Combinatorics based binary sequence
$begingroup$
We all know the standard base-$2$ representation of integers and many of you will know of Gray codes. Does anybody know of a name (or something I can use to find a good reference) for a method of sequencing a binary representation so that numbers higher in the sequence have more bits set? For example,
$$000, 001, 010, 100, 011, 101, 110, 111.$$
We set $i$ bits until all $n choose i$ combinations are used, then increment $i$. Does this sequence have a name?
combinatorics binary
edited Dec 12 '18 at 10:07
Klangen
1,74811334
asked Mar 25 '13 at 15:26
KevinKevin
61
$endgroup$
add a comment |
$begingroup$
We all know the standard base-$2$ representation of integers and many of you will know of Gray codes. Does anybody know of a name (or something I can use to find a good reference) for a method of sequencing a binary representation so that numbers higher in the sequence have more bits set? For example,
$$000, 001, 010, 100, 011, 101, 110, 111.$$
We set $i$ bits until all $n choose i$ combinations are used, then increment $i$. Does this sequence have a name?
combinatorics binary
edited Dec 12 '18 at 10:07
Klangen
1,74811334
asked Mar 25 '13 at 15:26
KevinKevin
61
$endgroup$
add a comment |
$begingroup$
We all know the standard base-$2$ representation of integers and many of you will know of Gray codes. Does anybody know of a name (or something I can use to find a good reference) for a method of sequencing a binary representation so that numbers higher in the sequence have more bits set? For example,
$$000, 001, 010, 100, 011, 101, 110, 111.$$
We set $i$ bits until all $n choose i$ combinations are used, then increment $i$. Does this sequence have a name?
combinatorics binary
edited Dec 12 '18 at 10:07
Klangen
1,74811334
asked Mar 25 '13 at 15:26
KevinKevin
61
$endgroup$
We all know the standard base-$2$ representation of integers and many of you will know of Gray codes. Does anybody know of a name (or something I can use to find a good reference) for a method of sequencing a binary representation so that numbers higher in the sequence have more bits set? For example,
$$000, 001, 010, 100, 011, 101, 110, 111.$$
We set $i$ bits until all $n choose i$ combinations are used, then increment $i$. Does this sequence have a name?
combinatorics binary
combinatorics binary
edited Dec 12 '18 at 10:07
Klangen
1,74811334
asked Mar 25 '13 at 15:26
KevinKevin
61
edited Dec 12 '18 at 10:07
Klangen
1,74811334
asked Mar 25 '13 at 15:26
KevinKevin
61
edited Dec 12 '18 at 10:07
Klangen
1,74811334
edited Dec 12 '18 at 10:07
Klangen
1,74811334
edited Dec 12 '18 at 10:07
Klangen
1,74811334
1,74811334
asked Mar 25 '13 at 15:26
KevinKevin
61
asked Mar 25 '13 at 15:26
KevinKevin
61
asked Mar 25 '13 at 15:26
KevinKevin
61
61
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
This would be ordering binary words by bit count (sideways sum, Hamming weight) and lexicographically within one class. Don't know of any more standard name for this ordering.
There is actually a fairly fast algorithm to find the lexicographically next word with the same bit count if it exists. It is a nice exercice to describe it.
answered Mar 25 '13 at 15:56
Marc van LeeuwenMarc van Leeuwen
87.2k5108223
$endgroup$
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%2f340724%2fcombinatorics-based-binary-sequence%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
$begingroup$
This would be ordering binary words by bit count (sideways sum, Hamming weight) and lexicographically within one class. Don't know of any more standard name for this ordering.
There is actually a fairly fast algorithm to find the lexicographically next word with the same bit count if it exists. It is a nice exercice to describe it.
answered Mar 25 '13 at 15:56
Marc van LeeuwenMarc van Leeuwen
87.2k5108223
$endgroup$
add a comment |
$begingroup$
This would be ordering binary words by bit count (sideways sum, Hamming weight) and lexicographically within one class. Don't know of any more standard name for this ordering.
There is actually a fairly fast algorithm to find the lexicographically next word with the same bit count if it exists. It is a nice exercice to describe it.
answered Mar 25 '13 at 15:56
Marc van LeeuwenMarc van Leeuwen
87.2k5108223
$endgroup$
add a comment |
$begingroup$
This would be ordering binary words by bit count (sideways sum, Hamming weight) and lexicographically within one class. Don't know of any more standard name for this ordering.
There is actually a fairly fast algorithm to find the lexicographically next word with the same bit count if it exists. It is a nice exercice to describe it.
answered Mar 25 '13 at 15:56
Marc van LeeuwenMarc van Leeuwen
87.2k5108223
$endgroup$
This would be ordering binary words by bit count (sideways sum, Hamming weight) and lexicographically within one class. Don't know of any more standard name for this ordering.
There is actually a fairly fast algorithm to find the lexicographically next word with the same bit count if it exists. It is a nice exercice to describe it.
answered Mar 25 '13 at 15:56
Marc van LeeuwenMarc van Leeuwen
87.2k5108223
answered Mar 25 '13 at 15:56
Marc van LeeuwenMarc van Leeuwen
87.2k5108223
answered Mar 25 '13 at 15:56
Marc van LeeuwenMarc van Leeuwen
87.2k5108223
answered Mar 25 '13 at 15:56
Marc van LeeuwenMarc van Leeuwen
87.2k5108223
87.2k5108223
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%2f340724%2fcombinatorics-based-binary-sequence%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
