Sign behaviour of sums of weighted numbers of partitions

Abstract

Let A be a subset of positive integers. By A-partition of n, we understand the representation of n as a sum of elements from the set A. For given $i, n\in \mathbb {N}$, by $c_{A}(i,n)$, we denote the number of A-partitions of n with exactly i parts. In the paper, we obtain several result concerning sign behaviour of the sequence $S_{A,k}(n)=\sum _{i=0}^{n}(-1)^{i}i^{k}c_{A}(i,n)$, where $k\in \mathbb {N}$ is fixed. In particular, we prove that for a broad class $\mathcal {A}$ of subsets of $\mathbb {N}_{+}$, we have that for each $A\in \mathcal {A}$, we have $(-1)^{n}S_{A,k}(n)\ge 0$ for each $n, k\in \mathbb {N}$.