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1 What is a generating function? - MIT Mathematics

Lecture notesMarch 1, 2015 generating FunctionsLecturer: Michel GoemansWe are going to discuss enumeration problems, and how to solve them using a powerful tool: generating functions. What is anenumeration problem? That s trying to determine the number ofobjects of sizensatisfying a certain definition. For instance,what is the number of permutations of{1,2,..,n}?(answer:n!), orwhat is the number of binary sequences of lengthn?(answer: 2n).Ok, now let us introduce some tools to answer more difficult enumerative What is a generating function?

nis the number of objects of size nin the class. Note that the variable xin generating functions doesn’t stand for anything but serves as a placeholder for keeping track of the coe cients of xn. Example 1. The generating function associated to the class of binary sequences (where the size of a sequence is its length) is A(x) = P n 0 2 nxn ...

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