All Questions
Tagged with combinatorics-on-words co.combinatorics
76 questions
4
votes
1
answer
189
views
Equation in the conjugacy class of a free group
I will pose the question in the form in which it originally appeared to me:
Let $a,b,c,d$ be different letters in a finite alphabet $\mathcal{Z}$. Let $Q$ and $R$ be finite words with letters from $\...
- 111
6
votes
1
answer
388
views
What is the max number of self-segregating words of length n?
A set of words S is called self-segregating if you don't need whitespaces to read them. It means that for any two words from S no new words from S arise between them.
For example the set ab, bc, ac, ...
- 125
10
votes
1
answer
467
views
Elegant proof for $xy < yx \Leftrightarrow x^\mathbb{N} < y^\mathbb{N}$
Let $x, y$ be finite words over totally ordered alphabet and $<$ denote the lexicographical order, i.e for two not necessarily finite words we say $x < y$ iff one of the following holds
There ...
- 163
1
vote
0
answers
169
views
A function $g : \{0,1\}^m \to \{0,1\}^{4m}$ such that the “circular discrepancy” between $g(x_1)$ and $g(x_2)$ is $\geq m$ for any $x_1 \neq x_2$
In this question, the term “word” implies a binary word, i.e. a sequence of bits.
Let $W(x)$ denote the number of non-zero bits in a word $x$.
Assuming that $x$ is an $s$-bit word and $0 \le k < s$,...
- 959
2
votes
0
answers
101
views
Combinatorics on non-associative words
In my P.h.d research, I deal (among other things) with non-associative words, which we call monomials, and we need to consider two types of operations with these monomials.
The first one is simply ...
6
votes
1
answer
279
views
A Sauer-Shelah-like lermma for prefix tree
I proved a variant of the Sauer-Shelah lemma and I was wondering if something like that is already known.
Let $S \subseteq \{0,1\}^n $. We say that a set of coordinates $K \subseteq [n]$ is shattered ...
- 419
2
votes
1
answer
168
views
Is there an efficient algorithm that allows to construct a binary word with particular properties related to its horizontal and vertical “subwords”?
Let $w$ denote an $mn$-bit word (i.e. a binary word of length $mn$). Assuming that $b_{i,j}$ denote individual bits, we can represent $w$ in the “rectangular” form as follows:
$$\begin{array}{l}
b_{1....
- 959
4
votes
1
answer
260
views
Word combinatorics terminology question
I'm looking for the name of what I suspect must be a standard property, and also for a possible statement about that property.
First the property: $W=a_0\ldots a_{n-1}$ has this property if for all $1\...
- 23.2k
1
vote
1
answer
121
views
Is there an efficient generalized algorithm to generate a set of binary words satisfying a particular cross-correlation property?
In this question, the term “word” implies a binary word, i.e. a sequence of bits.
Let $W(w)$ denote the number of non-zero bits in a word $w$.
Assuming that $l \geq 2$ is even, an $l$-bit word $w$ is ...
- 959
7
votes
2
answers
319
views
Is there an efficient generalized algorithm to find at least one binary word with the maximum rotational imbalance and the full $\{0, 1\}$-balance?
Assuming that $x$ is a sequence of $l$ bits (i.e. a binary word of length $l$) and $0 \le m < l$, let $R(x, m)$ denote the result of the left bitwise rotation (i.e. the left circular shift) of $x$ ...
- 959
3
votes
1
answer
261
views
Words with critical exponent $< \frac 73$
In a comment made by Gjergji Zaimi to this older question, it is conjectured that $\frac 73$ is the threshold separating countability and uncountability of the sets of infinite binary words having a ...
- 4,459
5
votes
1
answer
310
views
In the Oldenburger-Kolakoski sequence, is #1s = #2s infinitely many times?
The Oldenburger-Kolakoski sequence, $OK$, is the unique sequence of $1$s and $2$s that starts with $1$ and is its own runlength sequence:
$$OK = (1,2,2,1,1,2,1,2,2,1,2,2,1,1,2,1,1,2,2,1,2,1,1,\ldots).$...
