  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.
