Analytic combinatorics: a multidimensional approach/ Marni Mishna

Enumerative combinatorics --
A Primer on combinatorial calculus -- Combinatorial parameters -- Derived and transcendental classes --
Methods for asympotic analysis --
Generating for asympotic analysis -- Parallel taxonomies -- Singularities of multivariable rational functions -- Integration multivariable coefficient asymptotics -- Multiple points -- Partitions

This book is written in a reader-friendly fashion to better facilitate the understanding of the subject. Naturally, it is a firm introduction to the concept of analytic combinatorics and is a valuable tool to help readers better understand the structure and large-scale behavior of discrete objects. Primarily, the textbook is a gateway to the interactions between complex analysis and combinatorics. The study will lead readers through connections to number theory, algebraic geometry, probability and formal language theory.

The textbook starts by discussing objects that can be enumerated using generating functions, such as tree classes and lattice walks. It also introduces multivariate generating functions including the topics of the kernel method, and diagonal constructions. The second part explains methods of counting these objects, which involves deep mathematics coming from outside combinatorics, such as complex analysis and geometry.