***************************************************************************** * 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: St001614 ----------------------------------------------------------------------------- Collection: Skew partitions ----------------------------------------------------------------------------- Description: The cyclic permutation representation number of a skew partition. This is the size of the largest cyclic group $C$ such that the fake degree is the character of a permutation representation of $C$. See [[St001527]] for the restriction of this statistic to integer partitions. ----------------------------------------------------------------------------- References: [1] Alexandersson, P., Amini, N. The cone of cyclic sieving phenomena [[MathSciNet:3922156]] ----------------------------------------------------------------------------- Code: def fdeg(p): R. = QQ[] f = prod([ 1-q*q^j for j in range(p.size()) ]) b = sum([ j*k for j, k in enumerate(p) ]) return (q^b)*(f/p.hook_polynomial(q,q)) def fake_degree(a): R. = QQ[] if a == 0: return R(0) else: z = SymmetricFunctions(QQ).schur()(a) return R(sum([z.coefficient(p)*fdeg(p) for p in z.support()])) def standard_CSP(p, r): P = parent(p) q = P.gen() PP. = PolynomialRing(QQ) R. = PP.quo(q^r-1) return R(p).lift() def orbit_lengths(p, r): p = standard_CSP(p, r) p = {k: p[k] for k in range(r)} divs = sorted(divisors(r)) O = dict() for d in divs: count = p[r-d] if count: O[r//d] = count if count: for k in range(0, r, d): p[k] -= count assert all(c == 0 for c in p.values()), "%s" % p return O def has_cyclic_permutation_representation(p, r): if p == 0: return True try: lo = orbit_lengths(p, r) except AssertionError: return False return all(v in ZZ and ZZ(v) >= 0 for v in lo.values()) def statistic(la): s = SymmetricFunctions(QQ).schur() f = fake_degree(s(la)) for r in reversed(divisors(la.size())): if has_cyclic_permutation_representation(f, r): return r # alternative implementation using Theorem 2.7 def has_cyclic_permutation_representation_slow(p, r): for k in divisors(r): if sum(moebius(k//j) * p.subs(q=QQbar.zeta(r)^j) for j in divisors(k)) < 0: return False return True def statistic_slow(la): s = SymmetricFunctions(QQ).schur() f = fake_degree(s(la)) for r in reversed(divisors(la.size())): if has_cyclic_permutation_representation_slow(f, r): return r ----------------------------------------------------------------------------- Statistic values: [[1],[]] => 1 [[2],[]] => 2 [[1,1],[]] => 1 [[2,1],[1]] => 2 [[3],[]] => 3 [[2,1],[]] => 1 [[3,1],[1]] => 3 [[2,2],[1]] => 1 [[3,2],[2]] => 3 [[1,1,1],[]] => 3 [[2,2,1],[1,1]] => 3 [[2,1,1],[1]] => 3 [[3,2,1],[2,1]] => 3 [[4],[]] => 4 [[3,1],[]] => 1 [[4,1],[1]] => 4 [[2,2],[]] => 4 [[3,2],[1]] => 2 [[4,2],[2]] => 4 [[2,1,1],[]] => 1 [[3,2,1],[1,1]] => 4 [[3,1,1],[1]] => 1 [[4,2,1],[2,1]] => 4 [[3,3],[2]] => 1 [[4,3],[3]] => 4 [[2,2,1],[1]] => 4 [[3,3,1],[2,1]] => 4 [[3,2,1],[2]] => 4 [[4,3,1],[3,1]] => 4 [[2,2,2],[1,1]] => 1 [[3,3,2],[2,2]] => 1 [[3,2,2],[2,1]] => 4 [[4,3,2],[3,2]] => 4 [[1,1,1,1],[]] => 2 [[2,2,2,1],[1,1,1]] => 4 [[2,2,1,1],[1,1]] => 4 [[3,3,2,1],[2,2,1]] => 4 [[2,1,1,1],[1]] => 4 [[3,2,2,1],[2,1,1]] => 4 [[3,2,1,1],[2,1]] => 4 [[4,3,2,1],[3,2,1]] => 4 [[5],[]] => 5 [[4,1],[]] => 1 [[5,1],[1]] => 5 [[3,2],[]] => 5 [[4,2],[1]] => 1 [[5,2],[2]] => 5 [[3,1,1],[]] => 5 [[4,2,1],[1,1]] => 5 [[4,1,1],[1]] => 5 [[5,2,1],[2,1]] => 5 [[3,3],[1]] => 5 [[4,3],[2]] => 1 [[5,3],[3]] => 5 [[2,2,1],[]] => 5 [[3,3,1],[1,1]] => 5 [[3,2,1],[1]] => 5 [[4,3,1],[2,1]] => 5 [[4,2,1],[2]] => 5 [[5,3,1],[3,1]] => 5 [[3,2,2],[1,1]] => 5 [[4,3,2],[2,2]] => 5 [[4,2,2],[2,1]] => 5 [[5,3,2],[3,2]] => 5 [[2,1,1,1],[]] => 1 [[3,2,2,1],[1,1,1]] => 5 [[3,2,1,1],[1,1]] => 5 [[4,3,2,1],[2,2,1]] => 5 [[3,1,1,1],[1]] => 5 [[4,2,2,1],[2,1,1]] => 5 [[4,2,1,1],[2,1]] => 5 [[5,3,2,1],[3,2,1]] => 5 [[4,4],[3]] => 1 [[5,4],[4]] => 5 [[3,3,1],[2]] => 5 [[4,4,1],[3,1]] => 5 [[4,3,1],[3]] => 5 [[5,4,1],[4,1]] => 5 [[2,2,2],[1]] => 5 [[3,3,2],[2,1]] => 5 [[4,4,2],[3,2]] => 5 [[3,2,2],[2]] => 5 [[4,3,2],[3,1]] => 5 [[5,4,2],[4,2]] => 5 [[2,2,1,1],[1]] => 1 [[3,3,2,1],[2,1,1]] => 5 [[3,3,1,1],[2,1]] => 5 [[4,4,2,1],[3,2,1]] => 5 [[3,2,1,1],[2]] => 