On partitions with even parts below odd parts

TL;DR

This paper provides a combinatorial proof for the generating function of partitions with even parts below odd parts, explores the largest even part, and introduces a weighted overpartition generalization.

Contribution

It offers a new combinatorial proof for a specific partition generating function and extends the study to overpartition generalizations.

Findings

Combinatorial proof of the generating function identity

Analysis of the largest even part in such partitions

Introduction of a weighted overpartition generalization

Abstract

Recently, Andrews gave a detailed study of partitions with even parts below odd parts in which only the largest even part appears an odd number of times. In this paper, we provide a combinatorial proof of the generating function identity of such partitions. We also have a further investigation on the largest even part. Finally, we give an interesting weighted overpartition generalization.