Refining overlined parts in overpartitions via residue classes: Bijections, generating functions, and congruences

Abstract

Over the past several years, numerous authors have studied properties of the combinatorial objects known as overpartitions (which are natural generalizations of integer partitions). In this paper, we consider various classes of overpartitions where the "overlined parts" belong to certain residue classes modulo a positive integer m. We state new identities between such restricted overpartitions and standard partition functions. Finally, we prove a number of Ramanujan-like congruences for many of the restricted overpartition functions using elementary generating function manipulations.