5 [[4,3,2,1],[3,1,1]] => 5 [[4,3,1,1],[3,1]] => 5 [[5,4,2,1],[4,2,1]] => 5 [[3,3,3],[2,2]] => 5 [[4,4,3],[3,3]] => 5 [[4,3,3],[3,2]] => 5 [[5,4,3],[4,3]] => 5 [[2,2,2,1],[1,1]] => 1 [[3,3,3,1],[2,2,1]] => 5 [[3,3,2,1],[2,2]] => 5 [[4,4,3,1],[3,3,1]] => 5 [[3,2,2,1],[2,1]] => 5 [[4,3,3,1],[3,2,1]] => 5 [[4,3,2,1],[3,2]] => 5 [[5,4,3,1],[4,3,1]] => 5 [[2,2,2,2],[1,1,1]] => 1 [[3,3,3,2],[2,2,2]] => 5 [[3,3,2,2],[2,2,1]] => 5 [[4,4,3,2],[3,3,2]] => 5 [[3,2,2,2],[2,1,1]] => 5 [[4,3,3,2],[3,2,2]] => 5 [[4,3,2,2],[3,2,1]] => 5 [[5,4,3,2],[4,3,2]] => 5 [[1,1,1,1,1],[]] => 5 [[2,2,2,2,1],[1,1,1,1]] => 5 [[2,2,2,1,1],[1,1,1]] => 5 [[3,3,3,2,1],[2,2,2,1]] => 5 [[2,2,1,1,1],[1,1]] => 5 [[3,3,2,2,1],[2,2,1,1]] => 5 [[3,3,2,1,1],[2,2,1]] => 5 [[4,4,3,2,1],[3,3,2,1]] => 5 [[2,1,1,1,1],[1]] => 5 [[3,2,2,2,1],[2,1,1,1]] => 5 [[3,2,2,1,1],[2,1,1]] => 5 [[4,3,3,2,1],[3,2,2,1]] => 5 [[3,2,1,1,1],[2,1]] => 5 [[4,3,2,2,1],[3,2,1,1]] => 5 [[4,3,2,1,1],[3,2,1]] => 5 [[5,4,3,2,1],[4,3,2,1]] => 5 [[6],[]] => 6 [[5,1],[]] => 1 [[6,1],[1]] => 6 [[4,2],[]] => 6 [[5,2],[1]] => 2 [[6,2],[2]] => 6 [[4,1,1],[]] => 3 [[5,2,1],[1,1]] => 6 [[5,1,1],[1]] => 3 [[6,2,1],[2,1]] => 6 [[3,3],[]] => 3 [[4,3],[1]] => 6 [[5,3],[2]] => 3 [[6,3],[3]] => 6 [[3,2,1],[]] => 2 [[4,3,1],[1,1]] => 6 [[4,2,1],[1]] => 2 [[5,3,1],[2,1]] => 6 [[5,2,1],[2]] => 2 [[6,3,1],[3,1]] => 6 [[4,2,2],[1,1]] => 1 [[5,3,2],[2,2]] => 3 [[5,2,2],[2,1]] => 2 [[6,3,2],[3,2]] => 6 [[3,1,1,1],[]] => 3 [[4,2,2,1],[1,1,1]] => 6 [[4,2,1,1],[1,1]] => 6 [[5,3,2,1],[2,2,1]] => 6 [[4,1,1,1],[1]] => 6 [[5,2,2,1],[2,1,1]] => 6 [[5,2,1,1],[2,1]] => 6 [[6,3,2,1],[3,2,1]] => 6 [[4,4],[2]] => 6 [[5,4],[3]] => 2 [[6,4],[4]] => 6 [[3,3,1],[1]] => 3 [[4,4,1],[2,1]] => 6 [[4,3,1],[2]] => 3 [[5,4,1],[3,1]] => 6 [[5,3,1],[3]] => 3 [[6,4,1],[4,1]] => 6 [[2,2,2],[]] => 6 [[3,3,2],[1,1]] => 6 [[4,4,2],[2,2]] => 6 [[3,2,2],[1]] => 6 [[4,3,2],[2,1]] => 6 [[5,4,2],[3,2]] => 6 [[4,2,2],[2]] => 6 [[5,3,2],[3,1]] => 6 [[6,4,2],[4,2]] => 6 [[2,2,1,1],[]] => 3 [[3,3,2,1],[1,1,1]] => 6 [[3,3,1,1],[1,1]] => 3 [[4,4,2,1],[2,2,1]] => 6 [[3,2,1,1],[1]] => 1 [[4,3,2,1],[2,1,1]] => 6 [[4,3,1,1],[2,1]] => 3 [[5,4,2,1],[3,2,1]] => 6 [[4,2,1,1],[2]] => 3 [[5,3,2,1],[3,1,1]] => 6 [[5,3,1,1],[3,1]] => 3 [[6,4,2,1],[4,2,1]] => 6 [[4,3,3],[2,2]] => 1 [[5,4,3],[3,3]] => 2 [[5,3,3],[3,2]] => 3 [[6,4,3],[4,3]] => 6 [[3,2,2,1],[1,1]] => 3 [[4,3,3,1],[2,2,1]] => 6 [[4,3,2,1],[2,2]] => 6 [[5,4,3,1],[3,3,1]] => 6 [[4,2,2,1],[2,1]] => 6 [[5,3,3,1],[3,2,1]] => 6 [[5,3,2,1],[3,2]] => 6 [[6,4,3,1],[4,3,1]] => 6 [[3,2,2,2],[1,1,1]] => 3 [[4,3,3,2],[2,2,2]] => 3 [[4,3,2,2],[2,2,1]] => 6 [[5,4,3,2],[3,3,2]] => 6 [[4,2,2,2],[2,1,1]] => 3 [[5,3,3,2],[3,2,2]] => 3 [[5,3,2,2],[3,2,1]] => 6 [[6,4,3,2],[4,3,2]] => 6 [[2,1,1,1,1],[]] => 2 [[3,2,2,2,1],[1,1,1,1]] => 6 [[3,2,2,1,1],[1,1,1]] => 6 [[4,3,3,2,1],[2,2,2,1]] => 6 [[3,2,1,1,1],[1,1]] => 2 [[4,3,2,2,1],[2,2,1,1]] => 6 [[4,3,2,1,1],[2,2,1]] => 6 [[5,4,3,2,1],[3,3,2,1]] => 6 [[3,1,1,1,1],[1]] => 6 [[4,2,2,2,1],[2,1,1,1]] => 6 [[4,2,2,1,1],[2,1,1]] => 6 [[5,3,3,2,1],[3,2,2,1]] => 6 [[4,2,1,1,1],[2,1]] => 6 [[5,3,2,2,1],[3,2,1,1]] => 6 [[5,3,2,1,1],[3,2,1]] => 6 [[6,4,3,2,1],[4,3,2,1]] => 6 [[5,5],[4]] => 1 [[6,5],[5]] => 6 [[4,4,1],[3]] => 6 [[5,5,1],[4,1]] => 6 [[5,4,1],[4]] => 6 [[6,5,1],[5,1]] => 6 [[3,3,2],[2]] => 3 [[4,4,2],[3,1]] => 3 [[5,5,2],[4,2]] => 3 [[4,3,2],[3]] => 6 [[5,4,2],[4,1]] => 6 [[6,5,2],[5,2]] => 6 [[3,3,1,1],[2]] => 2 [[4,4,2,1],[3,1,1]] => 6 [[4,4,1,1],[3,1]] => 6 [[5,5,2,1],[4,2,1]] => 6 [[4,3,1,1],[3]] => 6 [[5,4,2,1],[4,1,1]] => 6 [[5,4,1,1],[4,1]] => 6 [[6,5,2,1],[5,2,1]] => 6 [[3,3,3],[2,1]] => 2 [[4,4,3],[3,2]] => 2 [[5,5,3],[4,3]] => 2 [[4,3,3],[3,1]] => 6 [[5,4,3],[4,2]] => 6 [[6,5,3],[5,3]] => 6 [[2,2,2,1],[1]] => 6 [[3,3,3,1],[2,1,1]] => 6 [[3,3,2,1],[2,1]] => 3 [[4,4,3,1],[3,2,1]] => 6 [[4,4,2,1],[3,2]] => 6 [[5,5,3,1],[4,3,1]] => 6 [[3,2,2,1],[2]] => 6 [[4,3,3,1],[3,1,1]] => 6 [[4,3,2,1],[3,1]] => 6 [[5,4,3,1],[4,2,1]] => 6 [[5,4,2,1],[4,2]] => 6 [[6,5,3,1],[5,3,1]] => 6 [[3,3,2,2],[2,1,1]] => 3 [[4,4,3,2],[3,2,2]] => 6 [[4,4,2,2],[3,2,1]] => 6 [[5,5,3,2],[4,3,2]] => 6 [[4,3,2,2],[3,1,1]] => 6 [[5,4,3,2],[4,2,2]] => 6 [[5,4,2,2],[4,2,1]] => 6 [[6,5,3,2],[5,3,2]] => 6 [[2,2,1,1,1],[1]] => 