Arithmetic Properties of Andrews' Singular Overpartitions

Abstract

In a very recent work, G. E. Andrews defined the combinatorial objects which he called {\it singular overpartitions} with the goal of presenting a general theorem for overpartitions which is analogous to theorems of Rogers--Ramanujan type for ordinary partitions with restricted successive ranks. As a small part of his work, Andrews noted two congruences modulo 3 which followed from elementary generating function manipulations. In this work, we prove that Andrews' results modulo 3 are two examples of an infinite family of congruences modulo 3 which hold for that particular function. We also expand the consideration of such arithmetic properties to other functions which are part of Andrews' framework for singular overpartitions.