Partitions with unique largest part and their generating functions

TL;DR

This paper introduces a new class of integer partitions where the largest part appears exactly once, derives their generating functions, and explores various related partition variants.

Contribution

It presents the concept of partitions with a unique largest part and provides explicit generating functions for these and related variants.

Findings

Derived generating functions for the new partition class

Analyzed several variations with their generating functions

Enhanced understanding of partition structures with unique largest parts

Abstract

The paper introduce a new type of partitions where the largest part appears exactly once, and the remaining parts constitute a partition of that largest part. We derive the generating function associated with these partitions and subsequently explore several variations, providing the corresponding generating functions for each variant.