Tamari lattices and noncrossing partitions in type B and beyond

الأصل · EN

The usual, or type Aₙ, Tamari lattice is a partial order on Tₙᵃ, the triangulations of an (n+3)-gon. We define a partial order on Tₙᵇ, 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 Tₙᵇ and the non-crossing partitions of type Bₙ defined by Reiner. For S any subset of [n], Reiner defined a pseudo-type BDˢₙ, to which is associated a subset of the noncrossing partitions of type Bₙ. We show that the elements of Tᵇₙ which correspond to the noncrossing partitions of type BDˢₙ posess a lattice structure induced from their inclusion in Tᵇₙ.