Related papers: Universal Partial Words over Non-Binary Alphabets
A de Bruijn sequence of order $k$ over a finite alphabet is a cyclic sequence with the property that it contains every possible $k$-sequence as a substring exactly once. Orthogonal de Bruijn sequences are collections of de Bruijn sequences…
In this paper, we find a criterium for universal equivalence of partially commutative Lie algebras whose defining graphs are trees. Besides, we obtain bases for partially commutative metabelian Lie algebras.
A word $w$ is called a reaching word of a subset $S$ of states in a deterministic finite automaton (DFA) if $S$ is the image of $Q$ under the action of $w$. A DFA is called completely reachable if every non-empty subset of the state set has…
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…
A square is the concatenation of a nonempty word with itself. A word has period p if its letters at distance p match. The exponent of a nonempty word is the quotient of its length over its smallest period. In this article we give a proof of…
In this paper, we consider partially commutative metabelian Lie algebras whose defining graphs are cycles. We show that such algebras are universally equivalent iff the corresponding cycles have the same length. Moreover, we give an example…
Various kinds of infinitary operations satisfying forms of associativity have been considered in the literature by various authors, including A. Tarski, C. Karp, J. H. Conway, D. Krob, N. Bedon, and C. Rispal. Applications include the…
A graph $G=(V,E)$ is word-representable if there exists a word $w$ over the alphabet $V$ such that letters $x$ and $y$ alternate in $w$ if and only if $xy\in E$. For integers $n>k>0 $, the shift graph $G(n,k)$ is the graph whose vertex set…
Let $Di\langle X\rangle$ be the free dialgebra over a field generated by a set $X$. Let $S$ be a monic subset of $Di\langle X\rangle$. A Composition-Diamond lemma for dialgebras is firstly established by Bokut, Chen and Liu in 2010…
In [X. Droubay et al, Episturmian words and some constructions of de Luca and Rauzy, Theoret. Comput. Sci. 255 (2001)], it was proved that every word w has at most |w|+1 many distinct palindromic factors, including the empty word. The…
We prove that for every integer $n > 0$ and for every alphabet $\Sigma_k$ of size $k \geq 3$, there exists a necklace of length $n$ whose Burrows-Wheeler Transform (BWT) is completely unclustered, i.e., it consists of exactly $n$ runs with…
The notion of a two-dimensional word arises naturally in the study of combinatorics on words, while the iterative construction of pedal triangles results in a rich dynamical system in the study of geometry. At first, these two classes of…
The notion of Abelian complexity of infinite words was recently used by the three last authors to investigate various Abelian properties of words. In particular, using van der Waerden's theorem, they proved that if a word avoids Abelian…
If w is a word in d>1 letters and G is a finite group, evaluation of w on a uniformly randomly chosen d-tuple in G gives a random variable with values in G, which may or may not be uniform. It is known that if G ranges over finite simple…
Shallit and Wang studied deterministic automatic complexity of words. They showed that the automatic Hausdorff dimension $I(\mathbf t)$ of the infinite Thue word satisfies $1/3\le I(\mathbf t)\le 2/3$. We improve that result by showing that…
We present sufficient conditions for when an ordering of universal cycles $\alpha_1, \alpha_2, \ldots, \alpha_m$ for disjoint sets $\mathbf{S}_1, \mathbf{S}_2, \ldots , \mathbf{S}_m$ can be concatenated together to obtain a universal cycle…
In this work, we introduce a new notion for representing graph classes with formal languages. In contrast to the seminal work by Kitaev and Pyatkin to represent graphs by words, we use formal binary languages in order to have a set of…
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…
The Fibonacci word $W$ on an infinite alphabet was introduced in [Zhang et al., Electronic J. Combinatorics 2017 24(2), 2-52] as a fixed point of the morphism $2i\rightarrow (2i)(2i+1)$, $(2i+1) \rightarrow (2i+2)$, $i\geq 0$. Here, for any…
Inspired by OEIS sequence A377912, which consists of the nonnegative integers in which every even digit (except possibly the last) is immediately followed by a strictly larger digit, we define even-up and odd-up words over an alphabet of…