Permutation of boxes with two colors
Given a row with $n$ black boxes and $n$ white boxes, how many permutations exist where each box can be adjacent to at most one other box of the same color?
In other words, three or more consecutive boxes with the same color are not allowed.
For $n=3$, I count $14$ permutations and for $n=4$, I get $34$ permutations, but what's the general rule?
permutations recreational-mathematics
add a comment |
Given a row with $n$ black boxes and $n$ white boxes, how many permutations exist where each box can be adjacent to at most one other box of the same color?
In other words, three or more consecutive boxes with the same color are not allowed.
For $n=3$, I count $14$ permutations and for $n=4$, I get $34$ permutations, but what's the general rule?
permutations recreational-mathematics
for n= 3, I count 12 permutations.
– Doug M
Jan 4 at 5:01
Similar question, generalized for $k$ colors, but without the restriction that there be equal number of each color: painting fence...
– Daniel Mathias
Jan 4 at 11:53
add a comment |
Given a row with $n$ black boxes and $n$ white boxes, how many permutations exist where each box can be adjacent to at most one other box of the same color?
In other words, three or more consecutive boxes with the same color are not allowed.
For $n=3$, I count $14$ permutations and for $n=4$, I get $34$ permutations, but what's the general rule?
permutations recreational-mathematics
Given a row with $n$ black boxes and $n$ white boxes, how many permutations exist where each box can be adjacent to at most one other box of the same color?
In other words, three or more consecutive boxes with the same color are not allowed.
For $n=3$, I count $14$ permutations and for $n=4$, I get $34$ permutations, but what's the general rule?
permutations recreational-mathematics
permutations recreational-mathematics
asked Jan 4 at 4:47
JensJens
3,7702928
3,7702928
for n= 3, I count 12 permutations.
– Doug M
Jan 4 at 5:01
Similar question, generalized for $k$ colors, but without the restriction that there be equal number of each color: painting fence...
– Daniel Mathias
Jan 4 at 11:53
add a comment |
for n= 3, I count 12 permutations.
– Doug M
Jan 4 at 5:01
Similar question, generalized for $k$ colors, but without the restriction that there be equal number of each color: painting fence...
– Daniel Mathias
Jan 4 at 11:53
for n= 3, I count 12 permutations.
– Doug M
Jan 4 at 5:01
for n= 3, I count 12 permutations.
– Doug M
Jan 4 at 5:01
Similar question, generalized for $k$ colors, but without the restriction that there be equal number of each color: painting fence...
– Daniel Mathias
Jan 4 at 11:53
Similar question, generalized for $k$ colors, but without the restriction that there be equal number of each color: painting fence...
– Daniel Mathias
Jan 4 at 11:53
add a comment |
1 Answer
1
active
oldest
votes
This is A177790 “Number of paths from (0,0) to (n,n) avoiding 3 or more consecutive east steps and 3 or more consecutive north steps” on OEIS. The closest thing to a formula given there is
$$
a(n) = sum_{i=0}^{lfloor n/2rfloor} 2binom{n-i}{i}^2 + binom{n-i}{i} binom{n-i-1}{i+1} + binom{n-i}{i}binom{n-i+1}{i-1}.
$$
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%2f3061326%2fpermutation-of-boxes-with-two-colors%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
This is A177790 “Number of paths from (0,0) to (n,n) avoiding 3 or more consecutive east steps and 3 or more consecutive north steps” on OEIS. The closest thing to a formula given there is
$$
a(n) = sum_{i=0}^{lfloor n/2rfloor} 2binom{n-i}{i}^2 + binom{n-i}{i} binom{n-i-1}{i+1} + binom{n-i}{i}binom{n-i+1}{i-1}.
$$
add a comment |
This is A177790 “Number of paths from (0,0) to (n,n) avoiding 3 or more consecutive east steps and 3 or more consecutive north steps” on OEIS. The closest thing to a formula given there is
$$
a(n) = sum_{i=0}^{lfloor n/2rfloor} 2binom{n-i}{i}^2 + binom{n-i}{i} binom{n-i-1}{i+1} + binom{n-i}{i}binom{n-i+1}{i-1}.
$$
add a comment |
This is A177790 “Number of paths from (0,0) to (n,n) avoiding 3 or more consecutive east steps and 3 or more consecutive north steps” on OEIS. The closest thing to a formula given there is
$$
a(n) = sum_{i=0}^{lfloor n/2rfloor} 2binom{n-i}{i}^2 + binom{n-i}{i} binom{n-i-1}{i+1} + binom{n-i}{i}binom{n-i+1}{i-1}.
$$
This is A177790 “Number of paths from (0,0) to (n,n) avoiding 3 or more consecutive east steps and 3 or more consecutive north steps” on OEIS. The closest thing to a formula given there is
$$
a(n) = sum_{i=0}^{lfloor n/2rfloor} 2binom{n-i}{i}^2 + binom{n-i}{i} binom{n-i-1}{i+1} + binom{n-i}{i}binom{n-i+1}{i-1}.
$$
answered Jan 4 at 5:01
Anders KaseorgAnders Kaseorg
47339
47339
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.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- 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.
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%2f3061326%2fpermutation-of-boxes-with-two-colors%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
for n= 3, I count 12 permutations.
– Doug M
Jan 4 at 5:01
Similar question, generalized for $k$ colors, but without the restriction that there be equal number of each color: painting fence...
– Daniel Mathias
Jan 4 at 11:53