Separable integer partition classes and partitions with congruence conditions

TL;DR

This paper explores partitions with parts congruent to specific residues modulo k, introduces a new class of overpartitions with congruence restrictions, and derives their generating functions, extending existing partition theory concepts.

Contribution

It introduces the concept of $(k,r)$-overpartitions, linking them to overpartition counts and extending separable integer partition classes to overpartitions with congruence conditions.

Findings

Number of $(k,k)$-overpartitions of n equals certain overpartition counts.

Derived generating functions for $(k,r)$-modulo overpartitions.

Extended separable integer partition classes to overpartitions with congruence restrictions.

Abstract

In this article, we first investigate the partitions whose parts are congruent to $a$ or $b$ modulo $k$ with the aid of separable integer partition classes with modulus $k$ introduced by Andrews. Then, we introduce the $(k,r)$-overpartitions in which only parts equivalent to $r$ modulo $k$ may be overlined and we will show that the number of $(k,k)$-overpartitions of $n$ equals the number of partitions of $n$ such that the $k$-th occurrence of a part may be overlined. Finally, we extend separable integer partition classes with modulus $k$ to overpartitions and then give the generating function for $(k,r)$-modulo overpartitions, which are the $(k,r)$-overpartitions satisfying certain congruence conditions.