The ordinary Bell polynomials can be expressed in the terms of exponential Bell polynomials:
In general, Bell polynomial refers to the exponential Bell polynomial, unless otherwise explicitly stated.
Combinatorial meaningedit
The exponential Bell polynomial encodes the information related to the ways a set can be partitioned. For example, if we consider a set {A, B, C}, it can be partitioned into two non-empty, non-overlapping subsets, which are also referred to as parts or blocks, in 3 different ways:
{{A}, {B, C}}
{{B}, {A, C}}
{{C}, {B, A}}
Thus, we can encode the information regarding these partitions as
Here, the subscripts of B3,2 tell us that we are considering the partitioning of a set with 3 elements into 2 blocks. The subscript of each xi indicates the presence of a block with i elements (or block of size i) in a given partition. So here, x2 indicates the presence of a block with two elements. Similarly, x1 indicates the presence of a block with a single element. The exponent of xij indicates that there are j such blocks of size i in a single partition. Here, the fact that both x1 and x2 have exponent 1 indicates that there is only one such block in a given partition. The coefficient of the monomial indicates how many such partitions there are. Here, there are 3 partitions of a set with 3 elements into 2 blocks, where in each partition the elements are divided into two blocks of sizes 1 and 2.
Since any set can be divided into a single block in only one way, the above interpretation would mean that Bn,1 = xn. Similarly, since there is only one way that a set with n elements be divided into n singletons, Bn,n = x1n.
As a more complicated example, consider
This tells us that if a set with 6 elements is divided into 2 blocks, then we can have 6 partitions with blocks of size 1 and 5, 15 partitions with blocks of size 4 and 2, and 10 partitions with 2 blocks of size 3.
The sum of the subscripts in a monomial is equal to the total number of elements. Thus, the number of monomials that appear in the partial Bell polynomial is equal to the number of ways the integer n can be expressed as a summation of k positive integers. This is the same as the integer partition of n into k parts. For instance, in the above examples, the integer 3 can be partitioned into two parts as 2+1 only. Thus, there is only one monomial in B3,2. However, the integer 6 can be partitioned into two parts as 5+1, 4+2, and 3+3. Thus, there are three monomials in B6,2. Indeed, the subscripts of the variables in a monomial are the same as those given by the integer partition, indicating the sizes of the different blocks. The total number of monomials appearing in a complete Bell polynomial Bn is thus equal to the total number of integer partitions of n.
Also the degree of each monomial, which is the sum of the exponents of each variable in the monomial, is equal to the number of blocks the set is divided into. That is, j1 + j2 + ... = k . Thus, given a complete Bell polynomial Bn, we can separate the partial Bell polynomial Bn,k by collecting all those monomials with degree k.
Finally, if we disregard the sizes of the blocks and put all xi = x, then the summation of the coefficients of the partial Bell polynomial Bn,k will give the total number of ways that a set with n elements can be partitioned into k blocks, which is the same as the Stirling numbers of the second kind. Also, the summation of all the coefficients of the complete Bell polynomial Bn will give us the total number of ways a set with n elements can be partitioned into non-overlapping subsets, which is the same as the Bell number.
In general, 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.
Examplesedit
For example, we have
because the ways to partition a set of 6 elements as 2 blocks 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 the ways to partition a set of 6 elements as 3 blocks 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.
Table of valuesedit
Below is a triangular array of the incomplete Bell polynomials :
k
n
0
1
2
3
4
5
6
0
1
2
3
4
5
6
Propertiesedit
Generating functionedit
The exponential partial Bell polynomials can be defined by the double series expansion of its generating function:
In other words, by what amounts to the same, by the series expansion of the k-th power:
The complete exponential Bell polynomial is defined by , or in other words:
Thus, the n-th complete Bell polynomial is given by
Likewise, the ordinary partial Bell polynomial can be defined by the generating function
Or, equivalently, by series expansion of the k-th power:
Similarly, a power-series version of Faà di Bruno's formula may be stated using Bell polynomials as follows. Suppose
Then
In particular, the complete Bell polynomials appear in the exponential of a formal power series:
which also represents the exponential generating function of the complete Bell polynomials on a fixed sequence of arguments .
