Partitioning into sets of bounded cardinality

Abstract

We show that the partitions of an n-element set into k members of a given set family can be counted in time O((2-ε)n), where ε > 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 O(poly(n)φn), where φ = 1.618... is the golden ratio; this improves a previous bound based on fast matrix multiplication. © 2009 Springer-Verlag.