Combinatorial number theory through diagramming and gesture

Within combinatorial number theory, we study a variety of problems about whole numbers that include enumerative, diagrammatic, or computational elements. We present results motivated by two different areas within combinatorial number theory: the study of partitions and the study of digital representations of integers. We take the perspective that mathematics research is mathematics learning; existing research from mathematics education on mathematics learning and problem solving can be applied to mathematics research. We illustrate this by focusing on the concept of diagramming and gesture as mathematical practice. The mathematics presented is viewed through this lens throughout the document. Joint with H. E. Burson and A. Straub, motivated by recent results working toward classifying $(s, t)$-core partitions into distinct parts, we present results on certain abaci diagrams. We give a recurrence (on $s$) for generating polynomials for $s$-core abaci diagrams with spacing $d$ and maximum position strictly less than $ms-r$ for positive integers $s$, $d$, $m$, and $r$. In the case $r =1$, this implies a recurrence for $(s, ms-1)$-core partitions into $d$-distinct parts, generalizing several recent results. We introduce the sets $Q(b;\{d_1, d_2, \ldots, d_k\})$ to be integers that can be represented as quotients of integers that can be written in base $b$ using only digits from the set $\{d_1, \ldots, d_k\}$. We explore in detail the sets $Q(b;\{d_1, d_2, \ldots, d_k\})$ where $d_1 = 0$ and the remaining digits form proper subsets of the set $\{1, 2, \ldots, b-1\}$ for the cases $b =3$, $b=4$ and $b=5$. We introduce modified multiplication transducers as a computational tool for studying these sets. We conclude with discussion of $Q(b; \{d_1, \ldots d_k\})$ for general $b$ and digit sets including $\{-1, 0, 1\}$. Sections of this dissertation are written for a nontraditional audience (outside of the academic mathematics research community)

On integer partitions corresponding to numerical semigroups

Numerical semigroups are cofinite additive submonoids of the natural numbers. In 2011, Keith and Nath illustrated an injection from numerical semigroups to integer partitions. We explore this connection between partitions and numerical semigroups with a focus on classifying the partitions that appear in the image of the injection from numerical semigroups. In particular, we count the number of partitions that correspond to numerical semigroups in terms of genus, Frobenius number, and multiplicity, with some restrictions

Representations of integers as quotients of sums of distinct powers of three

Which integers can be written as a quotient of sums of distinct powers of three? We outline our first steps toward an answer to this question, beginning with a necessary and almost sufficient condition. Then we discuss an algorithm that indicates whether it is possible to represent a given integer as a quotient of sums of distinct powers of three. When the given integer is representable, this same algorithm generates all possible representations. We develop a categorization of representations based on their connections to $0,1$-polynomials and give a complete description of the types of representations for all integers up to 364. Finally, we discuss in detail the representations of 7, 22, 34, 64, and 100, as well as some infinite families of integers