Reversion of seriesedit
Let two functions f and g be expressed in formal power series as
such that g is the compositional inverse of f defined by g(f(w)) = w or f(g(z)) = z. If f0 = 0 and f1 ≠ 0, then an explicit form of the coefficients of the inverse can be given in term of Bell polynomials as[8]
with and is the rising factorial, and
Asymptotic expansion of Laplace-type integralsedit
Consider the integral of the form
where (a,b) is a real (finite or infinite) interval, λ is a large positive parameter and the functions f and g are continuous. Let f have a single minimum in [a,b] which occurs at x = a. Assume that as x → a+,
with α > 0, Re(β) > 0; and that the expansion of f can be term wise differentiated. Then, Laplace–Erdelyi theorem states that the asymptotic expansion of the integral I(λ) is given by
where the coefficients cn are expressible in terms of an and bn using partial ordinary Bell polynomials, as given by Campbell–Froman–Walles–Wojdylo formula:
These formulae allow one to express the coefficients of monic polynomials in terms of the Bell polynomials of its zeroes. For instance, together with Cayley–Hamilton theorem they lead to expression of the determinant of a n × n square matrix A in terms of the traces of its powers:
Cycle index of symmetric groupsedit
The cycle index of the symmetric group can be expressed in terms of complete Bell polynomials as follows:
Moments and cumulantsedit
The sum
is the nth raw moment of a probability distribution whose first ncumulants are κ1, ..., κn. In other words, the nth moment is the nth complete Bell polynomial evaluated at the first n cumulants. Likewise, the nth cumulant can be given in terms of the moments as
where xi = 0 for all i > 2; thus allowing for a combinatorial interpretation of the coefficients of the Hermite polynomials. This can be seen by comparing the generating function of the Hermite polynomials
with that of Bell polynomials.
Representation of polynomial sequences of binomial typeedit
For any sequence a1, a2, …, an of scalars, let
Then this polynomial sequence is of binomial type, i.e. it satisfies the binomial identity
^Chou, W.-S.; Hsu, Leetsch C.; Shiue, Peter J.-S. (2006-06-01). "Application of Faà di Bruno's formula in characterization of inverse relations". Journal of Computational and Applied Mathematics. 190 (1–2): 151–169. doi:10.1016/j.cam.2004.12.041.
^Chu, Wenchang (2021-11-19). "Bell Polynomials and Nonlinear Inverse Relations". The Electronic Journal of Combinatorics. 28 (4). doi:10.37236/10390. ISSN 1077-8926.
Boyadzhiev, K. N. (2009). "Exponential Polynomials, Stirling Numbers, and Evaluation of Some Gamma Integrals". Abstract and Applied Analysis. 2009: 1–18. arXiv:0909.0979. Bibcode:2009AbApA2009....1B. doi:10.1155/2009/168672. S2CID 1608664. (contains also elementary review of the concept Bell-polynomials)
Charalambides, C. A. (2002). Enumerative Combinatorics. Chapman & Hall / CRC. p. 632. ISBN 9781584882909.
Comtet, L. (1974). Advanced Combinatorics: The Art of Finite and Infinite Expansions. Dordrecht, Holland / Boston, U.S.: Reidel Publishing Company. Archived from the original on 2017-06-01. Retrieved 2019-07-02.
Cvijović, D. (2011). "New identities for the partial Bell polynomials" (PDF). Applied Mathematics Letters. 24 (9): 1544–1547. doi:10.1016/j.aml.2011.03.043. S2CID 45311678. Archived (PDF) from the original on 2020-03-09. Retrieved 2020-06-05.
Griffiths, M. (2012). "Families of sequences from a class of multinomial sums". Journal of Integer Sequences. 15: Article 12.1.8. MR 2872465. Archived from the original on 2014-05-02. Retrieved 2012-06-27.
Kruchinin, V. V. (2011). "Derivation of Bell Polynomials of the Second Kind". arXiv:1104.5065 [math.CO].
Noschese, S.; Ricci, P. E. (2003). "Differentiation of Multivariable Composite Functions and Bell Polynomials". Journal of Computational Analysis and Applications. 5 (3): 333–340. doi:10.1023/A:1023227705558. S2CID 118361207.
Voinov, V. G.; Nikulin, M. S. (1994). "On power series, Bell polynomials, Hardy–Ramanujan–Rademacher problem and its statistical applications". Kybernetika. 30 (3): 343–358. ISSN 0023-5954.