Abstract. We define noncrossing partitions of a marked surface without punctures (interior marked points). We show that the natural partial order on noncrossing partitions is a graded lattice and describe its rank function topologically. Lower intervals in the lattice are isomorphic to products of noncrossing partition lattices of other surfaces. We similarly define noncrossing partitions of a symmetric marked surface with double points and prove some of the analogous results. The combination of symmetry and double points plays a role that one might have expected to be played by punctures.

We consider two problems that appear at first sight to be unrelated. The first problem is to count certain diagrams consisting of noncrossing arcs in the plane. The second problem concerns the weak order on the symmetric group. Each permutation $x$ has a canonical join representation: a unique lowest set of permutations joining to $x$. The second problem is to determine which sets of permutations appear as canonical join representations. The two problems turn out to be closely related because the noncrossing arc diagrams provide a combinatorial model for canonical join representations. The same considerations apply to more generally to lattice quotients of the weak order. Considering quotients produces, for example, a new combinatorial object counted by the Baxter numbers and an analogous new object in bijection with generic rectangulations.

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.