This course covers the mathematics of discrete structures and counting. Topics progress from foundational enumeration techniques to graph theory and extremal combinatorics. Each core topic concludes with a brief application segment, demonstrating how the relevant combinatorial tools apply to advanced areas of mathematics or computer science. Topics:

• Foundational Enumeration: Pigeonhole Principle, basic counting methods, permutations, and combinations.
• Advanced Counting: Binomial Theorem, double counting techniques, occupancy problems, the Principle of Inclusion-Exclusion, and discrete probability.
• Algebraic Methods for Enumeration: Ordinary generating functions, exponential generating functions, and techniques for setting up and solving recurrence relations.
• Graph Theory: Properties of trees, Prüfer codes, graph colorings, and the chromatic number.
• Extremal Combinatorics: Ramsey theory, Turán's theorem, Erdős-Stone theorem, Erdős-Ko-Rado theorem, Sperner's theorem, and the LYM inequality.
  (Note: Coverage of these final topics is tentative and depends on the pace of the class.)