Bulletin of the London Mathematical Society



Papers

GENERALIZED CATALAN NUMBERS, WEYL GROUPS AND ARRANGEMENTS OF HYPERPLANES


CHRISTOS A. ATHANASIADIS a1
a1 Department of Mathematics, University of Crete, 71409 Heraklion, Crete, Greece caa@math.uoc.gr

Article author query
athanasiadis c   [Google Scholar] 
 

Abstract

For an irreducible, crystallographic root system $\Phi$ in a Euclidean space $V$ and a positive integer $m$, the arrangement of hyperplanes in $V$ given by the affine equations $(\alpha, x)\,{=}\,k$, for $\alpha\,{\in}\,\Phi$ and $k\,{=}\,0, 1,\dots,m$, is denoted here by ${\mathcal A}_{\Phi}^m$. The characteristic polynomial of ${\mathcal A}_{\Phi}^m$ is related in the paper to that of the Coxeter arrangement ${\mathcal A}_{\Phi}$ (corresponding to $m\,{=}\,0$), and the number of regions into which the fundamental chamber of ${\mathcal A}_{\Phi}$ is dissected by the hyperplanes of ${\mathcal A}_{\Phi}^m$ is deduced to be equal to the product $\prod_{i=1}^{\ell} ({e_i\,{+}\,m h\,{+}\,1})/({e_i\,{+}\,1})$, where $e_1, e_2,\dots,e_\ell$ are the exponents of $\Phi$ 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 $\Phi$ are included.

(Received November 26 2002)
(Revised June 16 2003)

Maths Classification

20F55 (primary); 05A15; 52C35 (secondary).