Arithmetic Properties For $(r,s)$-Regular Partition Functions With Distinct Parts

Abstract

For any relatively prime integers $r$ and $s$, let $a_{r,s}(n)$ denote the number of $(r,s)$-regular partitions of a positive integer of $n$ into distinct parts. Prasad and Prasad (2018) proved many infinite families of congruences modulo 2 for $a_{3,5}(n)$. In this paper, we establish families of congruences modulo 2 and 4 for $a_{r,s}(n)$ with $(r,s)\in$ \{(2,5), (2,7), (4,5), (4,9)\}. For example, we show that for all $\beta \ge 0$ and $n\ge 0,$ we have $$a_{2,5}(4\cdot 5^{2\beta +1}n+\frac{37\cdot 5^{2\beta }-1}{6})\equiv 0(\mathrm{mod}4).$$