New congruences for Andrews' singular overpartitions

Abstract

Recently, Andrews defined the combinatorial objects which he called singular overpartitions and proved that these singular overpartitions, which depend on two parameters k and i, can be enumerated by the function [Formula: see text] which gives the number of overpartitions of n in which no part is divisible by k and only parts ≡ ±i ( mod k) may be overlined. He also proved that [Formula: see text]. Chen, Hirschhorn and Sellers then found infinite families of congruences modulo 3 and modulo powers of 2 for [Formula: see text], [Formula: see text] and [Formula: see text]. In this paper, we find new congruences for [Formula: see text] modulo 4, 18 and 36, infinite families of congruences modulo 2 and 4 for [Formula: see text], congruences modulo 2 and 3 for [Formula: see text], [Formula: see text], and congruences modulo 2 for [Formula: see text] and [Formula: see text]. We use simple p-dissections of Ramanujan's theta functions.