How many letters suffice to construct words with no repetition?Formula 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
Are cabin dividers used to "hide" the flex of the airplane?
Extreme, but not acceptable situation and I can't start the work tomorrow morning
Why is the design of haulage companies so “special”?
New order #4: World
Why did the Germans forbid the possession of pet pigeons in Rostov-on-Don in 1941?
Copycat chess is back
A poker game description that does not feel gimmicky
Why was the "bread communication" in the arena of Catching Fire left out in the movie?
How could a lack of term limits lead to a "dictatorship?"
Lied on resume at previous job
Is it wise to focus on putting odd beats on left when playing double bass drums?
Could a US political party gain complete control over the government by removing checks & balances?
Is there a familial term for apples and pears?
"listening to me about as much as you're listening to this pole here"
What causes the sudden spool-up sound from an F-16 when enabling afterburner?
What do the Banks children have against barley water?
Where else does the Shulchan Aruch quote an authority by name?
How to move the player while also allowing forces to affect it
Why airport relocation isn't done gradually?
Is ipsum/ipsa/ipse a third person pronoun, or can it serve other functions?
I’m planning on buying a laser printer but concerned about the life cycle of toner in the machine
Is it wise to hold on to stock that has plummeted and then stabilized?
Can I find out the caloric content of bread by dehydrating it?
Typesetting a double Over Dot on top of a symbol
How many letters suffice to construct words with no repetition?
Formula 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 hours ago
Andrés E. Caicedo
65.9k8160252
$endgroup$
migrated from mathoverflow.net 2 hours 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 hours ago
Andrés E. Caicedo
65.9k8160252
$endgroup$
migrated from mathoverflow.net 2 hours 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 hours 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 hours ago
Andrés E. Caicedo
65.9k8160252
edited 2 hours ago
Andrés E. Caicedo
65.9k8160252
edited 2 hours ago
Andrés E. Caicedo
65.9k8160252
edited 2 hours ago
Andrés E. Caicedo
65.9k8160252
edited 2 hours ago
Andrés E. Caicedo
65.9k8160252
65.9k8160252
asked 11 hours ago
PiCo
migrated from mathoverflow.net 2 hours ago
This question came from our site for professional mathematicians.
migrated from mathoverflow.net 2 hours ago
This question came from our site for professional mathematicians.
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
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 11 hours ago
user44191user44191
25114
$endgroup$
add a comment |
Your Answer
StackExchange.ifUsing("editor", function ()
return StackExchange.using("mathjaxEditing", function ()
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix)
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
);
);
, "mathjax-editing");
StackExchange.ready(function()
var channelOptions =
tags: "".split(" "),
id: "69"
;
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function()
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled)
StackExchange.using("snippets", function()
createEditor();
);
else
createEditor();
);
function createEditor()
StackExchange.prepareEditor(
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader:
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
,
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
);
);
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3180466%2fhow-many-letters-suffice-to-construct-words-with-no-repetition%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
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 11 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 11 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 11 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 11 hours ago
user44191user44191
25114
answered 11 hours ago
user44191user44191
25114
answered 11 hours ago
user44191user44191
25114
answered 11 hours ago
user44191user44191
25114
25114
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3180466%2fhow-many-letters-suffice-to-construct-words-with-no-repetition%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
