Some Comments on Regular Overpartitions modulo $2^k$

Abstract

For coprime integers $\ell,μ\ge 2$, Alanazi, Munagi, and Saikia (2026) studied $\overline{R}_{\ell,μ}(n)$, the number of overpartitions of $n$ in which no part is divisible by $\ell$ or by $μ$, together with the single-modulus analogue $\overline{R}_{\ell (n)$. We record a simple combinatorial mechanism that determines both functions modulo every power of $2$ in terms of the number of distinct part sizes of the underlying ordinary partition. We also deduce a clean characterization of $\overline{R}_{\ell}(n)$ and $\overline{R}_{\ell,μ}(n)$ modulo $4$ in terms of perfect squares.