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Abstract
In 2016 Bessenrodt–Ono discovered an inequality addressing additive and multiplicative properties of the partition function. Generalizations 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 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 by Chern–Fu–Tang and its polynomization. Finally, we also obtain a new result.
Highlights
1 Introduction and main results Bessenrodt and Ono [3] discovered an interesting inequality for partition numbers p(n)
In this paper we offer a new proof for the Bessenrodt–Ono inequality for partition numbers
For a → ∞ we can immediately observe that this is positive since the sum is a polynomial of degree 4 in a which grows faster than 48a2 (1 + ln (2a))
Summary
2.2 Right sum R The dominant term is related to k = 1 appearing in the right sum R. Note that the induction hypothesis cannot be applied in general to all terms. We decompose the right sum R into three parts. 2.2.1 The sum R1 The first sum related to k = 1 is simplified by the induction hypothesis:. 2.2.2 The sum R2 The second sum, using again the induction hypothesis, can be estimated from below with 0. This will be sufficient for our purpose: R2 > 0
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