# Questions tagged [combinatorics-on-words]

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

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### 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 ...
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### 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 $x$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ($a\to h(a)$):...
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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 $w'... 0answers 380 views ### Words and ranks Let me state two problems that look very much alike. The first one can be solved putting together answers that different people have given to some questions I asked here a few weeks ago. The second ... 2answers 268 views ### Uniqueness of “Limit” of Cyclic Binary Strings Set-up: By abuse, let$\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......
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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 $\frac{d(a_1a_2...a_n,a_{k+1}a_{k+2}...a_{k+n})}{n}<\epsilon$ ? ($d(u,v)$...
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### 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 ...
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### 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 $\{1,\ldots, n\}$...
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### 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"), ...
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### 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 $W(\Sigma; v)$. ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $m,k$...
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### 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-...
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### 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, $\ell_{001}(0001110010011001)=2$...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $\ell_n - 3^n = \ell_{n-1} \sim \ell_n/3$...
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### 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 $$w_0=0^{m-1}d_0, w_1=w_0d_1, w_2=w_1d_2, \ldots, w_k = w_{k-1}d_k$$ ...
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### 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 ...