Preface
Notation
Chapter 1. Prologue: Compositions of an integer
1.1. Counting compositions
1.2. The Fibonacci numbers from a combinatorial perspective
1.3. Weak compositions
1.4. Compositions with arbitrarily restricted parts
1.5. The fundamental theorem of composition enumeration
1.6. Basic tools for manipulating finite sums
Exercises
Chapter 2. Sets, functions, and relations
2.0. Notation and terminology
2.1. Functions
2.2. Finite sets
2.3. Cartesian products and their subsets
2.4. Counting surjections: A recursive formula
2.5. The domain partition induced by a function
2.6. The pigeonhole principle for functions
2.7. Relations
2.8. The matrix representation of a relation
2.9. Equivalence relations and partitions
References
Exercises
Project
2.A
Chapter 3. Binomial coefficients
3.1. Subsets of a finite set
3.2. Distributions, words, and lattice paths
3.3. Binomial inversion and the sieve formula
3.4. Problème des ménages
3.5. An inversion formula for set functions
3.6. *The Bonferroni inequalities
References
Exercises
Chapter 4. Multinomial coefficients and ordered partitions
4.1. Multinomial coefficients and ordered partitions
4.2. Ordered partitions and preferential rankings
4.3. Generating functions for ordered partitions
Reference
Exercises
Chapter 5. Graphs and trees
5.1. Graphs
5.2. Connected graphs
5.3. Trees
5.4. *Spanning trees
5.5. *Ramsey theory
5.6. *The probabilistic method
References
Exercises
Project
5.A
Chapter 6. Partitions: Stirling, Lah, and cycle numbers
6.1. Stirling numbers of the second kind
6.2. Restricted growth functions
6.3. The numbers 𝜎(𝑛,𝑘) and 𝑆(𝑛,𝑘) as connection constants
6.4. Stirling numbers of the first kind
6.5. Ordered occupancy: Lah numbers
6.6. Restricted ordered occupancy: Cycle numbers
6.7. Balls and boxes: The twenty-fold way
References
Exercises
Projects
6.A
6.B
Chapter 7. Intermission: Some unifying themes
7.1. The exponential formula
7.2. Comtet’s theorem
7.3. Lancaster’s theorem
References
Exercises
Project
7.A
Chapter 8. Combinatorics and number theory
8.1. Arithmetic functions
8.2. Circular words
8.3. Partitions of an integer
8.4. *Power sums
8.5. 𝑝-orders and Legendre’s theorem
8.6. Lucas’s congruence for binomial coefficients
8.7. *Restricted sums of binomial coefficients
References
Exercises
Project
8.A
Chapter 9. Differences and sums
9.1. Finite difference operators
9.2. Polynomial interpolation
9.3. The fundamental theorem of the finite difference calculus
9.4. The snake oil method
9.5. * The harmonic numbers
9.6. Linear homogeneous difference equations with constant coefficients
9.7. Constructing visibly real-valued solutions to difference equations with obviously real-valued solutions
9.8. The fundamental theorem of rational generating functions
9.9. Inefficient recursive formulae
9.10. Periodic functions and polynomial functions
9.11. A nonlinear recursive formula: The Catalan numbers
References
Exercises
Project
9.A
Chapter 10. Enumeration under group action
10.1. Permutation groups and orbits
10.2. Pólya’s first theorem
10.3. The pattern inventory: Pólya’s second theorem
10.4. Counting isomorphism classes of graphs
10.5. 𝐺-classes of proper subsets of colorings / group actions
10.6. De Bruijn’s generalization of Pólya theory
10.7. Equivalence classes of boolean functions
References
Exercises
Chapter 11. Finite vector spaces
11.1. Vector spaces over finite fields
11.2. Linear spans and linear independence
11.3. Counting subspaces
11.4. The 𝑞-binomial coefficients are Comtet numbers
11.5. 𝑞-binomial inversion
11.6. The 𝑞-Vandermonde identity
11.7. 𝑞-multinomial coefficients of the first kind
11.8. 𝑞-multinomial coefficients of the second kind
11.9. The distribution polynomials of statistics on discrete structures
11.10. Knuth’s analysis
11.11. The Galois numbers
References
Exercises
Projects
11.A
11.B
Chapter 12. Ordered sets
12.1. Total orders and their generalizations
12.2. *Quasi-orders and topologies
12.3. *Weak orders and ordered partitions
12.4. *Strict orders
12.5. Partial orders: basic terminology and notation
12.6. Chains and antichains
12.7. Matchings and systems of distinct representatives
12.8. *Unimodality and logarithmic concavity
12.9. Rank functions and Sperner posets
12.10. Lattices
References
Exercises
Projects
12.A
12.B
12.C
12.D
Chapter 13. Formal power series
13.1. Semigroup algebras
13.2. The Cauchy algebra
13.3. Formal power series and polynomials over ℂ
13.4. Infinite sums in ℂ^{ℕ}
13.5. Summation interchange
13.6. Formal derivatives
13.7. The formal logarithm
13.8. The formal exponential function
References
Exercises
Projects
13.A
13.B
13.C
Chapter 14. Incidence algebra: The grand unified theory of enumerative combinatorics
14.1. The incidence algebra of a locally finite poset
14.2. Infinite sums in ℂ^{Int (ℙ)}
14.3. The zeta function and the enumeration of chains
14.4. The chi function and the enumeration of maximal chains
14.5. The Möbius function
14.6. Möbius inversion formulas
14.7. The Möbius functions of four classical posets
14.8. Graded posets and the Jordan–Dedekind chain condition
14.9. Binomial posets
14.10. The reduced incidence algebra of a binomial poset
14.11. Modular binomial lattices
References
Exercises
Projects
14.A
14.B
Appendix A. Analysis review
A.1. Infinite series
A.2. Power series
A.3. Double sequences and series
References
Appendix B. Topology review
B.1. Topological spaces and their bases
B.2. Metric topologies
B.3. Separation axioms
B.4. Product topologies
B.5. The topology of pointwise convergence
References
Appendix C. Abstract algebra review
C.1. Algebraic structures with one composition
C.2. Algebraic structures with two compositions
C.3. 𝑅-algebraic structures
C.4. Substructures
C.5. Isomorphic structures
References
Index