Arithmetic Properties of Andrews' Singular Overpartitions

TL;DR

This paper explores the arithmetic properties of Andrews' singular overpartitions, proving an infinite family of congruences modulo 3 and extending these properties to related functions within Andrews' framework.

Contribution

It establishes an infinite family of modulo 3 congruences for Andrews' singular overpartitions and related functions, expanding understanding of their arithmetic properties.

Findings

Proved infinite family of modulo 3 congruences

Extended congruence results to related functions

Connected congruences to Andrews' framework for 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.