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
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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
$endgroup$
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
$endgroup$
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 54 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 |
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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).
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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).
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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).
$endgroup$
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).
$endgroup$
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
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