Tamari lattices and noncrossing partitions in type B

The usual, or type AnAn, Tamari lattice is a partial order on TnA, the triangulations of an (n+3)(n+3)-gon. We define a partial order on TnB, the set of centrally symmetric triangulations of a (2n+2)(2n+2)-gon. We show that it is a lattice, and that it shares certain other nice properties of the AnAn Tamari lattice, and therefore that it deserves to be considered the BnBn Tamari lattice. We also define a bijection between TnB and the noncrossing partitions of type BnBn defined by Reiner.