Separable overpartition classes and the $r$-chain excludant sizes of an overpartition

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