A Note On Certain Minimal Excludants Over Overpartitions

Let $\sigma \mathrm{M}\mathrm{e}\mathrm{x}(n)$ and ${\sigma }_e\mathrm{M}\mathrm{e}\mathrm{x}(n)$ denote the sum of minimal excludants and sum of even minimal excludants over all overpartitions respectively. In this work, we study these two combinatorial objects from an arithmetic perspective. We prove that, for $n\ge 0$, ${\sigma }_e\mathrm{M}\mathrm{e}\mathrm{x}(n)\equiv \sigma \mathrm{M}\mathrm{e}\mathrm{x}(n)-{\bar{p}}_{\ge 2}(n)(\mathrm{m}\mathrm{o}\mathrm{d}{2}^2),$ where ${\bar{p}}_{\ge 2}(n)$ denotes the number of overpartitions of $n$ with all parts at least $2$. Furthermore, we prove the asymptotic behavior of $\sigma \mathrm{M}\mathrm{e}\mathrm{x}(n)$, ${\sigma }_e\mathrm{M}\mathrm{e}\mathrm{x}(n)$, and ${\bar{p}}_{\ge 2}(n)$ as $n\to \mathrm{\infty }$. In particular, we prove that $\sigma \mathrm{M}\mathrm{e}\mathrm{x}(n)\sim 2{\sigma }_e\mathrm{M}\mathrm{e}\mathrm{x}(n)\mathrm{.}$