Research in Number Theory

, 5:32 | Cite as

Variants of an partition inequality of Bessenrodt–Ono

  • Bernhard HeimEmail author
  • Markus Neuhauser


In this paper we generalize and refine a partition inequality of Bessenrodt–Ono. We introduce and study the m shifted inequality
$$\begin{aligned} p(a) \, p(b) \ge p(a+b+m-1) \end{aligned}$$
where p(n) is the nth partition number, and \(a,b,m \in \mathbb {N}_0\) with ab positive. The inequality was first studied by Bessenrodt–Ono for \(m=1\). We finally suggest another generalization involving polynomials, which dictate the vanishing properties of the Fourier coefficients of powers of the Dedekind \(\eta \)-function.


Asymptotic analysis Dedekind eta-function Partitions Partition inequality Polynomials 

Mathematics Subject Classification

Primary 05A17 11P82 Secondary 05A20 



The authors thank the two reviewers for their very helpful suggestions. We thank the RWTH Aachen University and the Graduate School: Experimental and constructive algebra, and the German University of Technology in Oman for their hospitality and generous support.


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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.German University of Technology in OmanMuscatSultanate of Oman

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