Tamari lattices and noncrossing partitions in type B and D

The usual, or type An, Tamari lattice is a partial order on T A n , the triangulations of an (n+3)-gon. We define a partial order on T B n , the set of centrally symmetric triangulations of a (2n + 2)-gon. We show that it is a lattice, and that it shares certain other nice properties of the An Tamari lattice, and therefore that it deserves to be considered the Bn Tamari lattice. We define a bijection between T B n and the non-crossing partitions of type Bn defined by Reiner. Reiner has also defined the noncrossing partitions of type Dn as a subset of those of type Bn. We show that the elements of T B n which correspond to the noncrossing partitions of type Dn form a lattice under the order induced from their inclusion in T B n , which therefore deserves to be called the Dn Tamari lattice.