Polynomization of the Chern--Fu--Tang conjecture

Proof of the Bessenrodt--Ono inequality by Induction

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 A…

Fractional partitions and conjectures of Chern-Fu-Tang and Heim-Neuhauser

Many papers have studied inequalities for partition functions. Recently, a number of papers have considered mixtures between additive and multiplicative behavior in such inequalities. In particular, Chern-Fu-Tang and Heim-Neuhauser gave conjectures on inequalities for coefficients of powers of the generating partition function. These conjectures were posed in the context of colored partitions and the Nekrasov-Okounkov formula. Here, we study the precise size of differences of products of two such coefficients. This allows us to prove the Chern-Fu-Tang conjecture and to show the Heim-Neuhauser …

$k$-色划分的严格对数凹性

Strict Log-concavity of $k$-coloured Partitions

本文研究了$k$-色划分函数的严格对数凹性，通过引入递推序列和分数划分函数的严格不等式，证明了当$k≥2$时，$k$-色划分函数严格对数凹。