Date of Completion
5-3-2024
Degree Type
Honors Thesis
Discipline
Mathematics (MATH)
First Advisor
Josh Hallam
Abstract
A partially ordered set, or poset, is governed by an ordering that may or may not relate
any pair of objects in the set. Both the bonds of a graph and the partitions of a set are
partially ordered, and their poset structure can be depicted visually in a Hasse diagram. The
partitions of {1, 2, ..., n} form a particularly important poset known as the partition lattice
Πn. It is isomorphic to the bond lattice of the complete graph Kn, making it a special case
of the family of bond lattices.
Dowling and Wilson’s 1975 Top-Heavy Conjecture states that every bond lattice has at
least as many elements in its upper half as in its lower half. The existing proof of this
conjecture by Huh et al. in 2017 relies heavily on algebraic geometry. In this paper, we
provide 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 abstract
simplicial complex ∆n out of the edge sets of these graphs. Then, we show that ∆n is a
shellable complex for all n, and we use this result to prove that Πn is a top-heavy lattice.
Recommended Citation
Macdonald, Brian and Hallam, Josh, "An Alternate Proof for the Top-Heavy Conjecture on Partition Lattices Using Shellability" (2024). Honors Thesis. 505.
https://digitalcommons.lmu.edu/honors-thesis/505
Cover sheet for my thesis
