Is it faster to switch to using the mobius_function_matrix if I only want
the intervals starting from zero?
I'm also confused about what it means by the linear extension.
The part of my code in question is
F = []
for i in range(M.rank()+1):
F.extend(M.flats(i))
P = Poset((F, lambda x, y: x <= y))
nu = {}
for i in range(M.rank() + 1):
for e in M.flats(i):
nu[e] = P.mobius_function(P[0], e)
print nu
print P.mobius_function_matrix()
prints:
{frozenset([2]): -1, frozenset([0, 1, 2]): 2, frozenset([]): 1,
frozenset([0]): -1, frozenset([1]): -1}
[ 1 -1 -1 -1 2]
[ 0 1 0 0 -1]
[ 0 0 1 0 -1]
[ 0 0 0 1 -1]
[ 0 0 0 0 1]
Don't worry about flats or M, it has to do with matroids.
I was wondering if it would be faster to switch to using the
mobius_function_matrix, and how do I index into it properly? What's the
mapping from my sets to the indices?
Also, the relevant code from sage:
def mobius_function(self,i,j): # dumb algorithm
r"""
Returns the value of the M\"obius function of the poset
on the elements ``i`` and ``j``.
EXAMPLES::
sage: P = Poset([[1,2,3],[4],[4],[4],[]])
sage: H = P._hasse_diagram
sage: H.mobius_function(0,4)
2
sage: for u,v in P.cover_relations_iterator():
... if P.mobius_function(u,v) != -1:
... print "Bug in mobius_function!"
"""
try:
return self._mobius_function_values[(i,j)]
except AttributeError:
self._mobius_function_values = {}
return self.mobius_function(i,j)
except KeyError:
if i == j:
self._mobius_function_values[(i,j)] = 1
elif i > j:
self._mobius_function_values[(i,j)] = 0
else:
ci = self.closed_interval(i,j)
if len(ci) == 0:
self._mobius_function_values[(i,j)] = 0
else:
self._mobius_function_values[(i,j)] = \
-sum([self.mobius_function(i,k) for k in ci[:-1]])
return self._mobius_function_values[(i,j)]
def mobius_function_matrix(self):
r"""
Returns the matrix of the mobius function of this poset
This returns the sparse matrix over ZZ whose ``(x, y)`` entry
is the value of the M\"obius function of ``self`` evaluated on
``x`` and ``y``, and redefines :meth:`mobius_function` to use
it
.. note::
The result is cached in ``self._mobius_function_matrix``.
EXAMPLES::
sage: from sage.combinat.posets.hasse_diagram import
HasseDiagram
sage: H =
HasseDiagram({0:[1,3,2],1:[4],2:[4,5,6],3:[6],4:[7],5:[7],6:[7],7:[]})
sage: H.mobius_function_matrix()
[ 1 -1 -1 -1 1 0 1 0]
[ 0 1 0 0 -1 0 0 0]
[ 0 0 1 0 -1 -1 -1 2]
[ 0 0 0 1 0 0 -1 0]
[ 0 0 0 0 1 0 0 -1]
[ 0 0 0 0 0 1 0 -1]
[ 0 0 0 0 0 0 1 -1]
[ 0 0 0 0 0 0 0 1]
TESTS::
sage: H.mobius_function_matrix().is_immutable()
True
sage: hasattr(H,'_mobius_function_matrix')
True
sage: H.mobius_function == H._mobius_function_from_matrix
True
"""
if not hasattr(self,'_mobius_function_matrix'):
self._mobius_function_matrix =
self.lequal_matrix().inverse().change_ring(ZZ)
self._mobius_function_matrix.set_immutable()
self.mobius_function = self._mobius_function_from_matrix
return self._mobius_function_matrix
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