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.

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How can we have a circular sequence of $0$s and $1$s of length $d$ that are not periodic?

From: A Course in Combinatorics by van Lint / Wilson $M(d)$ is the number of circular sequences of length $d$ that are not periodic. How is this possible? If $d=1$, then the sequence $1$ is periodic. It has period $1$. If $d=2$, then $11$, $10$,…
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Permutations without repetition pattern (algorithm)

Disclaimer: I haven't done math in a decade. Sorry if this is not very scientific Hello, I am writing a computer program, that is supposed to return the number of all permutations of an input string, excluding the permutations where a letter is…
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How many finite sequences with exactly k different elements?

How many different sequences/strings of length $\ell$ contain exactly $k$ (out of $n$) different elements? Or, to put it differently, how many functions from $\{1,\dots,\ell\}$ to $\{1,\ldots,n\}$ have the property that their image is of size…
Fibonacci
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Generalisation of Binomial Coefficient (Combinatorics on words)

So, when trying to find subwords from a bigger word: $\binom{abracadabra}{ab} = 5$ with $ABracadabra$, $AbracadaBra$, $abrAcadaBra$, $abracAdaBra$, $abracadABra$. I have noticed that it doesn't go back (like first $a$ then $b$ in $abracadaBrA$) and…
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How many parallelograms can be formed when a parallelogram is cut by $2$ sets of $n$ parallel lines?

A parallelogram is cut by two sets of n parallel lines parallel to the sides of the parallelogram. The number of parallelogram thus formed is..?? I think we can do it by combinatorics.. But I'm not quite sure... Help me out please.
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What is the number of words of length $h$ in a sequence of subsets of words?

Let $L=\{0,1\}^*$ (the set of binary words on $0$ and $1$), Given an integer $k$, and $S$ a finite subset of $L$ define recursively the following sequence of subsets of $L$: $$\begin{align} A_1 &=S\cup \{\epsilon\} \\ A_{n+1}…
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Counting the number of 'good' tuples of a given length

Fix a length $L$ and consider an $L$-tuple of $a$ and $b$ that obey the following rules: $Rule$ $1:$ $\textbf{IF}$ there is a consecutive string of $b$'s in the $L$-tuple such that the string does not reach the end of the $L$-tuple, then there must…
Frog will do
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How many $6$-letter words can be formed from (A,B,C,C,D,E,F,G), where the Cs cant be near each other and the other letters can be the same?

How many $6$-letter words (meaning doesn't matter) can be formed from (A,B,C,C,D,E,F,G), where the Cs cant be near each other and the other letters can be the same? Also there must be $2$ Cs in each of them. I tried to solve it like this, but I'm…
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How many “words” of any length can be made from the letters in word: MAMMA?

Iam not sure how to solve this question because the word MAMMA has 3 M and 2 A. Thanks in advance
Meos
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How many n-letter words are there, such that number of letters "a" is even?

How many n-letter words (made of letters from 25-letter english alphabet) are there, such that number of letters "a" is even? ("a" appears even number of times in a word). I'm trying to create recursive formula, but with no success.
leller
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How to recursively define $w^i$ for $i≥0$

Given a string $w$, we denote with $w^i$ the string obtained by concatenating $i$ times $w$. How can I recursively define $w^i$ for $i≥0$? First of all, what does "concatenate" mean in this context?
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Periodicity of words

If we have a non-periodic word $u$. Is it possible to have another word $\beta = \gamma u^{i-1}$ with $i>1$ and $\gamma \neq u^*$ so that $\beta$ is periodic. it's intuitive to say that it is not possible, but I don't know how to prove or disprove…
ultrainstinct
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How many combinations are possible: English alphabet A-Z and digits 0-9 in a set of 12.

How many combinations can be made from a 12 set string of letters and number that can be repeated and used more than once in any order. Ie- TD3GD3BK6K7T and would be different than T7K6KB3DG3DT or KKTT733DDGB6.
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How many times can a word appear in a random phrase?

I'm taking about finding a word in a larger set of characters. Lets say, what is the probability/chance of finding the word 'math' in a random 8 length phrase. (For example gjbmdlep does not have math in dhhjmath does!) I assume it's something like…
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How many $8$-digit numbers do not contain the string $55555$?

I'm having a little trouble figuring out this exercise. Consider an $8$-digit number. How many numbers do not contain even numbers? How many numbers do not contain the string $55555$? For the first part I've worked out that since there are only $5$…
user865587
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