Theses and Dissertations

Date of Award

5-1-2026

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics

First Advisor

Brandt Kronholm

Second Advisor

Timothy Huber

Third Advisor

Josef Sifuentes

Abstract

In 2007, Kronholm established The Interval Theorem, infinite families of congruences in arithmetic progression, modulo any prime ��, for ��(��, ��), the function enumerating the partitions of �� into parts whose sizes come from the set {1, 2, … , ��}. In 2022, Eichhorn, Kronholm, and Larsen proved there are combinatorial statistics described in terms of the multiplicities of the part sizes that witness Kronholm’s Interval Theorem. Here, “witness" means given a congruence of the form ��(��, ��) ≡ 0 (mod ��), we can use these statistics to classify the set of partitions of �� into �� equally sized subsets by directly inspecting the partitions themselves based on that statistic.

It is known that the sequence of ��(��, ��) is periodic with period ��������(��), where ������(��) is the least common multiple of numbers in the set {1, 2, … , ��}. The congruences in The Interval Theorem happen exclusively at the end of the period. In this dissertation we consider the partition numbers that are not part of “The Interval Theorem" and prove the same multiplicity based statistics (MB statistics) that witness The Interval Theorem also witness an infinite family of congruences of the form

��(��, ��) ± ��(��′, ��) ≡ 0 (mod ��), where ��′ is determined by �� and neither �� nor ��′ need be at the end of any period.

Eichhorn, Kronholm, and Larsen considered the difference of two generating functions and showed it reduced to a polynomial thereby establishing the combinatorial witnesses for The Interval Theorem. In order to prove the same MB statistics witness this congruence, we do a close analysis of the same polynomial. Moreover, this result coupled with The Interval Theorem’s MB statistics allows us to resolve a conjecture of theirs on a different infinite family of partition congruences discovered by Kronholm in 2005.

Prior to this proof, there was little evidence that a combinatorial witness might exist for this congruence. Python programs were created to produce large amounts of data that allowed us to make conjectures that the same MB statistics that witnessed Kronholm’s Interval Theorem also witnessed this congruence. We conclude with a brief combinatorial discussion of why certain MB statistics witness congruences and others don’t.

Comments

Copyright 2026 Jena Gregory. All Rights Reserved. https://proquest.com/docview/3371202703

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