Related papers: Constructing Premaximal Binary Cube-free Words of …
A word is cubefree if it contains no non-empty subword of the form xxx. A morphism h : Sigma^* -> Sigma^* is k-uniform if h(a) has length k for all a in Sigma. A morphism is cubefree if it maps cubefree words to cubefree words. We show that…
We investigate the number of sets of words that can be formed from a finite alphabet, counted by the total length of the words in the set. An explicit expression for the counting sequence is derived from the generating function, and…
It is known that the number of overlap-free binary words of length n grows polynomially, while the number of cubefree binary words grows exponentially. We show that the dividing line between polynomial and exponential growth is 7/3. More…
Twins in a finite word are formed by a pair of identical subwords placed at disjoint sets of positions. We investigate the maximum length of twins in a random word over a $k$-letter alphabet. The obtained lower bounds for small values of…
We start by considering binary words containing the minimum possible numbers of squares and antisquares (where an antisquare is a word of the form $x \overline{x}$), and we completely classify which possibilities can occur. We consider…
A square-free word $w$ over a fixed alphabet $\Sigma$ is extremal if every word obtained from $w$ by inserting a single letter from $\Sigma$ (at any position) contains a square. Grytczuk et al. recently introduced the concept of extremal…
We consider questions related to the structure of infinite words (over an integer alphabet) with bounded additive complexity, i.e., words with the property that the number of distinct sums exhibited by factors of the same length is bounded…
A tower between two regular languages is a sequence of strings such that all strings on odd positions belong to one of the languages, all strings on even positions belong to the other language, and each string can be embedded into the next…
We re-examine previous constructions of infinite binary words containing few distinct squares with the goal of finding the "simplest", in a certain sense. We exhibit several new constructions. Rather than using tedious case-based arguments…
Cross-bifix-free sets are sets of words such that no prefix of any word is a sufix of any other word. In this paper, we introduce a general constructive method for the sets of cross-bifix-free q-ary words of fixed length. It enables us to…
We consider the class of groups whose word problem is poly-context-free; that is, an intersection of finitely many context-free languages. We show that any group which is virtually a finitely generated subgroup of a direct product of free…
We consider words over a binary alphabet. A word $w$ is overlap-free if it does not have factors (blocks of consecutive letters) of the form $uvuvu$ for nonempty $u$. Let $M(w)$ denote the number of positions that are middle positions of…
We investigate the structural relationship between prefix-free codes over the binary alphabet and a class of unlabeled rooted trees, which we call \emph{symmetric} trees. We establish a canonical correspondence between prefix-free codes and…
A bidimensional bifix (in short bibifix) of a square matrix T is a square submatrix of T which occurs in the top-left and bottom-right corners of T. This allows us to extend the definition of bifix-free words and cross-bifix-free set of…
In this paper we study the maximal pattern complexity of infinite words up to Abelian equivalence. We compute a lower bound for the Abelian maximal pattern complexity of infinite words which are both recurrent and aperiodic by projection.…
The avoidability, or unavoidability of patterns in words over finite alphabets has been studied extensively. A word (pattern) over a finite set is said to be unavoidable if, for all but finitely many words, there exists a morphism mapping…
A word is square-free if it does not contain any square (a word of the form $XX$), and is extremal square-free if it cannot be extended to a new square-free word by inserting a single letter at any position. Grytczuk, Kordulewski, and…
A longstanding question of Gromov asks whether every one-ended word-hyperbolic group contains a subgroup isomorphic to the fundamental group of a closed hyperbolic surface. An infinite family of word-hyperbolic groups can be obtained by…
Much work has been done to identify which binary codes can be represented by collections of open convex or closed convex sets. While not all binary codes can be realized by such sets, here we prove that every binary code can be realized by…
We introduced the notation of a set of prohibitions and give definitions of a complete set and a crucial word with respect to a given set of prohibitions. We consider 3 particular sets which appear in different areas of mathematics and for…