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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,
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