On the Uniqueness Conjecture for the Maximum Stirling Numbers of the Second Kind

Abstract

The Stirling numbers of the second kind S(n, k) satisfyA long standing conjecture asserts that there exists no n ≥ 3 such that S(n, k n ) = S(n, k n + 1). In this note, we give a characterization of this conjecture in terms of multinomial probabilities, as well as sufficient conditions on n ensuring that S(n, k n ) > S(n, k n + 1). KeywordsStirling number of the second kind • uniqueness conjecture • multinomial law