Introductory Combinatorics

Richard A. Brualdi is Bascom Professor of Mathematics, Emeritus at the University of Wisconsin-Madison. He served as Chair of the Department of Mathematics from 1993-1999. His research interests lie in matrix theory and combinatorics/graph theory. Professor Brualdi is the author or co-author of six books, and has published extensively. He is one of the editors-in-chief of the journal "Linear Algebra and its Applications" and of the journal "Electronic Journal of Combinatorics." He is a member of the American Mathematical Society, the Mathematical Association of America, the International Linear Algebra Society, and the Institute for Combinatorics and its Applications. He is also a Fellow of the Society for Industrial and Applied Mathematics.

1. What is Combinatorics?

1.1 Example: Perfect Covers of Chessboards

1.2 Example: Magic Squares

1.3 Example: The Four-Color Problem

1.4 Example: The Problem of the 36 Officers

1.5 Example: Shortest-Route Problem

1.6 Example: Mutually Overlapping Circles

1.7 Example: The Game of Nim

 

2. The Pigeonhole Principle

2.1 Pigeonhole Principle: Simple Form

2.2 Pigeonhole Principle: Strong Form

2.3 A Theorem of Ramsay

 

3. Permutations and Combinations

3.1 Four Basic Counting Principles

3.2 Permutations of Sets

3.3 Combinations of Sets

3.4 Permutations of Multisets

3.5 Combinations of Multisets

3.6 Finite Probability

 

4. Generating Permutations and Combinations

4.1 Generating Permutations

4.2 Inversions in Permutations

4.3 Generating Combinations

4.4 Generating r-Combinations

4.5 Partial Orders and Equivalence Relations

 

5. The Binomial Coefficients

5.1 Pascal's Formula

5.2 The Binomial Theorem

5.3 Unimodality of Binomial Coefficients

5.4 The Multinomial Theorem

5.5 Newton's Binomial Theorem

5.6 More on Partially Ordered Sets

 

6. The Inclusion-Exclusion Principle and Applications

6.1 The Inclusion-Exclusion Principle

6.2 Combinations with Repetition

6.3 Derangements

6.4 Permutations with Forbidden Positions

6.5 Another Forbidden Position Problem

6.6 Möbius Inversion

 

7. Recurrence Relations and Generating Functions

7.1 Some Number Sequences

7.2 Generating Functions

7.3 Exponential Generating Functions

7.4 Solving Linear Homogeneous Recurrence Relations

7.5 Nonhomogeneous Recurrence Relations

7.6 A Geometry Example

 

8. Special Counting Sequences

8.1 Catalan Numbers

8.2 Difference Sequences and Stirling Numbers

8.3 Partition Numbers

8.4 A Geometric Problem

8.5 Lattice Paths and Schröder Numbers

 

9. Systems of Distinct Representatives

9.1 General Problem Formulation

9.2 Existence of SDRs

9.3 Stable Marriages

 

10. Combinatorial Designs

10.1 Modular Arithmetic

10.2 Block Designs

10.3 Steiner Triple Systems

10.4 Latin Squares

 

11. Introduction to Graph Theory

11.1 Basic Properties

11.2 Eulerian Trails

11.3 Hamilton Paths and Cycles

11.4 Bipartite Multigraphs

11.5 Trees

11.6 The Shannon Switching Game

11.7 More on Trees

 

12. More on Graph Theory

12.1 Chromatic Number

12.2 Plane and Planar Graphs

12.3 A 5-color Theorem

12.4 Independence Number and Clique Number

12.5 Matching Number

12.6 Connectivity

 

13. Digraphs and Networks

13.1 Digraphs

13.2 Networks

13.3 Matching in Bipartite Graphs Revisited

 

14. Pólya Counting

14.1 Permutation and Symmetry Groups

14.2 Burnside's Theorem

14.3 Pólya's Counting formula