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 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 \(\alpha \ge 0\) $\alpha \ge 0$ and \(n\ge 0\) $n\ge 0$ , \(\begin{aligned} S\left( 2^{5+2\alpha }n+\left( 2^{5+2\alpha }-\frac{2^{2+2\alpha }-1}{3}\right) \right) \equiv 0 \pmod {16}. \end{aligned}\) $$\begin{array}{c}\hfill S\left(2^{5+2\alpha },n,+,\left(2^{5+2\alpha },-,\frac{2^{2+2\alpha }-1}{3}\right)\right)\equiv 0(mod16).\end{array}$$ 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.