Errata and other pertinent information are available on the book’s website
Bijective proofs are some of the most elegant and powerful techniques in all of mathematics. Suitable for readers without prior background in algebra or combinatorics, Bijective Combinatorics presents a general introduction to enumerative and algebraic combinatorics that emphasizes bijective methods.
The text systematically develops the mathematical tools, such as basic counting rules, recursions, inclusion-exclusion techniques, generating functions, bijective proofs, and linear-algebraic methods, needed to solve enumeration problems. These tools are used to analyze many combinatorial structures, including words, permutations, subsets, functions, compositions, integer partitions, graphs, trees, lattice paths, multisets, rook placements, set partitions, Eulerian tours, derangements, posets, tilings, and abaci. The book also delves into algebraic aspects of combinatorics, offering detailed treatments of formal power series, symmetric groups, group actions, symmetric polynomials, determinants, and the combinatorial calculus of tableaux. Each chapter includes summaries and extensive problem sets that review and reinforce the material.
Lucid, engaging, yet fully rigorous, this text describes a host of combinatorial techniques to help solve complicated enumeration problems. It covers the basic principles of enumeration, giving due attention to the role of bijective proofs in enumeration theory.
Combinatorial Identities and Recursions
Counting Problems in Graph Theory
Inclusion-Exclusion and Related Techniques
Ranking and Unranking
Counting Weighted Objects
Formal Power Series
The Combinatorics of Formal Power Series
Permutations and Group Actions
Tableaux and Symmetric Polynomials
Abaci and Antisymmetric Polynomials
Abaci and Integer Partitions
Jacobi Triple Product Identity
Ribbons and k-Cores
k-Quotients and Hooks
Pieri Rule for pk
Pieri Rule for ek
Pieri Rule for hk
Antisymmetric Polynomials and Schur Polynomials
Abaci and Tableaux
Skew Schur Polynomials
Inverse Kostka Matrix
Schur Expansion of Skew Schur Polynomials
Products of Schur Polynomials
Cyclic Shifting of Paths
Rook-Equivalence of Ferrers Boards
Parking Functions and Trees
Möbius Inversion and Field Theory
Quantum Binomial Coefficients and Subspaces
Tangent and Secant Numbers
Tournaments and the Vandermonde Determinant
Pfaffians and Perfect Matchings
Domino Tilings of Rectangles
Answers and Hints to Selected Exercises
A Summary and Exercises appear at the end of each chapter.
Nicholas A. Loehr teaches in the Department of Mathematics at Virginia Tech. His research interests include enumerative and algebraic combinatorics; symmetric and quasisymmetric functions; integer partitions, lattice paths, parking functions, and tableaux; bijective methods; and algorithm analysis.
This textbook, aimed at beginning graduate students, is the first to survey the subject emphasizing the role of bijections. … The final chapter contains a potpourri of delightful results … The exposition is careful and deliberate, and leaves no stone unturned … a welcome addition to the literature and a very nice book.
—David Callan, Mathematical Reviews, 2012d
A rule I have found to be true is that any book claiming to be suitable for beginners and yet leading to the frontiers of unsolved research problems does neither well. This book is the exception to that rule. … I found this book engaging. The proofs are very clear, and in many cases several proofs are offered. … This book could serve several purposes. By focusing on the first half of the book, it could be an excellent choice for a first course in combinatorics for senior undergraduates. By selecting topics and/or moving quickly, it could work well for a more mature audience. … it also makes a great reference for people who use combinatorics but are not specialists. … This is a very nice book that deserves serious consideration.
—Peter Rabinovitch, MAA Reviews, September 2011
This book is a comprehensive treatment of the combinatorics of counting … The book would be suitable for advanced undergraduates or as a graduate text. It would also be a good book for a computer scientist who wants to learn about enumeration. The book begins with an enchanting introductory chapter. It details a series of interesting motivational problems … The book is clearly written and packed with examples and problems (it provides answers and hints to selected problems at the end). The organization is superior, with helpful tables of notation and definitions at the end of each chapter, along with a point-by-point summary of the chief topics. … the book is an excellent one, and is a comprehensive and welcome addition to the area.
—Angele M. Hamel, Computing Reviews, August 2011