On the number of l-regular overpartitions

Abstract

An [Formula: see text]-regular overpartition of [Formula: see text] is an overpartition of [Formula: see text] into parts not divisible by [Formula: see text]. Let [Formula: see text] be the number of [Formula: see text]-regular overpartitions of [Formula: see text]. Andrews defined singular overpartitions counted by the partition function [Formula: see text]. It denotes the number of overpartitions of [Formula: see text] in which no part is divisible by [Formula: see text] and only parts[Formula: see text] may be overlined. He proved that [Formula: see text] and [Formula: see text] are divisible by [Formula: see text]. In this paper, we aim to introduce a crank of [Formula: see text]-regular overpartitions for [Formula: see text] to investigate the partition function [Formula: see text]. We give combinatorial interpretations for some congruences of [Formula: see text] including infinite families of congruences for [Formula: see text] modulo [Formula: see text] and [Formula: see text] as well as the congruences of Andrews for [Formula: see text] and [Formula: see text].