Roots of descent polynomials and an algebraic inequality on hook lengths

TL;DR

This paper proves a conjecture about the roots of descent polynomials by establishing an algebraic inequality related to hook lengths, advancing understanding of combinatorial polynomial roots.

Contribution

It introduces the 'Slice and Push Inequality' and applies it to bound the roots of descent polynomials, connecting algebraic inequalities with combinatorial formulas.

Findings

Bounded the roots of descent polynomials.

Established the 'Slice and Push Inequality'.

Linked hook-length formulas to root bounds.

Abstract

We prove a conjecture by Diaz-Lopez et al. that bounds the roots of descent polynomials. To do so, we prove an algebraic inequality, which we refer to as the "Slice and Push Inequality." This inequality compares expressions that come from Naruse's hook-length formula for the number of standard Young tableaux of a skew shape.