## Analytic CombinatoricsAnalytic combinatorics aims to enable precise quantitative predictions of the properties of large combinatorial structures. The theory has emerged over recent decades as essential both for the analysis of algorithms and for the study of scientific models in many disciplines, including probability theory, statistical physics, computational biology, and information theory. With a careful combination of symbolic enumeration methods and complex analysis, drawing heavily on generating functions, results of sweeping generality emerge that can be applied in particular to fundamental structures such as permutations, sequences, strings, walks, paths, trees, graphs and maps. This account is the definitive treatment of the topic. The authors give full coverage of the underlying mathematics and a thorough treatment of both classical and modern applications of the theory. The text is complemented with exercises, examples, appendices and notes to aid understanding. The book can be used for an advanced undergraduate or a graduate course, or for self-study. |

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quite a good book specially the part on words and generating functions..........

### Contents

3 | |

15 | |

Labelled Structures and Exponential Generating Functions | 95 |

Combinatorial Parameters and Multivariate Generating Functions | 151 |

Complex Analysis Rational and Meromorphic Asymptotics | 223 |

Applications of Rational and Meromorphic Asymptotics | 289 |

Singularity Analysis of Generating Functions | 375 |

Applications of Singularity Analysis | 439 |

Auxiliary Elementary Notions | 721 |

Basic Complex Analysis | 739 |

Holonomic functions | 748 |

Laplaces method | 755 |

Mellin transforms | 762 |

Concepts of Probability Theory | 769 |

779 | |

795 | |

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### Common terms and phrases

algebraic function algorithm analytic combinatorics analytic function aperiodic applies approximation asymptotic expansion basic binary trees binomial bounds Cayley trees Chapter coefficients combinatorial class complex compositions computable Consider construction contour corresponding cycles defined derived determined differential discrete distribution dominant singularity enumeration equivalent estimate Example exponential factor Figure finite fixed formal power series formula functional equation Gaussian graphs instance integer partitions integral inversion irreducible labelled length limit law linear logarithmic mappings matrix Mellin transform meromorphic multisets nodes non-negative number of components obtained parameter paths pattern permutations plane pole polynomial polyominoes power series probability problem proof properties Proposition radius of convergence random variable recurrence root saddle-point method satisfies schema Section set partitions simple varieties singular expansion singularity analysis solution specification square-root Subsection summands surjections symbolic method Theorem theory tion transforms unlabelled words