 
Summary: TRANSACTIONS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 357, Number 1, Pages 179196
S 00029947(04)035482
Article electronically published on March 23, 2004
ON A REFINEMENT OF THE GENERALIZED CATALAN
NUMBERS FOR WEYL GROUPS
CHRISTOS A. ATHANASIADIS
Abstract. Let be an irreducible crystallographic root system with Weyl
group W , coroot lattice Q and Coxeter number h, spanning a Euclidean space
V , and let m be a positive integer. It is known that the set of regions into
which the fundamental chamber of W is dissected by the hyperplanes in V
of the form (, x) = k for and k = 1, 2, . . . , m is equinumerous to the
set of orbits of the action of W on the quotient Q/ (mh + 1) Q. A bijection
between these two sets, as well as a bijection to the set of certain chains of order
ideals in the root poset of , are described and are shown to preserve certain
natural statistics on these sets. The number of elements of these sets and
their corresponding refinements generalize the classical Catalan and Narayana
numbers, which occur in the special case m = 1 and = An1.
1. Introduction and results
