# 4: Generating Functions

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• 4.1: The Idea of Generating Functions
In the theory of generating functions, we associate variables or polynomials or even power series with members of a set. There is no standard language describing how we associate variables with members of a set, so we shall invent some. By a picture of a member of a set we will mean a variable, or perhaps a product of powers of variables (or even a sum of products of powers of variables). A function that assigns a picture P(s) to each member s of a set S will be called a picture function.
• 4.2: Generating Functions for Integer Partitions
In the previous section (Section 4.1), we discussed how to visualize variables in a function using images as well as different methods to help us generate functions. In this section, we will explore how to generate functions for the number of partitions of an integer into parts of different sizes.
• 4.3: Generating Functions and Recurrence Relations
Algebraic manipulations with generating functions can sometimes reveal the solutions to a recurrence relation.
• 4.4: Generating Functions (Exercises)
This section contains the supplementary problems related to the materials discussed in Chapter 4.