The use of
exponential generating function
In mathematics, a generating function is a way of encoding an infinite sequence of numbers () by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinary ser ...
s (EGFs) to study the properties of
Stirling numbers
In mathematics, Stirling numbers arise in a variety of analytic and combinatorial problems. They are named after James Stirling, who introduced them in a purely algebraic setting in his book ''Methodus differentialis'' (1730). They were redisco ...
is a classical exercise in
combinatorial mathematics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many ap ...
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.
This article uses the coefficient extraction operator