SUMmER 2017 Results

Research in Combinatorics

Project Mentors: Tom Edgar and Tyler Ball

If x and y are two whole numbers, we say that y dominates x if every digit in y is at least as large as its corresponding digit in x, meaning the ones digit of y is at least as large as the ones digit in x; the tens digit of is at least as large as the tens digit of x; and so on for the hundreds digits, thousands digits, etc. If y dominates x, we write x<<y.

However, there is nothing special about the way we choose to represent numbers, where each digit represents a corresponding power of 10. We could also consider digital dominance of natural numbers using representations in terms of different bases - powers of 2, 3, etc. - where each digit represents the corresponding powers of the base.

Furthermore, we can consider more exotic representations of integers such as the factorial base, where allowable digits change in each position and represent the factorials, or the Zeckendorf representation where we write numbers as a sum of distinct non-consecutive Fibonacci numbers. We explored the arithmetic in these two settings and investigated connections to the corresponding digital dominance orders. In particular, we encountered, described, and classified certain combinatorial structures arising from these nonstandard positional representations.

Combinatorics of Zeckendorf Representations

T. Ball, R. Chaiser*, D. Dustin*, T. Edgar, P. Lagarde*, ``Combinatorics of Zeckendorf Representations," in preparation, 2018.

Abstract: We acknowledge the idea that natural numbers exist regardless of their representation. We explore the properties of a representation that utilizes the Fibonacci sequence as the base, which is called the Zeckendorf representation. We examine the combinatorics arising from the arithmetic of these representations, with a particular emphasis on understanding the ``Zeckendorf tree'' that encodes them. We introduce several new results related to the tree, allowing us to develop a partial analog to Kummer's classical theorem.

Combinatorics of Factorial Base Representations

T. Ball, J. Beckford*, P. Dalenberg*, T. Edgar, T. Rajabi*, ``Combinatorics of Factorial Base Representations," in preparation, 2018.

Abstract: Every nonnegative integer can be written using what is known as the factorial base representation. We define this notion and explore certain combinatorial structures arising from the arithmetic of these representations. In particular, we will investigate the sum-of-digits function, carry sequences, and a partial order referred to as digital dominance. Finally, we describe three analogs of a classical theorem due to Kummer that relate the combinatorial objects of interest by constructing a variety of new integer sequences.

P. Dalenberg* and T. Edgar, ``Consecutive factorial base Niven numbers," to appear in Fibonacci Quarterly, 2018.

Abstract: We prove that there are no consecutive runs of five or more factorial base Niven numbers. Moreover, we construct an infinite family of collections of four consecutive factorial base Niven numbers.


Research in Geometry

Project MentorsEric Bahuaud and Dylan Helliwell

A Closer look at Taxicab Geometry: The Parallelogram Law, Conic Sections, and Apollonian Sets

E. Bahuaud, D. Helliwell, A. Miller*, F. Nungaray*, S. Shergill*, in preparation
Abstract: What happens when we change how we think about distance? We explore this question by taking Euclidean geometric ideas and analyzing them in terms of the taxicab metric.  We study the extent to which the parallelogram law holds.  We study the familiar conic sections in terms of distance formulas and compare them to the ones found by slicing cones. We delve into the exploration of Apollonian sets.

Downtown Distances and Taxi Hubs: Triangle Centers in Taxicab Geometry

E. Bahuaud, D. Helliwell, S. Crawford*, A. Fish*, J. Tiffay*, N.Velez*, in preparation
Abstract: The Euclidean metric is not the only way to measure distance.  Using the Taxicab metric, which measures distance as the sum of the horizontal and vertical displacements, we can develop an entirely different geometry. Here we explore a synthetic approach to some triangle centers common in Euclidean geometry. We look for centers analogous to the circumcenter, incenter, Miquel points, and Fermat point, and develop tools for a synthetic analysis of Taxicab geometry.