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ON NONCROSSING AND NONNESTING PARTITIONS FOR CLASSICAL REFLECTION GROUPS
 

Summary: ON NONCROSSING AND NONNESTING PARTITIONS FOR
CLASSICAL REFLECTION GROUPS
CHRISTOS A. ATHANASIADIS
Abstract. The number of noncrossing partitions of {1, 2, . . . , n} with fixed block
sizes has a simple closed form, given by Kreweras, and coincides with the corre-
sponding number for nonnesting partitions. We show that a similar statement is
true for the analogues of such partitions for root systems B and C, defined recently
by Reiner in the noncrossing case and Postnikov in the nonnesting case. Some of
our tools come from the theory of hyperplane arrangements.
Submitted: January 30, 1998; Accepted: September 10, 1998
1. Introduction
A noncrossing partition of the set [n] = {1, 2, . . . , n} is a set partition of [n] such
that if a < b < c < d and a, c are contained in a block B of , while b, d are contained
in a block B of , then B = B . Noncrossing partitions are classical combinatorial
objects with an extensive literature, see [7, 9, 11, 12, 13, 17, 18, 19, 22]. Natural
analogues of noncrossing partitions for the classical reflection groups of type B, C
and D were introduced by Reiner [16] and were shown to have similar enumerative and
structural properties with those of the noncrossing partitions, which are associated
to the reflection groups of type A.
Nonnesting partitions were recently defined by Postnikov (see [16, Remark 2]) in

  

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

 

Collections: Mathematics