Doubled Patterns with Reversal and Square-Free Doubled Patterns

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