Here is a purely algebraic statement, as a first introduction to the combinatorial use of the formula.
For any formal power series of the form
we have
where
and the index runs through all partitions of the set
. (When the product is empty and by definition equals .)
One can write the formula in the following form:
and thus
where is the th complete Bell polynomial.
Alternatively, the exponential formula can also be written using the cycle index of the symmetric group, as follows:
where stands for the cycle index polynomial for the symmetric group , defined as:
and denotes the number of cycles of of size . This is a consequence of the general relation between and Bell polynomials: