Analytic Combinatorics is a self-contained treatment of the mathematics underlying the analysis of discrete structures, which has emerged over the past several decades as an essential tool in the understanding of properties of computer programs and scientific models with applications in physics, biology and chemistry. Thorough treatment of a large number of classical applications is an essential aspect of the presentation. Written by the leaders in the field of analytic combinatorics, this text is certain to become the definitive reference on the topic. The text is complemented with exercises, examples, appendices and notes to aid understanding therefore, it can be used as the basis for an advanced undergraduate or a graduate course on the subject, or for self-study.

Analytic Combinatorics

12 Large singularities of analytic functions 113 12.1 The saddle point 13 Multivariate generating functions 128 14 Mellin and other integral transforms 134 15 Functional equations, recurrences, and combinations of methods 137 15.1 Implicit functions, graphical enumeration, and related

Asymptotic enumeration methods

Flajolet and Soria established several central limit theorems for the parameter `number of components? in a wide class of combinatorial structures. In this paper, we shall prove a simple theorem which applies to characterize the convergence rates in their central limit theorems. This theorem is also applicable to arithmetical functions. Moreover, asymptotic expressions are derived for moments of integral order. Many examples from different applications are discussed.

http://algo.stat.sinica.edu.tw/hk/files/2005_07/pdf/On_convergence_rates_in_the_central_limit_theorems.pdf

On Convergence Rates in the Central Limit Theorems for Combinatorial Structures

Probabilistic Bounds on the Coefficients of Polynomials with Only Real Zeros

This book is the first to treat the analytic aspects of combinatorial enumeration from a multivariate perspective. Analytic combinatorics is a branch of enumeration that uses analytic techniques to estimate combinatorial quantities: generating functions are defined and their coefficients are then estimated via complex contour integrals. The multivariate case involves techniques well known in other areas of mathematics but not in combinatorics. Aimed at graduate students and researchers in enumerative combinatorics, the book contains all the necessary background, including a review of the uses of generating functions in combinatorial enumeration as well as chapters devoted to saddle point analysis, Groebner bases, Laurent series and amoebas, and a smattering of differential and algebraic topology. All software along with other ancillary material can be located via the book Web site, http://www.cs.auckland.ac.nz/~mcw/Research/mvGF/asymultseq/ACSVbook/.

/pdf/analytic-combinatorics-in-several-variables-50mw13zytm.pdf

Zhicheng Gao

Papers

Central and local limit theorems applied to asymptotic enumeration IV: multivariate generating functions