************************************************************************
* www.FindStat.org - The Combinatorial Statistic Finder *
* *
* Copyright (C) 2013 The FindStatCrew <info@findstat.org> *
* *
* 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. *
************************************************************************
------------------------------------------------------------------------
Map identifier: Mp00322
------------------------------------------------------------------------
Map name: Loehr-Warrington
------------------------------------------------------------------------
Domain: Integer partitions
------------------------------------------------------------------------
Codomain: Integer partitions
------------------------------------------------------------------------
Description: Return a partition whose diagonal inversion number is the length of the preimage.
------------------------------------------------------------------------
References: [1] Loehr, N. A., Warrington, G. S. A continuous family of partition statistics equidistributed with length [[MathSciNet:2475023]]
------------------------------------------------------------------------
Code:
from collections import defaultdict
def Bdy(mu, r, s, n):
"""
Return the corresponding 0-1 sequence (in French notation).
EXAMPLES::
sage: mu = Partition([12,12,10,8,7,4,1,1,1])
sage: w = Bdy(mu, 3, 2, 5); w
[1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0]
sage: Partitions().from_zero_one([1-e for e in w])
[12, 12, 10, 8, 7, 4, 1, 1, 1]
"""
mu = Partition(mu)
D = list(mu.to_dyck_word())
rn_deficit = r*n - len(D) // 2
sn_deficit = s*n - len(D) // 2
if rn_deficit > 0:
D = D + [0] * rn_deficit
elif rn_deficit < 0:
assert all(e == 0 for e in D[rn_deficit:])
D = D[:rn_deficit]
if sn_deficit > 0:
D = [1] * sn_deficit + D
elif sn_deficit < 0:
assert all(e == 1 for e in D[:-sn_deficit])
D = D[-sn_deficit:]
return D
def VBdy(mu, r, s, n):
"""
EXAMPLES::
sage: mu = Partition([12,12,10,8,7,4,1,1,1])
sage: VBdy(mu, 3, 2, 5)
[(0, 0),
(0, 1),
(1, 1),
(1, 2),
(1, 3),
(1, 4),
(2, 4),
(3, 4),
(4, 4),
(4, 5),
(5, 5),
(6, 5),
(7, 5),
(7, 6),
(8, 6),
(8, 7),
(9, 7),
(10, 7),
(10, 8),
(11, 8),
(12, 8),
(12, 9),
(12, 10),
(13, 10),
(14, 10),
(15, 10)]
"""
D = Bdy(mu, r, s, n)
l = [(0, 0)]
for e in D:
x, y = l[-1]
if e == 0:
l.append((x + 1, y))
else:
l.append((x, y + 1))
return l
def eulerian_tour(mu, r, s, n):
"""
EXAMPLES::
sage: mu = Partition([12,12,10,8,7,4,1,1,1])
sage: eulerian_tour(mu, 3, 2, 5)
[0, 3, 1, 4, 7, 10, 8, 6, 4, 7, 5, 3, 1, 4, 2, 5, 3, 1, 4, 2, 0, 3, 6, 4, 2, 0]
"""
return [r*y - s*x for x, y in VBdy(mu, r, s, n)]
def from_eulerian_tour(e, r, s, n):
"""
Return the partition corresponding to the Eulerian tour.
EXAMPLES::
sage: mu = Partition([12,12,10,8,7,4,1,1,1])
sage: e = eulerian_tour(mu, 3, 2, 5); e
[0, 3, 1, 4, 7, 10, 8, 6, 4, 7, 5, 3, 1, 4, 2, 5, 3, 1, 4, 2, 0, 3, 6, 4, 2, 0]
sage: from_eulerian_tour(e, 3, 2, 5)
[12, 12, 10, 8, 7, 4, 1, 1, 1]
"""
w = [1 if e[i] - e[i-1] == r else 0 for i in range(1, len(e))]
return Partitions().from_zero_one([1 - e for e in w])
def phi_a(mu, r, s, n):
"""
EXAMPLES::
sage: mu = Partition([12,10,10,8,7,6,1,1,1])
sage: dict(phi_a(mu, 3, 2, 5))
{3: (1, 1, 0, 0),
1: (0, 0, 0),
4: (1, 0, 1, 1, 0),
7: (1, 1),
10: (1,),
8: (0,),
6: (0, 1),
2: (0, 0, 0),
0: (0, 0),
5: (1, 0)}
"""
e = eulerian_tour(mu, r, s, n)
d = defaultdict(tuple)
for i in range(1, len(e)):
v = e[i]
d[v] += (ZZ(v - e[i-1] == r),)
return d
def from_phi_a(ws, r, s, n):
"""
EXAMPLES::
sage: mu = Partition([12,10,10,8,7,6,1,1,1])
sage: ws = phi_a(mu, 3, 2, 5)
sage: from_phi_a(ws, 3, 2, 5) == mu
True
"""
last = {v: len(w)-1 for v, w in ws.items()}
e = [0]
u = 0
while True:
w = ws[u]
i = last[u]
if i < 0:
assert all(i < 0 for i in last.values())
break
d = w[i]
last[u] -= 1
u = u - r if d == 1 else u + s
e.append(u)
return from_eulerian_tour(e[::-1], r, s, n)
def distances(T):
"""
Return a dictionary mapping vertices to distances from the 0
vertex.
EXAMPLES::
sage: distances([(3, 1), (4, 2), (0, 3), (1, 4), (7, 5), (8, 6), (4, 7), (10, 8), (7, 10)])
{0: 0, 3: 1, 1: 2, 4: 3, 7: 4, 10: 5, 2: 4, 5: 5, 8: 6, 6: 7}
"""
d = dict()
d[0] = 0
while len(d) <= len(T):
for u, v in T:
if u in d:
d[v] = d[u] + 1
return d
def involution(mu, r, s, n):
"""
EXAMPLES::
sage: mu = Partition([12,12,10,8,7,4,1,1,1])
sage: involution(mu, 3, 2, 5)
[12, 10, 10, 8, 7, 6, 1, 1, 1]
sage: m=20; all((mu == involution(involution(mu, 1, 1, 2*m), 1, 1, 2*m)) for mu in Partitions(m))
True
sage: m=20; all((mu == involution(involution(mu, 1, 2, 2*m), 1, 2, 2*m)) for mu in Partitions(m))
True
"""
ws = phi_a(mu, r, s, n)
T = [(v-r, v) if w[0] == 1 else (v+s, v)
for v, w in ws.items() if v]
d = distances(T)
for v in ws:
if (v and v + s in ws and v - r in ws
and d[v+s] == d[v-r] == d[v] - 1):
# green vertices
ws[v] = ws[v][::-1]
else:
# blue and red vertices
w = ws[v]
ws[v] = (w[0],) + w[1:][::-1]
return from_phi_a(ws, r, s, n)
@cached_function
def critical(n):
if n == 0:
return [0, oo]
return sorted(set(critical(n-1) + [ZZ(a)/(n-a) for a in range(1, n)]))
def mapping(mu):
"""
Return a partition whode diagonal inversion number is the length of mu.
EXAMPLES::
sage: s = findstat(378); s
St000378: The diagonal inversion number of an integer partition.
