Handbook of Enumerative Combinatorics

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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

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

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

Introduction

Basic Notions

Generating Functions

Unlabeled Trees

Labeled Trees

Selected Topics on Trees

What is a Map?

Counting Tree-Rooted Maps

Counting Planar Maps

Beyond Planar Maps, an Even Shorter Account

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

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;

Introduction

Preliminaries

Conjugacy

Lyndon words

Eulerian Graphs and De Bruijn Cycles

Unavoidable Sets

The Burrows-Wheeler Transform

The Gessel-Reutenauer Bijection

Suffix Arrays

Tilings

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

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

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

Introduction

Growth Rates of Principal Classes

Notions of Structure

The Set of All Growth Rates

Parking Functions

Introduction

Parking Functions and Labeled Trees

Many Faces of Parking Functions

Generalized Parking Functions

Parking Functions Associated with Graphs

Final Remarks

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

Introduction

Computer Algebra Essentials

Counting Algorithms

Symbolic Summation

The Guess-and-Prove Paradigm

Index

Miklos Bona 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. Miklos Bona 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 Bona (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 Miklos Bona 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

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