Roots of descent polynomials and an algebraic inequality on hook lengths (Q6117262)

Summary: By reinterpreting the descent polynomial as a function enumerating standard Young tableaux of a ribbon shape, we use Naruse's hook-length formula to express the descent polynomial as a product of two polynomials: one is a trivial part which is a product of linear factors, and the other comes from the excitation factor of Naruse's formula. We expand the excitation factor positively in a Newton basis which arises naturally from Naruse's formula. Under this expansion, each coefficient is the weight of a certain combinatorial object, which we introduce in this paper. We introduce and prove the ``Slice and Push Inequality'', which compares the weights of such combinatorial objects. As a consequence, we establish a proof of a conjecture by \textit{A. Diaz-Lopez} et al. [Discrete Math. 342, No. 6, 1674--1686 (2019; Zbl 1414.05009)] that bounds the roots of descent polynomials.