Lattice paths and Rogers--Ramanujan--Gordon type overpartitions

Abstract

In this paper, we establish a connection between Rogers-Ramanujan-Gordon type overpartitions to lattice paths with four kinds of unitary steps. By establishing the bijective relationship between overpartitions and lattice paths, we demonstrate that the theorems provided by Chen, Sang and Shi can be formulated in the form of lattice paths. Subsequently, inspired by Andrews' work on parity in partition identities and its related implications, we impose parity constraints on lattice paths and present some novel discoveries. By leveraging the parity outcomes within lattice paths, we also revisit overpartitions to derive results pertaining to overpartitions with parity considerations.