Related papers: Computing the $k$-binomial complexity of the Thue-…

Trapezoidal words are words having at most $n+1$ distinct factors of length $n$ for every $n\ge 0$. They therefore encompass finite Sturmian words. We give combinatorial characterizations of trapezoidal words and exhibit a formula for their…

We show that classical and quantum Kolmogorov complexity of binary strings agree up to an additive constant. Both complexities are defined as the minimal length of any (classical resp. quantum) computer program that outputs the…

Vincular or dashed patterns resemble classical patterns except that some of the letters within an occurrence are required to be adjacent. We prove several infinite families of Wilf-equivalences for k-ary words involving vincular patterns…

Following a recent paper of Anselmo et al., we consider $m \times n$ rectangular matrices formed from the Fibonacci word, and we show that their balance properties can be solved with a finite automaton. We also generalize the result to…

Covering arrays for words of length $t$ over a $d$ letter alphabet are $k \times n$ arrays with entries from the alphabet so that for each choice of $t$ columns, each of the $d^t$ $t$-letter words appears at least once among the rows of the…

We prove that every $n$-letter word over $k$-letter alphabet contains some word as a subsequence in at least $k^{n/4k(1+o(1))}$ many ways, and that this is sharp as $k\to\infty$. For fixed $k$, we show that the analogous number deviates…

In this paper we prove that for any infinite word W whose set of factors is closed under reversal, the following conditions are equivalent: (I) all complete returns to palindromes are palindromes; (II) P(n) + P(n+1) = C(n+1) - C(n) + 2 for…

Let ${\mathbb F}_q$ be the finite field of $q$ elements, where $q=p^r$ is a power of the prime $p$, and $\left(\beta_1, \beta_2, \dots, \beta_r \right)$ be an ordered basis of ${\mathbb F}_q$ over ${\mathbb F}_p$. For…

We investigate the least number of palindromic factors in an infinite word. We first consider general alphabets, and give answers to this problem for periodic and non-periodic words, closed or not under reversal of factors. We then…

Bisimulation equivalence (or bisimilarity) of first-order grammars is decidable, as follows from the decidability result by Senizergues (1998, 2005) that has been given in an equivalent framework of equational graphs with finite out-degree,…

A subsequence of a word $w$ is a word $u$ such that $u = w[i_1] w[i_2] , \dots w[i_{|u|}]$, for some set of indices $1 \leq i_1 < i_2 < \dots < i_k \leq |w|$. A word $w$ is $k$-subsequence universal over an alphabet $\Sigma$ if every word…

Recently, Fici, Restivo, Silva, and Zamboni defined a $k$-anti-power to be a word of the form $w_1w_2\cdots w_k$, where $w_1,w_2,\ldots,w_k$ are distinct words of the same length. They defined $AP(x,k)$ to be the set of all positive…

In this paper we introduce and study a new complexity measure for finite words. For positive integer $d$ special scattered subwords, called super-$d$-subwords, in which the gaps are of length at least $(d-1)$, are defined. We give methods…

We study the relative complexity of equivalence relations and preorders from computability theory and complexity theory. Given binary relations $R, S$, a componentwise reducibility is defined by $ R\le S \iff \ex f \, \forall x, y \, [xRy…

In the past few years powerful generalizations to the Euclidean k-means problem have been made, such as Bregman clustering [7], co-clustering (i.e., simultaneous clustering of rows and columns of an input matrix) [9,18], and tensor…

This thesis investigates three biologically inspired operations: prefix-suffix duplication, bounded prefix-suffix duplication, and prefix-suffix-square completion. Duplication, a common genetic mutation, involves repeating DNA sequences and…

We investigate the complexity of isomorphism relations for classes of finitely generated and n-generated computably enumerable (c.e.) algebras, presented via c.e. presentations -- that is, as quotients of term algebras over decidable sets…

A word $u$ is a subsequence of another word $w$ if $u$ can be obtained from $w$ by deleting some of its letters. The word $w$ with alph$(w)=\Sigma$ is called $k$-subsequence universal if the set of subsequences of length $k$ of $w$ contains…

The downward closure of a language is the set of all (not necessarily contiguous) subwords of its members. It is well-known that the downward closure of every language is regular. Moreover, recent results show that downward closures are…

A subsequence of a word $w$ is a word $u$ such that $u = w[i_1] w[i_2] \dots w[i_{k}]$, for some set of indices $1 \leq i_1 < i_2 < \dots < i_k \leq \lvert w\rvert$. A word $w$ is $k$-subsequence universal over an alphabet $\Sigma$ if every…