Define $[n,k]$ as the number of $(n+k)$-lists of the form $a_1+a_2+..+a_i$ where n of the elements are $1$s...
$begingroup$
Define $[n,k]$ as the number of $(n+k)$-lists of the form $(a_1+a_2+..+a_i)$ where n of the elements are $1$s and k of them are $-1$s and where for all i , the sum of the first i entries is non-negative: $(a_1+a_2+..+a_i geq$ for all $i$ belongs to $(n+k)$. Please help me find $[3,2]$. I can find it manually to be $5$. But was wondering how I can do that using combinatorics formulas.
combinatorics discrete-mathematics
asked Dec 11 '18 at 1:49
Lauren SmithLauren Smith
291
$endgroup$
add a comment |
$begingroup$
Define $[n,k]$ as the number of $(n+k)$-lists of the form $(a_1+a_2+..+a_i)$ where n of the elements are $1$s and k of them are $-1$s and where for all i , the sum of the first i entries is non-negative: $(a_1+a_2+..+a_i geq$ for all $i$ belongs to $(n+k)$. Please help me find $[3,2]$. I can find it manually to be $5$. But was wondering how I can do that using combinatorics formulas.
combinatorics discrete-mathematics
asked Dec 11 '18 at 1:49
Lauren SmithLauren Smith
291
$endgroup$
$begingroup$
These are known as ballot numbers, look at page $533$ here citeseerx.ist.psu.edu/viewdoc/…
$endgroup$
– Donald Splutterwit
Dec 11 '18 at 1:58
add a comment |
$begingroup$
Define $[n,k]$ as the number of $(n+k)$-lists of the form $(a_1+a_2+..+a_i)$ where n of the elements are $1$s and k of them are $-1$s and where for all i , the sum of the first i entries is non-negative: $(a_1+a_2+..+a_i geq$ for all $i$ belongs to $(n+k)$. Please help me find $[3,2]$. I can find it manually to be $5$. But was wondering how I can do that using combinatorics formulas.
combinatorics discrete-mathematics
asked Dec 11 '18 at 1:49
Lauren SmithLauren Smith
291
$endgroup$
Define $[n,k]$ as the number of $(n+k)$-lists of the form $(a_1+a_2+..+a_i)$ where n of the elements are $1$s and k of them are $-1$s and where for all i , the sum of the first i entries is non-negative: $(a_1+a_2+..+a_i geq$ for all $i$ belongs to $(n+k)$. Please help me find $[3,2]$. I can find it manually to be $5$. But was wondering how I can do that using combinatorics formulas.
combinatorics discrete-mathematics
combinatorics discrete-mathematics
asked Dec 11 '18 at 1:49
Lauren SmithLauren Smith
291
asked Dec 11 '18 at 1:49
Lauren SmithLauren Smith
291
asked Dec 11 '18 at 1:49
Lauren SmithLauren Smith
291
asked Dec 11 '18 at 1:49
Lauren SmithLauren Smith
291
asked Dec 11 '18 at 1:49
Lauren SmithLauren Smith
291
291
$begingroup$
These are known as ballot numbers, look at page $533$ here citeseerx.ist.psu.edu/viewdoc/…
$endgroup$
– Donald Splutterwit
Dec 11 '18 at 1:58
add a comment |
$begingroup$
These are known as ballot numbers, look at page $533$ here citeseerx.ist.psu.edu/viewdoc/…
$endgroup$
– Donald Splutterwit
Dec 11 '18 at 1:58
$begingroup$
These are known as ballot numbers, look at page $533$ here citeseerx.ist.psu.edu/viewdoc/…
$endgroup$
– Donald Splutterwit
Dec 11 '18 at 1:58
$begingroup$
These are known as ballot numbers, look at page $533$ here citeseerx.ist.psu.edu/viewdoc/…
$endgroup$
– Donald Splutterwit
Dec 11 '18 at 1:58
add a comment |
0
active
oldest
votes
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%2f3034765%2fdefine-n-k-as-the-number-of-nk-lists-of-the-form-a-1a-2-a-i-where%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
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%2f3034765%2fdefine-n-k-as-the-number-of-nk-lists-of-the-form-a-1a-2-a-i-where%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
$begingroup$
These are known as ballot numbers, look at page $533$ here citeseerx.ist.psu.edu/viewdoc/…
$endgroup$
– Donald Splutterwit
Dec 11 '18 at 1:58