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

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