Math 570, Combinatorics

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. Far from being exhaustive, the list below contains some of the most important topics in combinatorics.

Syllabus: Enumeration (permutations and combinations, the principle of inclusion-exclusion, generating functions, special counting sequences, construction of bijections, enumeration under group action). Composition of power series (the exponential formula, enumeration of trees, Lagrange inversion). Graph theory (basic concepts, network flows, matchings, algebraic graph theory, topological graph theory). Partially ordered sets (important examples, partitions into chains and antichains).

There are no specific prerequisites for this course, but prior experience with abstraction and proofs is helpful. Furthermore, the successful completion of a calculus course and an elementary algebra course (linear algebra, groups) is also helpful.

Textbook: Peter Cameron, Combinatorics. Topics, Techniques, Algorithms, Cambridge University Press, 1994, ISBN 0-521-45761.

Cristian Lenart, Department of Mathematics, ES 118, SUNY at Albany