A product of integer partitions

TL;DR

This paper introduces a bijection on integer partitions that yields recursive formulas, generating functions, and a new associative product, revealing invariants and algebraic structures related to Ferrers diagrams.

Contribution

It presents a novel bijection on partitions that defines a free associative algebra structure and introduces difference sets with invariant properties.

Findings

Derived recursive expressions and generating functions for partition sets

Established an associative product on partitions with a natural grading

Identified invariants related to difference sets in partitions

Abstract

I present a bijection on integer partitions that leads to recursive expressions, closed formulae and generating functions for the cardinality of certain sets of partitions of a positive integer $n$. The bijection leads also to a product on partitions that is associative with a natural grading thus defining a free associative algebra on the set of integer partitions. As an outcome of the computations, certain sets of integers appear that I call difference sets and the product of the integers in a difference set is an invariant for a family of sets of partitions. The main combinatorial objects used in these constructions are the central hooks of the Ferrers diagrams of partitions.