Abstract
Many combinatorial problems can be expressed in terms of the theory of hyper- plane arrangements, which relates the geometry of collections of hyperplanes in a vector space to the combinatorial data about how those hyperplanes intersect. In a seminal paper in 2000, Postnikov and Stanley studied the combinatorics of sev- eral families of hyperplane arrangements; among these was the Linial arrangement. Postnikov and Stanley related the combinatorics of the Linial arrangement to the combinatorics of a certain class of tournaments. Recently, Hetyei introduced a new hyperplane arrangement called the homogenized Linial arrangement to study a vari- ant of the tournaments considered by Postnikov and Stanley. This dissertation is structured around the further study of the homogenized Linial arrangements and several related combinatorial objects. In the first part of this dissertation, we study the homogenized Linial arrangement. Hetyei showed that the number of regions of the homogenized Linial arrangement is a median Genocchi number. The Genocchi numbers count a class of permutations known as Dumont permutations, and the median Genocchi numbers count the de- rangements among the Dumont permutations. We refine Hetyei’s result by providing a combinatorial interpretation for the coefficients of the characteristic polynomial of the intersection lattice of this arrangement. Moreover, we show that the Möbius in- variant of this lattice is a (nonmedian) Genocchi number. We show that the signless coefficents of the characteristic polynomial count Dumont-like permutations with a given number of cycles. This allows us to derive generating function formulas for the characteristic polynomial, which reduce to known generating functions for the median and nonmedian Genocchi numbers. As a byproduct of this work, we obtain new models for the Genocchi and median Genocchi numbers. In the second part of this dissertation, we generalize the results of the first part of the dissertation to the study of a new complex hyperplane arrangement that we call the homogenized Linial–Dowling arrangement. The intersection lattice of homoge- nized Linial–Dowling arrangement is a subposet of the Dowling Lattice for the cyclic group of order m, and the homogenized Linial–Dowling arrangement specializes to complexification of the homogenized Linial arrangement when m = 1. We compute the coefficients of the characteristic polynomial of this arrangement in terms of dec- orated versions of the permutations considered in the first part of this dissertation. This allows us to compute generating function formulas for the characteristic poly- nomial of the intersection lattice that reduce to the generating functions from the homogenized Linial arrangement. In the third part of this dissertation, we study a different generalization of the homogenized Linial arrangement that we call the homogenized ν-arrangement. These new arrangements are indexed by integer compositions, and specialize to the homoge- nized Linial arrangement when ν = (1,1,...,1). During our study of the homogenized Linial arrangement, we show that the intersection lattice of the homogenized Linial arrangement is isomorphic to the bond lattice of a certain Ferrers graph. Ferrers graphs are a family of graphs introduced by Ehrenborg and van Willigenburg that are indexed by integer partitions. We show that the bond lattice of any Ferrers graph is isomorphic to the intersection lattice of a homogenized ν-arrangement. Moreover, by generalizing some of the enumerative results from our study of the Dumont-like permutations, we are able to compute generating function formulas for the character- istic polynomials of the intersection lattice of certain infinite families of homogenized ν-arrangements