where the sum is taken over all sequences j1, j2, j3, ..., jn−k+1 of non-negative integers such that
- 1 Complete Bell polynomials
- 2 Combinatorial meaning
- 3 Properties
- 4 Applications of Bell polynomials
- 5 Software
- 6 See also
- 7 References
Complete Bell polynomials
is sometimes called the nth complete Bell polynomial. In order to contrast them with complete Bell polynomials, the polynomials Bn,k are sometimes called partial or incomplete Bell polynomials.
The complete Bell polynomials satisfy the following identity
If the integer n is partitioned into a sum in which "1" appears j1 times, "2" appears j2 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.
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.
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 Bn,k(x1,x2,...) on the sequence of ones equals a Stirling number of the second kind:
The sum of these values gives the value of the complete Bell polynomial on the sequence of ones:
which is the nth Bell number.
Touchard polynomial can be expressed as the value of the complete Bell polynomial on all arguments being x:
For sequences xn, yn, 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
For example, let us compute . We have
Applications of Bell polynomials
Faà di Bruno's formula
Faà di Bruno's formula may be stated in terms of Bell polynomials as follows:
Similarly, a power-series version of Faà di Bruno's formula may be stated using Bell polynomials as follows. Suppose
In particular, the complete Bell polynomials appear in the exponential of a formal power series:
Moments and cumulants
Representation of polynomial sequences of binomial type
For any sequence a1, a2, …, an of scalars, let
Then this polynomial sequence is of binomial type, i.e. it satisfies the binomial identity
- Example: For a1 = … = an = 1, the polynomials are called Touchard polynomials.
More generally, we have this result:
- Theorem: All polynomial sequences of binomial type are of this form.
If we define a formal power series
then for all n,
Bell polynomials are implemented in:
- Eric Temple Bell (1927–1928). "Partition Polynomials". Annals of Mathematics 29 (1/4): 38–46. doi:10.2307/1967979. JSTOR 1967979. MR 1502817.
- 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.
- Vassily G. Voinov, Mikhail S. Nikulin (1994). "On power series, Bell polynomials, Hardy-Ramanujan-Rademacher problem and its statistical applications". Kybernetika 30 (3): 343–358. ISSN 0023-5954.
- Andrews, George E. (1998). The Theory of Partitions. Cambridge Mathematical Library (1st pbk ed.). Cambridge University Press. pp. 204–211. ISBN 0-521-63766-X.
- 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.
- Abbas, Moncef; Bouroubi, Sadek (2005). "On new identities for Bell's polynomial". Disc. Math (293): 5–10. doi:10.1016/j.disc.2004.08.023. MR 2136048.
- 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. (contains also elementary review of the concept Bell-polynomials)
- V. V. Kruchinin (2011). "Derivation of Bell Polynomials of the Second Kind". arXiv:1104.5065.
- Griffiths, Martin (2012). "Families of sequences from a class of multinomial sums". Journal of Integer Sequences 15. MR 2872465.