sage: all(len(mu) == s(mapping(mu)) for n in range(10) for mu in Partitions(n))
True
"""
n = mu.size()
for q in critical(n):
if not q: continue
if q > 1: break
r = q.numer()
s = q.denom()
mu = involution(mu, r, s, ceil((r+s)/(r*s)*n))
return mu
------------------------------------------------------------------------
Map images:
[1] => [1]
[2] => [1,1]
[1,1] => [2]
[3] => [1,1,1]
[2,1] => [3]
[1,1,1] => [2,1]
[4] => [1,1,1,1]
[3,1] => [2,1,1]
[2,2] => [4]
[2,1,1] => [2,2]
[1,1,1,1] => [3,1]
[5] => [1,1,1,1,1]
[4,1] => [2,1,1,1]
[3,2] => [5]
[3,1,1] => [4,1]
[2,2,1] => [2,2,1]
[2,1,1,1] => [3,1,1]
[1,1,1,1,1] => [3,2]
[6] => [1,1,1,1,1,1]
[5,1] => [2,1,1,1,1]
[4,2] => [2,2,1,1]
[4,1,1] => [3,1,1,1]
[3,3] => [6]
[3,2,1] => [5,1]
[3,1,1,1] => [3,3]
[2,2,2] => [2,2,2]
[2,2,1,1] => [4,1,1]
[2,1,1,1,1] => [4,2]
[1,1,1,1,1,1] => [3,2,1]
[7] => [1,1,1,1,1,1,1]
[6,1] => [2,1,1,1,1,1]
[5,2] => [2,2,1,1,1]
[5,1,1] => [3,1,1,1,1]
[4,3] => [7]
[4,2,1] => [5,1,1]
[4,1,1,1] => [3,2,1,1]
[3,3,1] => [6,1]
[3,2,2] => [2,2,2,1]
[3,2,1,1] => [5,2]
[3,1,1,1,1] => [3,2,2]
[2,2,2,1] => [4,1,1,1]
[2,2,1,1,1] => [4,3]
[2,1,1,1,1,1] => [3,3,1]
[1,1,1,1,1,1,1] => [4,2,1]
[8] => [1,1,1,1,1,1,1,1]
[7,1] => [2,1,1,1,1,1,1]
[6,2] => [2,2,1,1,1,1]
[6,1,1] => [3,1,1,1,1,1]
[5,3] => [2,2,2,1,1]
[5,2,1] => [4,1,1,1,1]
[5,1,1,1] => [3,2,1,1,1]
[4,4] => [8]
[4,3,1] => [7,1]
[4,2,2] => [6,1,1]
[4,2,1,1] => [4,4]
[4,1,1,1,1] => [5,2,1]
[3,3,2] => [2,2,2,2]
[3,3,1,1] => [6,2]
[3,2,2,1] => [3,2,2,1]
[3,2,1,1,1] => [3,3,1,1]
[3,1,1,1,1,1] => [4,2,1,1]
[2,2,2,2] => [5,1,1,1]
[2,2,2,1,1] => [5,3]
[2,2,1,1,1,1] => [3,3,2]
[2,1,1,1,1,1,1] => [4,2,2]
[1,1,1,1,1,1,1,1] => [4,3,1]
[9] => [1,1,1,1,1,1,1,1,1]
[8,1] => [2,1,1,1,1,1,1,1]
[7,2] => [2,2,1,1,1,1,1]
[7,1,1] => [3,1,1,1,1,1,1]
[6,3] => [2,2,2,1,1,1]
[6,2,1] => [4,1,1,1,1,1]
[6,1,1,1] => [3,2,1,1,1,1]
[5,4] => [9]
[5,3,1] => [3,2,2,1,1]
[5,2,2] => [6,1,1,1]
[5,2,1,1] => [3,3,1,1,1]
[5,1,1,1,1] => [4,2,1,1,1]
[4,4,1] => [8,1]
[4,3,2] => [7,1,1]
[4,3,1,1] => [7,2]
[4,2,2,1] => [6,3]
[4,2,1,1,1] => [6,2,1]
[4,1,1,1,1,1] => [4,4,1]
[3,3,3] => [2,2,2,2,1]
[3,3,2,1] => [3,2,2,2]
[3,3,1,1,1] => [5,2,1,1]
[3,2,2,2] => [5,1,1,1,1]
[3,2,2,1,1] => [3,3,3]
[3,2,1,1,1,1] => [5,2,2]
[3,1,1,1,1,1,1] => [5,3,1]
[2,2,2,2,1] => [5,4]
[2,2,2,1,1,1] => [3,3,2,1]
[2,2,1,1,1,1,1] => [4,2,2,1]
[2,1,1,1,1,1,1,1] => [4,3,1,1]
[1,1,1,1,1,1,1,1,1] => [4,3,2]
[10] => [1,1,1,1,1,1,1,1,1,1]
[9,1] => [2,1,1,1,1,1,1,1,1]
[8,2] => [2,2,1,1,1,1,1,1]
[8,1,1] => [3,1,1,1,1,1,1,1]
[7,3] => [2,2,2,1,1,1,1]
[7,2,1] => [4,1,1,1,1,1,1]
[7,1,1,1] => [3,2,1,1,1,1,1]
[6,4] => [2,2,2,2,1,1]
[6,3,1] => [3,2,2,1,1,1]
[6,2,2] => [5,1,1,1,1,1]
[6,2,1,1] => [3,3,1,1,1,1]
[6,1,1,1,1] => [4,2,1,1,1,1]
[5,5] => [10]
[5,4,1] => [9,1]
[5,3,2] => [7,1,1,1]
[5,3,1,1] => [6,2,1,1]
[5,2,2,1] => [3,3,2,1,1]
[5,2,1,1,1] => [4,2,2,1,1]
[5,1,1,1,1,1] => [4,3,1,1,1]
[4,4,2] => [8,1,1]
[4,4,1,1] => [8,2]
[4,3,3] => [2,2,2,2,2]
[4,3,2,1] => [7,3]
[4,3,1,1,1] => [7,2,1]
[4,2,2,2] => [5,5]
[4,2,2,1,1] => [3,3,2,2]
[4,2,1,1,1,1] => [6,3,1]
[4,1,1,1,1,1,1] => [4,3,3]
[3,3,3,1] => [3,2,2,2,1]
[3,3,2,2] => [6,1,1,1,1]
[3,3,2,1,1] => [6,2,2]
[3,3,1,1,1,1] => [5,4,1]
[3,2,2,2,1] => [5,2,1,1,1]
[3,2,2,1,1,1] => [4,2,2,2]
[3,2,1,1,1,1,1] => [4,4,1,1]
[3,1,1,1,1,1,1,1] => [4,4,2]
[2,2,2,2,2] => [6,4]
[2,2,2,2,1,1] => [3,3,3,1]
[2,2,2,1,1,1,1] => [5,2,2,1]
[2,2,1,1,1,1,1,1] => [5,3,1,1]
[2,1,1,1,1,1,1,1,1] => [5,3,2]
[1,1,1,1,1,1,1,1,1,1] => [4,3,2,1]
[11] => [1,1,1,1,1,1,1,1,1,1,1]
[10,1] => [2,1,1,1,1,1,1,1,1,1]
[9,2] => [2,2,1,1,1,1,1,1,1]
[9,1,1] => [3,1,1,1,1,1,1,1,1]
[8,3] => [2,2,2,1,1,1,1,1]
[8,2,1] => [4,1,1,1,1,1,1,1]
[8,1,1,1] => [3,2,1,1,1,1,1,1]
[7,4] => [2,2,2,2,1,1,1]
[7,3,1] => [3,2,2,1,1,1,1]
[7,2,2] => [5,1,1,1,1,1,1]
[7,2,1,1] => [3,3,1,1,1,1,1]
[7,1,1,1,1] => [4,2,1,1,1,1,1]
[6,5] => [11]
[6,4,1] => [3,2,2,2,1,1]
[6,3,2] => [7,1,1,1,1]
[6,3,1,1] => [5,2,1,1,1,1]
[6,2,2,1] => [3,3,2,1,1,1]
[6,2,1,1,1] => [4,2,2,1,1,1]
[6,1,1,1,1,1] => [4,3,1,1,1,1]
[5,5,1] => [10,1]
[5,4,2] => [9,1,1]
[5,4,1,1] => [9,2]
[5,3,3] => [8,1,1,1]
[5,3,2,1] => [7,2,1,1]
[5,3,1,1,1] => [5,5,1]
[5,2,2,2] => [7,4]
[5,2,2,1,1] => [6,2,2,1]
[5,2,1,1,1,1] => [6,3,1,1]
[5,1,1,1,1,1,1] => [4,3,2,1,1]
[4,4,3] => [2,2,2,2,2,1]
[4,4,2,1] => [8,3]
[4,4,1,1,1] => [8,2,1]
[4,3,3,1] => [3,2,2,2,2]
[4,3,2,2] => [3,3,2,2,1]
[4,3,2,1,1] => [7,2,2]
[4,3,1,1,1,1] => [7,3,1]
[4,2,2,2,1] => [3,3,3,1,1]
[4,2,2,1,1,1] => [5,2,2,1,1]
[4,2,1,1,1,1,1] => [6,3,2]
[4,1,1,1,1,1,1,1] => [4,3,2,2]
[3,3,3,2] => [6,1,1,1,1,1]
[3,3,3,1,1] => [4,2,2,2,1]
[3,3,2,2,1] => [6,2,1,1,1]
[3,3,2,1,1,1] => [4,4,1,1,1]
[3,3,1,1,1,1,1] => [4,4,3]
[3,2,2,2,2] => [6,5]
[3,2,2,2,1,1] => [6,4,1]
[3,2,2,1,1,1,1] => [5,3,1,1,1]
[3,2,1,1,1,1,1,1] => [5,3,3]
[3,1,1,1,1,1,1,1,1] => [4,3,3,1]
[2,2,2,2,2,1] => [3,3,3,2]
[2,2,2,2,1,1,1] => [5,2,2,2]
[2,2,2,1,1,1,1,1] => [5,4,1,1]
[2,2,1,1,1,1,1,1,1] => [5,4,2]
[2,1,1,1,1,1,1,1,1,1] => [4,4,2,1]
[1,1,1,1,1,1,1,1,1,1,1] => [5,3,2,1]
[12] => [1,1,1,1,1,1,1,1,1,1,1,1]
[11,1] => [2,1,1,1,1,1,1,1,1,1,1]
[10,2] => [2,2,1,1,1,1,1,1,1,1]
[10,1,1] => [3,1,1,1,1,1,1,1,1,1]
[9,3] => [2,2,2,1,1,1,1,1,1]
[9,2,1] => [4,1,1,1,1,1,1,1,1]
[9,1,1,1] => [3,2,1,1,1,1,1,1,1]
[8,4] => [2,2,2,2,1,1,1,1]
[8,3,1] => [3,2,2,1,1,1,1,1]
[8,2,2] => [5,1,1,1,1,1,1,1]
[8,2,1,1] => [3,3,1,1,1,1,1,1]
[8,1,1,1,1] => [4,2,1,1,1,1,1,1]
[7,5] => [2,2,2,2,2,1,1]
