  In the combinatorics of finite finite Coxeter groups, there is a simple
formula giving the number of maximal chains of noncrossing partitions. It is a
reinterpretation of a result by Deligne which is due to Chapoton, and the goal
of this article is to refine the formula. First, we prove a one-parameter
generalization, by the considering enumeration of noncrossing chains where we
put a weight on some relations. Second, we consider an equivalence relation on
noncrossing chains coming from the natural action of the group on set
partitions, and we show that each equivalence class has a simple generating
function. Using this we recover Postnikov's hook length formula in type A and
obtain a variant in type B.

Combinatorics