- 479
3
votes
1
answer
349
views
Is the number of words finite, when you don't know how to count?
This question is inspired by this one:
Can you do math without knowing how to count?
Let $M_2$ be the set of words constructed by concatenation of the letters $a_1$ and $a_2$, with :
(*) : for any $x$ ...
- 4,074
4
votes
1
answer
245
views
Hausdorff dimension and critical exponent of words
What is the Hausdorff dimension of the subset $S_c \subset [0,1]$ of points such that the critical exponent of their binary expansion is $c$? It's clear that $\dim_H S_{\infty}=1$, but what can be ...
- 4,459
12
votes
1
answer
427
views
Subwords of the infinite Fibonacci word
Let $W = 01001010010010 \ldots$ be the infinite Fibonacci word, A003849 in the OEIS. Let $B(m)$ be the set of $m+1$ subwords of $W$ that have length $m$, and for each such subword $u$, let $p(u)$ be ...
- 479
2
votes
1
answer
121
views
Binary words starting with arbitrarily long squares
What is the measure of the following set of infinite binary words?
$S=\{w\in\{0,1\}^\omega\ \text{such that},\ \text{for every}\ N\in\mathbb{N},\, w\ \text{has a prefix of the form}\ pp\ \text{with}\ ...
- 4,459
33
votes
0
answers
2k
views
The easily bored sequence
If we want to compare the repetitiveness of two finite words, it looks reasonable, first of all, to consider more repetitive the word repeating more times one of its factors, and secondarily to ...
- 4,459
2
votes
3
answers
639
views
The critical exponent function
It is a known fact [1] that, for every $c\in (1,\infty]$, it is possible to find a finite alphabet $\mathcal{A}$ and a word $w\in \mathcal{A}^\omega$ such that $w$ has critical exponent $c$. It looks ...
- 4,459
9
votes
0
answers
467
views
Measuring the randomness of texts
The question concerns statistic properties of random words in a finite alphabet $A$.
By $A^{<\omega}$ we denote the set of all words in the alphabet $A$, i.e. finite sequences of elements of $A$.
...
- 42k
4
votes
0
answers
145
views
Words that give rise to an enumeration of elements of the symmetric group
Let $\mathbb{S}_m$ be the symmetric group on $m$ letters. Let $n=m-1$. Let $\mathbf{w}=a_1\cdots a_r$ be a word on the alphabet $\{1,\ldots,n\}$. We say that $\mathbf{w}$ gives rise to an enumeration ...
- 1,066
2
votes
0
answers
189
views
$V$-like actions of $V$
This continues my question about prefix-continuous bijections (since the answer was "yes").
Notation and conventions: Let $A$ be a finite alphabet and $L \subset A^*$ a language. Let $G$ be a group. ...
- 6,652
6
votes
1
answer
193
views
Is there a prefix-continuous bijection between finite words and eventually zero words?
Let
$$ X = \{x \in \{0,1\}^{\omega} \;|\; \exists m: \forall i \geq m: x_i = 0\} $$
(one-way infinite eventually zero words). Let $\{0,1\}^*$ denote the finite (not necessarily nonempty) words over $\{...
- 6,652
9
votes
2
answers
383
views
A cubefree-preserving morphism from 5 to 2?
A word is cubefree if it cannot be written as $xyyyz$ where $y$ has positive length.
Let $h$ be the morphism from $\{0,1,2,3,4\}^*$ to $\{0,1\}^*$ given for words of length 1 as follows ($a\to h(a)$):...
- 24.8k
1
vote
1
answer
110
views
Cliques in overlap graphs for words
Let $\Sigma$ be a finite alphabet, and consider the free monoid $\Sigma^*$. Given $w, w' \in \Sigma^*$ we say that $w$ overlaps $w'$ if there exist non-empty words $u, v, u'$ such that $w = uv$ and $w'...
- 435
10
votes
0
answers
399
views
Words and ranks
Let me state two problems that look very much alike. The first one can be solved putting together answers that different people have given to some questions I asked here a few weeks ago. The second ...
