Introductory Combinatorics

1. AN INTRODUCTION TO ENUMERATION. Elementary Counting Principles. Functions and the Pigeonhole Principle. Subsets. Using Binomial Coefficients. Mathematical Induction. 2. EQUIVALENCE, RELATIONS, PARTITIONS, AND MULTISETS. Equivalence Relations. Distributions and Multisets. Partitions and Stirling Numbers. Partitions of Integers. 3. ALGEBRAIC COUNTING TECHNIQUES. The Principle of Inclusion and Exclusion. The Concept of a Generating Function. Applications to Partitions and Inclusion--Exclusion. Recurrence Relations and Generating Functions. Exponential Generating Functions. 4. GRAPH THEORY. Eulerian Walks and the Idea of Graphs. Trees. Shortest Paths and Search Trees. Isomorphism and Planarity. Digraphs. Coloring. Graphs and Matrices. 5. MATCHING AND OPTIMIZATION. Matching Theory. The Greedy Algorithm. Network Flows. Flows, Connectivity, and Matching. 6. COMBINATORIAL DESIGNS. Latin Squares and Graeco-Latin Squares. Block Designs. Construction and Resolvability of Designs. Affine and Projective Planes. Codes and Designs. 7. ORDERED SETS. Partial Orderings. Linear Extensions and Chains. Lattices. Boolean Algebras. Mobius Functions. Products of Orderings. 8. ENUMERATION UNDER GROUP ACTION. Permutation Groups. Groups Acting on Sets. Polya's Enumeration Theorem.