Box Polynomials of Lattice Simplices

Abstract: The box polynomial of a lattice simplex is a variant of the more well-known h∗-polynomial, where the open fundamental parallelepiped is considered instead of the half-open. Box polynomials are connected to h∗-polynomials by a theorem of Betke and McMullen from 1985. This theorem can be used to prove certain properties of h∗-polynomials, such as unimodality and symmetry. In this thesis, we investigate box polynomials of a certain family of simplices, called s-lecture hall simplices. The h∗-polynomials of these simplices are a generalization of Eulerian polynomials, and were proven to be real-rooted by Savage and Visontai in 2015. We use a modified version of their proof to prove that the box polynomials are also real-rooted, and show that they are a generalization of derangement polynomials. We then use these results to partially answer a conjecture by Brändén and Leander regarding unimodality of h∗-polynomials of s-lecture hall order polytopes.