MATH501 Introduction to Combinatorial Theory, Fall 2009


This course is an introduction to Combinatorics aimed at first year graduate students. The goal is to learn about both methods and structures.
Methods includes things like counting tools and procedures like the lexicographical ordering, but also includes permutation groups and algorithms.
Structures includes objects that arise often in combinatorics (sets, set-systems, bit-strings, graphs a.k.a. networks, trees, partitions, examples of partial orderings, lattices, Latin Squares, designs, codes, projective planes).
Often it is the interplay between the structures and the methods that make combinatorics interesting and fun. Because of that, it is reasonable to study methods and structures somewhat intertwined (as opposed to say, completely separate). This is what we will do this time around.

General Information


Week 1: Binomial coefficients, bitstrings, power set, Binomial Theorem.
Week 2: The lexicographic order, ranking unranking, power set.
Week 3: Projective planes (mostly finite), Desargues, projective geometry (mostly finite).
Week 4: More on binomial coefficients, arrangements and selections, multinomial coefficients. Pigeonhole principle.
Weeks 5 and 6: Fibonacci and Catalan, Generating functions.
Week 7: Permutations, order tree, Dijkstra's algorithm for the lexicographic successor.
Week 8: Derangements, Inclusion-Exclusion, Euler's totient function.
Weeks 9 and 10: Graph theory: Euler, Petersen, Hamming, Relations, digraphs, graphs and tournaments. Landau, Payley. Trees. Bipartite graphs. Hamiltonian graphs.
Week 11: Automorphism groups. Degree sequence. Labelled trees (Cayley's Theorem), Pruefer code. Havel/Hakimi. Matchings. Cayley graphs.
Weeks 12 and 13: Strongly regular graphs, parameters, bounds. Existence and Nonexistence
Weeks 14 and 15: Network algorithms: Shortest path, minimum cost spanning tree, Floyd Warshall, max-flow-min-cut, travelling salesman problem.


Your final grade will be determined from a score of 500:
Homework                   100
Midterm 100
Project 100
Final 200
Total 500


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On 14 Oct 2009, 13:53.