Elementary Proofs of Arithmetic Properties for Schur-Type Overpartitions Modulo Small Powers of 2

TL;DR

This paper establishes new elementary proofs of infinite families of congruences modulo powers of 2 for the overpartition function of Schur-type, extending previous results and employing classical q-series identities.

Contribution

It provides novel elementary proofs of Ramanujan-like congruences for Schur-type overpartition counts modulo small powers of 2.

Findings

Proves infinite families of congruences modulo 16 for S(n).

Extends previous work on overpartition congruences by Broudy and Lovejoy.

Uses elementary q-series and generating function techniques.

Abstract

In 2022, Broudy and Lovejoy extensively studied the function $S(n)$ which counts the number of overpartitions of \emph{Schur-type}. In particular, they proved a number of congruences satisfied by $S(n)$ modulo $2$, $4$, and $5$. In this work, we extend their list of arithmetic properties satisfied by $S(n)$ by focusing on moduli which are small powers of 2. In particular, we prove the following infinite family of Ramanujan-like congruences: For all $\alpha\geq 0$ and $n\geq 0$, $$ S\left(2^{5+2\alpha}n+\left(2^{5+2\alpha}-\frac{2^{2+2\alpha}-1}{3}\right)\right)\equiv 0 \pmod{16}. $$ All of the proof techniques used herein are elementary, relying on classical $q$-series identities and generating function manipulations as well as the parameterization work popularized by Alaca, Alaca, and Williams.