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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…
Consider the set of those binary words with no non-empty factors of the form $xxx^R$. Du, Mousavi, Schaeffer, and Shallit asked whether this set of words grows polynomially or exponentially with length. In this paper, we demonstrate the…
A \itbf{cover} of a string $x = x[1..n]$ is a proper substring $u$ of $x$ such that $x$ can be constructed from possibly overlapping instances of $u$. A recent paper \cite{FIKPPST13} relaxes this definition --- an \itbf{enhanced cover} $u$…
The numbers we study in this paper are of the form $B_{n, p}(k)$, which is the number of binary words of length $n$ that contain the word $p$ (as a subsequence) exactly $k$ times. Our motivation comes from the analogous study of pattern…
Let $q, n, m \in \mathbb{N}$ be such that $q$ is a prime power and $a, b \in \mathbb{F}$. In this article we establish a sufficient condition for the existence of a primitive normal pair $(\alpha, f(\alpha)) \in \mathbb{F}_{q^m}$ over…
The sequence of partial sums of Fibonacci numbers, beginning with $2$, $4$, $7$, $12$, $20$, $33,\dots$, has several combinatorial interpretations (OEIS A000071). For instance, the $n$-th term in this sequence is the number of length-$n$…
We analyze the convergence order of an algorithm producing the digits of an absolutely normal number. Furthermore, we introduce a stronger concept of absolute normality by allowing Pisot numbers as bases, which leads to expansions with…
A word on $q$ symbols is a sequence of letters from a fixed alphabet of size $q$. For an integer $k\ge 1$, we say that a word $w$ is $k$-universal if, given an arbitrary word of length $k$, one can obtain it by removing entries from $w$. It…
We refine a uniform algebraic approach for deriving upper bounds on reset thresholds of synchronizing automata. We express the condition that an automaton is synchronizing in terms of linear algebra, and obtain upper bounds for the reset…
Given a pattern $P$ and a text $T$, both strings over a binary alphabet, the binary jumbled string matching problem consists in telling whether any permutation of $P$ occurs in $T$. The indexed version of this problem, i.e., preprocessing a…
We define generalized de Bruijn words as those words having a Burrows-Wheeler transform that is a concatenation of permutations of the alphabet. We show that generalized de Bruijn words are in 1-to-1 correspondence with Hamiltonian cycles…
A binary word is Sturmian if the occurrences of each letter are balanced, in the sense that in any two factors of the same length, the difference between the number of occurrences of the same letter is at most 1. In digital geometry,…
A pattern p (i.e., a string of variables and terminals) matches a word w, if w can be obtained by uniformly replacing the variables of p by terminal words. The respective matching problem, i.e., deciding whether or not a given pattern…
We study the binary Ehrenfeucht Mycielski sequence seeking a balance between the number of occurrences of different binary strings. There have been numerous attempts to prove the balance conjecture of the sequence, which roughly states that…
The sequence $(x_n)_{n\in\mathbb N} = (2,5,15,51,187,\dots)$ given by the rule $x_n=(2^n+1)(2^{n-1}+1)/3$ appears in several seemingly unrelated areas of mathematics. For example, $x_n$ is the density of a language of words of length $n$…
We consider the number of occurrences of subwords (non-consecutive sub-sequences) in a given word. We first define the notion of subword entropy of a given word that measures the maximal number of occurrences among all possible subwords. We…
The complement $\overline{x}$ of a binary word $x$ is obtained by changing each $0$ in $x$ to $1$ and vice versa. We study infinite binary words $\bf w$ that avoid sufficiently large complementary factors; that is, if $x$ is a factor of…
The deviation of the observed frequency of a word $w$ from its expected frequency in a given sequence $x$ is used to determine whether or not the word is avoided. This concept is particularly useful in DNA linguistic analysis. The value of…
A normally regular digraph with parameters $(v,k,\lambda,\mu)$ is a directed graph on $v$ vertices whose adjacency matrix $A$ satisfies the equation $AA^t=k I+\lambda (A+A^t)+\mu(J-I-A-A^t)$. This means that every vertex has out-degree $k$,…
A double occurrence word (DOW) is a word in which every symbol appears exactly twice; two DOWs are equivalent if one is a symbol-to-symbol image of the other. We consider the so called repeat pattern ($\alpha\alpha$) and the return pattern…