Deformations of Coxeter hyperplane arrangements

Abstract

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 x_i - x_j = 1, 1 \leq i<j \leq 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 Poincar&#39;e polynomial of certain finite subarrangements of the affine Coxeter arrangement of type A_{n-1}. These formulae enable us to prove a &#34;Riemann hypothesis&#34; on the location of zeros of the Poincar&#39;e polynomial. We also consider some generic deformations of Coxeter arrangements of type A_{n-1}.