Catalan triangulations of the Möbius band

A Catalan triangulation of the Möbius band is an abstract simplicial complex triangulating the Möbius band which uses no interior vertices, and has vertices labelled 1, 2, ..., n in order as one traverses the boundary. We prove two results about the structure of this set, analogous to well-known results for Catalan triangulations of the disk. The first is a generating function for Catalan triangulations of M having n vertices, and the second is that any two such triangulations are connected by a sequence of diagonal-flips.