A partial Bell polynomial 
is a polynomial
that enumerates set partitions of an 
-element set into exactly 
nonempty blocks, with a set
block of size 
assigned weight 
. The exponential convention
used here is characterized by the exponential
generating function
 |
(1)
|
where 
is a nonnegative integer. The boundary values
are 
, 
for 
, and 
for 
.
Equivalently, for a positive integer 
,
 |
(2)
|
where the sum runs over nonnegative integers 
, ..., 
.
When 
, the largest argument that can occur is
. Each monomial
has total degree 
. Replacing every 
by 
multiplies each monomial
by 
. For example,
 |  |  |
(3)
|
 |  |  |
(4)
|
 |  |  |
(5)
|
 |  |  |
(6)
|
Suppressing the arguments, a recurrence relation is
 |
(7)
|
where 
is a nonnegative integer and 
a positive integer. Summing
the partial Bell polynomials over 
gives the complete
Bell polynomials.
Several counting sequences are special values,
 |  |  |
(8)
|
 |  |  |
(9)
|
 |  |  |
(10)
|
where 
is a Stirling number of the second
kind, the second row gives the Lah number, and
the third gives the idempotent number, with
in the last two rows (Di Nardo and Guarino 2022).
Partial Bell polynomials express the derivatives of a function composition through Faà di Bruno's formula,
 |
(11)
|
where 
is a positive integer and the necessary derivatives
exist. For a random variable whose moment-generating
function is defined near 0, they also express cumulants
in terms of raw moments,
 |
(12)
|
where 
is a positive integer, 
are the raw moments, and
the 
th
cumulant. This follows by applying Faà
di Bruno's formula to the logarithm of the moment-generating
function near 0.
The ordinary normalization omits the factorial divisors in the generating series,
 |
(13)
|
It is related to the exponential normalization by
 |
(14)
|
(Di Nardo and Guarino 2022). For example, 
,
whereas 
. The name de
Moivre polynomial is used for the ordinary normalization by O'Sullivan (2022).
The exponential partial Bell polynomials are implemented in the Wolfram
Language as BellY[n,
k, {x1, x2, ...}].
