Lyndon words and Fibonacci numbers

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 $\lceil {\log }_2(n)\rceil +1$ 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 $\lceil {\log }_{\varphi }(n)\rceil +1$, where ϕ 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 ${\mathcal{L}}_{\mathbf{x}}$ 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 ${\mathcal{L}}_{\mathbf{x}}⩾{\mathcal{L}}_{\mathbf{f}}$, where f is the Fibonacci infinite word, with equality if and only if x is in the shift orbit closure of f.