Discrete Mathematics and Combinatorics Commons^™

Facets Of The Union-Closed Polytope, Daniel Gallagher

Doctoral Dissertations

In the haze of the 1970s, a conjecture was born to unknown parentage...the union-closed sets conjecture. Given a family of sets $\FF$, we say that $\FF$ is union-closed if for every two sets $S, T \in \FF$, we have $S \cup T \in \FF$. The union-closed sets conjecture states that there is an element in at least half of the sets of any (non-empty) union-closed family. In 2016, Pulaj, Raymond, and Theis reinterpreted the conjecture as an optimization problem that could be formulated as an integer program. This thesis is concerned with the study of the polytope formed by taking …

An Inquiry Into Lorentzian Polynomials, Tomás Aguilar-Fraga

HMC Senior Theses

In combinatorics, it is often desirable to show that a sequence is unimodal. One method of establishing this is by proving the stronger yet easier-to-prove condition of being log-concave, or even ultra-log-concave. In 2019, Petter Brändén and June Huh introduced the concept of Lorentzian polynomials, an exciting new tool which can help show that ultra-log-concavity holds in specific cases. My thesis investigates these Lorentzian polynomials, asking in which situations they are broadly useful. It covers topics such as matroid theory, discrete convexity, and Mason’s conjecture, a long-standing open problem in matroid theory. In addition, we discuss interesting applications to known …

Parking Garage Functions, Felicia Elizabeth Flores

Senior Projects Spring 2023

Senior Project submitted to The Division of Science, Mathematics and Computing of Bard College.

This project is about a generalization of parking functions called parking garage functions. Parking functions have been well studied, but the concept of parking garage functions is new and introduced in the project. Parking garage functions are sequences that represent the parking garage level preferences of cars which lead to all cars parking on a level after a systematic placement. We found a recursive formula for the number of sequences that are a parking garage function. We also found a closed formula for a subset of …

Minimal Sets, Union-Closed Families, And Frankl's Conjecture, Christopher S. Flippen

Theses and Dissertations

The most common statement of Frankl's conjecture is that for every finite family of sets closed under the union operation, there is some element which belongs to at least half of the sets in the family. Despite its apparent simplicity, Frankl's conjecture has remained open and highly researched since its first mention in 1979. In this paper, we begin by examining the history and previous attempts at solving the conjecture. Using these previous ideas, we introduce the concepts of minimal sets and minimally-generated families, some ideas related to viewing union-closed families as posets, and some constructions of families involving poset-defined …

Counting Spanning Trees On Triangular Lattices, Angie Wang

CMC Senior Theses

This thesis focuses on finding spanning tree counts for triangular lattices and other planar graphs comprised of triangular faces. This topic has applications in redistricting: many proposed algorithmic methods for detecting gerrymandering involve spanning trees, and graphs representing states/regions are often triangulated. First, we present and prove Kirchhoff’s Matrix Tree Theorem, a well known formula for computing the number of spanning trees of a multigraph. Then, we use combinatorial methods to find spanning tree counts for chains of triangles and 3 × n triangular lattices (some limiting formulas exist, but they rely on higher level mathematics). For a chain of …