On the problem of uniqueness for the maximum Stirling number(s) of the second kind (Q2768455)

It has been known that the Stirling numbers of the second kind, \(S(n,k)\), have an increasing interval and decreasing interval, when \(1 \leq k \leq n\). It follows that a maximum value is taken for one or two values of \(k\). The only known example of two maximum places is \(n=2\), when \(S(2,1)=S(2,2)=1\). H. Wegner conjectured that there are no more examples. Notwithstanding a confusing review in the literature, Wegner's conjecture is still open. Let \(E(x)\) denote the number of integer \(n\)'s (\(2\leq n \leq x \)), for which \(S(n,k)\) takes its maximum value twice. The main result of the paper is the bound \(E(x)=O(x^{3/5+ \varepsilon})\), and a heuristic argument, why \(E(x)\) is expected to be bounded. There is computer-based proof for \(E(10^6)=1\). (Unlike in many other computer-based proofs, sufficient details are given for those who want to repeat the calculations).