Description: Standard tools of enumerative combinatorics including partitions and compositions of integers, set partitions, generating functions, permutations with restricted positions, inclusion-exclusion, partially ordered sets. Permission of the instructor required to enroll.
Term: Fall 2026
Instructor: Prof. Sarah Brauner, braunsar@sas.upenn.edu
Classes: Tuesday and Thursday from 1:45-3:14pm in DRLB 4C6
Office hours: Tuesdays from 3:45-5:45 and by appointment
Course requirements and grading: There will be three homework assignments that you will complete but not turn in. There will be three in-class (closed note) exams based on these homework assignments (i.e. you will be asked very similar questions to those completed on the homework). You are also expected to come to class, and are strongly encouraged to attend office hours. The grade breakdown is as follows:
25%: Exam 1, based on Homework 1 (Tuesday, September 29)
25%: Exam 2, based on Homework 2 (Tuesday, November 3)
25%: Exam 3, based on Homework 3 (Thursday, December 3)
25%: Class participation
In class attendance and engagement (ask questions, work on problems collaboratively; 3 unexcused absences permitted)
Office hour attendance (at least once per homework assignment)
A note on homework: you are encouraged to work with your other classmates on homework. You are strongly discouraged from using AI to complete assignments; the goal of this graduate course is to prepare you to do research in combinatorics, and the best way to do this is to challenge yourself by thinking deeply about difficult problems. Generating an AI solution and memorizing it for exams will simply not prepare you to be a strong researcher.
The main themes are the course will be:
Generating functions (ordinary, exponential)
Determinantal formulas
Partially ordered sets (posets) and lattices
Along the way, we will see topics such as..
Permutation statistics
Lagrange inversion
Inclusion-Exclusion
Matrix-Tree theorem
Time permitting: Polya theory
Main texts:
R.P. Stanley, Enumerative combinatorics, Vol. I, Cambridge University Press. (Here is the book's errata)
F. Ardila, Algebraic and geometric methods in enumerative combinatorics, Part I, Handbook of enumerative combinatorics
Supplemental text:
B. Sagan, Combinatorics: the art of counting, American Mathematical Society
All problems will be from Enumerative Combinatorics, Volume 1 (colloquially, EC 1). See "A Note about the Exercises" on Page 114 of EC1 for context on problem difficulty.
Homework 1 (Related Exam on Tuesday, September 29):
From EC1, Chapter 1:
[2-] 66, 113
[2] 5, 21, 26, 29, 69
[2+] 175
Homework 2 (Related Exam on Thursday, October 29):
Homework 3 (Related Exam on Thursday, December 3):