def to_noncrossing_set_partition(self):
r"""
Return the noncrossing set partition (on half as many elements)
corresponding to the perfect matching if the perfect matching is
noncrossing, and otherwise gives an error.
OUTPUT:
The realization of ``self`` as a noncrossing set partition.
EXAMPLES::
sage: PerfectMatching([[1,3], [4,2]]).to_noncrossing_set_partition()
Traceback (most recent call last):
...
ValueError: matching must be non-crossing
sage: PerfectMatching([[1,4], [3,2]]).to_noncrossing_set_partition()
{{1, 2}}
sage: PerfectMatching([]).to_noncrossing_set_partition()
{}
"""
if not self.is_noncrossing():
raise ValueError("matching must be non-crossing")
else:
perm = self.to_permutation()
perm2 = Permutation(
[perm[2 * i] // 2 for i in range(len(perm) // 2)])
return SetPartition(perm2.cycle_tuples())
def __getitem__(self, i):
"""
Return the basis element indexed by ``i``.
INPUT:
- ``i`` -- a set partition or a list of list of integers
EXAMPLES::
sage: w = SymmetricFunctionsNonCommutingVariables(QQ).dual().w()
sage: w[[1], [2,3]]
w{{1}, {2, 3}}
sage: w[{1}, (2,3)]
w{{1}, {2, 3}}
sage: w[[]]
w{}
"""
if isinstance(i, SetPartition):
return self.monomial(i)
if i == []:
return self.one()
if not isinstance(i, tuple):
i = (i, )
return self.monomial(SetPartition(i))
def standardization(self):
"""
Return the standardization of ``self``.
See :meth:`SetPartition.standardization` for details.
EXAMPLES::
sage: n = PerfectMatching([('c','b'),('d','f'),('e','a')])
sage: n.standardization()
[(1, 5), (2, 3), (4, 6)]
"""
P = PerfectMatchings(2 * len(self))
return P(SetPartition.standardization(self))
def standardization(self):
"""
Return the standardization of ``self``.
See :meth:`SetPartition.standardization` for details.
EXAMPLES::
sage: n = PerfectMatching([('c','b'),('d','f'),('e','a')])
sage: n.standardization()
[(1, 5), (2, 3), (4, 6)]
"""
P = PerfectMatchings(2*len(self))
return P(SetPartition.standardization(self))
def one_basis(self):
r"""
Return the index of the basis element containing `1`.
OUTPUT:
- The empty set partition
EXAMPLES::
sage: m = SymmetricFunctionsNonCommutingVariables(QQ).m()
sage: m.one_basis()
{}
sage: w = SymmetricFunctionsNonCommutingVariables(QQ).dual().w()
sage: w.one_basis()
{}
"""
return SetPartition([])
def _element_constructor_(self, x):
"""
Construct an element of ``self``.
INPUT:
- ``x`` -- a set partition or list of lists of integers
EXAMPLES::
sage: m = SymmetricFunctionsNonCommutingVariables(QQ).m()
sage: m([[1,3],[2]])
m{{1, 3}, {2}}
sage: m(SetPartition([[1,3],[2]]))
m{{1, 3}, {2}}
"""
if isinstance(x, (list, tuple)):
x = SetPartition(x)
return super(NCSymBasis_abstract, self)._element_constructor_(x)
def decomposition(self):
"""
Find the decomposition of the matroid as a direct sum of indecomposable matroids.
Return a partition of the groundset.
Uses the algorithm of [PP19, Section 7].
"""
B = self.basis()
new_groundset = list(B) + list(self.groundset().difference(B))
# construct matrix with permuted columns
columns = [vector(self._A[:,self._groundset_to_index[e]]) for e in new_groundset]
A = matrix(ZZ, columns).transpose().echelon_form()
uf = DisjointSet(self.groundset())
for i in xrange(A.nrows()):
for j in xrange(i+1, A.ncols()):
if A[i,j] != 0:
uf.union(new_groundset[i], new_groundset[j])
return SetPartition(uf)