def StandardSkewTableaux(skp=None):
"""
Returns the combinatorial class of standard skew tableaux of shape
skp (where skp is a skew partition).
EXAMPLES::
sage: StandardSkewTableaux([[3, 2, 1], [1, 1]]).list()
[[[None, 1, 2], [None, 3], [4]],
[[None, 1, 2], [None, 4], [3]],
[[None, 1, 3], [None, 2], [4]],
[[None, 1, 4], [None, 2], [3]],
[[None, 1, 3], [None, 4], [2]],
[[None, 1, 4], [None, 3], [2]],
[[None, 2, 3], [None, 4], [1]],
[[None, 2, 4], [None, 3], [1]]]
"""
if skp is None:
return StandardSkewTableaux_all()
elif isinstance(skp, (int, Integer)):
return StandardSkewTableaux_size(skp)
elif skp in skew_partition.SkewPartitions():
return StandardSkewTableaux_skewpartition(
skew_partition.SkewPartition(skp))
else:
raise TypeError
def spin_polynomial_square(part, weight, length):
"""
Returns the spin polynomial associated with part, weight, and
length, with the substitution t -> t^2 made.
EXAMPLES::
sage: from sage.combinat.ribbon_tableau import spin_polynomial_square
sage: spin_polynomial_square([6,6,6],[4,2],3)
t^12 + t^10 + 2*t^8 + t^6 + t^4
sage: spin_polynomial_square([6,6,6],[4,1,1],3)
t^12 + 2*t^10 + 3*t^8 + 2*t^6 + t^4
sage: spin_polynomial_square([3,3,3,2,1], [2,2], 3)
t^7 + t^5
sage: spin_polynomial_square([3,3,3,2,1], [2,1,1], 3)
2*t^7 + 2*t^5 + t^3
sage: spin_polynomial_square([3,3,3,2,1], [1,1,1,1], 3)
3*t^7 + 5*t^5 + 3*t^3 + t
sage: spin_polynomial_square([5,4,3,2,1,1,1], [2,2,1], 3)
2*t^9 + 6*t^7 + 2*t^5
sage: spin_polynomial_square([[6]*6, [3,3]], [4,4,2], 3)
3*t^18 + 5*t^16 + 9*t^14 + 6*t^12 + 3*t^10
"""
R = ZZ['t']
t = R.gen()
if part in partition.Partitions():
part = skew_partition.SkewPartition([part,partition.Partition_class([])])
elif part in skew_partition.SkewPartitions():
part = skew_partition.SkewPartition(part)
if part == [[],[]] and weight == []:
return t.parent()(1)
return R(graph_implementation_rec(part, weight, length, functools.partial(spin_rec,t))[0])
def __iter__(self):
"""
EXAMPLES::
sage: SemistandardSkewTableaux(2,[1,1]).list() # indirect doctest
[[[1, 2]], [[1], [2]], [[None, 2], [1]], [[None, 1], [2]]]
"""
for p in skew_partition.SkewPartitions(self.n):
for ssst in SemistandardSkewTableaux_shape_weight(p, self.mu):
yield ssst
def cardinality(self):
"""
EXAMPLES::
sage: SemistandardSkewTableaux(2).cardinality()
8
"""
count = 0
for p in skew_partition.SkewPartitions(self.n):
count += SemistandardSkewTableaux_shape(p).cardinality()
return count
def cardinality(self):
"""
EXAMPLES::
sage: SemistandardSkewTableaux(2,[1,1]).cardinality()
4
"""
count = 0
for p in skew_partition.SkewPartitions(self.n):
count += SemistandardSkewTableaux_pmu(p, self.mu).cardinality()
return count
def SemistandardSkewTableaux(p=None, mu=None):
"""
Returns a combinatorial class of semistandard skew tableaux.
EXAMPLES::
sage: SemistandardSkewTableaux()
Semistandard skew tableaux
::
sage: SemistandardSkewTableaux(3)
Semistandard skew tableaux of size 3
::
sage: SemistandardSkewTableaux([[2,1],[]])
Semistandard skew tableaux of shape [[2, 1], []]
::
sage: SemistandardSkewTableaux([[2,1],[]],[2,1])
Semistandard skew tableaux of shape [[2, 1], []] and weight [2, 1]
::
sage: SemistandardSkewTableaux(3, [2,1])
Semistandard skew tableaux of size 3 and weight [2, 1]
"""
if p is None and mu is None:
return SemistandardSkewTableaux_all()
if p is None:
raise ValueError, "you must specify either a size or shape"
if isinstance(p, (int, Integer)):
if mu is None:
return SemistandardSkewTableaux_size(p)
else:
return SemistandardSkewTableaux_size_weight(p, mu)
if p in skew_partition.SkewPartitions():
p = skew_partition.SkewPartition(p)
if mu is None:
return SemistandardSkewTableaux_shape(p)
else:
return SemistandardSkewTableaux_shape_weight(p, mu)
def cardinality(self):
"""
EXAMPLES::
sage: StandardSkewTableaux(1).cardinality()
1
sage: StandardSkewTableaux(2).cardinality()
4
sage: StandardSkewTableaux(3).cardinality()
24
sage: StandardSkewTableaux(4).cardinality()
194
"""
count = 0
for skp in skew_partition.SkewPartitions(self.n):
count += StandardSkewTableaux_skewpartition(skp).cardinality()
return count
def __iter__(self):
"""
EXAMPLES::
sage: StandardSkewTableaux(2).list() #indirect doctest
[[[1, 2]], [[1], [2]], [[None, 1], [2]], [[None, 2], [1]]]
sage: StandardSkewTableaux(3).list() #indirect doctest
[[[1, 2, 3]],
[[1, 2], [3]], [[1, 3], [2]],
[[None, 1, 2], [3]], [[None, 1, 3], [2]],
[[None, 2, 3], [1]],
[[None, 1], [2, 3]], [[None, 2], [1, 3]],
[[None, None, 1], [2, 3]], [[None, None, 2], [1, 3]], [[None, None, 3], [1, 2]],
[[1], [2], [3]],
[[None, 1], [None, 2], [3]], [[None, 1], [None, 3], [2]], [[None, 2], [None, 3], [1]],
[[None, 1], [2], [3]], [[None, 2], [1], [3]], [[None, 3], [1], [2]],
[[None, None, 1], [None, 2], [3]], [[None, None, 1], [None, 3], [2]],
[[None, None, 2], [None, 1], [3]], [[None, None, 3], [None, 1], [2]],
[[None, None, 2], [None, 3], [1]], [[None, None, 3], [None, 2], [1]]]
"""
for skp in skew_partition.SkewPartitions(self.n):
for sst in StandardSkewTableaux_skewpartition(skp):
yield sst
