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

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

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.

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  1. 1.

    This formula was also obtained concurrently by Westbury (see Proposition 6.1 and 6.2 of [11]) and Han (see [3]).

  2. 2.

    D’Arcais defined the polynomial \(P_n(z) = Q_n(z - 1)\) via the infinite product, not the hook number expression.

  3. 3.

    Throughout this paper, we define \(A_{n,k}\) or \(c_{n,k}\) to be 0 for all undefined subscripts.

  4. 4.

    Note that p(n) is monotone increasing, thus all the terms are positive.


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We would like to thank Professor Ken Ono for the proposal of the project and his guidance. We thank Professor Bernhard Heim and Professor Markus Neuhauser for carefully reviewing the transcript and providing helpful feedbacks. We also thank Jonas Iskander for many helpful discussions. The research was supported by the generosity of the National Science Foundation under grant DMS-2002265, the National Security Agency under grant H98230-20-1-0012, the Templeton World Charity Foundation, and the Thomas Jefferson Fund at the University of Virginia.

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Correspondence to Letong Hong.

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Hong, L., Zhang, S. Towards Heim and Neuhauser’s unimodality conjecture on the Nekrasov–Okounkov polynomials. Res. number theory 7, 17 (2021).

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