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 $$\sum _{n\ge 1}{Q}_n(z)q^n:=\prod _{m=1}^{\infty }(1-q^m{)}^{-z-1}.$$ 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 $n\ge 1$. 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.

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