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In [[combinatorics|combinatorial]] mathematics, '''Dobiński’s formula'''<ref>G. Dobiński, "Summirung<!-- "Summirung" is an archaic spelling, and it is the spelling that was used in this title. --> der Reihe <math>\textstyle\sum\frac{n^m}{n!}</math> für ''m''&nbsp;=&nbsp;1,&nbsp;2,&nbsp;3,&nbsp;4,&nbsp;5,&nbsp;…", ''Grunert's Archiv'', volume 61, 1877, pages 333–336 (Internet Archive: [http://www.archive.org/stream/archivdermathem88unkngoog#page/n346]).</ref> states that the number of [[partition of a set|partitions of a set]] of ''n'' members is
== being upturned neck point sleepy ==


:<math>{1 \over e}\sum_{k=0}^\infty {k^n \over k!}.</math>
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The formula is named after G. Dobiński, who published it in 1877. The number on both sides of the formula has come to be called the ''n''th [[Bell number]] ''B''<sub>''n''</sub>, after the later work of [[Eric Temple Bell]].
== 'club certainly provide this occasion ==


The above formula can be seen as a particular case, for <math>x=0</math>, of the more general relation:
Xue Fei driving on.<br><br>'club certainly provide this occasion, [http://www.dmwai.com/webalizer/kate-spade-0.html ケイトスペード バッグ] but they [http://www.dmwai.com/webalizer/kate-spade-9.html kate spade 財布 ゴールド] sound good.' backseat [http://www.dmwai.com/webalizer/kate-spade-13.html ケイトスペード 人気バッグ] Lapa said.<br><br>'Oh, I really have not been.' I sin without blinking an eye lied.<br><br>'That guy, where you've been, or you pick one?' Xue Fei Tao, I sin-off with the Lord, [http://www.dmwai.com/webalizer/kate-spade-9.html ケイトスペード リボン バッグ] but such a clear sky, the way is still Zhunv curiosity aroused, three girl bite what ear said, chuckling again soon.<br><br>'few, what smile?' I asked the crime.<br><br>'We guess you are what capacity.' Xue Fei smile.<br><br>'guessed it? Tell me.' I sin asked.<br><br>'ah, producer ...... certainly filmmakers, otherwise Lanjie not so got the idea.' Xue Fei Road.<br><br>'wrong, who come.' I sin laughed.<br><br>'is the owner of it ...... there [http://www.dmwai.com/webalizer/kate-spade-12.html ケイトスペード ハンドバッグ] to see you this tie taste you know, the more low-key now more Tyrant ah.' Lapa envy authentic, Lanjie body
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:<math>{1 \over e}\sum_{k=x}^\infty {k^n \over (k-x)!} = \sum_{k=0}^n {n \choose k} B_{k} x^{n-k}</math>
== do not inquire about Nima chaos.' I am sorry to say sin ==


==Probabilistic content==
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Those familiar with [[probability theory]] will recognize the expression given by Dobinski's formula as the ''n''th [[moment (mathematics)|moment]] of the [[Poisson distribution]] with [[expected value]] 1.  Today, Dobinski's formula is sometimes stated by saying the number of partitions of a set of size ''n'' equals the ''n''th moment of that distribution.
== let so many honest Excellent drops ==


==A proof==
. See envious of the team for quite a while, until the two bachelor brothers Resentment winter authentic with: 'mouse you are wrong, this is not a weakness, which [http://www.dmwai.com/webalizer/kate-spade-15.html ケイトスペード クラッチバッグ] is an advantage, he was so cheap goods are 脚踩两只船, let so many honest Excellent drops, still remain a bachelor nirvana, right, Ge Jige early anxious to sell himself, nobody wanted not!? '<br>After<br>a jealousy envy hate everyone, but is [http://www.dmwai.com/webalizer/kate-spade-15.html ケイトスペード マザーズバッグ] sadly endless winter of two brothers, then deep that .........<br><br>third volume thief arena Chapter 46 tears laughing blossoms<br><br>first cup filled wine is drained Lin Yu Jing She put the cup on the table Dayton forthright manner to [http://www.dmwai.com/webalizer/kate-spade-11.html ケイトスペードのバッグ] the sentence: filling!<br><br>I sin to jump. saw Lin Yu Jing one. again filling up. thirty-eight degrees Fen A large cup of two hundred thirty-two - [http://www.dmwai.com/webalizer/kate-spade-5.html ケイトスペード 財布 セール] so ordinary people are not going to stand in Lin Yu Jing and drink [http://www.dmwai.com/webalizer/kate-spade-10.html ケイトスペード バッグ ショルダー] half of this . before Shu-been big eyes wide open to the general looked more than sin strange to ask:... '? You do not persuade me drink less anxious I was drunk is not'<br><br>'Do not drink and plan a drunk thing. drink Well advised me to
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The proof given here is an adaptation to probabilistic language, of the proof given by [[Gian-Carlo Rota|Rota]].<ref>* [[Gian-Carlo Rota]], [https://umdrive.memphis.edu/ccrousse/public/MATH%207029/rota.pdf "The Number of Partitions of a Set"], ''[[American Mathematical Monthly]]'', volume 71, number 5, May 1964, pages 498–504.</ref>
== ' I sin stunned ==


