Partitions of Finite Nonnegative Integer Sets with Identical Representation Functions

Abstract

Let $${\mathbb {N}}$$ be the set of all nonnegative integers. For $$S\subseteq {\mathbb {N}}$$ and $$n\in {\mathbb {N}}$$ , let the representation function $$R_{S}(n)$$ denote the number of solutions of the equation $$n=s+s'$$ with $$s, s'\in S$$ and $$s<s'$$ . In this paper, we determine the structure of $$C, D\subseteq {\mathbb {N}}$$ with $$C\cup D=[0, m]$$ , $$C\cap D=\{r_{1}, r_{2}\}$$ , $$r_{1}<r_{2}$$ and $$2\mid r_{1}$$ such that $$R_{C}(n)=R_{D}(n)$$ for any nonnegative integer n.