The combinatorics of identities involving overpartitions with distinct parts

TL;DR

This paper refines and provides combinatorial proofs for identities involving overpartitions with distinct parts, extending recent discoveries related to q-series formulas and partition identities.

Contribution

It introduces refinements based on the number of parts and offers combinatorial proofs for new identities involving overpartitions with distinct parts.

Findings

Refined identities involving overpartitions with distinct parts

Combinatorial proofs for the new identities

Extensions of recent q-series formulas

Abstract

Recently, Andrews and EI Bachraoui discovered several companions for some famous $q$-series formulas, and derived some new identities involving partitions and overpartitions with distinct parts. In this paper, we shall refine their results by the number of parts of partitions and furthermore, we will also provide the combinatorial proofs for those partition identities.