Combinatorialists use the [[Pochhammer symbol]] (''x'')<sub>''n''</sub> to denote the falling factorial
Things, brought the car, walked a little guilty, [http://www.dmwai.com/webalizer/kate-spade-7.html ケイトスペード 財布 通販] at most, is strange to read more I sin glances, telling him where to find him, I have to be to ask a few sins, Xu Pingqiu is stopping: 'The people old, energy die ...... seriously debilitating brain, so he take a break to go to ask it. '<br>They return with<br>down the steps, 许平秋 side still pensive eyes glanced over sin, the initiative asked: '?? Are you ready the [http://www.dmwai.com/webalizer/kate-spade-3.html ケイトスペード 財布 新作] night before dawn dry or dry.'<br><br>'This is a comment that you? or directly under the command?' I asked the crime, and pocketing the captain speaking tone than choking more.<br><br>'No, I guess you'll [http://www.dmwai.com/webalizer/kate-spade-8.html マザーズバッグ ケイトスペード] dry overnight, even if I do not go under [http://www.dmwai.com/webalizer/kate-spade-0.html ハンドバッグ ケイトスペード] the command can expel you, you know why?' Xu Pingqiu proud authentic.<br><br>'Why?' I sin stunned, really ready to dry overnight.<br><br>'Because you curious mind, fit suspense overnight ......' Xu Pingqiu smile, turned and walked slowly to his knowledge, all of the detection is started from the curious, once [http://www.dmwai.com/webalizer/kate-spade-10.html ケイトスペード 財布 セール] addicted, will not be able to extricate themselves, he
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:<math>(x)_n = x(x-1)(x-2)\cdots(x-n+1)\,</math>
== sad reminder that ==


(whereas, in the theory of [[special function]]s, the same notation denotes the ''rising'' factorial). If ''x'' and ''n'' are nonnegative integers, 0&nbsp;≤&nbsp;''n''&nbsp;≤&nbsp;''x'', then (''x'')<sub>''n''</sub> is the number of [[one-to-one function]]s that map a size-''n'' set into a size-''x'' set.
All away, to [http://www.dmwai.com/webalizer/kate-spade-4.html ケイトスペード バッグ 新作] the discipline of the person to write [http://www.dmwai.com/webalizer/kate-spade-3.html 財布 kate spade] a detailed inspection ...... things through and others may leave school up today. '<br><br>dryly said a hundred more than learning to go third, that pathetic, the few remaining teams see is so bleak, no one to go, are watched by disciplinary team with students go, good burst warrior to Come Nevermore desolate feeling.<br><br>Curiously, the ranks of the accident found that I actually did not sin, Jiang Xiaoyuan underground [http://www.dmwai.com/webalizer/kate-spade-0.html ハンドバッグ ケイトスペード] passage with this kid [http://www.dmwai.com/webalizer/kate-spade-15.html ケイトスペード アウトレット バッグ] is smart, as long as something happens, there must be him, as long as the offender, certainly not him.<br><br>sad reminder that, in addition to I sin, the whole network involved in the fight Douzhu, and Zheng Zhongliang walking and watching the Lord of fellow sufferers, murmured the sentence:<br><br>'I really accurate count, today is really to be afraid of crossing the robbery ...... this robbery [http://www.dmwai.com/webalizer/kate-spade-10.html ケイトスペード バッグ ショルダー] is make life difficult.'<br><br>Volume rookie elite baked Story Chapter 22<br><br>discipline at the school is located in the back of police equipment warehouse, designed to prepare students for the mistakes, a pedestrian and opened into a double-leaf doors, floor equipment
 
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Let ''&fnof;'' be any function from a size-''n'' set ''A'' into a size-''x'' set ''B''.  For any ''u''&nbsp;∈&nbsp;''B'', let ''&fnof;''<sup> &minus;1</sup>(''u'') = {''v''&nbsp;∈&nbsp;''A'' : ''&fnof;''(''v'') = ''u''}. Then {''&fnof;''<sup> &minus;1</sup>(''u'') : ''u''&nbsp;∈&nbsp;''B''} is a partition of ''A'', coming from the [[equivalence relation]] of "being in the same [[fiber (mathematics)|fiber]]".  This equivalence relation is called the "[[Kernel (set theory)|kernel]]" of the function ''&fnof;''.  Any function from ''A'' into ''B'' factors into
<ul>
* one function that maps a member of ''A'' to that part of the kernel to which it belongs, and
 
* another function, which is necessarily one-to-one, that maps the kernel into ''B''.
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The first of these two factors is completely determined by the partition π that is the kernel. The number of one-to-one functions from π into ''B'' is (''x'')<sub>|π|</sub>, where |π| is the number of parts in the partition π.  Thus the total number of functions from a size-''n'' set ''A'' into a size-''x'' set ''B'' is
 
 
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the index π running through the set of all partitions of ''A''.  On the other hand, the number of functions from ''A'' into ''B'' is clearly ''x''<sup>''n''</sup>. Therefore we have
 
 
</ul>
:<math>x^n = \sum_\pi (x)_{|\pi|}.\,</math>
 
If ''X'' is a [[Poisson distribution|Poisson-distributed]] [[random variable]] with [[expected value]] 1, then we get that the ''n''th moment of this probability distribution is
 
:<math>E(X^n) = \sum_\pi E((X)_{|\pi|}).\,</math>
 
But all of the [[factorial moment]]s E((''X'')<sub>''k''</sub>) of this probability distribution are equal to 1. Therefore
 
:<math>E(X^n) = \sum_\pi 1,\,</math>
 
and this is just the number of partitions of the set ''A''. Q.E.D.
 
==Notes and references==
{{reflist}}
 
[[Category:Combinatorics]]
[[Category:Probability theory]]
[[Category:Articles containing proofs]]

Latest revision as of 09:28, 7 October 2014

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