Chains in the noncrossing partition lattice

Abstract

We establish recursions counting various classes of chains in the noncrossing partition lattice of a finite Coxeter group. The recursions specialize a general relation which is proven uniformly (i.e. without appealing to the classification of finite Coxeter groups) using basic facts about noncrossing partitions. We solve these recursions for each finite Coxeter group in the classification. Among other results, we obtain a simpler proof of a known uniform formula for the number of maximal chains of noncrossing partitions and a new uniform formula for the number of edges in the noncrossing partition lattice. All of our results extend to the m-divisible noncrossing partition lattice.

Version 2: Several expository changes made, including changes in the abstract, thanks to helpful suggestions from several readers of Version 1. (See the Acknowledgments section of the paper.) Version 3: Minor changes to belatedly bring the arXiv version more into line with the last pre-publication version

References in corpus (6)

• Cluster algebras II: Finite type classification
• Clusters, Coxeter-sortable elements and noncrossing partitions
• Generalized cluster complexes and Coxeter combinatorics
• Enumerative properties of generalized associahedra
• Lattices in finite real reflection groups
• Faces of generalized cluster complexes and noncrossing partitions

Cited by in corpus (13)

• Noncrossing partitions and the shard intersection order
• Counting factorizations of Coxeter elements into products of reflections
• Orbites d'Hurwitz des factorisations primitives d'un élément de Coxeter
• Counting chains in the noncrossing partition lattice via the W-Laplacian
• Doppelgängers: Bijections of Plane Partitions
• Decomposition numbers for finite Coxeter groups and generalised non-crossing partitions
• Symmetric Decompositions and the Strong Sperner Property for Noncrossing Partition Lattices
• Tamari Lattices for Parabolic Quotients of the Symmetric Group
• Groupes de réflexion, géométrie du discriminant et partitions non-croisées
• Lyashko-Looijenga morphisms and submaximal factorisations of a Coxeter element
• Refined enumeration of noncrossing chains and hook formulas
• Lyashko-Looijenga morphisms and primitive factorizations of the Coxeter element
• A generalization of Euler numbers to finite Coxeter groups