Partitioning into Sets of Bounded Cardinality

We show that the partitions of an n-element set into A; members of a given set family can be counted in time O((2-epsilon)(n)), where epsilon > 0 depends only on the maximum size among the members of the family. Specifically, we give a simple combinatorial algorithm that counts the perfect matchings in a given graph on n vertices in time 0(poly(n)phi(n)), where phi = 1.618 ... is the golden ratio; this improves a previous bound based on fast matrix multiplication.