Separable overpartition classes and excludant sizes of an overpartition

TL;DR

This paper explores various properties of overpartitions, including excludant sizes, mex sequences, and introduces new classes called $L_k$- and $F_k$-overpartitions, advancing combinatorial understanding.

Contribution

It introduces the concepts of separable overpartition classes and analyzes excludant sizes and mex sequences, providing new insights into overpartition structures.

Findings

Characterization of $r$-chain minimal and maximal excludant sizes

Analysis of second minimal excludant and mex sequences

Introduction of $L_k$- and $F_k$-overpartition classes

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 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.