UNIVERSITY OF MINNESOTA 
SCHOOL OF MATHEMATICS

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

Spring 2022

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: 2:30-3:20pm, Mon, Wed, Fri, in-person in Vincent Hall 364 
COVID policy: Here is the explanation of University COVID policy from the University Senate.
Office hours: 5:20-6:00pm on Mondays, Wednesdays in-person in VinH 256,
and Tuesdays 9:05-9:55am at this Zoom link.  
Discord server: Use our class Discord server to ask questions, form study groups, etc.
Course content: This is a continuation of Math 8668, taught by Prof. Pylyavskyy in Fall 2021. Here are the planned topics, in roughly the order listed below.
  • Determinantal formulas, such as
    • Transfer matrix method
    • Gessel-Viennot-Lindstrom lemma
    • Matrix-tree and BEST theorem
    • (MacMahon's "master" theorem; we'll probably skip this)
    • (permanent-determinant method; we'll also probably skip this)
  • (Non-modular) Representation theory of finite groups
  • Representations of symmetric groups, emphasizing their relation to ...
    • symmetric functions,
    • partitions, Young tableux, and
    • general linear group representations (if time permits).
Prerequisites: Abstract algebra (groups, rings, modules, fields), and either Math 8668 or some combinatorics experience.  
Main texts R.P. Stanley, Enumerative combinatorics, Vols. I and II, Cambridge University Press
We have access to Vol II with our library credentials
Among other things, we will do a bit of Chapters 4 (Vol I) and 5 (Vol II) , and much of Chapter 7 (Vol II).
Other useful sources
Course requirements and grading There will be 3 homeworks during the semester, to be handed in at the course Canvas site. 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. 4 Hand in any four of these:
Stanley EC Vol I Chap. 4: #67, 68, 69, 73
Stanley EC Vol II Chap 5: #69, 70, 74
HW #2 Friday, Mar. 18 Hand in any four of these
group representation problems
HW #3 Friday, Apr. 29 Hand in any four of these:
Stanley EC Vol II Chap 7: #8, 12, 28, 30, 36, 47(a,c,d,e), 57, 58, 72, 73
Back to Reiner's Homepage.