Overpartitions and singular overpartitions

TL;DR

This paper provides a combinatorial proof of a result by Andrews on singular overpartitions, clarifying their structure and relationships within overpartition subclasses, and addresses an open question from Andrews' work.

Contribution

It offers a new combinatorial proof of Andrews' result on singular overpartitions, resolving an open problem and deepening understanding of their combinatorial properties.

Findings

Established a combinatorial proof of Andrews' result

Connected singular overpartitions with specific overpartition subclasses

Resolved an open question posed by Andrews

Abstract

Singular overpartitions, which are defined by George Andrews, are overpartitions whose Frobenius symbols have at most one overlined entry in each row. In his paper, Andrews obtained interesting combinatorial results on singular overpartitions, one of which relates a certain type of singular overpartitions with a subclass of overpartitions. In this paper, we provide a combinatorial proof of Andrews' result, which answers to one of his open questions.