Chains in the lattice of noncrossing partitions

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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.

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This work was carried out in part while R.S. was visiting the Institute for Mathematics and its Applications and was partly supported through NSF Grant CCR-8707539.

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Supported in part by NSF Grant DMS-8700995.