Derangements, Ehrhart Theory, and Local h-polynomials

TL;DR

This paper explores the deep connections between derangement polynomials, Ehrhart theory, and local h-polynomials, revealing new real-rootedness results and unifying concepts across combinatorics, algebra, and geometry.

Contribution

It establishes that derangement polynomials are generalized by local h*-polynomials of s-lecture hall simplices and links local h-polynomials of subdivisions to these polynomials.

Findings

Derangement polynomials are generalized by local h*-polynomials of s-lecture hall simplices.

All these polynomials are shown to be real-rooted.

Connections are made between local h-polynomials and local h*-polynomials, addressing open questions.

Abstract

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.