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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.
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