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

Cited by in corpus (1)

• Noncrossing partitions and the shard intersection order