Congruences for Andrews singular overpartition pairs

Abstract

Andrews defined the combinatorial objects called singular overpartitions denoted by [Formula: see text], which counts 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. In this paper, we investigate the arithmetic properties of Andrews singular overpartition pairs. Let [Formula: see text] be the number of overpartition pairs of [Formula: see text] in which no part is divisible by [Formula: see text] and only parts [Formula: see text] may be overlined. We will prove a number of Ramanujan like congruences and infinite families of congruences for [Formula: see text] modulo [Formula: see text] and [Formula: see text], infinite families of congruences for [Formula: see text] modulo [Formula: see text] and [Formula: see text], infinite families of congruences for [Formula: see text] modulo [Formula: see text] and [Formula: see text].