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-----------------------------------------------------------------------------
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.<q> = 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.<q> = 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.<q> = PolynomialRing(QQ)
R.<q> = 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
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Created: Sep 28, 2020 at 11:17 by Martin Rubey
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Last Updated: Sep 28, 2020 at 11:17 by Martin Rubey