PARTITIONS OF NATURAL NUMBERS AND THEIR ORDERED REPRESENTATION FUNCTIONS

Abstract Let $\mathbb {N}$ be the set of all nonnegative integers. For a set $A\subseteq \mathbb {N}$ , let $R_2(A,n)$ and $R_3(A,n)$ be the number of solutions of the equation $n=a_1+a_2$ with $a_1<a_2, a_1,a_2\in A$ and with $a_1\le a_2, a_1,a_2\in A$ , respectively. If $-N\le g\le N$ , Yan [‘On the structure of partition which the difference of their representation function is a constant’, Period. Math. Hungar. 82 (2021), 149–152] showed that there is a set $A\subseteq \mathbb {N}$ such that $R_i(A,n)-R_i(\mathbb {N}\setminus A,n)=g$ for all integers $n\ge 2N-1$ , where N is a positive integer. In this paper, we prove that if $g_1,g_2$ are nonnegative integers with $g_1\neq g_2$ , then there does not exist $A\subseteq \mathbb {N}$ such that $R_i(A,2n)-R_i(\mathbb {N}\setminus A,2n)=g_1$ and $R_i(A,2n+1)-R_i(\mathbb {N}\setminus A,2n+1)=g_2$ for all sufficiently large integers n .