def Ish(self, n, K=QQ, names=None):
r"""
Return the Ish arrangement.
INPUT:
- ``n`` -- integer
- ``K`` -- field (default:``QQ``)
- ``names`` -- tuple of strings or ``None`` (default); the
variable names for the ambient space
OUTPUT:
The Ish arrangement, which is the set of `n(n-1)` hyperplanes.
.. MATH::
\{ x_i - x_j = 0 : 1 \leq i \leq j \leq n \}
\cup
\{ x_1 - x_j = i : 1 \leq i \leq j \leq n \}.
EXAMPLES::
sage: a = hyperplane_arrangements.Ish(3); a
Arrangement of 6 hyperplanes of dimension 3 and rank 2
sage: a.characteristic_polynomial()
x^3 - 6*x^2 + 9*x
sage: b = hyperplane_arrangements.Shi(3)
sage: b.characteristic_polynomial()
x^3 - 6*x^2 + 9*x
TESTS::
sage: a.characteristic_polynomial.clear_cache() # long time
sage: a.characteristic_polynomial() # long time
x^3 - 6*x^2 + 9*x
REFERENCES:
.. [AR] D. Armstrong, B. Rhoades
"The Shi arrangement and the Ish arrangement"
:arxiv:`1009.1655`
"""
H = make_parent(K, n, names)
x = H.gens()
hyperplanes = []
for i in range(n):
for j in range(i+1, n):
hyperplanes.append(x[i] - x[j])
hyperplanes.append(x[0] - x[j] - (i+1))
A = H(*hyperplanes)
x = polygen(QQ, 'x')
charpoly = x * sum([(-1)**k * stirling_number2(n, n-k) *
prod([(x - 1 - j) for j in range(k, n-1)]) for k in range(0, n)])
A.characteristic_polynomial.set_cache(charpoly)
return A
def cardinality(self):
"""
EXAMPLES::
sage: OrderedSetPartitions(4,2).cardinality()
14
sage: OrderedSetPartitions(4,1).cardinality()
1
"""
set = self.s
n = self.n
return factorial(n)*stirling_number2(len(set),n)
def cardinality(self):
"""
The Stirling number of the second kind is the number of partitions
of a set of size `n` into `k` blocks.
EXAMPLES::
sage: SetPartitions(5, 3).cardinality()
25
sage: stirling_number2(5,3)
25
"""
return stirling_number2(len(self._set), self.n)
def cardinality(self):
"""
Return the cardinality of ``self``.
The number of ordered partitions of a set of size `n` into `k`
parts is equal to `k! S(n,k)` where `S(n,k)` denotes the Stirling
number of the second kind.
EXAMPLES::
sage: OrderedSetPartitions(4,2).cardinality()
14
sage: OrderedSetPartitions(4,1).cardinality()
1
"""
return factorial(self.n)*stirling_number2(len(self._set), self.n)
def semiorder(self, n, K=QQ, names=None):
r"""
Return the semiorder arrangement.
INPUT:
- ``n`` -- integer
- ``K`` -- field (default: `\QQ`)
- ``names`` -- tuple of strings or ``None`` (default); the
variable names for the ambient space
OUTPUT:
The semiorder arrangement, which is the set of `n(n-1)`
hyperplanes `\{ x_i - x_j = -1,1 : 1 \leq i \leq j \leq n\}`.
EXAMPLES::
sage: hyperplane_arrangements.semiorder(4)
Arrangement of 12 hyperplanes of dimension 4 and rank 3
TESTS::
sage: h = hyperplane_arrangements.semiorder(5)
sage: h.characteristic_polynomial()
x^5 - 20*x^4 + 180*x^3 - 790*x^2 + 1380*x
sage: h.characteristic_polynomial.clear_cache() # long time
sage: h.characteristic_polynomial() # long time
x^5 - 20*x^4 + 180*x^3 - 790*x^2 + 1380*x
"""
H = make_parent(K, n, names)
x = H.gens()
hyperplanes = []
for i in range(n):
for j in range(i + 1, n):
for k in [-1, 1]:
hyperplanes.append(x[i] - x[j] - k)
A = H(*hyperplanes)
x = polygen(QQ, 'x')
charpoly = x * sum([
stirling_number2(n, k) * prod([x - k - i for i in range(1, k)])
for k in range(1, n + 1)
])
A.characteristic_polynomial.set_cache(charpoly)
return A
def Shi(self, n, K=QQ, names=None):
r"""
Return the Shi arrangement.
INPUT:
- ``n`` -- integer
- ``K`` -- field (default:``QQ``)
- ``names`` -- tuple of strings or ``None`` (default); the
variable names for the ambient space
OUTPUT:
The Shi arrangement is the set of `n(n-1)` hyperplanes: `\{ x_i - x_j
= 0,1 : 1 \leq i \leq j \leq n \}`.
