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Generating functions

Generating functions are used to describe families of combinatorial objects. Let F {\displaystyle {\mathcal {F}}} denote the family of objects and let F (x) be its generating function. Then

where f n {\displaystyle f_{n}} denotes the number of combinatorial objects of size n. The number of combinatorial objects of size n is therefore given by the coefficient of x n {\displaystyle x^{n}}. Some common operation on families of combinatorial objects and its effect on the generating function will now be developed.The exponential generating function is also sometimes used. In this case it would have the form

Once determined, the generating function yields the information given by the previous approaches. In addition, the various natural operations on generating functions such as addition, multiplication, differentiation, etc., have a combinatorial significance; this allows one to extend results from one combinatorial problem in order to solve others.