How many letters suffice to construct words with no repetition?Bits and orbitsExtension of Tao-Green TheoremA property of unimodal sequencesPartitioning finite ordered setsRepresentability of sets of infinite sequences sharing common prefixes and factors (i.e. infixes)“Nyldon words”: understanding a class of words factorizing the free monoid increasinglyProbability of no $k$ 1's in arithmetic progression in binary sequence of length $n$Reference for one-sided subshiftsSequences with 3 LettersMinimum number of operations necessary to arrive at any configuration
How many letters suffice to construct words with no repetition?
Bits and orbitsExtension of Tao-Green TheoremA property of unimodal sequencesPartitioning finite ordered setsRepresentability of sets of infinite sequences sharing common prefixes and factors (i.e. infixes)“Nyldon words”: understanding a class of words factorizing the free monoid increasinglyProbability of no $k$ 1's in arithmetic progression in binary sequence of length $n$Reference for one-sided subshiftsSequences with 3 LettersMinimum number of operations necessary to arrive at any configuration
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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?
co.combinatorics symbolic-dynamics
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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?
co.combinatorics symbolic-dynamics
edited 1 hour ago
YCor
29k486140
asked 3 hours ago
PiCoPiCo
262
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PiCo is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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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?
co.combinatorics symbolic-dynamics
edited 1 hour ago
YCor
29k486140
asked 3 hours ago
PiCoPiCo
262
New contributor
PiCo is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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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?
co.combinatorics symbolic-dynamics
co.combinatorics symbolic-dynamics
edited 1 hour ago
YCor
29k486140
asked 3 hours ago
PiCoPiCo
262
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PiCo is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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PiCoPiCo
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YCor
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Wikipedia has some examples of square-free words 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 3 hours ago
user44191user44191
3,13511633
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Wikipedia has some examples of square-free words 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 3 hours ago
user44191user44191
3,13511633
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add a comment |
$begingroup$
Wikipedia has some examples of square-free words 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 3 hours ago
user44191user44191
3,13511633
$endgroup$
add a comment |
$begingroup$
Wikipedia has some examples of square-free words 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 3 hours ago
user44191user44191
3,13511633
$endgroup$
Wikipedia has some examples of square-free words 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 3 hours ago
user44191user44191
3,13511633
answered 3 hours ago
user44191user44191
3,13511633
answered 3 hours ago
user44191user44191
3,13511633
answered 3 hours ago
user44191user44191
3,13511633
3,13511633
add a comment |
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