# 5. Analytic Combinatorics

This chapter introduces *analytic combinatorics*, a modern approach to
the study of combinatorial structures of the sort that we encounter
frequently in the analysis of algorithms. The approach is predicated
on the idea that combinatorial structures are typically defined by
simple formal rules that are the key to learning their properties.
One eventual outgrowth of this observation is that a relatively small set of
*transfer theorems* ultimately yields accurate approximations of the
quantities that we seek. Figure 5.1 gives an general overview of the process.
Generating functions are the central objects of study in analytic
combinatorics. In the first place, we directly translate formal
definitions of combinatorial objects
into definitions of generating functions that enumerate objects or
describe their properties. In the second place, we use classical
mathematical analysis to extract estimates of
generating function coefficients.

## 5.1 Formal Basis

## 5.2 Symbolic Method for Unlabelled Classes

## 5.3 Symbolic Method for Labelled Classes

## 5.4 Symbolic Method for Parameters