UNIVERSITY OF MINNESOTA 
SCHOOL OF MATHEMATICS

Math 8669: Combinatorial theory
(Intro grad combinatorics, 2nd semester)

Spring 2003

Instructor: Victor Reiner (You can call me "Vic"). 
Office: Vincent Hall 256
Telephone (with voice mail): 625-6682
E-mail: reiner@math.umn.edu 
Classes: Mon-Wed-Fri 2:30-3:20pm, Vincent Hall 2. 
Office hours: Mon-Wed 3:35 PM; or by appointment.
(For the first few weeks, starting 1/28, the Wed office hour will
be an extra lecture slot to make up for missed lectures:
3/14, 4/7, 4/9, 4/11, 5/5, 5/7) 
Course content: This is a continuation of Math 8668, taught by Prof. Goldman in the Fall. We will pick up where he left off, pursuing the following topics, in something like the order listed below.
  • Further Lattice theory:
    - Distributive lattices
    - Modular and semimodular lattices
    - Geometric lattices and matroid theory
  • Some Sperner theory of posets
  • Rudimentary hypergeometric and q-series
  • (Complex) representation theory of finite groups
  • The combinatorics and representation theory
    of the symmetric group and symmetric functions:
    - Bases for symmetric functions and their "meaning"
    - The Specht construction of irreducibles for the symmetric group
    - Combinatorics of tableaux
    (e.g. Robinson-Schensted-Knuth algorithm, jeu-de-taquin, hook-length formula)
Prerequisites: Abstract algebra (groups, rings, modules, fields), and either Math 8668 or some combinatorics experience.  
Required text R.P. Stanley, Enumerative combinatorics, Vol. II, Cambridge University Press.
Available at the University bookstore. We will not really be using this book, however, until we get to the symmetric groups, symmetric functions, etc. Also, in the early part of the course, some of Stanley's Enumerative combinatorics, Vol. I may still appear, but is not required.
Other useful sources Symmetric group, symmetric functions, representations, etc.
I.G. Macdonald, Symmetric functions and Hall polynomials.
B. E. Sagan, The symmetric group: its representations, combinatorial algorithms, and symmetric functions.
G. James and A. Kerber, The representation theory of the symmetric group.
J.-P. Serre, Linear representations of finite groups
Lattice theory and matroid theory
M. Aigner, Combinatorial theory
G. Graetzer, General lattice theory
D. Welsh, Matroid theory
J. Oxley, Matroid theory
N. White's 3 volumes:
Combinatorial geometries , Theory of matroids , Matroid applications .
Hypergeometric and q-series
G. Gasper and M. Rahman, Basic hypergeometric series .
M. Petkvosek and D. Zeilberger, A=B
R. Graham, D. Knuth, and O. Patashnik, Concrete mathematics
Course requirements and grading There will be homeworks given out approximately every three weeks, and you will be expected to at least try every problem. Grades will be based both on the quality and quantity of homework turned in.
I encourage collaboration on the homework, as long as each person understands the solutions, writes them up in their own words, and indicates on the homework page with whom they have collaborated.

Homework assignments
Assignment Due date Problems
(from Stanley's E.C. II, unless otherwise specified)
HW #1 2/7/2003 HW 1 in PostScript, PDF
HW #2 2/28/2003 HW 2 in PostScript, PDF
HW #3 3/28/2003 HW 3 in PostScript, PDF
HW #4 4/23/2003 HW 4 in PostScript, PDF
HW #5 5/9/2003 HW 5 in PostScript, PDF
Back to Reiner's Homepage.