Partition Identities Using Modular Young Diagrams and Congruences Among Fractional Partition Functions
Open Access
Author:
Bevilacqua, Erin
Area of Honors:
Mathematics
Degree:
Bachelor of Science
Document Type:
Thesis
Thesis Supervisors:
Ae Ja Yee, Thesis Supervisor Leonid N Vaserstein, Thesis Honors Advisor
Keywords:
Partitions combinatorics number theory
Abstract:
In this work we look at the properties of the partition function in two different lights. In
the second chapter, we look at proving partition identities using modular Young diagrams, which
give each block a weight. We prove a result which generalizes a famous identity of Euler, and
look to generalize further for multiple residue classes modulus 3. We also introduce the Lecture
Hall Theorem and how modular diagrams may be applicable to this result. In the third chapter,
we use results about generating functions and modular forms to look at the fractional partition
functions, which are a generalization of the usual partition function. We prove results about `-
balanced congruences for the fractional partition function, finding conditions that guarantee the
existance of these congruences. Furthermore, we look for necessary conditions, showing cases
where congruences cannot exist.
