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NONCROSSING PARTITIONS FOR THE GROUP Dn CHRISTOS A. ATHANASIADIS AND VICTOR REINER
 

Summary: NONCROSSING PARTITIONS FOR THE GROUP Dn

CHRISTOS A. ATHANASIADIS AND VICTOR REINER
SIAM J. DISCRETE MATH. c 2004 Society for Industrial and Applied Mathematics
Vol. 18, No. 2, pp. 397­417
Dedicated to the memory of Rodica Simion
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 Sn, 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 Dn, 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 M¨obius 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 (case-
by-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.
Key words. noncrossing partition, nonnesting partition, reflection group, root poset, antichain,

  

Source: Athanasiadis, Christos - Department of Mathematics, University of Athens

 

Collections: Mathematics