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

Abstract

Abstract Let

$$Q_n(z)$$

Q n

( z )

be the polynomials associated with the Nekrasov–Okounkov formula

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

∑

n ≥ 1

Q n

( z )

q n

: =

∏

m = 1

∞

( 1 -

q m

)

- z - 1

.

In this paper we partially answer a conjecture of Heim and Neuhauser, which asks if

$$Q_n(z)$$

Q n

( z )

is unimodal, or stronger, log-concave for all

$$n \ge 1$$

n ≥ 1

. Through a new recursive formula, we show that if

$$A_{n,k}$$

A

n , k

is the coefficient of

$$z^k$$

z k

in

$$Q_n(z)$$

Q n

( z )

, then

$$A_{n,k}$$

A

n , k

is log-concave in k for

$$k \ll n^{1/6}/\log n$$

k ≪

n

1 / 6

/ log n

and monotonically decreasing for

$$k \gg \sqrt{n}\log n$$

k ≫

n

log n

. We also propose a conjecture that can potentially close the gap.