},{2,3},{4}} => 2 {{1},{2,3,5},{4}} => 1 {{1},{2,3},{4,5}} => 1 {{1},{2,3},{4},{5}} => 1 {{1,4,5},{2},{3}} => 3 {{1,4},{2,5},{3}} => 2 {{1,4},{2},{3,5}} => 1 {{1,4},{2},{3},{5}} => 1 {{1,5},{2,4},{3}} => 2 {{1},{2,4,5},{3}} => 1 {{1},{2,4},{3,5}} => 1 {{1},{2,4},{3},{5}} => 1 {{1,5},{2},{3,4}} => 2 {{1},{2,5},{3,4}} => 1 {{1},{2},{3,4,5}} => 1 {{1},{2},{3,4},{5}} => 1 {{1,5},{2},{3},{4}} => 2 {{1},{2,5},{3},{4}} => 1 {{1},{2},{3,5},{4}} => 1 {{1},{2},{3},{4,5}} => 1 {{1},{2},{3},{4},{5}} => 1 {{1,2,3,4,5,6}} => 6 {{1,2,3,4,5},{6}} => 5 {{1,2,3,4,6},{5}} => 5 {{1,2,3,4},{5,6}} => 4 {{1,2,3,4},{5},{6}} => 4 {{1,2,3,5,6},{4}} => 5 {{1,2,3,5},{4,6}} => 4 {{1,2,3,5},{4},{6}} => 3 {{1,2,3,6},{4,5}} => 4 {{1,2,3},{4,5,6}} => 3 {{1,2,3},{4,5},{6}} => 3 {{1,2,3,6},{4},{5}} => 4 {{1,2,3},{4,6},{5}} => 3 {{1,2,3},{4},{5,6}} => 3 {{1,2,3},{4},{5},{6}} => 3 {{1,2,4,5,6},{3}} => 5 {{1,2,4,5},{3,6}} => 4 {{1,2,4,5},{3},{6}} => 2 {{1,2,4,6},{3,5}} => 4 {{1,2,4},{3,5,6}} => 3 {{1,2,4},{3,5},{6}} => 3 {{1,2,4,6},{3},{5}} => 3 {{1,2,4},{3,6},{5}} => 2 {{1,2,4},{3},{5,6}} => 2 {{1,2,4},{3},{5},{6}} => 2 {{1,2,5,6},{3,4}} => 4 {{1,2,5},{3,4,6}} => 3 {{1,2,5},{3,4},{6}} => 2 {{1,2,6},{3,4,5}} => 3 {{1,2},{3,4,5,6}} => 2 {{1,2},{3,4,5},{6}} => 2 {{1,2,6},{3,4},{5}} => 3 {{1,2},{3,4,6},{5}} => 2 {{1,2},{3,4},{5,6}} => 2 {{1,2},{3,4},{5},{6}} => 2 {{1,2,5,6},{3},{4}} => 4 {{1,2,5},{3,6},{4}} => 3 {{1,2,5},{3},{4,6}} => 2 {{1,2,5},{3},{4},{6}} => 2 {{1,2,6},{3,5},{4}} => 3 {{1,2},{3,5,6},{4}} => 2 {{1,2},{3,5},{4,6}} => 2 {{1,2},{3,5},{4},{6}} => 2 {{1,2,6},{3},{4,5}} => 3 {{1,2},{3,6},{4,5}} => 2 {{1,2},{3},{4,5,6}} => 2 {{1,2},{3},{4,5},{6}} => 2 {{1,2,6},{3},{4},{5}} => 3 {{1,2},{3,6},{4},{5}} => 2 {{1,2},{3},{4,6},{5}} => 2 {{1,2},{3},{4},{5,6}} => 2 {{1,2},{3},{4},{5},{6}} => 2 {{1,3,4,5,6},{2}} => 5 {{1,3,4,5},{2,6}} => 4 {{1,3,4,5},{2},{6}} => 1 {{1,3,4,6},{2,5}} => 4 {{1,3,4},{2,5,6}} => 3 {{1,3,4},{2,5},{6}} => 3 {{1,3,4,6},{2},{5}} => 2 {{1,3,4},{2,6},{5}} => 1 {{1,3,4},{2},{5,6}} => 1 {{1,3,4},{2},{5},{6}} => 1 {{1,3,5,6},{2,4}} => 4 {{1,3,5},{2,4,6}} => 3 {{1,3,5},{2,4},{6}} => 2 {{1,3,6},{2,4,5}} => 3 {{1,3},{2,4,5,6}} => 2 {{1,3},{2,4,5},{6}} => 2 {{1,3,6},{2,4},{5}} => 3 {{1,3},{2,4,6},{5}} => 2 {{1,3},{2,4},{5,6}} => 2 {{1,3},{2,4},{5},{6}} => 2 {{1,3,5,6},{2},{4}} => 3 {{1,3,5},{2,6},{4}} => 2 {{1,3,5},{2},{4,6}} => 1 {{1,3,5},{2},{4},{6}} => 1 {{1,3,6},{2,5},{4}} => 2 {{1,3},{2,5,6},{4}} => 1 {{1,3},{2,5},{4,6}} => 1 {{1,3},{2,5},{4},{6}} => 1 {{1,3,6},{2},{4,5}} => 2 {{1,3},{2,6},{4,5}} => 1 {{1,3},{2},{4,5,6}} => 1 {{1,3},{2},{4,5},{6}} => 1 {{1,3,6},{2},{4},{5}} => 2 {{1,3},{2,6},{4},{5}} => 1 {{1,3},{2},{4,6},{5}} => 1 {{1,3},{2},{4},{5,6}} => 1 {{1,3},{2},{4},{5},{6}} => 1 {{1,4,5,6},{2,3}} => 4 {{1,4,5},{2,3,6}} => 3 {{1,4,5},{2,3},{6}} => 1 {{1,4,6},{2,3,5}} => 3 {{1,4},{2,3,5,6}} => 2 {{1,4},{2,3,5},{6}} => 2 {{1,4,6},{2,3},{5}} => 2 {{1,4},{2,3,6},{5}} => 1 {{1,4},{2,3},{5,6}} => 1 {{1,4},{2,3},{5},{6}} => 1 {{1,5,6},{2,3,4}} => 3 {{1,5},{2,3,4,6}} => 2 {{1,5},{2,3,4},{6}} => 1 {{1,6},{2,3,4,5}} => 2 {{1},{2,3,4,5,6}} => 1 {{1},{2,3,4,5},{6}} => 1 {{1,6},{2,3,4},{5}} => 2 {{1},{2,3,4,6},{5}} => 1 {{1},{2,3,4},{5,6}} => 1 {{1},{2,3,4},{5},{6}} => 1 {{1,5,6},{2,3},{4}} => 3 {{1,5},{2,3,6},{4}} => 2 {{1,5},{2,3},{4,6}} => 1 {{1,5},{2,3},{4},{6}} => 1 {{1,6},{2,3,5},{4}} => 2 {{1},{2,3,5,6},{4}} => 1 {{1},{2,3,5},{4,6}} => 1 {{1},{2,3,5},{4},{6}} => 1 {{1,6},{2,3},{4,5}} => 2 {{1},{2,3,6},{4,5}} => 1 {{1},{2,3},{4,5,6}} => 1 {{1},{2,3},{4,5},{6}} => 1 {{1,6},{2,3},{4},{5}} => 2 {{1},{2,3,6},{4},{5}} => 1 {{1},{2,3},{4,6},{5}} => 1 {{1},{2,3},{4},{5,6}} => 1 {{1},{2,3},{4},{5},{6}} => 1 {{1,4,5,6},{2},{3}} => 4 {{1,4,5},{2,6},{3}} => 3 {{1,4,5},{2},{3,6}} => 1 {{1,4,5},{2},{3},{6}} => 1 {{1,4,6},{2,5},{3}} => 3 {{1,4},{2,5,6},{3}} => 2 {{1,4},{2,5},{3,6}} => 2 {{1,4},{2,5},{3},{6}} => 1 {{1,4,6},{2},{3,5}} => 2 {{1,4},{2,6},{3,5}} => 1 {{1,4},{2},{3,5,6}} => 1 {{1,4},{2},{3,5},{6}} => 1 {{1,4,6},{2},{3},{5}} => 2 {{1,4},{2,6},{3},{5}} => 1 {{1,4},{2},{3,6},{5}} => 1 {{1,4},{2},{3},{5,6}} => 1 {{1,4},{2},{3},{5},{6}} => 1 {{1,5,6},{2,4},{3}} => 3 {{1,5},{2,4,6},{3}} => 2 {{1,5},{2,4},{3,6}} => 1 {{1,5},{2,4},{3},{6}} => 1 {{1,6},{2,4,5},{3}} => 2 {{1},{2,4,5,6},{3}} => 1 {{1},{2,4,5},{3,6}} => 1 {{1},{2,4,5},{3},{6}} => 1 {{1,6},{2,4},{3,5}} => 2 {{1},{2,4,6},{3,5}} => 1 {{1},{2,4},{3,5,6}} => 1 {{1},{2,4},{3,5},{6}} => 1 {{1,6},{2,4},{3},{5}} => 2 {{1},{2,4,6},{3},{5}} => 1 {{1},{2,4},{3,6},{5}} => 1 {{1},{2,4},{3},{5,6}} => 1 {{1},{2,4},{3},{5},{6}} => 1 {{1,5,6},{2},{3,4}} => 3 {{1,5},{2,6},{3,4}} => 2 {{1,5},{2},{3,4,6}} => 1 {{1,5},{2},{3,4},{6}} => 1 {{1,6},{2,5},{3,4}} => 2 {{1},{2,5,6},{3,4}} => 1 {{1},{2,5},{3,4,6}} => 1 {{1},{2,5},{3,4},{6}} => 1 {{1,6},{2},{3,4,5}} => 2 {{1},{2,6},{3,4,5}} => 1 {{1},{2},{3,4,5,6}} => 1 {{1},{2},{3,4,5},{6}} => 1 {{1,6},{2},{3,4},{5}} => 2 {{1},{2,6},{3,4},{5}} => 1 {{1},{2},{3,4,6},{5}} => 1 {{1},{2},{3,4},{5,6}} => 1 {{1},{2},{3,4},{5},{6}} => 1 {{1,5,6},{2},{3},{4}} => 3 {{1,5},{2,6},{3},{4}} => 2 {{1,5},{2},{3,6},{4}} => 1 {{1,5},{2},{3},{4,6}} => 1 {{1,5},{2},{3},{4},{6}} => 1 {{1,6},{2,5},{3},{4}} => 2 {{1},{2,5,6},{3},{4}} => 1 {{1},{2,5},{3,6},{4}} => 1 {{1},{2,5},{3},{4,6}} => 1 {{1},{2,5},{3},{4},{6}} => 1 {{1,6},{2},{3,5},{4}} => 2 {{1},{2,6},{3,5},{4}} => 1 {{1},{2},{3,5,6},{4}} => 1 {{1},{2},{3,5},{4,6}} => 1 {{1},{2},{3,5},{4},{6}} => 1 {{1,6},{2},{3},{4,5}} => 2 {{1},{2,6},{3},{4,5}} => 1 {{1},{2},{3,6},{4,5}} => 1 {{1},{2},{3},{4,5,6}} => 1 {{1},{2},{3},{4,5},{6}} => 1 {{1,6},{2},{3},{4},{5}} => 2 {{1},{2,6},{3},{4},{5}} => 1 {{1},{2},{3,6},{4},{5}} => 1 {{1},{2},{3},{4,6},{5}} => 1 {{1},{2},{3},{4},{5,6}} => 1 {{1},{2},{3},{4},{5},{6}} => 1 {{1,2,3,4,5,6,7}} => 7 {{1,2,3,4,5,6},{7}} => 6 {{1,2,3,4,5,7},{6}} => 6 {{1,2,3,4,5},{6,7}} => 5 {{1,2,3,4,5},{6},{7}} => 5 {{1,2,3,4,6,7},{5}} => 6 {{1,2,3,4,6},{5,7}} => 5 {{1,2,3,4,6},{5},{7}} => 4 {{1,2,3,4,7},{5,6}} => 5 {{1,2,3,4},{5,6,7}} => 4 {{1,2,3,4},{5,6},{7}} => 4 {{1,2,3,4,7},{5},{6}} => 5 {{1,2,3,4},{5,7},{6}} => 4 {{1,2,3,4},{5},{6,7}} => 4 {{1,2,3,4},{5},{6},{7}} => 4 {{1,2,3,5,6,7},{4}} => 6 {{1,2,3,5,6},{4,7}} => 5 {{1,2,3,5,6},{4},{7}} => 3 {{1,2,3,5,7},{4,6}} => 5 {{1,2,3,5},{4,6,7}} => 4 {{1,2,3,5},{4,6},{7}} => 4 {{1,2,3,5,7},{4},{6}} => 4 {{1,2,3,5},{4,7},{6}} => 3 {{1,2,3,5},{4},{6,7}} => 3 {{1,2,3,5},{4},{6},{7}} => 3 {{1,2,3,6,7},{4,5}} => 5 {{1,2,3,6},{4,5,7}} => 4 {{1,2,3,6},{4,5},{7}} => 3 {{1,2,3,7},{4,5,6}} => 4 {{1,2,3},{4,5,6,7}} => 3 {{1,2,3},{4,5,6},{7}} => 3 {{1,2,3,7},{4,5},{6}} => 4 {{1,2,3},{4,5,7},{6}} => 3 {{1,2,3},{4,5},{6,7}} => 3 {{1,2,3},{4,5},{6},{7}} => 3 {{1,2,3,6,7},{4},{5}} => 5 {{1,2,3,6},{4,7},{5}} => 4 {{1,2,3,6},{4},{5,7}} => 3 {{1,2,3,6},{4},{5},{7}} => 3 {{1,2,3,7},{4,6},{5}} => 4 {{1,2,3},{4,6,7},{5}} => 3 {{1,2,3},{4,6},{5,7}} => 3 {{1,2,3},{4,6},{5},{7}} => 3 {{1,2,3,7},{4},{5,6}} => 4 {{1,2,3},{4,7},{5,6}} => 3 {{1,2,3},{4},{5,6,7}} => 3 {{1,2,3},{4},{5,6},{7}} => 3 {{1,2,3,7},{4},{5},{6}} => 4 {{1,2,3},{4,7},{5},{6}} => 3 {{1,2,3},{4},{5,7},{6}} => 3 {{1,2,3},{4},{5},{6,7}} => 3 {{1,2,3},{4},{5},{6},{7}} => 3 {{1,2,4,5,6,7},{3}} => 6 {{1,2,4,5,6},{3,7}} => 5 {{1,2,4,5,6},{3},{7}} => 2 {{1,2,4,5,7},{3,6}} => 5 {{1,2,4,5},{3,6,7}} => 4 {{1,2,4,5},{3,6},{7}} => 4 {{1,2,4,5,7},{3},{6}} => 3 {{1,2,4,5},{3,7},{6}} => 2 {{1,2,4,5},{3},{6,7}} => 2 {{1,2,4,5},{3},{6},{7}} => 2 {{1,2,4,6,7},{3,5}} => 5 {{1,2,4,6},{3,5,7}} => 4 {{1,2,4,6},{3,5},{7}} => 3 {{1,2,4,7},{3,5,6}} => 4 {{1,2,4},{3,5,6,7}} => 3 {{1,2,4},{3,5,6},{7}} => 3 {{1,2,4,7},{3,5},{6}} => 4 {{1,2,4},{3,5,7},{6}} => 3 {{1,2,4},{3,5},{6,7}} => 3 {{1,2,4},{3,5},{6},{7}} => 3 {{1,2,4,6,7},{3},{5}} => 4 {{1,2,4,6},{3,7},{5}} => 3 {{1,2,4,6},{3},{5,7}} => 2 {{1,2,4,6},{3},{5},{7}} => 2 {{1,2,4,7},{3,6},{5}} => 3 {{1,2,4},{3,6,7},{5}} => 2 {{1,2,4},{3,6},{5,7}} => 2 {{1,2,4},{3,6},{5},{7}} => 2 {{1,2,4,7},{3},{5,6}} => 3 {{1,2,4},{3,7},{5,6}} => 2 {{1,2,4},{3},{5,6,7}} => 2 {{1,2,4},{3},{5,6},{7}} => 2 {{1,2,4,7},{3},{5},{6}} => 3 {{1,2,4},{3,7},{5},{6}} => 2 {{1,2,4},{3},{5,7},{6}} => 