# Questions tagged [combinatorics-on-words]

A branch of combinatorics that focuses on the study of words and formal languages

88
questions

**4**

votes

**0**answers

53 views

### Computability of the “free envelope rank” of an endomorphism of a free group

Let $F$ be a free group freely generated by the finite set $S$ and $\sigma\colon F\to F$ be a group morphism. We define the free envelope rank of $\sigma$, written $r(\sigma)$, as the smallest $k$ for ...

**2**

votes

**1**answer

279 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 $x$ ...

**1**

vote

**0**answers

80 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 ...

**2**

votes

**1**answer

85 views

### What is the cardinality of the set of Dyck natural numbers of semilength $k$?

In arXiv:2102.02777 ("Recursive Prime Factorizations: Dyck Words as Numbers"), I show that there is a 1:1 correspondence between $\mathbb{N} = \{0,1,2,3,4,\ldots\}$ and $\mathcal{D}_{r_{\...

**-1**

votes

**1**answer

108 views

### Prove using Dyck naturals: for $n \in \mathbb{N}_{+}$ and big enough $k \in \mathbb{N}_{+}$, $p_{k-1} < \cdots < np_{k-a_{n}}$ (a is A073093)

While conducting research in connection with arXiv:2102.02777 ("Recursive Prime Factorizations: Dyck Words as Numbers"), I noticed certain interesting patterns, one of which inspired the ...

**1**

vote

**0**answers

59 views

### Words with finite critical exponent

Let $\mathcal{A}$ be a finite set. Is there a nice characterization of the subset of $S\subset \mathcal{A}^\omega$ such that every $w\in S$ has finite critical exponent? Of course $S$ has measure zero ...

**1**

vote

**1**answer

92 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}\ ...

**16**

votes

**0**answers

827 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 ...

**2**

votes

**3**answers

577 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 ...

**3**

votes

**1**answer

170 views

### Binary words that are nonconstant on long arithmetic progressions

Let $w=x_0 x_1 x_2 \ldots$ be an infinite word, where each $x_i\in \{0,1\}$. For each positive integer $k$ (thought of as the jump size of an arithmetic progression) and each residue $0\leq a \leq k-...

**10**

votes

**0**answers

258 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 $A$.
...

**4**

votes

**0**answers

97 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 ...

**2**

votes

**0**answers

175 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**

votes

**1**answer

160 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 $\{...

**9**

votes

**2**answers

350 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 ($a\to h(a)$):...

**1**

vote

**1**answer

91 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'...

**10**

votes

**0**answers

380 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 ...

**6**

votes

**2**answers

268 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......

**4**

votes

**1**answer

215 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 $\frac{d(a_1a_2...a_n,a_{k+1}a_{k+2}...a_{k+n})}{n}<\epsilon$ ?
($d(u,v)$...

**7**

votes

**1**answer

171 views

### Is the density of 1's in the Fibonacci word uniform?

The Fibonacci word is the limit of the sequence of words starting with $0$ and satisfying rules $0 \to 01, 1 \to 0$. Equivalently, it is obtained from the recursion $S_n= S_{n-1}S_{n-2}$ under ...

**8**

votes

**1**answer

170 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 $\{1,\ldots, n\}$...

**19**

votes

**3**answers

1k views

### What is the fairest order for stage-striking (and is it the Thue-Morse sequence)?

Here's a fair-sequencing problem that doesn't quite match the usual fair-division problems. I think that, like those, the answer should also be the Thue-Morse sequence ("balanced alternation"), ...

**3**

votes

**1**answer

113 views

### Partition theorems for located words

In this paper Bergelson, Blass, and Hindman prove the following
Theorem 1.2 Let $W(\Sigma; v)$ be colored with finitely may colors and let $\bar s$ be an infinite sequence from $W(\Sigma; v)$. ...

