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

Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 License.


Department of Mathematical Sciences

Committee Chair

Tricia Brown

Committee Member 1

Andrew Sills

Committee Member 2

Daniel Gray


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.

Research Data and Supplementary Material