Signs behaviour of sums of weighted numbers of partitions

TL;DR

This paper investigates the sign behavior of a sequence derived from weighted counts of partitions of integers into elements of a subset, establishing positivity results for broad classes of such subsets.

Contribution

It introduces new results on the sign patterns of sums involving weighted partition counts for various subsets of positive integers.

Findings

For broad classes of subsets, the sequence's sign pattern is determined by the parity of n.

The paper proves that for each subset in the class, (-1)^n times the sequence is non-negative.

Results extend understanding of sign behavior in partition enumeration sequences.

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 $i, n\in\N$, 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 $k\in\N$ is fixed. In particular, we prove that for a broad class $\cal{A}$ of subsets of $\N_{+}$ we have that for each $A\in \cal{A}$ we have $(-1)^{n}S_{A,k}(n)\geq 0$ for each $n, k\in\N$.