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