This material has been published in Europ. J. Combin. 32 (2011), 1253-1281, the only definitive repository of the content that has been certified and accepted after peer review. Copyright and all rights therein are retained by Elsevier B.V. This material may not be copied or reposted without explicit permission.

Gábor Hetyei and Christian Krattenthaler

The poset of bipartitions

(36 pages)

Abstract. Bipartitional relations were introduced by Foata and Zeilberger in their characterization of relations which give rise to equidistribution of the associated inversion statistic and major index. We consider the natural partial order on bipartitional relations given by inclusion. We show that, with respect to this partial order, the bipartitional relations on a set of size n form a graded lattice of rank 3n-2. Moreover, we prove that the order complex of this lattice is homotopy equivalent to a sphere of dimension n-2. Each proper interval in this lattice has either a contractible order complex, or it is isomorphic to the direct product of Boolean lattices and smaller lattices of bipartitional relations. As a consequence, we obtain that the Möbius function of every interval is 0, 1, or -1. The main tool in the proofs is discrete Morse theory as developed by Forman, and an application of this theory to order complexes of graded posets, designed by Babson and Hersh, in the extended form of Hersh and Welker.

The following versions are available:

Back to Christian Krattenthaler's home page.