derivation of the generating series for the Stirling numbers of the second kind
The derivation of the generating series is much simpler if one makes use of the composition lemma for exponential generating series. We are looking for the generating series for sets of nonempty sets, so in the notation of Jackson and Goulden, we have the set decomposition:
where is the set of all canonical unordered sets, is the set which we are interested in counting, and is star-composition of sets of labelled combinatorial objects.
The set has one object in it of each weight, and so has exponential generating series:
The set then has generating series:
So, by the star composition lemma and the above decomposition,
By tensoring the weight function with a weight function counting the number of parts each set partition contains, we get
using a derivation similar to the one above.
|Title||derivation of the generating series for the Stirling numbers of the second kind|
|Date of creation||2013-03-22 15:13:35|
|Last modified on||2013-03-22 15:13:35|
|Last modified by||cgibbard (959)|