College
College of Engineering and Polymer Science
Date of Last Revision
2026-05-01 13:56:05
Major
Applied Mathematics
Honors Course
MATH 498
Number of Credits
2
Degree Name
Bachelor of Science
Date of Expected Graduation
Spring 2026
Abstract
In this paper, we study Catalan numbers and their generalization, hyper-Catalan numbers, and explore how these sequences arise naturally in the context of solving polynomial equations using infinite power series. We begin by introducing the Catalan numbers through their combinatorial interpretation as triangulations of convex polygons. Using this geometric definition, we derive a relation whose recursive structure leads to a quadratic functional equation. Interpreting this relation as a formal power series equation allows us to express solutions to quadratic equations as infinite power series whose coefficients are given by the Catalan numbers. This framework is then extended by allowing polygon dissections into larger polygonal faces (such as quadrilaterals and pentagons), leading to the hyper-Catalan numbers. We define the corresponding generating function and determine an explicit formula for the hyper-Catalan numbers and demonstrate how the resulting generating series provides formal power series solutions to polynomial equations of arbitrary degree.
Research Sponsor
James Cossey
First Reader
Jeffrey Riedl
Second Reader
Stefan Forcey
Honors Faculty Advisor
James Cossey
Proprietary and/or Confidential Information
No
Community Engaged Scholarship
No
Recommended Citation
Natale, Leilani, "Catalan and Hyper-Catalan Numbers: Combinatorial Applications to Polynomial Equations" (2026). Williams Honors College, Honors Research Projects. 2158.
https://ideaexchange.uakron.edu/honors_research_projects/2158
Included in
Algebra Commons, Algebraic Geometry Commons, Analysis Commons, Discrete Mathematics and Combinatorics Commons, Other Mathematics Commons