Related papers: On Generating Binary Words Palindromically
The notion of palindromic length of a finite word, as well as an infinite word, was first introduced by Frid, Puzynina and Zamboni\cite{FRID2013737}. They conjectured that if the palindromic length of an infinite word is bounded, then this…
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…
These lecture notes provide an introduction to combinatorics on words and its interactions with dynamics, algebra, and arithmetic. The central theme is the notion of low factor complexity for infinite words. We investigate the following…
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…
We consider decision problems for relations over finite and infinite words defined by finite automata. We prove that the equivalence problem for binary deterministic rational relations over infinite words is undecidable in contrast to the…
In this paper, we characterize by lexicographic order all finite Sturmian and episturmian words, i.e., all (finite) factors of such infinite words. Consequently, we obtain a characterization of infinite episturmian words in a "wide sense"…
Let S=Sym(\Omega) be the group of all permutations of an infinite set \Omega. Extending an argument of Macpherson and Neumann, it is shown that if U is a generating set for S as a group, respectively as a monoid, then there exists a…
Sturmian words are infinite binary words with many equivalent definitions: They have a minimal factor complexity among all aperiodic sequences; they are balanced sequences (the labels 0 and 1 are as evenly distributed as possible) and they…
An infinite word has the property $R_m$ if every factor has exactly $m$ return words. Vuillon showed that $R_2$ characterizes Sturmian words. We prove that a word satisfies $R_m$ if its complexity function is $(m-1)n+1$ and if it contains…
We implement a decision procedure for answering questions about a class of infinite words that might be called (for lack of a better name) "Fibonacci-automatic". This class includes, for example, the famous Fibonacci word f = 01001010...,…
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…
A gapped repeat (respectively, palindrome) occurring in a word $w$ is a factor $uvu$ (respectively, $u^Rvu$) of $w$. In such a repeat (palindrome) $u$ is called the arm of the repeat (respectively, palindrome), while $v$ is called the gap.…
This paper classifies binary morphisms that map to ultimately periodic words. In particular, if a morphism h maps an infinite non-ultimately periodic word to an ultimately periodic word then it must be true that h(0) commutes with h(1).
A palindromic periodicity is a factor of an infinite word $(ps)^\omega$ where $p$ and $s$ are palindromes and the factor has length at least $|ps|$, for example, $accabaccab$. In this paper we describe several ways in which a palindromic…
In this article, we show that the Kamae-Xue complexity function for an infinite sequence classifies eventual periodicity completely. We prove that an infinite binary word $x_1x_2 \cdots $ is eventually periodic if and only if…
A prefix normal word is a binary word with the property that no substring has more $1$s than the prefix of the same length. By proving that the set of prefix normal words is a bubble language, we can exhaustively list all prefix normal…
A finite word $u$ is called closed if its longest repeated prefix has exactly two occurrences in $u,$ once as a prefix and once as a suffix. We study the function $f_x^c:\mathbb N \rightarrow \mathbb N$ which counts the number of closed…
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 this paper, we survey the rich theory of infinite episturmian words which generalize to any finite alphabet, in a rather resembling way, the well-known family of Sturmian words on two letters. After recalling definitions and basic…
We study the equality problem for infinite words obtained by iterating morphisms. In particular, we give a practical algorithm to decide whether or not two words generated by primitive morphisms are equal.