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arXiv · 2108.00191

Proof of the Bessenrodt--Ono inequality by Induction

Bernhard Heim·Markus Neuhauseer
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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.

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Bernhard Heim, Markus Neuhauseer. 2021-07-31. Proof of the Bessenrodt--Ono inequality by Induction. https://arxiv.org/abs/2108.00191

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