Overpartitions with separated overlined parts and non-overlined parts

TL;DR

This paper explores a new class of overpartitions where overlined and non-overlined parts are separated by size constraints, extending the concept of parity-separated partitions.

Contribution

It introduces and analyzes overpartitions with separated overlined and non-overlined parts based on size relations, inspired by parity-separated partitions.

Findings

Characterization of the new overpartition class

Connections to existing partition identities

Potential combinatorial interpretations

Abstract

Recently, Andrews considered the partitions with parts separated by parity, in which parts of a given parity are all smaller than those of the other parity. Inspired from the partitions with parts separated by parity, we investigate the overpartitions with separated overlined parts and non-overlined parts, in which the sizes of overlined parts (resp. non-overlined parts) are greater than or equal to those of non-overlined parts (resp. overlined parts).