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Cover -- Half-title -- Title -- Copyright -- Contents -- Preface -- An Invitation to Analytic Combinatorics -- Part A SYMBOLIC METHODS -- I Combinatorial Structures and Ordinary Generating Functions -- I. 1. Symbolic enumeration methods -- I. 2. Admissible constructions and specifications -- I. 3. Integer compositions and partitions -- I. 4. Words and regular languages -- I. 5. Tree structures -- I. 6. Additional constructions -- I. 7. Perspective -- II Labelled Structures and Exponential Generating Functions -- II. 1. Labelled classes -- II. 2. Admissible labelled constructions -- II. 3. Surjections, set partitions, and words -- II. 4. Alignments, permutations, and related structures -- II. 5. Labelled trees, mappings, and graphs -- II. 6. Additional constructions -- II. 7. Perspective -- III Combinatorial Parameters and Multivariate Generating Functions -- III. 1. An introduction to bivariate generating functions (BGFs) -- III. 2. Bivariate generating functions and probability distributions -- III. 3. Inherited parameters and ordinary MGFs -- III. 4. Inherited parameters and exponential MGFs -- III. 5. Recursive parameters -- III. 6. Complete generating functions and discrete models -- III. 7. Additional constructions -- III. 8. Extremal parameters -- III. 9. Perspective -- Part B COMPLEX ASYMPTOTICS -- IV Complex Analysis, Rational and Meromorphic Asymptotics -- IV. 1. Generating functions as analytic objects -- IV. 2. Analytic functions and meromorphic functions -- IV. 3. Singularities and exponential growth of coefficients -- IV. 4. Closure properties and computable bounds -- IV. 5. Rational and meromorphic functions -- IV. 6. Localization of singularities -- IV. 7. Singularities and functional equations -- IV. 8. Perspective -- V Applications of Rational and Meromorphic Asymptotics -- V. 1. A roadmap to rational and meromorphic asymptotics.
V. 2. The supercritical sequence schema -- V. 3. Regular specifications and languages -- V. 4. Nested sequences, lattice paths, and continued fractions -- V. 5. Paths in graphs and automata -- V. 6. Transfer matrix models -- V. 7. Perspective -- VI Singularity Analysis of Generating Functions -- VI. 1. A glimpse of basic singularity analysis theory -- VI. 2. Coefficient asymptotics for the standard scale -- VI. 3. Transfers -- VI. 4. The process of singularity analysis -- VI. 5. Multiple singularities -- VI. 6. Intermezzo: functions amenable to singularity analysis -- VI. 7. Inverse functions -- VI. 8. Polylogarithms -- VI. 9. Functional composition -- VI. 10. Closure properties -- VI. 11. Tauberian theory and Darboux's method -- VI. 12. Perspective -- VII Applications of Singularity Analysis -- VII. 1. A roadmap to singularity analysis asymptotics -- VII. 2. Sets and the exp-log schema -- VII. 3. Simple varieties of trees and inverse functions -- VII. 4. Tree-like structures and implicit functions -- VII. 5. Unlabelled non-plane trees and Polya operators -- VII. 6. Irreducible context-free structures -- VII. 7. The general analysis of algebraic functions -- VII. 8. Combinatorial applications of algebraic functions -- VII. 9. Ordinary differential equations and systems -- VII. 10. Singularity analysis and probability distributions -- VII. 11. Perspective -- VIII Saddle-point Asymptotics -- VIII. 1. Landscapes of analytic functions and saddle-points -- VIII. 2. Saddle-point bounds -- VIII. 3. Overview of the saddle-point method -- VIII. 4. Three combinatorial examples -- VIII. 5. Admissibility -- VIII. 6. Integer partitions -- VIII. 7. Saddle-points and linear differential equations -- VIII. 8. Large powers -- VIII. 9. Saddle-points and probability distributions -- VIII. 10. Multiple saddle-points -- VIII. 11. Perspective -- Part C RANDOM STRUCTURES.
IX Multivariate Asymptotics and Limit Laws -- IX. 1. Limit laws and combinatorial structures -- IX. 2. Discrete limit laws -- IX. 3. Combinatorial instances of discrete laws -- IX. 4. Continuous limit laws -- IX. 5. Quasi-powers and Gaussian limit laws -- IX. 6. Perturbation of meromorphic asymptotics -- IX. 7. Perturbation of singularity analysis asymptotics -- IX. 8. Perturbation of saddle-point asymptotics -- IX. 9. Local limit laws -- IX. 10. Large deviations -- IX. 11. Non-Gaussian continuous limits -- IX. 12. Multivariate limit laws -- IX. 13. Perspective -- Part D APPENDICES -- APPENDIX A Auxiliary Elementary Notions -- A.1. Arithmetical functions -- A.2. Asymptotic notations -- A.3. Combinatorial probability -- A.4. Cycle construction -- A.5. Formal power series -- A.6. Lagrange inversion -- A.7. Regular languages -- A.8. Stirling numbers -- A.9. Tree concepts -- APPENDIX B Basic Complex Analysis -- B.1. Algebraic elimination -- B.2. Equivalent definitions of analyticity -- B.3. Gamma function -- B.4. Holonomic functions -- B.5. Implicit Function Theorem -- B.6. Laplace's method -- B.7. Mellin transforms -- B.8. Several complex variables -- APPENDIX C Concepts of Probability Theory -- C.1. Probability spaces and measure -- C.2. Random variables -- C.3. Transforms of distributions -- C.4. Special distributions -- C.5. Convergence in law -- Bibliography -- Index.
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