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

Abstract

The Stirling numbers of the second kind S(n, k) satisfy $$\begin{aligned} S(n,0) \cdots >S(n,n). \end{aligned}$$ A long standing conjecture asserts that there exists no $$n\ge 3$$ 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 $$S(n,k_n)>S(n,k_n+1)$$ .