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

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.

Theorems & Definitions (14)

• Lemma 1.1
• Theorem 1.2
• Theorem 1.3
• Remark
• Lemma 2.1: Theorem 1.64 and 1.65 of webofmod
• Lemma 2.2: Theorem 2.58 of webofmod
• Lemma 3.1
• proof
• Lemma 3.2
• proof