1 [[3,3,2,2,1],[2,1,1,1]] => 6 [[3,3,2,1,1],[2,1,1]] => 3 [[4,4,3,2,1],[3,2,2,1]] => 6 [[3,3,1,1,1],[2,1]] => 2 [[4,4,2,2,1],[3,2,1,1]] => 6 [[4,4,2,1,1],[3,2,1]] => 6 [[5,5,3,2,1],[4,3,2,1]] => 6 [[3,2,1,1,1],[2]] => 6 [[4,3,2,2,1],[3,1,1,1]] => 6 [[4,3,2,1,1],[3,1,1]] => 6 [[5,4,3,2,1],[4,2,2,1]] => 6 [[4,3,1,1,1],[3,1]] => 6 [[5,4,2,2,1],[4,2,1,1]] => 6 [[5,4,2,1,1],[4,2,1]] => 6 [[6,5,3,2,1],[5,3,2,1]] => 6 [[4,4,4],[3,3]] => 3 [[5,5,4],[4,4]] => 3 [[5,4,4],[4,3]] => 6 [[6,5,4],[5,4]] => 6 [[3,3,3,1],[2,2]] => 2 [[4,4,4,1],[3,3,1]] => 6 [[4,4,3,1],[3,3]] => 6 [[5,5,4,1],[4,4,1]] => 6 [[4,3,3,1],[3,2]] => 6 [[5,4,4,1],[4,3,1]] => 6 [[5,4,3,1],[4,3]] => 6 [[6,5,4,1],[5,4,1]] => 6 [[2,2,2,2],[1,1]] => 3 [[3,3,3,2],[2,2,1]] => 1 [[4,4,4,2],[3,3,2]] => 3 [[3,3,2,2],[2,2]] => 3 [[4,4,3,2],[3,3,1]] => 3 [[5,5,4,2],[4,4,2]] => 3 [[3,2,2,2],[2,1]] => 6 [[4,3,3,2],[3,2,1]] => 6 [[5,4,4,2],[4,3,2]] => 6 [[4,3,2,2],[3,2]] => 6 [[5,4,3,2],[4,3,1]] => 6 [[6,5,4,2],[5,4,2]] => 6 [[2,2,2,1,1],[1,1]] => 6 [[3,3,3,2,1],[2,2,1,1]] => 6 [[3,3,3,1,1],[2,2,1]] => 6 [[4,4,4,2,1],[3,3,2,1]] => 6 [[3,3,2,1,1],[2,2]] => 6 [[4,4,3,2,1],[3,3,1,1]] => 6 [[4,4,3,1,1],[3,3,1]] => 6 [[5,5,4,2,1],[4,4,2,1]] => 6 [[3,2,2,1,1],[2,1]] => 6 [[4,3,3,2,1],[3,2,1,1]] => 6 [[4,3,3,1,1],[3,2,1]] => 6 [[5,4,4,2,1],[4,3,2,1]] => 6 [[4,3,2,1,1],[3,2]] => 6 [[5,4,3,2,1],[4,3,1,1]] => 6 [[5,4,3,1,1],[4,3,1]] => 6 [[6,5,4,2,1],[5,4,2,1]] => 6 [[3,3,3,3],[2,2,2]] => 3 [[4,4,4,3],[3,3,3]] => 6 [[4,4,3,3],[3,3,2]] => 6 [[5,5,4,3],[4,4,3]] => 6 [[4,3,3,3],[3,2,2]] => 6 [[5,4,4,3],[4,3,3]] => 6 [[5,4,3,3],[4,3,2]] => 6 [[6,5,4,3],[5,4,3]] => 6 [[2,2,2,2,1],[1,1,1]] => 1 [[3,3,3,3,1],[2,2,2,1]] => 6 [[3,3,3,2,1],[2,2,2]] => 2 [[4,4,4,3,1],[3,3,3,1]] => 6 [[3,3,2,2,1],[2,2,1]] => 3 [[4,4,3,3,1],[3,3,2,1]] => 6 [[4,4,3,2,1],[3,3,2]] => 6 [[5,5,4,3,1],[4,4,3,1]] => 6 [[3,2,2,2,1],[2,1,1]] => 6 [[4,3,3,3,1],[3,2,2,1]] => 6 [[4,3,3,2,1],[3,2,2]] => 6 [[5,4,4,3,1],[4,3,3,1]] => 6 [[4,3,2,2,1],[3,2,1]] => 6 [[5,4,3,3,1],[4,3,2,1]] => 6 [[5,4,3,2,1],[4,3,2]] => 6 [[6,5,4,3,1],[5,4,3,1]] => 6 [[2,2,2,2,2],[1,1,1,1]] => 2 [[3,3,3,3,2],[2,2,2,2]] => 6 [[3,3,3,2,2],[2,2,2,1]] => 2 [[4,4,4,3,2],[3,3,3,2]] => 6 [[3,3,2,2,2],[2,2,1,1]] => 6 [[4,4,3,3,2],[3,3,2,2]] => 6 [[4,4,3,2,2],[3,3,2,1]] => 6 [[5,5,4,3,2],[4,4,3,2]] => 6 [[3,2,2,2,2],[2,1,1,1]] => 6 [[4,3,3,3,2],[3,2,2,2]] => 6 [[4,3,3,2,2],[3,2,2,1]] => 6 [[5,4,4,3,2],[4,3,3,2]] => 6 [[4,3,2,2,2],[3,2,1,1]] => 6 [[5,4,3,3,2],[4,3,2,2]] => 6 [[5,4,3,2,2],[4,3,2,1]] => 6 [[6,5,4,3,2],[5,4,3,2]] => 6 [[1,1,1,1,1,1],[]] => 3 [[2,2,2,2,2,1],[1,1,1,1,1]] => 6 [[2,2,2,2,1,1],[1,1,1,1]] => 3 [[3,3,3,3,2,1],[2,2,2,2,1]] => 6 [[2,2,2,1,1,1],[1,1,1]] => 6 [[3,3,3,2,2,1],[2,2,2,1,1]] => 6 [[3,3,3,2,1,1],[2,2,2,1]] => 6 [[4,4,4,3,2,1],[3,3,3,2,1]] => 6 [[2,2,1,1,1,1],[1,1]] => 3 [[3,3,2,2,2,1],[2,2,1,1,1]] => 6 [[3,3,2,2,1,1],[2,2,1,1]] => 3 [[4,4,3,3,2,1],[3,3,2,2,1]] => 6 [[3,3,2,1,1,1],[2,2,1]] => 6 [[4,4,3,2,2,1],[3,3,2,1,1]] => 6 [[4,4,3,2,1,1],[3,3,2,1]] => 6 [[5,5,4,3,2,1],[4,4,3,2,1]] => 6 [[2,1,1,1,1,1],[1]] => 6 [[3,2,2,2,2,1],[2,1,1,1,1]] => 6 [[3,2,2,2,1,1],[2,1,1,1]] => 6 [[4,3,3,3,2,1],[3,2,2,2,1]] => 6 [[3,2,2,1,1,1],[2,1,1]] => 6 [[4,3,3,2,2,1],[3,2,2,1,1]] => 6 [[4,3,3,2,1,1],[3,2,2,1]] => 6 [[5,4,4,3,2,1],[4,3,3,2,1]] => 6 [[3,2,1,1,1,1],[2,1]] => 6 [[4,3,2,2,2,1],[3,2,1,1,1]] => 6 [[4,3,2,2,1,1],[3,2,1,1]] => 6 [[5,4,3,3,2,1],[4,3,2,2,1]] => 6 [[4,3,2,1,1,1],[3,2,1]] => 6 [[5,4,3,2,2,1],[4,3,2,1,1]] => 6 [[5,4,3,2,1,1],[4,3,2,1]] => 6 [[6,5,4,3,2,1],[5,4,3,2,1]] => 6 [[7],[]] => 7 [[6,1],[]] => 1 [[7,1],[1]] => 7 [[5,2],[]] => 7 [[6,2],[1]] => 1 [[7,2],[2]] => 7 [[5,1,1],[]] => 7 [[6,2,1],[1,1]] => 7 [[6,1,1],[1]] => 7 [[7,2,1],[2,1]] => 7 [[4,3],[]] => 7 [[5,3],[1]] => 7 [[6,3],[2]] => 1 [[7,3],[3]] => 7 [[4,2,1],[]] => 7 [[5,3,1],[1,1]] => 7 [[5,2,1],[1]] => 7 [[6,3,1],[2,1]] => 7 [[6,2,1],[2]] => 7 [[7,3,1],[3,1]] => 7 [[5,2,2],[1,1]] => 7 [[6,3,2],[2,2]] => 7 [[6,2,2],[2,1]] => 7 [[7,3,2],[3,2]] => 7 [[4,1,1,1],[]] => 1 [[5,2,2,1],[1,1,1]] => 7 [[5,2,1,1],[1,1]] => 7 [[6,3,2,1],[2,2,1]] => 7 [[5,1,1,1],[1]] => 7 [[6,2,2,1],[2,1,1]] => 7 [[6,2,1,1],[2,1]] => 7 [[7,3,2,1],[3,2,1]] => 7 [[4,4],[1]] => 7 [[5,4],[2]] => 7 [[6,4],[3]] => 1 [[7,4],[4]] => 7 [[3,3,1],[]] => 7 [[4,4,1],[1,1]] => 7 [[4,3,1],[1]] => 7 [[5,4,1],[2,1]] => 7 [[5,3,1],[2]] => 7 [[6,4,1],[3,1]] => 7 [[6,3,1],[3]] => 7 [[7,4,1],[4,1]] => 7 [[3,2,2],[]] => 7 [[4,3,2],[1,1]] => 7 [[5,4,2],[2,2]] => 7 [[4,2,2],[1]] => 7 [[5,3,2],[2,1]] => 7 [[6,4,2],[3,2]] => 7 [[5,2,2],[2]] => 7 [[6,3,2],[3,1]] => 7 [[7,4,2],[4,2]] => 7 [[3,2,1,1],[]] => 7 [[4,3,2,1],[1,1,1]] => 7 [[4,3,1,1],[1,1]] => 7 [[5,4,2,1],[2,2,1]] => 7 [[4,2,1,1],[1]] => 1 [[5,3,2,1],[2,1,1]] => 7 [[5,3,1,1],[2,1]] => 7 [[6,4,2,1],[3,2,1]] => 7 [[5,2,1,1],[2]] => 7 [[6,3,2,1],[3,1,1]] => 7 [[6,3,1,1],[3,1]] => 7 [[7,4,2,1],[4,2,1]] => 7 [[5,3,3],[2,2]] => 7 [[6,4,3],[3,3]] => 7 [[6,3,3],[3,2]] => 7 [[7,4,3],[4,3]] => 7 [[4,2,2,1],[1,1]] => 1 [[5,3,3,1],[2,2,1]] => 7 [[5,3,2,1],[2,2]] => 7 [[6,4,3,1],[3,3,1]] => 7 [[5,2,2,1],[2,1]] => 7 [[6,3,3,1],[3,2,1]] => 7 [[6,3,2,1],[3,2]] => 7 [[7,4,3,1],[4,3,1]] => 7 [[4,2,2,2],[1,1,1]] => 1 [[5,3,3,2],[2,2,2]] => 7 [[5,3,2,2],[2,2,1]] => 7 [[6,4,3,2],[3,3,2]] => 7 [[5,2,2,2],[2,1,1]] => 7 [[6,3,3,2],[3,2,2]] => 7 [[6,3,2,2],[3,2,1]] => 7 [[7,4,3,2],[4,3,2]] => 7 [[3,1,1,1,1],[]] => 7 [[4,2,2,2,1],[1,1,1,1]] => 7 [[4,2,2,1,1],[1,1,1]] => 7 [[5,3,3,2,1],[2,2,2,1]] => 7 [[4,2,1,1,1],[1,1]] => 7 [[5,3,2,2,1],[2,2,1,1]] => 7 [[5,3,2,1,1],[2,2,1]] => 7 [[6,4,3,2,1],[3,3,2,1]] => 7 [[4,1,1,1,1],[1]] => 7 [[5,2,2,2,1],[2,1,1,1]] => 7 [[5,2,2,1,1],[2,1,1]] => 7 [[6,3,3,2,1],[3,2,2,1]] => 7 [[5,2,1,1,1],[2,1]] => 7 [[6,3,2,2,1],[3,2,1,1]] => 7 [[6,3,2,1,1],[3,2,1]] => 7 [[7,4,3,2,1],[4,3,2,1]] => 7 [[5,5],[3]] => 7 [[6,5],[4]] => 1 [[7,5],[5]] => 7 [[4,4,1],[2]] => 7 [[5,5,1],[3,1]] => 7 [[5,4,1],[3]] => 7 [[6,5,1],[4,1]] => 7 [[6,4,1],[4]] => 7 [[7,5,1],[5,1]] => 7 [[3,3,2],[1]] => 7 [[4,4,2],[2,1]] => 7 [[5,5,2],[3,2]] => 7 [[4,3,2],[2]] => 7 [[5,4,2],[3,1]] => 7 [[6,5,2],[4,2]] => 7 [[5,3,2],[3]] => 7 [[6,4,2],[4,1]] => 7 [[7,5,2],[5,2]] => 7 [[3,3,1,1],[1]] => 7 [[4,4,2,1],[2,1,1]] => 7 [[4,4,1,1],[2,1]] => 7 [[5,5,2,1],[3,2,1]] => 7 [[4,3,1,1],[2]] => 1 [[5,4,2,1],[3,1,1]] => 7 [[5,4,1,1],[3,1]] => 7 [[6,5,2,1],[4,2,1]] => 7 [[5,3,1,1],[3]] => 7 [[6,4,2,1],[4,1,1]] => 7 [[6,4,1,1],[4,1]] => 7 [[7,5,2,1],[5,2,1]] => 7 [[3,3,3],[1,1]] => 7 [[4,4,3],[2,2]] => 7 [[5,5,3],[3,3]] => 7 [[4,3,3],[2,1]] => 7 [[5,4,3],[3,2]] => 7 [[6,5,3],[4,3]] => 7 [[5,3,3],[3,1]] => 7 [[6,4,3],[4,2]] => 7 [[7,5,3],[5,3]] => 7 [[2,2,2,1],[]] => 7 [[3,3,3,1],[1,1,1]] => 7 [[3,3,2,1],[1,1]] => 7 [[4,4,3,1],[2,2,1]] => 7 [[4,4,2,1],[2,2]] => 7 [[5,5,3,1],[3,3,1]] => 7 [[3,2,2,1],[1]] => 7 [[4,3,3,1],[2,1,1]] => 7 [[4,3,2,1],[2,1]] => 1 [[5,4,3,1],[3,2,1]] => 7 [[5,4,2,1],[3,2]] => 7 [[6,5,3,1],[4,3,1]] => 7 [[4,2,2,1],[2]] => 7 [[5,3,3,1],[3,1,1]] => 7 [[5,3,2,1],[3,1]] => 7 [[6,4,3,1],[4,2,1]] => 7 [[6,4,2,1],[4,2]] => 7 [[7,5,3,1],[5,3,1]] => 7 [[3,3,2,2],[1,1,1]] => 7 [[4,4,3,2],[2,2,2]] => 7 [[4,4,2,2],[2,2,1]] => 7 [[5,5,3,2],[3,3,2]] => 7 [[4,3,2,2],[2,1,1]] => 1 [[5,4,3,2],[3,2,2]] => 7 [[5,4,2,2],[3,2,1]] => 7 [[6,5,3,2],[4,3,2]] => 7 [[5,3,2,2],[3,1,1]] => 7 [[6,4,3,2],[4,2,2]] => 7 [[6,4,2,2],[4,2,1]] => 7 [[7,5,3,2],[5,3,2]] => 7 [[2,2,1,1,1],[]] => 7 [[3,3,2,2,1],[1,1,1,1]] => 7 [[3,3,2,1,1],[1,1,1]] => 7 [[4,4,3,2,1],[2,2,2,1]] => 7 [[3,3,1,1,1],[1,1]] => 7 [[4,4,2,2,1],[2,2,1,1]] => 7 [[4,4,2,1,1],[2,2,1]] => 7 [[5,5,3,2,1],[3,3,2,1]] => 7 [[3,2,1,1,1],[1]] => 7 [[4,3,2,2,1],[2,1,1,1]] => 7 [[4,3,2,1,1],[2,1,1]] => 7 [[5,4,3,2,1],[3,2,2,1]] => 7 [[4,3,1,1,1],[2,1]] => 7 [[5,4,2,2,1],[3,2,1,1]] => 7 [[5,4,2,1,1],[3,2,1]] => 7 [[6,5,3,2,1],[4,3,2,1]] => 7 [[4,2,1,1,1],[2]] => 7 [[5,3,2,2,1],[3,1,1,1]] => 7 [[5,3,2,1,1],[3,1,1]] => 7 [[6,4,3,2,1],[4,2,2,1]] => 7 [[5,3,1,1,1],[3,1]] => 7 [[6,4,2,2,1],[4,2,1,1]] => 7 [[6,4,2,1,1],[4,2,1]] => 7 [[7,5,3,2,1],[5,3,2,1]] => 7 [[5,4,4],[3,3]] => 7 [[6,5,4],[4,4]] => 7 [[6,4,4],[4,3]] => 7 [[7,5,4],[5,4]] => 7 [[4,3,3,1],[2,2]] => 1 [[5,4,4,1],[3,3,1]] => 7 [[5,4,3,1],[3,3]] => 7 [[6,5,4,1],[4,4,1]] => 7 [[5,3,3,1],[3,2]] => 7 [[6,4,4,1],[4,3,1]] => 7 [[6,4,3,1],[4,3]] => 7 [[7,5,4,1],[5,4,1]] => 7 [[3,2,2,2],[1,1]] => 7 [[4,3,3,2],[2,2,1]] => 1 [[5,4,4,2],[3,3,2]] => 7 [[4,3,2,2],[2,2]] => 7 [[5,4,3,2],[3,3,1]] => 7 [[6,5,4,2],[4,4,2]] => 7 [[4,2,2,2],[2,1]] => 7 [[5,3,3,2],[3,2,1]] => 7 [[6,4,4,2],[4,3,2]] => 7 [[5,3,2,2],[3,2]] => 7 [[6,4,3,2],[4,3,1]] => 7 [[7,5,4,2],[5,4,2]] => 7 [[3,2,2,1,1],[1,1]] => 7 [[4,3,3,2,1],[2,2,1,1]] => 7 [[4,3,3,1,1],[2,2,1]] => 7 [[5,4,4,2,1],[3,3,2,1]] => 7 [[4,3,2,1,1],[2,2]] => 7 [[5,4,3,2,1],[3,3,1,1]] => 7 [[5,4,3,1,1],[3,3,1]] => 7 [[6,5,4,2,1],[4,4,2,1]] => 7 [[4,2,2,1,1],[2,1]] => 7 [[5,3,3,2,1],[3,2,1,1]] => 7 [[5,3,3,1,1],[3,2,1]] => 7 [[6,4,4,2,1],[4,3,2,1]] => 7 [[5,3,2,1,1],[3,2]] => 7 [[6,4,3,2,1],[4,3,1,1]] => 7 [[6,4,3,1,1],[4,3,1]] => 7 [[7,5,4,2,1],[5,4,2,1]] => 7 [[4,3,3,3],[2,2,2]] => 1 [[5,4,4,3],[3,3,3]] => 7 [[5,4,3,3],[3,3,2]] => 7 [[6,5,4,3],[4,4,3]] => 7 [[5,3,3,3],[3,2,2]] => 7 [[6,4,4,3],[4,3,3]] => 7 [[6,4,3,3],[4,3,2]] => 7 [[7,5,4,3],[5,4,3]] => 7 [[3,2,2,2,1],[1,1,1]] => 7 [[4,3,3,3,1],[2,2,2,1]] => 7 [[4,3,3,2,1],[2,2,2]] => 7 [[5,4,4,3,1],[3,3,3,1]] => 7 [[4,3,2,2,1],[2,2,1]] => 7 [[5,4,3,3,1],[3,3,2,1]] => 7 [[5,4,3,2,1],[3,3,2]] => 7 [[6,5,4,3,1],[4,4,3,1]] => 7 [[4,2,2,2,1],[2,1,1]] => 7 [[5,3,3,3,1],[3,2,2,1]] => 7 [[5,3,3,2,1],[3,2,2]] => 7 [[6,4,4,3,1],[4,3,3,1]] => 7 [[5,3,2,2,1],[3,2,1]] => 7 [[6,4,3,3,1],[4,3,2,1]] => 7 [[6,4,3,2,1],[4,3,2]] => 7 [[7,5,4,3,1],[5,4,3,1]] => 7 [[3,2,2,2,2],[1,1,1,1]] => 7 [[4,3,3,3,2],[2,2,2,2]] => 7 [[4,3,3,2,2],[2,2,2,1]] => 7 [[5,4,4,3,2],[3,3,3,2]] => 7 [[4,3,2,2,2],[2,2,1,1]] => 7 [[5,4,3,3,2],[3,3,2,2]] => 7 [[5,4,3,2,2],[3,3,2,1]] => 7 [[6,5,4,3,2],[4,4,3,2]] => 7 [[4,2,2,2,2],[2,1,1,1]] => 7 [[5,3,3,3,2],[3,2,2,2]] => 7 [[5,3,3,2,2],[3,2,2,1]] => 7 [[6,4,4,3,2],[4,3,3,2]] => 7 [[5,3,2,2,2],[3,2,1,1]] => 7 [[6,4,3,3,2],[4,3,2,2]] => 7 [[6,4,3,2,2],[4,3,2,1]] => 7 [[7,5,4,3,2],[5,4,3,2]] => 7 [[2,1,1,1,1,1],[]] => 1 [[3,2,2,2,2,1],[1,1,1,1,1]] => 7 [[3,2,2,2,1,1],[1,1,1,1]] => 7 [[4,3,3,3,2,1],[2,2,2,2,1]] => 7 [[3,2,2,1,1,1],[1,1,1]] => 7 [[4,3,3,2,2,1],[2,2,2,1,1]] => 7 [[4,3,3,2,1,1],[2,2,2,1]] => 7 [[5,4,4,3,2,1],[3,3,3,2,1]] => 7 [[3,2,1,1,1,1],[1,1]] => 7 [[4,3,2,2,2,1],[2,2,1,1,1]] => 7 [[4,3,2,2,1,1],[2,2,1,1]] => 7 [[5,4,3,3,2,1],[3,3,2,2,1]] => 7 [[4,3,2,1,1,1],[2,2,1]] => 7 [[5,4,3,2,2,1],[3,3,2,1,1]] => 7 [[5,4,3,2,1,1],[3,3,2,1]] => 7 [[6,5,4,3,2,1],[4,4,3,2,1]] => 7 [[3,1,1,1,1,1],[1]] => 7 [[4,2,2,2,2,1],[2,1,1,1,1]] => 7 [[4,2,2,2,1,1],[2,1,1,1]] => 7 [[5,3,3,3,2,1],[3,2,2,2,1]] => 7 [[4,2,2,1,1,1],[2,1,1]] => 7 [[5,3,3,2,2,1],[3,2,2,1,1]] => 7 [[5,3,3,2,1,1],[3,2,2,1]] => 7 [[6,4,4,3,2,1],[4,3,3,2,1]] => 7 [[4,2,1,1,1,1],[2,1]] => 7 [[5,3,2,2,2,1],[3,2,1,1,1]] => 7 [[5,3,2,2,1,1],[3,2,1,1]] => 7 [[6,4,3,3,2,1],[4,3,2,2,1]] => 7 [[5,3,2,1,1,1],[3,2,1]] => 7 [[6,4,3,2,2,1],[4,3,2,1,1]] => 7 [[6,4,3,2,1,1],[4,3,2,1]] => 7 [[7,5,4,3,2,1],[5,4,3,2,1]] => 7 [[6,6],[5]] => 1 [[7,6],[6]] => 7 [[5,5,1],[4]] => 7 [[6,6,1],[5,1]] => 7 [[6,5,1],[5]] => 7 [[7,6,1],[6,1]] => 7 [[4,4,2],[3]] => 7 [[5,5,2],[4,1]] => 7 [[6,6,2],[5,2]] => 7 [[5,4,2],[4]] => 7 [[6,5,2],[5,1]] => 7 [[7,6,2],[6,2]] => 7 [[4,4,1,1],[3]] => 1 [[5,5,2,1],[4,1,1]] => 7 [[5,5,1,1],[4,1]] => 7 [[6,6,2,1],[5,2,1]] => 7 [[5,4,1,1],[4]] => 7 [[6,5,2,1],[5,1,1]] => 7 [[6,5,1,1],[5,1]] => 7 [[7,6,2,1],[6,2,1]] => 7 [[3,3,3],[2]] => 7 [[4,4,3],[3,1]] => 7 [[5,5,3],[4,2]] => 7 [[6,6,3],[5,3]] => 7 [[4,3,3],[3]] => 7 [[5,4,3],[4,1]] => 7 [[6,5,3],[5,2]] => 7 [[7,6,3],[6,3]] => 7 [[3,3,2,1],[2]] => 7 [[4,4,3,1],[3,1,1]] => 7 [[4,4,2,1],[3,1]] => 1 [[5,5,3,1],[4,2,1]] => 7 [[5,5,2,1],[4,2]] => 7 [[6,6,3,1],[5,3,1]] => 7 [[4,3,2,1],[3]] => 7 [[5,4,3,1],[4,1,1]] => 7 [[5,4,2,1],[4,1]] => 7 [[6,5,3,1],[5,2,1]] => 7 [[6,5,2,1],[5,2]] => 7 [[7,6,3,1],[6,3,1]] => 7 [[4,4,2,2],[3,1,1]] => 1 [[5,5,3,2],[4,2,2]] => 7 [[5,5,2,2],[4,2,1]] => 7 [[6,6,3,2],[5,3,2]] => 7 [[5,4,2,2],[4,1,1]] => 7 [[6,5,3,2],[5,2,2]] => 7 [[6,5,2,2],[5,2,1]] => 7 [[7,6,3,2],[6,3,2]] => 7 [[3,3,1,1,1],[2]] => 7 [[4,4,2,2,1],[3,1,1,1]] => 7 [[4,4,2,1,1],[3,1,1]] => 7 [[5,5,3,2,1],[4,2,2,1]] => 7 [[4,4,1,1,1],[3,1]] => 7 [[5,5,2,2,1],[4,2,1,1]] => 7 [[5,5,2,1,1],[4,2,1]] => 7 [[6,6,3,2,1],[5,3,2,1]] => 7 [[4,3,1,1,1],[3]] => 7 [[5,4,2,2,1],[4,1,1,1]] => 7 [[5,4,2,1,1],[4,1,1]] => 7 [[6,5,3,2,1],[5,2,2,1]] => 7 [[5,4,1,1,1],[4,1]] => 7 [[6,5,2,2,1],[5,2,1,1]] => 7 [[6,5,2,1,1],[5,2,1]] => 7 [[7,6,3,2,1],[6,3,2,1]] => 7 [[4,4,4],[3,2]] => 7 [[5,5,4],[4,3]] => 