Towards Heim and Neuhauser’s unimodality conjecture on the Nekrasov–Okounkov polynomials
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Author(s)
Hong, Letong
Zhang, Shengtong
Date Issued
February 19, 2021
Publisher
Springer International Publishing
Citation
Research in Number Theory. 2021 Feb 19;7(1):17
Version
Author's final manuscript
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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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DOI of Published Version
https://doi.org/10.1007/s40993-021-00244-2
