Bell polynomials

In combinatorial mathematics, the Bell polynomials, named in honor of Eric Temple Bell, are given by

<math>B_{n,k}(x_1,x_2,\dots,x_{n-k+1})<math>

<math>=\sum{n! \over j_1!j_2!\cdots j_{n-k+1}!}

\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>

the sum extending over all sequences j₁, j₂, j₃, ..., j_n−k+1 of positive integers such that

<math>j_1+j_2+\cdots = k\quad\mbox{and}\quad j_1+2j_2+3j_3+\cdots=n.<math>

Contents

1 "Complete" Bell polynomials 2 Combinatorial meaning 2.1 Examples 2.2 Stirling numbers and Bell numbers 3 Where do Bell polynomials occur? 3.1 Composition of formal power series and Fàa di Bruno's formula 3.2 Moments and cumulants 3.3 Representation of polynomial sequences of binomial type 4 References

"Complete" Bell polynomials

The sum

<math>B_n(x_1,\dots,x_n)=\sum_{k=1}^n B_{n,k}(x_1,x_2,\dots)<math>

is sometimes called the nth complete Bell polynomial. That polynomial satisfies the following identity

<math>B_n(x_1,\dots,x_n) = \det\left[\begin{matrix}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 \\ \\

-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} \\ \\ 0 & -1 & x_1 & {n-3 \choose 1} x_2 & {n-3 \choose 2} x_3 & \cdots & \cdots & x_{n-2} \\ \\ 0 & 0 & -1 & x_1 & {n-4 \choose 1} x_2 & \cdots & \cdots & x_{n-3} \\ \\ 0 & 0 & 0 & -1 & x_1 & \cdots & \cdots & x_{n-4} \\ \\ \vdots & \vdots & \vdots & \vdots & \vdots & \ddots & & \vdots \\ \\ 0 & 0 & 0 & 0 & 0 & \cdots & -1 & x_1 \end{matrix}\right]<math>

Combinatorial meaning

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.

Examples

For example, we have

<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 of set of 6 as 5+1,

15 ways to partition of set of 6 as 4+2, and

10 ways to partition a set of 6 as 3+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+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(x₁,x₂,...) when all xs are equal to 1 is a Stirling number of the second kind:

<math>B_{n,k}(1,1,\dots)=S(n,k)=\left\{\begin{matrix} n \\ k \end{matrix}\right\}.<math>

The sum

<math>\sum_{k=1}^n B_{n,k}(1,1,1,\dots)<math>

is the nth Bell number, which is the number of partitions of a set of size n.

Where do Bell polynomials occur?

Composition of formal power series and Fàa di Bruno's formula

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>

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)

= 1 + \sum_{n=1}^\infty {B_n(a_1,\dots,a_n) \over n!} x^n.<math>

See also exponential formula.

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 nth moment of a probability distribution whose first n cumulants are κ₁, ..., 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 a₁, a₂, a₃, ... 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>

References

• Eric Temple Bell, Partition Polynomials, Annals of Mathematics, volume 29, 1927, pages 38 - 46.