- 20.2k
6
votes
2
answers
319
views
Uniqueness of "Limit" of Cyclic Binary Strings
Set-up: By abuse, let $\sigma$ represent both the left shift operator on infinite bi-infinite strings and the cyclic left shift operator on finite strings. (Thus, for example, $\sigma(...01\bar{0}10......
- 115
4
votes
1
answer
231
views
Conjecture about infinite word
Let $w=a_1a_2a_3...$ be an infinite word over a finite alphabet and $\epsilon>0$. Do there exist integers $n,k$ such that $\frac{d(a_1a_2...a_n,a_{k+1}a_{k+2}...a_{k+n})}{n}<\epsilon$ ?
($d(u,v)$...
user136612
8
votes
1
answer
213
views
Minimum number of permutations of $\{1,\ldots, n\}$ that together contain every $k$-subpermutation
Define a $k$-permutation of $\{1,\ldots, n\}$ to be a word $\tau_1 \ldots \tau_k$ such that $\{\tau_1,\ldots,\tau_k\}$ is a $k$-subset of $\{1,\ldots, n\}$. Thus an $n$-permutation of $\{1,\ldots, n\}$...
- 11.2k
12
votes
1
answer
415
views
"Bisecting" a free subgroup with respect to word length
My broad question is regarding the lengths of (reduced) words in a subgroup of a free group.
As motivation, consider the free group $Gp(S)$ where $|S|=n$, that is, a free group of rank $n$. Let $S=\{...
- 949
4
votes
1
answer
164
views
Covering sequences of words
(If anyone has a better title please change it!)
Given two finite words $v,w$ in the alphabet $\{a,b\}$, define the $v$-proportion of $w$ to be the largest number of letters in $w$ which can be ...
- 743
0
votes
1
answer
159
views
How many words are there such that some word $X$ is subsequence of them?
Let's define subsequence of the word as part of the word created by deleting some of its letters, for example aetics is a subsequence of mathematics.
QUESTION.
Given a $3$-letter word (let's call it ...
17
votes
0
answers
536
views
Question about combinatorics on words
Let $\{a_1,a_2,...,a_n\}$ be an alphabet and let $\{u_1,...,u_n\}$ be words in this alphabet, and $a_i\mapsto u_i$ be a substitution $\phi$.
Question: Is there an algorithm to check if for some $m,k$...
user6976
5
votes
1
answer
399
views
Number of Lyndon words of given weight
Consider the alphabet consisting of two letters $a$ and $b$, and put the lexicographic order in which $a<b$.
We say that a non-empty word $w$ in this alphabet is a Lyndon word if, for any non-...
- 51
3
votes
1
answer
282
views
Longest runs and concentration of measure
Consider the longest runs $\ell_\sigma(x)$ of the pattern $\sigma$ for $\sigma\in \{0, 1, 01, 10, 001,\dots\}$ etc. in a binary sequence $x=x_1\dots x_n$.
For example, $\ell_{001}(0001110010011001)=2$...
- 24.8k
0
votes
0
answers
185
views
Sum of unit vectors always has a binary span after constrained permutations
Conjecture:
Let $e_1 = (1,0,\ldots,0), \ldots , e_{m_1+m_2} = (0,\ldots,0,1)$ be the unit vectors of the standard basis $E$ of $\mathbb{R}^{m_1+m_2}$.
An enumeration $ E \cup -E = \{f_1, \ldots, ...
- 121
1
vote
1
answer
105
views
Weighted counting of circular codes
Given a circular code $X$ (for example: $X=\{ w,b \}$) with generating function $u(z)=\sum\limits_{k=0}^{\infty}{u_k z^k}$ (in this example : $u(z)=2z$), the generating function $p(z)=\sum\limits_{k=0}...
- 26.5k
1
vote
1
answer
220
views
Building the string on $\{0,1\}$ alphabet with $\Omega(n^{2})$ different substrings [closed]
As we know the number of different substrings has the upper bound $O(n^{2})$.
