Introduction to Enumerative Combinatorics

ForewordPrefaceAcknowledgmentsI How: Methods1 Basic Methods1.1 When We Add and When We Subtract1.1.1 When We Add1.1.2 When We Subtract1.2 When We Multiply1.2.1 The Product Principle1.2.2 Using Several Counting Principles1.2.3 When Repetitions Are Not Allowed1.3 When We Divide1.3.1 The Division Principle1.3.2 Subsets1.4 Applications of Basic Counting Principles1.4.1 Bijective Proofs1.4.2 Properties of Binomial Coefficients1.4.3 Permutations With Repetition1.5 The Pigeonhole Principle1.6 Notes1.7 Chapter Review1.8 Exercises1.9 Solutions to Exercises1.10 Supplementary Exercises2 Direct Applications of Basic Methods2.1 Multisets and Compositions2.1.1 Weak Compositions2.1.2 Compositions2.2 Set Partitions2.2.1 Stirling Numbers of the Second Kind2.2.2 Recurrence Relations for Stirling Numbers of the Second Kind2.2.3 When the Number of Blocks Is Not Fixed2.3 Partitions of Integers2.3.1 Nonincreasing Finite Sequences of Integers2.3.2 Ferrers Shapes and Their Applications2.3.3 Excursion: Euler’s Pentagonal Number Theorem2.4 The Inclusion-Exclusion Principle2.4.1 Two Intersecting Sets2.4.2 Three Intersecting Sets2.4.3 Any Number of Intersecting Sets2.5 The Twelvefold Way2.6 Notes2.7 Chapter Review2.8 Exercises2.9 Solutions to Exercises2.10 Supplementary Exercises3 Generating Functions3.1 Power Series3.1.1 Generalized Binomial Coefficients3.1.2 Formal Power Series3.2 Warming Up: Solving Recursions3.2.1 Ordinary Generating Functions3.2.2 Exponential Generating Functions3.3 Products of Generating Functions3.3.1 Ordinary Generating Functions3.3.2 Exponential Generating Functions3.4 Excursion: Composition of Two Generating Functions3.4.1 Ordinary Generating Functions3.4.2 Exponential Generating Functions3.5 Excursion: A Different Type of Generating Function3.6 Notes3.7 Chapter Review3.8 Exercises3.9 Solutions to Exercises3.10 Supplementary ExercisesII What: Topics4 Counting Permutations4.1 Eulerian Numbers4.2 The Cycle Structure of Permutations4.2.1 Stirling Numbers of the First Kind4.2.2 Permutations of a Given Type4.3 Cycle Structure and Exponential Generating Functions4.4 Inversions4.4.1 Counting Permutations with Respect to Inversions4.5 Notes4.6 Chapter Review4.7 Exercises4.8 Solutions to Exercises4.9 Supplementary Exercises5 Counting Graphs5.1 Counting Trees and Forests5.1.1 Counting Trees5.2 The Notion of Graph Isomorphisms5.3 Counting Trees on Labeled Vertices5.3.1 Counting Forests5.4 Graphs and Functions5.4.1 Acyclic Functions5.4.2 Parking Functions5.5 When the Vertices Are Not Freely Labeled5.5.1 Rooted Plane Trees5.5.2 Binary Plane Trees5.6 Excursion: Graphs on Colored Vertices5.6.1 Chromatic Polynomials5.6.2 Counting k-colored Graphs5.7 Graphs and Generating Functions5.7.1 Generating Functions of Trees5.7.2 Counting Connected Graphs5.7.3 Counting Eulerian Graphs5.8 Notes5.9 Chapter Review5.10 Exercises5.11 Solutions to Exercises5.12 Supplementary Exercises6 Extremal Combinatorics6.1 Extremal Graph Theory6.1.1 Bipartite Graphs6.1.2 Tur´an’s Theorem6.1.3 Graphs Excluding Cycles6.1.4 Graphs Excluding Complete Bipartite Graphs6.2 Hypergraphs6.2.1 Hypergraphs with Pairwise Intersecting Edges6.2.2 Hypergraphs with Pairwise Incomparable Edges6.3 Something Is More Than Nothing: Existence Proofs6.3.1 Property B6.3.2 Excluding Monochromatic Arithmetic Progressions6.3.3 Codes Over Finite Alphabets6.4 Notes6.5 Chapter Review6.6 Exercises6.7 Solutions to Exercises6.8 Supplementary ExercisesIII What Else: Special Topics7 Symmetric Structures7.1 Hypergraphs with Symmetries7.2 Finite Projective Planes7.2.1 Excursion: Finite Projective Planes of Prime Power Order7.3 Error-Correcting Codes7.3.1 Words Far Apart7.3.2 Codes from Hypergraphs7.3.3 Perfect Codes7.4 Counting Symmetric Structures7.5 Notes7.6 Chapter Review7.7 Exercises7.8 Solutions to Exercises7.9 Supplementary Exercises8 Sequences in Combinatorics8.1 Unimodality8.2 Log-Concavity8.2.1 Log-Concavity Implies Unimodality8.2.2 The Product Property8.2.3 Injective Proofs8.3 The Real Zeros Property8.4 Notes8.5 Chapter Review8.6 Exercises8.7 Solutions to Exercises8.8 Supplementary Exercises9 Counting Magic Squares and Magic Cubes9.1 An Interesting Distribution Problem9.2 Magic Squares of Fixed Size9.2.1 The Case of n = 39.2.2 The Function Hn(r) for Fixed n9.3 Magic Squares of Fixed Line Sum9.4 Why Magic Cubes Are Different9.5 Notes9.6 Chapter Review9.7 Exercises9.8 Supplementary ExercisesA The Method of Mathematical InductionA.1 Weak InductionA.2 Strong InductionBibliographyIndexFrequently Used Notation