On the enumeration of positive cells in generalized cluster complexes and Catalan hyperplane arrangements

@article{Athanasiadis2006OnTE,
  title={On the enumeration of positive cells in generalized cluster complexes and Catalan hyperplane arrangements},
  author={Christos A. Athanasiadis and Eleni Tzanaki},
  journal={Journal of Algebraic Combinatorics},
  year={2006},
  volume={23},
  pages={355-375}
}
Let Φ be an irreducible crystallographic root system with Weyl group W and coroot lattice $$\check{Q}$$, spanning a Euclidean space V. Let m be a positive integer and $${\mathcal A}^{m}_{\Phi}$$ be the arrangement of hyperplanes in V of the form $$(\alpha, x) = k$$ for $$\alpha \in \Phi$$ and $$k = 0, 1,\dots,m$$. It is known that the number $$N^+ (\Phi, m)$$ of bounded dominant regions of $${\mathcal A}^{m}_{\Phi}$$ is equal to the number of facets of the positive part $$\Delta^m_+ (\Phi)$$ of… 
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