Log-Concavity and Related Properties of the Cycle Index Polynomials

Log-Concavity and Related Properties of the Cycle Index Polynomials

Let An denote the n-th 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 logconcave, then the resulting quantities An satisfy the two inequalities An−1An+1 ≤ An ≤ ( n+1 n ) An−1An+1. This implies that the coefficients of the formal power series exp(g(u)) are log-concave whenever tho...

Log - Concavity and Related Properties of the Cycle Index

Log - Concavity and Related Properties of the Cycle Index Polynomials Edward

Let An denote the n-th 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 logconcave, then the resulting quantities An satisfy the two inequalities An 1An+1 A2n n+1 n An 1An+1: This implies that the coe cients of the formal power series exp(g(u)) are log-concave whenever those of g(u...

Unimodality and Log-Concavity of Polynomials

A polynomial is unimodal if its sequence of coefficients are increasing up to an index, and then are decreasing after that index. A polynomial is logconcave if the sequence of the logarithms of the coefficients is concave. We prove that if P (x) is a polynomial with nonnegative non-decreasing coefficients then P (x+z) is unimodal for any natural z. Furthermore, we prove that if P (x) is a log-c...

Expansions of Chromatic Polynomials and Log-concavity

In this paper we present several results and open problems about logconcavity properties of sequences associated with graph colorings. Five polynomials intimately related to the chromatic polynomial of a graph are introduced and their zeros, combinatorial and log-concavity properties are studied. Four of these polynomials have never been considered before in the literature and some yield new ex...