Math 274: Combinatorics II

(Graduate combinatorics, 2nd semester)

Spring 2022


Instructor: Sam Hopkins (call me "Sam")
Office: Academic Support Bldg B - 214 (temporary)
E-mail: samuelfhopkins@gmail.com 
Classes: Mon-Wed-Fri, 11:10am-12:00pm
Course format: Online (via Zoom: link will be emailed)
Office hours: By appointment (email me to set up a time)
Course content: This is the 2nd semester of graduate combinatorics (click here for the 1st semester.).
We will continue to study the enumeration of discrete structures, with a new focus on connections to algebra (a.k.a. symmetries!). We will try to cover the following:
  • Group actions on combinatorial sets:
    • group actions, orbits, stabilizers, etc.,
    • Burnside's Lemma,
    • Pólya-Redfield enumeration,
    • the cyclic sieving phenomenon,
    • (maybe) groups acting on posets
  • Symmetric functions:
    • the ring of symmetric functions and its bases,
    • Schur functions and semistandard Young tableaux,
    • the Lindström-Gessel-Viennot lemma and the Jacobi-Trudi formula,
    • reverse plane partitions and the Hillman-Grassl correspondence,
    • standard Young tableaux and the hook length formula,
    • the Robinson-Schensted-Knuth correspondence,
    • (maybe) quasisymmetric functions,
    • (maybe) representations of the symmetric group,
    • (maybe) representations of the general linear group
But if there are any topics you are especially interested in (or not interested in), please let me know! I am happy to tailor this course to the interests of the students. Also, I will try to review all the necessary background (either from algebra or from combinatorics) as we go along.
Prerequisites: Calculus, linear algebra, undergraduate algebra (groups, rings, fields).
It's helpful to have taken the 1st semester of grad combinatorics, but is not strictly necessary.
Main text: B. Sagan, Combinatorics: the Art of Counting.
In the 1st semester we covered material from Chapters 1-5 (even though this was not the main text for that course). In this semester we will mostly study material from Chapters 6-8.
Other nice sources: Chapter 7 of R.P. Stanley, Enumerative combinatorics, Vol. II.
B. Sagan, The Symmetric Group.
Class notes: Class notes will be posted periodically below.
[More to be posted]
Grading: There will be 3 homework assignments for the semester.
The grading of the assignments will depend on both the quality and quantity of homework turned in. Beyond that, I expect you to show up to class and be engaged. Collaboration on the homework is encouraged, 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: Problems will mostly be exercises from Sagan, plus some problems I come up with.
Assignment Due date Problems
HW #1 2/11
TBD
HW #2 3/25 TBD
HW #3 4/22 TBD