Exponential Generating Function

Main reference Baidu Encyclopedia

The common generating function is mainly to find the number of combinations, while the exponential generating function is to find the number of multiple permutations.

Exponential generating function problem:
　　Suppose there are n elements, among which a1, a2, ···, an are different from each other, and by performing full permutations, n! different permutations can be obtained. If one of the elements a1 is repeated n1 times, there must be repeated elements in all arrangements, and the number of truly different arrangements should be n!/n1!, that is, the repetition degree is n1!

For the same reason, a1 is repeated n1 times, a2 is repeated n2 times, ..., ak is repeated nk times, n1+n2+...+nk=n. For such a full arrangement of n elements, the number of different arrangements that can be obtained is actually

definition

edit

For the sequence a0, a1, a2, ... define

Ge(x)=

   

+

 

for sequence {

   

The exponential generating function of }

Example:

Let n be a positive integer. Determine the exponential generating function for the following series:

p(n,0),p(n,1),p(n,2),…,p(n,n)

where p(n,k) denotes the number of k permutations of n elements, so for k=0,1,2,...,n, the number of permutations is n!/(nk)!. So the exponential generating function is:

 

therefore

   

is the exponential generating function of the sequence p(n,0), p(n,1), p(n,2),...,p(n,n).