Arithmetic Properties of Overpartition Pairs

TL;DR

This paper explores the arithmetic properties of overpartition pairs, deriving identities, congruences, and combinatorial interpretations, extending the understanding of their divisibility and congruence patterns.

Contribution

It introduces new Ramanujan-type identities, explicit congruences, and combinatorial ranks for overpartition pairs, expanding the theoretical framework and understanding of their arithmetic properties.

Findings

Derived two Ramanujan-type identities for overpartition pairs.

Established explicit congruences modulo 3, 5, and 9.

Identified three ranks as combinatorial interpretations of divisibility by three.

Abstract

Bringmann and Lovejoy introduced a rank for overpartition pairs and investigated its role in congruence properties of $\bar{pp}(n)$, the number of overpartition pairs of n. In particular, they applied the theory of Klein forms to show that there exist many Ramanujan-type congruences for the number $\bar{pp}(n)$. In this paper, we shall derive two Ramanujan-type identities and some explicit congruences for $\bar{pp}(n)$. Moreover, we find three ranks as combinatorial interpretations of the fact that $\bar{pp}(n)$ is divisible by three for any n. We also construct infinite families of congruences for $\bar{pp}(n)$ modulo 3, 5, and 9.