Restricted partition functions and the $r$-log-concavity of quasi-polynomial-like functions

Abstract

Let $\mathcal{A}={\left(a_i\right)}_{i=1}^{\infty }$ be a weakly increasing sequence of positive integers and let $k$ be a fixed positive integer. For an arbitrary integer $n$, the restricted partition $p_{\mathcal{A}}(n,k)$ enumerates all the partitions of $n$ whose parts belong to the multiset $\{a_1,a_2,\dots ,a_k\}$. In this paper we investigate some generalizations of the log-concavity of $p_{\mathcal{A}}(n,k)$. We deal with both some basic extensions like, for instance, the strong log-concavity and a more intriguing challenge that is the $r$-log-concavity of both quasi-polynomial-like functions in general, and the restricted partition function in particular. For each of the problems, we present an efficient solution.

arXiv comment24 pages, 12 figures