Infinite smooth Lyndon words

TL;DR

This paper characterizes all infinite smooth Lyndon words, showing they are limited to specific minimal smooth words over certain numerical alphabets, thus answering a natural open question in the field.

Contribution

It provides a complete classification of infinite smooth Lyndon words, identifying exactly which ones satisfy both properties, expanding understanding of their structure and properties.

Findings

Only specific minimal smooth words are infinite smooth Lyndon words.

The classification includes words over even and odd numerical alphabets.

The results connect smooth words with Lyndon words through lexicographic order.

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 $\Delta^{-1}_1(m_{\{1<b\}})$, with $b$ odd, where $m_\A$ is the minimal infinite smooth word with respect to the lexicographic order over a numerical alphabet $\A$ and $\Delta$ is the run-length encoding function.