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 ⌈ log ϕ ( 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 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 x is in the shift orbit closure of f.