| |
| |
Preface | |
| |
| |
An Invitation to Analytic Combinatorics | |
| |
| |
| |
Symbolic Methods | |
| |
| |
| |
Combinatorial Structures and Ordinary Generating Functions | |
| |
| |
| |
Symbolic enumeration Methods | |
| |
| |
| |
Admissible constructions and specifications | |
| |
| |
| |
Integer compositions and partitions | |
| |
| |
| |
Words and regular languages | |
| |
| |
| |
Tree structures | |
| |
| |
| |
Additional constructions | |
| |
| |
| |
Perspective | |
| |
| |
| |
Labelled Structures and Exponential Generating Functions | |
| |
| |
| |
Labelled classes | |
| |
| |
| |
Admissible labelled constructions | |
| |
| |
| |
Surjections, set partitions, and words | |
| |
| |
| |
Alignments, permutations, and related structures | |
| |
| |
| |
Labelled trees, mapping, and graphs | |
| |
| |
| |
Additional constructions | |
| |
| |
| |
Perspective | |
| |
| |
| |
Combinatorial Parameters and Multivariate Generating Functions | |
| |
| |
| |
An introduction to bivariate generating functions (BGFs) | |
| |
| |
| |
Bivariate generating functions and probability distributions | |
| |
| |
| |
Inherited parameters and ordinary MGFs | |
| |
| |
| |
Inherited parameters and exponential MGFs | |
| |
| |
| |
Recursive parameters | |
| |
| |
| |
Complete generating functions and discrete models | |
| |
| |
| |
Additional constructions | |
| |
| |
| |
Extremal parameters | |
| |
| |
| |
Perspective | |
| |
| |
| |
Complex Asymptotics | |
| |
| |
| |
Complex Analysis, Rational and Meromorphic Asymptotics | |
| |
| |
| |
Generating functions as analytic objects | |
| |
| |
| |
Analytic functions and meromorphic functions | |
| |
| |
| |
Singularities and exponential growth of coefficients | |
| |
| |
| |
Closure properties and computable bounds | |
| |
| |
| |
Rational and meromorphic functions | |
| |
| |
| |
Localization of singularities | |
| |
| |
| |
Singularities and functional equations | |
| |
| |
| |
Perspective | |
| |
| |
| |
Applications Of Rational and Meromorphic Asymptotics | |
| |
| |
| |
A roadmap to rational and meromorphic asymptotics | |
| |
| |
| |
The supercritical sequence schema | |
| |
| |
| |
Regular specifications and languages | |
| |
| |
| |
Nested sequences, lattice paths, and continued fractions | |
| |
| |
| |
Paths in graphs and automata | |
| |
| |
| |
Transfer matrix models | |
| |
| |
| |
Perspective | |
| |
| |
| |
Singularity Analysis Of Generating Functions | |
| |
| |
| |
A glimpse of basic singularity analysis theory | |
| |
| |
| |
Coefficient asymptotics for the standard scale | |
| |
| |
| |
Transfers | |
| |
| |
| |
The process of singularity analysis | |
| |
| |
| |
Multiple singularities | |
| |
| |
| |
Intermezzo: functions amenable to singularity analysis | |
| |
| |
| |
Inverse functions | |
| |
| |
| |
Polylogarithms | |
| |
| |
| |
Functional composition | |
| |
| |
| |
Closure properties | |
| |
| |
| |
Tauberian theory and Darboux's method | |
| |
| |
| |
Perspective | |
| |
| |
| |
Applications of Singularity Analysis | |
| |
| |
| |
A roadmap to singularity analysis asymptotics | |
| |
| |
| |
Sets and the exp-log schema | |
| |
| |
| |
Simple varieties of trees and inverse functions | |
| |
| |
| |
Tree-like structures and implicit functions | |
| |
| |
| |
Unlabelled non-plane trees and P�lya operators | |
| |
| |
| |
Irreducible context-free structures | |
| |
| |
| |
The general analysis of algebraic functions | |
| |
| |
| |
Combinatorial applications of algebraic functions | |
| |
| |
| |
Ordinary differential equations and systems | |
| |
| |
| |
Singularity analysis and probability distributions | |
| |
| |
| |
Perspective | |
| |
| |
| |
Saddle-Point Asymptotics | |
| |
| |
| |
Landscapes of analytic functions and saddle-points | |
| |
| |
| |
Saddle-point bounds | |
| |
| |
| |
Overview of the saddle-point method | |
| |
| |
| |
Three combinatorial examples | |
| |
| |
| |
Admissibility | |
| |
| |
| |
Integer partitions | |
| |
| |
| |
Saddle-points and linear differential equations | |
| |
| |
| |
Large powers | |
| |
| |
| |
Saddle-points and probability distributions | |
| |
| |
| |
Multiple saddle-points | |
| |
| |
| |
Perspective | |
| |
| |
| |
Random Structures | |
| |
| |
| |
Multivariate Asymptotics and Limit Laws | |
| |
| |
| |
Limit laws and combinatorial structures | |
| |
| |
| |
Discrete limit laws | |
| |
| |
| |
Combinatorial instances of discrete laws | |
| |
| |
| |
Continuous limit laws | |
| |
| |
| |
Quasi-powers and Guassian limit laws | |
| |
| |
| |
Perturbation of meromorphic asymptotics | |
| |
| |
| |
Pertubation of singularity analysis asymptotics | |
| |
| |
| |
Perturbation of saddle-point asymptotics | |
| |
| |
| |
Local limit laws | |
| |
| |
| |
Large deviations | |
| |
| |
| |
Non-Gaussian continuous limits | |
| |
| |
| |
Multivariate limit laws | |
| |
| |
| |
Perspective | |
| |
| |
| |
Appendices | |
| |
| |
| |
Auxiliary Elementary Notions | |
| |
| |
| |
Arithmetical functions | |
| |
| |
| |
Asymptotic notations | |
| |
| |
| |
Combinatorial probability | |
| |
| |
| |
Cycle construction | |
| |
| |
| |
Formal power series | |
| |
| |
| |
Lagrange inversion | |
| |
| |
| |
Regular languages | |
| |
| |
| |
Stirling numbers. | |
| |
| |
| |
Tree Concepts | |
| |
| |
| |
Basic Complex Analysis | |
| |
| |
| |
Algebraic elimination | |
| |
| |
| |
Equivalent definitions of analyticity | |
| |
| |
| |
Gamma function | |
| |
| |
| |
Holonomic functions | |
| |
| |
| |
Implicit Function Theorem | |
| |
| |
| |
Laplace's method | |
| |
| |
| |
Mellin transforms | |
| |
| |
| |
Several complex variables | |
| |
| |
| |
Concepts Of Probability Theory | |
| |
| |
| |
Probability spaces and measure | |
| |
| |
| |
Random variables | |
| |
| |
| |
Transforms of distributions | |
| |
| |
| |
Special distributions | |
| |
| |
| |
Convergence in law | |
| |
| |
Bibliography | |
| |
| |
Index | |