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