Unique maximum property of the Stirling numbers of the second kind

Abstract

Letting f(n) and (n) be the first and last maxim of the graph S(n,k); k = 1, 2, ... , n, Kanold [J. Reine Angew. Math 230 (1968), 211-212] shows that, for sufficiently large n, n/log n < f(n) = (n) = n h(n)/log n with h(n) subject only to h(n) [to infinity] as n [to infinity]. This result was subsequently improved by Harborth {J. Reine Angew. Math 230 (1968), 213-214} to yield lim f(n)nﾹlog n = lim (n)nﾹlog n = 1. Together with Harper's result , it is concluded that S(n,k) have, asymptotically, a single maximum. Lieb [J. of Comb. Theory 5 (1968), 203-206] shows that Stirling numbers of the second kind possess the property of Strong Logarithmic Concavity, and thus are unimodal. Dobsn [J. of Comb. Theory 5 (1968). 212-214 and 7 (1969), 116-121] shows a similar result in a stronger form. However, the classical problem of establishing that S(n,k) possesses a "unique" maximum for all n >/= 3 remains unsolved. It is the purpose of this paper to provide the complete solution of this classical problem