How many letters suffice to construct words with no repetition?
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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 59 mins ago
Andrés E. Caicedo
65.9k8160252
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migrated from mathoverflow.net 1 hour ago
This question came from our site for professional mathematicians.
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$begingroup$
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 59 mins ago
Andrés E. Caicedo
65.9k8160252
$endgroup$
migrated from mathoverflow.net 1 hour 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 59 mins 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 59 mins ago
Andrés E. Caicedo
65.9k8160252
edited 59 mins ago
Andrés E. Caicedo
65.9k8160252
edited 59 mins ago
Andrés E. Caicedo
65.9k8160252
edited 59 mins ago
Andrés E. Caicedo
65.9k8160252
edited 59 mins ago
Andrés E. Caicedo
65.9k8160252
65.9k8160252
asked 10 hours ago
PiCo
migrated from mathoverflow.net 1 hour ago
This question came from our site for professional mathematicians.
migrated from mathoverflow.net 1 hour ago
This question came from our site for professional mathematicians.
add a comment |
add a comment |
1 Answer
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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 10 hours ago
user44191user44191
25114
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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 10 hours ago
user44191user44191
25114
$endgroup$
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 10 hours ago
user44191user44191
25114
$endgroup$
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 10 hours ago
user44191user44191
25114
$endgroup$
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 10 hours ago
user44191user44191
25114
answered 10 hours ago
user44191user44191
25114
answered 10 hours ago
user44191user44191
25114
answered 10 hours ago
user44191user44191
25114
25114
add a comment |
add a comment |
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