Tutte seminar - Bruce Richmond

On the Maximum of the Stirling Numbers

Abstract:

Say an integer n is exceptional if the maximum Stirling number of the second kind S(n, k) occurs for two(of necessity consecutive) values of k. We with Graeme Kemkes and Donatella Merlini have shown that the number of exceptional integers less than or equal to x is O(x^{1/2+\e}) for any \e > 0. We derive a similar result for partitions of n into exactly k parts p(n, k). The estimates of Bombieri and Pila for lattice points on convex curves and the asymptotic behaviour of S(n, k) and p(n, k) are required.