7 [[6,6,4],[5,4]] => 7 [[5,4,4],[4,2]] => 7 [[6,5,4],[5,3]] => 7 [[7,6,4],[6,4]] => 7 [[3,3,3,1],[2,1]] => 7 [[4,4,4,1],[3,2,1]] => 7 [[4,4,3,1],[3,2]] => 1 [[5,5,4,1],[4,3,1]] => 7 [[5,5,3,1],[4,3]] => 7 [[6,6,4,1],[5,4,1]] => 7 [[4,3,3,1],[3,1]] => 7 [[5,4,4,1],[4,2,1]] => 7 [[5,4,3,1],[4,2]] => 7 [[6,5,4,1],[5,3,1]] => 7 [[6,5,3,1],[5,3]] => 7 [[7,6,4,1],[6,4,1]] => 7 [[2,2,2,2],[1]] => 7 [[3,3,3,2],[2,1,1]] => 7 [[4,4,4,2],[3,2,2]] => 7 [[3,3,2,2],[2,1]] => 7 [[4,4,3,2],[3,2,1]] => 1 [[5,5,4,2],[4,3,2]] => 7 [[4,4,2,2],[3,2]] => 7 [[5,5,3,2],[4,3,1]] => 7 [[6,6,4,2],[5,4,2]] => 7 [[3,2,2,2],[2]] => 7 [[4,3,3,2],[3,1,1]] => 7 [[5,4,4,2],[4,2,2]] => 7 [[4,3,2,2],[3,1]] => 7 [[5,4,3,2],[4,2,1]] => 7 [[6,5,4,2],[5,3,2]] => 7 [[5,4,2,2],[4,2]] => 7 [[6,5,3,2],[5,3,1]] => 7 [[7,6,4,2],[6,4,2]] => 7 [[2,2,2,1,1],[1]] => 7 [[3,3,3,2,1],[2,1,1,1]] => 7 [[3,3,3,1,1],[2,1,1]] => 7 [[4,4,4,2,1],[3,2,2,1]] => 7 [[3,3,2,1,1],[2,1]] => 7 [[4,4,3,2,1],[3,2,1,1]] => 7 [[4,4,3,1,1],[3,2,1]] => 7 [[5,5,4,2,1],[4,3,2,1]] => 7 [[4,4,2,1,1],[3,2]] => 7 [[5,5,3,2,1],[4,3,1,1]] => 7 [[5,5,3,1,1],[4,3,1]] => 7 [[6,6,4,2,1],[5,4,2,1]] => 7 [[3,2,2,1,1],[2]] => 7 [[4,3,3,2,1],[3,1,1,1]] => 7 [[4,3,3,1,1],[3,1,1]] => 7 [[5,4,4,2,1],[4,2,2,1]] => 7 [[4,3,2,1,1],[3,1]] => 7 [[5,4,3,2,1],[4,2,1,1]] => 7 [[5,4,3,1,1],[4,2,1]] => 7 [[6,5,4,2,1],[5,3,2,1]] => 7 [[5,4,2,1,1],[4,2]] => 7 [[6,5,3,2,1],[5,3,1,1]] => 7 [[6,5,3,1,1],[5,3,1]] => 7 [[7,6,4,2,1],[6,4,2,1]] => 7 [[4,4,3,3],[3,2,2]] => 1 [[5,5,4,3],[4,3,3]] => 7 [[5,5,3,3],[4,3,2]] => 7 [[6,6,4,3],[5,4,3]] => 7 [[5,4,3,3],[4,2,2]] => 7 [[6,5,4,3],[5,3,3]] => 7 [[6,5,3,3],[5,3,2]] => 7 [[7,6,4,3],[6,4,3]] => 7 [[3,3,2,2,1],[2,1,1]] => 7 [[4,4,3,3,1],[3,2,2,1]] => 7 [[4,4,3,2,1],[3,2,2]] => 7 [[5,5,4,3,1],[4,3,3,1]] => 7 [[4,4,2,2,1],[3,2,1]] => 7 [[5,5,3,3,1],[4,3,2,1]] => 7 [[5,5,3,2,1],[4,3,2]] => 7 [[6,6,4,3,1],[5,4,3,1]] => 7 [[4,3,2,2,1],[3,1,1]] => 7 [[5,4,3,3,1],[4,2,2,1]] => 7 [[5,4,3,2,1],[4,2,2]] => 7 [[6,5,4,3,1],[5,3,3,1]] => 7 [[5,4,2,2,1],[4,2,1]] => 7 [[6,5,3,3,1],[5,3,2,1]] => 7 [[6,5,3,2,1],[5,3,2]] => 7 [[7,6,4,3,1],[6,4,3,1]] => 7 [[3,3,2,2,2],[2,1,1,1]] => 7 [[4,4,3,3,2],[3,2,2,2]] => 7 [[4,4,3,2,2],[3,2,2,1]] => 7 [[5,5,4,3,2],[4,3,3,2]] => 7 [[4,4,2,2,2],[3,2,1,1]] => 7 [[5,5,3,3,2],[4,3,2,2]] => 7 [[5,5,3,2,2],[4,3,2,1]] => 7 [[6,6,4,3,2],[5,4,3,2]] => 7 [[4,3,2,2,2],[3,1,1,1]] => 7 [[5,4,3,3,2],[4,2,2,2]] => 7 [[5,4,3,2,2],[4,2,2,1]] => 7 [[6,5,4,3,2],[5,3,3,2]] => 7 [[5,4,2,2,2],[4,2,1,1]] => 7 [[6,5,3,3,2],[5,3,2,2]] => 7 [[6,5,3,2,2],[5,3,2,1]] => 7 [[7,6,4,3,2],[6,4,3,2]] => 7 [[2,2,1,1,1,1],[1]] => 1 [[3,3,2,2,2,1],[2,1,1,1,1]] => 7 [[3,3,2,2,1,1],[2,1,1,1]] => 7 [[4,4,3,3,2,1],[3,2,2,2,1]] => 7 [[3,3,2,1,1,1],[2,1,1]] => 7 [[4,4,3,2,2,1],[3,2,2,1,1]] => 7 [[4,4,3,2,1,1],[3,2,2,1]] => 7 [[5,5,4,3,2,1],[4,3,3,2,1]] => 7 [[3,3,1,1,1,1],[2,1]] => 7 [[4,4,2,2,2,1],[3,2,1,1,1]] => 7 [[4,4,2,2,1,1],[3,2,1,1]] => 7 [[5,5,3,3,2,1],[4,3,2,2,1]] => 7 [[4,4,2,1,1,1],[3,2,1]] => 7 [[5,5,3,2,2,1],[4,3,2,1,1]] => 7 [[5,5,3,2,1,1],[4,3,2,1]] => 7 [[6,6,4,3,2,1],[5,4,3,2,1]] => 7 [[3,2,1,1,1,1],[2]] => 7 [[4,3,2,2,2,1],[3,1,1,1,1]] => 7 [[4,3,2,2,1,1],[3,1,1,1]] => 7 [[5,4,3,3,2,1],[4,2,2,2,1]] => 7 [[4,3,2,1,1,1],[3,1,1]] => 7 [[5,4,3,2,2,1],[4,2,2,1,1]] => 7 [[5,4,3,2,1,1],[4,2,2,1]] => 7 [[6,5,4,3,2,1],[5,3,3,2,1]] => 7 [[4,3,1,1,1,1],[3,1]] => 7 [[5,4,2,2,2,1],[4,2,1,1,1]] => 7 [[5,4,2,2,1,1],[4,2,1,1]] => 7 [[6,5,3,3,2,1],[5,3,2,2,1]] => 7 [[5,4,2,1,1,1],[4,2,1]] => 7 [[6,5,3,2,2,1],[5,3,2,1,1]] => 7 [[6,5,3,2,1,1],[5,3,2,1]] => 7 [[7,6,4,3,2,1],[6,4,3,2,1]] => 7 [[5,5,5],[4,4]] => 7 [[6,6,5],[5,5]] => 7 [[6,5,5],[5,4]] => 7 [[7,6,5],[6,5]] => 7 [[4,4,4,1],[3,3]] => 1 [[5,5,5,1],[4,4,1]] => 7 [[5,5,4,1],[4,4]] => 7 [[6,6,5,1],[5,5,1]] => 7 [[5,4,4,1],[4,3]] => 7 [[6,5,5,1],[5,4,1]] => 7 [[6,5,4,1],[5,4]] => 7 [[7,6,5,1],[6,5,1]] => 7 [[3,3,3,2],[2,2]] => 7 [[4,4,4,2],[3,3,1]] => 1 [[5,5,5,2],[4,4,2]] => 7 [[4,4,3,2],[3,3]] => 7 [[5,5,4,2],[4,4,1]] => 7 [[6,6,5,2],[5,5,2]] => 7 [[4,3,3,2],[3,2]] => 7 [[5,4,4,2],[4,3,1]] => 7 [[6,5,5,2],[5,4,2]] => 7 [[5,4,3,2],[4,3]] => 7 [[6,5,4,2],[5,4,1]] => 7 [[7,6,5,2],[6,5,2]] => 7 [[3,3,3,1,1],[2,2]] => 7 [[4,4,4,2,1],[3,3,1,1]] => 7 [[4,4,4,1,1],[3,3,1]] => 