Consider the strings on $\{0,1\}$ alphabet. Can I build a string with $\Omega(n^{2})$ different substrings?
Actually I ...
- 137
5
votes
1
answer
447
views
Periodic strings
I wish to ask a problem in periodic strings, it might be well-known but I am a beginner in this subject, so I am very glad if someones can show me. My problem is that can we add some string to the end ...
- 51
2
votes
0
answers
115
views
Zero-one links: how many, and how to produce?
For $m \geq 1$, define a link to be a zero-one word $w=d_0d_1 \ldots d_k$, where $d_0=0$ and $k=2^m-1$ , such that the words
$$ w_0=0^{m-1}d_0, w_1=w_0d_1, w_2=w_1d_2, \ldots, w_k = w_{k-1}d_k $$
...
- 3,443
5
votes
2
answers
203
views
Existence of an infinite word with a predetermined asymptotic for the word complexity
Let $w$ be an infinite binary word, for example: $$1010100001 0010011000 0001001110 0101011011 \dots$$
Let $N_w(k)$ be the set of distinct subwords of $w$ of length $k$, and $n_w(k)$ the cardinal of ...
3
votes
1
answer
147
views
Number and asymptotic for cyclic sequences
Cyclic sequence is equivalence class of cyclic shift action.
If $a = (a_1, ... , a_i)_c$ is cyclic sequence then $(a_1, a_2, \ldots a_{i-1}, a_i)_c = (a_2, a_3, \ldots, a_i, a_1)_c = \ldots = (a_i, ...
- 123
2
votes
0
answers
187
views
Words with Local plus Global Constraints
While doing estimates on the complexity of an algorithm I have run into a word-combinatorial problem with both a local and a global constraint.
This seems to be a rather general situation and I'm ...
- 121
3
votes
1
answer
274
views
Counting triple-free sequences
The Thue-Morse sequence is a triple-free element $(a_i)\in\{0,1\}^\mathbb{N}$ which can be constructed by iterating the transformations $0\to 01$ and $1\to 10$ starting from $0$.
The beginning of the ...
- 743
4
votes
1
answer
301
views
Strings with no long runs from proper subalphabets
Let $R_{n,k,b}$ be the number of $b$-ary strings of length $n$ that contain some run of length at least $k$ from some $(b-1)$-ary subalphabet. Let $N_{n,k,b}=b^n-R_{n,k,b}$ be the size of the ...
- 24.8k
19
votes
5
answers
1k
views
Three-halves-free words (analogous to square-free)
A square-free word
is a string of symbols (a "word") that avoids the pattern $XX$, where $X$ is any
consecutive sequence of symbols in the string.
For alphabets of two symbols, the longest square-free ...
- 151k
15
votes
1
answer
558
views
Combinatorics of palindromic decompositions
This is sort of a companion to my question Number of trivializations of a trivial word in the free group (which in turn is motivated by my earlier question here). It turns out that that question may ...
- 18.4k
13
votes
1
answer
543
views
Number of trivializations of a trivial word in the free group
Let $M$ be the free monoid on $2n$ generators $x_1,X_1,...,x_n,X_n$ and consider the set $T$ of all those elements of $M$ which map to 1 of the free group on $x_1,...,x_n$ under the homomorphism $\pi$ ...
- 18.4k
15
votes
0
answers
487
views
Word complexity of primes mod 4
For an infinite binary word $w$, the word complexity $f_w(n)$ is defined as the number of different subwords of length $n$. The asymptotic behavior of this function is an important parameter of the ...
- 17.1k
3
votes
1
answer
614
views
Combinatorics problem involving counting the number of certain substrings
I'm not sure if this question is suited for MO, but it does seem quite challenging to me, and is required for a research problem in chemistry I'm working on. I did try getting help from elsewhere (...
- 285
0
votes
1
answer
530
views
Is it possible to create an infinite sequence in which no subsequence is repeated 3 times in a row?
In Chess, there is the Threefold Repetition rule where if a sequence of moves is repeated 3 times in a row, either player can claim a draw.
Say two players wanted to play a legal, infinite game of ...
- 133