Families of Congruences for Partitions with $k$-colored odd parts

TL;DR

This paper investigates infinite families of congruences modulo 3 for partitions where odd parts can be in k colors and even parts are restricted, extending classical partition congruence results.

Contribution

It introduces new infinite families of congruences for colored odd-part partitions, expanding understanding of partition congruences beyond classical cases.

Findings

Identifies infinite families of congruences modulo 3

Extends classical partition congruence results

Proposes open questions for future research

Abstract

The study of integer partitions and their congruences dates back to 1919 when Ramanujan discovered his famous congruences for the partition function, $p(n)$. Since then, many other kinds of partition functions have been discovered, as well as their respective congruences. Recently, Hirschorn and Sellers have consider partitions in which the odd parts may appear in $k$ colors and the even parts are restricted to at most one color. It turns out that these partitions exhibit fascinating families of congruences. In this paper, we look at a set of congruences that give rise to infinite families modulo 3. We also give some questions at the end that could aid further research into these partitions.