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?










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    9












    $begingroup$


    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?










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    migrated from mathoverflow.net 1 hour ago


    This question came from our site for professional mathematicians.





















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      9








      9





      $begingroup$


      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?










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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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      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.
























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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).







          share|cite|improve this answer









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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).







            share|cite|improve this answer









            $endgroup$


















              15












              $begingroup$

              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).







              share|cite|improve this answer









              $endgroup$
















                15












                15








                15





                $begingroup$

                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).







                share|cite|improve this answer









                $endgroup$



                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).








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                answered 10 hours ago









                user44191user44191

                25114




                25114






























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