Congruences modulo 9 for singular overpartitions

Abstract

In a recent work, Andrews introduced the new combinatorial objects called singular overpartitions. He proved that these singular overpartitions can be enumerated by the partition function [Formula: see text] which 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. In this paper, we consider the function [Formula: see text] from an arithmetical point of view. We establish a number of Ramanujan-like congruences and a congruence relation modulo [Formula: see text] for [Formula: see text] by employing some generating function dissection techniques. With the aid of these congruences and the relation, we obtain two new infinite families of congruences modulo [Formula: see text] satisfied by [Formula: see text].