The poset of bipartitions

TL;DR

This paper studies the structure of bipartitional relations ordered by inclusion, revealing it forms a graded lattice with topological properties and computing its Möbius function using discrete Morse theory.

Contribution

It characterizes the lattice structure, homotopy type, and Möbius function of bipartitional relations, applying advanced topological and combinatorial methods.

Findings

The bipartitional relations form a graded lattice of rank 3n-2.

The order complex is homotopy equivalent to an (n-2)-sphere.

Möbius function values are always 0, 1, or -1.

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 M\"obius 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.