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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>
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| 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
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| :<math>j_1+j_2+\cdots = k\quad\mbox{and}\quad j_1+2j_2+3j_3+\cdots=n.</math>
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| ==Complete Bell polynomials== | |
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| 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>
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| is sometimes called the ''n''th '''complete Bell polynomial'''. In order to contrast them with complete Bell polynomials, the polynomials ''B''<sub>''n'', ''k''</sub> defined above are sometimes called "partial" Bell polynomials.
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| The complete Bell polynomials satisfy the following identity
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| :<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 \\ \\
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| -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} \\ \\
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| 0 & -1 & x_1 & {n-3 \choose 1} x_2 & {n-3 \choose 2} x_3 & \cdots & \cdots & x_{n-2} \\ \\
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| 0 & 0 & -1 & x_1 & {n-4 \choose 1} x_2 & \cdots & \cdots & x_{n-3} \\ \\
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| 0 & 0 & 0 & -1 & x_1 & \cdots & \cdots & x_{n-4} \\ \\
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| 0 & 0 & 0 & 0 & -1 & \cdots & \cdots & x_{n-5} \\ \\
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| \vdots & \vdots & \vdots & \vdots & \vdots & \ddots & \ddots & \vdots \\ \\
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| 0 & 0 & 0 & 0 & 0 & \cdots & -1 & x_1 \end{bmatrix}.</math>
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| ==Combinatorial meaning==
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| 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.
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| ===Examples=== | |
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| For example, we have
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| :<math>B_{6,2}(x_1,x_2,x_3,x_4,x_5)=6x_5x_1+15x_4x_2+10x_3^2</math>
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| because there are
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| :6 ways to partition a set of 6 as 5 + 1,
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| :15 ways to partition a set of 6 as 4 + 2, and
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| :10 ways to partition a set of 6 as 3 + 3.
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| Similarly,
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| :<math>B_{6,3}(x_1,x_2,x_3,x_4)=15x_4x_1^2+60x_3x_2x_1+15x_2^3</math>
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| because there are
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| :15 ways to partition a set of 6 as 4 + 1 + 1,
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| :60 ways to partition a set of 6 as 3 + 2 + 1, and
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| :15 ways to partition a set of 6 as 2 + 2 + 2.
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| ==Properties== | |
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| * <math>B_{n,k}(1!,2!,\dots,(n-k+1)!) = \binom{n}{k}\binom{n-1}{k-1} (n-k)!</math>
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| ===Stirling numbers and Bell numbers===
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| 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]]:
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| :<math>B_{n,k}(1,1,\dots)=S(n,k)=\left\{{n\atop k}\right\}.</math>
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| The sum
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| :<math>\sum_{k=1}^n B_{n,k}(1,1,1,\dots) = \sum_{k=1}^n \left\{{n\atop k}\right\}</math>
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| is the ''n''th [[Bell number]], which is the number of partitions of a set of size ''n''.
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| ===Convolution identity===
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| For sequences ''x''<sub>''n''</sub>, ''y''<sub>''n''</sub>, ''n'' = 1, 2, ..., define a sort of [[convolution]] by:
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| :<math>(x \diamondsuit y)_n = \sum_{j=1}^{n-1} {n \choose j} x_j y_{n-j}</math>.
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| Note that the bounds of summation are 1 and ''n'' − 1, not 0 and ''n'' .