7 [[5,5,5,2,1],[4,4,2,1]] => 7 [[4,4,3,1,1],[3,3]] => 7 [[5,5,4,2,1],[4,4,1,1]] => 7 [[5,5,4,1,1],[4,4,1]] => 7 [[6,6,5,2,1],[5,5,2,1]] => 7 [[4,3,3,1,1],[3,2]] => 7 [[5,4,4,2,1],[4,3,1,1]] => 7 [[5,4,4,1,1],[4,3,1]] => 7 [[6,5,5,2,1],[5,4,2,1]] => 7 [[5,4,3,1,1],[4,3]] => 7 [[6,5,4,2,1],[5,4,1,1]] => 7 [[6,5,4,1,1],[5,4,1]] => 7 [[7,6,5,2,1],[6,5,2,1]] => 7 [[3,3,3,3],[2,2,1]] => 7 [[4,4,4,3],[3,3,2]] => 1 [[5,5,5,3],[4,4,3]] => 7 [[4,4,3,3],[3,3,1]] => 7 [[5,5,4,3],[4,4,2]] => 7 [[6,6,5,3],[5,5,3]] => 7 [[4,3,3,3],[3,2,1]] => 7 [[5,4,4,3],[4,3,2]] => 7 [[6,5,5,3],[5,4,3]] => 7 [[5,4,3,3],[4,3,1]] => 7 [[6,5,4,3],[5,4,2]] => 7 [[7,6,5,3],[6,5,3]] => 7 [[2,2,2,2,1],[1,1]] => 7 [[3,3,3,3,1],[2,2,1,1]] => 7 [[3,3,3,2,1],[2,2,1]] => 7 [[4,4,4,3,1],[3,3,2,1]] => 7 [[4,4,4,2,1],[3,3,2]] => 7 [[5,5,5,3,1],[4,4,3,1]] => 7 [[3,3,2,2,1],[2,2]] => 7 [[4,4,3,3,1],[3,3,1,1]] => 7 [[4,4,3,2,1],[3,3,1]] => 7 [[5,5,4,3,1],[4,4,2,1]] => 7 [[5,5,4,2,1],[4,4,2]] => 7 [[6,6,5,3,1],[5,5,3,1]] => 7 [[3,2,2,2,1],[2,1]] => 7 [[4,3,3,3,1],[3,2,1,1]] => 7 [[4,3,3,2,1],[3,2,1]] => 7 [[5,4,4,3,1],[4,3,2,1]] => 7 [[5,4,4,2,1],[4,3,2]] => 7 [[6,5,5,3,1],[5,4,3,1]] => 7 [[4,3,2,2,1],[3,2]] => 7 [[5,4,3,3,1],[4,3,1,1]] => 7 [[5,4,3,2,1],[4,3,1]] => 7 [[6,5,4,3,1],[5,4,2,1]] => 7 [[6,5,4,2,1],[5,4,2]] => 7 [[7,6,5,3,1],[6,5,3,1]] => 7 [[3,3,3,2,2],[2,2,1,1]] => 7 [[4,4,4,3,2],[3,3,2,2]] => 7 [[4,4,4,2,2],[3,3,2,1]] => 7 [[5,5,5,3,2],[4,4,3,2]] => 7 [[4,4,3,2,2],[3,3,1,1]] => 7 [[5,5,4,3,2],[4,4,2,2]] => 7 [[5,5,4,2,2],[4,4,2,1]] => 7 [[6,6,5,3,2],[5,5,3,2]] => 7 [[4,3,3,2,2],[3,2,1,1]] => 7 [[5,4,4,3,2],[4,3,2,2]] => 7 [[5,4,4,2,2],[4,3,2,1]] => 7 [[6,5,5,3,2],[5,4,3,2]] => 7 [[5,4,3,2,2],[4,3,1,1]] => 7 [[6,5,4,3,2],[5,4,2,2]] => 7 [[6,5,4,2,2],[5,4,2,1]] => 7 [[7,6,5,3,2],[6,5,3,2]] => 7 [[2,2,2,1,1,1],[1,1]] => 1 [[3,3,3,2,2,1],[2,2,1,1,1]] => 7 [[3,3,3,2,1,1],[2,2,1,1]] => 7 [[4,4,4,3,2,1],[3,3,2,2,1]] => 7 [[3,3,3,1,1,1],[2,2,1]] => 7 [[4,4,4,2,2,1],[3,3,2,1,1]] => 7 [[4,4,4,2,1,1],[3,3,2,1]] => 7 [[5,5,5,3,2,1],[4,4,3,2,1]] => 7 [[3,3,2,1,1,1],[2,2]] => 7 [[4,4,3,2,2,1],[3,3,1,1,1]] => 7 [[4,4,3,2,1,1],[3,3,1,1]] => 7 [[5,5,4,3,2,1],[4,4,2,2,1]] => 7 [[4,4,3,1,1,1],[3,3,1]] => 7 [[5,5,4,2,2,1],[4,4,2,1,1]] => 7 [[5,5,4,2,1,1],[4,4,2,1]] => 7 [[6,6,5,3,2,1],[5,5,3,2,1]] => 7 [[3,2,2,1,1,1],[2,1]] => 7 [[4,3,3,2,2,1],[3,2,1,1,1]] => 7 [[4,3,3,2,1,1],[3,2,1,1]] => 7 [[5,4,4,3,2,1],[4,3,2,2,1]] => 7 [[4,3,3,1,1,1],[3,2,1]] => 7 [[5,4,4,2,2,1],[4,3,2,1,1]] => 7 [[5,4,4,2,1,1],[4,3,2,1]] => 7 [[6,5,5,3,2,1],[5,4,3,2,1]] => 7 [[4,3,2,1,1,1],[3,2]] => 7 [[5,4,3,2,2,1],[4,3,1,1,1]] => 7 [[5,4,3,2,1,1],[4,3,1,1]] => 7 [[6,5,4,3,2,1],[5,4,2,2,1]] => 7 [[5,4,3,1,1,1],[4,3,1]] => 7 [[6,5,4,2,2,1],[5,4,2,1,1]] => 7 [[6,5,4,2,1,1],[5,4,2,1]] => 7 [[7,6,5,3,2,1],[6,5,3,2,1]] => 7 [[4,4,4,4],[3,3,3]] => 1 [[5,5,5,4],[4,4,4]] => 7 [[5,5,4,4],[4,4,3]] => 7 [[6,6,5,4],[5,5,4]] => 7 [[5,4,4,4],[4,3,3]] => 7 [[6,5,5,4],[5,4,4]] => 7 [[6,5,4,4],[5,4,3]] => 7 [[7,6,5,4],[6,5,4]] => 7 [[3,3,3,3,1],[2,2,2]] => 7 [[4,4,4,4,1],[3,3,3,1]] => 7 [[4,4,4,3,1],[3,3,3]] => 7 [[5,5,5,4,1],[4,4,4,1]] => 7 [[4,4,3,3,1],[3,3,2]] => 7 [[5,5,4,4,1],[4,4,3,1]] => 7 [[5,5,4,3,1],[4,4,3]] => 7 [[6,6,5,4,1],[5,5,4,1]] => 7 [[4,3,3,3,1],[3,2,2]] => 7 [[5,4,4,4,1],[4,3,3,1]] => 7 [[5,4,4,3,1],[4,3,3]] => 7 [[6,5,5,4,1],[5,4,4,1]] => 7 [[5,4,3,3,1],[4,3,2]] => 7 [[6,5,4,4,1],[5,4,3,1]] => 7 [[6,5,4,3,1],[5,4,3]] => 7 [[7,6,5,4,1],[6,5,4,1]] => 7 [[2,2,2,2,2],[1,1,1]] => 7 [[3,3,3,3,2],[2,2,2,1]] => 7 [[4,4,4,4,2],[3,3,3,2]] => 7 [[3,3,3,2,2],[2,2,2]] => 7 [[4,4,4,3,2],[3,3,3,1]] => 7 [[5,5,5,4,2],[4,4,4,2]] => 7 [[3,3,2,2,2],[2,2,1]] => 7 [[4,4,3,3,2],[3,3,2,1]] => 7 [[5,5,4,4,2],[4,4,3,2]] => 7 [[4,4,3,2,2],[3,3,2]] => 7 [[5,5,4,3,2],[4,4,3,1]] => 7 [[6,6,5,4,2],[5,5,4,2]] => 7 [[3,2,2,2,2],[2,1,1]] => 7 [[4,3,3,3,2],[3,2,2,1]] => 7 [[5,4,4,4,2],[4,3,3,2]] => 7 [[4,3,3,2,2],[3,2,2]] => 7 [[5,4,4,3,2],[4,3,3,1]] => 7 [[6,5,5,4,2],[5,4,4,2]] => 7 [[4,3,2,2,2],[3,2,1]] => 7 [[5,4,3,3,2],[4,3,2,1]] => 7 [[6,5,4,4,2],[5,4,3,2]] => 7 [[5,4,3,2,2],[4,3,2]] => 7 [[6,5,4,3,2],[5,4,3,1]] => 7 [[7,6,5,4,2],[6,5,4,2]] => 7 [[2,2,2,2,1,1],[1,1,1]] => 1 [[3,3,3,3,2,1],[2,2,2,1,1]] => 7 [[3,3,3,3,1,1],[2,2,2,1]] => 