Chains in the lattice of noncrossing partitions

Abstract

The lattice of noncrossing set partitions is known to admit an R-labeling. Under this labeling, maximal chains give rise to permutations. We discuss structural and enumerative properties of the lattice of noncrossing partitions, which pertain to a new permutation statistic, m(σ), defined as the number of maximal chains labeled by σ. Möbius inversion results and related facts about the lattice of unrestricted set partitions are also presented.