Lyndon words and Fibonacci numbers

TL;DR

This paper establishes a precise lower bound on the number of Lyndon factors in words using Fibonacci numbers and characterizes aperiodic infinite words through a Fibonacci-based Lyndon factor counting method.

Contribution

It introduces a new lower bound for Lyndon factors based on the golden ratio and characterizes aperiodic infinite words via Fibonacci Lyndon words.

Findings

The lower bound for Lyndon factors is log_{phi}(n) + 1.

Fibonacci Lyndon words attain this lower bound.

Aperiodic infinite words are characterized by Lyndon factor counts related to the Fibonacci word.

Abstract

It is a fundamental property of non-letter Lyndon words that they can be expressed as a concatenation of two shorter Lyndon words. This leads to a naive lower bound log_{2}(n)} + 1 for the number of distinct Lyndon factors that a Lyndon word of length n must have, but this bound is not optimal. In this paper we show that a much more accurate lower bound is log_{phi}(n) + 1, where phi denotes the golden ratio (1 + sqrt{5})/2. We show that this bound is optimal in that it is attained by the Fibonacci Lyndon words. We then introduce a mapping L_x that counts the number of Lyndon factors of length at most n in an infinite word x. We show that a recurrent infinite word x is aperiodic if and only if L_x >= L_f, where f is the Fibonacci infinite word, with equality if and only if f is in the shift orbit closure of f.