Questions tagged [combinatorics-on-words]

combinatorial properties of strings of symbols from a finite alphabet. Also includes sequences such as the Thue-Morse and Rudin-Shapiro sequence.

332 questions
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The Soup Problem: how to asymptotically fairly split a geometric series and a constant one using a single pattern?

Literally every time I'm serving some soup I'm thinking of this little mathematical problem I devised. Imagine you have a very large (= infinite, for the purposes of the actual problem) bowl of soup and want to divide it into two halves using a…
17
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Does there exist infinite words using the alphabet $\{A,B,C,D\}$ that avoids patterns $XX,\ XAX,\ XBX,\ XCX,\ XDX$?

Another form of this question is: Does there exist a gap-1 square-free infinite word using the alphabet {A,B,C,D}? Normally square-free in this context means that there are no sub-words twice in a row that follows the pattern XX. For example the…
quantus14
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Does the fraction of distinct substrings in prefixes of the Thue–Morse sequence of length $2^n$ tend to $73/96$?

Recall that the Thue–Morse sequence$^{[1]}$$\!^{[2]}$$\!^{[3]}$ is an infinite binary sequence that begins with $\,t_0 = 0,$ and whose each prefix $p_n$ of length $2^n$ is immediately followed by its bitwise complement (i.e. obtained by flipping…
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Asymptotic length of reduced word on free group with replacements

This seems to be an elementary question, but it's proving hard for me to just Google. Suppose you have a sequence which picks elements out of $\{a, a^{-1}, b, b^{-1}, c, c^{-1}\}$ with equal probability. After, say, seven steps you'll get words like…
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How many permutations of a multiset have a run of length k?

Background $\newcommand\ms[1]{\mathsf #1}\def\msP{\ms P}\def\msS{\ms S}\def\mfS{\mathfrak S}$Suppose I have $n$ marbles of $c$ colors, where $c≤n$. Let $n_i$ denote the number of marbles of color $i$. Let $\msP=(1^{n_1} 2^{n_2} \dots c^{n_c})$ be…
9
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Prefixes of a word multiplying to the identity in a free group

Let $A$ be a finite alphabet, and let $w \in (A \cup A^{-1})^\ast$ be a freely reduced word over the alphabet $A$ and formal inverse symbols $A^{-1}$. Suppose $w$ is non-empty. Can there ever be non-empty prefixes $p_i \in (A \cup A^{-1})^\ast$ of…
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Average number of strings with edit distance at most 3 (larger alphabet)

Consider a string of length $n \geq 3$ over an alphabet $\{1,\dots, \sigma\}$. An edit operation is a single symbol insert, deletion or substitution. The edit distance between two strings is the minimum number of edit operations needed to…
graffe
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9
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Counting binary strings of length $n$ that contain no two adjacent blocks of 1s of the same length?

Is it possible to count exactly the number of binary strings of length $n$ that contain no two adjacent blocks of 1s of the same length? More precisely, if we represent the string as $0^{x_1}1^{y_1}0^{x_2}1^{y_2}\cdots 0^{x_{k-1}}1^{y_{k-1}}0^{x_k}$…
Nocturne
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USAMO 2017 -TSTST P2: Which words can Ana pick?

Ana and Banana are playing a game. First Ana picks a word, which is defined to be a nonempty sequence of capital English letters. (The word does not need to be a valid English word.) Then Banana picks a nonnegative integer $k$ and challenges Ana to…
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How many letters suffice to construct words with no repetition?

Given a finite set $A=\{a_1,\ldots , a_k\}$, consider the sequences of any length that can be constructed using the elements of $A$ and which contain no repetition, a repetition being a pair of consecutive subsequences (of any length) that are…
PiCo
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What is the subword complexity function of this infinite word?

Let $w_{0}$ denote the finite word $01$ in the free monoid $\{ 0, 1 \}^{\ast}$, and for $i \in \mathbb{N}$ define $w_{i}$ as the word obtained by adjoining the first $\left\lfloor \frac{\ell(w_{i-1})}{2} \right\rfloor$ entries in $w_{i-1}$ to the…
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Is concatenation of digit-strings transitive?

Given two digit-strings $A$ and $B$, let $AB$ be their concatenation. So for example, if $A = ``102"$ and $B = ``101"$, $AB = ``102101"$. We then say $AB \geq BA$ since $102101 \geq 101102$. Now given digit strings $A$, $B$, and $C$, is it true that…
7
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Chance letter a next to b in circle with whole alphabet such that no vowels next to each other

Here's a question from a book on probability I'm working through: If the $26$ letters of the alphabet are written down in a ring so that no two vowels come together, what is the chance that a is next to b? Here's what I did. Let's fix a. Since b…
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How many words of length $k$ are there such that no symbol in the alphabet $\Sigma$ occur exactly once?

Introduction Given an alphabet $\Sigma$ of size $s$, I want to find a way of counting words $w$ of length $k$ which obey the rule: No symbol occurs exactly once in $w$. We'll call this number $Q^s_k$. I am particularly interested in closed-form…
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Interesting question regarding generating function on words

Let $W_n$ be the set of all words of length n, on alephbet {a,b,c}. Let $L$ be the maximal length of consecutive $a$ letters in a word. A. Find the generating function of the number of words in $W_n$ such as $L<3$. B. Find the generating function of…
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