Separable overpartition classes and excludant sizes of an overpartition

An overpartition is a partition such that the first occurrence (equivalently, the last occurrence) of a number may be overlined. In this article, we investigate three contents of overpartitions. We first consider the $r$-chain minimal and maximal excludant sizes of an overpartition. Then, we study the second minimal excludant and mex sequence of an overpartition. Finally, we introduce $L_k$-overpartitions and $F_k$-overpartitions, which are separable overpartition classes.