Unimodality Problems in Ehrhart Theory

Ehrhart theory is the study of sequences recording the number of integer points in non-negative integral dilates of rational polytopes. For a given lattice polytope, this sequence is encoded in a finite vector called the Ehrhart $h^*$-vector. Ehrhart $h^*$-vectors have connections to many areas of mathematics, including commutative algebra and enumerative combinatorics. In this survey we discuss what is known about unimodality for Ehrhart $h^*$-vectors and highlight open questions and problems.Comment: Published in Recent Trends in Combinatorics, Beveridge, A., et al. (eds), Springer, 2016, pp 687-711, doi 10.1007/978-3-319-24298-9_27. This version updated October 2017 to correct an error in the original versio

Lattice Point Generating Functions and Symmetric Cones

We show that a recent identity of Beck-Gessel-Lee-Savage on the generating function of symmetrically contrained compositions of integers generalizes naturally to a family of convex polyhedral cones that are invariant under the action of a finite reflection group. We obtain general expressions for the multivariate generating functions of such cones, and work out the specific cases of a symmetry group of type A (previously known) and types B and D (new). We obtain several applications of the special cases in type B, including identities involving permutation statistics and lecture hall partitions.Comment: 19 page

Polyhedral geometry for lecture hall partitions

Lecture hall partitions are a fundamental combinatorial structure which have been studied extensively over the past two decades. These objects have produced new results, as well as reinterpretations and generalizations of classicial results, which are of interest in combinatorial number theory, enumerative combinatorics, and convex geometry. In a recent survey of Savage \cite{Savage-LHP-Survey}, a wide variety of these results are nicely presented. However, since the publication of this survey, there have been many new developments related to the polyhedral geometry and Ehrhart theory arising from lecture hall partitions. Subsequently, in this survey article, we focus exclusively on the polyhedral geometric results in the theory of lecture hall partitions in an effort to showcase these new developments. In particular, we highlight results on lecture hall cones, lecture hall simplices, and lecture hall order polytopes. We conclude with an extensive list of open problems and conjectures in this area.Comment: 20 pages; To appear in to proceedings of the 2018 Summer Workshop on Lattice Polytopes at Osaka Universit