Fifty or so years ago Tamari defined a lattice, now known by his name. It has Catalan number of vertices — its Hasse diagram, as an undirected graph, is the one-skeleton of the associahedron — and it is a quotient of the weak Bruhat order. Much is known about this remarkable lattice, but, as Knuth says, the number of maximal chains "remains mysterious". This is especially odd, because the numbers of maximal chains in many closely related lattices are known. I will describe recent and current work, joint with my student Luke Nelson, on enumerating maximum length chains. We hope it is progress toward finding the number of maximal chains.
