Related papers: Dyck Words, Pattern Avoidance, and Automatic Seque…

Overlap-free words are words over the binary alphabet $A=\{a, b\}$ that do not contain factors of the form $xvxvx$, where $x \in A$ and $v \in A^*$. We analyze the asymptotic growth of the number $u_n$ of overlap-free words of length $n$ as…

We exhibit a recurrence on the number of discrete line segments joining two integer points in the plane using an encoding of such segments as balanced words of given length and height over the two-letter alphabet $\{0,1\}$. We give…

We consider sets of factors that can be avoided in square-free words on two-generator free groups. The elements of the group are presented in terms of 0,1,2,3 such that 0 and 2 (resp.,1 and 3) are inverses of each other so that 02, 20, 13…

For $p \in (0,1)$, sample a binary sequence from the infinite product measure of Bernoulli$(p)$ distributions. It is known that for $p=1/2$, almost every binary sequence is Poisson generic in the sense of Peres and Weiss, a property that…

Matrices of rank at most k are defined by the vanishing of polynomials of degree k + 1 in their entries (namely, their (k + 1)-times-(k + 1)-subdeterminants), regardless of the size of the matrix. We prove a qualitative analogue of this…

In combinatorics of words, a concatenation of $k$ consecutive equal blocks is called a power of order $k$. In this paper we take a different point of view and define an anti-power of order $k$ as a concatenation of $k$ consecutive pairwise…

Given a real number $\lambda > 1$, we say that $d|n$ is a $\lambda$-middle divisor of $n$ if $$ \sqrt{\frac{n}{\lambda}} < d \leq \sqrt{\lambda n}. $$ We will prove that there are integers having an arbitrarily large number of…

We say that a word $w$ on a totally ordered alphabet avoids the word $v$ if there are no subsequences in $w$ order-equivalent to $v$. In this paper we suggest a new approach to the enumeration of words on at most $k$ letters avoiding a…

A word is square-free if it does not contain nonempty factors of the form $XX$. In 1906 Thue proved that there exist arbitrarily long square-free words over a $3$-letter alphabet. It was proved recently [7] that among these words there are…

Given a random text over a finite alphabet, we study the frequencies at which fixed-length words occur as subsequences. As the data size grows, the joint distribution of word counts exhibits a rich asymptotic structure. We investigate all…

The Ces\`aro limit - the asymptotic average of a sequence of real numbers - is an operator of fundamental importance in probability, statistics and mathematical analysis. To better understand sequences with Ces\`aro limits, this paper…

Word matches are often used in sequence comparison methods, either as a measure of sequence similarity or in the first search steps of algorithms such as BLAST or BLAT. The D2 statistic is the number of matches of words of k letters between…

It is often claimed that Bayesian methods, in particular Bayes factor methods for hypothesis testing, can deal with optional stopping. We first give an overview, using elementary probability theory, of three different mathematical meanings…

An overlap-free (or $\beta$-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 an overlap (or a factor of exponent at least $\beta$,…

We show how to enumerate words in $1^{m_1} \dots n^{m_n}$ that avoid the increasing consecutive pattern $12 \dots r$ for any $r \geq 2$. Our approach yields an $O(n^{s+1})$ algorithm to enumerate words in $1^s \dots n^s$, avoiding the…

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…

We examine deterministic and nondeterministic state complexities of regular operations on prefix-free languages. We strengthen several results by providing witness languages over smaller alphabets, usually as small as possible. We next…

Let $s_n$ be the number of words consisting of the ternary alphabet consisting of the digits 0, 1, and 2 such that no subword (or factor) is a square (a word concatenated with itself, e.g., $11$, $1212$, or $102102$). From computational…

We present a strongly exponential lower bound that applies both to the subset synchronization threshold for binary deterministic automata and to the careful synchronization threshold for binary partial automata. In the later form, the…

It is an open problem whether $ \binom{2n}{n} $ is divisible by 4 or 9 for all $n>256$. In connection with this, we prove that for a fixed uneven $m$ the asymptotic density of $k$'s such that $ m \nmid \binom{2^{k+1}}{2^{k}} $ is 0. To do…