Lyndon words and Fibonacci numbers

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 ⌈logφ(n)⌉ + 1, where φ denotes the golden ratio (1 + √ 5)/2. We show that this bound is optimal in that it is attained by the Fibonacci Lyndon words. We then introduce a mapping Lx 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 Lx ≥ Lf , where f is the Fibonacci infinite word, with equality if and only if x is in the shift orbit closure of f .