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

TL;DR

This paper studies the distribution of the number of parts modulo m in m-ary partitions of integers, proving equidistribution on certain subsets and providing new insights and proofs for existing results.

Contribution

It establishes conditions for equidistribution of parts modulo m in m-ary partitions and offers an alternative proof of a recent related theorem.

Findings

Number of parts is equidistributed modulo m on specific subsets of m-ary partitions.

Conditions identified for equidistribution on the entire set of partitions.

Provides an alternative proof of a recent result by Andrews, Fraenkel, and Sellers.

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 about the number of $m$-ary partitions modulo $m$.