 
Summary: J Algebr Comb (2006) 23: 355375
DOI 10.1007/s1080100683488
On the enumeration of positive cells in generalized cluster
complexes and Catalan hyperplane arrangements
Christos A. Athanasiadis · Eleni Tzanaki
Received: April 5, 2005 / Revised: October 3, 2005 / Accepted: October 12, 2005
C Springer Science + Business Media, LLC 2006
Abstract Let be an irreducible crystallographic root system with Weyl group W and
coroot lattice Q, spanning a Euclidean space V . Let m be a positive integer and Am
be the
arrangement of hyperplanes in V of the form (, x) = k for and k = 0, 1, . . . , m. It
is known that the number N+
( , m) of bounded dominant regions of Am
is equal to the
number of facets of the positive part m
+( ) of the generalized cluster complex associated to
the pair ( , m) by S. Fomin and N. Reading.
We define a statistic on the set of bounded dominant regions of Am
and conjecture that the
corresponding refinement of N+
