Related papers: Generating a Gray code for prefix normal words in …
We investigate the behavior of the periods and border lengths of random words over a fixed alphabet. We show that the asymptotic probability that a random word has a given maximal border length $k$ is a constant, depending only on $k$ and…
Ulam words are binary words defined recursively as follows: the length-$1$ Ulam words are $0$ and $1$, and a binary word of length $n$ is Ulam if and only if it is expressible uniquely as a concatenation of two shorter, distinct Ulam words.…
The observed frequency of the longest proper prefix, the longest proper suffix, and the longest infix of a word $w$ in a given sequence $x$ can be used for classifying $w$ as avoided or overabundant. The definitions used for the expectation…
We investigate certain word-construction games with variable turn orders. In these games, Alice and Bob take turns on choosing consecutive letters of a word of fixed length, with Alice winning if the result lies in a predetermined target…
A word is said to be bordered if it contains a nonempty proper prefix that is also a suffix. A pair of words $(u, v)$ is said to be mutually bordered if there exists a word that is a nonempty proper prefix of $u$ and suffix of $v$, and…
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…
We introduce a new construction for the balancing of non-binary sequences that make use of Gray codes for prefix coding. Our construction provides full encoding and decoding of sequences, including the prefix. This construction is based on…
We propose a method to improve traditional character-based PPM text compression algorithms. Consider a text file as a sequence of alternating words and non-words, the basic idea of our algorithm is to encode non-words and prefixes of words…
A word $w$ over an alphabet $\Sigma$ is a Lyndon word if there exists an order defined on $\Sigma$ for which $w$ is lexicographically smaller than all of its conjugates (other than itself). We introduce and study \emph{universal Lyndon…
It is well-known that, given a probability distribution over $n$ characters, in the worst case it takes (\Theta (n \log n)) bits to store a prefix code with minimum expected codeword length. However, in this paper we first show that, for…
Consider a random word $X^n=(X_1,\ldots ,X_n)$ in an alphabet consisting of $4$ letters, with the letters viewed either as $A$, $U$, $G$ and $C$ (i.e., nucleotides in an RNA sequence) or $\alpha$, $\bar{\alpha}$, $\beta$ and $\bar{\beta}$…
Free words are elements of a free monoid, generated over an alphabet via the binary operation of concatenation. Casually speaking, a free word is a finite string of letters. Henceforth, we simply refer to them as words. Motivated by recent…
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…
Cross-bifix-free sets are sets of words such that no prefix of any word is a suffix of any other word. In this paper, we introduce a general constructive method for the sets of cross-bifix-free binary words of fixed length. It enables us to…
A word of length $n$ is rich if it contains $n$ nonempty palindromic factors. An infinite word is rich if all of its finite factors are rich. Baranwal and Shallit produced an infinite binary rich word with critical exponent $2+\sqrt{2}/2$…
An absent word of a word y of length n is a word that does not occur in y. It is a minimal absent word if all its proper factors occur in y. Minimal absent words have been computed in genomes of organisms from all domains of life; their…
The combinatorics of RNA plays a central role in biology. Mathematical biologists have several commonly-used models for RNA: words in a fixed alphabet (representing the primary sequence of nucleotides) and plane trees (representing the…
As neural language models approach human performance on NLP benchmark tasks, their advances are widely seen as evidence of an increasingly complex understanding of syntax. This view rests upon a hypothesis that has not yet been empirically…
The (bitwise) complement $\overline{x}$ of a binary word $x$ is obtained by changing each $0$ in $x$ to $1$ and vice versa. An $\textit{antisquare}$ is a nonempty word of the form $x\, \overline{x}$. In this paper, we study infinite binary…
We discuss the notion of privileged word, recently introduced by Peltomaki. A word w is privileged if it is of length <=1, or has a privileged border that occurs exactly twice in w. We prove the following results: (1) if w^k is privileged…