### A characterization of infinite smooth Lyndon words

*Geneviève Paquin*

#### Abstract

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 Δ^{-1}_{1}(m_{{1<b}}), with b odd, where m_{A}is the minimal infinite smooth word with respect to the lexicographic order over a numerical alphabet A and Δ is the run-length encoding function.Full Text: PDF PostScript