Math 8668: Combinatorial theory
(Intro grad combinatorics, 1st semester)

Fall 2015

Instructor: Victor Reiner (You can call me "Vic"). 
Office: Vincent Hall 256
Telephone (with voice mail): 625-6682
Classes: Mon-Wed-Fri 11:15-12:05pm, Vincent Hall 206. 
Office hours: Mon 12:20-1:10pm, Tues 4:40-5:30pm,
and by appointment.  
Course content: This is the first semester of the 2-semester Math 8668-69 sequence;
Math 8669 will be taught by Prof. Stanton in Spring 2016.

We will study basic combinatorial objects (e.g., subsets, multisets, permutations, set/number partitions, compositions, graphs, trees), their enumeration, and other properties, such as graphical or partially ordered set structures. Roughly speaking this is what we ended up discussing in the first semester:

  • Generating functions (ordinary, exponential)
  • Polya theory
  • Permutation statistics
  • Lagrange inversion
  • (skipped Hypergeometric notation and identities-- next semester??)
  • Inclusion-exclusion
  • Determinantal formulas
    • Matrix-tree and BEST theorem
    • permanent-determinant, pfaffian-hafnian method
    • Gessel-Viennot-Lindstrom lemma,
    • MacMahon's "master" theorem,
    • Transfer matrix method
  • Partially ordered sets and lattices
    • Distributive lattices, Birkhoff's Theorem.
    • Moebius functions and Moebius inversion
    • (skipped Sperner theory of posets)

    Here are scans of my lecture notes: Batch 1, Batch 2, Batch 3, Batch 4, Batch 5
    (Note: not guaranteed to conform with what actually happened in lecture!)
Prerequisites: Calculus, linear algebra, undergraduate algebra (groups, rings, fields).
Main text: R.P. Stanley, Enumerative combinatorics, Vol. I, Cambridge University Press.
Available from our library as an online resource.
Here are the book's errata.
Other useful sources: F. Ardila, Algebraic and geometric methods in enumerative combinatorics, Part I.
H. Wilf, generatingfunctionology
D. Stanton and D. White, Constructive combinatorics
N. Loehr, Bijective combinatorics
M. Aigner, Combinatorial theory
J.H. Van Lint and R. Wilson, A course in combinatorics

Course requirements and grading: There will be 3 homeworks during the semester.
Grades will be based both on the quality and quantity of homework turned in.
I also expect you to show up to class, and ask questions (not graded!)
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.

Homework assignments
Assignment Due date Problems
HW #1 Friday, Oct. 9
(note change!)
Any 8 of these Stanley Ch. 1 exercises:
[2-]: #66, 113
[2 ]: #5, 20, 21, 26, 29, 47(a,b), 54, 68, 69, 102(a,b)
[2+]: #12, 18, 47(c), 175, 178, 102(c)
HW #2 Wed, Nov. 11
(note change!)
All of these Stanley Ch. 2 exercises:
[2-]: #2
[2 ]: #25
[2+]: #14 (at least A1(n), A2(n), A3(n))
HW #3 Wed, Dec. 16 Any 6 from either of
-- these Stanley Ch. 4 transfer-matrix method exercises
[2]: #67, 68, 69, 73
-- or these Stanley Ch. 3 exercises:
[2-]: #10(a)
[2 ]: #34, 42(a,b), 45(a), 46(a,c), 47(a,b,c), 53, 70(a,b)
[2+]: #10(b), 14(a,b), 46(b), 85
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