def cardinality(self):
"""
Return the number of Lyndon words with the evaluation e.
EXAMPLES::
sage: LyndonWords([]).cardinality()
0
sage: LyndonWords([2,2]).cardinality()
1
sage: LyndonWords([2,3,2]).cardinality()
30
Check to make sure that the count matches up with the number of
Lyndon words generated::
sage: comps = [[],[2,2],[3,2,7],[4,2]] + Compositions(4).list()
sage: lws = [LyndonWords(comp) for comp in comps]
sage: all(lw.cardinality() == len(lw.list()) for lw in lws)
True
"""
evaluation = self._e
le = list(evaluation)
if not evaluation:
return Integer(0)
n = sum(evaluation)
return sum(
moebius(j) * multinomial([ni // j for ni in evaluation])
for j in divisors(gcd(le))) // n
def cardinality(self):
r"""
Return the cardinality of ``self``.
The number of ordered set partitions of a set of length `k` with
composition shape `\mu` is equal to
.. MATH::
\frac{k!}{\prod_{\mu_i \neq 0} \mu_i!}.
EXAMPLES::
sage: OrderedSetPartitions(5,[2,3]).cardinality()
10
sage: OrderedSetPartitions(0, []).cardinality()
1
sage: OrderedSetPartitions(0, [0]).cardinality()
1
sage: OrderedSetPartitions(0, [0,0]).cardinality()
1
sage: OrderedSetPartitions(5, [2,0,3]).cardinality()
10
"""
return multinomial(self.c)