Tamari lattices and noncrossing partitions in type B and beyond

Abstract

The usual, or type A_n, Tamari lattice is a partial order on T_n^A, the triangulations of an (n+3)-gon. We define a partial order on T_n^B, 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 A_n Tamari lattice, and therefore that it deserves to be considered the B_n Tamari lattice. We define a bijection between T_n^B and the non-crossing partitions of type B_n defined by Reiner. For S any subset of [n], Reiner defined a pseudo-type BD^S_n, to which is associated a subset of the noncrossing partitions of type B_n. We show that the elements of T^B_n which correspond to the noncrossing partitions of type BD^S_n posess a lattice structure induced from their inclusion in T^B_n.