New combinatorial descriptions of the triangulations of cyclic polytopes and the second higher Stasheff--Tamari posets
type: Item
instance of: scholarly article
title: New combinatorial descriptions of the triangulations of cyclic polytopes and the second higher Stasheff--Tamari posets
zbMATH Open document ID: 1026.52009
DOI: 10.1023/A:1022851226643
author: Hugh Thomas
published in: Order
publication date: 2003-06-09T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 52B05
Mathematics Subject Classification ID: 52B12
Mathematics Subject Classification ID: 06A07
zbMATH DE Number: 1924894
zbMATH Keywords: cyclic polytopes
zbMATH Keywords: triangulations
zbMATH Keywords: higher Stasheff-Tamari posets
zbMATH Keywords: 2-dimension
reviewed by: Johann Linhart
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1924894