Proof of the Bessenrodt–Ono Inequality by Induction

@article{Heim2021ProofOT,
title={Proof of the Bessenrodt–Ono Inequality by Induction},
author={Bernhard Heim and Markus Neuhauser},
journal={Research in Number Theory},
year={2021}
}
• Published 31 July 2021
• Mathematics
• Research in Number Theory
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… Expand

References

SHOWING 1-10 OF 27 REFERENCES
Polynomization of the Chern–Fu–Tang conjecture
• Mathematics
• 2020
Bessenrodt and Ono's work on additive and multiplicative properties of the partition function and DeSalvo and Pak's paper on the log-concavity of the partition function have generated many beautifulExpand
Polynomization of the Bessenrodt–Ono Inequality
• Mathematics
• 2019
In this paper we investigate the generalization of the Bessenrodt--Ono inequality by following Gian-Carlo Rota's advice in studying problems in combinatorics and number theory in terms of roots ofExpand
The Nekrasov-Okounkov hook length formula: refinement, elementary proof, extension and applications
The paper is devoted to the derivation of the expansion formula for the powers of the Euler Product in terms of partition hook lengths, discovered by Nekrasov and Okounkov in their study of theExpand
Jensen polynomials for the Riemann zeta function and other sequences
• Mathematics, Medicine
• Proceedings of the National Academy of Sciences
• 2019
The Pólya–Jensen criterion for the Riemann hypothesis asserts that RH is equivalent to the hyperbolicity of certain Jensen polynomials for all degrees d≥1 and all shifts n, and this criterion is confirmed for all sufficiently large shifts n. Expand
Combinatorial Proof of a Partition Inequality of Bessenrodt-Ono
• Mathematics
• 2017
We provide a combinatorial proof of the inequality \$\${p(a)p(b) > p(a+b)}\$\$p(a)p(b)>p(a+b), where p(n) is the partition function and a, \$\${b > 1}\$\$b>1 are integers satisfying \$\${a+b > 9}\$\$a+b>9. ThisExpand
Some inequalities for k-colored partition functions
• Mathematics
• 2017
Motivated by a partition inequality of Bessenrodt and Ono, we obtain analogous inequalities for k-colored partition functions \$\$p_{-k}(n)\$\$p-k(n) for all \$\$k\ge 2\$\$k≥2. This enables us to extend theExpand
Multiplicative Properties of the Number of k-Regular Partitions
• Mathematics
• 2014
In a previous paper of the second author with K. Ono, surprising multiplicative properties of the partition function were presented. Here, we deal with k-regular partitions. Extending the generatingExpand
Exact formulae for the fractional partition functions
• Mathematics
• 2019
The partition function p ( n ) has been a testing ground for applications of analytic number theory to combinatorics. In particular, Hardy and Ramanujan invented the “circle method” to estimate theExpand
Log-concavity of the partition function
• Mathematics
• 2013
We prove that the partition function \$\$p(n)\$\$p(n) is log-concave for all \$\$n>25\$\$n>25. We then extend the results to resolve two related conjectures by Chen and one by Sun. The proofs are based onExpand
The Dedekind eta function and D’Arcais-type polynomials
• Mathematics
• 2020
D’Arcais-type polynomials encode growth and non-vanishing properties of the coefficients of powers of the Dedekind eta function. They also include associated Laguerre polynomials. We prove growthExpand