class AlternatingSignMatrices(Parent, UniqueRepresentation):
r"""
Class of all `n \times n` alternating sign matrices.
An alternating sign matrix of size `n` is an `n \times n` matrix of `0`s,
`1`s and `-1`s such that the sum of each row and column is `1` and the
non-zero entries in each row and column alternate in sign.
Alternating sign matrices of size `n` are in bijection with
:class:`monotone triangles <MonotoneTriangles>` with `n` rows.
INPUT:
- `n` -- an integer, the size of the matrices.
- ``use_monotone_triangle`` -- (Default: ``True``) If ``True``, the
generation of the matrices uses monotone triangles, else it will use the
earlier and now obsolete contre-tableaux implementation;
must be ``True`` to generate a lattice (with the ``lattice`` method)
EXAMPLES:
This will create an instance to manipulate the alternating sign
matrices of size 3::
sage: A = AlternatingSignMatrices(3)
sage: A
Alternating sign matrices of size 3
sage: A.cardinality()
7
Notably, this implementation allows to make a lattice of it::
sage: L = A.lattice()
sage: L
Finite lattice containing 7 elements
sage: L.category()
Category of facade finite lattice posets
"""
def __init__(self, n, use_monotone_triangles=True):
r"""
Initialize ``self``.
TESTS::
sage: A = AlternatingSignMatrices(4)
sage: TestSuite(A).run()
sage: A == AlternatingSignMatrices(4, use_monotone_triangles=False)
False
"""
self._n = n
self._matrix_space = MatrixSpace(ZZ, n)
self._umt = use_monotone_triangles
Parent.__init__(self, category=FiniteEnumeratedSets())
def _repr_(self):
r"""
Return a string representation of ``self``.
TESTS::
sage: A = AlternatingSignMatrices(4); A
Alternating sign matrices of size 4
"""
return "Alternating sign matrices of size %s" % self._n
def _repr_option(self, key):
"""
Metadata about the :meth:`_repr_` output.
See :meth:`sage.structure.parent._repr_option` for details.
EXAMPLES::
sage: A = AlternatingSignMatrices(3)
sage: A._repr_option('element_ascii_art')
True
"""
return self._matrix_space._repr_option(key)
def __contains__(self, asm):
"""
Check if ``asm`` is in ``self``.
TESTS::
sage: A = AlternatingSignMatrices(3)
sage: [[0,1,0],[1,0,0],[0,0,1]] in A
True
sage: [[0,1,0],[1,-1,1],[0,1,0]] in A
True
sage: [[0, 1],[1,0]] in A
False
sage: [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]] in A
False
sage: [[-1, 1, 1],[1,-1,1],[1,1,-1]] in A
False
"""
if isinstance(asm, AlternatingSignMatrix):
return asm._matrix.nrows() == self._n
try:
asm = self._matrix_space(asm)
except (TypeError, ValueError):
return False
for row in asm:
pos = False
for val in row:
if val > 0:
if pos:
return False
else:
pos = True
elif val < 0:
if pos:
pos = False
else:
return False
if not pos:
return False
if any(sum(row) != 1 for row in asm.columns()):
return False
return True
def _element_constructor_(self, asm):
"""
Construct an element of ``self``.
EXAMPLES::
sage: A = AlternatingSignMatrices(3)
sage: elt = A([[1, 0, 0],[0, 1, 0],[0, 0, 1]]); elt
[1 0 0]
[0 1 0]
[0 0 1]
sage: elt.parent() is A
True
sage: A([[3, 2, 1], [2, 1], [1]])
[1 0 0]
[0 1 0]
[0 0 1]
"""
if isinstance(asm, AlternatingSignMatrix):
if asm.parent() is self:
return asm
raise ValueError("Cannot convert between alternating sign matrices of different sizes")
if asm in MonotoneTriangles(self._n):
return self.from_monotone_triangle(asm)
return self.element_class(self, self._matrix_space(asm))
Element = AlternatingSignMatrix
def _an_element_(self):
"""
Return an element of ``self``.
"""
return self.element_class(self, self._matrix_space.identity_matrix())
def from_monotone_triangle(self, triangle):
r"""
Return an alternating sign matrix from a monotone triangle.
