Related papers: Palindromic factorization of rich words
In this paper we consider two problems concerning string factorisation. Specifically given a string $w$ and an integer $k$ find a factorisation of $w$ where each factor has length bounded by $k$ and has the minimum (the FmD problem) or the…
A prefix normal word is a binary word whose prefixes contain at least as many 1s as any of its factors of the same length. Introduced by Fici and Lipt\'ak in 2011 the notion of prefix normality is so far only defined for words over the…
We prove that, for any pure morphic word $w$, if the frequencies of all letters in $w$ exist, then the frequencies of all factors in $w$ exist as well. This result answers a question of Saari in his doctoral thesis.
We study the palindrome complexity of infinite sequences on finite alphabets, i.e., the number of palindromic factors (blocks) of given length occurring in a given sequence. We survey the known results and obtain new results for some…
We study the asymptotics and fine-scale behavior of quantitative combinatorial measures of infinite words and related dynamical and algebraic structures. We construct infinite recurrent words $w$ whose complexity functions $p_w(n)$ are…
We study infinite binary words that contain few distinct palindromes. In particular, we classify such words according to their critical exponents. This extends results by Fici and Zamboni [TCS 2013]. Interestingly, the words with 18 and 20…
We characterize the infinite words determined by indexed languages. An infinite language $L$ determines an infinite word $\alpha$ if every string in $L$ is a prefix of $\alpha$. If $L$ is regular or context-free, it is known that $\alpha$…
A reconstruction problem of words from scattered factors asks for the minimal information, like multisets of scattered factors of a given length or the number of occurrences of scattered factors from a given set, necessary to uniquely…
Two results on palindromicity of bi-infinite words in a finite alphabet are presented. The first is a simple, but efficient criterion to exclude palindromicity of minimal sequences and applies, in particular, to the Rudin-Shapiro sequence.…
A regular continuant is the denominator $K$ of a terminating regular continued fraction, interpreted as a function of the partial quotients. We regard $K$ as a function defined on the set of all finite words on the alphabet $1<2<3<\dots$…
Given two functions $\mathbf{a}\!:\! [n] \rightarrow [n]$ and $\mathbf{b}\!:\! [n] \rightarrow [n]$ chosen uniformly at random, any word $w=w_1w_2\dots w_k\in \{a,b\}^k$ induces a random function $\mathbf{w}\!:\! [n] \rightarrow [n]$ by…
The probabilistic Waring problem for finite simple groups asks whether every word of the form $w_1w_2$, where $w_1$ and $w_2$ are non-trivial words in disjoint sets of variables, induces almost uniform distribution on finite simple groups…
A narrow connection between infinite binary words rich in classical palindromes and infinite binary words rich simultaneously in palindromes and pseudopalindromes (the so-called $H$-rich words) is demonstrated. The correspondence between…
Given a group word $w$ and a group $G$, the set of $w$-values in $G$ is denoted by $G_w$ and the verbal subgroup $w(G)$ is the one generated by $G_w$. In the present paper we consider profinite groups admitting a word $w$ such that the…
A nonzero element of an integral domain (or commutative cancellative monoid) is called atomic if it can be written as a finite product of irreducible elements (also called atoms). In this paper, we introduce and investigate an unrestricted…
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…
To every integer monic polynomial of degree m can be associated a `replacement rule' that generates a word W* from another word W consisting of symbols belonging to a finite `alphabet' of size 2m. This rule applied iteratively on almost any…
A word $w$ over an alphabet $\Sigma$ is a Lyndon word if there exists an order defined on $\Sigma$ for which $w$ is lexicographically smaller than all of its conjugates (other than itself). We introduce and study \emph{universal Lyndon…
Lambda words are sequences obtained by encoding the differences between ordered elements of the form i+j\theta, where i and j are non-negative integers and 1 < \theta <2. Lambda words are right-infinite words defined over an infinite…
In this paper we explore a new hierarchy of classes of languages and infinite words and its connection with complexity classes. Namely, we say that a language belongs to the class $L_k$ if it is a subset of the catenation of $k$ languages…