**2**

votes

**1**answer

188 views

### Unique(ish) infinite string avoiding a set of patterns

Let $\Sigma$ be a finite alphabet of size at least 2. A (possibly infinite) string $s$ over alphabet $\Sigma$ encounters a pattern $p \in \mathbb{N}^*$ iff there is a non-erasing morphism $f: \mathbb{...

**4**

votes

**1**answer

127 views

### Subshifts with a free semigroup

Let $X$ be a subshift on a finite alphabet. I'm interested in the following property: there exist words $s,t\in\mathcal L(X)$ (the language of $X$) such that $\{s,t\}^*\subset \mathcal L(X)$. That is, ...

**1**

vote

**0**answers

35 views

### Latent Dirichlet Allocation on Contrived Data

I am doing a project that seems like it might be susceptible to Latent Dirichlet Allocation. However, my data is highly contrived (both in test cases and use cases) and my "words" don't come close to ...

**12**

votes

**1**answer

312 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=\{...

**11**

votes

**1**answer

302 views

### Unique words in dihedral groups

Suppose $x$ is a word over the alphabet $\{0,1\}$.
Let $a$, $b$ be elements of the group Dih$_k$ for some $k$.
Let $\varphi=\varphi_{a,b,k}$ be the map from words over $\{0,1\}$ to elements of the ...

**3**

votes

**0**answers

232 views

### Cayley Graphs and Cyclically reduced words [closed]

Let $G$ be a finite group and $S$ be a symmetric generating set for $G$. (EDIT: Assume $S$ does not contain involutions!) Cyclically reduced words can be thought of as minimal length representatives ...

**4**

votes

**1**answer

155 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 ...

**0**

votes

**1**answer

145 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 ...

**5**

votes

**0**answers

691 views

### The functional equation $f(x) = qx + qxf(x) - f(x^2)$

A word (i.e., ordered string of letters) is bifix-free provided it has no proper initial string and terminal string that are identical. For example, the word $ingratiating$ has bifix $ing$, but the ...

**17**

votes

**0**answers

503 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 $m,k$...

**5**

votes

**1**answer

299 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-...

**3**

votes

**1**answer

213 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, $\ell_{001}(0001110010011001)=2$...

**0**

votes

**0**answers

177 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, ...

**1**

vote

**1**answer

101 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}...

**4**

votes

**2**answers

129 views

### Sturmian subword whose reverse is not a subword

Let ${\cal L}_n$ be the set of all subwords of length $n$ of a biinfinite Sturmian sequence, induced by a rotation coding with irrational angle $\theta$.
Take a word $w \in {\cal L}_{2^n}$ and write ...

**1**

vote

**1**answer

176 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 ...

**4**

votes

**1**answer

380 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 ...

**3**

votes

**0**answers

198 views

### Repartition of 1's in the “Chacon word”

Consider the "Chacon words": $B_0=0$ and $B_{n+1} = B_nB_n1B_n$. The word $B_n$ has $\ell_n := \frac{3^{n+1}-1}{2}$ digits and the number of $1$'s in $B_n$ is $\ell_n - 3^n = \ell_{n-1} \sim \ell_n/3$...

**2**

votes

**0**answers

112 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
$$ w_0=0^{m-1}d_0, w_1=w_0d_1, w_2=w_1d_2, \ldots, w_k = w_{k-1}d_k $$
...

**4**

votes

**1**answer

140 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

127 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, ...

**2**

votes

**0**answers

178 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 ...

**3**

votes

**1**answer

232 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 ...

**1**

vote

**1**answer

121 views

### Terminology for set of infinite strings with a certain prefix

Let $\mathcal{A}$ be a finite alphabet, and let $C$ be the Cantor space $\mathcal{A}^\omega$ under the product topology.
Given a finite string $s \in \mathcal{A}^*$, let $C(s)$ be the set of all ...

**4**

votes

**1**answer

294 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 ...

**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 ...

**15**

votes

**1**answer

487 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 ...