ENGELS INEQUALITY FOR BELL NUMBERS

K. Engel has conjectured that the average number of blocks in a partit ion of an rt-set is a concave function of n. The average in question i s a quotient of two Bell numbers less 1, and we prove Engel's conjectu re for all n sufficiently large by an extension of the Moser-Wyman asy mptotic formula for the Bell numbers. We also give a general theorem w hich specializes to an inequality about Bell numbers less complex than Engel's, in that fewer terms of the asymptotic expansion are needed t o verify it for all sufficiently large n. (C) 1995 Academic Press, Inc .