def __iter__(self):
"""
TESTS::
sage: [ p for p in OrderedSetPartitions([1,2,3,4], [2,1,1]) ]
[[{1, 2}, {3}, {4}],
[{1, 2}, {4}, {3}],
[{1, 3}, {2}, {4}],
[{1, 4}, {2}, {3}],
[{1, 3}, {4}, {2}],
[{1, 4}, {3}, {2}],
[{2, 3}, {1}, {4}],
[{2, 4}, {1}, {3}],
[{3, 4}, {1}, {2}],
[{2, 3}, {4}, {1}],
[{2, 4}, {3}, {1}],
[{3, 4}, {2}, {1}]]
sage: len(OrderedSetPartitions([1,2,3,4], [1,1,1,1]))
24
sage: [ x for x in OrderedSetPartitions([1,4,7], [3]) ]
[[{1, 4, 7}]]
sage: [ x for x in OrderedSetPartitions([1,4,7], [1,2]) ]
[[{1}, {4, 7}], [{4}, {1, 7}], [{7}, {1, 4}]]
sage: [ p for p in OrderedSetPartitions([], []) ]
[[]]
sage: [ p for p in OrderedSetPartitions([1], [1]) ]
[[{1}]]
Let us check that it works for large size (:trac:`16646`)::
sage: OrderedSetPartitions(42).first()
[{1}, {2}, {3}, {4}, {5}, {6}, {7}, {8}, {9}, {10}, {11}, {12},
{13}, {14}, {15}, {16}, {17}, {18}, {19}, {20}, {21}, {22}, {23},
{24}, {25}, {26}, {27}, {28}, {29}, {30}, {31}, {32}, {33}, {34},
{35}, {36}, {37}, {38}, {39}, {40}, {41}, {42}]
"""
comp = self.c
lset = [x for x in self._set]
l = len(self.c)
dcomp = [-1] + comp.descents(final_descent=True)
p = []
for j in range(l):
p += [j + 1] * comp[j]
from sage.combinat.permutation import Permutations_mset
for x in Permutations_mset(p):
res = permutation.to_standard(x).inverse()
res = [lset[x - 1] for x in res]
yield self.element_class(
self,
[Set(res[dcomp[i] + 1:dcomp[i + 1] + 1]) for i in range(l)])
def __iter__(self):
"""
TESTS::
sage: [ p for p in OrderedSetPartitions([1,2,3,4], [2,1,1]) ]
[[{1, 2}, {3}, {4}],
[{1, 2}, {4}, {3}],
[{1, 3}, {2}, {4}],
[{1, 4}, {2}, {3}],
[{1, 3}, {4}, {2}],
[{1, 4}, {3}, {2}],
[{2, 3}, {1}, {4}],
[{2, 4}, {1}, {3}],
[{3, 4}, {1}, {2}],
[{2, 3}, {4}, {1}],
[{2, 4}, {3}, {1}],
[{3, 4}, {2}, {1}]]
sage: len(OrderedSetPartitions([1,2,3,4], [1,1,1,1]))
24
sage: [ x for x in OrderedSetPartitions([1,4,7], [3]) ]
[[{1, 4, 7}]]
sage: [ x for x in OrderedSetPartitions([1,4,7], [1,2]) ]
[[{1}, {4, 7}], [{4}, {1, 7}], [{7}, {1, 4}]]
sage: [ p for p in OrderedSetPartitions([], []) ]
[[]]
sage: [ p for p in OrderedSetPartitions([1], [1]) ]
[[{1}]]
Let us check that it works for large size (:trac:`16646`)::
sage: OrderedSetPartitions(42).first()
[{1}, {2}, {3}, {4}, {5}, {6}, {7}, {8}, {9}, {10}, {11}, {12},
{13}, {14}, {15}, {16}, {17}, {18}, {19}, {20}, {21}, {22}, {23},
{24}, {25}, {26}, {27}, {28}, {29}, {30}, {31}, {32}, {33}, {34},
{35}, {36}, {37}, {38}, {39}, {40}, {41}, {42}]
"""
comp = self.c
lset = [x for x in self._set]
l = len(self.c)
dcomp = [-1] + comp.descents(final_descent=True)
p = []
for j in range(l):
p += [j + 1] * comp[j]
from sage.combinat.permutation import Permutations_mset
for x in Permutations_mset(p):
res = permutation.to_standard(x).inverse()
res = [lset[x - 1] for x in res]
yield self.element_class(self, [Set(res[dcomp[i]+1:dcomp[i+1]+1])
for i in range(l)])
def spin_rec(t, nexts, current, part, weight, length):
"""
Routine used for constructing the spin polynomial.
