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