Chains in the noncrossing partition lattice

TL;DR

This paper develops recursive formulas to count chains in the noncrossing partition lattice of finite Coxeter groups, providing new uniform formulas and extending results to m-divisible cases.

Contribution

It introduces a uniform recursive approach to count chains in noncrossing partition lattices across all finite Coxeter groups, simplifying proofs and extending to m-divisible cases.

Findings

Derived recursive formulas for counting chains in noncrossing partition lattices.

Provided a simpler proof for the known formula of maximal chains.

Established a new uniform formula for the number of edges in the 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.