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

Spring 2010

Instructor: Victor Reiner (You can call me "Vic"). 
Office: Vincent Hall 256
Telephone (with voice mail): 625-6682
Classes: Mon-Wed-Fri 1:25-2:15pm, Vincent Hall 211. 
Office hours: To be determined,
and by appointment.  
Course content: This is a continuation of Math 8668, taught by Prof. Dennis White in Fall 2009. The general theme is to study extra structure on basic combinatorial objects, beyond enumerating them. We will pursue the following topics, in roughly the order listed below.
  • Partially ordered sets, including...
    • theory of incidence algebras and Moebius function,
    • some Sperner theory, and
    • lattices (distributive, modular, semimodular, geometric, matroids)
  • (Non-modular) Representation theory of finite groups
  • Representations of symmetric groups, emphasizing their relation to ...
    • symmetric functions,
    • partitions, Young tableux, and
    • general linear groups (if time permits).
Prerequisites: Abstract algebra (groups, rings, modules, fields), and either Math 8668 or some combinatorics experience.  
Main text(s) R.P. Stanley, Enumerative combinatorics, Vols. I and II,
Cambridge University Press.
We will be doing, among other things,
  • much of Vol .I, Chapter 3,
  • perhaps a tiny bit of Vol. I. Chapter 4,
  • much of Vol. II, Chapter 7
  • Other useful sources General
    J.H. Van Lint and R. Wilson, A course in combinatorics
    D. Stanton and D. White, Constructive combinatorics
    Posets, lattice and matroid theory
    M. Aigner, Combinatorial theory
    J. Oxley, Matroid theory
    Some lectures on matroids from a 2005 summer school in Vienna
    Symmetric group, symmetric functions, representations, etc.
    B. E. Sagan, The symmetric group: its representations, combinatorial algorithms, and symmetric functions.
    I.G. Macdonald, Symmetric functions and Hall polynomials.
    W. Fulton, Young tableaux
    W. Fulton and J. Harris, Representation theory: a first course
    Course requirements and grading There will be 3 or 4 homeworks during the semester. 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.
    Since homework problems that come from the volumes by Stanley have some solutions in the book, students must explain them more fully on their homework.

    Homework assignments
    Assignment Due date Problems
    HW #1 Friday, Feb. 26 HW 1 in PDF
    HW #2 Friday, April 9 HW 2 in PDF
    HW #3 Friday, May 7 HW 3 in PDF
    Back to Reiner's Homepage.