ENGELS INEQUALITY FOR BELL NUMBERS

[摘要] K. Engel has conjectured that the average number of blocks in a partition of an rt-set is a concave function of n. The average in question is a quotient of two Bell numbers less 1, and we prove Engel's conjecture for all n sufficiently large by an extension of the Moser-Wyman asymptotic formula for the Bell numbers. We also give a general theorem which specializes to an inequality about Bell numbers less complex than Engel's, in that fewer terms of the asymptotic expansion are needed to verify it for all sufficiently large n. (C) 1995 Academic Press, Inc.

[发布日期] 1995-10-01