## MATH501 Introduction to Combinatorial Theory, Fall 2009

### Objectives

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

• Call Number 63396
• Instructor: Anton Betten, room 207, Weber building. Email lastname at math dot colostate dot edu
• Course website: http://www.math.colostate.edu/ betten/courses/MATH501/FA09/501_syllabus.html http://www.math.colostate.edu/~betten/courses/MATH501/FA09/501_syllabus.html
• Credits: 3
• Class: M  W  F     3 - 3:50 pm,     ENGRG E 206
• Prerequisites: none
• Text: Combinatorial Methods with Computer Applications, Jonathan Gross, Chapmann and Hall / CRC. ISBN13: 978-1-58488-743-0 (Hardcover).
• Homework: Assigned bi-weekly, due Wednesdays
• Projects: You will do one project that either involves programming or a more theoretical part with write-up.
• Midterm: 10/14/09
• Final Exam: As scheduled by the registrar's office.
• Office hours: Monday 1pm, Friday 1pm

### Syllabus:

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.

 Homework 100 Midterm 100 Project 100 Final 200 Total 500

### Homework

Homework assignment #1
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Homework assignment #3
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Homework assignment #10

File translated from TEX by TTH, version 3.81.
On 14 Oct 2009, 13:53.