Journal of Computational and Applied Mathematics
Linear inequalities for flags in graded partially ordered sets
Journal of Combinatorial Theory Series A
Communications of the ACM
Lexicographic Shellability for Balanced Complexes
Journal of Algebraic Combinatorics: An International Journal
The Möbius function of a composition poset
Journal of Algebraic Combinatorics: An International Journal
Enumeration in Convex Geometries and Associated Polytopal Subdivisions of Spheres
Discrete & Computational Geometry
Enumerative Combinatorics: Volume 1
Enumerative Combinatorics: Volume 1
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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 is isomorphic to the direct product of Boolean lattices and smaller lattices of bipartitional relations. As a consequence, we obtain that the Mobius 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.