New Combinatorial Descriptions of the Triangulations of Cyclic Polytopes and the Second Higher Stasheff–Tamari Posets

This paper is concerned with the d-dimensional cyclic polytope with n vertices, C(n,d), and the set of its triangulations, S(n,d). We show that there is a bijection between S(n,d) and certain partitions of the set of increasing d-tuples on the integers 1 to n–1. We give a combinatorial characterization of the second higher Stasheff–Tamari poset, which is a partial ordering of S(n,d), and we determine its 2-dimension. There is a well-known representation of triangulations of an n-gon by right bracket vectors. We generalize this to cyclic polytopes of higher dimensions.