## Abstract

The poset of noncrossing partitions can be naturally defined for any finite Coxeter group W. It is a self-dual, graded lattice which reduces to the classical lattice of noncrossing partitions of {1,2,n} defined by Kreweras in 1972 when W is the symmetric group S _{n}, and to its type B analogue defined by the second author in 1997 when W is the hyperoctahedral group. We give a combinatorial description of this lattice in terms of noncrossing planar graphs in the case of the Coxeter group of type D _{n}, thus answering a question of Bessis. Using this description, we compute a number of fundamental enumerative invariants of this lattice, such as the rank sizes, number of maximal chains, and Möbius function. We also extend to the type D case the statement that noncrossing partitions are equidistributed to nonnesting partitions by block sizes, previously known for types A, B, and C. This leads to a (caseby-case) proof of a theorem valid for all root systems: the noncrossing and nonnesting subspaces within the intersection lattice of the Coxeter hyperplane arrangement have the same distribution according to W-orbits.

Original language | English (US) |
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Pages (from-to) | 397-417 |

Number of pages | 21 |

Journal | SIAM Journal on Discrete Mathematics |

Volume | 18 |

Issue number | 2 |

DOIs | |

State | Published - Oct 2004 |

## Keywords

- Antichain
- Catalan number
- Garside structure
- Narayana numbers
- Noncrossing partition
- Nonnesting partition
- Reflection group
- Root poset
- Type D

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