2 {{1,2,4},{3},{5},{6,7}} => 2 {{1,2,4},{3},{5},{6},{7}} => 2 {{1,2,5,6,7},{3,4}} => 5 {{1,2,5,6},{3,4,7}} => 4 {{1,2,5,6},{3,4},{7}} => 2 {{1,2,5,7},{3,4,6}} => 4 {{1,2,5},{3,4,6,7}} => 3 {{1,2,5},{3,4,6},{7}} => 3 {{1,2,5,7},{3,4},{6}} => 3 {{1,2,5},{3,4,7},{6}} => 2 {{1,2,5},{3,4},{6,7}} => 2 {{1,2,5},{3,4},{6},{7}} => 2 {{1,2,6,7},{3,4,5}} => 4 {{1,2,6},{3,4,5,7}} => 3 {{1,2,6},{3,4,5},{7}} => 2 {{1,2,7},{3,4,5,6}} => 3 {{1,2},{3,4,5,6,7}} => 2 {{1,2},{3,4,5,6},{7}} => 2 {{1,2,7},{3,4,5},{6}} => 3 {{1,2},{3,4,5,7},{6}} => 2 {{1,2},{3,4,5},{6,7}} => 2 {{1,2},{3,4,5},{6},{7}} => 2 {{1,2,6,7},{3,4},{5}} => 4 {{1,2,6},{3,4,7},{5}} => 3 {{1,2,6},{3,4},{5,7}} => 2 {{1,2,6},{3,4},{5},{7}} => 2 {{1,2,7},{3,4,6},{5}} => 3 {{1,2},{3,4,6,7},{5}} => 2 {{1,2},{3,4,6},{5,7}} => 2 {{1,2},{3,4,6},{5},{7}} => 2 {{1,2,7},{3,4},{5,6}} => 3 {{1,2},{3,4,7},{5,6}} => 2 {{1,2},{3,4},{5,6,7}} => 2 {{1,2},{3,4},{5,6},{7}} => 2 {{1,2,7},{3,4},{5},{6}} => 3 {{1,2},{3,4,7},{5},{6}} => 2 {{1,2},{3,4},{5,7},{6}} => 2 {{1,2},{3,4},{5},{6,7}} => 2 {{1,2},{3,4},{5},{6},{7}} => 2 {{1,2,5,6,7},{3},{4}} => 5 {{1,2,5,6},{3,7},{4}} => 4 {{1,2,5,6},{3},{4,7}} => 2 {{1,2,5,6},{3},{4},{7}} => 2 {{1,2,5,7},{3,6},{4}} => 4 {{1,2,5},{3,6,7},{4}} => 3 {{1,2,5},{3,6},{4,7}} => 3 {{1,2,5},{3,6},{4},{7}} => 2 {{1,2,5,7},{3},{4,6}} => 3 {{1,2,5},{3,7},{4,6}} => 2 {{1,2,5},{3},{4,6,7}} => 2 {{1,2,5},{3},{4,6},{7}} => 2 {{1,2,5,7},{3},{4},{6}} => 3 {{1,2,5},{3,7},{4},{6}} => 2 {{1,2,5},{3},{4,7},{6}} => 2 {{1,2,5},{3},{4},{6,7}} => 2 {{1,2,5},{3},{4},{6},{7}} => 2 {{1,2,6,7},{3,5},{4}} => 4 {{1,2,6},{3,5,7},{4}} => 3 {{1,2,6},{3,5},{4,7}} => 2 {{1,2,6},{3,5},{4},{7}} => 2 {{1,2,7},{3,5,6},{4}} => 3 {{1,2},{3,5,6,7},{4}} => 2 {{1,2},{3,5,6},{4,7}} => 2 {{1,2},{3,5,6},{4},{7}} => 2 {{1,2,7},{3,5},{4,6}} => 3 {{1,2},{3,5,7},{4,6}} => 2 {{1,2},{3,5},{4,6,7}} => 2 {{1,2},{3,5},{4,6},{7}} => 2 {{1,2,7},{3,5},{4},{6}} => 3 {{1,2},{3,5,7},{4},{6}} => 2 {{1,2},{3,5},{4,7},{6}} => 2 {{1,2},{3,5},{4},{6,7}} => 2 {{1,2},{3,5},{4},{6},{7}} => 2 {{1,2,6,7},{3},{4,5}} => 4 {{1,2,6},{3,7},{4,5}} => 3 {{1,2,6},{3},{4,5,7}} => 2 {{1,2,6},{3},{4,5},{7}} => 2 {{1,2,7},{3,6},{4,5}} => 3 {{1,2},{3,6,7},{4,5}} => 2 {{1,2},{3,6},{4,5,7}} => 2 {{1,2},{3,6},{4,5},{7}} => 2 {{1,2,7},{3},{4,5,6}} => 3 {{1,2},{3,7},{4,5,6}} => 2 {{1,2},{3},{4,5,6,7}} => 2 {{1,2},{3},{4,5,6},{7}} => 2 {{1,2,7},{3},{4,5},{6}} => 3 {{1,2},{3,7},{4,5},{6}} => 2 {{1,2},{3},{4,5,7},{6}} => 2 {{1,2},{3},{4,5},{6,7}} => 2 {{1,2},{3},{4,5},{6},{7}} => 2 {{1,2,6,7},{3},{4},{5}} => 4 {{1,2,6},{3,7},{4},{5}} => 3 {{1,2,6},{3},{4,7},{5}} => 2 {{1,2,6},{3},{4},{5,7}} => 2 {{1,2,6},{3},{4},{5},{7}} => 2 {{1,2,7},{3,6},{4},{5}} => 3 {{1,2},{3,6,7},{4},{5}} => 2 {{1,2},{3,6},{4,7},{5}} => 2 {{1,2},{3,6},{4},{5,7}} => 2 {{1,2},{3,6},{4},{5},{7}} => 2 {{1,2,7},{3},{4,6},{5}} => 3 {{1,2},{3,7},{4,6},{5}} => 2 {{1,2},{3},{4,6,7},{5}} => 2 {{1,2},{3},{4,6},{5,7}} => 2 {{1,2},{3},{4,6},{5},{7}} => 2 {{1,2,7},{3},{4},{5,6}} => 3 {{1,2},{3,7},{4},{5,6}} => 2 {{1,2},{3},{4,7},{5,6}} => 2 {{1,2},{3},{4},{5,6,7}} => 2 {{1,2},{3},{4},{5,6},{7}} => 2 {{1,2,7},{3},{4},{5},{6}} => 3 {{1,2},{3,7},{4},{5},{6}} => 2 {{1,2},{3},{4,7},{5},{6}} => 2 {{1,2},{3},{4},{5,7},{6}} => 2 {{1,2},{3},{4},{5},{6,7}} => 2 {{1,2},{3},{4},{5},{6},{7}} => 2 {{1,3,4,5,6,7},{2}} => 6 {{1,3,4,5,6},{2,7}} => 5 {{1,3,4,5,6},{2},{7}} => 1 {{1,3,4,5,7},{2,6}} => 5 {{1,3,4,5},{2,6,7}} => 4 {{1,3,4,5},{2,6},{7}} => 4 {{1,3,4,5,7},{2},{6}} => 2 {{1,3,4,5},{2,7},{6}} => 1 {{1,3,4,5},{2},{6,7}} => 1 {{1,3,4,5},{2},{6},{7}} => 1 {{1,3,4,6,7},{2,5}} => 5 {{1,3,4,6},{2,5,7}} => 4 {{1,3,4,6},{2,5},{7}} => 3 {{1,3,4,7},{2,5,6}} => 4 {{1,3,4},{2,5,6,7}} => 3 {{1,3,4},{2,5,6},{7}} => 3 {{1,3,4,7},{2,5},{6}} => 4 {{1,3,4},{2,5,7},{6}} => 3 {{1,3,4},{2,5},{6,7}} => 3 {{1,3,4},{2,5},{6},{7}} => 3 {{1,3,4,6,7},{2},{5}} => 3 {{1,3,4,6},{2,7},{5}} => 2 {{1,3,4,6},{2},{5,7}} => 1 {{1,3,4,6},{2},{5},{7}} => 1 {{1,3,4,7},{2,6},{5}} => 2 {{1,3,4},{2,6,7},{5}} => 1 {{1,3,4},{2,6},{5,7}} => 1 {{1,3,4},{2,6},{5},{7}} => 