Towards Heim and Neuhauser’s unimodality conjecture on the Nekrasov–Okounkov polynomials

Abstract

Let $Q_n(z)$ be the polynomials associated with the Nekrasov–Okounkov formula

$$\begin{aligned} \sum _{n\ge 1} Q_n(z) q^n := \prod _{m = 1}^\infty (1 - q^m)^{-z - 1}. \end{aligned}$$

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 $n \ge 1$. 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.