Related papers: Palindromic factorization of rich words

A word $u$ defined over an alphabet $\mathcal{A}$ is $c$-balanced ($c\in\mathbb{N}$) if for all pairs of factors $v$, $w$ of $u$ of the same length and for all letters $a\in\mathcal{A}$, the difference between the number of letters $a$ in…

A factor $u$ of a word $w$ is a cover of $w$ if every position in $w$ lies within some occurrence of $u$ in $w$. A word $w$ covered by $u$ thus generalizes the idea of a repetition, that is, a word composed of exact concatenations of $u$.…

A word $u$ is a scattered factor of $w$ if $u$ can be obtained from $w$ by deleting some of its letters. That is, there exist the (potentially empty) words $u_1,u_2,..., u_n$, and $v_0,v_1,..,v_n$ such that $u = u_1u_2...u_n$ and $w =…

We introduce and study a complexity function on words $c_x(n),$ called \emph{cyclic complexity}, which counts the number of conjugacy classes of factors of length $n$ of an infinite word $x.$ We extend the well-known Morse-Hedlund theorem…

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…

We consider the class ${\cal P}_1$ of all infinite words $x\in A^\omega$ over a finite alphabet $A$ admitting a prefixal factorization, i.e., a factorization $x= U_0 U_1U_2 \cdots $ where each $U_i$ is a non-empty prefix of $x.$ With each…

Let $F= < a,b>$ be a rank two free group. A word $W(a,b)$ in $F$ is {\sl primitive} if it, along with another group element, generates the group. It is a {\sl palindrome} (with respect to $a$ and $b$) if it reads the same forwards and…

Let $P$ be a partially ordered set and consider the free monoid $P^*$ of all words over $P$. If $w,w'\in P^*$ then $w'$ is a factor of $w$ if there are words $u,v$ with $w=uw'v$. Define generalized factor order on $P^*$ by letting $u\le w$…

We study how much injective morphisms can increase the repetitiveness of a given word. This question has a few possible variations depending on the meaning of ``repetitiveness''. We concentrate on fractional exponents of finite words and…

We prove that the property of being closed (resp., palindromic, rich, privileged trapezoidal, balanced) is expressible in first-order logic for automatic (and some related) sequences. It therefore follows that the characteristic function of…

We construct an infinite word $w$ over the $5$-letter alphabet such that for every factor $f$ of $w$ of length at least two, there exists a cyclic permutation of $f$ that is not a factor of $w$. In other words, $w$ does not contain a…

The palindromic length of the finite word $v$ is equal to the minimal number of palindromes whose concatenation is equal to $v$. It was conjectured in 2013 that for every infinite aperiodic word $x$, the palindromic length of its factors is…

We show that there exists an uniformly recurrent infinite word whose set of factors is closed under reversal and which has only finitely many palindromic factors.

A \emph{power} is a word of the form $\underbrace{uu...u}_{k \; \text{times}}$, where $u$ is a word and $k$ is a positive integer; the power is also called a {\em $k$-power} and $k$ is its {\em exponent}. We prove that for any $k \ge 2$,…

The block reversal of a word $w$, denoted by $\mathtt{BR}(w)$, is a generalization of the concept of the reversal of a word, obtained by concatenating the blocks of the word in the reverse order. We characterize non-binary and binary words…

Let $L_{k,\alpha}^{\mathbb{Z}}$ denote the set of all bi-infinite $\alpha$-power free words over an alphabet with $k$ letters, where $\alpha$ is a positive rational number and $k$ is positive integer. We prove that if $\alpha\geq 5$, $k\geq…

A morphic word is obtained by iterating a morphism to generate an infinite word, and then applying a coding. We characterize morphic words with polynomial growth in terms of a new type of infinite word called a $\textit{zigzag word}$. A…

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 word of length $n$ is rich if it contains $n$ nonempty palindromic factors. An infinite word is rich if all of its finite factors are rich. Baranwal and Shallit produced an infinite binary rich word with critical exponent $2+\sqrt{2}/2$…

Free words are elements of a free monoid, generated over an alphabet via the binary operation of concatenation. Casually speaking, a free word is a finite string of letters. Henceforth, we simply refer to them as words. Motivated by recent…