Department of University of California, Berkeley Mathematics 249 Algebraic Vera Serganova, Fall 2006

My office hours are 11:00-12:00 on Tuesdays and Thursdays, in 709 Evans Hall. I can be reached by telephone at (64)2-2150 and electronic mail at serganov@math. You are welcome to ask questions by email. Homework assignments and other stuff related to this class can be found on the web http:// math.berkeley.edu/ ˜ serganov/249. First homework assignment is due on Thursday, September 7 The text for this class is Richard Stanley, Enumerative Combinatorics, vol.1,2. Good references are Macdonald, Symmetric functions and Hall polynomials, Fulton, Young tableaux, Graham, Knuth, Patashnik, Concrete mathematics To understand this course you need basic knowledge of . Math 250 is more than sufficient. Each Thursday I will post a problem assignment (3-4 problems) on the material of the week lectures. The homework will be collected the next Thursday. The grade will be computed according to the following proportions: 50% for your homework and 50% for the take home final. But if you solve all problems in your final (there will be hard ones in it) you get A for the course.

Course outline

• Generating functions and special numbers (Stirling numbers, Bernoulli numbers, q-binomial coefficients e.t.c.) • Permutation statistics • Inclusion-exclusion principle and its generalizations • Posets and lattices. Semimodular and distributive lattices, Moebius function and its applica- tions • Combinatorics of hyperplane arrangements, • Rational generating functions and quasi-polynomials • Linear Diophantine equations. The Ehrhart quasi-polynomial of a • Symmetric functions, Schur polynomials, Young tableaux, Kostka numbers • The RSK algorithm and the Cauchy identity • The Littlewood-Richardson rule • The Jacobi-Trudy identity • The Murnagahan-Nakayama rule • Symmetric functions and representations of symmetric groups and GL(n) • Plane partitions

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