Stirling numbers and exponential generating functions in symbolic combinatorics

The use of <a href="/facts/Exponential_generating_function/K7UVjnjL">exponential generating functions</a> (EGFs) to study the properties of <a href="/facts/Stirling_numbers/MC05d5pF">Stirling numbers</a> is a classical exercise in <a href="/facts/Combinatorics/k14xUoKT">combinatorial mathematics</a> and possibly the canonical example of how <a href="/facts/Symbolic_combinatorics/Sq6Q7JgX">symbolic combinatorics</a> 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 
 
 
 
 [
 
 z
 
 n
 
 
 ]
 
 
 {\displaystyle [z^{n}]}
 
 for <a href="/facts/Formal_power_series/CNmaG0Vk">formal power series</a>, as well as the (labelled) operators 
 
 
 
 
 
 C
 
 
 
 
 {\displaystyle {\mathfrak {C}}}
 
 (for cycles) and 
 
 
 
 
 
 P
 
 
 
 
 {\displaystyle {\mathfrak {P}}}
 
 (for sets) on combinatorial classes, which are explained on the page for <a href="/facts/Symbolic_combinatorics/Sq6Q7JgX">symbolic combinatorics</a>. 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

<ul><li><a href="/facts/Permutation/JSw9ukmv">permutations</a> (for unsigned Stirling numbers of the first kind):</li></ul>

P
          
        
        =
        SET
        ⁡
        (
        CYC
        ⁡
        (
        
          
            Z
          
        
        )
        )
        ,
      
    
    {\displaystyle {\mathcal {P}}=\operatorname {SET} (\operatorname {CYC} ({\mathcal {Z}})),}

and

<ul><li><a href="/facts/Set_partition/VeP9g8CA">set partitions</a> into non-empty subsets (for Stirling numbers of the second kind):</li></ul>

B
          
        
        =
        SET
        ⁡
        (
        
          SET
          
            ≥
            1
          
        
        ⁡
        (
        
          
            Z
          
        
        )
        )
        ,
      
    
    {\displaystyle {\mathcal {B}}=\operatorname {SET} (\operatorname {SET} _{\geq 1}({\mathcal {Z}})),}

where 
 
 
 
 
 
 Z
 
 
 
 
 {\displaystyle {\mathcal {Z}}}
 
 is the singleton class.
Warning: The notation used here for the Stirling numbers is not that of the Wikipedia articles on Stirling numbers; square brackets denote the signed Stirling numbers here.

Stirling numbers and exponential generating functions in symbolic combinatorics open-in-new

Stirling numbers and exponential generating functions in symbolic combinatorics