Bell polynomials - Definition

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>

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 indistinguisbable 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.

Bell numbers

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.

Relation to Faà di Bruno's formula

The coefficients in these polynomials are the Faà di Bruno coefficients, occurring in Faà di Bruno's formula for the nth derivative of a composition of two functions.

Moments and cumulants

The sum

<math>\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.

References

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