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

TL;DR

This paper investigates congruence properties of $(r,s)$-regular partition functions with distinct parts, extending known results to new pairs and establishing infinite families of congruences modulo 2 and 4.

Contribution

It introduces new infinite families of congruences modulo 2 and 4 for specific $(r,s)$-regular partition functions with distinct parts, generalizing previous work.

Findings

Established congruences modulo 2 and 4 for $a_{r,s}(n)$ with specific $(r,s)$ pairs.

Extended known congruence results to new pairs $(2,7)$, $(4,5)$, $(4,9)$.

Provided explicit formulas for infinite families of congruences.

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 $\beta \geq 0$ and $n \geq 0,$ we have $$a_{2,5}\Big(4\cdot 5^{2\beta+1}n+\dfrac{37\cdot5^{2\beta}-1}{6}\Big)\equiv0 \pmod4. $$