Congruences modulo powers of 3 for 2-color partition triples

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*}