This dissertation presents recent contributions to Ehrhart theory and its applications in Combinatorics. It investigates the enumeration and structure of integer points subject to linear inequalities from a geometric perspective.
We give an introduction in Chapter 1 and background on polyhedral geometry and combinatorial structures used in this work in Chapter 2.
In Chapter 3 we use Ehrhart polynomials to count combinatorial and geometric data in generalized permutahedra and hypergraphs. Generalized permutahedra are a class of polytopes with many interesting combinatorial subclasses. We introduce pruned inside-out polytopes, a generalization of inside-out polytopes introduced by Beck–Zaslavsky (2006), which have many applications such as recovering the famous reciprocity result for graph colorings by Stanley. We study the integer point count of pruned inside-out polytopes by applying classical Ehrhart polynomials and Ehrhart–Macdonald reciprocity. This yields a geometric perspective on and a generalization of a combinatorial reciprocity theorem for generalized permutahedra by Aguiar–Ardila (2023), Billera–Jia–Reiner (2009), and Karaboghossian (2022). Applying this reciprocity theorem to hypergraphic polytopes allows us to give a geometric proof of a combinatorial reciprocity theorem for hypergraph colorings by Aval–Karaboghossian–Tanasa (2020). Aside from the reciprocity for generalized permutahedra, this proof relies only on elementary geometric and combinatorial properties of hypergraphs and their associated polytopes.
In Chapter 4, which is joint work with Eleonore Bach and Matthias Beck, we investigate the coefficients of the Ehrhart polynomial for special classes of zonotopes associated with signed graphs. There is a well-established dictionary between zonotopes, hyperplane arrangements, and (oriented) matroids. Arguably one of the most famous examples is the class of graphical zonotopes, also called acyclotopes, which encode subzonotopes of the type-A root polytope, the permutahedron. Stanley gave a general interpretation of the coefficients of the Ehrhart polynomial (integer-point counting function for a polytope) of a zonotope via linearly independent subsets of its generators. Applying this to the graphical case shows that Ehrhart coefficients count (labeled) forests of the graph of fixed sizes. Our first goal is to extend and popularize this story to other root systems, which on the combinatorial side is encoded by signed graphs analogously to the work by Greene–Zaslavsky (1983). We compute the Ehrhart polynomial of the acyclotope in the signed case, and we give a matroid-dual construction. This gives rise to tocyclotopes and we compute their Ehrhart polynomials. Applying the same duality construction to a general integral matrix leads to a lattice Gale zonotope, whose face structure was studied by McMullen (1971). We describe the Ehrhart polynomials of lattice Gale zonotopes in terms of the given matrix.
Chapter 5 is joint work with Matthias Beck and Sophia Elia. Here, we extend Ehrhart theory to consider rational dilates of polytopes. The Ehrhart quasipolynomial of a rational polytope P encodes fundamental arithmetic data of P, namthe number of integer lattice points in positive integral dilates of P. The enumerative theory of lattice points in rational (equivalently, real) dilates of rational polytopes is much younger, starting with work by Linke (2011), Baldoni–Berline–Köppe–Vergne (2013), and Stapledon (2017). We introduce a generating-function ansatz for rational Ehrhart quasipolynomials, which unifies several known results in classical and rational Ehrhart theory. In particular, we define γ-rational Gorenstein polytopes, which extend the classical notion to the rational setting and encompass the generalized reflexive polytopes studied by Fiset–Kasprzyk (2008) and Kasprzyk–Nill (2012).
In Chapter 6, which is joint work with Alexander E. Black and Raman Sanyal we study poset permutahedra, an interesting class of polytopes arising as monotone path polytopes of order polytopes. Poset permutahedra are an amalgamation of order polytopes and permutahedra. We show that poset permutahedra give a unifying perspective on several recent classes of polytopes that occurred, for example, in connection with colorful subdivisions of polygons and Hessenberg varieties. As with order polytopes, the geometry and the combinatorics of poset permutahedra can be completely described in terms of the underlying poset. As applications of our results, we give a combinatorial description of the h-vectors of the partitioned permutahedra of Horiguchi–Masuda–Shareshian–Song (2024) and poset generalizations of Landau’s score sequences of tournaments. To prove our results, we show that poset permutahedra arise from order polytopes via the fiber polytope construction of Billera–Sturmfels (1992).
