The lecture presents generalizations of the classical Catalan and Narayana
numbers in the context of Weyl groups and several combinatorial
interpretations of them. The first interpretation is through the regions
of a certain deformation of the Coxeter arrangement and leads to a nice product
formula for the generalized Catalan numbers in terms of the exponents and
the Coxeter number. Other interpretations involve certain chains of ideals
in the root poset, the action of the Weyl group on a certain quotient of
the coroot lattice, f-vectors and h-vectors of generalized associahedra and
the lattice of noncrossing partitions. Several open problems are discussed.

Keywords:

Hyperplane arrangements; Weyl groups; coexeter arrangements; Catalan number; Narayana number; root poset; ideals; chains of ideals; coroot lattice; noncrossing partitions.

Lecture #10727

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