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 $\{a_1,a_2,\ldots,a_k\}$. 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.