Stirling numbers and exponential generating functions

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.

This article uses the coefficient extraction operator ${\displaystyle [z^{n}]}$ for formal power series, as well as the (labelled) operators ${\displaystyle {\mathfrak {C}}}$ (for cycles) and ${\displaystyle {\mathfrak {P}}}$ (for sets) on combinatorial classes, which are explained on the page for symbolic combinatorics. Given a combinatorial class, the cycle operator creates the class obtained by placing objects from the source class along a cycle of some length, where cyclical symmetries are taken into account, and the set operator creates the class obtained by placing objects from the source class in a set (symmetries from the symmetric group, i.e. an "unstructured bag".) The two combinatorial classes (shown without additional markers) are

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