EXAMPLES::
sage: A = AlternatingSignMatrices(3)
sage: A.from_monotone_triangle([[3, 2, 1], [2, 1], [1]])
[1 0 0]
[0 1 0]
[0 0 1]
sage: A.from_monotone_triangle([[3, 2, 1], [3, 2], [3]])
[0 0 1]
[0 1 0]
[1 0 0]
"""
n = len(triangle)
if n != self._n:
raise ValueError("Incorrect size")
asm = []
prev = [0]*n
for line in reversed(triangle):
v = [1 if j+1 in reversed(line) else 0 for j in range(n)]
row = [a-b for (a, b) in zip(v, prev)]
asm.append(row)
prev = v
return self.element_class(self, self._matrix_space(asm))
def size(self):
r"""
Return the size of the matrices in ``self``.
TESTS::
sage: A = AlternatingSignMatrices(4)
sage: A.size()
4
"""
return self._n
def cardinality(self):
r"""
Return the cardinality of ``self``.
The number of `n \times n` alternating sign matrices is equal to
.. MATH::
\prod_{k=0}^{n-1} \frac{(3k+1)!}{(n+k)!} = \frac{1! 4! 7! 10!
\cdots (3n-2)!}{n! (n+1)! (n+2)! (n+3)! \cdots (2n-1)!}
EXAMPLES::
sage: [AlternatingSignMatrices(n).cardinality() for n in range(0, 11)]
[1, 1, 2, 7, 42, 429, 7436, 218348, 10850216, 911835460, 129534272700]
"""
return Integer(prod( [ factorial(3*k+1)/factorial(self._n+k)
for k in range(self._n)] ))
def matrix_space(self):
"""
Return the underlying matrix space.
EXAMPLES::
sage: A = AlternatingSignMatrices(3)
sage: A.matrix_space()
Full MatrixSpace of 3 by 3 dense matrices over Integer Ring
"""
return self._matrix_space
def __iter__(self):
r"""
Iterator on the alternating sign matrices of size `n`.
If defined using ``use_monotone_triangles``, this iterator
will use the iteration on the monotone triangles. Else, it
will use the iteration on contre-tableaux.
TESTS::
sage: A = AlternatingSignMatrices(4)
sage: len(list(A))
42
"""
if self._umt:
for t in MonotoneTriangles(self._n):
yield self.from_monotone_triangle(t)
else:
for c in ContreTableaux(self._n):
yield from_contre_tableau(c)
def _lattice_initializer(self):
r"""
Return a 2-tuple to use in argument of ``LatticePoset``.
For more details about the cover relations, see
``MonotoneTriangles``. Notice that the returned matrices are
made immutable to ensure their hashability required by
``LatticePoset``.
EXAMPLES:
Proof of the lattice property for alternating sign matrices of
size 3::
sage: A = AlternatingSignMatrices(3)
sage: P = Poset(A._lattice_initializer())
sage: P.is_lattice()
True
"""
assert(self._umt)
(mts, rels) = MonotoneTriangles(self._n)._lattice_initializer()
bij = dict((t, self.from_monotone_triangle(t)) for t in mts)
asms, rels = bij.itervalues(), [(bij[a], bij[b]) for (a,b) in rels]
return (asms, rels)
def cover_relations(self):
r"""
Iterate on the cover relations between the alternating sign
matrices.
EXAMPLES::
sage: A = AlternatingSignMatrices(3)
sage: for (a,b) in A.cover_relations():
... eval('a, b')
...
(
[1 0 0] [0 1 0]
[0 1 0] [1 0 0]
[0 0 1], [0 0 1]
)
(
[1 0 0] [1 0 0]
[0 1 0] [0 0 1]
[0 0 1], [0 1 0]
)
(
[0 1 0] [ 0 1 0]
[1 0 0] [ 1 -1 1]
[0 0 1], [ 0 1 0]
)
(
[1 0 0] [ 0 1 0]
[0 0 1] [ 1 -1 1]
[0 1 0], [ 0 1 0]
)
(
[ 0 1 0] [0 0 1]
[ 1 -1 1] [1 0 0]
[ 0 1 0], [0 1 0]
)
(
[ 0 1 0] [0 1 0]
[ 1 -1 1] [0 0 1]
[ 0 1 0], [1 0 0]
)
(
[0 0 1] [0 0 1]
[1 0 0] [0 1 0]
[0 1 0], [1 0 0]
)
(
[0 1 0] [0 0 1]
[0 0 1] [0 1 0]
[1 0 0], [1 0 0]
)
"""
(_, rels) = self._lattice_initializer()
return (_ for _ in rels)
def lattice(self):
r"""
Return the lattice of the alternating sign matrices of size
`n`, created by ``LatticePoset``.
EXAMPLES::
sage: A = AlternatingSignMatrices(3)
sage: L = A.lattice()
sage: L
Finite lattice containing 7 elements
"""
return LatticePoset(self._lattice_initializer(), cover_relations=True)
class AlternatingSignMatrices(Parent, UniqueRepresentation):
r"""
Class of all `n \times n` alternating sign matrices.
