Congruences in Arithmetic Progression for Coefficients of Gaussian Polynomials and Crank Statistics

Author

Joselyne Aniceto, The University of Texas Rio Grande Valley

Date of Award

5-2025

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics

First Advisor

Brandt Kronholm

Second Advisor

Nicolas Smoot

Third Advisor

Timothy Huber

Abstract

The study of partition congruences, inspired by Ramanujan’s discoveries for ��(��) over a century ago, remains a central topic in this field. This dissertation examines congruence properties in two restricted partition functions: ��(��,��), which counts partitions of �� into at most �� parts, and ��(��,��,��), which further limits the size of the largest part to be at most ��. Building on Kronholm’s 2007 result, now known as the Interval Theorem, and a recent result by Eichhorn, Engle, and Kronholm, we establish new infinite families of congruences for ��(��,��,��). This dissertation extends not only the recent results of Eichhorn, Engle, and Kronholm, but also reveals deeper and unexpected aspects of Kronholm’s 2007 results. A key component in our approach is a two-colored partition function which leads to additional congruences.

We also investigate combinatorial witnesses, otherwise known as cranks for these congruences. We establish polynomial formulas for certain partition functions which are then used to test whether certain statistics evenly distribute partitions among residue classes. These results expand and reinforce a conjecture of Eichhorn, Engle, and Kronholm on cranks.

Comments

Copyright 2025 Joselyne Aniceto. https://proquest.com/docview/3240611341

Recommended Citation

Aniceto, J. (2025). Congruences in Arithmetic Progression for Coefficients of Gaussian Polynomials and Crank Statistics [Doctoral dissertation, The University of Texas Rio Grande Valley]. ScholarWorks @ UTRGV. https://scholarworks.utrgv.edu/etd/1717