18.315 Combinatorial Theory

Topic for fall 1996: symmetric functions.

Course meets: Tuesdays and Thursdays, 1-2:30, Room 4-145.

Lecturer: Professor Sergey Fomin, Room 2-363B, 253-1713, fomin@math.mit.edu

Text: R.P.Stanley, Enumerative combinatorics, vol.2, to appear circa 1998.

Reference texts:

I.G.Macdonald, Symmetric functions and Hall polynomials, 2nd edition, Oxford University Press, 1995.

B.E.Sagan, The symmetric group, Wadsworth and Brooks/Cole, 1991.

R.P.Stanley, Enumerative combinatorics, vol.1, Wadsworth and Brooks/Cole, 1986. Second edition to be published by Oxford University Press.

The ring of symmetric functions and its various bases. The Schur functions

Identities involving symmetric functions

Young tableaux. The Robinson-Schensted-Knuth correspondence

Noncommutative Schur functions

The Littlewood-Richardson and Murnaghan-Nakayama rules

Irreducible representations of the symmetric group. The Frobenius map

Enumeration of plane partitions

Quasi-symmetric functions. Ribbon Schur functions

"Jeu de taquin." Knuth equivalence. Greene's invariants.