An alternating sign matrix of size `n` is an `n \times n` matrix of `0`'s,
`1`'s and `-1`'s such that the sum of each row and column is `1` and the
non-zero entries in each row and column alternate in sign.
Alternating sign matrices of size `n` are in bijection with
:class:`monotone triangles <MonotoneTriangles>` with `n` rows.
INPUT:
- `n` -- an integer, the size of the matrices.
- ``use_monotone_triangle`` -- (Default: ``True``) If ``True``, the
generation of the matrices uses monotone triangles, else it will use the
earlier and now obsolete contre-tableaux implementation;
must be ``True`` to generate a lattice (with the ``lattice`` method)
EXAMPLES:
This will create an instance to manipulate the alternating sign
matrices of size 3::
sage: A = AlternatingSignMatrices(3)
sage: A
Alternating sign matrices of size 3
sage: A.cardinality()
7
Notably, this implementation allows to make a lattice of it::
sage: L = A.lattice()
sage: L
Finite lattice containing 7 elements
sage: L.category()
Category of facade finite lattice posets
"""
def __init__(self, n, use_monotone_triangles=True):
r"""
Initialize ``self``.
TESTS::
sage: A = AlternatingSignMatrices(4)
sage: TestSuite(A).run()
sage: A == AlternatingSignMatrices(4, use_monotone_triangles=False)
False
"""
self._n = n
self._matrix_space = MatrixSpace(ZZ, n)
self._umt = use_monotone_triangles
Parent.__init__(self, category=FiniteEnumeratedSets())
def _repr_(self):
r"""
Return a string representation of ``self``.
TESTS::
sage: A = AlternatingSignMatrices(4); A
Alternating sign matrices of size 4
"""
return "Alternating sign matrices of size %s" % self._n
def _repr_option(self, key):
"""
Metadata about the :meth:`_repr_` output.
See :meth:`sage.structure.parent._repr_option` for details.
EXAMPLES::
sage: A = AlternatingSignMatrices(3)
sage: A._repr_option('element_ascii_art')
True
"""
return self._matrix_space._repr_option(key)
def __contains__(self, asm):
"""
Check if ``asm`` is in ``self``.
TESTS::
sage: A = AlternatingSignMatrices(3)
sage: [[0,1,0],[1,0,0],[0,0,1]] in A
True
sage: [[0,1,0],[1,-1,1],[0,1,0]] in A
True
sage: [[0, 1],[1,0]] in A
False
sage: [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]] in A
False
sage: [[-1, 1, 1],[1,-1,1],[1,1,-1]] in A
False
"""
if isinstance(asm, AlternatingSignMatrix):
return asm._matrix.nrows() == self._n
try:
asm = self._matrix_space(asm)
except (TypeError, ValueError):
return False
for row in asm:
pos = False
for val in row:
if val > 0:
if pos:
return False
else:
pos = True
elif val < 0:
if pos:
pos = False
else:
return False
if not pos:
return False
if any(sum(row) != 1 for row in asm.columns()):
return False
return True
def _element_constructor_(self, asm):
"""
Construct an element of ``self``.
EXAMPLES::
sage: A = AlternatingSignMatrices(3)
sage: elt = A([[1, 0, 0],[0, 1, 0],[0, 0, 1]]); elt
[1 0 0]
[0 1 0]
[0 0 1]
sage: elt.parent() is A
True
sage: A([[3, 2, 1], [2, 1], [1]])
[1 0 0]
[0 1 0]
[0 0 1]
"""
if isinstance(asm, AlternatingSignMatrix):
if asm.parent() is self:
return asm
raise ValueError("Cannot convert between alternating sign matrices of different sizes")
if asm in MonotoneTriangles(self._n):
return self.from_monotone_triangle(asm)
return self.element_class(self, self._matrix_space(asm))
Element = AlternatingSignMatrix
def _an_element_(self):
"""
Return an element of ``self``.
"""
return self.element_class(self, self._matrix_space.identity_matrix())
def from_monotone_triangle(self, triangle):
r"""
Return an alternating sign matrix from a monotone triangle.
