Congruences modulo powers of 3 for 2-color partition triples

TL;DR

This paper establishes new infinite families of congruences modulo powers of 3 for 2-color partition triples with specific restrictions, expanding understanding of their divisibility properties.

Contribution

It proves novel infinite families of congruences modulo powers of 3 for 2-color partition triples with parts constrained by multiples of k, for k=1, 3, and 9.

Findings

Proves that certain partition counts are divisible by increasing powers of 3.

Establishes explicit congruences for all n and alpha.

Extends known results to new classes of partition functions.

Abstract

Let $p_{k,3}(n)$ enumerate the number of 2-color partition triples of $n$ where one of the colors appears only in parts that are multiples of $k$. In this paper, we prove several infinite families of congruences modulo powers of 3 for $p_{k,3}(n)$ with $k=1, 3$, and $9$. For example, for all integers $n\geq0$ and $\alpha\geq1$, we prove that \begin{align*} p_{3,3}\left(3^{\alpha}n+\dfrac{3^{\alpha}+1}{2}\right) &\equiv0\pmod{3^{\alpha+1}} \end{align*} and \begin{align*} p_{3,3}\left(3^{\alpha+1}n+\dfrac{5\times3^{\alpha}+1}{2}\right) &\equiv0\pmod{3^{\alpha+4}}. \end{align*}