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In [[combinatorics|combinatorial]] [[mathematics]], the '''Bell polynomials''', named in honor of [[Eric Temple Bell]], are a [[triangular array]] of polynomials given by
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:<math>B_{n,k}(x_1,x_2,\dots,x_{n-k+1})</math>
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:<math>=\sum{n! \over j_1!j_2!\cdots j_{n-k+1}!}
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\left({x_1\over 1!}\right)^{j_1}\left({x_2\over 2!}\right)^{j_2}\cdots\left({x_{n-k+1} \over (n-k+1)!}\right)^{j_{n-k+1}},</math>


where the sum is taken over all sequences ''j''<sub>1</sub>, ''j''<sub>2</sub>, ''j''<sub>3</sub>, ..., ''j''<sub>''n''−''k''+1</sub> of non-negative integers such that
'''MathML'''
:<math forcemathmode="mathml">E=mc^2</math>


:<math>j_1+j_2+\cdots = k\quad\mbox{and}\quad j_1+2j_2+3j_3+\cdots=n.</math>
<!--'''PNG'''  (currently default in production)
:<math forcemathmode="png">E=mc^2</math>


==Complete Bell polynomials==
'''source'''
:<math forcemathmode="source">E=mc^2</math> -->


The sum
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:<math>B_n(x_1,\dots,x_n)=\sum_{k=1}^n B_{n,k}(x_1,x_2,\dots,x_{n-k+1})</math>
==Demos==


is sometimes called the ''n''th '''complete Bell polynomial'''. In order to contrast them with complete Bell polynomials, the polynomials ''B''<sub>''n'',&nbsp;''k''</sub> defined above are sometimes called "partial" Bell polynomials.
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:


The complete Bell polynomials satisfy the following identity


:<math>B_n(x_1,\dots,x_n) = \det\begin{bmatrix}x_1 & {n-1 \choose 1} x_2 & {n-1 \choose 2}x_3 & {n-1 \choose 3} x_4 & {n-1 \choose 4} x_5 & \cdots & \cdots & x_n \\  \\
* accessibility:
-1 & x_1 & {n-2 \choose 1} x_2 & {n-2 \choose 2} x_3 & {n-2 \choose 3} x_4 & \cdots & \cdots & x_{n-1} \\  \\
** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]]
0 & -1 & x_1 & {n-3 \choose 1} x_2 & {n-3 \choose 2} x_3 & \cdots & \cdots & x_{n-2} \\  \\
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Audio-Windows7-InternetExplorer.ogg Internet Explorer + MathPlayer (audio)]
0 & 0 & -1 & x_1 & {n-4 \choose 1} x_2 & \cdots  & \cdots & x_{n-3} \\  \\
** [https://commons.wikimedia.org/wiki/File:MathPlayer-SynchronizedHighlighting-WIndows7-InternetExplorer.png Internet Explorer + MathPlayer (synchronized highlighting)]
0 & 0 & 0 & -1 & x_1 & \cdots & \cdots & x_{n-4} \\  \\
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Braille-Windows7-InternetExplorer.png Internet Explorer + MathPlayer (braille)]
0 & 0 & 0 & 0 & -1 & \cdots & \cdots & x_{n-5} \\  \\
** NVDA+MathPlayer: [[File:Nvda-mathml-example-1.wav|thumb|Nvda-mathml-example-1]], [[File:Nvda-mathml-example-2.wav|thumb|Nvda-mathml-example-2]], [[File:Nvda-mathml-example-3.wav|thumb|Nvda-mathml-example-3]], [[File:Nvda-mathml-example-4.wav|thumb|Nvda-mathml-example-4]], [[File:Nvda-mathml-example-5.wav|thumb|Nvda-mathml-example-5]], [[File:Nvda-mathml-example-6.wav|thumb|Nvda-mathml-example-6]], [[File:Nvda-mathml-example-7.wav|thumb|Nvda-mathml-example-7]].
\vdots & \vdots & \vdots &  \vdots & \vdots & \ddots & \ddots & \vdots  \\  \\
** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]].
0 & 0 & 0 & 0 & 0 & \cdots & -1 & x_1  \end{bmatrix}.</math>
** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode.


==Combinatorial meaning==
==Test pages ==


If the integer ''n'' is [[integer partition|partitioned]] into a sum in which "1" appears ''j''<sub>1</sub> times, "2" appears ''j''<sub>2</sub> times, and so on, then the number of [[partition of a set|partitions of a set]] of size ''n'' that collapse to that partition of the integer ''n'' when the members of the set become indistinguishable is the corresponding coefficient in the polynomial.
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:
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===Examples===
*[[Inputtypes|Inputtypes (private Wikis only)]]
 
*[[Url2Image|Url2Image (private Wikis only)]]
For example, we have
==Bug reporting==
 
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .
:<math>B_{6,2}(x_1,x_2,x_3,x_4,x_5)=6x_5x_1+15x_4x_2+10x_3^2</math>
 
because there are
 
:6 ways to partition a set of 6 as 5&nbsp;+&nbsp;1,
:15 ways to partition a set of 6 as 4&nbsp;+&nbsp;2, and
:10 ways to partition a set of 6 as 3&nbsp;+&nbsp;3.
 
