The first positive rank and crank moments for overpartitions

TL;DR

This paper extends the study of rank and crank moments from ordinary partitions to overpartitions, proving a strict inequality for their first moments and offering a new combinatorial interpretation.

Contribution

It introduces the first positive rank and crank moments for overpartitions and establishes a strict inequality, along with a novel combinatorial perspective.

Findings

Proved strict inequality for first overpartition rank and crank moments

Established a new combinatorial interpretation for overpartition moments

Extended classical results from ordinary partitions to overpartitions

Abstract

In 2003, Atkin and Garvan initiated the study of rank and crank moments for ordinary partitions. These moments satisfy a strict inequality. We prove that a strict inequality also holds for the first rank and crank moments of overpartitions and consider a new combinatorial interpretation in this setting.