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

Abstract

Let A = (ai)∞ i=1 be a weakly increasing sequence of positive integers and let k be a fixed positive integer. For an arbitrary integer n, the restricted partition pA(n, k) enumerates all the partitions of n whose parts belong to the multiset {a1, a2, . . . , ak}. In this paper we investigate some generalizations of the log-concavity of pA(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.