Proof of the Bessenrodt–Ono Inequality by Induction

  title={Proof of the Bessenrodt–Ono Inequality by Induction},
  author={Bernhard Heim and Markus Neuhauser},
  journal={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… 

Tables from this paper


Polynomization of the Chern–Fu–Tang conjecture
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 beautiful
Polynomization of the Bessenrodt–Ono Inequality
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 of
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 the
Jensen polynomials for the Riemann zeta function and other sequences
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.
Combinatorial Proof of a Partition Inequality of Bessenrodt-Ono
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. This
Some inequalities for k-colored partition functions
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 the
Multiplicative Properties of the Number of k-Regular Partitions
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 generating
Exact formulae for the fractional partition functions
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 the
Log-concavity of the partition function
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 on
The Dedekind eta function and D’Arcais-type polynomials
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 growth