Combinatorics /
Nicholas A. Loehr.
- Second edition.
- 1 online resource (xxiv, 618 pages)
Previous edition: Bijective combinatorics / Nicholas A. Loehr (Boca Raton, FL : Chapman and Hall/CRC, c2011).
part 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 Mo¨bius Inversion -- chapter 5 Generating Functions -- chapter 6 Ranking, Unranking, and Successor Algorithms -- part 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.
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, the book presents an introduction to enumerative and algebraic combinatorics emphasizing bijective methods. The text develops mathematical tools, such as basic counting rules, recursions, inclusion-exclusion techniques, generating functions, bijective proofs, and linear-algebraic methods to solve enumeration problems. The tools are used to analyze combinatorial structures, words, permutations, subsets, functions, compositions, integer partitions, graphs, trees, lattice paths, multisets, rook placements, and set partitions. -- 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.