Related papers: Rich square-free words
We focus on $\Theta$-rich and almost $\Theta$-rich words over a finite alphabet $\mathcal{A}$, where $\Theta$ is an involutive antimorphism over $\mathcal{A}^*$. We show that any recurrent almost $\Theta$-rich word $\uu$ is an image of a…
We discuss the notion of privileged word, recently introduced by Peltomaki. A word w is privileged if it is of length <=1, or has a privileged border that occurs exactly twice in w. We prove the following results: (1) if w^k is privileged…
Circular words are cyclically ordered finite sequences of letters. We give a computer-free proof of the following result by Currie: square-free circular words over the ternary alphabet exist for all lengths $l$ except for 5, 7, 9, 10, 14,…
Entringer, Jackson, and Schatz conjectured in 1974 that every infinite cubefree binary word contains arbitrarily long squares. In this paper we show this conjecture is false: there exist infinite cubefree binary words avoiding all squares…
A word is level if each letter appears in it the same number of times, plus or minus 1. We give a complete characterization of the lengths for which level ternary circular square-free words exist. Key words: combinatorics on words, circular…
In this paper we answer two recent questions from Charlier et al. and Harju about self-shuffling words. An infinite word $w$ is called self-shuffling, if $w=\prod_{i=0}^\infty U_iV_i=\prod_{i=0}^\infty U_i=\prod_{i=0}^\infty V_i$ for some…
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…
A graph $G = (V, E)$ is word-representable, if there exists a word w over the alphabet V such that for letters ${x, y} \in V$ , $x$ and $y$ alternate in $w$ if and only if $xy \in E$. In this paper, we prove that any non-empty…
We study the set of finite words with zero palindromic defect, i.e., words rich in palindromes. This set is factorial, but not recurrent. We focus on description of pairs of rich words which cannot occur simultaneously as factors of a…
We prove that every concatenation of $10$ or more binary squares contains an overlap. The bound $10$ is best possible. In contrast, over a ternary alphabet, there are infinitely long overlap-free words that consist of a concatenation of…
A square is a word of the form $xx$ for a non-empty word $x$. Brlek and Li [Comb. Theory, 2025] proved that the number of distinct squares in a word $w$ of length $n$ is at most $n - \sigma$, where $\sigma$ is the number of letters used in…
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,…
A word $\bar{w} = \bar{u}\bar{u}$ is a $long$ $square$ if $\bar{u}$ is of length at least 3; a word $\bar{w}$ is $long$-$square$-$free$ if $\bar{w}$ contains no sub-word that is a long square. We can use words to generate graph colorings; a…
We consider sets of factors that can be avoided in square-free words on two-generator free groups. The elements of the group are presented in terms of 0,1,2,3 such that 0 and 2 (resp.,1 and 3) are inverses of each other so that 02, 20, 13…
In [A. Frid, S. Puzynina, L.Q. Zamboni, \textit{On palindromic factorization of words}, Adv. in Appl. Math. 50 (2013), 737-748], it was conjectured that any infinite word whose palindromic lengths of factors are bounded is ultimately…
A square is a factor $S = (S_1; S_2)$ where $S_1$ and $S_2$ have the same pattern, and a permutation is said to be square-free if it contains no non-trivial squares. The permutation is further said to be bicrucial if every extension to the…
An efficient, when compared to exhaustive enumeration, algorithm for computing the number of square-free words of length $n$ over the alphabet $\{a, b, c\}$ is presented.
We show that the number of length-n words over a k-letter alphabet having no even palindromic prefix is the same as the number of length-n unbordered words, by constructing an explicit bijection between the two sets. A slightly different…
A word $u=u_1\dots u_n$ is a scattered factor of a word $w$ if $u$ can be obtained from $w$ by deleting some of its letters: there exist the (potentially empty) words $v_0,v_1,..,v_n$ such that $w = v_0u_1v_1...u_nv_n$. The set of all…
We enumerate all ternary length-l square-free words, which are words avoiding squares of words up to length l, for l<=24. We analyse the singular behaviour of the corresponding generating functions. This leads to new upper entropy bounds…