Handbook of Enumerative Combinatorics 1st Edition by Miklos Bona ISBN 1482220865 9781482220865

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ISBN 10: 1482220865 

ISBN 13: 9781482220865

Author: Miklós Bóna

Presenting the state of the art, the Handbook of Enumerative Combinatorics brings together the work of today’s most prominent researchers. The contributors survey the methods of combinatorial enumeration along with the most frequent applications of these methods.

This important new work is edited by Miklós Bóna of the University of Florida where he is a member of the Academy of Distinguished Teaching Scholars. He received his Ph.D. in mathematics at Massachusetts Institute of Technology in 1997. Miklós is the author of four books and more than 65 research articles, including the award-winning Combinatorics of Permutations. Miklós Bóna is an editor-in-chief for the Electronic Journal of Combinatorics and Series Editor of the Discrete Mathematics and Its Applications Series for CRC Press/Chapman and Hall.

The first two chapters provide a comprehensive overview of the most frequently used methods in combinatorial enumeration, including algebraic, geometric, and analytic methods. These chapters survey generating functions, methods from linear algebra, partially ordered sets, polytopes, hyperplane arrangements, and matroids. Subsequent chapters illustrate applications of these methods for counting a wide array of objects.

The contributors for this book represent an international spectrum of researchers with strong histories of results. The chapters are organized so readers advance from the more general ones, namely enumeration methods, towards the more specialized ones.

Topics include coverage of asymptotic normality in enumeration, planar maps, graph enumeration, Young tableaux, unimodality, log-concavity, real zeros, asymptotic normality, trees, generalized Catalan paths, computerized enumeration schemes, enumeration of various graph classes, words, tilings, pattern avoidance, computer algebra, and parking functions.

This book will be beneficial to a wide audience. It will appeal to experts on the topic interested in learning more about the finer points, readers interested in a systematic and organized treatment of the topic, and novices who are new to the field.

Table of contents:

Algebraic and Geometric Methods in Enumerative Combinatorics
Introduction
What is a Good Answer?
Generating Functions
Linear Algebra Methods
Posets
Polytopes
Hyperplane Arrangements
Matroids
Acknowledgments

Analytic Methods; Helmut Prodinger
Introduction
Combinatorial Constructions and Associated Ordinary Generating Functions
Combinatorial Constructions and Associated Exponential Generating Functions
Partitions and Q-Series
Some Applications of the Adding a Slice Technique
Lagrange Inversion Formula
Lattice Path Enumeration: The Continued Fraction Theorem
Lattice Path Enumeration: The Kernel Method
Gamma and Zeta Function
Harmonic Numbers and Their Generating Functions
Approximation of Binomial Coefficients
Mellin Transform and Asymptotics of Harmonic Sums
The Mellin-Perron Formula
Mellin-Perron Formula: Divide-and-Conquer Recursions
Rice’s Method
Approximate Counting
Singularity Analysis of Generating Functions
Longest Runs in Words
Inversions in Permutations and Pumping Moments
Tree Function
The Saddle Point Method
Hwang’s Quasi-Power Theorem

TOPICS

Asymptotic Normality in Enumeration; E. Rodney Canfield
The Normal Distribution
Method 1: Direct Approach
Method 2: Negative Roots
Method 3: Moments
Method 4: Singularity Analysis
Local Limit Theorems
Multivariate Asymptotic Normality
Normality in Service to Approximate Enumeration

Trees; Michael Drmota
Introduction
Basic Notions
Generating Functions
Unlabeled Trees
Labeled Trees
Selected Topics on Trees

Planar maps; Gilles Schaeffer
What is a Map?
Counting Tree-Rooted Maps
Counting Planar Maps
Beyond Planar Maps, an Even Shorter Account

Graph Enumeration; Marc Noy
Introduction
Graph Decompositions
Connected Graphs with Given Excess
Regular Graphs
Monotone and Hereditary Classes
Planar Graphs
Graphs on Surfaces and Graph Minors
Digraphs
Unlabelled Graphs

Unimodality, Log-Concavity, Real–Rootedness and Beyond; Petter Brándén
Introduction
Probabilistic Consequences of Real–Rootedness
Unimodality and G-Nonnegativity
Log–Concavity and Matroids
Infinite Log-Concavity
The Neggers–Stanley Conjecture
Preserving Real–Rootedness
Common Interleavers
Multivariate Techniques
Historical Notes

Words; Dominique Perrin and Antonio Restivo
Introduction
Preliminaries
Conjugacy
Lyndon words
Eulerian Graphs and De Bruijn Cycles
Unavoidable Sets
The Burrows-Wheeler Transform
The Gessel-Reutenauer Bijection
Suffix Arrays

Tilings; James Propp
Introduction and Overview
The Transfer Matrix Method
Other Determinant Methods
Representation-Theoretic Methods
Other Combinatorial Methods
Related Topics, and an Attempt at History
Some Emergent Themes
Software
Frontiers

Lattice Path Enumeration; Christian Krattenthaler
Introduction
Lattice Paths Without Restrictions
Linear Boundaries of Slope 1
Simple Paths with Linear Boundaries of Rational Slope, I
Simple Paths with Linear Boundaries with Rational Slope, II
Simple Paths with a Piecewise Linear Boundary
Simple Paths with General Boundaries
Elementary Results on Motzkin and Schroder Paths
A continued Fraction for the Weighted Counting of Motzkin Paths
Lattice Paths and Orthogonal Polynomials
Motzkin Paths in a Strip
Further Results for Lattice Paths in the Plane
Non-Intersecting Lattice Paths
Lattice Paths and Their Turns
Multidimensional Lattice Paths
Multidimensional Lattice Paths Bounded by a Hyperplane
Multidimensional Paths With a General Boundary
The Reflection Principle in Full Generality
Q-Counting Of Lattice Paths and Rogers–Ramanujan Identities
Self-Avoiding Walks

Catalan Paths and q; t-enumeration; James Haglund
Introduction to q-Analogues and Catalan Numbers
The q; t-Catalan Numbers
Parking Functions and the Hilbert Series
The q; t-Schr¿der Polynomial
Rational Catalan Combinatorics

Permutation Classes; Vincent Vatter
Introduction
Growth Rates of Principal Classes
Notions of Structure
The Set of All Growth Rates

Parking Functions; Catherine H. Yan
Introduction
Parking Functions and Labeled Trees
Many Faces of Parking Functions
Generalized Parking Functions
Parking Functions Associated with Graphs
Final Remarks

Standard Young Tableaux; Ron Adin and Yuval Roichman
Introduction
Preliminaries
Formulas for Thin Shapes
Jeu de taquin and the RS Correspondence
Formulas for Classical Shapes
More Proofs of the Hook Length Formula
Formulas for Skew Strips
Truncated and Other Non-Classical Shapes
Rim Hook and Domino Tableaux
q-Enumeration
Counting Reduced Words
Appendix 1: Representation Theoretic Aspects
Appendix 2: Asymptotics and Probabilistic Aspects

Computer Algebra; Manuel Kauers
Introduction
Computer Algebra Essentials
Counting Algorithms
Symbolic Summation
The Guess-and-Prove Paradigm
Index

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Tags: Miklos Bona, Handbook, Enumerative, Combinatorics