Topics to be discussed
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Topics to be Covered The course will cover a selection of the following topics, mostly from van Lint and Wilson’s A course in combinatorics, in roughly the order indicated. While I would like to cover all of them, this syllabus is deliberately a little bit overambitious. The actual topics discussed will depend on how things go and on student interest. Part 1: Graph Theory 1. An introduction to graph theory, starting with the famous Bridges of KoĢnigsberg problem. Eulerian and Hamiltonian circuits; trees and their basic properties. 2. Planarity and coloring. A graph is planar if it can be drawn on a sheet of paper without any edges crossing when they shouldn’t, and we will prove that certain graphs aren’t planar. A graph coloring has the property that connected vertices can’t be the same color, and we will discuss results on which graphs can be colored by n colors. An introduction to Ramsey theory, and the recent proof of the Four Color Theorem. 3. Turan’s theorem: How many edges must a graph contain to guarantee a triangle? – and related questions. Matchings and Hall’s “marriage theorem”. 4. Dilworth’s theorem and extremal set theory; possibly topics from Chapters 7-9 in the book. Part 2: Enumerative combinatorics 1. Introduction and “basic counting”. Counting problems under a variety of hypotheses. Binomial coefficients, Stirling numbers, and Stanley’s“twelvefold way”. 2. The method of inclusion-exclusion. Introduction to some number-theoretic functions. Derangements. 3. Recurrence relations and generating functions. Fibonacci numbers and the Catalan numbers. Solving recurrences, and two proofs of the formula for Catalan numbers. 4. The partition function. Ferrers diagrams, Durfee squares, and the Jacobi triple product identity. Variations and identities. The hook-length formula. Ranks, cranks, and Ramanujan’s congruences. Using generating functions to prove inequalities, and a brief introduction to more advanced theory. 5. Polya counting; i.e., counting under symmetry. Relations to group theory, and the Rubik’s cube.
