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)