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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
##### Keywords:
Engel’s inequality; Bell number; partition; $$n$$-set
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##### References:
 [1] {\scE. A. Bender and E. R. Canfield}, Log concavity and a related property of the cycle index polynomials, preprint. · Zbl 0853.05013 [2] {\scE. R. Canfield and L. H. Harper}, A simplified guide to large antichains in the partition lattice, Congr. Numer., to appear. · Zbl 0833.06003 [3] Comtet, L, Advanced combinatories, (1974), Reidel Dordrecht [4] Engel, K, On the average rank of an element in a filter of the partition lattice, J. combin. theory ser. A, 64, 67-78, (1994) · Zbl 0795.05051 [5] {\scJ. R. Griggs}, personal communication. [6] Harper, L.H, Stirling behavior is asymptotically normal, Ann. math. stat., 38, 410-414, (1967) · Zbl 0154.43703 [7] Moser, L; Wyman, M, An asymptotic formula for the Bell numbers, Trans. royal soc. Canada III, 49, 49-54, (1955) · Zbl 0066.31001 [8] Wilf, H.S, Generating functionology, (1990), Academic Press New York
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