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

Towards Heim and Neuhauser's Unimodality Conjecture on the Nekrasov-Okounkov polynomials

Let $Q_n(z)$ be the polynomials associated with the Nekrasov-Okounkov formula $$\sum_{n\geq 1} Q_n(z) q^n := \prod_{m = 1}^\infty (1 - q^m)^{-z - 1}.$$ In this paper we partially answer a conjecture of Heim and Neuhauser, which asks if $Q_n(z)$ is unimodal, or stronger, log-concave for all $n \geq 1$. Through a new recursive formula, we show that if $A_{n,k}$ is the coefficient of $z^k$ in $Q_n(z)$, then $A_{n,k}$ is log-concave in $k$ for $k \ll n^{1/6}/\log n$ and monotonically decreasing for $k \gg \sqrt{n}\log n$. We also propose a conjecture that can potentially close the gap.

Turán inequalities for the plane partition function

Heim, Neuhauser, and Tröger recently established some inequalities for MacMahon's plane partition function $\mathrm{PL}(n)$ that generalize known results for Euler's partition function $p(n)$. They also conjectured that $\mathrm{PL}(n)$ is log-concave for all $n\geq 12.$ We prove this conjecture. Moreover, for every $d\geq 1$, we prove their speculation that $\mathrm{PL}(n)$ satisfies the degree $d$ Turán inequality for sufficiently large $n$. The case where $d=2$ is the case of log-concavity.

关于 D'Arcais 多项式对数凹性的最小反例

On the Smallest Counterexample to the Log-Concavity of the D'Arcais Polynomials

通过改进渐近方法，确定了 D'Arcais 多项式对数凹性猜想的最小反例为 λ=65,214,507,758,400，并研究了反例的渐近密度。

小k时d'Arcais数的渐进行为

Asymptotics of the d'Arcais Numbers at Small $k$

研究小k时d'Arcais数的渐进行为，推导出其渐近表达式，并验证关于数列比值的猜想在不同k值下的成立情况。