Term of Award
Summer 2024
Degree Name
Master of Science in Mathematics (M.S.)
Document Type and Release Option
Thesis (open access)
Copyright Statement / License for Reuse
This work is licensed under a Creative Commons Attribution 4.0 License.
Department
Department of Mathematical Sciences
Committee Chair
Tricia Brown
Committee Member 1
Andrew Sills
Committee Member 2
Daniel Gray
Abstract
Lattice path enumeration, through the lens of Catalan numbers, plays a crucial role in combinatorics. This thesis delves into enumerations of some of the most common lattice paths – north-east paths, up-down paths, and Dyck paths – with restrictions applied. The first restriction is counting north-east lattice paths that only cross the diagonal line, y=x, once. The second form of lattice paths with restrictions is up-down paths that cross the x-axis exactly once and fall to a fixed depth of k. While working through this module, a novel proof for a known integer sequence was used, then applied to generate a formula to produce a new sequence. These results are demonstrated through visual examples and rigorous proofs, utilizing mathematical induction. This exploration provides a comprehensive understanding of the Catalan numbers and their applications in enumerating restricted lattice paths, laying the groundwork for further combinatorial investigations.
Recommended Citation
White, Vince, "Enumeration of Lattice Paths with Restrictions" (2024). Electronic Theses and Dissertations. 2799.
https://digitalcommons.georgiasouthern.edu/etd/2799
Research Data and Supplementary Material
No