On the uniqueness conjecture for the maximum Stirling numbers of the second kind (Q2038842)

It is known that the sequence \(S(n,k)\) of Stirling numbers of the second kind is unimodal in \(k\). Wegner conjectured that the mode \(k_n\) of \(S(n,k)\) is unique for \(n \geq 3\) in \(1973\). This paper gives a characterization of Wegner's conjecture, by using of the probabilistic representation of \(S(n,k)\). Furthermore, the sufficient conditions of \(n\) ensuring that \(S(n,k_n) > S(n,k_n+1)\) are derived.