Information Processing Letters
Some properties of the singular words of the Fibonacci word
European Journal of Combinatorics
Sturmian words, Lyndon words and trees
Theoretical Computer Science
Periodicity and the golden ratio
Theoretical Computer Science - Special issue: papers dedicated to the memory of Marcel-Paul Schützenberger
Fine and Wilf's theorem for three periods and a generalization of Sturmian words
Theoretical Computer Science
Automatic Sequences: Theory, Applications, Generalizations
Automatic Sequences: Theory, Applications, Generalizations
Palindromic prefixes and episturmian words
Journal of Combinatorial Theory Series A
Fine and Wilf words for any periods II
Theoretical Computer Science
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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"@f(n)@?+1, where @f 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.