Related papers: Topological invariants for words of linear factor …
A finite word $w$ is called \emph{rich} if it contains $\vert w\vert+1$ distinct palindromic factors including the empty word. For every finite rich word $w$ there are distinct nonempty palindromes $w_1, w_2,\dots,w_p$ such that…
The subword complexity of a finite word $w$ of length $N$ is a function which associates to each $n\le N$ the number of all distinct subwords of $w$ having the length $n$. We define the \emph{maximal complexity} C(w) as the maximum of the…
Let $w$ be a finite word of length $n$. In this paper, we study the maximum possible number of distinct rational power factors in a finite word. A rational power is a word of the form $u=p^kp'$, where $p$ is a nonempty finite word, $k$ is…
We introduce the space function $s(n)$ of a finitely presented semigroup $S =<A\mid R>.$ To define $s(n)$ we consider pairs of words $w,w'$ over $A$ of length at most $n$ equal in $S$ and use relations from $R$ for the transformations…
In this paper, we study the fibers of "automorphic word maps", a certain generalization of word maps, on finite groups and on nonabelian finite simple groups in particular. As an application, we derive a structural restriction on finite…
We study equations in groups G with unique m-th roots for each positive integer m. A word equation in two letters is an expression of the form w(X,A) = B, where w is a finite word in the alphabet {X,A}. We think of A,B in G as fixed…
In this paper we investigate local to global phenomena for a new family of complexity functions of infinite words indexed by $k \in \Ni \cup \{+\infty\}$ where $\Ni$ denotes the set of positive integers. Two finite words $u$ and $v$ in…
Partial words are sequences over a finite alphabet that may contain wildcard symbols, called holes, which match or are compatible with all letters; partial words without holes are said to be full words (or simply words). Given an infinite…
Let S be a finite set of words over an alphabet Sigma. The set S is said to be complete if every word w over the alphabet Sigma is a factor of some element of S*, i.e. w belongs to Fact(S*). Otherwise if S is not complete, we are interested…
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 begin with a new analysis of formal words. Let w be a formal word in letters g_1,...,g_k. The word map associated with w maps the permutations s_1,...,s_k in S_n to the permutation obtained by replacing for each i, every occurrence of…
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…
Let $w$ be a word in the free group of rank $n \in \mathbb{N}$ and let $\mathcal{V}(w)$ be the variety of groups defined by the law $w=1$. Define $\mathcal{V}(w^*)$ to be the class of all groups $G$ in which for any infinite subsets $X_1,…
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 word is called closed if it has a prefix which is also its suffix and there is no internal occurrences of this prefix in the word. In this paper we study words that are rich in closed factors, i.e., which contain the maximal possible…
The symbolic complexity of an infinite word $W$ is the function $p_W(l)$ counting the number of different subwords in $W$ of length $l$. In this paper our main purpose is to study the complexity for a class of topological dynamical systems,…
Let $f_W(n)$ be the number of different factors of length $n$ appearing in $W$. A classical result of Morse and Hedlund, stated in 1938, asserts that an infinite word $W$ is ultimately periodic if and only if $f_W(n)\leq n$ for some $n\in…
A word $u=u_1\dots u_n$ is a scattered factor of a word $w$ if $u$ can be obtained from $w$ by deleting some of its letters: there exist the (potentially empty) words $v_0,v_1,..,v_n$ such that $w = v_0u_1v_1...u_nv_n$. The set of all…
A finite word $w$ is called \emph{rich} if it contains $\vert w\vert+1$ distinct palindromic factors including the empty word. Let $q\geq 2$ be the size of the alphabet. Let $R(n)$ be the number of rich words of length $n$. Let $d>1$ be a…
Let $A_q$ be a $q$-letter alphabet and $w$ be a right infinite word on this alphabet. A subword of $w$ is a block of consecutive letters of $w$. The subword complexity function of $w$ assigns to each positive integer $n$ the number $f_w(n)$…