An Alternate Proof for the Top-Heavy Conjecture on Partition Lattices Using Shellability

A partially ordered set, or poset, is governed by an ordering that may or may not relateany pair of objects in the set. Both the bonds of a graph and the partitions of a set arepartially ordered, and their poset structure can be depicted visually in a Hasse diagram. Thepartitions of {1, 2, ..., n} form a particularly important poset known as the partition latticeΠn. It is isomorphic to the bond lattice of the complete graph Kn, making it a special caseof the family of bond lattices.Dowling and Wilson’s 1975 Top-Heavy Conjecture states that every bond lattice has atleast as many elements in its upper half as in its lower half. The existing proof of thisconjecture by Huh et al. in 2017 relies heavily on algebraic geometry. In this paper, weprovide an alternate combinatorial proof for the Top-Heavy Conjecture on partition lattices.To do this, we define a specific class of forests on n vertices and construct an abstractsimplicial complex ∆n out of the edge sets of these graphs. Then, we show that ∆n is ashellable complex for all n, and we use this result to prove that Πn is a top-heavy lattice.