A refinement of a result of Andrews and Newman on the sum of minimal excludants

TL;DR

This paper refines a known result relating minimal excludants in partitions to colored partitions, discovers new congruences, explores moments of minimal excludants, and offers an alternative proof of a related identity.

Contribution

It provides a refined version of Andrews and Newman's result, introduces new congruences, studies moments of minimal excludants, and presents an alternate proof of an existing identity.

Findings

Sum of minimal excludants equals the count of two-colored distinct partitions.

Established congruences modulo 4 and 8 for related functions.

Proposed conjectures for additional congruences.

Abstract

In this article, we refine a result of Andrews and Newman, that is, the sum of minimal excludants over all the partitions of a number $n$ equals the number of partitions of $n$ into distinct parts with two colors. As a consequence, we find congruences modulo 4 and 8 for the functions appearing in this refinement. We also conjecture three further congruences for these functions. In addition, we also initiate the study of $k^{th}$ moments of minimal excludants. At the end, we also provide an alternate proof of a beautiful identity due to Hopkins, Sellers and Stanton.