UC Davis

We investigate the face structure and Ehrhart polynomials of several families of polytopes that have been central to the author’s research journey throughout graduate study, including generalized type A and type B permutohedra, hive polytopes associated with Littlewood–Richardson coefficients, and parking function polytopes. In Chapter 1, we present essential background on the foundational concepts that underpin the results of this dissertation. This includes an introduction to polytopes, their face structures and normal fans, as well as an overview of Ehrhart theory, which concerns the enumeration of lattice points in rational polytopes. Later in the chapter, we introduce type A generalized and type B permutohedra, and provide existing formulas for their Ehrhart polynomials. These core concepts and definitions provide the groundwork for the more specialized constructions and results developed in the chapters that follow.In Chapter 2, we derive a formula for the Ehrhart polynomials of type B generalized permutohedra, providing a concise alternative to a recent formula obtained by Eur, Fink, Larson, and Spink from a study of delta-matroids. Our approach builds on the techniques and tools introduced by Postnikov around two decades ago in his work on type A generalized permutohedra—a rich family of polytopes deeply connected to many mathematical concepts such as matroids, graphs, and Weyl groups. Our approach suggests that many combinatorial models originally developed for type A generalized permutohedra may be used directly or suitably adapted to investigate the combinatoral properties of their type B analogues. We conclude the chapter by proposing several questions for future work on how the combinatorial properties of type B generalized permutohedra could be deduced from those of their type A counterparts.In Chapter 3, we provide an alternative proof of the conjecture by King, Tollu, and Toumazet that stretched Littlewood-Richardson coefficient ctνtλ,tµ is a polynomial function in t. Note that the conjecture was shown to be true by Derksen and Weyman using semi-invariants of quivers. Later, Rassart used Steinberg’s formula, the hive conditions, and the Kostant partition function to show a stronger result that cνλ,µ is indeed a polynomial in variables ν, λ, µ provided they lie in certain polyhedral cones. Motivated by Rassart’s approach, we give a short alternative proof of the polynomiality of ctνtλ,tµ using Steinberg’s formula and a simple argument about the chamber complex of the Kostant partition function. The main idea of our proof uses the hive conditions to realize ctνtλ,tµ as the number of lattice points in the t-dilation of a rational polytope, which implies, by Ehrhart theory, that ctνtλ,tµ is a quasi-polynomial. We then employ Steinberg’s formula and the chamber complex of the Kostant partition function to deduce that ctνtλ,tµ is a polynomial for sufficiently large t, and hence must be a polynomial for t ≥ 0. Additionally, we discuss a connection to flow polytopes and outline potential research problems for future work regarding the coefficients of the polynomial ctνtλ,tµ.In Chapter 4, we extended the concept of parking function polytopes and investigated their normal fans, face posets, h-polynomials, and connections to other families of polytopes. This extension broadens the family to encompass several additional combinatorial types of polytopes. To describe their face structures and normal fans, we introduced generalized forms of ordered set partitions, which we refer to as binary partitions and skewed binary partitions. Using properties of preorder cones, we developed tools to characterize the family of skewed binary partitions that correspond bijectively to the normal fan–and thus the face poset–of a parking function polytope. This framework leads to several related findings, including the insight that the combinatorial type of a parking function polytope depends solely on its defining multiplicity vector, and a characterization of simple parking function polytopes.Later in the chapter, we present a formula for the h-polynomials of simple parking function polytopes in terms of generalized Eulerian polynomials, and further refine it to express the h-polynomials as sums of products of classical Eulerian polynomials. At the end of the chapter, we discuss how parking function polytopes can be realized as polymatroids, type B generalized permutohedra, and type A generalized permutohedra. These connections allow us to derive formulas for the volume and Ehrhart polynomials of parking function polytopes.