[7,4,1] => [3,2,2,2,1,1,1]
[7,3,2] => [6,1,1,1,1,1,1]
[7,3,1,1] => [5,2,1,1,1,1,1]
[7,2,2,1] => [3,3,2,1,1,1,1]
[7,2,1,1,1] => [4,2,2,1,1,1,1]
[7,1,1,1,1,1] => [4,3,1,1,1,1,1]
[6,6] => [12]
[6,5,1] => [11,1]
[6,4,2] => [3,3,2,2,1,1]
[6,4,1,1] => [4,2,2,2,1,1]
[6,3,3] => [8,1,1,1,1]
[6,3,2,1] => [7,2,1,1,1]
[6,3,1,1,1] => [4,4,1,1,1,1]
[6,2,2,2] => [3,3,3,1,1,1]
[6,2,2,1,1] => [5,2,2,1,1,1]
[6,2,1,1,1,1] => [5,3,1,1,1,1]
[6,1,1,1,1,1,1] => [4,3,2,1,1,1]
[5,5,2] => [10,1,1]
[5,5,1,1] => [10,2]
[5,4,3] => [9,1,1,1]
[5,4,2,1] => [9,3]
[5,4,1,1,1] => [9,2,1]
[5,3,3,1] => [8,2,1,1]
[5,3,2,2] => [6,6]
[5,3,2,1,1] => [7,4,1]
[5,3,1,1,1,1] => [4,4,4]
[5,2,2,2,1] => [7,2,2,1]
[5,2,2,1,1,1] => [5,5,1,1]
[5,2,1,1,1,1,1] => [5,5,2]
[5,1,1,1,1,1,1,1] => [6,3,2,1]
[4,4,4] => [2,2,2,2,2,2]
[4,4,3,1] => [3,2,2,2,2,1]
[4,4,2,2] => [8,4]
[4,4,2,1,1] => [8,2,2]
[4,4,1,1,1,1] => [8,3,1]
[4,3,3,2] => [3,3,2,2,2]
[4,3,3,1,1] => [4,2,2,2,2]
[4,3,2,2,1] => [6,2,2,1,1]
[4,3,2,1,1,1] => [6,3,1,1,1]
[4,3,1,1,1,1,1] => [7,3,2]
[4,2,2,2,2] => [3,3,3,3]
[4,2,2,2,1,1] => [6,2,2,2]
[4,2,2,1,1,1,1] => [4,3,3,1,1]
[4,2,1,1,1,1,1,1] => [4,4,2,1,1]
[4,1,1,1,1,1,1,1,1] => [5,3,2,1,1]
[3,3,3,3] => [7,1,1,1,1,1]
[3,3,3,2,1] => [6,2,1,1,1,1]
[3,3,3,1,1,1] => [4,3,2,2,1]
[3,3,2,2,2] => [7,5]
[3,3,2,2,1,1] => [7,3,1,1]
[3,3,2,1,1,1,1] => [6,3,3]
[3,3,1,1,1,1,1,1] => [4,3,3,2]
[3,2,2,2,2,1] => [6,5,1]
[3,2,2,2,1,1,1] => [6,4,1,1]
[3,2,2,1,1,1,1,1] => [6,4,2]
[3,2,1,1,1,1,1,1,1] => [4,4,2,2]
[3,1,1,1,1,1,1,1,1,1] => [5,3,2,2]
[2,2,2,2,2,2] => [3,3,3,2,1]
[2,2,2,2,2,1,1] => [5,2,2,2,1]
[2,2,2,2,1,1,1,1] => [5,4,1,1,1]
[2,2,2,1,1,1,1,1,1] => [5,4,3]
[2,2,1,1,1,1,1,1,1,1] => [4,4,3,1]
[2,1,1,1,1,1,1,1,1,1,1] => [5,3,3,1]
[1,1,1,1,1,1,1,1,1,1,1,1] => [5,4,2,1]
[13] => [1,1,1,1,1,1,1,1,1,1,1,1,1]
[12,1] => [2,1,1,1,1,1,1,1,1,1,1,1]
[11,2] => [2,2,1,1,1,1,1,1,1,1,1]
[11,1,1] => [3,1,1,1,1,1,1,1,1,1,1]
[10,3] => [2,2,2,1,1,1,1,1,1,1]
[10,2,1] => [4,1,1,1,1,1,1,1,1,1]
[10,1,1,1] => [3,2,1,1,1,1,1,1,1,1]
[9,4] => [2,2,2,2,1,1,1,1,1]
[9,3,1] => [3,2,2,1,1,1,1,1,1]
[9,2,2] => [5,1,1,1,1,1,1,1,1]
[9,2,1,1] => [3,3,1,1,1,1,1,1,1]
[9,1,1,1,1] => [4,2,1,1,1,1,1,1,1]
[8,5] => [2,2,2,2,2,1,1,1]
[8,4,1] => [3,2,2,2,1,1,1,1]
[8,3,2] => [6,1,1,1,1,1,1,1]
[8,3,1,1] => [5,2,1,1,1,1,1,1]
[8,2,2,1] => [3,3,2,1,1,1,1,1]
[8,2,1,1,1] => [4,2,2,1,1,1,1,1]
[8,1,1,1,1,1] => [4,3,1,1,1,1,1,1]
[7,6] => [13]
[7,5,1] => [3,2,2,2,2,1,1]
[7,4,2] => [3,3,2,2,1,1,1]
[7,4,1,1] => [4,2,2,2,1,1,1]
[7,3,3] => [8,1,1,1,1,1]
[7,3,2,1] => [6,2,1,1,1,1,1]
[7,3,1,1,1] => [4,4,1,1,1,1,1]
[7,2,2,2] => [3,3,3,1,1,1,1]
[7,2,2,1,1] => [5,2,2,1,1,1,1]
[7,2,1,1,1,1] => [5,3,1,1,1,1,1]
[7,1,1,1,1,1,1] => [4,3,2,1,1,1,1]
[6,6,1] => [12,1]
[6,5,2] => [11,1,1]
[6,5,1,1] => [11,2]
[6,4,3] => [9,1,1,1,1]
[6,4,2,1] => [7,2,2,1,1]
[6,4,1,1,1] => [4,3,2,2,1,1]
[6,3,3,1] => [8,2,1,1,1]
[6,3,2,2] => [3,3,3,2,1,1]
[6,3,2,1,1] => [7,3,1,1,1]
[6,3,1,1,1,1] => [4,3,3,1,1,1]
[6,2,2,2,1] => [5,2,2,2,1,1]
[6,2,2,1,1,1] => [5,4,1,1,1,1]
[6,2,1,1,1,1,1] => [4,4,2,1,1,1]
[6,1,1,1,1,1,1,1] => [5,3,2,1,1,1]
[5,5,3] => [10,1,1,1]
[5,5,2,1] => [10,3]
[5,5,1,1,1] => [10,2,1]
[5,4,4] => [2,2,2,2,2,2,1]
[5,4,3,1] => [9,2,1,1]
[5,4,2,2] => [9,4]
[5,4,2,1,1] => [9,2,2]
[5,4,1,1,1,1] => [9,3,1]
[5,3,3,2] => [8,5]
[5,3,3,1,1] => [8,4,1]
[5,3,2,2,1] => [6,6,1]
[5,3,2,1,1,1] => [7,3,3]
[5,3,1,1,1,1,1] => [7,4,2]
[5,2,2,2,2] => [8,2,2,1]
[5,2,2,2,1,1] => [7,4,1,1]
[5,2,2,1,1,1,1] => [5,4,4]
[5,2,1,1,1,1,1,1] => [4,4,4,1]
[5,1,1,1,1,1,1,1,1] => [5,5,2,1]
[4,4,4,1] => [3,2,2,2,2,2]
[4,4,3,2] => [3,3,2,2,2,1]
[4,4,3,1,1] => [4,2,2,2,2,1]
[4,4,2,2,1] => [6,2,2,1,1,1]
[4,4,2,1,1,1] => [8,3,1,1]
[4,4,1,1,1,1,1] => [8,3,2]
[4,3,3,3] => [7,1,1,1,1,1,1]
[4,3,3,2,1] => [7,2,2,2]
[4,3,3,1,1,1] => [4,3,2,2,2]
[4,3,2,2,2] => [3,3,3,2,2]
[4,3,2,2,1,1] => [6,5,1,1]
[4,3,2,1,1,1,1] => [6,3,2,1,1]
[4,3,1,1,1,1,1,1] => [6,3,2,2]
[4,2,2,2,2,1] => [5,2,2,2,2]
[4,2,2,2,1,1,1] => [5,5,1,1,1]
[4,2,2,1,1,1,1,1] => [7,3,2,1]
[4,2,1,1,1,1,1,1,1] => [6,3,3,1]
[4,1,1,1,1,1,1,1,1,1] => [6,4,2,1]
[3,3,3,3,1] => [7,2,1,1,1,1]
[3,3,3,2,2] => [7,6]
[3,3,3,2,1,1] => [6,3,1,1,1,1]
[3,3,3,1,1,1,1] => [4,3,3,3]
[3,3,2,2,2,1] => [7,5,1]
[3,3,2,2,1,1,1] => [6,5,2]
[3,3,2,1,1,1,1,1] => [4,4,2,2,1]
[3,3,1,1,1,1,1,1,1] => [5,3,2,2,1]
[3,2,2,2,2,2] => [3,3,3,3,1]
[3,2,2,2,2,1,1] => [4,3,3,2,1]
[3,2,2,2,1,1,1,1] => [5,5,3]
[3,2,2,1,1,1,1,1,1] => [4,4,3,1,1]
[3,2,1,1,1,1,1,1,1,1] => [5,3,3,1,1]
[3,1,1,1,1,1,1,1,1,1,1] => [5,4,2,1,1]
[2,2,2,2,2,2,1] => [6,2,2,2,1]
[2,2,2,2,2,1,1,1] => [6,4,1,1,1]
[2,2,2,2,1,1,1,1,1] => [6,4,3]
[2,2,2,1,1,1,1,1,1,1] => [4,4,3,2]
[2,2,1,1,1,1,1,1,1,1,1] => [5,3,3,2]
[2,1,1,1,1,1,1,1,1,1,1,1] => [5,4,2,2]
[1,1,1,1,1,1,1,1,1,1,1,1,1] => [5,4,3,1]
[14] => [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
[13,1] => [2,1,1,1,1,1,1,1,1,1,1,1,1]
[12,2] => [2,2,1,1,1,1,1,1,1,1,1,1]
[12,1,1] => [3,1,1,1,1,1,1,1,1,1,1,1]
[11,3] => [2,2,2,1,1,1,1,1,1,1,1]
[11,2,1] => [4,1,1,1,1,1,1,1,1,1,1]
[11,1,1,1] => [3,2,1,1,1,1,1,1,1,1,1]
[10,4] => [2,2,2,2,1,1,1,1,1,1]
[10,3,1] => [3,2,2,1,1,1,1,1,1,1]
[10,2,2] => [5,1,1,1,1,1,1,1,1,1]
[10,2,1,1] => [3,3,1,1,1,1,1,1,1,1]
[10,1,1,1,1] => [4,2,1,1,1,1,1,1,1,1]
[9,5] => [2,2,2,2,2,1,1,1,1]
[9,4,1] => [3,2,2,2,1,1,1,1,1]
[9,3,2] => [6,1,1,1,1,1,1,1,1]
[9,3,1,1] => [5,2,1,1,1,1,1,1,1]
[9,2,2,1] => [3,3,2,1,1,1,1,1,1]