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| Let <math>x_n^{k\diamondsuit}\,</math> be the ''n''th term of the sequence
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| :<math>\displaystyle\underbrace{x\diamondsuit\cdots\diamondsuit x}_{k\ \mathrm{factors}}.\,</math>
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| Then
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| :<math>B_{n,k}(x_1,\dots,x_{n-k+1}) = {x_{n}^{k\diamondsuit} \over k!}.\,</math>
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| For example, let us compute <math> B_{4,3}(x_1,x_2) </math>. We have
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| :<math> x = ( x_1 \ , \ x_2 \ , \ x_3 \ , \ x_4 \ , \dots ) </math>
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| :<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>
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| :<math> x \diamondsuit x \diamondsuit x = ( 0 \ ,\ 0 \ , \ 6 x_1^3 \ , \ 36 x_1^2 x_2 \ , \dots ) </math>
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| and thus,
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| :<math> B_{4,3}(x_1,x_2) = \frac{ ( x \diamondsuit x \diamondsuit x)_4 }{3!} = 6 x_1^2 x_2. </math>
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| ==Applications of Bell polynomials==
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| ===Faà di Bruno's formula===
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| {{main|Faà di Bruno's formula}}
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| [[Faà di Bruno's formula]] may be stated in terms of Bell polynomials as follows:
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| :<math>{d^n \over dx^n} f(g(x)) = \sum_{k=1}^n f^{(k)}(g(x)) B_{n,k}\left(g'(x),g''(x),\dots,g^{(n-k+1)}(x)\right).</math>
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| Similarly, a power-series version of Faà di Bruno's formula may be stated using Bell polynomials as follows. Suppose
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| :<math>f(x)=\sum_{n=1}^\infty {a_n \over n!} x^n \qquad
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| \mathrm{and} \qquad g(x)=\sum_{n=1}^\infty {b_n \over n!} x^n.</math>
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| Then
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| :<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>
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| In particular, the complete Bell polynomials appear in the exponential of a [[formal power series]]:
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| :<math>\exp\left(\sum_{n=1}^\infty {a_n \over n!} x^n \right)
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| = \sum_{n=0}^\infty {B_n(a_1,\dots,a_n) \over n!} x^n.</math>
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| ===Moments and cumulants===
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| The sum
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| :<math>B_n(\kappa_1,\dots,\kappa_n)=\sum_{k=1}^n B_{n,k}(\kappa_1,\dots,\kappa_{n-k+1})</math> | |
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| 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.
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| ===Representation of polynomial sequences of binomial type===
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| For any sequence ''a''<sub>1</sub>, ''a''<sub>2</sub>, ''a''<sub>3</sub>, ... of scalars, let
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| :<math>p_n(x)=\sum_{k=1}^n B_{n,k}(a_1,\dots,a_{n-k+1}) x^k.</math> | |
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| Then this polynomial sequence is of [[binomial type]], i.e. it satisfies the binomial identity
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| :<math>p_n(x+y)=\sum_{k=0}^n {n \choose k} p_k(x) p_{n-k}(y)</math>
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| for ''n'' ≥ 0. In fact we have this result:
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| :'''Theorem:''' All polynomial sequences of binomial type are of this form.
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| If we let
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| :<math>h(x)=\sum_{n=1}^\infty {a_n \over n!} x^n</math>
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| taking this power series to be purely formal, then for all ''n'',
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| :<math>h^{-1}\left( {d \over dx}\right) p_n(x) = n p_{n-1}(x).</math>
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| ==Software==
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| * Bell polynomials, complete Bell polynomials and generalized Bell polynomials are implemented in [[Mathematica]] as [http://reference.wolfram.com/mathematica/ref/BellY.html BellY], in [[Maple (software)|Maple]] as [http://www.maplesoft.com/support/help/Maple/view.aspx?path=BellB BellB], and in [[Sage (mathematics software)|Sage]] as [http://www.sagemath.org/doc/reference/combinat/sage/combinat/combinat.html#sage.combinat.combinat.bell_polynomial bell_polynomial].
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| ==See also==
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| * [[Bell matrix]]
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| * [[Exponential formula]]
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| ==References==
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| * {{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 }}
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| * {{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}}
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| * {{cite book |author=[[Steven Roman]] |title=''The Umbral Calculus'' |publisher=[[Dover Publications]]}}
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| * {{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}}
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| * {{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 }}
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| * {{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}}
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| *{{ cite journal|first1=Moncef
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| |last1=Abbas
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| |first2=Sadek
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| |last2=Bouroubi
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| |title=On new identities for Bell's polynomial
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| |journal=Disc. Math
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| |year=2005
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| |number=293
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| |pages=5–10
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| |mr=2136048
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| |doi=10.1016/j.disc.2004.08.023
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| }}
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| * {{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)
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| * {{cite arxiv| author=V. V. Kruchinin
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| |year=2011|eprint=1104.5065
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| |title=Derivation of Bell Polynomials of the Second Kind}}
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| * {{cite journal
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| |first1=Martin
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| |last1=Griffiths
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| |title=Families of sequences from a class of multinomial sums
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| |journal=Journal of Integer Sequences
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| |url=http://www.cs.uwaterloo.ca/journals/JIS/VOL15/Griffiths/griffiths20.html
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| |year=2012
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| |mr=2872465
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| |volume=15
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| }}
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| {{DEFAULTSORT:Bell Polynomials}}
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| [[Category:Enumerative combinatorics]]
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| [[Category:Polynomials]]
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