# A Hardy-Ramanujan-Rademacher-type formula for $(r,s)$-regular partitions

## TL;DR

This paper derives a Hardy-Ramanujan-Rademacher-type infinite series formula for counting partitions of integers into parts avoiding multiples of two square-free, coprime integers, expanding the analytical tools for such partition functions.

## Contribution

It introduces a novel infinite series formula for $(r,s)$-regular partitions, generalizing classical partition formulas to new parameter settings.

## Key findings

- Derived an explicit series for $p_{r,s}(n)$
- Extended classical methods to new partition constraints
- Provided analytical tools for $(r,s)$-regular partitions

## Abstract

Let $p_{r,s}(n)$ denote the number of partitions of a positive integer $n$ into parts containing no multiples of $r$ or $s$, where $r>1$ and $s>1$ are square-free, relatively prime integers. We use classical methods to derive a Hardy-Ramanujan-Rademacher-type infinite series for $p_{r,s}(n)$.