Math 566: Combinatorial Theory

Professor: Jonah Blasiak

Winter 2012

Monday, Wednesday, Friday 11:00-12:00, 553 Dennison

Course Description: Algebraic combinatorics is the study of combinatorial objects that arise in group theory and representation theory (like tableaux) and also involves using algebraic tools (like generating functions) to study combinatorial objects. The first half of the course will be a sample of topics from algebraic combinatorics including generating functions, permutation statistics, q-counting, and posets. The second half of the course will be an in-depth study of symmetric functions and tableaux, following Chapter 7 of Stanley. Problem solving will be emphasized.
Prerequisites: linear algebra and some exposure to proofs and abstract mathematics. Familiarity with abstract algebra (412 or 512) would be useful but is not required.
Level: mixed undergraduate and graduate.
Office Hours: 3831 East Hall, Monday 12-1:30 pm.
Office Hours/Problem Session: 2nd floor commons or nearby classroom - East Hall, Thursday 5:30-7:30 pm.
Recommended texts:
  • Enumerative combinatorics, vol.1, R. P. Stanley
  • Enumerative combinatorics, vol.2, R. P. Stanley
  • Other references:
  • Generating functionology, vol.2, Herbert S. Wilf (pdf available here)
  • Homework Policy: You may consult each other, the library, the internet and any other source for aid provided (1) you list all people and sources who aided you, or whom you aided and (2) you write-up the solutions independently, in your own language. It is likely that you will be able to find solutions to some of the problems if you look hard enough. Being able to search and read through literature is a useful skill, but is not the main focus of this class. It is recommended that you reserve extensive literature searches for only the hardest problems.
    Guidelines for problem set writeups
    Another useful guide, though more geared toward contest problems: How to Write a Solution - by Richard Rusczyk & Mathew Crawford
    Take home exam policy: You may not consult with other people or outside sources; you may consult your notes and the textbooks as well as any handouts I provide.
    Take home midterm to be handed out Feb. 17, due Feb. 24 in class.
    Take home final to be handed out on the last day of class (April 16) and due 3:30pm on Tuesday, April 24. The final will cover material from the entire class, but more questions will be related to material from the second half.
    Grading policy:
  • 30% homework
  • 30% midterm
  • 40% final

  • Tentative Syllabus

    January 4, 6: Hook length formula advertisement, binomial coefficients, permutation statistics (1.2-1.3 of Stanley)
    Problem Set 1, Due: Friday, January 13

    January 9, 11, 13: Permutation statistics, compositions, Fibonacci numbers (1.2-1.3 of Stanley)
    Problem Set 2, Due: Friday, January 20

    January 18, 20: Introduction to generating functions, Stirling numbers (1.1, 1.4 of Stanley, 1.1-2.3 of Wilf)
    Problem Set 3, Due: Friday, January 27

    January 23, 25, 27: Stirling numbers, Catalan numbers, exponential formula (1.4 of Stanley, 3.1-3.12 of Wilf)
    Problem Set 4, Due: Friday, February 3

    January 30, February 1, 3: parking functions, q-binomial coefficients (1.7 of Stanley 2nd edition)
    Problem Set 5, Due: Friday, February 10

    February 6, 8, 10: hyperplane arrangements, posets, Mobius function (Chapters 1 and 2 of Stanley's notes)
    Problem Set 6, Due: Friday, February 17. Some comments on this homework set.

    February 13, 15, 17: hyperplane arrangements, graphical arrangements, Shi arrangement (Chapter 2 of Stanley's notes)
    Midterm, Due: Friday, February 24 at the beginning of class

    February 20, 22, 24: partitions, dominance order, tableaux, five bases for symmetric functions (Stanley 1.8, 7.1-7.7)
    Office hours changed this week for the midterm: 12-1:30 Monday,Wednesday, 5-6 Thursday
    Problem Set 7, Due: Friday, March 9

    March 5, 7, 9: the involution w, power sum symmetric functions, specializations (Stanley 7.5-7.8)
    Problem Set 8, Due: Friday, March 16

    March 12, 14, 16: Schur functions, tableaux, RSK algorithm (Stanley 7.10-7.13)
    Problem Set 9, Due: Friday, March 23

    March 19, 21, 23: RSK algorithm, growth diagrams, inner product on symmetric functions (Stanley 7.9-7.14)
    Problem Set 10, Due: Friday, March 30

    March 26, 28, 30: Classical definition of Schur functions, Pieri rule (Stanley 7.14-7.16)
    Problem Set 11, Due: Friday, April 6

    April 2, 4, 6: skew Schur functions, Jacobi-Trudi identity, hooklength formula (Stanley 7.15-7.16, 7.21)
    Problem Set 12, Do not turn in

    April 9, 11, 13: Knuth equivalence, Greene's theorem (Stanley A1.1)
    Office hours this Thursday will be in my office from 5:30 - 7:30

    April 16: Review, mainly symmetric functions.
    Please fill out course evaluations on ctools by April 18.
    Office hours will be as usual this week, except Thursday will be in my office: Mon 16th 12-1:30, Th 19th 5:30-7:30, Mon 23, 12-1:30