Doubled patterns with reversal and square-free doubled patterns - LIRMM - Laboratoire d’Informatique, de Robotique et de Microélectronique de Montpellier Access content directly
Preprints, Working Papers, ... Year : 2023

Doubled patterns with reversal and square-free doubled patterns

Antoine Domenech
  • Function : Author
  • PersonId : 1294594
Pascal Ochem

Abstract

In combinatorics on words, a word $w$ over an alphabet $\Sigma$ is said to avoid a pattern $p$ over an alphabet $\Delta$ if there is no factor $f$ of $w$ such that $f=h(p)$ where $h:\Delta^*\to\Sigma^*$ is a non-erasing morphism. A pattern $p$ is said to be $k$-avoidable if there exists an infinite word over a $k$-letter alphabet that avoids $p$. A pattern is \emph{doubled} if every variable occurs at least twice. Doubled patterns are known to be $3$-avoidable. Currie, Mol, and Rampersad have considered a generalized notion which allows variable occurrences to be reversed. That is, $h(V^R)$ is the mirror image of $h(V)$ for every $V\in\Delta$. We show that doubled patterns with reversal are $3$-avoidable. We also conjecture that (classical) doubled patterns that do not contain a square are $2$-avoidable. We confirm this conjecture for patterns with at most 4 variables. This implies that for every doubled pattern $p$, the growth rate of ternary words avoiding $p$ is at least the growth rate of ternary square-free words. A previous version of this paper containing only the first result has been presented at WORDS 2021.
Fichier principal
Vignette du fichier
2105.04673.pdf (139.63 Ko) Télécharger le fichier
Origin Files produced by the author(s)
Licence

Dates and versions

lirmm-03799694 , version 1 (17-10-2023)

Licence

Identifiers

Cite

Antoine Domenech, Pascal Ochem. Doubled patterns with reversal and square-free doubled patterns. 2023. ⟨lirmm-03799694⟩
25 View
7 Download

Altmetric

Share

Gmail Mastodon Facebook X LinkedIn More