[9,2,1,1,1] => [4,2,2,1,1,1,1,1,1]
[9,1,1,1,1,1] => [4,3,1,1,1,1,1,1,1]
[8,6] => [2,2,2,2,2,2,1,1]
[8,5,1] => [3,2,2,2,2,1,1,1]
[8,4,2] => [3,3,2,2,1,1,1,1]
[8,4,1,1] => [4,2,2,2,1,1,1,1]
[8,3,3] => [7,1,1,1,1,1,1,1]
[8,3,2,1] => [6,2,1,1,1,1,1,1]
[8,3,1,1,1] => [4,4,1,1,1,1,1,1]
[8,2,2,2] => [3,3,3,1,1,1,1,1]
[8,2,2,1,1] => [5,2,2,1,1,1,1,1]
[8,2,1,1,1,1] => [5,3,1,1,1,1,1,1]
[8,1,1,1,1,1,1] => [4,3,2,1,1,1,1,1]
[7,7] => [14]
[7,6,1] => [13,1]
[7,5,2] => [3,3,2,2,2,1,1]
[7,5,1,1] => [4,2,2,2,2,1,1]
[7,4,3] => [9,1,1,1,1,1]
[7,4,2,1] => [6,2,2,1,1,1,1]
[7,4,1,1,1] => [4,3,2,2,1,1,1]
[7,3,3,1] => [8,2,1,1,1,1]
[7,3,2,2] => [3,3,3,2,1,1,1]
[7,3,2,1,1] => [6,3,1,1,1,1,1]
[7,3,1,1,1,1] => [4,3,3,1,1,1,1]
[7,2,2,2,1] => [5,2,2,2,1,1,1]
[7,2,2,1,1,1] => [5,4,1,1,1,1,1]
[7,2,1,1,1,1,1] => [4,4,2,1,1,1,1]
[7,1,1,1,1,1,1,1] => [5,3,2,1,1,1,1]
[6,6,2] => [12,1,1]
[6,6,1,1] => [12,2]
[6,5,3] => [11,1,1,1]
[6,5,2,1] => [11,3]
[6,5,1,1,1] => [11,2,1]
[6,4,4] => [10,1,1,1,1]
[6,4,3,1] => [9,2,1,1,1]
[6,4,2,2] => [8,2,2,1,1]
[6,4,2,1,1] => [6,6,1,1]
[6,4,1,1,1,1] => [7,3,2,1,1]
[6,3,3,2] => [7,7]
[6,3,3,1,1] => [8,3,1,1,1]
[6,3,2,2,1] => [4,3,3,2,1,1]
[6,3,2,1,1,1] => [4,4,2,2,1,1]
[6,3,1,1,1,1,1] => [5,3,2,2,1,1]
[6,2,2,2,2] => [7,2,2,2,1]
[6,2,2,2,1,1] => [7,4,1,1,1]
[6,2,2,1,1,1,1] => [4,4,3,1,1,1]
[6,2,1,1,1,1,1,1] => [5,3,3,1,1,1]
[6,1,1,1,1,1,1,1,1] => [5,4,2,1,1,1]
[5,5,4] => [2,2,2,2,2,2,2]
[5,5,3,1] => [10,2,1,1]
[5,5,2,2] => [10,4]
[5,5,2,1,1] => [10,2,2]
[5,5,1,1,1,1] => [10,3,1]
[5,4,4,1] => [3,2,2,2,2,2,1]
[5,4,3,2] => [9,5]
[5,4,3,1,1] => [9,4,1]
[5,4,2,2,1] => [9,2,2,1]
[5,4,2,1,1,1] => [9,3,1,1]
[5,4,1,1,1,1,1] => [9,3,2]
[5,3,3,3] => [3,3,3,2,2,1]
[5,3,3,2,1] => [8,5,1]
[5,3,3,1,1,1] => [8,3,3]
[5,3,2,2,2] => [3,3,3,3,1,1]
[5,3,2,2,1,1] => [8,4,1,1]
[5,3,2,1,1,1,1] => [8,4,2]
[5,3,1,1,1,1,1,1] => [7,3,3,1]
[5,2,2,2,2,1] => [5,5,1,1,1,1]
[5,2,2,2,1,1,1] => [7,4,3]
[5,2,2,1,1,1,1,1] => [4,4,3,3]
[5,2,1,1,1,1,1,1,1] => [7,4,2,1]
[5,1,1,1,1,1,1,1,1,1] => [5,4,4,1]
[4,4,4,2] => [3,3,2,2,2,2]
[4,4,4,1,1] => [4,2,2,2,2,2]
[4,4,3,3] => [8,1,1,1,1,1,1]
[4,4,3,2,1] => [5,2,2,2,2,1]
[4,4,3,1,1,1] => [4,3,2,2,2,1]
[4,4,2,2,2] => [8,2,2,2]
[4,4,2,2,1,1] => [7,5,1,1]
[4,4,2,1,1,1,1] => [6,3,2,1,1,1]
[4,4,1,1,1,1,1,1] => [8,3,2,1]
[4,3,3,3,1] => [7,2,1,1,1,1,1]
[4,3,3,2,2] => [7,2,2,1,1,1]
[4,3,3,2,1,1] => [6,6,2]
[4,3,3,1,1,1,1] => [7,3,2,2]
[4,3,2,2,2,1] => [4,3,3,2,2]
[4,3,2,2,1,1,1] => [5,5,4]
[4,3,2,1,1,1,1,1] => [6,5,2,1]
[4,3,1,1,1,1,1,1,1] => [5,5,2,1,1]
[4,2,2,2,2,2] => [6,2,2,2,1,1]
[4,2,2,2,2,1,1] => [6,4,1,1,1,1]
[4,2,2,2,1,1,1,1] => [4,4,4,1,1]
[4,2,2,1,1,1,1,1,1] => [4,4,4,2]
[4,2,1,1,1,1,1,1,1,1] => [5,5,2,2]
[4,1,1,1,1,1,1,1,1,1,1] => [5,5,3,1]
[3,3,3,3,2] => [8,6]
[3,3,3,3,1,1] => [7,3,1,1,1,1]
[3,3,3,2,2,1] => [7,6,1]
[3,3,3,2,1,1,1] => [4,4,2,2,2]
[3,3,3,1,1,1,1,1] => [6,3,2,2,1]
[3,3,2,2,2,2] => [3,3,3,3,2]
[3,3,2,2,2,1,1] => [7,5,2]
[3,3,2,2,1,1,1,1] => [5,3,2,2,2]
[3,3,2,1,1,1,1,1,1] => [6,3,3,1,1]
[3,3,1,1,1,1,1,1,1,1] => [6,4,2,1,1]
[3,2,2,2,2,2,1] => [4,3,3,3,1]
[3,2,2,2,2,1,1,1] => [6,4,4]
[3,2,2,2,1,1,1,1,1] => [5,3,3,3]
[3,2,2,1,1,1,1,1,1,1] => [6,3,3,2]
[3,2,1,1,1,1,1,1,1,1,1] => [6,4,2,2]
[3,1,1,1,1,1,1,1,1,1,1,1] => [6,4,3,1]
[2,2,2,2,2,2,2] => [6,2,2,2,2]
[2,2,2,2,2,2,1,1] => [6,5,1,1,1]
[2,2,2,2,2,1,1,1,1] => [6,5,3]
[2,2,2,2,1,1,1,1,1,1] => [4,4,3,2,1]
[2,2,2,1,1,1,1,1,1,1,1] => [5,3,3,2,1]
[2,2,1,1,1,1,1,1,1,1,1,1] => [5,4,2,2,1]
[2,1,1,1,1,1,1,1,1,1,1,1,1] => [5,4,3,1,1]
[1,1,1,1,1,1,1,1,1,1,1,1,1,1] => [5,4,3,2]
[15] => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
[14,1] => [2,1,1,1,1,1,1,1,1,1,1,1,1,1]
[13,2] => [2,2,1,1,1,1,1,1,1,1,1,1,1]
[13,1,1] => [3,1,1,1,1,1,1,1,1,1,1,1,1]
[12,3] => [2,2,2,1,1,1,1,1,1,1,1,1]
[12,2,1] => [4,1,1,1,1,1,1,1,1,1,1,1]
[12,1,1,1] => [3,2,1,1,1,1,1,1,1,1,1,1]
[11,4] => [2,2,2,2,1,1,1,1,1,1,1]
[11,3,1] => [3,2,2,1,1,1,1,1,1,1,1]
[11,2,2] => [5,1,1,1,1,1,1,1,1,1,1]
[11,2,1,1] => [3,3,1,1,1,1,1,1,1,1,1]
[11,1,1,1,1] => [4,2,1,1,1,1,1,1,1,1,1]
[10,5] => [2,2,2,2,2,1,1,1,1,1]
[10,4,1] => [3,2,2,2,1,1,1,1,1,1]
[10,3,2] => [6,1,1,1,1,1,1,1,1,1]
[10,3,1,1] => [5,2,1,1,1,1,1,1,1,1]
[10,2,2,1] => [3,3,2,1,1,1,1,1,1,1]
[10,2,1,1,1] => [4,2,2,1,1,1,1,1,1,1]
[10,1,1,1,1,1] => [4,3,1,1,1,1,1,1,1,1]
[9,6] => [2,2,2,2,2,2,1,1,1]
[9,5,1] => [3,2,2,2,2,1,1,1,1]
[9,4,2] => [3,3,2,2,1,1,1,1,1]
[9,4,1,1] => [4,2,2,2,1,1,1,1,1]
[9,3,3] => [7,1,1,1,1,1,1,1,1]
[9,3,2,1] => [6,2,1,1,1,1,1,1,1]
[9,3,1,1,1] => [4,4,1,1,1,1,1,1,1]
[9,2,2,2] => [3,3,3,1,1,1,1,1,1]
[9,2,2,1,1] => [5,2,2,1,1,1,1,1,1]
[9,2,1,1,1,1] => [5,3,1,1,1,1,1,1,1]
[9,1,1,1,1,1,1] => [4,3,2,1,1,1,1,1,1]
[8,7] => [15]
[8,6,1] => [3,2,2,2,2,2,1,1]
[8,5,2] => [3,3,2,2,2,1,1,1]
[8,5,1,1] => [4,2,2,2,2,1,1,1]
[8,4,3] => [9,1,1,1,1,1,1]
[8,4,2,1] => [6,2,2,1,1,1,1,1]
[8,4,1,1,1] => [4,3,2,2,1,1,1,1]
[8,3,3,1] => [7,2,1,1,1,1,1,1]
[8,3,2,2] => [3,3,3,2,1,1,1,1]
[8,3,2,1,1] => [6,3,1,1,1,1,1,1]
[8,3,1,1,1,1] => [4,3,3,1,1,1,1,1]
[8,2,2,2,1] => [5,2,2,2,1,1,1,1]
[8,2,2,1,1,1] => [5,4,1,1,1,1,1,1]
[8,2,1,1,1,1,1] => [4,4,2,1,1,1,1,1]
[8,1,1,1,1,1,1,1] => [5,3,2,1,1,1,1,1]
[7,7,1] => [14,1]
[7,6,2] => [13,1,1]
[7,6,1,1] => [13,2]
[7,5,3] => [3,3,3,2,2,1,1]
[7,5,2,1] => [5,2,2,2,2,1,1]
[7,5,1,1,1] => [4,3,2,2,2,1,1]
[7,4,4] => [10,1,1,1,1,1]
[7,4,3,1] => [9,2,1,1,1,1]
[7,4,2,2] => [8,2,2,1,1,1]
[7,4,2,1,1] => [5,5,1,1,1,1,1]
