Chapman and Hall/CRC
248 pages | 30 B/W Illus.
The text starts by discussing the objects that can be enumerated using multivariate generating functions, such as permutations, maps, and lattice walks. The author is an expert on the last example. She will also introduce multivariate generating functions, and have a section about the Kernel method (a topic so vaste that Thomas Prellberg is considering a book forus on it). She will also discuss diagonals. The second part explains the methods of counting these objects. This will involve deep mathematics coming from outside combinatorics, such as complex analysis and topology. It is the need for these tools that makes the topic so difficult, so here the presentation will be reader-friendly.
A Primer on Combinatorical Calculus
Derived and Transcendental Classes
Generating Functions as Analytic Objects
Singularities of Multvariable Rational Functions
Integration and Multivariable Coefficient Asymptotics
Case Study: Partitions