Log-concavity for unimodal sequences

TL;DR

This paper proves that the number of unimodal sequences of size n is log-concave, using an exact formula derived from mixed false modular forms, advancing understanding of their combinatorial and analytic properties.

Contribution

It establishes log-concavity for unimodal sequences via an exact formula, extending methods to other coefficients of mixed mock/false modular objects.

Findings

Number of unimodal sequences of size n is log-concave.

Method based on exact formulas can be applied to other modular form coefficients.

Advances understanding of combinatorial properties of unimodal sequences.

Abstract

In this paper, we prove that the number of unimodal sequences of size $n$ is log-concave. These are coefficients of a mixed false modular form and have a Rademacher-type exact formula due to recent work of the second author and Nazaroglu on false theta functions. Log-concavity and higher Tur\'{a}n inequalities have been well-studied for (restricted) partitions and coefficients of weakly holomorphic modular forms, and analytic proofs generally require precise asymptotic series with error term. In this paper, we proceed from the exact formula for unimodal sequences to carry out this calculation. We expect our method applies to other exact formulas for coefficients of mixed mock/false modular objects.