A characterization of infinite smooth Lyndon words

In a recent paper, Brlek, Jamet and Paquin showed that some extremal infinite smooth words are also infinite Lyndon words. This result raises a natural question: are they the only ones? If no, what do the infinite smooth words that are also Lyndon words look like? In this paper, we give the answer, proving that the only infinite smooth Lyndon words are m{a<b}, with a, b even, m{1<b} and -1 1 (m{1<b}), with b odd, where mA is the minimal infinite smooth word with respect to the lexicographic order over a numerical alphabet A and is the run-length encoding function. Key words: Lyndon words; Lyndon factorization; smooth words; extremal smooth words.

• Type: Journal
• Year: 2010
• Where: DMTCS
• Authors: Geneviève Paquin
• Keywords: DMTCS 2010, Infinite Smooth Word, Lyndon Words, Sensor Networks, Smooth Words