INPUT:
- ``weight`` -- list of non-negative integers
- ``length`` -- the length of the ribbons we're tiling with
- ``t`` -- the variable
EXAMPLES::
sage: from sage.combinat.ribbon_tableau import spin_rec
sage: sp = SkewPartition
sage: t = ZZ['t'].gen()
sage: spin_rec(t, [], [[[], [3, 3]]], sp([[2, 2, 2], []]), [2], 3)
[t^4]
sage: spin_rec(t, [[0], [t^4]], [[[2, 1, 1, 1, 1], [0, 3]], [[2, 2, 2], [3, 0]]], sp([[2, 2, 2, 2, 1], []]), [2, 1], 3)
[t^5]
sage: spin_rec(t, [], [[[], [3, 3, 0]]], sp([[3, 3], []]), [2], 3)
[t^2]
sage: spin_rec(t, [[t^4], [t^3], [t^2]], [[[2, 2, 2], [0, 0, 3]], [[3, 2, 1], [0, 3, 0]], [[3, 3], [3, 0, 0]]], sp([[3, 3, 3], []]), [2, 1], 3)
[t^6 + t^4 + t^2]
sage: spin_rec(t, [[t^5], [t^4], [t^6 + t^4 + t^2]], [[[2, 2, 2, 2, 1], [0, 0, 3]], [[3, 3, 1, 1, 1], [0, 3, 0]], [[3, 3, 3], [3, 0, 0]]], sp([[3, 3, 3, 2, 1], []]), [2, 1, 1], 3)
[2*t^7 + 2*t^5 + t^3]
"""
if not current:
return [parent(t).zero()]
tmp = []
partp = part[0].conjugate()
ell = len(partp)
#compute the contribution of the ribbons added at
#the current node
for val in current:
perms = val[1]
perm = [
partp[i] + ell - (i + 1) - perms[i] for i in reversed(range(ell))
]
perm = to_standard(perm)
tmp.append(weight[-1] * (length - 1) - perm.number_of_inversions())
if nexts:
return [
sum(
sum(t**tval * nval for nval in nexts[i])
for i, tval in enumerate(tmp))
]
else:
return [sum(t**val for val in tmp)]
def spin_rec(t, nexts, current, part, weight, length):
"""
Routine used for constructing the spin polynomial.
INPUT:
- ``weight`` -- list of non-negative integers
- ``length`` -- the length of the ribbons we're tiling with
- ``t`` -- the variable
EXAMPLES::
sage: from sage.combinat.ribbon_tableau import spin_rec
sage: sp = SkewPartition
sage: t = ZZ['t'].gen()
sage: spin_rec(t, [], [[[], [3, 3]]], sp([[2, 2, 2], []]), [2], 3)
[t^4]
sage: spin_rec(t, [[0], [t^4]], [[[2, 1, 1, 1, 1], [0, 3]], [[2, 2, 2], [3, 0]]], sp([[2, 2, 2, 2, 1], []]), [2, 1], 3)
[t^5]
sage: spin_rec(t, [], [[[], [3, 3, 0]]], sp([[3, 3], []]), [2], 3)
[t^2]
sage: spin_rec(t, [[t^4], [t^3], [t^2]], [[[2, 2, 2], [0, 0, 3]], [[3, 2, 1], [0, 3, 0]], [[3, 3], [3, 0, 0]]], sp([[3, 3, 3], []]), [2, 1], 3)
[t^6 + t^4 + t^2]
sage: spin_rec(t, [[t^5], [t^4], [t^6 + t^4 + t^2]], [[[2, 2, 2, 2, 1], [0, 0, 3]], [[3, 3, 1, 1, 1], [0, 3, 0]], [[3, 3, 3], [3, 0, 0]]], sp([[3, 3, 3, 2, 1], []]), [2, 1, 1], 3)
[2*t^7 + 2*t^5 + t^3]
"""
if not current:
return [parent(t).zero()]
tmp = []
partp = part[0].conjugate()
ell = len(partp)
#compute the contribution of the ribbons added at
#the current node
for val in current:
perms = val[1]
perm = [partp[i] + ell - (i + 1) - perms[i] for i in reversed(range(ell))]
perm = to_standard(perm)
tmp.append( weight[-1]*(length-1) - perm.number_of_inversions() )
if nexts:
return [ sum(sum(t**tval * nval for nval in nexts[i])
for i, tval in enumerate(tmp)) ]
else:
return [ sum(t**val for val in tmp) ]
