Overpartitions, lattice paths and Rogers-Ramanujan identities

TL;DR

This paper extends classical partition identities to overpartitions using lattice paths and generalized Durfee squares, unifying many Rogers-Ramanujan type identities and their generalizations.

Contribution

It introduces new overpartition concepts and relates them to classical identities, broadening the combinatorial framework for partition theory.

Findings

Derived new overpartition identities

Unified several Rogers-Ramanujan type identities

Extended classical partition theorems to overpartitions

Abstract

We extend partition-theoretic work of Andrews, Bressoud, and Burge to overpartitions, defining the notions of successive ranks, generalized Durfee squares, and generalized lattice paths, and then relating these to overpartitions defined by multiplicity conditions on the parts. This leads to many new partition and overpartition identities, and provides a unification of a number of well-known identities of the Rogers-Ramanujan type. Among these are Gordon's generalization of the Rogers-Ramanujan identities, Andrews' generalization of the G\"ollnitz-Gordon identities, and Lovejoy's ``Gordon's theorems for overpartitions."