Signs 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 $n,k\in \mathbb{N}$.

arXiv comment12 pages