Similarly,
 
:<math>B_{6,3}(x_1,x_2,x_3,x_4)=15x_4x_1^2+60x_3x_2x_1+15x_2^3</math>
 
because there are
 
:15 ways to partition a set of 6 as 4&nbsp;+&nbsp;1&nbsp;+&nbsp;1,
:60 ways to partition a set of 6 as 3&nbsp;+&nbsp;2&nbsp;+&nbsp;1, and
:15 ways to partition a set of 6 as 2&nbsp;+&nbsp;2&nbsp;+&nbsp;2.
 
==Properties==
 
* <math>B_{n,k}(1!,2!,\dots,(n-k+1)!) = \binom{n}{k}\binom{n-1}{k-1} (n-k)!</math>
 
===Stirling numbers and Bell numbers===
 
The value of the Bell polynomial ''B''<sub>''n'',''k''</sub>(''x''<sub>1</sub>,''x''<sub>2</sub>,...) when all ''x''s are equal to 1 is a [[Stirling number of the second kind]]:
 
:<math>B_{n,k}(1,1,\dots)=S(n,k)=\left\{{n\atop k}\right\}.</math>
 
The sum
 
:<math>\sum_{k=1}^n B_{n,k}(1,1,1,\dots) = \sum_{k=1}^n \left\{{n\atop k}\right\}</math>
 
is the ''n''th [[Bell number]], which is the number of partitions of a set of size ''n''.
 
===Convolution identity===
 
For sequences ''x''<sub>''n''</sub>, ''y''<sub>''n''</sub>, ''n'' = 1, 2, ..., define a sort of [[convolution]] by:
 
:<math>(x \diamondsuit y)_n = \sum_{j=1}^{n-1} {n \choose j} x_j y_{n-j}</math>.
 
Note that the bounds of summation are 1 and ''n''&nbsp;−&nbsp;1, not 0 and ''n'' .
 
Let <math>x_n^{k\diamondsuit}\,</math> be the ''n''th term of the sequence
 
:<math>\displaystyle\underbrace{x\diamondsuit\cdots\diamondsuit x}_{k\ \mathrm{factors}}.\,</math>
 
Then
 
:<math>B_{n,k}(x_1,\dots,x_{n-k+1}) = {x_{n}^{k\diamondsuit} \over k!}.\,</math>
 
For example, let us compute <math> B_{4,3}(x_1,x_2) </math>.  We have
 
:<math> x = ( x_1 \ , \ x_2 \ , \ x_3 \ , \ x_4 \ , \dots ) </math>
 
:<math> x \diamondsuit x = ( 0,\  2 x_1^2 \ ,\  6 x_1 x_2 \ , \  8 x_1 x_3 + 6 x_2^2 \ , \dots ) </math>
 
:<math> x \diamondsuit x \diamondsuit x = (  0 \ ,\ 0 \  , \ 6 x_1^3 \ , \ 36 x_1^2 x_2 \ , \dots ) </math>
 
and thus,
 
:<math> B_{4,3}(x_1,x_2) = \frac{ ( x \diamondsuit x \diamondsuit x)_4 }{3!} = 6 x_1^2 x_2. </math>
 
==Applications of Bell polynomials==
===Faà di Bruno's formula===
{{main|Faà di Bruno's formula}}
 
[[Faà di Bruno's formula]] may be stated in terms of Bell polynomials as follows:
 
:<math>{d^n \over dx^n} f(g(x)) = \sum_{k=0}^n f^{(k)}(g(x)) B_{n,k}\left(g'(x),g''(x),\dots,g^{(n-k+1)}(x)\right).</math>
 
Similarly, a power-series version of Faà di Bruno's formula may be stated using Bell polynomials as follows.  Suppose
 
:<math>f(x)=\sum_{n=1}^\infty {a_n \over n!} x^n \qquad
\mathrm{and} \qquad g(x)=\sum_{n=1}^\infty {b_n \over n!} x^n.</math>
 
Then
 
:<math>g(f(x)) = \sum_{n=1}^\infty
{\sum_{k=1}^{n} b_k B_{n,k}(a_1,\dots,a_{n-k+1}) \over n!} x^n.</math>
 
In particular, the complete Bell polynomials appear in the exponential of a [[formal power series]]:
 
:<math>\exp\left(\sum_{n=1}^\infty {a_n \over n!} x^n \right)
= \sum_{n=0}^\infty {B_n(a_1,\dots,a_n) \over n!} x^n.</math>
 
===Moments and cumulants===
 
The sum
 
:<math>B_n(\kappa_1,\dots,\kappa_n)=\sum_{k=1}^n B_{n,k}(\kappa_1,\dots,\kappa_{n-k+1})</math>
 
is the ''n''th [[moment (mathematics)|moment]] of a [[probability distribution]] whose first ''n'' [[cumulant]]s are κ<sub>1</sub>, ..., κ<sub>''n''</sub>.  In other words, the ''n''th moment is the ''n''th complete Bell polynomial evaluated at the first ''n'' cumulants.
 
