Log-concavity and related properties of the cycle index polynomials

Abstract

Let An denote the nth-cycle index polynomial, in the variables Xj, for the symmetric group on n letters. We show that if the variables Xj are assigned nonnegative real values which are log-concave, then the resulting quantities An satisfy the two inequalities An - 1 An + 1 ≤ A2n ≤ ((n + 1)/n)An - 1 An + 1. This implies that the coefficients of the formal power series exp(g(u)) are log-concave whenever those of g(u) satisfy a condition slightly weaker than log-concavity. The latter includes many familiar combinatorial sequences, only some of which were previously known to be log-concave. To prove the first inequality we show that in fact the difference A2n - A- 1 An +1 can be written as a polynomial with positive coefficients in the expressions Xj and XjXk - Xj - 1 Xk + 1, j ≤ k. The second inequality is proven combinatorially, by working with the notion of a marked permutation, which we introduce in this paper. The latter is a permutation each of whose cycles is assigned a subset of available markers {Mi, j}. Each marker has a weight, wt(Mi, j) = xj, and we relate the second inequality to properties of the weight enumerator polynomials. Finally, using asymptotic analysis, we show that the same inequalities hold for n sufficiently large when the Xj are fixed with only finite many nonzero values, with no additional assumption on the Xj. © 1996 Academic Press, Inc.