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

TL;DR

This paper investigates the unimodality and log-concavity of Nekrasov-Okounkov polynomials, providing partial results and recursive formulas that support the conjecture for large classes of coefficients.

Contribution

It introduces a new recursive formula for the polynomials and establishes partial log-concavity and monotonicity results, advancing understanding of the conjecture.

Findings

Coefficients are log-concave in certain ranges of k

Coefficients are decreasing for large k

Proposes a conjecture to bridge the gap in results

Abstract

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.