Title:  The lattice of bipartitions

Abstract:  Bipartitional relations were introduced by Foata and Zeilberger, who showed these are precisely the relations which give rise to equidistribution of the associated inversion statistic and major index. In this talk we consider the natural partial order on bipartitional relations given by inclusion, and prove that the Möbius function of each of is intervals is 0, 1, or -1. To achieve this goal we will explore the topology of the order complex. We will see that bipartitional relations on a set of size n form a graded lattice of rank 3n-2. 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. The main tool in the proofs of these results 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.

This is joint work with Christian Krattenthaler.