Related papers: Bounds on Zimin Word Avoidance
To any infinite word w over a finite alphabet A we can associate two infinite words min(w) and max(w) such that any prefix of min(w) (resp. max(w)) is the lexicographically smallest (resp. greatest) amongst the factors of w of the same…
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,…
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…
Two finite words $u,v$ are 2-binomially equivalent if, for all words $x$ of length at most 2, the number of occurrences of $x$ as a (scattered) subword of $u$ is equal to the number of occurrences of $x$ in $v$. This notion is a refinement…
We find finite-state recurrences to enumerate the words on the alphabet $[n]^r$ which avoid the patterns 123 and $1k(k-1)\dots2$, and, separately, the words which avoid the patterns 1234 and $1k(k-1)\dots2$.
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 x of w and no non-erasing morphism h from Delta^* to Sigma^* such that h(p) = x. Bell and Goh have recently…
Word $W$ is said to encounter word $V$ provided there is a homomorphism $\phi$ mapping letters to nonempty words so that $\phi(V)$ is a substring of $W$. For example, taking $\phi$ such that $\phi(h)=c$ and $\phi(u)=ien$, we see that…
A word is said to be \emph{bordered} if it contains a non-empty proper prefix that is also a suffix. We can naturally extend this definition to pairs of non-empty words. A pair of words $(u,v)$ is said to be \emph{mutually bordered} if…
We find generating functions for the number of words avoiding certain patterns or sets of patterns on at most 2 distinct letters and determine which of them are equally avoided. We also find the exact number of words avoiding certain…
We study the notion of quasiperiodicity, in the sense of "coverability", for biinfinite words. All previous work about quasiperiodicity focused on right infinite words, but the passage to the biinfinite case could help to prove stronger…
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…
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…
A closed word (a.k.a. periodic-like word or complete first return) is a word whose longest border does not have internal occurrences, or, equivalently, whose longest repeated prefix is not right special. We investigate the structure of…
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…
In this note we present a characterisation of all unary and binary patterns that do not only contain variables, but also reversals of their instances. These types of variables were studied recently in either more general or particular…
We consider the number of occurrences of subwords (non-consecutive sub-sequences) in a given word. We first define the notion of subword entropy of a given word that measures the maximal number of occurrences among all possible subwords. We…
A pattern $\alpha$ is a string of variables and terminal letters. We say that $\alpha$ matches a word $w$, consisting only of terminal letters, if $w$ can be obtained by replacing the variables of $\alpha$ by terminal words. The matching…
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…
In the literature two notions of the word problem for a variety occur. A variety has a decidable word problem if every finitely presented algebra in the variety has a decidable word problem. It has a uniformly decidable word problem if…
Let $w$ be an infinite word on an alphabet $A$. We denote by $(n_i)_{i \geq 1}$ the increasing sequence (assumed to be infinite) of all lengths of palindrome prefixes of $w$. In this text, we give an explicit construction of all words $w$…