Sign behavior of sums of weighted numbers of partitions

Abstract: Let $A$ be a subset of the positive integers. By an $A$-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 talk I will describe several results concerning the sign behaviour of the sequence $S_{A,k}(n) = \sum_{i=0}^n(-1)^i i^k c_A(i, n)$, for fixed $k\in \mathbb{N}$. I will focus on the periodicity of the sequence of signs for different forms of $A$. Finally, I will also mention some conjectures and questions that arose naturally during our research.

The talk is based on a joint work with Maciej Ulas (Jagiellonian University).

number theory

Audience: researchers in the discipline

( paper )

Combinatorial and additive number theory (CANT 2022)