[7,4,1,1,1,1] => [6,3,2,1,1,1,1]
[7,3,3,2] => [3,3,3,3,1,1,1]
[7,3,3,1,1] => [8,3,1,1,1,1]
[7,3,2,2,1] => [4,3,3,2,1,1,1]
[7,3,2,1,1,1] => [4,4,2,2,1,1,1]
[7,3,1,1,1,1,1] => [5,3,2,2,1,1,1]
[7,2,2,2,2] => [6,2,2,2,1,1,1]
[7,2,2,2,1,1] => [6,4,1,1,1,1,1]
[7,2,2,1,1,1,1] => [4,4,3,1,1,1,1]
[7,2,1,1,1,1,1,1] => [5,3,3,1,1,1,1]
[7,1,1,1,1,1,1,1,1] => [5,4,2,1,1,1,1]
[6,6,3] => [12,1,1,1]
[6,6,2,1] => [12,3]
[6,6,1,1,1] => [12,2,1]
[6,5,4] => [11,1,1,1,1]
[6,5,3,1] => [11,2,1,1]
[6,5,2,2] => [11,4]
[6,5,2,1,1] => [11,2,2]
[6,5,1,1,1,1] => [11,3,1]
[6,4,4,1] => [10,2,1,1,1]
[6,4,3,2] => [9,2,2,1,1]
[6,4,3,1,1] => [9,3,1,1,1]
[6,4,2,2,1] => [8,5,1,1]
[6,4,2,1,1,1] => [8,3,2,1,1]
[6,4,1,1,1,1,1] => [6,6,2,1]
[6,3,3,3] => [9,6]
[6,3,3,2,1] => [7,7,1]
[6,3,3,1,1,1] => [7,3,2,2,1]
[6,3,2,2,2] => [8,2,2,2,1]
[6,3,2,2,1,1] => [5,5,5]
[6,3,2,1,1,1,1] => [7,3,3,1,1]
[6,3,1,1,1,1,1,1] => [7,4,2,1,1]
[6,2,2,2,2,1] => [6,6,1,1,1]
[6,2,2,2,1,1,1] => [4,4,3,2,1,1]
[6,2,2,1,1,1,1,1] => [5,3,3,2,1,1]
[6,2,1,1,1,1,1,1,1] => [5,4,2,2,1,1]
[6,1,1,1,1,1,1,1,1,1] => [5,4,3,1,1,1]
[5,5,5] => [2,2,2,2,2,2,2,1]
[5,5,4,1] => [3,2,2,2,2,2,2]
[5,5,3,2] => [10,5]
[5,5,3,1,1] => [10,4,1]
[5,5,2,2,1] => [10,2,2,1]
[5,5,2,1,1,1] => [10,3,1,1]
[5,5,1,1,1,1,1] => [10,3,2]
[5,4,4,2] => [3,3,2,2,2,2,1]
[5,4,4,1,1] => [4,2,2,2,2,2,1]
[5,4,3,3] => [3,3,3,2,2,2]
[5,4,3,2,1] => [9,5,1]
[5,4,3,1,1,1] => [9,3,3]
[5,4,2,2,2] => [9,2,2,2]
[5,4,2,2,1,1] => [9,4,1,1]
[5,4,2,1,1,1,1] => [9,4,2]
[5,4,1,1,1,1,1,1] => [9,3,2,1]
[5,3,3,3,1] => [4,3,3,2,2,1]
[5,3,3,2,2] => [3,3,3,3,3]
[5,3,3,2,1,1] => [8,5,2]
[5,3,3,1,1,1,1] => [6,3,2,2,1,1]
[5,3,2,2,2,1] => [4,3,3,3,1,1]
[5,3,2,2,1,1,1] => [8,4,3]
[5,3,2,1,1,1,1,1] => [8,3,3,1]
[5,3,1,1,1,1,1,1,1] => [8,4,2,1]
[5,2,2,2,2,2] => [7,2,2,2,2]
[5,2,2,2,2,1,1] => [6,6,3]
[5,2,2,2,1,1,1,1] => [4,4,3,2,2]
[5,2,2,1,1,1,1,1,1] => [5,3,3,2,2]
[5,2,1,1,1,1,1,1,1,1] => [7,4,3,1]
[5,1,1,1,1,1,1,1,1,1,1] => [5,4,3,3]
[4,4,4,3] => [8,1,1,1,1,1,1,1]
[4,4,4,2,1] => [5,2,2,2,2,2]
[4,4,4,1,1,1] => [4,3,2,2,2,2]
[4,4,3,3,1] => [8,2,1,1,1,1,1]
[4,4,3,2,2] => [7,2,2,2,1,1]
[4,4,3,2,1,1] => [4,4,2,2,2,1]
[4,4,3,1,1,1,1] => [5,3,2,2,2,1]
[4,4,2,2,2,1] => [8,4,1,1,1]
[4,4,2,2,1,1,1] => [4,4,4,1,1,1]
[4,4,2,1,1,1,1,1] => [7,5,2,1]
[4,4,1,1,1,1,1,1,1] => [5,5,4,1]
[4,3,3,3,2] => [7,2,2,1,1,1,1]
[4,3,3,3,1,1] => [7,3,1,1,1,1,1]
[4,3,3,2,2,1] => [7,4,1,1,1,1]
[4,3,3,2,1,1,1] => [8,3,2,2]
[4,3,3,1,1,1,1,1] => [5,5,2,1,1,1]
[4,3,2,2,2,2] => [7,6,1,1]
[4,3,2,2,2,1,1] => [7,4,4]
[4,3,2,2,1,1,1,1] => [4,4,4,3]
[4,3,2,1,1,1,1,1,1] => [7,4,2,2]
[4,3,1,1,1,1,1,1,1,1] => [6,4,4,1]
[4,2,2,2,2,2,1] => [7,5,1,1,1]
[4,2,2,2,2,1,1,1] => [7,5,3]
[4,2,2,2,1,1,1,1,1] => [6,4,2,1,1,1]
[4,2,2,1,1,1,1,1,1,1] => [5,4,2,2,2]
[4,2,1,1,1,1,1,1,1,1,1] => [5,4,4,1,1]
[4,1,1,1,1,1,1,1,1,1,1,1] => [5,4,4,2]
[3,3,3,3,3] => [8,7]
[3,3,3,3,2,1] => [8,6,1]
[3,3,3,3,1,1,1] => [7,3,2,1,1,1]
[3,3,3,2,2,2] => [3,3,3,3,2,1]
[3,3,3,2,2,1,1] => [7,6,2]
[3,3,3,2,1,1,1,1] => [7,3,3,2]
[3,3,3,1,1,1,1,1,1] => [6,5,2,1,1]
[3,3,2,2,2,2,1] => [4,3,3,3,2]
[3,3,2,2,2,1,1,1] => [6,3,3,1,1,1]
[3,3,2,2,1,1,1,1,1] => [6,3,3,3]
[3,3,2,1,1,1,1,1,1,1] => [6,5,2,2]
[3,3,1,1,1,1,1,1,1,1,1] => [6,5,3,1]
[3,2,2,2,2,2,2] => [6,2,2,2,2,1]
[3,2,2,2,2,2,1,1] => [6,3,2,2,2]
[3,2,2,2,2,1,1,1,1] => [4,4,3,3,1]
[3,2,2,2,1,1,1,1,1,1] => [4,4,4,2,1]
[3,2,2,1,1,1,1,1,1,1,1] => [5,5,2,2,1]
[3,2,1,1,1,1,1,1,1,1,1,1] => [5,5,3,1,1]
[3,1,1,1,1,1,1,1,1,1,1,1,1] => [5,5,3,2]
[2,2,2,2,2,2,2,1] => [6,5,1,1,1,1]
[2,2,2,2,2,2,1,1,1] => [6,5,4]
[2,2,2,2,2,1,1,1,1,1] => [5,3,3,3,1]
[2,2,2,2,1,1,1,1,1,1,1] => [6,3,3,2,1]
[2,2,2,1,1,1,1,1,1,1,1,1] => [6,4,2,2,1]
[2,2,1,1,1,1,1,1,1,1,1,1,1] => [6,4,3,1,1]
[2,1,1,1,1,1,1,1,1,1,1,1,1,1] => [6,4,3,2]
[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => [5,4,3,2,1]
[16] => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
[15,1] => [2,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
[14,2] => [2,2,1,1,1,1,1,1,1,1,1,1,1,1]
[14,1,1] => [3,1,1,1,1,1,1,1,1,1,1,1,1,1]
[13,3] => [2,2,2,1,1,1,1,1,1,1,1,1,1]
[13,2,1] => [4,1,1,1,1,1,1,1,1,1,1,1,1]
[13,1,1,1] => [3,2,1,1,1,1,1,1,1,1,1,1,1]
[12,4] => [2,2,2,2,1,1,1,1,1,1,1,1]
[12,3,1] => [3,2,2,1,1,1,1,1,1,1,1,1]
[12,2,2] => [5,1,1,1,1,1,1,1,1,1,1,1]
[12,2,1,1] => [3,3,1,1,1,1,1,1,1,1,1,1]
[12,1,1,1,1] => [4,2,1,1,1,1,1,1,1,1,1,1]
[11,5] => [2,2,2,2,2,1,1,1,1,1,1]
[11,4,1] => [3,2,2,2,1,1,1,1,1,1,1]
[11,3,2] => [6,1,1,1,1,1,1,1,1,1,1]
[11,3,1,1] => [5,2,1,1,1,1,1,1,1,1,1]
[11,2,2,1] => [3,3,2,1,1,1,1,1,1,1,1]
[11,2,1,1,1] => [4,2,2,1,1,1,1,1,1,1,1]
[11,1,1,1,1,1] => [4,3,1,1,1,1,1,1,1,1,1]
[10,6] => [2,2,2,2,2,2,1,1,1,1]
[10,5,1] => [3,2,2,2,2,1,1,1,1,1]
[10,4,2] => [3,3,2,2,1,1,1,1,1,1]
[10,4,1,1] => [4,2,2,2,1,1,1,1,1,1]
[10,3,3] => [7,1,1,1,1,1,1,1,1,1]
[10,3,2,1] => [6,2,1,1,1,1,1,1,1,1]
[10,3,1,1,1] => [4,4,1,1,1,1,1,1,1,1]
[10,2,2,2] => [3,3,3,1,1,1,1,1,1,1]
[10,2,2,1,1] => [5,2,2,1,1,1,1,1,1,1]
[10,2,1,1,1,1] => [5,3,1,1,1,1,1,1,1,1]
[10,1,1,1,1,1,1] => [4,3,2,1,1,1,1,1,1,1]
[9,7] => [2,2,2,2,2,2,2,1,1]
[9,6,1] => [3,2,2,2,2,2,1,1,1]
[9,5,2] => [3,3,2,2,2,1,1,1,1]
[9,5,1,1] => [4,2,2,2,2,1,1,1,1]
[9,4,3] => [8,1,1,1,1,1,1,1,1]
[9,4,2,1] => [6,2,2,1,1,1,1,1,1]
[9,4,1,1,1] => [4,3,2,2,1,1,1,1,1]
[9,3,3,1] => [7,2,1,1,1,1,1,1,1]
[9,3,2,2] => [3,3,3,2,1,1,1,1,1]
[9,3,2,1,1] => [6,3,1,1,1,1,1,1,1]
[9,3,1,1,1,1] => [4,3,3,1,1,1,1,1,1]
