3
votes
1answer
110 views

Calculating the probability that all possible length $r$ subwords exists in a string, with or without overlaps allowed

Let $S$ be a length $L$ string, where each character in the string is chosen with uniform random probability over an alphabet with $q$ characters. For example, a binary string would imply $q = 2$, a ...
2
votes
1answer
107 views

Representability of sets of infinite sequences sharing common prefixes and factors (i.e. infixes)

Here we are concerned with the space $X^{\omega}$ of infinite sequences. Denote by $F_n(\xi)$ the set of factors (consecutive finite subsequences) of length $n$ and consider the set $$ K_n(\xi) = ...
13
votes
7answers
1k views

Two questions from combinatorics on words

Question 1. Assume that an infinite word $u\in\{0,1\}^{\mathbb Z}$ is not balanced. Is it true that there exists a finite 0-1 word $w$ such that $0w01w1$ or $1w10w0$ is a factor of $u$? Is it true ...
5
votes
1answer
108 views

What prefix and factors determine a ultimately periodic word uniquely

Let $\xi$ be an ultimately periodic sequence, i.e. there exists finite sequences $p, q \in X^*$ such that $\xi = pq^{\omega}$. Does there exists a $n > 0$ such that the prefix of length $n$ and all ...
3
votes
2answers
166 views

Ordering on words

What are the known computation-friendly well-orderings on words from $A^*$, where $A$ is a finite alphabet, except the standard weightlex and syllable-order?
20
votes
2answers
546 views

congruence on words: having the same (scattered) subwords of length at most n

For a fixed finite alphabet $A=\{a,b,...\}$, write $x \sim_n y$ if the two words $x$ and $y$ have the same (scattered) subwords of length at most $n$. The relation $\sim_n$ is a congruence of finite ...
4
votes
4answers
457 views

Subwords of cube-free binary words

I'm currently working on subwords of cube-free binary words. A binary word is one composed of letters from a two-letter alphabet such as $\{0,1\}$. A word $y$ is a subword of $w$ if there exist ...
10
votes
1answer
435 views

Analogues of the Knuth and Forgotten equivalences on permutations: have they been studied?

Consider a totally ordered alphabet $A$ of $n$ letters. Let $W$ be the set of all words over $A$ which have no two letters equal. Then, for example, we can define the Knuth equivalence on $W$ as the ...
21
votes
3answers
693 views

an operation on binary strings

Recently, as part of some joint research, Tom Roby was led to a curious operation on strings of L's and R's which he calls "bounce-reading": We start by reading the string at the left. When the ...
8
votes
2answers
576 views

Ubiquitous Zimin words

Let $w$ be a word in letters $x_1,...,x_n$. A value of $w$ is any word of the form $w(u_1,...,u_n)$ where $u_1,...,u_n$ are words. For example, $abaaba$ is a value of $x^2$. A word $u$ is called ...
18
votes
5answers
1k views

Are there uncountably many cube-free infinite binary words?

In Cube-free infinite binary words it was established that there are infinitely many cube-free infinite binary words (see the earlier question for definitions of terms). The construction given in ...
11
votes
3answers
1k views

Cube-free infinite binary words

A word $y$ is a subword of $w$ if there exist words $x$ and $z$ (possibly empty) such that $w=xyz$. Thus, $01$ is a subword of $0110$, but $00$ is not a subword of $0110$. I'm interested in ...
13
votes
5answers
1k views

subwords of the fibonacci word

The Fibonacci word is the limit of the sequence of words starting with "0" and satisfying rules $0 \to 01, 1 \to 0$. It's equivalent to have initial conditions $S_0 = 0, S_1 = 01$ and then recursion ...
3
votes
2answers
239 views

Maximal words (reloaded)

I have 3 more questions about maximal words (which are just another way of talking of necklaces). Let W be a finite word on a two symbol alphabet {0,1}; let us say that W is maximal if it is the last ...
3
votes
5answers
341 views

Strings and “co-subsequences”

Let $S$ be a string over some alphabet $\Sigma$. It is well known that a substring of $S$ is commonly defined as a sequence of contiguous elements from $S$, while a subsequence of $S$ is a sequence ...