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 $n\ge 0$ and $\alpha \ge 1$, we prove that

$$\begin{aligned} p_{3,3}\left( 3^{\alpha }n+\dfrac{3^{\alpha }+1}{2}\right)&\equiv 0\pmod {3^{\alpha +1}} \end{aligned}$$

and

$$\begin{aligned} p_{3,3}\left( 3^{\alpha +1}n+\dfrac{5\times 3^{\alpha }+1}{2}\right)&\equiv 0\pmod {3^{\alpha +4}}. \end{aligned}$$