Combinatorics of counting finite fuzzy subsets

Let F(n) denote the number of equivalence classes of fuzzy subsets of a finite set of n elements under a natural equivalence generalizing equality of subsets in the crisp case. Applying results from integer partitions, and partitions of a set, a formula was developed for F(n) recently using the ideas of flags, keychains and pinned-flags. In this paper, F(n) is re-examined through an interpretation of equivalence classes of fuzzy subsets as ordered partitions or chains in the Boolean Algebra of the power set of a set. We derive some recurrence relations, an infinite series as a closed form and a generating function for F(n) for any natural number n.