A sequence a,, a,, . . . , a, of real numbers is said to be unimodal if for some 0 s j _c n we have a, 5 a , 5 . . 5 ai 2 a,,, 2 . . 2 a,, and is said to be logarithmically concave (or log-concave for short) if a: 2 a, ,a ,+ , for all 1 5 i 5 n 1. Clearly a log-concave sequence of positive terms is unimodal. Let us say that the sequence a,, a,, . . . , a, has no internal zeros if there do not exist integers 0 5 i < j < k 5 n satisfying a, f 0, a, = 0, ak # 0. Then in fact a nonnegative log-concave sequence with no internal zeros is unimodal. The sequence a,, a , , . . . , a, is called symmetric if a, = a,-, for 0 5 i 5 n. Thus a symmetric unimodal sequence a,, a, , . . . , a,, has its maximum at the middle term ( n even) or middle two terms (n odd). We also say that a polynomial a, + a,4 +. . + an$ has a certain property (such as unimodal, log-concave, or symmetric) if its sequence a,, a,, . . . , a, of coefficients has that property. Our object here is to survey the surprisingly rich variety of methods for showing that a sequence is log-concave or unimodal. For each method we will give examples of its applicability to combinatorially defined sequences that arise naturally from problems in algebra, combinatorics, and geometry. We make no attempt, however, to give a comprehensive account of all work done in this area.

Log‐Concave and Unimodal Sequences in Algebra, Combinatorics, and Geometry

Un grand nombre de suites d'interet combinatoire sont unimodales ou log-concaves. Il s'agit de proprietes difficiles a montrer et non conservees par des transformations lineaires, en general. On peut affiner ces deux notions pour definir les suites de frequences de Polya qui possedent des proprietes plus convenables. On montre des applications combinatoires diverses, telle que la coloration de graphes, de ces suites

Unimodal Log Concave and Polya Frequency Sequences in Combinatorics

On the log-convexity of combinatorial sequences

This paper is devoted to the study of the log-convexity of combinatorial sequences. We show that the log-convexity is preserved under componentwise sum, under binomial convolution, and by the linear transformations given by the matrices of binomial coefficients and Stirling numbers of two kinds. We develop techniques for dealing with the log-convexity of sequences satisfying a three-term recurrence. We also introduce the concept of $q$-log-convexity and establish the connection with linear transformations preserving the log-convexity. As applications of our results, we prove the log-convexity and $q$-log-convexity of many famous combinatorial sequences of numbers and polynomials.

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David C Kurtz

Papers

A note on concavity properties of triangular arrays of numbers