The Density of Zero and One in Fibonacci Word for Subwords and Their Palindromes

TL;DR

This paper investigates the distribution of zeros and ones in subwords of the Fibonacci word and their palindromes, revealing insights into their combinatorial structure through computational analysis.

Contribution

It introduces a computational approach to analyze character densities in Fibonacci subwords and their palindromes, enhancing understanding of their combinatorial properties.

Findings

Density patterns differ between subwords and their palindromes.

Palindromic subwords exhibit distinct character distributions.

Results contribute to the combinatorial theory of Fibonacci words.

Abstract

This paper studies the density of zero and one in subwords of the Fibonacci word with lengths less than thirty and compares them to the densities of their corresponding palindromes. We used computational methods to produce a sufficiently large piece of the Fibonacci word, extract all unique subwords up to a predetermined length, and calculate their palindrome. The density of each character (0 and 1) was then determined for both the original subwords and their palindromic counterparts. This study contributes to a deeper understanding of the combinatorial properties of the Fibonacci word and the behavior of its constituent elements under reversal.