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Rational Catalan Combinatorics


Given a finite Coxeter group W and a Coxeter element c, the W-noncrossing partitions are given by [1,c], the interval between 1 and c in W under the absolute order. When W is the symmetric group S_a, the noncrossing partitions turn out to be classical noncrossing partitions of [a] counted by the Catalan numbers. By attaching an additional integral paramenter b which is coprime to a, we define a set NC(a,b) of rational noncrossing partitions, a subset of the ordinary noncrossing partitions of [b-1]. We study the poset structure this set inherits from the poset of classical noncrossing partitions, ordered by refinement. We prove that NC(a,b) is closed under a dihedral action and that the rotation action on NC(a,b) exhibits the cyclic sieving phenomenon. We also generalize noncrossing parking functions to the rational setting and provide a character formula for the action of S_a X Z_{b-1} on a,b-noncrossing parking functions. Finally, we give a group-theoretic interpretation in type A for NC(a,b) in terms of compatible sequences.

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