Some Comments on Regular Overpartitions modulo $2^k$

Abstract

For coprime integers $\ell ,\hat{I}¼\ge 2$, Alanazi, Munagi, and Saikia (2026) studied ${\overline{R}}_{\ell ,\hat{I}¼}(n)$, the number of overpartitions of $n$ in which no part is divisible by $\ell$ or by $\hat{I}¼$, 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 ${\overline{R}}_{\ell ,\hat{I}¼}(n)$ modulo $4$ in terms of perfect squares.

5 pages, comments are welcome