EXAMPLES::
sage: A = AlternatingSignMatrices(3)
sage: A.from_monotone_triangle([[3, 2, 1], [2, 1], [1]])
[1 0 0]
[0 1 0]
[0 0 1]
sage: A.from_monotone_triangle([[3, 2, 1], [3, 2], [3]])
[0 0 1]
[0 1 0]
[1 0 0]
"""
n = len(triangle)
if n != self._n:
raise ValueError("Incorrect size")
asm = []
prev = [0]*n
for line in reversed(triangle):
v = [1 if j+1 in reversed(line) else 0 for j in range(n)]
row = [a-b for (a, b) in zip(v, prev)]
asm.append(row)
prev = v
return self.element_class(self, self._matrix_space(asm))
def size(self):
r"""
Return the size of the matrices in ``self``.
TESTS::
sage: A = AlternatingSignMatrices(4)
sage: A.size()
4
"""
return self._n
def cardinality(self):
r"""
Return the cardinality of ``self``.
The number of `n \times n` alternating sign matrices is equal to
.. MATH::
\prod_{k=0}^{n-1} \frac{(3k+1)!}{(n+k)!} = \frac{1! 4! 7! 10!
\cdots (3n-2)!}{n! (n+1)! (n+2)! (n+3)! \cdots (2n-1)!}
EXAMPLES::
sage: [AlternatingSignMatrices(n).cardinality() for n in range(0, 11)]
[1, 1, 2, 7, 42, 429, 7436, 218348, 10850216, 911835460, 129534272700]
"""
return Integer(prod( [ factorial(3*k+1)/factorial(self._n+k)
for k in range(self._n)] ))
def matrix_space(self):
"""
Return the underlying matrix space.
EXAMPLES::
sage: A = AlternatingSignMatrices(3)
sage: A.matrix_space()
Full MatrixSpace of 3 by 3 dense matrices over Integer Ring
"""
return self._matrix_space
def __iter__(self):
r"""
Iterator on the alternating sign matrices of size `n`.
If defined using ``use_monotone_triangles``, this iterator
will use the iteration on the monotone triangles. Else, it
will use the iteration on contre-tableaux.
TESTS::
sage: A = AlternatingSignMatrices(4)
sage: len(list(A))
42
"""
if self._umt:
for t in MonotoneTriangles(self._n):
yield self.from_monotone_triangle(t)
else:
for c in ContreTableaux(self._n):
yield from_contre_tableau(c)
def _lattice_initializer(self):
r"""
Return a 2-tuple to use in argument of ``LatticePoset``.
For more details about the cover relations, see
``MonotoneTriangles``. Notice that the returned matrices are
made immutable to ensure their hashability required by
``LatticePoset``.
EXAMPLES:
Proof of the lattice property for alternating sign matrices of
size 3::
sage: A = AlternatingSignMatrices(3)
sage: P = Poset(A._lattice_initializer())
sage: P.is_lattice()
True
"""
assert(self._umt)
(mts, rels) = MonotoneTriangles(self._n)._lattice_initializer()
bij = dict((t, self.from_monotone_triangle(t)) for t in mts)
asms, rels = bij.itervalues(), [(bij[a], bij[b]) for (a,b) in rels]
return (asms, rels)
def cover_relations(self):
r"""
Iterate on the cover relations between the alternating sign
matrices.
EXAMPLES::
sage: A = AlternatingSignMatrices(3)
sage: for (a,b) in A.cover_relations():
... eval('a, b')
...
(
[1 0 0] [0 1 0]
[0 1 0] [1 0 0]
[0 0 1], [0 0 1]
)
(
[1 0 0] [1 0 0]
[0 1 0] [0 0 1]
[0 0 1], [0 1 0]
)
(
[0 1 0] [ 0 1 0]
[1 0 0] [ 1 -1 1]
[0 0 1], [ 0 1 0]
)
(
[1 0 0] [ 0 1 0]
[0 0 1] [ 1 -1 1]
[0 1 0], [ 0 1 0]
)
(
[ 0 1 0] [0 0 1]
[ 1 -1 1] [1 0 0]
[ 0 1 0], [0 1 0]
)
(
[ 0 1 0] [0 1 0]
[ 1 -1 1] [0 0 1]
[ 0 1 0], [1 0 0]
)
(
[0 0 1] [0 0 1]
[1 0 0] [0 1 0]
[0 1 0], [1 0 0]
)
(
[0 1 0] [0 0 1]
[0 0 1] [0 1 0]
[1 0 0], [1 0 0]
)
"""
(_, rels) = self._lattice_initializer()
return (_ for _ in rels)
def lattice(self):
r"""
Return the lattice of the alternating sign matrices of size
`n`, created by ``LatticePoset``.
EXAMPLES::
sage: A = AlternatingSignMatrices(3)
sage: L = A.lattice()
sage: L
Finite lattice containing 7 elements
"""
return LatticePoset(self._lattice_initializer(), cover_relations=True)
