Handbook of Enumerative Combinatorics

METHODS

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-Schrӧder 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

Biography

Miklós Bóna received his Ph.D. in mathematics at Massachusetts Institute of Technology in 1997. Since 1999, he has taught at the University of Florida, where in 2010 he was inducted in the Academy of Distinguished Teaching Scholars. He is the author of four books and more than 65 research articles, mostly focusing on enumerative and analytic combinatorics. His book Combinatorics of Permutations won the Outstanding Title Award from Choice, the journal of the American Library Association. He has mentored numerous graduate and undergraduate students. Miklós Bóna is an editor-in-chief for the Electronic Journal of Combinatorics, and for two book series at CRC Press.

"Mathematical handbooks are among the most essential library resources, providing compilations of formulas, tables, graphs, etc. Traditional handbooks speak equally to experts and casual users of mathematics. Other handbooks, such as the current work, are really encyclopedic compendiums of survey articles primarily addressing readers who make mathematics their main business. They supplement systematic monographs that develop subjects methodically but require extreme reader commitment and journal literature that provides quick access to specific results for those with prerequisite knowledge. Researchers will benefit from rapid authoritative citations to newer or lesser-known results. Students, undergraduate and graduate, will find accessible, systematic snapshots of whole subjects, helping them discover what they most wish to learn and, equally, what they will then need to learn on the way. Enumerative combinatorics means counting problems, so that subject begins classically with permutations and combinations but is active now with connections to probability, graph theory, statistical mechanics, geometry, representation theory, analysis, and computer science. Chapters here divide between general counting methods, both exact and approximate, and special classes of objects for counting via any suitable means. The volume, part of the 'Discrete Mathematics and Its Applications' series, is well edited by Bóna (Univ. of Florida), who successfully pools the expertise of leaders in the field. Summing up: Recommended. Upper-division undergraduates through professionals/practitioners."
—D. V. Feldman, University of New Hampshire, Durham, USA, for CHOICE, March 2016

"I cannot think of any topic that I would like to have seen presented here that the book omits. The chapters discuss not only methods in the study of enumerative combinatorics, but also objects that lend themselves to study along these lines. … accessible to a wide audience … this will clearly be a book that anybody with a serious interest in combinatorics will want to have on his or her bookshelf, and of course it belongs in any self-respecting university library. Having seen firsthand what it takes to edit a handbook like this, I know that Miklós Bóna must have invested a great deal of time and effort in the creation of this volume, as did the authors of the individual chapters. Their efforts have not been in vain; this is a valuable book."
—MAA Reviews, July 2015