1 {{1,3,4,7},{2},{5,6}} => 2 {{1,3,4},{2,7},{5,6}} => 1 {{1,3,4},{2},{5,6,7}} => 1 {{1,3,4},{2},{5,6},{7}} => 1 {{1,3,4,7},{2},{5},{6}} => 2 {{1,3,4},{2,7},{5},{6}} => 1 {{1,3,4},{2},{5,7},{6}} => 1 {{1,3,4},{2},{5},{6,7}} => 1 {{1,3,4},{2},{5},{6},{7}} => 1 {{1,3,5,6,7},{2,4}} => 5 {{1,3,5,6},{2,4,7}} => 4 {{1,3,5,6},{2,4},{7}} => 2 {{1,3,5,7},{2,4,6}} => 4 {{1,3,5},{2,4,6,7}} => 3 {{1,3,5},{2,4,6},{7}} => 3 {{1,3,5,7},{2,4},{6}} => 3 {{1,3,5},{2,4,7},{6}} => 2 {{1,3,5},{2,4},{6,7}} => 2 {{1,3,5},{2,4},{6},{7}} => 2 {{1,3,6,7},{2,4,5}} => 4 {{1,3,6},{2,4,5,7}} => 3 {{1,3,6},{2,4,5},{7}} => 2 {{1,3,7},{2,4,5,6}} => 3 {{1,3},{2,4,5,6,7}} => 2 {{1,3},{2,4,5,6},{7}} => 2 {{1,3,7},{2,4,5},{6}} => 3 {{1,3},{2,4,5,7},{6}} => 2 {{1,3},{2,4,5},{6,7}} => 2 {{1,3},{2,4,5},{6},{7}} => 2 {{1,3,6,7},{2,4},{5}} => 4 {{1,3,6},{2,4,7},{5}} => 3 {{1,3,6},{2,4},{5,7}} => 2 {{1,3,6},{2,4},{5},{7}} => 2 {{1,3,7},{2,4,6},{5}} => 3 {{1,3},{2,4,6,7},{5}} => 2 {{1,3},{2,4,6},{5,7}} => 2 {{1,3},{2,4,6},{5},{7}} => 2 {{1,3,7},{2,4},{5,6}} => 3 {{1,3},{2,4,7},{5,6}} => 2 {{1,3},{2,4},{5,6,7}} => 2 {{1,3},{2,4},{5,6},{7}} => 2 {{1,3,7},{2,4},{5},{6}} => 3 {{1,3},{2,4,7},{5},{6}} => 2 {{1,3},{2,4},{5,7},{6}} => 2 {{1,3},{2,4},{5},{6,7}} => 2 {{1,3},{2,4},{5},{6},{7}} => 2 {{1,3,5,6,7},{2},{4}} => 4 {{1,3,5,6},{2,7},{4}} => 3 {{1,3,5,6},{2},{4,7}} => 1 {{1,3,5,6},{2},{4},{7}} => 1 {{1,3,5,7},{2,6},{4}} => 3 {{1,3,5},{2,6,7},{4}} => 2 {{1,3,5},{2,6},{4,7}} => 2 {{1,3,5},{2,6},{4},{7}} => 1 {{1,3,5,7},{2},{4,6}} => 2 {{1,3,5},{2,7},{4,6}} => 1 {{1,3,5},{2},{4,6,7}} => 1 {{1,3,5},{2},{4,6},{7}} => 1 {{1,3,5,7},{2},{4},{6}} => 2 {{1,3,5},{2,7},{4},{6}} => 1 {{1,3,5},{2},{4,7},{6}} => 1 {{1,3,5},{2},{4},{6,7}} => 1 {{1,3,5},{2},{4},{6},{7}} => 1 {{1,3,6,7},{2,5},{4}} => 3 {{1,3,6},{2,5,7},{4}} => 2 {{1,3,6},{2,5},{4,7}} => 1 {{1,3,6},{2,5},{4},{7}} => 1 {{1,3,7},{2,5,6},{4}} => 2 {{1,3},{2,5,6,7},{4}} => 1 {{1,3},{2,5,6},{4,7}} => 1 {{1,3},{2,5,6},{4},{7}} => 1 {{1,3,7},{2,5},{4,6}} => 2 {{1,3},{2,5,7},{4,6}} => 1 {{1,3},{2,5},{4,6,7}} => 1 {{1,3},{2,5},{4,6},{7}} => 1 {{1,3,7},{2,5},{4},{6}} => 2 {{1,3},{2,5,7},{4},{6}} => 1 {{1,3},{2,5},{4,7},{6}} => 1 {{1,3},{2,5},{4},{6,7}} => 1 {{1,3},{2,5},{4},{6},{7}} => 1 {{1,3,6,7},{2},{4,5}} => 3 {{1,3,6},{2,7},{4,5}} => 2 {{1,3,6},{2},{4,5,7}} => 1 {{1,3,6},{2},{4,5},{7}} => 1 {{1,3,7},{2,6},{4,5}} => 2 {{1,3},{2,6,7},{4,5}} => 1 {{1,3},{2,6},{4,5,7}} => 1 {{1,3},{2,6},{4,5},{7}} => 1 {{1,3,7},{2},{4,5,6}} => 2 {{1,3},{2,7},{4,5,6}} => 1 {{1,3},{2},{4,5,6,7}} => 1 {{1,3},{2},{4,5,6},{7}} => 1 {{1,3,7},{2},{4,5},{6}} => 2 {{1,3},{2,7},{4,5},{6}} => 1 {{1,3},{2},{4,5,7},{6}} => 1 {{1,3},{2},{4,5},{6,7}} => 1 {{1,3},{2},{4,5},{6},{7}} => 1 {{1,3,6,7},{2},{4},{5}} => 3 {{1,3,6},{2,7},{4},{5}} => 2 {{1,3,6},{2},{4,7},{5}} => 1 {{1,3,6},{2},{4},{5,7}} => 1 {{1,3,6},{2},{4},{5},{7}} => 1 {{1,3,7},{2,6},{4},{5}} => 2 {{1,3},{2,6,7},{4},{5}} => 1 {{1,3},{2,6},{4,7},{5}} => 1 {{1,3},{2,6},{4},{5,7}} => 1 {{1,3},{2,6},{4},{5},{7}} => 1 {{1,3,7},{2},{4,6},{5}} => 2 {{1,3},{2,7},{4,6},{5}} => 1 {{1,3},{2},{4,6,7},{5}} => 1 {{1,3},{2},{4,6},{5,7}} => 1 {{1,3},{2},{4,6},{5},{7}} => 1 {{1,3,7},{2},{4},{5,6}} => 2 {{1,3},{2,7},{4},{5,6}} => 1 {{1,3},{2},{4,7},{5,6}} => 1 {{1,3},{2},{4},{5,6,7}} => 1 {{1,3},{2},{4},{5,6},{7}} => 1 {{1,3,7},{2},{4},{5},{6}} => 2 {{1,3},{2,7},{4},{5},{6}} => 1 {{1,3},{2},{4,7},{5},{6}} => 1 {{1,3},{2},{4},{5,7},{6}} => 1 {{1,3},{2},{4},{5},{6,7}} => 1 {{1,3},{2},{4},{5},{6},{7}} => 1 {{1,4,5,6,7},{2,3}} => 5 {{1,4,5,6},{2,3,7}} => 4 {{1,4,5,6},{2,3},{7}} => 1 {{1,4,5,7},{2,3,6}} => 4 {{1,4,5},{2,3,6,7}} => 3 {{1,4,5},{2,3,6},{7}} => 3 {{1,4,5,7},{2,3},{6}} => 2 {{1,4,5},{2,3,7},{6}} => 1 {{1,4,5},{2,3},{6,7}} => 1 {{1,4,5},{2,3},{6},{7}} => 1 {{1,4,6,7},{2,3,5}} => 4 {{1,4,6},{2,3,5,7}} => 3 {{1,4,6},{2,3,5},{7}} => 2 {{1,4,7},{2,3,5,6}} => 3 {{1,4},{2,3,5,6,7}} => 2 {{1,4},{2,3,5,6},{7}} => 2 {{1,4,7},{2,3,5},{6}} => 3 {{1,4},{2,3,5,7},{6}} => 2 {{1,4},{2,3,5},{6,7}} => 2 {{1,4},{2,3,5},{6},{7}} => 2 {{1,4,6,7},{2,3},{5}} => 