Recursive constructions for the higher Stasheff—Tamari orders in dimension three using the Outer Tamari and Tamari Block posets

Abstract

We define a poset called the Outer Tamari poset, and show it is isomorphic to a subposet of the Tamari lattice introduced by Pallo (1986), and studied further and called the Comb poset by Csar, Sengupta, and Suksompong (2014). We use the Outer Tamari poset to develop recursive formulas for the number of triangulations of the 3-dimensional cyclic polytopes. These triangulations can be viewed as elements of both the higher Stasheff–Tamari orders in dimension three and the Tamari Block lattices we defined in a previous article. So our work here can also be seen as constructing recursive enumerations of these posets.