(Ebook) Analytic combinatorics in several variables 1st Edition by Pemantle R, Wilson M C ISBN 3513533543 9783513533548

(Ebook) Analytic combinatorics in several variables 1st Edition by Pemantle R, Wilson M C ISBN 3513533543 9783513533548

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ISBN 10: 3513533543 
ISBN 13: 9783513533548
Author: Pemantle R, Wilson M C

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's website, http://www.cs.auckland.ac.nz/~mcw/Research/mvGF/asymultseq/ACSVbook/.

(Ebook) Analytic combinatorics in several variables 1st Table of contents:

PART I: Combinatorial Enumeration
1 Introduction
1.1 Arrays of Numbers
1.2 Generating Functions and Asymptotics
1.3 New Multivariate Methods
1.4 Outline of the Remaining Chapters
Notes
Exercises
2 Generating Functions
2.1 Formal Power Series
2.2 Rational Operations on Generating Functions
2.3 Algebraic Generating Functions
2.4 Generating Functions
2.5 Exponentiation: Set Partitions
Notes
Exercises
3 Univariate Asymptotics
3.1 Rational Functions: An Explicit Formula
3.2 Saddle Point Methods
3.3 Circle Methods
3.4 Transfer Theorems
Notes
Exercises
PART II: Mathematical Background
4 Fourier-Laplace Integrals in One Variable
4.1 Statement of Main Result
4.2 Real Integrands
4.3 Complex Phase
4.4 Classical Methods: Steepest Descent (Saddle Point)
4.5 Analytic Versus C Category
Notes
Exercises
5 Fourier-Laplace Integrals in More than One Variable
5.1 Overview
5.2 Standard Phase
5.3 Real Part of Phase Has a Strict Minimum
5.4 Localization
5.5 Examples and Extensions
Notes
Exercises
6 Techniques of Symbolic Computation via Gröbner Bases
6.1 Solving Systems of Polynomial Equations
6.2 Examples of Gröbner Basis Computation
6.3 D-Modules: Computing with D-Finite Functions
Notes
Exercises
7 Cones, Laurent Series, and Amoebas
7.1 Cones and Dual Cones
7.2 Laurent Series
7.3 Amoebas
Notes
Exercises
PART III: Multivariate Enumeration
8 Overview of Analytic Methods for Multivariate Generating Functions
8.1 Exponential Rate
8.2 Morse Theory Redux
8.3 Critical Points
8.4 Minimal Points
8.5 Describing the Quasi-Local Cycles
8.6 Evaluating the Integral on a Quasi-Local Cycle
Notes
Exercises
9 Smooth Point Asymptotics
9.1 Smooth Points
9.2 The Smooth Point Integral via Surgery
9.3 The Smooth Point Integral via Residue Forms
9.4 Effectively Computing the Intersection Class in Two Cases
9.5 Explicit Formulae for the Leading Term
9.6 Limit Laws from Probability Theory
Notes
Exercises
10 Multiple Point Asymptotics
10.1 Multiple Points
10.2 Iterated Residues
10.3 Formulae for Coefficients
10.4 The Base Case via Surgery
Notes
Exercises
11 Cone Point Asymptotics
11.1 Cones, Hyperbolicity, and Deformations
11.2 Evaluating the Asymptotics
11.3 Examples and Consequences
Notes
Exercises
12 Worked Examples
12.1 Four Independent Examples
12.2 Powers, Quasi-Powers, and Riordan Arrays
12.3 Lagrange Inversion
12.4 Transfer Matrices
Notes
Exercises
13 Extensions
13.1 The Diagonal Method
13.2 Algebraic Generating Functions
13.3 Higher-Order Asymptotics
13.4 Phase Transitions
13.5 Conclusion

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Tags: Pemantle R, Wilson M C, combinatorics, Analytic