Date of Award
2025
Document Type
Open Access Dissertation
Degree Name
Doctor of Philosophy in Mathematical Sciences (PhD)
Administrative Home Department
Department of Mathematical Sciences
Advisor 1
William Keith
Committee Member 1
Brandt Kronholm
Committee Member 2
Robert Schneider
Committee Member 3
Fabrizio Zanello
Abstract
Identities of integer partitions generally state that two dissimilar appearing families of partitions are in fact equinumerous when both are restricted to any fixed size. Euler's theorem is a classic example of such an identity, which equates the number of partitions with odd parts to the number of partitions with distinct parts. Lately, analogs of known partition identities involving weights other than size have begun to attract research interest. This dissertation is an investigation of two such weights. In Chapter 2, we study Schmidt weights, which count only parts with indices belonging to some given subset of the positive integers. This is motivated by recent work of Andrews and Keith, which produced a highly general partition identity involving Schmidt weights that allowed new q-series sum-product identities to be discovered, including an identity that is a generating function for 2-colored partitions. We further generalize their partition identity, and use a special case of this new result to derive a companion to their 2-colored identity for overpartitions. Then, in Chapter 3 we prove an infinite family of q-series identities which generalizes the 2-colored identity to t colors, for any t at least 2. Finally in Chapter 4, we study the partition perimeter, which is defined to be one less than the sum of the largest part and number of parts. A purely combinatorial proof for a perimeter analog of a classic identity of Glaisher recently found by Amdeberhan, Andrews, and Ballantine is given. We use similar methods to prove a generalization of a perimeter identity of Fu and Tang as well.
Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 License.
Recommended Citation
Waldron, Hunter, "The Combinatorics of Integer Partitions Enumerated by some Exotic Weights", Open Access Dissertation, Michigan Technological University, 2025.