3 {{1,4,6},{2,3,7},{5}} => 2 {{1,4,6},{2,3},{5,7}} => 1 {{1,4,6},{2,3},{5},{7}} => 1 {{1,4,7},{2,3,6},{5}} => 2 {{1,4},{2,3,6,7},{5}} => 1 {{1,4},{2,3,6},{5,7}} => 1 {{1,4},{2,3,6},{5},{7}} => 1 {{1,4,7},{2,3},{5,6}} => 2 {{1,4},{2,3,7},{5,6}} => 1 {{1,4},{2,3},{5,6,7}} => 1 {{1,4},{2,3},{5,6},{7}} => 1 {{1,4,7},{2,3},{5},{6}} => 2 {{1,4},{2,3,7},{5},{6}} => 1 {{1,4},{2,3},{5,7},{6}} => 1 {{1,4},{2,3},{5},{6,7}} => 1 {{1,4},{2,3},{5},{6},{7}} => 1 {{1,5,6,7},{2,3,4}} => 4 {{1,5,6},{2,3,4,7}} => 3 {{1,5,6},{2,3,4},{7}} => 1 {{1,5,7},{2,3,4,6}} => 3 {{1,5},{2,3,4,6,7}} => 2 {{1,5},{2,3,4,6},{7}} => 2 {{1,5,7},{2,3,4},{6}} => 2 {{1,5},{2,3,4,7},{6}} => 1 {{1,5},{2,3,4},{6,7}} => 1 {{1,5},{2,3,4},{6},{7}} => 1 {{1,6,7},{2,3,4,5}} => 3 {{1,6},{2,3,4,5,7}} => 2 {{1,6},{2,3,4,5},{7}} => 1 {{1,7},{2,3,4,5,6}} => 2 {{1},{2,3,4,5,6,7}} => 1 {{1},{2,3,4,5,6},{7}} => 1 {{1,7},{2,3,4,5},{6}} => 2 {{1},{2,3,4,5,7},{6}} => 1 {{1},{2,3,4,5},{6,7}} => 1 {{1},{2,3,4,5},{6},{7}} => 1 {{1,6,7},{2,3,4},{5}} => 3 {{1,6},{2,3,4,7},{5}} => 2 {{1,6},{2,3,4},{5,7}} => 1 {{1,6},{2,3,4},{5},{7}} => 1 {{1,7},{2,3,4,6},{5}} => 2 {{1},{2,3,4,6,7},{5}} => 1 {{1},{2,3,4,6},{5,7}} => 1 {{1},{2,3,4,6},{5},{7}} => 1 {{1,7},{2,3,4},{5,6}} => 2 {{1},{2,3,4,7},{5,6}} => 1 {{1},{2,3,4},{5,6,7}} => 1 {{1},{2,3,4},{5,6},{7}} => 1 {{1,7},{2,3,4},{5},{6}} => 2 {{1},{2,3,4,7},{5},{6}} => 1 {{1},{2,3,4},{5,7},{6}} => 1 {{1},{2,3,4},{5},{6,7}} => 1 {{1},{2,3,4},{5},{6},{7}} => 1 {{1,5,6,7},{2,3},{4}} => 4 {{1,5,6},{2,3,7},{4}} => 3 {{1,5,6},{2,3},{4,7}} => 1 {{1,5,6},{2,3},{4},{7}} => 1 {{1,5,7},{2,3,6},{4}} => 3 {{1,5},{2,3,6,7},{4}} => 2 {{1,5},{2,3,6},{4,7}} => 2 {{1,5},{2,3,6},{4},{7}} => 1 {{1,5,7},{2,3},{4,6}} => 2 {{1,5},{2,3,7},{4,6}} => 1 {{1,5},{2,3},{4,6,7}} => 1 {{1,5},{2,3},{4,6},{7}} => 1 {{1,5,7},{2,3},{4},{6}} => 2 {{1,5},{2,3,7},{4},{6}} => 1 {{1,5},{2,3},{4,7},{6}} => 1 {{1,5},{2,3},{4},{6,7}} => 1 {{1,5},{2,3},{4},{6},{7}} => 1 {{1,6,7},{2,3,5},{4}} => 3 {{1,6},{2,3,5,7},{4}} => 2 {{1,6},{2,3,5},{4,7}} => 1 {{1,6},{2,3,5},{4},{7}} => 1 {{1,7},{2,3,5,6},{4}} => 2 {{1},{2,3,5,6,7},{4}} => 1 {{1},{2,3,5,6},{4,7}} => 1 {{1},{2,3,5,6},{4},{7}} => 1 {{1,7},{2,3,5},{4,6}} => 2 {{1},{2,3,5,7},{4,6}} => 1 {{1},{2,3,5},{4,6,7}} => 1 {{1},{2,3,5},{4,6},{7}} => 1 {{1,7},{2,3,5},{4},{6}} => 2 {{1},{2,3,5,7},{4},{6}} => 1 {{1},{2,3,5},{4,7},{6}} => 1 {{1},{2,3,5},{4},{6,7}} => 1 {{1},{2,3,5},{4},{6},{7}} => 1 {{1,6,7},{2,3},{4,5}} => 3 {{1,6},{2,3,7},{4,5}} => 2 {{1,6},{2,3},{4,5,7}} => 1 {{1,6},{2,3},{4,5},{7}} => 1 {{1,7},{2,3,6},{4,5}} => 2 {{1},{2,3,6,7},{4,5}} => 1 {{1},{2,3,6},{4,5,7}} => 1 {{1},{2,3,6},{4,5},{7}} => 1 {{1,7},{2,3},{4,5,6}} => 2 {{1},{2,3,7},{4,5,6}} => 1 {{1},{2,3},{4,5,6,7}} => 1 {{1},{2,3},{4,5,6},{7}} => 1 {{1,7},{2,3},{4,5},{6}} => 2 {{1},{2,3,7},{4,5},{6}} => 1 {{1},{2,3},{4,5,7},{6}} => 1 {{1},{2,3},{4,5},{6,7}} => 1 {{1},{2,3},{4,5},{6},{7}} => 1 {{1,6,7},{2,3},{4},{5}} => 3 {{1,6},{2,3,7},{4},{5}} => 2 {{1,6},{2,3},{4,7},{5}} => 1 {{1,6},{2,3},{4},{5,7}} => 1 {{1,6},{2,3},{4},{5},{7}} => 1 {{1,7},{2,3,6},{4},{5}} => 2 {{1},{2,3,6,7},{4},{5}} => 1 {{1},{2,3,6},{4,7},{5}} => 1 {{1},{2,3,6},{4},{5,7}} => 1 {{1},{2,3,6},{4},{5},{7}} => 1 {{1,7},{2,3},{4,6},{5}} => 2 {{1},{2,3,7},{4,6},{5}} => 1 {{1},{2,3},{4,6,7},{5}} => 1 {{1},{2,3},{4,6},{5,7}} => 1 {{1},{2,3},{4,6},{5},{7}} => 1 {{1,7},{2,3},{4},{5,6}} => 2 {{1},{2,3,7},{4},{5,6}} => 1 {{1},{2,3},{4,7},{5,6}} => 1 {{1},{2,3},{4},{5,6,7}} => 1 {{1},{2,3},{4},{5,6},{7}} => 1 {{1,7},{2,3},{4},{5},{6}} => 2 {{1},{2,3,7},{4},{5},{6}} => 1 {{1},{2,3},{4,7},{5},{6}} => 1 {{1},{2,3},{4},{5,7},{6}} => 1 {{1},{2,3},{4},{5},{6,7}} => 1 {{1},{2,3},{4},{5},{6},{7}} => 1 {{1,4,5,6,7},{2},{3}} => 5 {{1,4,5,6},{2,7},{3}} => 4 {{1,4,5,6},{2},{3,7}} => 1 {{1,4,5,6},{2},{3},{7}} => 1 {{1,4,5,7},{2,6},{3}} => 4 {{1,4,5},{2,6,7},{3}} => 3 {{1,4,5},{2,6},{3,7}} => 3 {{1,4,5},{2,6},{3},{7}} => 1 {{1,4,5,7},{2},{3,6}} => 2 {{1,4,5},{2,7},{3,6}} => 1 {{1,4,5},{2},{3,6,7}} => 1 {{1,4,5},{2},{3,6},{7}} => 1 {{1,4,5,7},{2},{3},{6}} => 