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C 2020 by Simone Claire Sisneros-Thiry. All Rights Reserved. COMBINATORIAL NUMBER THEORY THROUGH DIAGRAMMING and GESTURE

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C 2020 by Simone Claire Sisneros-Thiry. All Rights Reserved. COMBINATORIAL NUMBER THEORY THROUGH DIAGRAMMING and GESTURE c 2020 by Simone Claire Sisneros-Thiry. All rights reserved. COMBINATORIAL NUMBER THEORY THROUGH DIAGRAMMING AND GESTURE BY SIMONE CLAIRE SISNEROS-THIRY DISSERTATION Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics in the Graduate College of the University of Illinois at Urbana-Champaign, 2020 Urbana, Illinois Doctoral Committee: Professor Bruce Berndt, Chair Professor Bruce Reznick, Director of Research Professor Matthew Ando Professor Rochelle Guti´errez Abstract 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; fd1; d2; : : : ; dkg) to be integers that can be represented as quotients of integers that can be written in base b using only digits from the set fd1; : : : ; dkg. We explore in detail the sets Q(b; fd1; d2; : : : ; dkg) where d1 = 0 and the remaining digits form proper subsets of the set f1; 2; : : : ; b − 1g 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; fd1; : : : dkg) for general b and digit sets including {−1; 0; 1g. Sections of this dissertation are written for a nontraditional audience (outside of the academic mathematics research community). ii To my students, past, present, and future, who inspire me to keep learning and growing as a mathematician. iii Acknowledgments This dissertation would not have been possible without the love, support, and encouragement of so many of my friends, family, and teachers. You are my village. First and foremost, I want to thank my advisor, Bruce Reznick. From the very beginning, you encouraged me to make the thesis my own and gave me the courage to step outside the box. None of this would have been possible without your guidance, reassurance, and unwavering support. To the members of my committee: Matt Ando, thank you for your boundless enthusiasm, for seeing the value in my process, and for asking essential questions about audience. Bruce Berndt, thank you for sharing stories and pictures, for introducing me to the fascinating world of partitions, and for connecting me to the welcoming community of mathematicians who love partitions. Rochelle Guti´errez,thank you for creating space for me to explore diagramming and gesture, for pushing me to think hard about context, and for always grounding me in what is most important. To the University of Maine at Farmington math department, especially Paul Gies, Lori Koban, and Nic Koban, thank you for building my confidence, curiosity, and commitment to both mathematics and math education. To the MSRI-UP 2013 community, especially Ive Rubio and Candice Price, thank you for making me believe that I belong in mathematics and for teaching me the value of mentoring and collaboration. Piper Harron, your thesis opened the door for me to dream of something different. Thank you for leading the way. Aditya Adiredja, thank you for helping me feel less alone in this work, and for hard advice at critical moments. Dennis Eichhorn, thank you for so many joyful and playful mathematical conversations. Armin Straub, thank you for sharing ideas and collaboration. To the Green Street Band, and especially my academic siblings Katie, Dana, and Gracie, thank you for listening, commiserating, and for words of encouragement. To my students, especially the Fall 2017 Math 347 Merit workshop and 2018-2019 IGL Outreach Team, thank you for letting me be a part of your mathematical journeys. Education Justice Project students and outside members, thank you for the intentionality of community building, intellectual engagement, and care. iv Thank you especially to the EJP Calculus sequence students for creating the most vibrant, curious, and inclusive mathematical community I have ever had the honor to be a part of. I have been so supported by so many friends within and outside the math community. Lauren Breton, thank you for being confident in my success and proud of my achievements, from the very beginning. Hannah Burson, I am so grateful to have had you alongside me as a collaborator and friend every step of the way. Cara