How many letters suffice to construct words with no repetition? The 2019 Stack Overflow Developer Survey Results Are InFormula for sub and super sequence length given 2 stringsThe number of sequences with k elements, containing a given elementMaximal Hamming distance$4$-element subsets of the set $1,2,3,ldots,10$ that do not contain any pair of consecutive numbersAn example showing that van der Waerden's theorem is not true for infinite arithmetic progressionsCounting the number of words made of $2n$ lettersThe number of procedures needed to make an arbitrary permutation to the identityIs there a string of all words without repetition?Recurrence for Number of Words of Length $r$ over $[n]$ with no three consecutive letters the sameCombinatorics - Sequences with repetition and restrictions
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How many letters suffice to construct words with no repetition?
The 2019 Stack Overflow Developer Survey Results Are InFormula for sub and super sequence length given 2 stringsThe number of sequences with k elements, containing a given elementMaximal Hamming distance$4$-element subsets of the set $1,2,3,ldots,10$ that do not contain any pair of consecutive numbersAn example showing that van der Waerden's theorem is not true for infinite arithmetic progressionsCounting the number of words made of $2n$ lettersThe number of procedures needed to make an arbitrary permutation to the identityIs there a string of all words without repetition?Recurrence for Number of Words of Length $r$ over $[n]$ with no three consecutive letters the sameCombinatorics - Sequences with repetition and restrictions
$begingroup$
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 equal. Is it true that $k = 4$ is the minimum number of elements in $A$ that allows the construction of sequences of any length containing no repetition? Can anyone indicate a reference for this result, if true?
combinatorics combinatorics-on-words
edited 2 days ago
Andrés E. Caicedo
65.9k8160252
$endgroup$
migrated from mathoverflow.net 2 days ago
This question came from our site for professional mathematicians.
add a comment |
$begingroup$
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 equal. Is it true that $k = 4$ is the minimum number of elements in $A$ that allows the construction of sequences of any length containing no repetition? Can anyone indicate a reference for this result, if true?
combinatorics combinatorics-on-words
edited 2 days ago
Andrés E. Caicedo
65.9k8160252
$endgroup$
migrated from mathoverflow.net 2 days ago
This question came from our site for professional mathematicians.
add a comment |
$begingroup$
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 equal. Is it true that $k = 4$ is the minimum number of elements in $A$ that allows the construction of sequences of any length containing no repetition? Can anyone indicate a reference for this result, if true?
combinatorics combinatorics-on-words
edited 2 days ago
Andrés E. Caicedo
65.9k8160252
$endgroup$
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 equal. Is it true that $k = 4$ is the minimum number of elements in $A$ that allows the construction of sequences of any length containing no repetition? Can anyone indicate a reference for this result, if true?
combinatorics combinatorics-on-words
combinatorics combinatorics-on-words
edited 2 days ago
Andrés E. Caicedo
65.9k8160252
edited 2 days ago
Andrés E. Caicedo
65.9k8160252
edited 2 days ago
Andrés E. Caicedo
65.9k8160252
edited 2 days ago
Andrés E. Caicedo
65.9k8160252
edited 2 days ago
Andrés E. Caicedo
65.9k8160252
65.9k8160252
asked 2 days ago
PiCo
migrated from mathoverflow.net 2 days ago
This question came from our site for professional mathematicians.
migrated from mathoverflow.net 2 days ago
This question came from our site for professional mathematicians.
add a comment |
add a comment |
1 Answer
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$begingroup$
Wikipedia has some examples of square-free sequences of infinite length (and therefore square-free words of arbitrary length) over alphabets with 3 letters.
https://en.wikipedia.org/wiki/Square-free_word
One example of an infinite square-free word over an alphabet of size 3 is the word over the alphabet 0,±1 obtained by taking the first difference of the Thue–Morse sequence.[6][7] That is, from the Thue–Morse sequence
0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, ...
one forms a new sequence in which each term is the difference of two consecutive terms of the Thue–Morse sequence. The resulting square-free word is
1, 0, −1, 1, −1, 0, 1, 0, −1, 0, 1, −1, 1, 0, −1, ... (sequence A029883 in the OEIS).
answered 2 days ago
user44191user44191
26114
$endgroup$
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1 Answer
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1 Answer
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$begingroup$
Wikipedia has some examples of square-free sequences of infinite length (and therefore square-free words of arbitrary length) over alphabets with 3 letters.
https://en.wikipedia.org/wiki/Square-free_word
One example of an infinite square-free word over an alphabet of size 3 is the word over the alphabet 0,±1 obtained by taking the first difference of the Thue–Morse sequence.[6][7] That is, from the Thue–Morse sequence
0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, ...
one forms a new sequence in which each term is the difference of two consecutive terms of the Thue–Morse sequence. The resulting square-free word is
1, 0, −1, 1, −1, 0, 1, 0, −1, 0, 1, −1, 1, 0, −1, ... (sequence A029883 in the OEIS).
answered 2 days ago
user44191user44191
26114
$endgroup$
add a comment |
$begingroup$
Wikipedia has some examples of square-free sequences of infinite length (and therefore square-free words of arbitrary length) over alphabets with 3 letters.
https://en.wikipedia.org/wiki/Square-free_word
One example of an infinite square-free word over an alphabet of size 3 is the word over the alphabet 0,±1 obtained by taking the first difference of the Thue–Morse sequence.[6][7] That is, from the Thue–Morse sequence
0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, ...
one forms a new sequence in which each term is the difference of two consecutive terms of the Thue–Morse sequence. The resulting square-free word is
1, 0, −1, 1, −1, 0, 1, 0, −1, 0, 1, −1, 1, 0, −1, ... (sequence A029883 in the OEIS).
answered 2 days ago
user44191user44191
26114
$endgroup$
add a comment |
$begingroup$
Wikipedia has some examples of square-free sequences of infinite length (and therefore square-free words of arbitrary length) over alphabets with 3 letters.
https://en.wikipedia.org/wiki/Square-free_word
One example of an infinite square-free word over an alphabet of size 3 is the word over the alphabet 0,±1 obtained by taking the first difference of the Thue–Morse sequence.[6][7] That is, from the Thue–Morse sequence
0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, ...
one forms a new sequence in which each term is the difference of two consecutive terms of the Thue–Morse sequence. The resulting square-free word is
1, 0, −1, 1, −1, 0, 1, 0, −1, 0, 1, −1, 1, 0, −1, ... (sequence A029883 in the OEIS).
answered 2 days ago
user44191user44191
26114
$endgroup$
Wikipedia has some examples of square-free sequences of infinite length (and therefore square-free words of arbitrary length) over alphabets with 3 letters.
https://en.wikipedia.org/wiki/Square-free_word
One example of an infinite square-free word over an alphabet of size 3 is the word over the alphabet 0,±1 obtained by taking the first difference of the Thue–Morse sequence.[6][7] That is, from the Thue–Morse sequence
0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, ...
one forms a new sequence in which each term is the difference of two consecutive terms of the Thue–Morse sequence. The resulting square-free word is
1, 0, −1, 1, −1, 0, 1, 0, −1, 0, 1, −1, 1, 0, −1, ... (sequence A029883 in the OEIS).
answered 2 days ago
user44191user44191
26114
answered 2 days ago
user44191user44191
26114
answered 2 days ago
user44191user44191
26114
answered 2 days ago
user44191user44191
26114
26114
add a comment |
add a comment |
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-combinatorics, combinatorics-on-words