[9,2,2,2,1] => [5,2,2,2,1,1,1,1,1]
[9,2,2,1,1,1] => [5,4,1,1,1,1,1,1,1]
[9,2,1,1,1,1,1] => [4,4,2,1,1,1,1,1,1]
[9,1,1,1,1,1,1,1] => [5,3,2,1,1,1,1,1,1]
[8,8] => [16]
[8,7,1] => [15,1]
[8,6,2] => [3,3,2,2,2,2,1,1]
[8,6,1,1] => [4,2,2,2,2,2,1,1]
[8,5,3] => [3,3,3,2,2,1,1,1]
[8,5,2,1] => [5,2,2,2,2,1,1,1]
[8,5,1,1,1] => [4,3,2,2,2,1,1,1]
[8,4,4] => [10,1,1,1,1,1,1]
[8,4,3,1] => [9,2,1,1,1,1,1]
[8,4,2,2] => [7,2,2,1,1,1,1,1]
[8,4,2,1,1] => [5,5,1,1,1,1,1,1]
[8,4,1,1,1,1] => [6,3,2,1,1,1,1,1]
[8,3,3,2] => [3,3,3,3,1,1,1,1]
[8,3,3,1,1] => [7,3,1,1,1,1,1,1]
[8,3,2,2,1] => [4,3,3,2,1,1,1,1]
[8,3,2,1,1,1] => [4,4,2,2,1,1,1,1]
[8,3,1,1,1,1,1] => [5,3,2,2,1,1,1,1]
[8,2,2,2,2] => [6,2,2,2,1,1,1,1]
[8,2,2,2,1,1] => [6,4,1,1,1,1,1,1]
[8,2,2,1,1,1,1] => [4,4,3,1,1,1,1,1]
[8,2,1,1,1,1,1,1] => [5,3,3,1,1,1,1,1]
[8,1,1,1,1,1,1,1,1] => [5,4,2,1,1,1,1,1]
[7,7,2] => [14,1,1]
[7,7,1,1] => [14,2]
[7,6,3] => [13,1,1,1]
[7,6,2,1] => [13,3]
[7,6,1,1,1] => [13,2,1]
[7,5,4] => [11,1,1,1,1,1]
[7,5,3,1] => [4,3,3,2,2,1,1]
[7,5,2,2] => [8,2,2,2,1,1]
[7,5,2,1,1] => [4,4,2,2,2,1,1]
[7,5,1,1,1,1] => [5,3,2,2,2,1,1]
[7,4,4,1] => [10,2,1,1,1,1]
[7,4,3,2] => [9,2,2,1,1,1]
[7,4,3,1,1] => [9,3,1,1,1,1]
[7,4,2,2,1] => [8,4,1,1,1,1]
[7,4,2,1,1,1] => [8,3,2,1,1,1]
[7,4,1,1,1,1,1] => [5,5,2,1,1,1,1]
[7,3,3,3] => [3,3,3,3,2,1,1]
[7,3,3,2,1] => [4,3,3,3,1,1,1]
[7,3,3,1,1,1] => [6,3,2,2,1,1,1]
[7,3,2,2,2] => [6,2,2,2,2,1,1]
[7,3,2,2,1,1] => [4,4,4,1,1,1,1]
[7,3,2,1,1,1,1] => [6,3,3,1,1,1,1]
[7,3,1,1,1,1,1,1] => [6,4,2,1,1,1,1]
[7,2,2,2,2,1] => [6,5,1,1,1,1,1]
[7,2,2,2,1,1,1] => [4,4,3,2,1,1,1]
[7,2,2,1,1,1,1,1] => [5,3,3,2,1,1,1]
[7,2,1,1,1,1,1,1,1] => [5,4,2,2,1,1,1]
[7,1,1,1,1,1,1,1,1,1] => [5,4,3,1,1,1,1]
[6,6,4] => [12,1,1,1,1]
[6,6,3,1] => [12,2,1,1]
[6,6,2,2] => [12,4]
[6,6,2,1,1] => [12,2,2]
[6,6,1,1,1,1] => [12,3,1]
[6,5,5] => [2,2,2,2,2,2,2,2]
[6,5,4,1] => [11,2,1,1,1]
[6,5,3,2] => [11,5]
[6,5,3,1,1] => [11,4,1]
[6,5,2,2,1] => [11,2,2,1]
[6,5,2,1,1,1] => [11,3,1,1]
[6,5,1,1,1,1,1] => [11,3,2]
[6,4,4,2] => [10,2,2,1,1]
[6,4,4,1,1] => [10,3,1,1,1]
[6,4,3,3] => [8,8]
[6,4,3,2,1] => [9,5,1,1]
[6,4,3,1,1,1] => [9,3,2,1,1]
[6,4,2,2,2] => [7,7,1,1]
[6,4,2,2,1,1] => [8,4,4]
[6,4,2,1,1,1,1] => [8,5,2,1]
[6,4,1,1,1,1,1,1] => [5,5,5,1]
[6,3,3,3,1] => [9,6,1]
[6,3,3,2,2] => [9,2,2,2,1]
[6,3,3,2,1,1] => [7,7,2]
[6,3,3,1,1,1,1] => [6,6,2,1,1]
[6,3,2,2,2,1] => [8,3,2,2,1]
[6,3,2,2,1,1,1] => [8,3,3,1,1]
[6,3,2,1,1,1,1,1] => [6,6,2,2]
[6,3,1,1,1,1,1,1,1] => [6,6,3,1]
[6,2,2,2,2,2] => [8,5,1,1,1]
[6,2,2,2,2,1,1] => [6,5,5]
[6,2,2,2,1,1,1,1] => [7,3,3,2,1]
[6,2,2,1,1,1,1,1,1] => [7,4,2,2,1]
[6,2,1,1,1,1,1,1,1,1] => [7,4,3,1,1]
[6,1,1,1,1,1,1,1,1,1,1] => [5,4,3,2,1,1]
[5,5,5,1] => [3,2,2,2,2,2,2,1]
[5,5,4,2] => [3,3,2,2,2,2,2]
[5,5,4,1,1] => [4,2,2,2,2,2,2]
[5,5,3,3] => [10,6]
[5,5,3,2,1] => [10,5,1]
[5,5,3,1,1,1] => [10,3,3]
[5,5,2,2,2] => [10,2,2,2]
[5,5,2,2,1,1] => [10,4,1,1]
[5,5,2,1,1,1,1] => [10,4,2]
[5,5,1,1,1,1,1,1] => [10,3,2,1]
[5,4,4,3] => [3,3,3,2,2,2,1]
[5,4,4,2,1] => [5,2,2,2,2,2,1]
[5,4,4,1,1,1] => [4,3,2,2,2,2,1]
[5,4,3,3,1] => [4,3,3,2,2,2]
[5,4,3,2,2] => [7,2,2,2,1,1,1]
[5,4,3,2,1,1] => [9,5,2]
[5,4,3,1,1,1,1] => [9,3,2,2]
[5,4,2,2,2,1] => [9,4,1,1,1]
[5,4,2,2,1,1,1] => [9,4,3]
[5,4,2,1,1,1,1,1] => [9,3,3,1]
[5,4,1,1,1,1,1,1,1] => [9,4,2,1]
[5,3,3,3,2] => [3,3,3,3,2,2]
[5,3,3,3,1,1] => [7,3,2,2,1,1]
[5,3,3,2,2,1] => [4,3,3,3,3]
[5,3,3,2,1,1,1] => [7,3,3,1,1,1]
[5,3,3,1,1,1,1,1] => [7,5,2,1,1]
[5,3,2,2,2,2] => [6,2,2,2,2,2]
[5,3,2,2,2,1,1] => [7,3,2,2,2]
[5,3,2,2,1,1,1,1] => [4,4,4,2,1,1]
[5,3,2,1,1,1,1,1,1] => [5,5,2,2,1,1]
[5,3,1,1,1,1,1,1,1,1] => [8,4,3,1]
[5,2,2,2,2,2,1] => [4,4,3,3,1,1]
[5,2,2,2,2,1,1,1] => [8,3,3,2]
[5,2,2,2,1,1,1,1,1] => [6,3,3,2,1,1]
[5,2,2,1,1,1,1,1,1,1] => [5,5,3,1,1,1]
[5,2,1,1,1,1,1,1,1,1,1] => [7,4,3,2]
[5,1,1,1,1,1,1,1,1,1,1,1] => [5,4,3,2,2]
[4,4,4,4] => [9,1,1,1,1,1,1,1]
[4,4,4,3,1] => [8,2,1,1,1,1,1,1]
[4,4,4,2,2] => [8,2,2,2,2]
[4,4,4,2,1,1] => [4,4,2,2,2,2]
[4,4,4,1,1,1,1] => [5,3,2,2,2,2]
[4,4,3,3,2] => [8,2,2,1,1,1,1]
[4,4,3,3,1,1] => [8,3,1,1,1,1,1]
[4,4,3,2,2,1] => [4,4,3,2,2,1]
[4,4,3,2,1,1,1] => [5,3,3,2,2,1]
[4,4,3,1,1,1,1,1] => [5,4,2,2,2,1]
[4,4,2,2,2,2] => [7,6,1,1,1]
[4,4,2,2,2,1,1] => [7,6,3]
[4,4,2,2,1,1,1,1] => [8,4,2,1,1]
[4,4,2,1,1,1,1,1,1] => [5,4,4,1,1,1]
[4,4,1,1,1,1,1,1,1,1] => [5,4,4,3]
[4,3,3,3,3] => [9,7]
[4,3,3,3,2,1] => [7,4,1,1,1,1,1]
[4,3,3,3,1,1,1] => [7,3,2,1,1,1,1]
[4,3,3,2,2,2] => [8,6,1,1]
[4,3,3,2,2,1,1] => [8,5,3]
[4,3,3,2,1,1,1,1] => [5,3,3,3,1,1]
[4,3,3,1,1,1,1,1,1] => [8,4,2,2]
[4,3,2,2,2,2,1] => [4,4,4,4]
[4,3,2,2,2,1,1,1] => [4,4,3,3,2]
[4,3,2,2,1,1,1,1,1] => [7,4,4,1]
[4,3,2,1,1,1,1,1,1,1] => [7,5,3,1]
[4,3,1,1,1,1,1,1,1,1,1] => [5,5,3,3]
[4,2,2,2,2,2,2] => [6,6,1,1,1,1]
[4,2,2,2,2,2,1,1] => [6,6,4]
[4,2,2,2,2,1,1,1,1] => [5,3,3,3,2]
[4,2,2,2,1,1,1,1,1,1] => [6,4,2,2,1,1]
[4,2,2,1,1,1,1,1,1,1,1] => [6,4,3,1,1,1]
[4,2,1,1,1,1,1,1,1,1,1,1] => [6,4,3,3]
[4,1,1,1,1,1,1,1,1,1,1,1,1] => [5,4,3,3,1]
[3,3,3,3,3,1] => [8,7,1]
[3,3,3,3,2,2] => [3,3,3,3,3,1]
[3,3,3,3,2,1,1] => [8,6,2]
[3,3,3,3,1,1,1,1] => [7,4,2,1,1,1]
[3,3,3,2,2,2,1] => [4,3,3,3,2,1]
[3,3,3,2,2,1,1,1] => [7,6,2,1]
[3,3,3,2,1,1,1,1,1] => [7,5,2,2]
[3,3,3,1,1,1,1,1,1,1] => [6,5,4,1]
[3,3,2,2,2,2,2] => [7,2,2,2,2,1]
[3,3,2,2,2,2,1,1] => [7,3,3,3]
[3,3,2,2,2,1,1,1,1] => [4,4,4,2,2]
[3,3,2,2,1,1,1,1,1,1] => [5,5,2,2,2]
