Math 273: Combinatorics I

(Graduate combinatorics, 1st semester)

Fall 2026


Instructor:

 Sam Hopkins (call me "Sam")
Office: Annex III (Graduate School) - #220
E-mail: sam.hopkins@howard.edu

Classes:

 Mon-Wed, 11:10am-12:30pm
Office: Annex III (Graduate School) - #224

Office hours:

 By appointment (email me to set up a time)

Course content:

 This is the 1st semester of graduate combinatorics.
We will study basic combinatorial objects (subsets, multisets, permutations, set/number partitions, compositions, graphs, trees, etc.), their enumeration, and additional structures they carry (such as partial orders). Roughly speaking we will cover the following:
  • Generating functions (ordinary and exponential)
  • Basic objects:
    • integer partitions and compositions,
    • subsets and multisets,
    • permutations,
    • set partitions,
    • the Catalan families
  • The 12-fold way
  • Combinatorial statistics and q-analogs:
    • q-binomial and q-multinomial coefficients,
    • permutation statistics (Mahonian and Eulerian)
  • Counting with signs:
    • the principle of inclusion-exclusion,
    • sign-reversing involutions
  • Determinantal/trace formulas:
    • matrix-tree theorem,
    • transfer matrix method
  • Partially ordered sets and lattices:
    • distributive lattices, Birkhoff's Theorem,
    • Möbius functions and Möbius inversion
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.

Prerequisites:

 Calculus, linear algebra, undergraduate algebra (groups, rings, fields)

Main texts:

 R.P. Stanley, Enumerative combinatorics, Vol. I, 2nd ed.
F. Ardila, Algebraic and geometric methods in enumerative combinatorics, Part 1.
Problems will come from Stanley, but the lectures will more closely follow Ardila.

Other nice sources:

 H. Wilf, generatingfunctionology.
B. Sagan, Combinatorics: the Art of Counting.

Syllabus:

 Click here for the class syllabus.

Class notes:

 The following notes are from the Fall 2021 iteration of the class, which we're closely following:
Batch 1; [more to be posted]
(Change "_bw.pdf" to "_gray.pdf" for grayscale versions.)]

Grading:

 Grading for the course will be based on Homeworks, a Midterm Exam, and the Final Exam. The grading scheme is:
  • HWs = 50% of grade
  • Midterm = 25% of grade
  • Final = 25% of grade
There will be 4 homework assignments for the semester. There will be an in person midterm exam, about halfway through. There will be an in person final exam, during finals week.
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. Collaboration on the exams is not allowed. Beyond that, I expect you to show up to class and be engaged. Grades will be posted to the Canvas site.

Assignments:

 [To be posted]