Log-balanced 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 connecti...

A sequence $\{ a_n \}_{n \ge 0}$ is said to be asymptotically $r$-log-convex if it is $r$-log-convex for $n$ sufficiently large. We present a criterion on the asymptotical $r$-log-convexity based on the asymptotic behavior of $a_n a_{n+2}/a_{n+1}^2$. As an application, we show that most P-recursive sequences are asymptotic $r$-log-convexity for any integer $r$ once they are log-convex. Moreover, for a concrete integer $r$, we present a systematic method to find the explicit i...

This is a survey on recent developments on unimodality, log-concavity and real-rootedness in combinatorics. Stanley and Brenti have written extensive surveys of various techniques that can be used to prove real-rootedness, log-concavity or unimodality. After a brief introduction, we will complement these surveys with a survey over some new techniques that have been developed, as well as problems and conjectures that have been solved. This is a draft of a chapter to appear in ...

Two general methods for establishing the logarithmic behavior of recursively defined sequences of real numbers are presented. One is the interlacing method, and the other one is based on calculus. Both methods are used to prove logarithmic behavior of some combinatorially relevant sequences, such as Motzkin and Schr\"oder numbers, sequences of values of some classic orthogonal polynomials, and many others. The calculus method extends also to two- (or more-) indexed sequences.

Why do natural and interesting sequences often turn out to be log-concave? We give one of many possible explanations, from the viewpoint of "standard conjectures". We illustrate with several examples from combinatorics.

Here we prove some conjectures on the monotony of combinatorial sequences from the recent preprint of Zhi--Wei Sun.

The q-Catalan numbers studied by Carlitz and Riordan are polynomials in q with nonnegative coefficients. They evaluate, at q=1, to the Catalan numbers: 1, 1, 2, 5, 14,..., a log-convex sequence. We use a combinatorial interpretation of these polynomials to prove a q-log-convexity result. The sequence of q-Catalan numbers is not q-log-convex in the narrow sense used by other authors, so our work suggests a more flexible definition of q-log convex be adopted.

We show that Stieltjes moment sequences are infinitely log-convex, which parallels a famous result that (finite) P\'olya frequency sequences are infinitely log-concave. We introduce the concept of $q$-Stieltjes moment sequences of polynomials and show that many well-known polynomials in combinatorics are such sequences. We provide a criterion for linear transformations and convolutions preserving Stieltjes moment sequences. Many well-known combinatorial sequences are shown to...

Using calculus we show how to prove some combinatorial inequalities of the type log-concavity or log-convexity. It is shown by this method that binomial coefficients and Stirling numbers of the first and second kinds are log-concave, and that Motzkin numbers and secondary structure numbers of rank 1 are log-convex. In fact, we prove via calculus a much stronger result that a natural continuous ``patchwork'' (i.e. corresponding dynamical systems) of Motzkin numbers and seconda...

A sequence $\{z_n\}_{n\geq0}$ is called ratio log-convex in the sense that the ratio sequence $\{\frac{z_{n+1}}{z_n}\}_{n\geq0}$ is log-convex. Based on a three-term recurrence for sequences, we develop techniques for dealing with the ratio log-convexity of ratio sequences. As applications, we prove that the ratio sequences of numbers, including the derangement numbers, the Motzkin numbers, the Fine numbers, Franel numbers and the Domb numbers are ratio log-convex, respective...