# Characterizing the number of coloured $m$-ary partitions modulo $m$, with and without gaps

## TL;DR

This paper extends previous work on the modulo properties of $m$-ary partitions to coloured $m$-ary partitions, providing explicit generating function expansions and new characterizations.

## Contribution

It introduces a novel proof method and explicit generating function expansions for coloured $m$-ary partitions modulo $m$, with and without gaps.

## Key findings

- Explicit expansions for generating functions modulo m
- Complete characterization of coloured m-ary partitions modulo m
- Extension of previous results to coloured partitions

## Abstract

In a pair of recent papers, Andrews, Fraenkel and Sellers provide a complete characterization for the number of $m$-ary partitions modulo $m$, with and without gaps. In this paper we extend these results to the case of coloured $m$-ary partitions, with and without gaps. Our method of proof is different, giving explicit expansions for the generating functions modulo $m$