Certain eta-quotients and arithmetic density of Andrews' singular overpartitions

Abstract

In order to give overpartition analogues of Rogers-Ramanujan type theorems for the ordinary partition function, Andrews defined the so-called singular overpartitions. Singular overpartition function C‾k,i(n) counts the number of overpartitions of n in which no part is divisible by k and only parts ≡±i(modk) may be overlined. Andrews also proved two beautiful Ramanujan type congruences modulo 3 satisfied by C‾3,1(n). Later on, Aricheta proved that for an infinite family of ℓ, C‾3ℓ,ℓ(n) is almost always divisible by 2. In this article, for an infinite subfamily of ℓ considered by Aricheta, we prove that C‾3ℓ,ℓ(n) is almost always divisible by arbitrary powers of 2. We also prove that C‾3ℓ,ℓ(n) is almost always divisible by arbitrary powers of 3 when ℓ=3,6,12,24. Proofs of our density results rely on the modularity of certain eta-quotients which arise naturally as generating functions for the Andrews' singular overpartition functions.