Some Separable integer partition classes

TL;DR

This paper explores separable integer partition classes, focusing on partitions separated by parity and extending these concepts to overpartitions, including their relation to Rogers-Ramanujan identities.

Contribution

It introduces the extension of separable integer partition classes to overpartitions and analyzes their properties and connections to classical identities.

Findings

Analysis of partitions separated by parity using separable classes

Extension of separable classes to overpartitions

Study of overpartition analogues of Rogers-Ramanujan identities

Abstract

Recently, Andrews introduced separable integer partition classes and analyzed some well-known theorems. In this paper, we investigate partitions with parts separated by parity introduced by Andrews with the aid of separable integer partition classes with modulus $2$. We also extend separable integer partition classes with modulus $1$ to overpartitions, called separable overpartition classes. We study overpartitions and the overpartition analogue of Rogers-Ramanujan identities, which are separable overpartition classes.