Tagged Questions

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

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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 ...
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Which criteria guarantee an orthogonal circuit in $\mathbb R^3$ to be rigid?

For $n\ge4$, define an orthogonal circuit or O-circuit as a closed circuit of $n$ unit segments in $\mathbb R^3$ such that any two neighboring segments form a right angle. (Physically this could be ...
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Do runs of every length occur in this sequence?

This is a repost from user r.e.s's unsolved Math Stack Exchange question: Do runs of every length occur in this string? That question was derived from my original question on the subject: Does this ...
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Über theorem on unavoidable patterns?

Let $A$ be an alphabet of $k$ symbols, and $p$ a pattern. An example of a pattern is $p=XX$, where $X$ is any finite string of symbols from $A^+$. Avoiding $p$ is avoiding any subword repeated twice ...
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Is there a name for infinite words containing every finite words?

Apparently, the closest thing I've found would be normal number http://mathworld.wolfram.com/NormalNumber.html But requiring that every finite words occurs is weaker than this property. So I'm ...
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Covariance matrix for number of powers in a word

A word over the alphabet $\{0,1\}$ of length $n$ may contain squares, cubes, and generally $k$th powers, where $2\le k\le n$. Let $O_k(w)$ denote the number of $k$th power occurrences in the word $w$. ...
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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 ...
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Probability that a word in the free group becomes (much) shorter?

Let $w$ be a word of length $2\ell$ chosen at random on the alphabet $\{x_1,x_1^{-1},x_2,x_2^{-1},\dotsc,x_k,x_k^{-1}\}$. By the reduction $\rho(w)$ I mean what you obtain by deleting substrings of ...
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Generalised de Bruijn Graph

I have encountered sets of the following type, consisting of words over a finite aphabet $A$. If $S$ is such a set, then $S$ is finite, No word in $S$ is part of another element of $S$, and every ...
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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 ...
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Avoidable words

Let $u(x_1,...,x_n)$ be a word, $k\in \mathbb{N}$. We say that $u$ is $k$-avoidable if there exists an infinite word in $k$ letters $\{a_1,...,a_k\}$ which does not contain values of $u$ (i.e. words ...
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Notation for ends of a string

I work now a lot with strings of characters and other finite sequences and found that I need many times a good notation for "cutting the end" a string. If $a$ is a finite sequence and $a'$ is its ...
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Software for Combinatorial Algebra sought

I am looking for software which helps me do straightforward tasks in combinatorial algebra. Let me give an example of what I mean by a straightforward task: I have two graded (generally ...
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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 ...
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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 ...
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Equivalent subshifts

Let $X$ be a finite set, $(X^{\mathbb Z}, T)$ is the shift, i.e. the Tikhonov topological space of all bi-infinite words in $X$, $T$ shifts the words one letter to the right. A subshift is a closed ...
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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 ...
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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 right-...
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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 ...