Graduate Courses S07010457 S07010458 S07010459 S07010460 S07010461 S07010462 S07010463 S07010464 Seminar

Course No: S07010459     Credit: 4        Periods: 80

Combinatorics

Required Textbook:   Li Qiao, Introduction to Combinatorics, Advance Educational Press, 1993.

Background of Course:  Combinatorics, also called combinatorial mathematics or combinatorial analysis, is a branch of mathematics. Like many branches of mathematics, its boundaries are not clearly defined, but the central problem may be considered that of arranging objects according to specified rules and finding out in how many ways this may be done. If the specified rules are very simple, then the chief emphasis is on the enumeration of the number of ways in which the arrangement may be made. If the rules are subtle or complicated, the chief problem is whether or  not such arrangements exist, and to find methods for constructing the arrangements.

There has been an explosive growth in combinatorics in recent years. One important reason for this growth has been the fundamental role that combinatorics plays as a tool in computer science and related areas. A further reason has been the prodigious effort, inaugurated by G. -C. Rota around 1964, to bring coherence and unity to the discipline of combinatorics, particularly enumeration, and to incorporate it into the mainstream of contemporary mathematics. Combinatorial enumeration has been greatly elucidated by this effort, as has its role in such areas of mathematics as finite group theory, representation theory, communicative algebra, algebraic geometry, and algebraic topology.

Course Description:  This course provides the most basic enumerative theory and methods of combinatorics. The course covers permutations and combinations,  partitions, generating functions and recursions, inversion formulae, Polya¨s Theorem, distinct representatives, Ramsey¨s theorem, some extremal problems, block designs, difference sets, orthogonal Latin squares.

### Audience: Beginning graduate Students (Masters)

Preparatory Courses:   Calculus, Liner Algebra, Abstract Algebra.

Test Form:  Written Examination

Reading List:  1. Shao Jiayu, Combinatorics, Tongji Unversity Press, 1991.

2.  Marshall. Hall, Jr., Combinatorial Theory (the second edition). John Wiley & Sons Inc., 1986

Suggestions for Further Reading:  Stanley, R. P., Enumerative Combinatorics, Volume 1. Cambridge  University Press, 1997.