Related papers: Abelian bordered factors and periodicity
We say that an infinite word w is weak abelian periodic if it can be factorized into finite words with the same frequencies of letters. In the paper we study properties of weak abelian periodicity, its relations with balance and frequency.…
The relationship between the length of a word and the maximum length of its unbordered factors is investigated in this paper. Consider a finite word w of length n. We call a word bordered, if it has a proper prefix which is also a suffix of…
A word is said to be bordered if it contains a nonempty proper prefix that is also a suffix. A pair of words $(u, v)$ is said to be mutually bordered if there exists a word that is a nonempty proper prefix of $u$ and suffix of $v$, and…
Deciding periodicity of infinite words generated by morphisms is a classical result in combinatorics on words from 80's by Harju, Linna and Pansiot. In this paper, we are interested in this question in the abelian setting. Two words are…
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…
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 non-empty word $w$ is a border of the word $u$ if $\vert w\vert<\vert u\vert$ and $w$ is both a prefix and a suffix of $u$. A word $u$ with the border $w$ is closed if $u$ has exactly two occurrences of $w$. A word $u$ is privileged if…
Call a reduced word $w$ multiplicity-bounding if and only if a finite group on which the word map of $w$ has a fiber of positive proportion $\rho$ can only contain each nonabelian finite simple group $S$ as a composition factor with…
Two finite words $u$ and $v$ are called abelian equivalent if each letter occurs equally many times in both $u$ and $v$. The abelian closure $\mathcal{A}(\mathbf{x})$ of an infinite word $\mathbf{x}$ is the set of infinite words…
Two finite words $u$ and $v$ are called Abelian equivalent if each letter occurs equally many times in both $u$ and $v$. The abelian closure $\mathcal{A}(\mathbf{x})$ of (the shift orbit closure of) an infinite word $\mathbf{x}$ is the set…
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…
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…
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 {\em square} is a word of the form $uu$. In this paper we prove that for a given finite word $w$, the number of distinct square factors of $w$ is bounded by $|w|-|\Alphabet(w)|+1$, where $|w|$ denotes the length of $w$ and…
We say that a finite factor $f$ of a word $w$ is \emph{imaged} if there exists a non-erasing morphism $m$, distinct from the identity, such that $w$ contains $m(f)$. We show that every infinite word contains an imaged factor of length at…
The notion of Abelian complexity of infinite words was recently used by the three last authors to investigate various Abelian properties of words. In particular, using van der Waerden's theorem, they proved that if a word avoids Abelian…
A finite word $w$ of length $n$ contains at most $n+1$ distinct palindromic factors. If the bound $n+1$ is attained, the word $w$ is called rich. An infinite word $w$ is called rich if every finite factor of $w$ is rich. Let $w$ be a word…
In this paper, we study the relation between periodicity of two-dimensional words and their abelian pattern complexity. A pattern $\cal{P}$ in $\mathbb{Z}^n$ is the set of all translations of some finite subset $F$ of $\mathbb{Z}^n$. An…
The abelian critical exponent of an infinite word $w$ is defined as the maximum ratio between the exponent and the period of an abelian power occurring in $w$. It was shown by Fici et al. that the set of finite abelian critical exponents of…
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…