7 [[4,4,4,4,2,1],[3,3,3,2,1]] => 7 [[3,3,3,2,1,1],[2,2,2]] => 7 [[4,4,4,3,2,1],[3,3,3,1,1]] => 7 [[4,4,4,3,1,1],[3,3,3,1]] => 7 [[5,5,5,4,2,1],[4,4,4,2,1]] => 7 [[3,3,2,2,1,1],[2,2,1]] => 7 [[4,4,3,3,2,1],[3,3,2,1,1]] => 7 [[4,4,3,3,1,1],[3,3,2,1]] => 7 [[5,5,4,4,2,1],[4,4,3,2,1]] => 7 [[4,4,3,2,1,1],[3,3,2]] => 7 [[5,5,4,3,2,1],[4,4,3,1,1]] => 7 [[5,5,4,3,1,1],[4,4,3,1]] => 7 [[6,6,5,4,2,1],[5,5,4,2,1]] => 7 [[3,2,2,2,1,1],[2,1,1]] => 7 [[4,3,3,3,2,1],[3,2,2,1,1]] => 7 [[4,3,3,3,1,1],[3,2,2,1]] => 7 [[5,4,4,4,2,1],[4,3,3,2,1]] => 7 [[4,3,3,2,1,1],[3,2,2]] => 7 [[5,4,4,3,2,1],[4,3,3,1,1]] => 7 [[5,4,4,3,1,1],[4,3,3,1]] => 7 [[6,5,5,4,2,1],[5,4,4,2,1]] => 7 [[4,3,2,2,1,1],[3,2,1]] => 7 [[5,4,3,3,2,1],[4,3,2,1,1]] => 7 [[5,4,3,3,1,1],[4,3,2,1]] => 7 [[6,5,4,4,2,1],[5,4,3,2,1]] => 7 [[5,4,3,2,1,1],[4,3,2]] => 7 [[6,5,4,3,2,1],[5,4,3,1,1]] => 7 [[6,5,4,3,1,1],[5,4,3,1]] => 7 [[7,6,5,4,2,1],[6,5,4,2,1]] => 7 [[3,3,3,3,3],[2,2,2,2]] => 7 [[4,4,4,4,3],[3,3,3,3]] => 7 [[4,4,4,3,3],[3,3,3,2]] => 7 [[5,5,5,4,3],[4,4,4,3]] => 7 [[4,4,3,3,3],[3,3,2,2]] => 7 [[5,5,4,4,3],[4,4,3,3]] => 7 [[5,5,4,3,3],[4,4,3,2]] => 7 [[6,6,5,4,3],[5,5,4,3]] => 7 [[4,3,3,3,3],[3,2,2,2]] => 7 [[5,4,4,4,3],[4,3,3,3]] => 7 [[5,4,4,3,3],[4,3,3,2]] => 7 [[6,5,5,4,3],[5,4,4,3]] => 7 [[5,4,3,3,3],[4,3,2,2]] => 7 [[6,5,4,4,3],[5,4,3,3]] => 7 [[6,5,4,3,3],[5,4,3,2]] => 7 [[7,6,5,4,3],[6,5,4,3]] => 7 [[2,2,2,2,2,1],[1,1,1,1]] => 1 [[3,3,3,3,3,1],[2,2,2,2,1]] => 7 [[3,3,3,3,2,1],[2,2,2,2]] => 7 [[4,4,4,4,3,1],[3,3,3,3,1]] => 7 [[3,3,3,2,2,1],[2,2,2,1]] => 7 [[4,4,4,3,3,1],[3,3,3,2,1]] => 7 [[4,4,4,3,2,1],[3,3,3,2]] => 7 [[5,5,5,4,3,1],[4,4,4,3,1]] => 7 [[3,3,2,2,2,1],[2,2,1,1]] => 7 [[4,4,3,3,3,1],[3,3,2,2,1]] => 7 [[4,4,3,3,2,1],[3,3,2,2]] => 7 [[5,5,4,4,3,1],[4,4,3,3,1]] => 7 [[4,4,3,2,2,1],[3,3,2,1]] => 7 [[5,5,4,3,3,1],[4,4,3,2,1]] => 7 [[5,5,4,3,2,1],[4,4,3,2]] => 7 [[6,6,5,4,3,1],[5,5,4,3,1]] => 7 [[3,2,2,2,2,1],[2,1,1,1]] => 7 [[4,3,3,3,3,1],[3,2,2,2,1]] => 7 [[4,3,3,3,2,1],[3,2,2,2]] => 7 [[5,4,4,4,3,1],[4,3,3,3,1]] => 7 [[4,3,3,2,2,1],[3,2,2,1]] => 7 [[5,4,4,3,3,1],[4,3,3,2,1]] => 7 [[5,4,4,3,2,1],[4,3,3,2]] => 7 [[6,5,5,4,3,1],[5,4,4,3,1]] => 7 [[4,3,2,2,2,1],[3,2,1,1]] => 7 [[5,4,3,3,3,1],[4,3,2,2,1]] => 7 [[5,4,3,3,2,1],[4,3,2,2]] => 7 [[6,5,4,4,3,1],[5,4,3,3,1]] => 7 [[5,4,3,2,2,1],[4,3,2,1]] => 7 [[6,5,4,3,3,1],[5,4,3,2,1]] => 7 [[6,5,4,3,2,1],[5,4,3,2]] => 7 [[7,6,5,4,3,1],[6,5,4,3,1]] => 7 [[2,2,2,2,2,2],[1,1,1,1,1]] => 1 [[3,3,3,3,3,2],[2,2,2,2,2]] => 7 [[3,3,3,3,2,2],[2,2,2,2,1]] => 7 [[4,4,4,4,3,2],[3,3,3,3,2]] => 7 [[3,3,3,2,2,2],[2,2,2,1,1]] => 7 [[4,4,4,3,3,2],[3,3,3,2,2]] => 7 [[4,4,4,3,2,2],[3,3,3,2,1]] => 7 [[5,5,5,4,3,2],[4,4,4,3,2]] => 7 [[3,3,2,2,2,2],[2,2,1,1,1]] => 7 [[4,4,3,3,3,2],[3,3,2,2,2]] => 7 [[4,4,3,3,2,2],[3,3,2,2,1]] => 7 [[5,5,4,4,3,2],[4,4,3,3,2]] => 7 [[4,4,3,2,2,2],[3,3,2,1,1]] => 7 [[5,5,4,3,3,2],[4,4,3,2,2]] => 7 [[5,5,4,3,2,2],[4,4,3,2,1]] => 7 [[6,6,5,4,3,2],[5,5,4,3,2]] => 7 [[3,2,2,2,2,2],[2,1,1,1,1]] => 7 [[4,3,3,3,3,2],[3,2,2,2,2]] => 7 [[4,3,3,3,2,2],[3,2,2,2,1]] => 7 [[5,4,4,4,3,2],[4,3,3,3,2]] => 7 [[4,3,3,2,2,2],[3,2,2,1,1]] => 7 [[5,4,4,3,3,2],[4,3,3,2,2]] => 7 [[5,4,4,3,2,2],[4,3,3,2,1]] => 7 [[6,5,5,4,3,2],[5,4,4,3,2]] => 7 [[4,3,2,2,2,2],[3,2,1,1,1]] => 7 [[5,4,3,3,3,2],[4,3,2,2,2]] => 7 [[5,4,3,3,2,2],[4,3,2,2,1]] => 7 [[6,5,4,4,3,2],[5,4,3,3,2]] => 7 [[5,4,3,2,2,2],[4,3,2,1,1]] => 7 [[6,5,4,3,3,2],[5,4,3,2,2]] => 7 [[6,5,4,3,2,2],[5,4,3,2,1]] => 7 [[7,6,5,4,3,2],[6,5,4,3,2]] => 7 [[1,1,1,1,1,1,1],[]] => 7 [[2,2,2,2,2,2,1],[1,1,1,1,1,1]] => 7 [[2,2,2,2,2,1,1],[1,1,1,1,1]] => 7 [[3,3,3,3,3,2,1],[2,2,2,2,2,1]] => 7 [[2,2,2,2,1,1,1],[1,1,1,1]] => 7 [[3,3,3,3,2,2,1],[2,2,2,2,1,1]] => 7 [[3,3,3,3,2,1,1],[2,2,2,2,1]] => 7 [[4,4,4,4,3,2,1],[3,3,3,3,2,1]] => 7 [[2,2,2,1,1,1,1],[1,1,1]] => 7 [[3,3,3,2,2,2,1],[2,2,2,1,1,1]] => 7 [[3,3,3,2,2,1,1],[2,2,2,1,1]] => 7 [[4,4,4,3,3,2,1],[3,3,3,2,2,1]] => 7 [[3,3,3,2,1,1,1],[2,2,2,1]] => 7 [[4,4,4,3,2,2,1],[3,3,3,2,1,1]] => 7 ----------------------------------------------------------------------------- Created: Sep 28, 2020 at 11:17 by Martin Rubey ----------------------------------------------------------------------------- Last Updated: Sep 28, 2020 at 11:17 by Martin Rubey