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Engel’s inequality for Bell numbers. (English) Zbl 0830.11010
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 \(n\).

MSC:
11B83 Special sequences and polynomials
11B73 Bell and Stirling numbers
05A18 Partitions of sets
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