Noncrossing partitions for the group D n

Christos A. Athanasiadis, Victor Reiner

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54 Scopus citations

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 languageEnglish (US)
Pages (from-to)397-417
Number of pages21
JournalSIAM Journal on Discrete Mathematics
Volume18
Issue number2
DOIs
StatePublished - Oct 2004

Keywords

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

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