The distribution of the number of parts of $m$-ary partitions modulo $m$

Abstract

We investigate the number of parts modulo~$m$ of $m$-ary partitions of a positive integer~$n$. We prove that the number of parts is equidistributed modulo~$m$ on a special subset of $m$-ary partitions. As consequences, we explain when the number of parts is equidistributed modulo~$m$ on the entire set of partitions, and we provide an alternate proof of a recent result of Andrews, Fraenkel and Sellers regarding the number of $m$-ary partitions modulo~$m$.