Deformations of Coxeter Hyperplane Arrangements

We investigate several hyperplane arrangements that can be viewed as deformations of Coxeter arrangements. In particular, we prove a conjecture of Linial and Stanley that the number of regions of the arrangement xi−xj=1, 1⩽i<j⩽n, is equal to the number of alternating trees on n+1 vertices. Remarkably, these numbers have several additional combinatorial interpretations in terms of binary trees, partially ordered sets, and tournaments. More generally, we give formulae for the number of regions and the Poincaré polynomial of certain finite subarrangements of the affine Coxeter arrangement of type An−1. These formulae enable us to prove a “Riemann hypothesis” on the location of zeros of the Poincaré polynomial. We give asymptotics of the Poincaré polynomials when n goes to infinity. We also consider some generic deformations of Coxeter arrangements of type An−1.