On Minimal Excludant over Overpartitions

TL;DR

This paper investigates new definitions of the minimal excludant in overpartitions, analyzing their combinatorial, asymptotic, and arithmetic properties to deepen understanding of overpartition structures.

Contribution

It introduces two novel definitions of the minimal excludant in overpartitions and studies their properties, expanding the theoretical framework of partition analysis.

Findings

Derived formulas for the $\sigma$-function in new cases

Established asymptotic behavior of the minimal excludant sums

Explored arithmetic properties related to overpartition structures

Abstract

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.