Combinatorics

Combinatorics, Second Edition is a well-rounded, general introduction to the subjects of enumerative, bijective, and algebraic combinatorics. The textbook emphasizes bijective proofs, which provide elegant solutions to counting problems by setting up one-to-one correspondences between two sets of combinatorial objects. The author has written the textbook to be accessible to readers without any prior background in abstract algebra or combinatorics.

Part I of the second edition develops an array of mathematical tools to solve counting problems: basic counting rules, recursions, inclusion-exclusion techniques, generating functions, bijective proofs, and linear algebraic methods. These tools are used to analyze combinatorial structures such as words, permutations, subsets, functions, graphs, trees, lattice paths, and much more.

Part II cover topics in algebraic combinatorics including group actions, permutation statistics, symmetric functions, and tableau combinatorics.

This edition provides greater coverage of the use of ordinary and exponential generating functions as a problem-solving tool. Along with two new chapters, several new sections, and improved exposition throughout, the textbook is brimming with many examples and exercises of various levels of difficulty.

Table of Contents

Part I Counting

Chapter 1 Basic Counting

Chapter 2 Combinatorial Identities and Recursions

Chapter 3 Counting Problems in Graph Theory

Chapter 4 Inclusion-exclusion, Involutions, and Mobius Inversion

Chapter 5 Generating Functions

Chapter 6 Ranking, Unranking, and Successor Algorithms

Part II Algebraic Combinatorics

Chapter 7 Groups, Permutations, and Group Actions

Chapter 8 Permutation Statistics and Q-analogues

Chapter 9 Tableaux and Symmetric Polynomials

Chapter 10 Abaci and Antisymmetric Polynomials

Chapter 11 Algebraic Aspects of Generating Functions

Chapter 12 Additional Topics

Appendix: Definitions from Algebra

About the Author

Dr. Nicholas Loehr currently teaches mathematics at Virginia Tech. He has published over 30 journal articles in combinatorics, abstract algebra, and related areas. His research interests include enumerative and algebraic combinatorics; symmetric and quasisymmetric polynomials; integer partitions, lattice paths, parking functions, and tableaux; bijective methods; and algorithm analysis. Dr. Loehr is also the recipient of four teaching awards.