Related papers: Circularly squarefree words and unbordered conjuga…
We revisit the question of classification of balanced circular words and focus on the case of a ternary alphabet. We propose a $3$-dimensional generalisation of the discrete approximation representation of Christoffel words. By considering…
We investigate the scattered palindromic subwords in a finite word. We start by characterizing the words with the least number of scattered palindromic subwords. Then, we give an upper bound for the total number of palindromic subwords in a…
A non-empty word $w$ is a \emph{border} of a word $u$ if $\vert w\vert<\vert u\vert$ and $w$ is both a prefix and a suffix of $u$. A word $u$ is \emph{privileged} if $\vert u\vert\leq 1$ or if $u$ has a privileged border $w$ that appears…
We consider the following novel variation on a classical avoidance problem from combinatorics on words: instead of avoiding repetitions in all factors of a word, we avoid repetitions in all factors where each individual factor is considered…
We regard a finite word $u=u_1u_2\cdots u_n$ up to word isomorphism as an equivalence relation on $\{1,2,\ldots, n\}$ where $i$ is equivalent to $j$ if and only if $x_i=x_j.$ Some finite words (in particular all binary words) are generated…
We answer a question of Harju: An infinite square-free ternary word with an $n$-stem factorization exists for any $n\ge 13$. We show that there are uniform ternary morphisms of length $k$ for every $k\ge 23$. This resolves almost completely…
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 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…
We review the recent progress in the investigation of powerfree words, with particular emphasis on binary cubefree and ternary squarefree words. Besides various bounds on the entropy, we provide bounds on letter frequencies and consider…
A permutation is said to be a square if it can be obtained by shuffling two order-isomorphic patterns. The definition is intended to be the natural counterpart to the ordinary shuffle of words and languages. In this paper, we tackle the…
A power is a word of the form $\underbrace{uu...u}_{k \; \text{times}}$, where $u$ is a word and $k$ is a positive integer and a square is a word of the form $uu$. Fraenkel and Simpson conjectured in 1998 that the number of distinct 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…
We investigate the behavior of the periods and border lengths of random words over a fixed alphabet. We show that the asymptotic probability that a random word has a given maximal border length $k$ is a constant, depending only on $k$ and…
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…
We consider words $w$ over the alphabet $\Sigma=\{0,1,2\}$. It is shown that there are irreducibly square-free words of all lengths $n$ except 4,5,7 and 12. Such a word is square-free (i.e., it has no repetitions $uu$ as factors), but by…
We completely characterize the words that can be avoided in infinite squarefree ternary words.
Upper bounds on the maximum number of codewords in a binary code of a given length and minimum Hamming distance are considered. New bounds are derived by a combination of linear programming and counting arguments. Some of these bounds…
An overlap-free (or $\beta$-free) word $w$ over a fixed alphabet $\Sigma$ is extremal if every word obtained from $w$ by inserting a single letter from $\Sigma$ at any position contains an overlap (or a factor of exponent at least $\beta$,…
We show that the conjugacy problem is solvable in [finitely generated free]-by-cyclic groups, by using a result of O. Maslakova that one can algorithmically find generating sets for the fixed subgroups of free group automorphisms, and one…
The avoidability of binary patterns by binary cube-free words is investigated and the exact bound between unavoidable and avoidable patterns is found. All avoidable patterns are shown to be D0L-avoidable. For avoidable patterns, the growth…