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{align*} p_{3,3}\left(3^αn+\dfrac{3^α+1}{2}\right) &\equiv0\pmod{3^{α+1}} \end{align*} and \begin{align*} p_{3,3}\left(3^{α+1}n+\dfrac{5\times3^α+1}{2}\right) &\equiv0\pmod{3^{α+4}}. \end{align*}

arXiv comment14 pages, to appear in Period. Math. Hungar