Related papers: Avoiding Square-Free Words on Free Groups
A finite or infinite word is called a $G$-word for a labelled graph $G$ on the vertex set $A_n = \{0,1,..., n-1\}$ if $w = i_1i_2...i_k \in A_n^*$, where each factor $i_ji_{j+1}$ is an edge of $E$, i.e, $w$ represents a walk in $G$. We show…
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 $f= (p)$ where $h: \Delta^*\to\Sigma^*$ is a non-erasing morphism. A pattern…
A double occurrence word $w$ over a finite alphabet $\Sigma$ is a word in which each alphabet letter appears exactly twice. Such words arise naturally in the study of topology, graph theory, and combinatorics. Recently, double occurrence…
An abelian square is the concatenation of two words that are anagrams of one another. A word of length $n$ can contain at most $\Theta(n^2)$ distinct factors, and there exist words of length $n$ containing $\Theta(n^2)$ distinct…
A word~$w$ has a border $u$ if $u$ is a non-empty proper prefix and suffix of $u$. A word~$w$ is said to be \emph{closed} if $w$ is of length at most $1$ or if $w$ has a border that occurs exactly twice in $w$. A word~$w$ is said to be…
The exponent of a word is the ratio of its length over its smallest period. The repetitive threshold r(a) of an a-letter alphabet is the smallest rational number for which there exists an infinite word whose finite factors have exponent at…
Carpi constructed an infinite word over a 4-letter alphabet that avoids squares in all subsequences indexed by arithmetic progressions of odd difference. We show a connection between Carpi's construction and the paperfolding words. We…
In the free group $F_k$, an element is said to be primitive if it belongs to a free generating set. In this paper, we describe what a generic primitive element looks like. We prove that up to conjugation, a random primitive word of length…
We extend the theory of minimal absent words to (rooted and unrooted) trees, having edges labeled by letters from an alphabet $\Sigma$ of cardinality $\sigma$. We show that the set $\text{MAW}(T)$ of minimal absent words of a rooted (resp.…
We prove that for every $n \in \mathbb{N}$ and $\delta>0$ there exists a word $w_n \in F_2$ of length $n^{2/3} \log(n)^{3+\delta}$ which is a law for every finite group of order at most $n$. This improves upon the main result of [A. Thom,…
Using a new approach based on automatic sequences, logic, and a decision procedure, we reprove some old theorems about circularly squarefree words and unbordered conjugates in a new and simpler way. Furthermore, we prove three new results…
Elements of the commutator subgroup of a free group can be presented as values of canonical forms, called Wicks forms. We show that, starting from sufficiently high genus g, there is a sequence of words w(g) which can be presented by f(g)…
We count the number of occurrences of certain patterns in given words. We choose these words to be the set of all finite approximations of a sequence generated by a morphism with certain restrictions. The patterns in our considerations are…
Given a word $w$, what is the maximum possible number of appearances of $w$ reading contiguously along any of the directions in $\{-1, 0, 1\}^d \setminus \{\mathbf{0}\}$ in a large $d$-dimensional grid (as in a word search)? Patchell and…
This paper concerns the avoidability of abelian and additive powers in infinite rich words. In particular, we construct an infinite additive $5$-power-free rich word over $\{0,1\}$ and an infinite additive $4$-power-free rich word over…
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…
In this paper we propose an algorithm to generate binary words with no more 0's than 1's having a fixed number of 1's and avoiding the pattern $(10)^j1$ for any fixed $j \geq 1$. We will prove that this generation is exhaustive, that is,…
A word contains a \emph{half-flip} if it contains non-empty factors $uv$ and $vu$ where $|u|=|v|$. Fici reports a non-constructive proof of the existence of an infinite word over a finite alphabet avoiding half-flips and asks for the size…
A word $w=w_1w_2\cdots w_n$ is alternating if either $w_1<w_2>w_3<w_4>\cdots$ (when the word is up-down) or $w_1>w_2<w_3>w_4<\cdots$ (when the word is down-up). In this paper, we initiate the study of (pattern-avoiding) alternating words.…
In this text, we consider random permutations which can be written as free words in several independent random permutations: firstly, we fix a non trivial word $w$ in letters $g_1,g_1^{-1},..., g_k,g_k^{-1}$, secondly, for all $n$, we…