Partitions of finite nonnegative integer sets with identical representation functions

Abstract

Let N be the set of all nonnegative integers. For S ⊆ N and n ∈ N, let the representation function RS(n) denote the number of solutions of the equation n = s + s ′ with s, s ′ ∈ S and s < s ′ . In this paper, we determine the structure of C, D ⊆ N with C ∪ D = [0, m] and |C ∩ D| = 2 such that RC(n) = RD(n) for any nonnegative integer n.