 Bell polynomials

In combinatorial mathematics, the Bell polynomials, named in honor of Eric Temple Bell, are a triangular array of polynomials given by
the sum extending over all sequences j_{1}, j_{2}, j_{3}, ..., j_{n−k+1} of nonnegative integers such that
Contents
Convolution identity
For sequences x_{n}, y_{n}, n = 1, 2, ..., define a sort of convolution by:
 .
Note that the bounds of summation are 1 and n − 1, not 0 and n .
Let be the nth term of the sequence
Then
For example, let us compute B_{4,3}(x_{1},x_{2}). We have
and thus,
Complete Bell polynomials
The sum
is sometimes called the nth complete Bell polynomial. In order to contrast them with complete Bell polynomials, the polynomials B_{n, k} defined above are sometimes called "partial" Bell polynomials. The complete Bell polynomials satisfy the following identity
Combinatorial meaning
If the integer n is partitioned into a sum in which "1" appears j_{1} times, "2" appears j_{2} times, and so on, then the number of 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.
Examples
For example, we have
because there are
 6 ways to partition a set of 6 as 5 + 1,
 15 ways to partition a set of 6 as 4 + 2, and
 10 ways to partition a set of 6 as 3 + 3.
Similarly,
because there are
 15 ways to partition a set of 6 as 4 + 1 + 1,
 60 ways to partition a set of 6 as 3 + 2 + 1, and
 15 ways to partition a set of 6 as 2 + 2 + 2.
Stirling numbers and Bell numbers
The value of the Bell polynomial B_{n,k}(x_{1},x_{2},...) when all xs are equal to 1 is a Stirling number of the second kind:
The sum
is the nth Bell number, which is the number of partitions of a set of size n.
Properties
Applications of Bell polynomials
Faà di Bruno's formula
Main article: Faà di Bruno's formulaFaà di Bruno's formula may be stated in terms of Bell polynomials as follows:
Similarly, a powerseries version of Faà di Bruno's formula may be stated using Bell polynomials as follows. Suppose
Then
The complete Bell polynomials appear in the exponential of a formal power series:
See also exponential formula.
Moments and cumulants
The sum
is the nth moment of a probability distribution whose first n cumulants are κ_{1}, ..., κ_{n}. In other words, the nth moment is the nth complete Bell polynomial evaluated at the first n cumulants.
Representation of polynomial sequences of binomial type
For any sequence a_{1}, a_{2}, a_{3}, ... of scalars, let
Then this polynomial sequence is of binomial type, i.e. it satisfies the binomial identity
for n ≥ 0. In fact we have this result:
 Theorem: All polynomial sequences of binomial type are of this form.
If we let
taking this power series to be purely formal, then for all n,
Software
 Bell polynomials, complete Bell polynomials and generalized Bell polynomials are implemented in Mathematica as BellY.
See also
 Bell matrix
References
 Eric Temple Bell (1927–1928). "Partition Polynomials". Annals of Mathematics 29 (1/4): 38–46. doi:10.2307/1967979. JSTOR 1967979. MR1502817.
 Louis Comtet (1974). Advanced Combinatorics: The Art of Finite and Infinite Expansions. Dordrecht, Holland / Boston, U.S.: Reidel Publishing Company.
 Steven Roman. The Umbral Calculus. Dover Publications.
 Khristo N. Boyadzhiev (2009). "Exponential Polynomials, Stirling Numbers, and Evaluation of Some Gamma Integrals". Abstract and Applied Analysis 2009: Article ID 168672. doi:10.1155/2009/168672. http://www.hindawi.com/journals/aaa/2009/168672.html. (contains also elementary review of the concept Bellpolynomials)
 Silvia Noschese, Paolo E. Ricci (2003). "Differentiation of Multivariable Composite Functions and Bell Polynomials". Journal of Computational Analysis and Applications 5 (3): 333–340. doi:10.1023/A:1023227705558. http://www.springerlink.com/content/m815387gx128r1w4/. '
 Vassily G.Voinov, Mikhail S.Nikulin (1994). "On power series, Bell polynomials, HardyRamanujanRademacher problem and its statistical applications". Kybernetika 30 (3): 343–358. ISSN 5954 0023 5954.
 Kruchinin, V.V., 2011 , Derivation of Bell Polynomials of the Second Kind(ArXiv)
Categories: Enumerative combinatorics
 Polynomials
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