Summary: European Journal of Combinatorics 28 (2007) 12081215
On some enumerative aspects of generalized
Christos A. Athanasiadis1
Department of Mathematics, University of Crete, 71409 Heraklion, Crete, Greece
Received 15 August 2005; accepted 5 February 2006
Available online 3 March 2006
We prove a conjecture of F. Chapoton relating certain enumerative invariants of (a) the cluster complex
associated by S. Fomin and A. Zelevinsky with a finite root system and (b) the lattice of noncrossing
partitions associated with the corresponding finite real reflection group.
c 2006 Elsevier Ltd. All rights reserved.
1. The result
Let be a finite root system spanning an n-dimensional Euclidean space V with
corresponding finite reflection group W. Let + be a positive system for with corresponding
simple system . The cluster complex () was introduced by Fomin and Zelevinsky within
the context of their theory of cluster algebras . It is a pure (n - 1)-dimensional
simplicial complex on the vertex set + (- ) which is homeomorphic to a sphere .
Although () was initially defined under the assumption that is crystallographic , its