Overpartition pairs and two classes of basic hypergeometric series

TL;DR

This paper explores the combinatorial structures of overpartition pairs and their relation to basic hypergeometric series, revealing new identities and connections with lattice paths and partition conditions.

Contribution

It introduces new combinatorial interpretations of hypergeometric series through overpartition pairs and derives multiple identities linking partitions, overpartitions, and lattice paths.

Findings

Hypergeometric series as generating functions for overpartition pairs

Specializations yield identities for partitions and overpartitions

Connections established with lattice path combinatorics

Abstract

We study the combinatorics of two classes of basic hypergeometric series. We first show that these series are the generating functions for certain overpartition pairs defined by frequency conditions on the parts. We then show that when specialized these series are also the generating functions for overpartition pairs with bounded successive ranks, overpartition pairs with conditions on their Durfee dissection, as well as certain lattice paths. When further specialized, the series become infinite products, leading to numerous identities for partitions, overpartitions, and overpartition pairs.