===Representation of polynomial sequences of binomial type===
 
For any sequence ''a''<sub>1</sub>, ''a''<sub>2</sub>, ''a''<sub>3</sub>, ... of scalars, let
 
:<math>p_n(x)=\sum_{k=1}^n B_{n,k}(a_1,\dots,a_{n-k+1}) x^k.</math>
 
Then this polynomial sequence is of [[binomial type]], i.e. it satisfies the binomial identity
 
:<math>p_n(x+y)=\sum_{k=0}^n {n \choose k} p_k(x) p_{n-k}(y)</math>
 
for ''n'' ≥ 0.  In fact we have this result:
 
:'''Theorem:''' All polynomial sequences of binomial type are of this form.
 
If we let
 
:<math>h(x)=\sum_{n=1}^\infty {a_n \over n!} x^n</math>
 
taking this power series to be purely formal, then for all ''n'',
 
:<math>h^{-1}\left( {d \over dx}\right) p_n(x) = n p_{n-1}(x).</math>
 
==Software==
 
* Bell polynomials, complete Bell polynomials and generalized Bell polynomials are implemented in [[Mathematica]] as [http://reference.wolfram.com/mathematica/ref/BellY.html BellY].
 
==See also==
 
* [[Bell matrix]]
* [[Exponential formula]]
 
==References==
* {{cite journal | author=[[Eric Temple Bell]] |title=Partition Polynomials | jstor=1967979 |journal=[[Annals of Mathematics]] |volume=29 |issue=1/4 |year=1927–1928 |pages=38–46 |doi=10.2307/1967979 | mr=1502817 }}
* {{cite book |author=Louis Comtet |title=Advanced Combinatorics: The Art of Finite and Infinite Expansions |publisher=Reidel Publishing Company |place=Dordrecht, Holland / Boston, U.S. |year=1974}}
* {{cite book |author=[[Steven Roman]] |title=''The Umbral Calculus'' |publisher=[[Dover Publications]]}}
* {{cite journal |author=Vassily G. Voinov, Mikhail S. Nikulin |title=On power series, Bell polynomials, Hardy-Ramanujan-Rademacher problem and its statistical applications |journal= Kybernetika | volume=30 |issue=3 |pages=343–358 |year=1994 |issn=00235954}}
* {{cite book | first=George E. | last=Andrews | authorlink=George Andrews (mathematician) | title=The Theory of Partitions | series=Cambridge Mathematical Library | publisher=[[Cambridge University Press]] | year=1998 | edition=1st pbk | isbn=0-521-63766-X | pages=204–211 }}
* {{cite journal |author=Silvia Noschese, Paolo E. Ricci |title=Differentiation of Multivariable Composite Functions and Bell Polynomials |journal=Journal of Computational Analysis and Applications | volume=5 |issue=3 |pages=333–340 |year=2003 |doi=10.1023/A:1023227705558}}
*{{ cite journal|first1=Moncef
|last1=Abbas
|first2=Sadek
|last2=Bouroubi
|title=On new identities for Bell's polynomial
|journal=Disc. Math
|year=2005
|number=293
|pages=5-10
|mr=2136048
|doi=10.1016/j.disc.2004.08.023
}}
* {{cite journal |author=Khristo N. Boyadzhiev |title=Exponential Polynomials, Stirling Numbers, and Evaluation of Some Gamma Integrals |journal=[[Abstract and Applied Analysis]] |volume=2009 |pages=Article ID 168672 |year=2009 |doi=10.1155/2009/168672}} (contains also elementary review of the concept Bell-polynomials)
* {{cite arxiv| author=V. V. Kruchinin
|year=2011|eprint=1104.5065
|title=Derivation of Bell Polynomials of the Second Kind}}
* {{cite journal
|first1=Martin
|last1=Griffiths
|title=Families of sequences from a class of multinomial sums
|journal=Journal of Integer Sequences
|url=http://www.cs.uwaterloo.ca/journals/JIS/VOL15/Griffiths/griffiths20.html
|year=2012
|mr=2872465
|volume=15
}}
 
{{DEFAULTSORT:Bell Polynomials}}
[[Category:Enumerative combinatorics]]
[[Category:Polynomials]]
 
[[fr:Polynômes de Bell]]
[[ru:Полиномы Белла]]

Latest revision as of 23:52, 15 September 2019

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