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

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 kth moments of minimal excludants. At the end, we also provide an alternate proof of a beautiful identity due to Hopkins, Sellers, and Stanton.