Vol. 12, No. 7, 2019

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Some combinatorics from Zeckendorf representations

Tyler Ball, Rachel Chaiser, Dean Dustin, Tom Edgar and Paul Lagarde

Vol. 12 (2019), No. 7, 1241–1260
Abstract

We explore some properties of the so-called Zeckendorf representations of integers, where we write an integer as a sum of distinct, nonconsecutive Fibonacci numbers. 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 some possibly new results related to the tree, allowing us to develop a partial analog to Kummer’s classical theorem about counting the number of “carries” involved in arithmetic. Finally, we finish with some conjectures and possible future projects related to the combinatorics of these representations.

Keywords
Fibonacci, Zeckendorf, digital dominance order
Mathematical Subject Classification 2010
Primary: 06A07, 11B39, 11B75, 11Y55
Milestones
Received: 28 March 2019
Accepted: 10 June 2019
Published: 12 October 2019

Communicated by Arthur T. Benjamin
Authors
Tyler Ball
Clover Park High School
Lakewood, WA
United States
Rachel Chaiser
University of Colorado Boulder
Boulder, CO
United States
Dean Dustin
University of Nebraska
Lincoln, NE
United States
Tom Edgar
Department of Mathematics
Pacific Lutheran University
Tacoma, WA
United States
Paul Lagarde
South Merrimack Christian Academy
Merrimack, NH
United States