2 {{1,4,5},{2,7},{3},{6}} => 1 {{1,4,5},{2},{3,7},{6}} => 1 {{1,4,5},{2},{3},{6,7}} => 1 {{1,4,5},{2},{3},{6},{7}} => 1 {{1,4,6,7},{2,5},{3}} => 4 {{1,4,6},{2,5,7},{3}} => 3 {{1,4,6},{2,5},{3,7}} => 2 {{1,4,6},{2,5},{3},{7}} => 1 {{1,4,7},{2,5,6},{3}} => 3 {{1,4},{2,5,6,7},{3}} => 2 {{1,4},{2,5,6},{3,7}} => 2 {{1,4},{2,5,6},{3},{7}} => 1 {{1,4,7},{2,5},{3,6}} => 3 {{1,4},{2,5,7},{3,6}} => 2 {{1,4},{2,5},{3,6,7}} => 2 {{1,4},{2,5},{3,6},{7}} => 2 {{1,4,7},{2,5},{3},{6}} => 2 {{1,4},{2,5,7},{3},{6}} => 1 {{1,4},{2,5},{3,7},{6}} => 1 {{1,4},{2,5},{3},{6,7}} => 1 {{1,4},{2,5},{3},{6},{7}} => 1 {{1,4,6,7},{2},{3,5}} => 3 {{1,4,6},{2,7},{3,5}} => 2 {{1,4,6},{2},{3,5,7}} => 1 {{1,4,6},{2},{3,5},{7}} => 1 {{1,4,7},{2,6},{3,5}} => 2 {{1,4},{2,6,7},{3,5}} => 1 {{1,4},{2,6},{3,5,7}} => 1 {{1,4},{2,6},{3,5},{7}} => 1 {{1,4,7},{2},{3,5,6}} => 2 {{1,4},{2,7},{3,5,6}} => 1 {{1,4},{2},{3,5,6,7}} => 1 {{1,4},{2},{3,5,6},{7}} => 1 {{1,4,7},{2},{3,5},{6}} => 2 {{1,4},{2,7},{3,5},{6}} => 1 {{1,4},{2},{3,5,7},{6}} => 1 {{1,4},{2},{3,5},{6,7}} => 1 {{1,4},{2},{3,5},{6},{7}} => 1 {{1,4,6,7},{2},{3},{5}} => 3 {{1,4,6},{2,7},{3},{5}} => 2 {{1,4,6},{2},{3,7},{5}} => 1 {{1,4,6},{2},{3},{5,7}} => 1 {{1,4,6},{2},{3},{5},{7}} => 1 {{1,4,7},{2,6},{3},{5}} => 2 {{1,4},{2,6,7},{3},{5}} => 1 {{1,4},{2,6},{3,7},{5}} => 1 {{1,4},{2,6},{3},{5,7}} => 1 {{1,4},{2,6},{3},{5},{7}} => 1 {{1,4,7},{2},{3,6},{5}} => 2 {{1,4},{2,7},{3,6},{5}} => 1 {{1,4},{2},{3,6,7},{5}} => 1 {{1,4},{2},{3,6},{5,7}} => 1 {{1,4},{2},{3,6},{5},{7}} => 1 {{1,4,7},{2},{3},{5,6}} => 2 {{1,4},{2,7},{3},{5,6}} => 1 {{1,4},{2},{3,7},{5,6}} => 1 {{1,4},{2},{3},{5,6,7}} => 1 {{1,4},{2},{3},{5,6},{7}} => 1 {{1,4,7},{2},{3},{5},{6}} => 2 {{1,4},{2,7},{3},{5},{6}} => 1 {{1,4},{2},{3,7},{5},{6}} => 1 {{1,4},{2},{3},{5,7},{6}} => 1 {{1,4},{2},{3},{5},{6,7}} => 1 {{1,4},{2},{3},{5},{6},{7}} => 1 {{1,5,6,7},{2,4},{3}} => 4 {{1,5,6},{2,4,7},{3}} => 3 {{1,5,6},{2,4},{3,7}} => 1 {{1,5,6},{2,4},{3},{7}} => 1 {{1,5,7},{2,4,6},{3}} => 3 {{1,5},{2,4,6,7},{3}} => 2 {{1,5},{2,4,6},{3,7}} => 2 {{1,5},{2,4,6},{3},{7}} => 1 {{1,5,7},{2,4},{3,6}} => 2 {{1,5},{2,4,7},{3,6}} => 1 {{1,5},{2,4},{3,6,7}} => 1 {{1,5},{2,4},{3,6},{7}} => 1 {{1,5,7},{2,4},{3},{6}} => 2 {{1,5},{2,4,7},{3},{6}} => 1 {{1,5},{2,4},{3,7},{6}} => 1 {{1,5},{2,4},{3},{6,7}} => 1 {{1,5},{2,4},{3},{6},{7}} => 1 {{1,6,7},{2,4,5},{3}} => 3 {{1,6},{2,4,5,7},{3}} => 2 {{1,6},{2,4,5},{3,7}} => 1 {{1,6},{2,4,5},{3},{7}} => 1 {{1,7},{2,4,5,6},{3}} => 2 {{1},{2,4,5,6,7},{3}} => 1 {{1},{2,4,5,6},{3,7}} => 1 {{1},{2,4,5,6},{3},{7}} => 1 {{1,7},{2,4,5},{3,6}} => 2 {{1},{2,4,5,7},{3,6}} => 1 {{1},{2,4,5},{3,6,7}} => 1 {{1},{2,4,5},{3,6},{7}} => 1 {{1,7},{2,4,5},{3},{6}} => 2 {{1},{2,4,5,7},{3},{6}} => 1 {{1},{2,4,5},{3,7},{6}} => 1 {{1},{2,4,5},{3},{6,7}} => 1 {{1},{2,4,5},{3},{6},{7}} => 1 {{1,6,7},{2,4},{3,5}} => 3 {{1,6},{2,4,7},{3,5}} => 2 {{1,6},{2,4},{3,5,7}} => 1 {{1,6},{2,4},{3,5},{7}} => 1 {{1,7},{2,4,6},{3,5}} => 2 {{1},{2,4,6,7},{3,5}} => 1 {{1},{2,4,6},{3,5,7}} => 1 {{1},{2,4,6},{3,5},{7}} => 1 {{1,7},{2,4},{3,5,6}} => 2 {{1},{2,4,7},{3,5,6}} => 1 {{1},{2,4},{3,5,6,7}} => 1 {{1},{2,4},{3,5,6},{7}} => 1 {{1,7},{2,4},{3,5},{6}} => 2 {{1},{2,4,7},{3,5},{6}} => 1 {{1},{2,4},{3,5,7},{6}} => 1 {{1},{2,4},{3,5},{6,7}} => 1 {{1},{2,4},{3,5},{6},{7}} => 1 {{1,6,7},{2,4},{3},{5}} => 3 {{1,6},{2,4,7},{3},{5}} => 2 {{1,6},{2,4},{3,7},{5}} => 1 {{1,6},{2,4},{3},{5,7}} => 1 {{1,6},{2,4},{3},{5},{7}} => 1 {{1,7},{2,4,6},{3},{5}} => 2 {{1},{2,4,6,7},{3},{5}} => 1 {{1},{2,4,6},{3,7},{5}} => 1 {{1},{2,4,6},{3},{5,7}} => 1 {{1},{2,4,6},{3},{5},{7}} => 1 {{1,7},{2,4},{3,6},{5}} => 2 {{1},{2,4,7},{3,6},{5}} => 1 {{1},{2,4},{3,6,7},{5}} => 1 {{1},{2,4},{3,6},{5,7}} => 1 {{1},{2,4},{3,6},{5},{7}} => 1 {{1,7},{2,4},{3},{5,6}} => 2 {{1},{2,4,7},{3},{5,6}} => 1 {{1},{2,4},{3,7},{5,6}} => 1 {{1},{2,4},{3},{5,6,7}} => 1 {{1},{2,4},{3},{5,6},{7}} => 1 {{1,7},{2,4},{3},{5},{6}} => 2 {{1},{2,4,7},{3},{5},{6}} => 1 {{1},{2,4},{3,7},{5},{6}} => 1 {{1},{2,4},{3},{5,7},{6}} => 1 {{1},{2,4},{3},{5},{6,7}} => 