Derangements, Ehrhart Theory, and Local h-polynomials

The Eulerian polynomials and derangement polynomials are two well-studied generating functions that frequently arise in combinatorics, algebra, and geometry. When one makes an appearance, the other often does so as well, and their corresponding generalizations are similarly linked. This is this case in the theory of subdivisions of simplicial complexes, where the Eulerian polynomial is an $h$-polynomial and the derangement polynomial is its local $h$-polynomial. Separately, in Ehrhart theory the Eulerian polynomials are generalized by the $h^\ast$-polynomials of $s$-lecture hall simplices. Here, we show that derangement polynomials are analogously generalized by the box polynomials, or local $h^\ast$-polynomials, of the $s$-lecture hall simplices, and that these polynomials are all real-rooted. We then connect the two theories by showing that the local $h$-polynomials of common subdivisions in algebra and topology are realized as local $h^\ast$-polynomials of $s$-lecture hall simplices. We use this connection to address some open questions on real-rootedness and unimodality of generating polynomials, some from each side of the story.