A note on avoidable words in squarefree ternary words

Doubled patterns with reversal and square-free doubled patterns

In combinatorics on words, a word $w$ over an alphabet $Σ$ is said to avoid a pattern $p$ over an alphabet $Δ$ if there is no factor $f$ of $w$ such that $f=h(p)$ where $h:Δ^*\toΣ^*$ 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…

Squarefree words with interior disposable factors

We give a partial answer to a problem of Harju by constructing an infinite ternary squarefree word $w$ with the property that for every $k \geq 3312$ there is an interior length-$k$ factor of $w$ that can be deleted while still preserving squarefreeness. We also examine Thue's famous squarefree word (generated by iterating the map $0 \to 012$, $1 \to 02$, $2 \to 1$) and characterize the positions $i$ for which deleting the symbol appearing at position $i$ preserves squarefreeness.

Lengths of Irreducible and Delicate Words

We study words that barely avoid repetitions, for several senses of "barely". A squarefree (respectively, overlap-free, cubefree) word is irreducible if removing any one of its interior letters creates a square (respectively, overlap, cube). A squarefree (respectively, overlap-free, cubefree) word is delicate if changing any one of its letters creates a square (respectively, overlap, cube). We classify the lengths of irreducible and delicate squarefree, overlap-free, and cubefree words over binary and ternary alphabets.

4-tangrams是4-可避免的

4-tangrams are 4-avoidable

本文研究tangram字的切割数可避免性，证明了$t(3)=t(4)=4$，回答了Dębski等人提出的问题。

An Improved Lower Bound for $n$-Brinkhuis $k$-Triples

Let $s_n$ be the number of words consisting of the ternary alphabet consisting of the digits 0, 1, and 2 such that no subword (or factor) is a square (a word concatenated with itself, e.g., $11$, $1212$, or $102102$). From computational evidence, $s_n$ grows exponentially at a rate of about $1.317277^n$. While known upper bounds are already relatively close to the conjectured rate, effective lower bounds are much more difficult to obtain. In this paper, we construct a $54$-Brinkhuis $952$-triple, which leads to an improved lower bound on the number of $n$-letter ternary squarefree words: $952^…