EXAMPLES::
sage: hyperplane_arrangements.Shi(4)
Arrangement of 12 hyperplanes of dimension 4 and rank 3
TESTS::
sage: h = hyperplane_arrangements.Shi(4)
sage: h.characteristic_polynomial()
x^4 - 12*x^3 + 48*x^2 - 64*x
sage: h.characteristic_polynomial.clear_cache() # long time
sage: h.characteristic_polynomial() # long time
x^4 - 12*x^3 + 48*x^2 - 64*x
"""
H = make_parent(K, n, names)
x = H.gens()
hyperplanes = []
for i in range(n):
for j in range(i + 1, n):
for const in [0, 1]:
hyperplanes.append(x[i] - x[j] - const)
A = H(*hyperplanes)
x = polygen(QQ, 'x')
charpoly = x * sum([(-1)**k * stirling_number2(n, n - k) *
prod([(x - 1 - j) for j in range(k, n - 1)])
for k in range(0, n)])
A.characteristic_polynomial.set_cache(charpoly)
return A
def Shi(self, n, K=QQ, names=None):
r"""
Return the Shi arrangement.
INPUT:
- ``n`` -- integer
- ``K`` -- field (default:``QQ``)
- ``names`` -- tuple of strings or ``None`` (default); the
variable names for the ambient space
OUTPUT:
The Shi arrangement is the set of `n(n-1)` hyperplanes: `\{ x_i - x_j
= 0,1 : 1 \leq i \leq j \leq n \}`.
EXAMPLES::
sage: hyperplane_arrangements.Shi(4)
Arrangement of 12 hyperplanes of dimension 4 and rank 3
TESTS::
sage: h = hyperplane_arrangements.Shi(4)
sage: h.characteristic_polynomial()
x^4 - 12*x^3 + 48*x^2 - 64*x
sage: h.characteristic_polynomial.clear_cache() # long time
sage: h.characteristic_polynomial() # long time
x^4 - 12*x^3 + 48*x^2 - 64*x
"""
H = make_parent(K, n, names)
x = H.gens()
hyperplanes = []
for i in range(n):
for j in range(i+1, n):
for const in [0, 1]:
hyperplanes.append(x[i] - x[j] - const)
A = H(*hyperplanes)
x = polygen(QQ, 'x')
charpoly = x * sum([(-1)**k * stirling_number2(n, n-k) *
prod([(x - 1 - j) for j in range(k, n-1)]) for k in range(0, n)])
A.characteristic_polynomial.set_cache(charpoly)
return A
def semiorder(self, n, K=QQ, names=None):
r"""
Return the semiorder arrangement.
INPUT:
- ``n`` -- integer
- ``K`` -- field (default: `\QQ`)
- ``names`` -- tuple of strings or ``None`` (default); the
variable names for the ambient space
OUTPUT:
The semiorder arrangement, which is the set of `n(n-1)`
hyperplanes `\{ x_i - x_j = -1,1 : 1 \leq i \leq j \leq n\}`.
EXAMPLES::
sage: hyperplane_arrangements.semiorder(4)
Arrangement of 12 hyperplanes of dimension 4 and rank 3
TESTS::
sage: h = hyperplane_arrangements.semiorder(5)
sage: h.characteristic_polynomial()
x^5 - 20*x^4 + 180*x^3 - 790*x^2 + 1380*x
sage: h.characteristic_polynomial.clear_cache() # long time
sage: h.characteristic_polynomial() # long time
x^5 - 20*x^4 + 180*x^3 - 790*x^2 + 1380*x
"""
H = make_parent(K, n, names)
x = H.gens()
hyperplanes = []
for i in range(n):
for j in range(i+1, n):
for k in [-1, 1]:
hyperplanes.append(x[i] - x[j] - k)
A = H(*hyperplanes)
x = polygen(QQ, 'x')
charpoly = x * sum([stirling_number2(n, k) * prod([x - k - i for i in range(1, k)])
for k in range(1, n+1)])
A.characteristic_polynomial.set_cache(charpoly)
return A
def cardinality(self):
"""
EXAMPLES::
sage: OrderedSetPartitions(0).cardinality()
1
sage: OrderedSetPartitions(1).cardinality()
1
sage: OrderedSetPartitions(2).cardinality()
3
sage: OrderedSetPartitions(3).cardinality()
13
sage: OrderedSetPartitions([1,2,3]).cardinality()
13
sage: OrderedSetPartitions(4).cardinality()
75
sage: OrderedSetPartitions(5).cardinality()
541
"""
return sum([factorial(k)*stirling_number2(len(self._set),k) for k in range(len(self._set)+1)])
def cardinality(self):
"""
EXAMPLES::
sage: OrderedSetPartitions(0).cardinality()
1
sage: OrderedSetPartitions(1).cardinality()
1
sage: OrderedSetPartitions(2).cardinality()
3
sage: OrderedSetPartitions(3).cardinality()
13
sage: OrderedSetPartitions([1,2,3]).cardinality()
13
sage: OrderedSetPartitions(4).cardinality()
75
sage: OrderedSetPartitions(5).cardinality()
541
"""
return sum([factorial(k)*stirling_number2(len(self._set),k) for k in range(len(self._set)+1)])
