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Arithmetic Properties of Andrews' Singular Overpartitions

Shi-Chao Chen, Michael D. Hirschhorn, James A. Sellers

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.

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.

Paper Structure

This paper contains 4 sections, 27 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. Results for $\overline{C}_{3,1}(n)$
  3. Results for $\overline{C}_{4,1}(n)$
  4. Results for $\overline{C}_{6,1}(n)$ and $\overline{C}_{6,2}(n)$

Key Result

Theorem \oldthetheorem

Let $N$ be a positive integer which is not expressible as the sum of two nonnegative squares. Then

Theorems & Definitions (48)

  • Theorem \oldthetheorem
  • proof
  • Corollary \oldthetheorem
  • Corollary \oldthetheorem
  • Corollary \oldthetheorem
  • Remark \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • Corollary \oldthetheorem
  • proof
  • ...and 38 more