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

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.