generatingfunctionology

Author : Herbert S. Wilf, Department of Mathematics, University of Pennsylvania
Publication Date : 1994

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Book Excerpts:

This book is about generating functions and some of their uses in discrete mathematics. The subject is so vast that there is no attempt to give a comprehensive discussion. Instead this book tries only to communicate some of the main ideas.

Generating functions are a bridge between discrete mathematics, on the one hand, and continuous analysis (particularly complex variable theory) on the other. It is possible to study them solely as tools for solving discrete problems. As such there is much that is powerful and magical in the way generating functions give unified methods for handling such problems. The reader who wished to omit the analytical parts of the subject would skip chapter 5 and portions of the earlier material. To omit those parts of the subject, however, is like listening to a stereo broadcast of, say, Beethoven’s Ninth Symphony, using only the left audio channel.

The full beauty of the subject of generating functions emerges only from tuning in on both channels: the discrete and the continuous. See how they make the solution of difference equations into child’s play. Then see how the theory of functions of a complex variable gives, virtually by inspection, the approximate size of the solution. The interplay between the two channels is vitally important for the appreciation of the music. In recent years there has been a vigorous trend in the direction of finding bijective proofs of combinatorial theorems. That is, if we want to prove that two sets have the same cardinality then we should be able to do it by exhibiting an explicit bijection between the sets. In many cases the fact that the two sets have the same cardinality was discovered in the first place by generating function arguments. Also, even though bijective arguments may be known, the generating function proofs may be shorter or more elegant.

The bijective proofs give one a certain satisfying feeling that one "really" understands why the theorem is true. The generating function arguments often give satisfying feelings of naturalness, and "oh, I could have thought of that," as well as usually offering the best route to finding exact or approximate formulas for the numbers in question.

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