Proof of the Bessenrodt--Ono inequality by Induction

TL;DR

This paper provides a new proof of the Bessenrodt--Ono inequality for partition functions using recursion, extending to related results and introducing a novel finding.

Contribution

It introduces a recursion-based proof for the Bessenrodt--Ono inequality and extends it to related partition function results, offering a new approach.

Findings

New recursion-based proof of the Bessenrodt--Ono inequality

Extension of the proof to Chern--Fu--Tang's result and polynomization

A new related result is obtained

Abstract

In 2016 Bessenrodt--Ono discovered an inequality addressing additive and multiplicative properties of the partition function. Generalization by several authors have been given; on partitions with rank in a given residue class by Hou--Jagadeesan and Males, on $k$-regular partitions by Beckwith--Bessenrodt, on $k$-colored partitions by Chern, Fu, Tang, and Heim--Neuhauser on their polynomization, and Dawsey--Masri on the Andrews ${\it spt}$-function. The proofs depend on non-trivial asymptotic formulas related to the circle method on one side, or a sophisticated combinatorial proof invented by Alanazi--Gagola--Munagi. We offer in this paper a new proof of the Bessenrodt--Ono inequality, which is built on a well-known recursion formula for partition numbers. We extend the proof to the result of Chern--Fu--Tang and its polynomization. Finally, we also obtain a new result.