The use of exponential generating functions (EGFs) to study the properties of Stirling numbers is a classical exercise in combinatorial mathematics and possibly the canonical example of how symbolic combinatorics is used.
It also illustrates the parallels in the construction of these two types of numbers, lending support to the binomial-style notation that is used for them.
for formal power series, as well as the (labelled) operators
(for sets) on combinatorial classes, which are explained on the page for symbolic combinatorics.
The two combinatorial classes (shown without additional markers) are and where
This decomposition is examined in some detail on the page on the statistics of random permutations.
Translating to generating functions we obtain the mixed generating function of the unsigned Stirling numbers of the first kind: Now the signed Stirling numbers of the first kind are obtained from the unsigned ones through the relation Hence the generating function
The Flajolet–Sedgewick fundamental theorem applies (labelled case).
of permutations from cycles, which is given by and yields the Stirling numbers of the first kind.
The decomposition is equivalent to the EGF Differentiate to obtain which implies that by convolution of exponential generating functions and because differentiating an EGF drops the first coefficient and shifts Bn+1 to z n/n!.
The EGF of the Stirling numbers of the second kind is obtained by marking every subset that goes into the partition with the term
, giving Translating to generating functions, we obtain This EGF yields the formula for the Stirling numbers of the second kind: or which simplifies to