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

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 $\alpha \ge 0$ and $n\ge 0$, $$S(2^{5+2\alpha }n+(2^{5+2\alpha }-\frac{2^{2+2\alpha }-1}{3}))\equiv 0(\mathrm{m}\mathrm{o}\mathrm{d}16)\mathrm{.}$$ 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.