## Dissertations, Master's Theses and Master's Reports

2021

#### Document Type

Open Access Dissertation

#### Degree Name

Doctor of Philosophy in Mathematical Sciences (PhD)

Department of Mathematical Sciences

Fabrizio Zanello

William J. Keith

#### Committee Member 3

Bryan J. Freyberg

#### Abstract

This dissertation focuses on problems related to integer partitions under various finiteness restrictions. Much of our work involves the collection of partitions fitting inside a fixed partition $\lambda$, and the associated generating function $G_{\lambda}$.

In Chapter 2, we discuss the flawlessness of such generating functions, as proved by Pouzet using the Multicolor Theorem. We give novel applications of the Multicolor Theorem to re-prove flawlessness of pure $O$-sequences, and show original flawlessness results for other combinatorial sequences. We also present a linear-algebraic generalization of the Multicolor Theorem that may have far-reaching applications.

In Chapter 3, we extend a technique due to Stanton to prove unimodality of $G_\lambda$ for certain infinite families of partitions $\lambda$ in 5 and 6 parts.

Our most substantial work is presented in Chapter 4, where we initiate the study of the novel poset $\ds{P_n=\{G_\lambda\,|\,\lambda\vdash n\}}$. We describe some general structural properties of this poset. Of greatest significance is our result that two balancing" operations on the principal hooks of a partition $\lambda$ produce generating functions at least as large as $G_{\lambda}$ (in the ordering of $P_n$), hereby imposing a strong necessary condition on the maxima of $P_n$. We conjecture an asymptotic value of $|P_n|$, and show that determining $|P_n|$ exactly appears to be nontrivial. This we demonstrate by providing an infinite family of non-conjugate pairs of partitions that have the same generating function. Finally, we prove asymptotic results on the number of maxima in this poset.

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