Simply construed, combinatorics entails the  counting and classifying of finite objects. Such objects vary from permutations and graphs, to posets and matroids; they have in common the idea of a device that represents discrete data. Enumerative combinatorics deals with the exact or asymptotic enumeration of this data. Rook theory is the enumerative study of non-attacking rook placements on a board and matroid theory is the combinatorial abstraction of the notion of linear independence in linear algebra. In this dissertation, we will describe a new connection between rook theory and matroid theory, and study some consequences of this connection. In a different direction, we will build new tools to study the combinatorial features of the set of all points equidistant from two fixed points, where the notion of distance comes from a polytope. Across this work, the following three perspectives are employed to understand combinatorial data: (a) discrete and convex geometric objects like polyhedra and polyhedral~complexes; (b) bijective methods; and (c) the geometry of polynomials.  

This thesis is divided into two parts. The first part consists of Paper A on the geometric combinatorial theory of bisectors. A polyhedral norm is a notion of distance arising from centrally symmetric polytopes; they have found application in modelling problems in areas ranging from algebraic statistics, topological data analysis, robotics, and crystallography.  The bisector associated to any polyhedral norm is a polyhedral complex whose maximal cells are labeled by pairs of facets of the polytope. We identify precisely which maximal cells are present, and in doing so, systematize the study of the bisection fan of a polytope, a tool that captures fundamental combinatorial information about the structure of the bisector. We focus on four fundamental notions of distance ---polygonal norms, || · ||1 and || · ||∞ norms, and the discrete Wasserstein norm. In each of these cases, we give an explicit combinatorial description of the bisection fan, thereby extending work of Criado, Joswig, and Santos (2022). 

The second part of this thesis --- spanned by Papers B,C, and D --- is concerned with novel enumerative and matroidal properties of rook placements. In particular, in Paper B, we introduce and define a new matroid whose bases are the set of non-nesting rook placements on a skew Ferrers board; this establishes the first known bridge between rook theory and matroid theory. The structure of rook matroids is interesting: they form a subclass of both transversal matroids and positroids. In this regard, and through the skew shape association, rook matroids bear a striking resemblance to lattice path matroids. We explore this connection in depth by (a) proving a precise criterion for when a rook matroid and lattice path matroid are isomorphic; (b) proving that despite the failure of isomorphism in general, rook matroids and lattice path matroids have the same Tutte polynomial; and (c) proving that every lattice path matroid is a certain contraction of a rook matroids, thereby obtaining a new perspective on results of de Mier--Bonin--Noy (2004) and Oh (2011). 

We then apply this matroid structure to two enumerative problems that bear no relation to one another at first glance. The first is determining the precise distributional property satisfied by the generating polynomial of the set of non-nesting rook placements on a skew shape. This question is motivated by the famous Heilmann--Lieb theorem on the real-rootedness of the matching polynomial. In contrast to the case of unrestricted rook placements, the polynomial in question satisfies ultra-log-concavity, but not real-rootedness. The second enumerative problem concerns the log-concavity consequence of the Neggers--Stanley conjecture. The (P, ω)-Eulerian polynomial  --- the descent-generating polynomial of the set of (inverses of) linear extensions of a labeled poset (P, ω) --- was introduced by Stanley (1972) in his PhD thesis and was conjectured to be real-rooted, first by Neggers (1978) and later by Stanley (1986) himself. It was eventually disproven, in one formulation by Brändén (2005) and in another by Stembridge (2007). The natural follow-up question, also a conjecture of Brenti (1989), is whether the (P, ω)-Eulerian polynomial is log-concave in general. We provide an affirmative answer to this conjecture in the case when (P, ω) is a naturally labeled poset of width two, by combining two ideas: a classical bijection known in the theory of distributive lattices, and the Brändén--Huh theory of Lorentzian polynomials.  

In Paper D, the positroidal structure of non-nesting rook placements is explored in greater depth. Some consequences include a new proof of the positroid structure and an inequality description of the base polytope of the rook matroid using only the combinatorics of the underlying skew shape. Answering a question of Lam (2024), we characterize rook matroids from the positroidal point of view. 

In Paper C, we establish the real-rootedness of the generating polynomial of complete rook placements on Ferrers boards, enumerated by the number of ascents. This set of rook placements is interesting in connection with the natural poset--theoretic structure that it has: the lower interval of 312-avoiding permutations in the Bruhat order. This polynomial also represents another yet another generalization of the classical Eulerian polynomials; it is similar to but distinct from the generalization introduced by Savage and Visontai (2015).