Infinite families of congruences modulo powers of 5 for 2-color partition

Abstract

In this work, we investigate the arithmetic properties of $p^{t}_{1,\ell}(n)$, which counts 2-color partitions of $n$ where one color appears only in parts that are not multiples of $t$, and the other color appears only in parts that are multiples of $\ell$.  By constructing generating functions for $p^{t}_{1,\ell}$ across specific arithmetic progressions, we establish Ramanujan-type infinite families of congruences modulo powers of $5$ for $p^{t}_{1,\ell}(n)$.