On Minimal Excludant over Overpartitions

A partition of a positive integer $n$ is a non-increasing sequence of positive integers which sum to $n$. A recently studied aspect of partitions is the minimal excludant of a partition, which is defined to be the smallest positive integer that is not a part of the partition. In 2024, Aricheta and Donato studied the minimal excludant of the non-overlined parts of an overpartition, where an overpartition of $n$ is a partition of $n$ in which the first occurrence of a number may be overlined. In this research, we explore two other definitions of the minimal excludant of an overpartition: (i) considering only the overlined parts, and (ii) considering both the overlined and non-overlined parts. We discuss the combinatorial, asymptotic, and arithmetic properties of the corresponding $\sigma$-function, which gives the sum of the minimal excludants over all overpartitions.