On $\ell -$regular and $2-$color partition triples modulo powers of $3$

Abstract

Let $T_{\ell }(n)$ denote the number of $\ell -$regular partition triples of $n$ and let $p_{\ell ,3}(n)$ enumerates the number of 2--color partition triples of $n$ where one of the colors appear only in parts that are multiples of $\ell$. In this paper, we prove several infinite families of congruences modulo powers of 3 for $T_{\ell }(n)$ and $p_{\ell ,3}(n)$, where $\ell \ge 1$ and $\equiv 0(\mathrm{mod}{3}^k)$, and $\equiv \pm 3^k(\mathrm{mod}{3}^{k+1})$.