Engel's inequality for Bell numbers (Q1899066)

\textit{K. Engel} [J. Comb. Theory, Ser. A 65, No. 1, 67-78 (1994; Zbl 0795.05051)]\ conjectured that \(\tau_n= (B_{n+ 1}/ B_n) -1\) (\(B_n\) the \(n\)-th Bell number; \(\tau_n\) the average number of blocks in a partition of an \(n\)-set) is concave. The author proves the conjecture for all \(n\) sufficiently large, using the asymptotic formula for Bell numbers of Moser and Wyman. He also shows that the average number of singleton blocks in a partition of an \(n\)-set is an increasing function of \(n\).