Proof Stirling equality
Proof that
$k^n = sum_{j=1}^{k}binom{k}{j}j!S(n,j)$
To justify this last equality you have to consider, firstly, that if $X$ and $Y$ are two sets with $n$ and $k$ elements, respectively, then there are $k^{n}$ applications of $X$ and $Y$. To reach the equality it have to use that these applications can be constructed in the following way: the images of the elements of X are a series of elements of Y, say j of them, where $1leq jleq k$ (the rest will not have preimage). Then, first, we decided that elements of Y have preimagenes and, once decided that the elements "arrives" the application, it only remains to build a suprayective application that has only these elements as an image. This is the process that should lead to the desired formula.
discrete-mathematics proof-verification stirling-numbers
asked Nov 29 at 9:07
Str0nger
11
add a comment |
Proof that
$k^n = sum_{j=1}^{k}binom{k}{j}j!S(n,j)$
To justify this last equality you have to consider, firstly, that if $X$ and $Y$ are two sets with $n$ and $k$ elements, respectively, then there are $k^{n}$ applications of $X$ and $Y$. To reach the equality it have to use that these applications can be constructed in the following way: the images of the elements of X are a series of elements of Y, say j of them, where $1leq jleq k$ (the rest will not have preimage). Then, first, we decided that elements of Y have preimagenes and, once decided that the elements "arrives" the application, it only remains to build a suprayective application that has only these elements as an image. This is the process that should lead to the desired formula.
discrete-mathematics proof-verification stirling-numbers
asked Nov 29 at 9:07
Str0nger
11
"applications"? Do you means maps? There is a typo in the header. And you can add the tag proof-verification :)
– Stockfish
Nov 29 at 9:09
Welcome to MSE. Please read this text about how to ask a good question.
– José Carlos Santos
Nov 29 at 9:11
Your proof is correct modulo small language issues.
– Michal Adamaszek
Nov 29 at 11:06
add a comment |
Proof that
$k^n = sum_{j=1}^{k}binom{k}{j}j!S(n,j)$
To justify this last equality you have to consider, firstly, that if $X$ and $Y$ are two sets with $n$ and $k$ elements, respectively, then there are $k^{n}$ applications of $X$ and $Y$. To reach the equality it have to use that these applications can be constructed in the following way: the images of the elements of X are a series of elements of Y, say j of them, where $1leq jleq k$ (the rest will not have preimage). Then, first, we decided that elements of Y have preimagenes and, once decided that the elements "arrives" the application, it only remains to build a suprayective application that has only these elements as an image. This is the process that should lead to the desired formula.
discrete-mathematics proof-verification stirling-numbers
asked Nov 29 at 9:07
Str0nger
11
Proof that
$k^n = sum_{j=1}^{k}binom{k}{j}j!S(n,j)$
To justify this last equality you have to consider, firstly, that if $X$ and $Y$ are two sets with $n$ and $k$ elements, respectively, then there are $k^{n}$ applications of $X$ and $Y$. To reach the equality it have to use that these applications can be constructed in the following way: the images of the elements of X are a series of elements of Y, say j of them, where $1leq jleq k$ (the rest will not have preimage). Then, first, we decided that elements of Y have preimagenes and, once decided that the elements "arrives" the application, it only remains to build a suprayective application that has only these elements as an image. This is the process that should lead to the desired formula.
discrete-mathematics proof-verification stirling-numbers
discrete-mathematics proof-verification stirling-numbers
asked Nov 29 at 9:07
Str0nger
11
asked Nov 29 at 9:07
Str0nger
11
edited Nov 29 at 9:12
asked Nov 29 at 9:07
Str0nger
11
asked Nov 29 at 9:07
Str0nger
11
asked Nov 29 at 9:07
Str0nger
11
11
"applications"? Do you means maps? There is a typo in the header. And you can add the tag proof-verification :)
– Stockfish
Nov 29 at 9:09
Welcome to MSE. Please read this text about how to ask a good question.
– José Carlos Santos
Nov 29 at 9:11
Your proof is correct modulo small language issues.
– Michal Adamaszek
Nov 29 at 11:06
add a comment |
"applications"? Do you means maps? There is a typo in the header. And you can add the tag proof-verification :)
– Stockfish
Nov 29 at 9:09
Welcome to MSE. Please read this text about how to ask a good question.
– José Carlos Santos
Nov 29 at 9:11
Your proof is correct modulo small language issues.
– Michal Adamaszek
Nov 29 at 11:06
"applications"? Do you means maps? There is a typo in the header. And you can add the tag proof-verification :)
– Stockfish
Nov 29 at 9:09
"applications"? Do you means maps? There is a typo in the header. And you can add the tag proof-verification :)
– Stockfish
Nov 29 at 9:09
Welcome to MSE. Please read this text about how to ask a good question.
– José Carlos Santos
Nov 29 at 9:11
Welcome to MSE. Please read this text about how to ask a good question.
– José Carlos Santos
Nov 29 at 9:11
Your proof is correct modulo small language issues.
– Michal Adamaszek
Nov 29 at 11:06
Your proof is correct modulo small language issues.
– Michal Adamaszek
Nov 29 at 11:06
add a comment |
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%2f3018376%2fproof-stirling-equality%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
active
oldest
votes
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.
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%2f3018376%2fproof-stirling-equality%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
"applications"? Do you means maps? There is a typo in the header. And you can add the tag proof-verification :)
– Stockfish
Nov 29 at 9:09
Welcome to MSE. Please read this text about how to ask a good question.
– José Carlos Santos
Nov 29 at 9:11
Your proof is correct modulo small language issues.
– Michal Adamaszek
Nov 29 at 11:06