New combinatorial descriptions of the triangulations of cyclic polytopes and the second higher Stasheff--Tamari posets
Let \(C(n,d)\) be a cyclic \(d\)-polytope with \(n\) vertices, and \(S(n,d)\) the set of its triangulations. The author describes a bijection between \(S(n,d)\) and certain partitions of the set of increasing \(d\)-tuples on the integers \(1\) to \(n-1\). This yields a new combinatorial characterization of the triangulations of \(C(n,d)\). However, the main result of the paper is a combinatorial characterization of a certain natural partial order on \(S(n,d)\), namely the second higher Stasheff-Tamari poset defined by \textit{P. Edelman} and \textit{V. Reiner} [Mathematika 43, 127-154 (1996; Zbl 0854.06003)]. There is a well-known representation of triangulations of an \(n\)-gon by right bracket vectors. This is generalized to cyclic polytopes of higher dimensions and leads to the determination of the \(2\)-dimension of \(S(n,d)\) with regard to the above mentioned partial order.
scientific article; zbMATH DE number 1924894