The maximum Stirling number(s) of the second kind
by
Rod Canfield
University of Georgia
Coauthors: Carl Pomerance (Bell Laboratories / Lucent Technologies)

Let S(n, k) be the Stirling number of the second kind; that is, the number of partitions of an n-set into k non empty pairwise disjoint blocks. It is well known that for each n these numbers are strictly log concave: S(n, k)² > S(n, k-1) S(m, k+1). Since the ratio S(n, k+1)/S(n, k) is therefore stictly decreasing, the maximum Stirling number occurs either for a unique value of k, or it occurs for two consecutive k. The only known case of the latter is when n=2. We will show that the number of n <= x such that S(n, k) has a repeated maximum is O(x^c) for some c < 1.

Date received: April 11, 2001