Prof. Jim Haglund, jhaglund@math.upenn.edu
Course webpage: http://www.math.upenn.edu/~jhaglund/580/

Office hours: Tu 3-4pm, Th 11am-12pm, F 1-2pm in DRL 4E2B.

Office Phone: (215) 573-9093.

Lecture: MW 12-1:30 in DRLB 3C4

Course:This course is meant to be an introduction to combinatorics suitable for advanced undergraduates or graduate students in mathematics. In particular, it covers certain core material that all students who take a Ph. D. oral exam in combinatorics must know. This semester our basic text will be Enumerative Combinatorics Volume I by Richard Stanley. We will be covering Chapters 1,2,and part of 3. In addition we will be using Chapter 1 of my book The q,t-Catalan numbers and the Space of Diagonal Harmonics. This contains some introductory material on q-analogues that will be a useful supplement to Stanleys book. I will also be supplementing the material from Chapter 2 of Stanley, on rook polynomials, with some results from recent research in this area. Specific topics to be covered in the course include:

Basic Objects in Enumeration: Generating Functions, Permutation Statistics, Eulerian Polynomials, Multiset Permutations, q-Binomial Coefficients, The Twelvefold Way, Stirling and Bell Numbers.

Involutions, Determinants.

Inclusion-Exclusion.

Rook Polynomials: Permutations with Restricted Position, q-Rook Polynomials, Zeros of Rook Polynomials and Matching Polynomials.

Posets: Mobius Functions.

The Transfer-Matrix Method.

Required Text:
Enumerative Combinatorics, Volume 1 by Richard P. Stanley (available in the Penn bookstore).
A good auxillary reference for this course I recommend is generatingfunctionology, by Herb Wilf. This can be downloaded for free from his webpage www.math.upenn.edu/~wilf. Both Stanley's and Wilf's books are on reserve in the Physics/Math Library on the 3rd floor of DRL.

Exams: There will be two hour exams and a final exam. Each hour exam counts 25% of your grade, and the final exam counts 40%. There will also be HW assignments (posted at the bottom of this web page) counting 10% of your grade. Hour exam 1 will be on Monday, Oct. 8 during lecture, and Hour exam 2 will be on Monday, Nov. 12, during lecture. The final exam is on Tuesday, December 18 from 12pm-2pm in DRL 4C2.

Midterm 2: Covers material in "Notes on Rook Polynomials" (see link above) and sections 2.3, 2.4, 2.7, 3.1, and parts of 4.5 (Jordan-Holder set, Thm. 4.5.4, Thm. 4.5.8, Thm. 4.5.14; in particular, connections between unitary compositions from rook theory notes and (P,\omega)-partitions of disjoint union of chains.

Important Dates:
Midterm 1: Oct. 8, 12-1:30pm
Last Day to Drop a Course: Friday, Oct. 12
Fall Break: Saturday, Oct. 13 - Tuesday, Oct.16.
Midterm 2: Nov. 12, 12-1:30pm
Thanksgiving Break: Thursday, Nov. 22 - Sunday, Nov. 25
Last Day of Class: Friday, Dec. 7
Final Exam: Cumulative. December 18, Tuesday, 12:00pm-2:00pm in DRL 4C2.

Homework Assignments:
HW1 (due 9/24): Do exercises 1.5, 1.7,1.10, 1.11 from Chapter 1 of "The q,t-Catalan Numbers ...".
HW2 (due 10/10): Do exercises 1.4, 1.5, 1.8 a) and b), 2.1, 2.8 a) thru e) from Stanley (here exercise m.k denotes exercise k of Chapter m).
HW3 (due 10/22) Do exercises 2.15, 2.16 (a) thru (e), 1.9 (a), 1.10, 1.16, 1.31.
HW4 (due 11/5) Try and prove the identity \sum_k {x+k \choose n}t_k(B) = PR(x,B) combinatorially as in the proof in Stanley of \sum_k x(x-1)\cdots (x-k+1)r_{n-k}(B) = PR(x,B). Here t_k(B) is the kth hit number of the Ferrers board B. Also, show how the Jacobi-Trudi identity (Stanley, volume II, p.342), which gives a formula for the Schur fundtion as a detetrminant, is a special case of Theorem 2.7.1 in Stanley.
HW5(due 11/19) Exercises 2.7, 2.10, 2.11 from "Notes on Rook Polynomials", and 4.24 parts (a),(b),(c).
HW6 (due 12/18): Do exercises 3.5, 3.23, 3.45, 3.63.