1 {{1},{2,4},{3},{5},{6},{7}} => 1 {{1,5,6,7},{2},{3,4}} => 4 {{1,5,6},{2,7},{3,4}} => 3 {{1,5,6},{2},{3,4,7}} => 1 {{1,5,6},{2},{3,4},{7}} => 1 {{1,5,7},{2,6},{3,4}} => 3 {{1,5},{2,6,7},{3,4}} => 2 {{1,5},{2,6},{3,4,7}} => 2 {{1,5},{2,6},{3,4},{7}} => 1 {{1,5,7},{2},{3,4,6}} => 2 {{1,5},{2,7},{3,4,6}} => 1 {{1,5},{2},{3,4,6,7}} => 1 {{1,5},{2},{3,4,6},{7}} => 1 {{1,5,7},{2},{3,4},{6}} => 2 {{1,5},{2,7},{3,4},{6}} => 1 {{1,5},{2},{3,4,7},{6}} => 1 {{1,5},{2},{3,4},{6,7}} => 1 {{1,5},{2},{3,4},{6},{7}} => 1 {{1,6,7},{2,5},{3,4}} => 3 {{1,6},{2,5,7},{3,4}} => 2 {{1,6},{2,5},{3,4,7}} => 1 {{1,6},{2,5},{3,4},{7}} => 1 {{1,7},{2,5,6},{3,4}} => 2 {{1},{2,5,6,7},{3,4}} => 1 {{1},{2,5,6},{3,4,7}} => 1 {{1},{2,5,6},{3,4},{7}} => 1 {{1,7},{2,5},{3,4,6}} => 2 {{1},{2,5,7},{3,4,6}} => 1 {{1},{2,5},{3,4,6,7}} => 1 {{1},{2,5},{3,4,6},{7}} => 1 {{1,7},{2,5},{3,4},{6}} => 2 {{1},{2,5,7},{3,4},{6}} => 1 {{1},{2,5},{3,4,7},{6}} => 1 {{1},{2,5},{3,4},{6,7}} => 1 {{1},{2,5},{3,4},{6},{7}} => 1 {{1,6,7},{2},{3,4,5}} => 3 {{1,6},{2,7},{3,4,5}} => 2 {{1,6},{2},{3,4,5,7}} => 1 {{1,6},{2},{3,4,5},{7}} => 1 {{1,7},{2,6},{3,4,5}} => 2 {{1},{2,6,7},{3,4,5}} => 1 {{1},{2,6},{3,4,5,7}} => 1 {{1},{2,6},{3,4,5},{7}} => 1 {{1,7},{2},{3,4,5,6}} => 2 {{1},{2,7},{3,4,5,6}} => 1 {{1},{2},{3,4,5,6,7}} => 1 {{1},{2},{3,4,5,6},{7}} => 1 {{1,7},{2},{3,4,5},{6}} => 2 {{1},{2,7},{3,4,5},{6}} => 1 {{1},{2},{3,4,5,7},{6}} => 1 {{1},{2},{3,4,5},{6,7}} => 1 {{1},{2},{3,4,5},{6},{7}} => 1 {{1,6,7},{2},{3,4},{5}} => 3 {{1,6},{2,7},{3,4},{5}} => 2 {{1,6},{2},{3,4,7},{5}} => 1 {{1,6},{2},{3,4},{5,7}} => 1 {{1,6},{2},{3,4},{5},{7}} => 1 {{1,7},{2,6},{3,4},{5}} => 2 {{1},{2,6,7},{3,4},{5}} => 1 {{1},{2,6},{3,4,7},{5}} => 1 {{1},{2,6},{3,4},{5,7}} => 1 {{1},{2,6},{3,4},{5},{7}} => 1 {{1,7},{2},{3,4,6},{5}} => 2 {{1},{2,7},{3,4,6},{5}} => 1 {{1},{2},{3,4,6,7},{5}} => 1 {{1},{2},{3,4,6},{5,7}} => 1 {{1},{2},{3,4,6},{5},{7}} => 1 {{1,7},{2},{3,4},{5,6}} => 2 {{1},{2,7},{3,4},{5,6}} => 1 {{1},{2},{3,4,7},{5,6}} => 1 {{1},{2},{3,4},{5,6,7}} => 1 {{1},{2},{3,4},{5,6},{7}} => 1 {{1,7},{2},{3,4},{5},{6}} => 2 {{1},{2,7},{3,4},{5},{6}} => 1 {{1},{2},{3,4,7},{5},{6}} => 1 {{1},{2},{3,4},{5,7},{6}} => 1 {{1},{2},{3,4},{5},{6,7}} => 1 {{1},{2},{3,4},{5},{6},{7}} => 1 {{1,5,6,7},{2},{3},{4}} => 4 {{1,5,6},{2,7},{3},{4}} => 3 {{1,5,6},{2},{3,7},{4}} => 1 {{1,5,6},{2},{3},{4,7}} => 1 {{1,5,6},{2},{3},{4},{7}} => 1 {{1,5,7},{2,6},{3},{4}} => 3 {{1,5},{2,6,7},{3},{4}} => 2 {{1,5},{2,6},{3,7},{4}} => 2 {{1,5},{2,6},{3},{4,7}} => 1 {{1,5},{2,6},{3},{4},{7}} => 1 {{1,5,7},{2},{3,6},{4}} => 2 {{1,5},{2,7},{3,6},{4}} => 1 {{1,5},{2},{3,6,7},{4}} => 1 {{1,5},{2},{3,6},{4,7}} => 1 {{1,5},{2},{3,6},{4},{7}} => 1 {{1,5,7},{2},{3},{4,6}} => 2 {{1,5},{2,7},{3},{4,6}} => 1 {{1,5},{2},{3,7},{4,6}} => 1 {{1,5},{2},{3},{4,6,7}} => 1 {{1,5},{2},{3},{4,6},{7}} => 1 {{1,5,7},{2},{3},{4},{6}} => 2 {{1,5},{2,7},{3},{4},{6}} => 1 {{1,5},{2},{3,7},{4},{6}} => 1 {{1,5},{2},{3},{4,7},{6}} => 1 {{1,5},{2},{3},{4},{6,7}} => 1 {{1,5},{2},{3},{4},{6},{7}} => 1 {{1,6,7},{2,5},{3},{4}} => 3 {{1,6},{2,5,7},{3},{4}} => 2 {{1,6},{2,5},{3,7},{4}} => 1 {{1,6},{2,5},{3},{4,7}} => 1 {{1,6},{2,5},{3},{4},{7}} => 1 {{1,7},{2,5,6},{3},{4}} => 2 {{1},{2,5,6,7},{3},{4}} => 1 {{1},{2,5,6},{3,7},{4}} => 1 {{1},{2,5,6},{3},{4,7}} => 1 {{1},{2,5,6},{3},{4},{7}} => 1 {{1,7},{2,5},{3,6},{4}} => 2 {{1},{2,5,7},{3,6},{4}} => 1 {{1},{2,5},{3,6,7},{4}} => 1 {{1},{2,5},{3,6},{4,7}} => 1 {{1},{2,5},{3,6},{4},{7}} => 1 {{1,7},{2,5},{3},{4,6}} => 2 {{1},{2,5,7},{3},{4,6}} => 1 {{1},{2,5},{3,7},{4,6}} => 1 {{1},{2,5},{3},{4,6,7}} => 1 {{1},{2,5},{3},{4,6},{7}} => 1 {{1,7},{2,5},{3},{4},{6}} => 2 {{1},{2,5,7},{3},{4},{6}} => 1 {{1},{2,5},{3,7},{4},{6}} => 1 {{1},{2,5},{3},{4,7},{6}} => 1 {{1},{2,5},{3},{4},{6,7}} => 1 {{1},{2,5},{3},{4},{6},{7}} => 1 {{1,6,7},{2},{3,5},{4}} => 3 {{1,6},{2,7},{3,5},{4}} => 2 {{1,6},{2},{3,5,7},{4}} => 1 {{1,6},{2},{3,5},{4,7}} => 1 {{1,6},{2},{3,5},{4},{7}} => 1 {{1,7},{2,6},{3,5},{4}} => 2 {{1},{2,6,7},{3,5},{4}} => 1 {{1},{2,6},{3,5,7},{4}} => 1 {{1},{2,6},{3,5},{4,7}} => 1 {{1},{2,6},{3,5},{4},{7}} => 1 {{1,7},{2},{3,5,6},{4}} => 2 {{1},{2,7},{3,5,6},{4}} => 1 {{1},{2},{3,5,6,7},{4}} => 1 {{1},{2},{3,5,6},{4,7}} => 1 {{1},{2},{3,5,6},{4},{7}} => 1 {{1,7},{2},{3,5},{4,6}} => 2 {{1},{2,7},{3,5},{4,6}} => 1 {{1},{2},{3,5,7},{4,6}} => 1 {{1},{2},{3,5},{4,6,7}} => 1 {{1},{2},{3,5},{4,6},{7}} => 1 {{1,7},{2},{3,5},{4},{6}} => 2 {{1},{2,7},{3,5},{4},{6}} => 1 {{1},{2},{3,5,7},{4},{6}} => 1 {{1},{2},{3,5},{4,7},{6}} => 1 {{1},{2},{3,5},{4},{6,7}} => 1 {{1},{2},{3,5},{4},{6},{7}} => 1 {{1,6,7},{2},{3},{4,5}} => 3 {{1,6},{2,7},{3},{4,5}} => 2 {{1,6},{2},{3,7},{4,5}} => 1 {{1,6},{2},{3},{4,5,7}} => 1 {{1,6},{2},{3},{4,5},{7}} => 1 {{1,7},{2,6},{3},{4,5}} => 2 {{1},{2,6,7},{3},{4,5}} => 1 {{1},{2,6},{3,7},{4,5}} => 1 {{1},{2,6},{3},{4,5,7}} => 1 {{1},{2,6},{3},{4,5},{7}} => 1 {{1,7},{2},{3,6},{4,5}} => 2 {{1},{2,7},{3,6},{4,5}} => 1 {{1},{2},{3,6,7},{4,5}} => 1 {{1},{2},{3,6},{4,5,7}} => 1 {{1},{2},{3,6},{4,5},{7}} => 1 {{1,7},{2},{3},{4,5,6}} => 2 {{1},{2,7},{3},{4,5,6}} => 1 {{1},{2},{3,7},{4,5,6}} => 1 {{1},{2},{3},{4,5,6,7}} => 1 {{1},{2},{3},{4,5,6},{7}} => 1 {{1,7},{2},{3},{4,5},{6}} => 2 {{1},{2,7},{3},{4,5},{6}} => 1 {{1},{2},{3,7},{4,5},{6}} => 1 {{1},{2},{3},{4,5,7},{6}} => 1 {{1},{2},{3},{4,5},{6,7}} => 1 {{1},{2},{3},{4,5},{6},{7}} => 1 {{1,6,7},{2},{3},{4},{5}} => 3 {{1,6},{2,7},{3},{4},{5}} => 2 {{1,6},{2},{3,7},{4},{5}} => 1 {{1,6},{2},{3},{4,7},{5}} => 1 {{1,6},{2},{3},{4},{5,7}} => 1 {{1,6},{2},{3},{4},{5},{7}} => 1 {{1,7},{2,6},{3},{4},{5}} => 2 {{1},{2,6,7},{3},{4},{5}} => 1 {{1},{2,6},{3,7},{4},{5}} => 1 {{1},{2,6},{3},{4,7},{5}} => 1 {{1},{2,6},{3},{4},{5,7}} => 1 {{1},{2,6},{3},{4},{5},{7}} => 1 {{1,7},{2},{3,6},{4},{5}} => 2 {{1},{2,7},{3,6},{4},{5}} => 1 {{1},{2},{3,6,7},{4},{5}} => 1 {{1},{2},{3,6},{4,7},{5}} => 1 {{1},{2},{3,6},{4},{5,7}} => 1 {{1},{2},{3,6},{4},{5},{7}} => 1 {{1,7},{2},{3},{4,6},{5}} => 2 {{1},{2,7},{3},{4,6},{5}} => 1 {{1},{2},{3,7},{4,6},{5}} => 1 {{1},{2},{3},{4,6,7},{5}} => 1 {{1},{2},{3},{4,6},{5,7}} => 1 {{1},{2},{3},{4,6},{5},{7}} => 1 {{1,7},{2},{3},{4},{5,6}} => 2 {{1},{2,7},{3},{4},{5,6}} => 1 {{1},{2},{3,7},{4},{5,6}} => 1 {{1},{2},{3},{4,7},{5,6}} => 1 {{1},{2},{3},{4},{5,6,7}} => 1 {{1},{2},{3},{4},{5,6},{7}} => 1 {{1,7},{2},{3},{4},{5},{6}} => 2 {{1},{2,7},{3},{4},{5},{6}} => 1 {{1},{2},{3,7},{4},{5},{6}} => 1 {{1},{2},{3},{4,7},{5},{6}} => 1 {{1},{2},{3},{4},{5,7},{6}} => 1 {{1},{2},{3},{4},{5},{6,7}} => 1 {{1},{2},{3},{4},{5},{6},{7}} => 1 {{1},{2},{3,4,5,6,7,8}} => 1 {{1},{2,4,5,6,7,8},{3}} => 1 {{1},{2,3,5,6,7,8},{4}} => 1 {{1},{2,3,4,6,7,8},{5}} => 1 {{1},{2,3,4,5,7,8},{6}} => 1 {{1},{2,3,4,5,6,7},{8}} => 1 {{1},{2,3,4,5,6,8},{7}} => 1 {{1},{2,3,4,5,6,7,8}} => 1 {{1,2},{3,4,5,6,7,8}} => 2 {{1,4,5,6,7,8},{2},{3}} => 6 {{1,3,5,6,7,8},{2},{4}} => 5 {{1,3,4,5,6,7,8},{2}} => 7 {{1,4,5,6,7,8},{2,3}} => 6 {{1,2,4,5,6,7,8},{3}} => 7 {{1,2,5,6,7,8},{3,4}} => 6 {{1,2,3,5,6,7,8},{4}} => 7 {{1,2,3,6,7,8},{4,5}} => 6 {{1,2,3,4,6,7,8},{5}} => 7 {{1,2,3,4,5,6},{7,8}} => 6 {{1,2,3,4,7,8},{5,6}} => 6 {{1,2,3,4,5,7,8},{6}} => 7 {{1,2,3,4,5,6,7},{8}} => 7 {{1,8},{2,3,4,5,6,7}} => 2 {{1,2,3,4,5,8},{6,7}} => 6 {{1,2,3,4,5,6,8},{7}} => 7 {{1,2,3,4,5,6,7,8}} => 8 {{1,3,5,6,7,8},{2,4}} => 6 {{1,3,4,6,7,8},{2,5}} => 6 {{1,2,4,6,7,8},{3,5}} => 6 {{1,3,4,5,7,8},{2,6}} => 6 {{1,2,4,5,7,8},{3,6}} => 6 {{1,2,3,5,7,8},{4,6}} => 6 {{1,3,4,5,6,8},{2,7}} => 6 {{1,2,4,5,6,8},{3,7}} => 6 {{1,2,3,5,6,8},{4,7}} => 6 {{1,2,3,4,6,8},{5,7}} => 6 {{1,3,4,5,6,7},{2,8}} => 6 {{1,2,4,5,6,7},{3,8}} => 6 {{1,2,3,5,6,7},{4,8}} => 6 {{1,2,3,4,6,7},{5,8}} => 6 {{1,2,3,4,5,7},{6,8}} => 6 {{1,3},{2,4,5,6,7,8}} => 2 {{1,4},{2,3,5,6,7,8}} => 2 {{1,5},{2,3,4,6,7,8}} => 2 {{1,6},{2,3,4,5,7,8}} => 2 {{1,7},{2,3,4,5,6,8}} => 2 ----------------------------------------------------------------------------- Created: May 30, 2017 at 09:43 by Martin Rubey ----------------------------------------------------------------------------- Last Updated: Nov 18, 2017 at 22:17 by Martin Rubey