[3,3,2,1,1,1,1,1,1,1,1] => [5,5,4,1,1]
[3,3,1,1,1,1,1,1,1,1,1,1] => [5,5,4,2]
[3,2,2,2,2,2,2,1] => [6,3,2,2,2,1]
[3,2,2,2,2,2,1,1,1] => [6,5,2,1,1,1]
[3,2,2,2,2,1,1,1,1,1] => [6,3,3,2,2]
[3,2,2,2,1,1,1,1,1,1,1] => [6,4,2,2,2]
[3,2,2,1,1,1,1,1,1,1,1,1] => [6,4,4,1,1]
[3,2,1,1,1,1,1,1,1,1,1,1,1] => [6,4,4,2]
[3,1,1,1,1,1,1,1,1,1,1,1,1,1] => [5,4,4,2,1]
[2,2,2,2,2,2,2,2] => [7,5,1,1,1,1]
[2,2,2,2,2,2,2,1,1] => [7,5,4]
[2,2,2,2,2,2,1,1,1,1] => [4,4,4,3,1]
[2,2,2,2,2,1,1,1,1,1,1] => [6,3,3,3,1]
[2,2,2,2,1,1,1,1,1,1,1,1] => [6,5,2,2,1]
[2,2,2,1,1,1,1,1,1,1,1,1,1] => [6,5,3,1,1]
[2,2,1,1,1,1,1,1,1,1,1,1,1,1] => [6,5,3,2]
[2,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => [5,5,3,2,1]
[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => [6,4,3,2,1]
[17] => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
[16,1] => [2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
[15,2] => [2,2,1,1,1,1,1,1,1,1,1,1,1,1,1]
[15,1,1] => [3,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
[14,3] => [2,2,2,1,1,1,1,1,1,1,1,1,1,1]
[14,2,1] => [4,1,1,1,1,1,1,1,1,1,1,1,1,1]
[14,1,1,1] => [3,2,1,1,1,1,1,1,1,1,1,1,1,1]
[13,4] => [2,2,2,2,1,1,1,1,1,1,1,1,1]
[13,3,1] => [3,2,2,1,1,1,1,1,1,1,1,1,1]
[13,2,2] => [5,1,1,1,1,1,1,1,1,1,1,1,1]
[13,2,1,1] => [3,3,1,1,1,1,1,1,1,1,1,1,1]
[13,1,1,1,1] => [4,2,1,1,1,1,1,1,1,1,1,1,1]
[12,5] => [2,2,2,2,2,1,1,1,1,1,1,1]
[12,4,1] => [3,2,2,2,1,1,1,1,1,1,1,1]
[12,3,2] => [6,1,1,1,1,1,1,1,1,1,1,1]
[12,3,1,1] => [5,2,1,1,1,1,1,1,1,1,1,1]
[12,2,2,1] => [3,3,2,1,1,1,1,1,1,1,1,1]
[12,2,1,1,1] => [4,2,2,1,1,1,1,1,1,1,1,1]
[12,1,1,1,1,1] => [4,3,1,1,1,1,1,1,1,1,1,1]
[11,6] => [2,2,2,2,2,2,1,1,1,1,1]
[11,5,1] => [3,2,2,2,2,1,1,1,1,1,1]
[11,4,2] => [3,3,2,2,1,1,1,1,1,1,1]
[11,4,1,1] => [4,2,2,2,1,1,1,1,1,1,1]
[11,3,3] => [7,1,1,1,1,1,1,1,1,1,1]
[11,3,2,1] => [6,2,1,1,1,1,1,1,1,1,1]
[11,3,1,1,1] => [4,4,1,1,1,1,1,1,1,1,1]
[11,2,2,2] => [3,3,3,1,1,1,1,1,1,1,1]
[11,2,2,1,1] => [5,2,2,1,1,1,1,1,1,1,1]
[11,2,1,1,1,1] => [5,3,1,1,1,1,1,1,1,1,1]
[11,1,1,1,1,1,1] => [4,3,2,1,1,1,1,1,1,1,1]
[10,7] => [2,2,2,2,2,2,2,1,1,1]
[10,6,1] => [3,2,2,2,2,2,1,1,1,1]
[10,5,2] => [3,3,2,2,2,1,1,1,1,1]
[10,5,1,1] => [4,2,2,2,2,1,1,1,1,1]
[10,4,3] => [8,1,1,1,1,1,1,1,1,1]
[10,4,2,1] => [6,2,2,1,1,1,1,1,1,1]
[10,4,1,1,1] => [4,3,2,2,1,1,1,1,1,1]
[10,3,3,1] => [7,2,1,1,1,1,1,1,1,1]
[10,3,2,2] => [3,3,3,2,1,1,1,1,1,1]
[10,3,2,1,1] => [6,3,1,1,1,1,1,1,1,1]
[10,3,1,1,1,1] => [4,3,3,1,1,1,1,1,1,1]
[10,2,2,2,1] => [5,2,2,2,1,1,1,1,1,1]
[10,2,2,1,1,1] => [5,4,1,1,1,1,1,1,1,1]
[10,2,1,1,1,1,1] => [4,4,2,1,1,1,1,1,1,1]
[10,1,1,1,1,1,1,1] => [5,3,2,1,1,1,1,1,1,1]
[9,8] => [17]
[9,7,1] => [3,2,2,2,2,2,2,1,1]
[9,6,2] => [3,3,2,2,2,2,1,1,1]
[9,6,1,1] => [4,2,2,2,2,2,1,1,1]
[9,5,3] => [3,3,3,2,2,1,1,1,1]
[9,5,2,1] => [5,2,2,2,2,1,1,1,1]
[9,5,1,1,1] => [4,3,2,2,2,1,1,1,1]
[9,4,4] => [10,1,1,1,1,1,1,1]
[9,4,3,1] => [8,2,1,1,1,1,1,1,1]
[9,4,2,2] => [7,2,2,1,1,1,1,1,1]
[9,4,2,1,1] => [5,5,1,1,1,1,1,1,1]
[9,4,1,1,1,1] => [6,3,2,1,1,1,1,1,1]
[9,3,3,2] => [3,3,3,3,1,1,1,1,1]
[9,3,3,1,1] => [7,3,1,1,1,1,1,1,1]
[9,3,2,2,1] => [4,3,3,2,1,1,1,1,1]
[9,3,2,1,1,1] => [4,4,2,2,1,1,1,1,1]
[9,3,1,1,1,1,1] => [5,3,2,2,1,1,1,1,1]
[9,2,2,2,2] => [6,2,2,2,1,1,1,1,1]
[9,2,2,2,1,1] => [6,4,1,1,1,1,1,1,1]
[9,2,2,1,1,1,1] => [4,4,3,1,1,1,1,1,1]
[9,2,1,1,1,1,1,1] => [5,3,3,1,1,1,1,1,1]
[9,1,1,1,1,1,1,1,1] => [5,4,2,1,1,1,1,1,1]
[8,8,1] => [16,1]
[8,7,2] => [15,1,1]
[8,7,1,1] => [15,2]
[8,6,3] => [3,3,3,2,2,2,1,1]
[8,6,2,1] => [5,2,2,2,2,2,1,1]
[8,6,1,1,1] => [4,3,2,2,2,2,1,1]
[8,5,4] => [11,1,1,1,1,1,1]
[8,5,3,1] => [4,3,3,2,2,1,1,1]
[8,5,2,2] => [7,2,2,2,1,1,1,1]
[8,5,2,1,1] => [4,4,2,2,2,1,1,1]
[8,5,1,1,1,1] => [5,3,2,2,2,1,1,1]
[8,4,4,1] => [10,2,1,1,1,1,1]
[8,4,3,2] => [9,2,2,1,1,1,1]
[8,4,3,1,1] => [9,3,1,1,1,1,1]
[8,4,2,2,1] => [7,4,1,1,1,1,1,1]
[8,4,2,1,1,1] => [7,3,2,1,1,1,1,1]
[8,4,1,1,1,1,1] => [5,5,2,1,1,1,1,1]
[8,3,3,3] => [3,3,3,3,2,1,1,1]
[8,3,3,2,1] => [4,3,3,3,1,1,1,1]
[8,3,3,1,1,1] => [6,3,2,2,1,1,1,1]
[8,3,2,2,2] => [6,2,2,2,2,1,1,1]
[8,3,2,2,1,1] => [4,4,4,1,1,1,1,1]
[8,3,2,1,1,1,1] => [6,3,3,1,1,1,1,1]
[8,3,1,1,1,1,1,1] => [6,4,2,1,1,1,1,1]
[8,2,2,2,2,1] => [6,5,1,1,1,1,1,1]
[8,2,2,2,1,1,1] => [4,4,3,2,1,1,1,1]
[8,2,2,1,1,1,1,1] => [5,3,3,2,1,1,1,1]
[8,2,1,1,1,1,1,1,1] => [5,4,2,2,1,1,1,1]
[8,1,1,1,1,1,1,1,1,1] => [5,4,3,1,1,1,1,1]
[7,7,3] => [14,1,1,1]
[7,7,2,1] => [14,3]
[7,7,1,1,1] => [14,2,1]
[7,6,4] => [13,1,1,1,1]
[7,6,3,1] => [13,2,1,1]
[7,6,2,2] => [13,4]
[7,6,2,1,1] => [13,2,2]
[7,6,1,1,1,1] => [13,3,1]
[7,5,5] => [12,1,1,1,1,1]
[7,5,4,1] => [11,2,1,1,1,1]
[7,5,3,2] => [9,2,2,2,1,1]
[7,5,3,1,1] => [8,3,2,2,1,1]
[7,5,2,2,1] => [4,4,3,2,2,1,1]
[7,5,2,1,1,1] => [5,3,3,2,2,1,1]
[7,5,1,1,1,1,1] => [5,4,2,2,2,1,1]
[7,4,4,2] => [10,2,2,1,1,1]
[7,4,4,1,1] => [10,3,1,1,1,1]
[7,4,3,3] => [10,7]
[7,4,3,2,1] => [9,4,1,1,1,1]
[7,4,3,1,1,1] => [9,3,2,1,1,1]
[7,4,2,2,2] => [7,7,1,1,1]
[7,4,2,2,1,1] => [4,4,3,3,1,1,1]
[7,4,2,1,1,1,1] => [8,4,2,1,1,1]
[7,4,1,1,1,1,1,1] => [5,4,4,1,1,1,1]
[7,3,3,3,1] => [4,3,3,3,2,1,1]
[7,3,3,2,2] => [8,2,2,2,2,1]
[7,3,3,2,1,1] => [8,3,3,1,1,1]
[7,3,3,1,1,1,1] => [6,5,2,1,1,1,1]
[7,3,2,2,2,1] => [6,3,2,2,2,1,1]
[7,3,2,2,1,1,1] => [5,3,3,3,1,1,1]
[7,3,2,1,1,1,1,1] => [5,5,2,2,1,1,1]
[7,3,1,1,1,1,1,1,1] => [5,5,3,1,1,1,1]
[7,2,2,2,2,2] => [8,5,1,1,1,1]
[7,2,2,2,2,1,1] => [4,4,4,2,1,1,1]
[7,2,2,2,1,1,1,1] => [6,3,3,2,1,1,1]
