Partitions of \(\mathbb{Z}_m\) with identical representation function

Authors

Keywords:

Partition, representation function

Abstract

For a given set $S\subseteq\mathbb{Z}_m$ and $\overline{n}\in\mathbb{Z}_m$, $R_S(\overline{n})$ is defined as the number of solutions of the equation $\overline{n}=\overline{s}+\overline{s'}$ with ordered pair $(\overline{s},\overline{s'})\in S^2$. In this paper, we determine the structure of $A,B\subseteq \mathbb{Z}_m$ with $|(A\cup B)\setminus(A\cap B)|=m-2$ such that $R_{A}(\overline{n})=R_{B}(\overline{n})$ for all $\overline{n}\in \mathbb{Z}_m$, where $m$ is even integer.

Published

2021-03-10

Issue

Section

Articles