Generating Functions
Generating functions are used to describe families of combinatorial objects. Let denote the family of objects and let F(x) be its generating function. Then:
Where denotes the number of combinatorial objects of size n. The number of combinatorial objects of size n is therefore given by the coefficient of . 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:
Read more about this topic: Enumerative Combinatorics
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“Empirical science is apt to cloud the sight, and, by the very knowledge of functions and processes, to bereave the student of the manly contemplation of the whole.”
—Ralph Waldo Emerson (1803–1882)