def cardinality(self):
"""
Returns the number of standard skew tableaux with shape of the skew
partition skp. This uses a formula due to Aitken
(see Cor. 7.16.3 of [Sta1999]_).
EXAMPLES::
sage: StandardSkewTableaux([[3, 2, 1], [1, 1]]).cardinality()
8
"""
outer, inner = self.skp
m = len(outer)
n = sum(outer) - sum(inner)
outer = list(outer)
inner = list(inner) + [0]*(m-len(inner))
a = zero_matrix(QQ, m)
for i in range(m):
for j in range(m):
v = outer[i] - inner[j] - i + j
if v < 0:
a[i,j] = 0
else:
a[i,j] = 1/factorial(v)
return ZZ(factorial(n) * a.det())
def cardinality(self):
"""
Returns the number of standard skew tableaux with shape of the skew
partition skp. This uses a formula due to Aitken
(see Cor. 7.16.3 of [Sta1999]_).
EXAMPLES::
sage: StandardSkewTableaux([[3, 2, 1], [1, 1]]).cardinality()
8
"""
outer, inner = self.skp
m = len(outer)
n = sum(outer) - sum(inner)
outer = list(outer)
inner = list(inner) + [0] * (m - len(inner))
a = zero_matrix(QQ, m)
for i in range(m):
for j in range(m):
v = outer[i] - inner[j] - i + j
if v < 0:
a[i, j] = 0
else:
a[i, j] = 1 / factorial(v)
return ZZ(factorial(n) * a.det())
def number_of_classes(self, invertible=False, q=None):
"""
Return the number of similarity classes of matrices of type ``self``.
IMPUT:
- ``invertible`` -- Boolean; return number of invertible classes if set
to ``True``
- ``q`` -- An integer or an indeterminate
EXAMPLES::
sage: tau = SimilarityClassType([[1, [1]], [1, [1]]])
sage: tau.number_of_classes()
1/2*q^2 - 1/2*q
"""
if q is None:
q = ZZ["q"].gen()
if self.size() == 0:
return q.parent().one()
list_of_degrees = [PT.degree() for PT in self]
maximum_degree = max(list_of_degrees)
numerator = prod(
[
prod([primitives(d + 1, invertible=invertible, q=q) - i for i in range(list_of_degrees.count(d + 1))])
for d in range(maximum_degree)
]
)
tau_list = list(self)
D = dict((i, tau_list.count(i)) for i in tau_list)
denominator = reduce(mul, [factorial(D[primary_type]) for primary_type in D.keys()])
return numerator / denominator
def number_of_classes(self, invertible=False, q=None):
"""
Return the number of similarity classes of matrices of type ``self``.
IMPUT:
- ``invertible`` -- Boolean; return number of invertible classes if set
to ``True``
- ``q`` -- An integer or an indeterminate
EXAMPLES::
sage: tau = SimilarityClassType([[1, [1]], [1, [1]]])
sage: tau.number_of_classes()
1/2*q^2 - 1/2*q
"""
if q is None:
q = ZZ['q'].gen()
if self.size() == 0:
return q.parent().one()
list_of_degrees = [PT.degree() for PT in self]
maximum_degree = max(list_of_degrees)
numerator = prod([
prod([
primitives(d + 1, invertible=invertible, q=q) - i
for i in range(list_of_degrees.count(d + 1))
]) for d in range(maximum_degree)
])
tau_list = list(self)
D = dict((i, tau_list.count(i)) for i in tau_list)
denominator = reduce(
mul, [factorial(D[primary_type]) for primary_type in D.keys()])
return numerator / denominator