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
  • 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,
    • increasing and decreasing subsequences,
    • the representation theory of the symmetric 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 study material from Chapters 6 and 7.

Other nice sources:

(for group actions) L. Shapiro, Finite Groups Acting on Sets with Applications.
(for symmetric functions) B. Sagan, The Symmetric Group.
(for both topics) Chapter 7 of R.P. Stanley, Enumerative combinatorics, Vol. II.
(for the "toggle" description of RSK) S. Hopkins, RSK via local transformations.

Class notes:

Batch 1; Batch 2; Batch 3; Batch 4; Batch 5; Batch 6; Batch 7; Batch 8

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 with whom they collaborated.

Homework assignments:

I will write the problems myself, possibly adapting Sagan's exercises.
Assignment Due date Problems
HW #1 Friday, 2/11
Monday, 2/14
Click here
HW #2 Friday, 3/25
Monday, 3/28
Click here
HW #3 Friday, 4/22 Click here