 
Summary: Bull. London Math. Soc. 36 (2004) 294302 Ce2004 London Mathematical Society
DOI: 10.1112/S0024609303002856
GENERALIZED CATALAN NUMBERS, WEYL GROUPS AND
ARRANGEMENTS OF HYPERPLANES
CHRISTOS A. ATHANASIADIS
Abstract
For an irreducible, crystallographic root system in a Euclidean space V and a positive integer
m, the arrangement of hyperplanes in V given by the affine equations (, x) = k, for and
k = 0, 1, . . . , m, is denoted here by Am
. The characteristic polynomial of Am
is related in the paper
to that of the Coxeter arrangement A (corresponding to m = 0), and the number of regions into
which the fundamental chamber of A is dissected by the hyperplanes of Am
is deduced to be
equal to the product i=1(ei + mh + 1)/(ei + 1), where e1, e2, . . . , e are the exponents of and h
is the Coxeter number. A similar formula for the number of bounded regions follows. Applications
to the enumeration of antichains in the root poset of are included.
1. Introduction and results
Let V be an dimensional Euclidean space, with inner product ( , ), and let be
an irreducible, crystallographic root system [12, Section 2.9] spanning V . For a
