TY - JOUR

T1 - Noncrossing partitions for the group D n

AU - Athanasiadis, Christos A.

AU - Reiner, Victor

PY - 2004/10

Y1 - 2004/10

N2 - 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.

AB - 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.

KW - Antichain

KW - Catalan number

KW - Garside structure

KW - Narayana numbers

KW - Noncrossing partition

KW - Nonnesting partition

KW - Reflection group

KW - Root poset

KW - Type D

UR - http://www.scopus.com/inward/record.url?scp=18844445398&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=18844445398&partnerID=8YFLogxK

U2 - 10.1137/S0895480103432192

DO - 10.1137/S0895480103432192

M3 - Article

AN - SCOPUS:18844445398

VL - 18

SP - 397

EP - 417

JO - SIAM Journal on Discrete Mathematics

JF - SIAM Journal on Discrete Mathematics

SN - 0895-4801

IS - 2

ER -