[7,2,2,1,1,1,1,1,1] => [6,4,2,2,1,1,1]
[7,2,1,1,1,1,1,1,1,1] => [6,4,3,1,1,1,1]
[7,1,1,1,1,1,1,1,1,1,1] => [5,4,3,2,1,1,1]
[6,6,5] => [2,2,2,2,2,2,2,2,1]
[6,6,4,1] => [12,2,1,1,1]
[6,6,3,2] => [12,5]
[6,6,3,1,1] => [12,4,1]
[6,6,2,2,1] => [12,2,2,1]
[6,6,2,1,1,1] => [12,3,1,1]
[6,6,1,1,1,1,1] => [12,3,2]
[6,5,5,1] => [3,2,2,2,2,2,2,2]
[6,5,4,2] => [11,2,2,1,1]
[6,5,4,1,1] => [11,3,1,1,1]
[6,5,3,3] => [11,6]
[6,5,3,2,1] => [11,5,1]
[6,5,3,1,1,1] => [11,3,3]
[6,5,2,2,2] => [11,2,2,2]
[6,5,2,2,1,1] => [11,4,1,1]
[6,5,2,1,1,1,1] => [11,4,2]
[6,5,1,1,1,1,1,1] => [11,3,2,1]
[6,4,4,3] => [3,3,3,3,2,2,1]
[6,4,4,2,1] => [10,5,1,1]
[6,4,4,1,1,1] => [10,3,2,1,1]
[6,4,3,3,1] => [8,8,1]
[6,4,3,2,2] => [9,6,1,1]
[6,4,3,2,1,1] => [9,4,4]
[6,4,3,1,1,1,1] => [9,5,2,1]
[6,4,2,2,2,1] => [9,5,1,1,1]
[6,4,2,2,1,1,1] => [4,4,3,3,3]
[6,4,2,1,1,1,1,1] => [8,4,4,1]
[6,4,1,1,1,1,1,1,1] => [8,5,3,1]
[6,3,3,3,2] => [10,2,2,2,1]
[6,3,3,3,1,1] => [9,6,2]
[6,3,3,2,2,1] => [9,3,2,2,1]
[6,3,3,2,1,1,1] => [7,7,2,1]
[6,3,3,1,1,1,1,1] => [6,5,5,1]
[6,3,2,2,2,2] => [6,6,1,1,1,1,1]
[6,3,2,2,2,1,1] => [8,5,2,1,1]
[6,3,2,2,1,1,1,1] => [8,5,2,2]
[6,3,2,1,1,1,1,1,1] => [5,5,5,1,1]
[6,3,1,1,1,1,1,1,1,1] => [5,5,5,2]
[6,2,2,2,2,2,1] => [8,5,4]
[6,2,2,2,2,1,1,1] => [8,3,3,2,1]
[6,2,2,2,1,1,1,1,1] => [6,6,2,2,1]
[6,2,2,1,1,1,1,1,1,1] => [6,6,3,1,1]
[6,2,1,1,1,1,1,1,1,1,1] => [6,6,3,2]
[6,1,1,1,1,1,1,1,1,1,1,1] => [7,4,3,2,1]
[5,5,5,2] => [3,3,2,2,2,2,2,1]
[5,5,5,1,1] => [4,2,2,2,2,2,2,1]
[5,5,4,3] => [3,3,3,2,2,2,2]
[5,5,4,2,1] => [5,2,2,2,2,2,2]
[5,5,4,1,1,1] => [4,3,2,2,2,2,2]
[5,5,3,3,1] => [10,6,1]
[5,5,3,2,2] => [8,2,2,2,1,1,1]
[5,5,3,2,1,1] => [10,5,2]
[5,5,3,1,1,1,1] => [10,3,2,2]
[5,5,2,2,2,1] => [10,4,1,1,1]
[5,5,2,2,1,1,1] => [10,4,3]
[5,5,2,1,1,1,1,1] => [10,3,3,1]
[5,5,1,1,1,1,1,1,1] => [10,4,2,1]
[5,4,4,4] => [9,1,1,1,1,1,1,1,1]
[5,4,4,3,1] => [4,3,3,2,2,2,1]
[5,4,4,2,2] => [6,2,2,2,2,2,1]
[5,4,4,2,1,1] => [4,4,2,2,2,2,1]
[5,4,4,1,1,1,1] => [5,3,2,2,2,2,1]
[5,4,3,3,2] => [9,2,2,2,2]
[5,4,3,3,1,1] => [8,3,2,2,2]
[5,4,3,2,2,1] => [7,3,2,2,1,1,1]
[5,4,3,2,1,1,1] => [9,3,3,1,1]
[5,4,3,1,1,1,1,1] => [9,4,2,1,1]
[5,4,2,2,2,2] => [8,6,1,1,1]
[5,4,2,2,2,1,1] => [9,5,3]
[5,4,2,2,1,1,1,1] => [9,3,3,2]
[5,4,2,1,1,1,1,1,1] => [9,4,2,2]
[5,4,1,1,1,1,1,1,1,1] => [9,4,3,1]
[5,3,3,3,3] => [3,3,3,3,3,1,1]
[5,3,3,3,2,1] => [4,3,3,3,2,2]
[5,3,3,3,1,1,1] => [7,6,2,1,1]
[5,3,3,2,2,2] => [7,2,2,2,2,1,1]
[5,3,3,2,2,1,1] => [8,3,3,3]
[5,3,3,2,1,1,1,1] => [7,6,2,2]
[5,3,3,1,1,1,1,1,1] => [6,6,4,1]
[5,3,2,2,2,2,1] => [6,3,2,2,2,2]
[5,3,2,2,2,1,1,1] => [7,3,3,2,2]
[5,3,2,2,1,1,1,1,1] => [8,4,2,2,1]
[5,3,2,1,1,1,1,1,1,1] => [7,4,4,1,1]
[5,3,1,1,1,1,1,1,1,1,1] => [5,4,4,2,1,1]
[5,2,2,2,2,2,2] => [6,6,5]
[5,2,2,2,2,2,1,1] => [4,4,4,4,1]
[5,2,2,2,2,1,1,1,1] => [7,3,3,3,1]
[5,2,2,2,1,1,1,1,1,1] => [7,5,2,2,1]
[5,2,2,1,1,1,1,1,1,1,1] => [8,4,3,2]
[5,2,1,1,1,1,1,1,1,1,1,1] => [5,5,3,2,1,1]
[5,1,1,1,1,1,1,1,1,1,1,1,1] => [6,4,3,2,1,1]
[4,4,4,4,1] => [9,2,1,1,1,1,1,1]
[4,4,4,3,2] => [8,2,2,1,1,1,1,1]
[4,4,4,3,1,1] => [8,3,1,1,1,1,1,1]
[4,4,4,2,2,1] => [4,4,3,2,2,2]
[4,4,4,2,1,1,1] => [5,3,3,2,2,2]
[4,4,4,1,1,1,1,1] => [5,4,2,2,2,2]
[4,4,3,3,3] => [9,8]
[4,4,3,3,2,1] => [8,4,1,1,1,1,1]
[4,4,3,3,1,1,1] => [8,3,2,1,1,1,1]
[4,4,3,2,2,2] => [7,7,3]
[4,4,3,2,2,1,1] => [7,3,3,2,1,1]
[4,4,3,2,1,1,1,1] => [7,4,2,2,1,1]
[4,4,3,1,1,1,1,1,1] => [5,4,3,2,2,1]
[4,4,2,2,2,2,1] => [4,4,3,3,2,1]
[4,4,2,2,2,1,1,1] => [5,3,3,3,3]
[4,4,2,2,1,1,1,1,1] => [7,6,3,1]
[4,4,2,1,1,1,1,1,1,1] => [8,4,3,1,1]
[4,4,1,1,1,1,1,1,1,1,1] => [5,4,3,3,2]
[4,3,3,3,3,1] => [9,7,1]
[4,3,3,3,2,2] => [8,7,1,1]
[4,3,3,3,2,1,1] => [7,3,3,1,1,1,1]
[4,3,3,3,1,1,1,1] => [7,4,2,1,1,1,1]
[4,3,3,2,2,2,1] => [8,6,3]
[4,3,3,2,2,1,1,1] => [4,4,4,2,2,1]
[4,3,3,2,1,1,1,1,1] => [7,4,3,1,1,1]
[4,3,3,1,1,1,1,1,1,1] => [5,4,3,3,1,1]
[4,3,2,2,2,2,2] => [7,5,1,1,1,1,1]
[4,3,2,2,2,2,1,1] => [6,6,2,1,1,1]
[4,3,2,2,2,1,1,1,1] => [6,3,3,2,2,1]
[4,3,2,2,1,1,1,1,1,1] => [7,4,3,3]
[4,3,2,1,1,1,1,1,1,1,1] => [7,4,4,2]
[4,3,1,1,1,1,1,1,1,1,1,1] => [5,4,4,2,2]
[4,2,2,2,2,2,2,1] => [7,5,5]
[4,2,2,2,2,2,1,1,1] => [4,4,4,3,1,1]
[4,2,2,2,2,1,1,1,1,1] => [5,5,4,1,1,1]
[4,2,2,2,1,1,1,1,1,1,1] => [7,5,3,1,1]
[4,2,2,1,1,1,1,1,1,1,1,1] => [7,5,3,2]
[4,2,1,1,1,1,1,1,1,1,1,1,1] => [5,5,3,2,2]
[4,1,1,1,1,1,1,1,1,1,1,1,1,1] => [6,4,3,2,2]
[3,3,3,3,3,2] => [3,3,3,3,3,2]
[3,3,3,3,3,1,1] => [8,7,2]
[3,3,3,3,2,2,1] => [4,3,3,3,3,1]
[3,3,3,3,2,1,1,1] => [8,6,2,1]
[3,3,3,3,1,1,1,1,1] => [7,4,2,2,2]
[3,3,3,2,2,2,2] => [7,2,2,2,2,2]
[3,3,3,2,2,2,1,1] => [5,3,3,3,2,1]
[3,3,3,2,2,1,1,1,1] => [5,4,4,4]
[3,3,3,2,1,1,1,1,1,1] => [7,5,4,1]
[3,3,3,1,1,1,1,1,1,1,1] => [5,5,4,3]
[3,3,2,2,2,2,2,1] => [7,3,2,2,2,1]
[3,3,2,2,2,2,1,1,1] => [5,5,2,2,2,1]
[3,3,2,2,2,1,1,1,1,1] => [6,4,2,2,2,1]
[3,3,2,2,1,1,1,1,1,1,1] => [6,4,4,1,1,1]
[3,3,2,1,1,1,1,1,1,1,1,1] => [6,4,4,3]
[3,3,1,1,1,1,1,1,1,1,1,1,1] => [5,4,4,3,1]
[3,2,2,2,2,2,2,2] => [7,6,1,1,1,1]
[3,2,2,2,2,2,2,1,1] => [7,5,2,1,1,1]
[3,2,2,2,2,2,1,1,1,1] => [6,3,3,3,1,1]
[3,2,2,2,2,1,1,1,1,1,1] => [6,5,2,2,1,1]
[3,2,2,2,1,1,1,1,1,1,1,1] => [6,5,3,1,1,1]
[3,2,2,1,1,1,1,1,1,1,1,1,1] => [6,5,3,3]
-----------------------------------------------------------------------------
Created: Mar 19, 2024 at 14:36 by Martin Rubey
-----------------------------------------------------------------------------
Last Updated: Aug 05, 2024 at 22:55 by Martin Rubey