  We establish recursions counting various classes of chains in the noncrossing
partition lattice of a finite Coxeter group. The recursions specialize a
general relation which is proven uniformly (i.e. without appealing to the
classification of finite Coxeter groups) using basic facts about noncrossing
partitions. We solve these recursions for each finite Coxeter group in the
classification. Among other results, we obtain a simpler proof of a known
uniform formula for the number of maximal chains of noncrossing partitions and
a new uniform formula for the number of edges in the noncrossing partition
lattice. All of our results extend to the m-divisible noncrossing partition
lattice.
