Counting Lyndon factors

TL;DR

This paper investigates the maximum and expected counts of Lyndon factors in words, explores bounds for Lyndon factors in Lyndon words, and challenges existing conjectures with new counterexamples and generalizations.

Contribution

It provides formulas for Lyndon factors, refutes a conjecture about Christoffel words, and extends results to Christoffel words from Fibonacci Lyndon words.

Findings

Maximum number of Lyndon factors in words of length n determined

Formulas for expected total and distinct Lyndon factors derived

Counterexample provided to Saari's conjecture on Christoffel words

Abstract

In this paper, we determine the maximum number of distinct Lyndon factors that a word of length $n$ can contain. We also derive formulas for the expected total number of Lyndon factors in a word of length $n$ on an alphabet of size $\sigma$, as well as the expected number of distinct Lyndon factors in such a word. The minimum number of distinct Lyndon factors in a word of length $n$ is $1$ and the minimum total number is $n$, with both bounds being achieved by $x^n$ where $x$ is a letter. A more interesting question to ask is what is the minimum number of distinct Lyndon factors in a Lyndon word of length $n$? In this direction, it is known (Saari, 2014) that an optimal lower bound for the number of distinct Lyndon factors in a Lyndon word of length $n$ is $\lceil\log_{\phi}(n) + 1\rceil$, where $\phi$ denotes the golden ratio $(1 + \sqrt{5})/2$. Moreover, this lower bound is attained by the so-called finite "Fibonacci Lyndon words", which are precisely the Lyndon factors of the well-known "infinite Fibonacci word" -- a special example of a "infinite Sturmian word". Saari (2014) conjectured that if $w$ is Lyndon word of length $n$, $n\ne 6$, containing the least number of distinct Lyndon factors over all Lyndon words of the same length, then $w$ is a Christoffel word (i.e., a Lyndon factor of an infinite Sturmian word). We give a counterexample to this conjecture. Furthermore, we generalise Saari's result on the number of distinct Lyndon factors of a Fibonacci Lyndon word by determining the number of distinct Lyndon factors of a given Christoffel word. We end with two open problems.