On conjectures regarding the Nekrasov--Okounkov hook length formula

TL;DR

This paper investigates three conjectures related to the Nekrasov--Okounkov hook length formula, proving one, refuting another with a counterexample, and confirming the third's unimodality up to degree 1000.

Contribution

It proves a refined Nekrasov--Okounkov formula, provides a counterexample to a conjecture on polynomial roots, and confirms unimodality for polynomials up to degree 1000.

Findings

Proof of the refined Nekrasov--Okounkov formula.

Counterexample to the conjecture on roots of D'Arcais polynomials.

Confirmation of unimodality up to degree 1000.

Abstract

The Nekrasov--Okounkov hook length formula provides a fundamental link between the theory of partitions and the coefficients of powers of the Dedekind eta function. In this paper we examine three conjectures presented by Amdeberhan. The first conjecture is a refined Nekrasov--Okounkov formula involving hooks with trivial legs. We prove the conjecture. The second conjecture is on properties of the roots of the underlying D'Arcais polynomials. We give a counterexample and present a new conjecture. The third conjecture is on the unimodality of the coefficients of the involved polynomials. We confirm the conjecture up to the polynomial degree $1000$.