Combinatorics Fall 2015
Class: AMAT 570-0001 (9999). Meets TTh in ES-153, between 8:45am--10:05am.
Instructor: Cristian Lenart, Office ES-116A, tel. 442-4635, e-mail: .
Overview. Combinatorics is a subject of increasing importance, owing to its links with other parts of pure and applied mathematics, as well as computer science. Combinatorics studies discrete structures arising both is abstract areas such as group theory and geometry, and applied areas such as optimization, networks, and statistics. Due to the advent of computers, which are ideally suited to manipulating discrete structures, combinatorics has become one of the fastest growing areas of mathematics. This course is a broad introduction to combinatorics, with emphasis on both theory and applications.
Syllabus. Generating functions. Special counting sequences. Composition of generating functions and enumeration of trees. Partially ordered sets. Enumeration under group action. Algebraic graph theory. Extremal combinatorics. A more advanced/specialized topic chosen from: symmetric functions and tableau combinatorics, combinatorial Hopf algebras, matroids, hyperplane arrangements, polytopes, topological combinatorics.
Prerequisite: an undergraduate course in abstract algebra or permission of instructor.
Text:
1. Miklos Bona, Introduction to Enumerative Combinatorics, McGraw-Hill, 2005, ISBN-13: 9780073125619 (required), available on half.com, amazon.com etc.
2. Richard Stanley, Enumerative Combinatorics I-II, Cambridge Univ. Press, ISBN 0521 66351 2 - vol. I, and 0521 78987 7 - vol. II (recommended).
3. Peter Cameron, Combinatorics. Topics, Techniques, Algorithms, Cambridge University Press, 1994, ISBN 0-521-45761 (recommended).
4. Other source(s) related to the more advanced/specialized topic.
Evaluation: Your grade in this course is based on solving the problems on the weekly problem sets.
Homework. You are allowed to work together on the problem sets (just discuss ideas, and everyone writes up his/her own solution). All problems assigned during one week will be collected at the end of the following week. Late homework is accepted for half credit. You are welcome to ask for help (during office hours, or in class in case of short questions) after you made some attempts at the problems.
Office hours: Tuesday between 10:10am--11:10am, 1:10pm--2:10pm, and Thursday between 10:10am--11:10am.
Attendance: Attendance in class is an integral part of the course, and hence is required. Every absence starting with the fourth will lower your overall score by 5%. Absences are only excused upon presentation of written medical excuse or comparable evidence of an emergency.
Important: You typically have to spend twice as much time or more on studying outside of class than you spend in class. In particular, it is very important to work regularly on exercises in order to test your understanding and in order to master the required techniques. Reading the book is considered part of the homework. The assigned homework represents a bare minimum amount of work needed, so you are strongly advised to work on other problems in the book. Those having difficulty are urged to contact the instructor as soon as possible (not just before the tests and exam), and to come to office hours. You are also strongly encouraged to actively participate in class and exchange ideas with the instructor and your colleagues.
Out of consideration for your fellow students’ efforts to learn, and your instructor’s efforts to teach, you are required to arrive on time for class and to remain seated (barring an emergency) until the class is finished.
For the same reasons, please turn off cell phones, do not surf the internet, play video games, read etc., or otherwise goof off and distract other people in the room. Loud eating or drinking, as well as repeated talking while the instructor or other students are talking are not allowed during the class. Repeated violations of any of the above rules shall be grounds for sanction or dismissal from the class.
Note. For those of you interested in finding out more about the interaction of combinatorics and other areas of pure mathematics, you can have a look at a recent survey I have written, and which can be found on my webpage: http://www.albany.edu/lenart.