Minimal Excludant over Overpartitions

Define the minimal excludant of an overpartition $\pi$, denoted $\overline{\text{mex}}(\pi)$, to be the smallest positive integer that is not a part of the non-overlined parts of $\pi$. For a positive integer $n$, the function $\sigma\overline{\text{mex}}(n)$ is the sum of the minimal excludants over all overpartitions of $n$. In this paper, we proved that the $\sigma\overline{\text{mex}}(n)$ equals the number of partitions of $n$ into distinct parts using three colors. We also provide an asymptotic formula for $\sigma\overline{\text{mex}}(n)$ and show that $\sigma\overline{\text{mex}}(n)$ is almost always even and is odd exactly when $n$ is a triangular number. Moreover, we generalize $\overline{\text{mex}}(\pi)$ using the least $r$-gaps, denoted $\overline{\text{mex}_r}(\pi)$, defined as the smallest part of the non-overlined parts of the overpartition $\pi$ appearing less than $r$ times. Similarly, for a positive integer $n$, the function $\sigma_r\overline{\text{mex}}(n)$ is the sum of the least $r$-gaps over all overpartitions of $n$. We derive a generating function and an asymptotic formula for $\sigma_r\overline{\text{mex}}(n)$ . Lastly, we study the arithmetic density of $\sigma_r\overline{\text{mex}}(n)$ modulo $2^k$, where $r=2^m\cdot3^n, m,n \in \mathbb{Z}_\geq 0.$