An Asymptotic Formula for the Number of Classes of Sets of n Indistinguishable Elements

This note addresses the following problem posed by S . M. Ulam. Find a concise expression for the number of classes of sets of n indistinguishable elements . Here, a class of sets is a set whose members are subsets of a given finite set of n elements, and referring to the elements as indistinguishable means that two classes are identical if some permutation on the elements transforms one class into the other . Harary [2] gives some formulas which, when applied together with Polya's well-known enumeration theorem [3], result in an exact expression for the number of classes. Unfortunately, this expression involves a complicated sum over either the permutations on n or the partitions on n . The problem may be generalized in the obvious way to the hierarchy of classes of classes of sets and so forth . The following results apply not only to classes of sets but to this hierarchy as well, giving simple asymptotic estimates .