Monical, thank you for for feelings, encouragement, and early morning breakfast dates. Jennifer McNeilly, thank you for being my role model, for accountability and solidarity, and for making a space for math education in our math department. Amelia Tebbe and Sarah Loeb, thank you for welcoming me into Altgeld 150, for mentoring and reassurance in the early years. Liz Tatum, thank you for late nights, Zoom study sessions, and listening to my rants. Marissa Loving and Vanessa Rivera Qui~nones,seeing you shine just a few steps ahead made all of the difference. Marissa, thank you for your dedication to mathematics and justice, and for pushing me to see myself as a mathematician. Vanessa, thank you for seeing all of me, for being my biggest cheerleader, and for advocating for my well being above all else. My family has been a constant source of support and encouragement, giving me the strength and confidence to pursue this degree. To my nieces and nephews, Ella, Andres, Kate, and Marcelo, thank you for playing math games with me, for cooking with me, and for sharing your enthusiasm. To my sister Danielle Thiry- Zaragoza, thank you for taking me with your family on adventures, for teaching me to prioritize fun in a busy life, and for your unwavering confidence. To my sister Monique Thiry-Zaragoza, thank you for hearing my feelings, for showing up for me in so many ways, and for growing with me. To my parents, thank you for valuing so many ways of thinking, learning, and knowing, and for instilling that value in me. To my dad, Pierre Thiry, thank you for your relentless optimism, joy in learning, and motivation to give time, energy and enthusiasm. To my mom Estella Sisneros, thank you for filling my life with art, for encouraging me to be creative, to feel deeply, and keep striving to be and do better. Thank you both for celebrating my achievements, for wiping away my tears, for everything. The best parts of who I am I get from you. Much love and thanks go to Indy the dog, Calliope the cat, and my many plants, who remind me to take joy in the little things every day, to sleep, to stretch, to play, and to get enough sun and water. To my partner, Derrek Yager, thank you for building a life with me that is full of warmth, learning, and growth. Thank you for prioritizing my finishing, for encouraging me to take risks, and for choosing me and us, over and over. You make me smile every day. Thank you for making a home with me, for loving me, and for taking such good care of our little family. Everything I do is better because of you. v I am grateful for financial support from the University of Illinois Graduate College. The work in this dissertation was funded by a Graduate College Fellowship and a Dissertation Completion Fellowship. I also want to thank the Illinois Counseling Center for creating the space, time, and structure of the Graduate Women's General Therapy Group. Members of that group (past and present) and the community it provided were an essential support network throughout this process. vi Table of Contents List of Figures . ix Notation . xi Chapter 1 Introduction . 1 1.1 Overview of the Chapters . .1 1.1.1 Diagramming and Gesture . .1 1.1.2 Simultaneous Core Partitions and Abaci . .2 1.1.3 Integer Quotients from Restricted Digit Sets . .5 1.2 Positioning Myself . .9 Chapter 2 Diagramming and Gesture . 11 2.1 The Front and the Back of Mathematics . 11 2.2 Embodied Cognition, Diagram, and Gesture . 15 2.2.1 Embodied Roots of Conceptual Mathematics . 15 2.2.2 Diagram and Gesture . 17 2.3 Diagramming and Gesture . 19 2.3.1 Diagramming and Gesture Techniques . 21 2.4 Applications to Research Mathematics . 25 Chapter 3 Introduction to Core Partitions and Abaci . 28 3.1 Definitions . 28 3.1.1 Families of Core Partitions . 33 3.2 Abacus Diagrams and Notation . 36 Chapter 4 Counting Abaci . 43 4.1 Preliminaries . 43 4.1.1 Some `Small' Proofs. 43 4.1.2 Abacus Diagrams and Core Partitions . 48 4.1.3 Building with Beads . 51 4.2 Enumerating s-core Abaci of Bounded Height . 59 4.2.1 Primary Results . 59 4.2.2 Initial Conditions and Observations . 75 Chapter 5 Quotients from Restricted Digit Sets and Multiplication Transducers . 80 5.1 Definitions and Notation . 80 5.1.1 Base b Numeration Systems .
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