Last updated 15 Jan 2026
This course provides a general introduction to this beautiful subject. Students will learn both classical methods for clever counting, as well as how to apply more modern methods to analyze recursions and sequences. Students will also learn fundamental techniques in graph theory. Throughout the course, students will encounter novel challenges that blur the lines between combinatorics and other subjects, such as number theory, probability, and geometry, so that they develop the skills to creatively combine seeming disparate areas of mathematics.
This course is structured around challenge. Lecture topics are hand-picked to reflect the rather advanced ability level of the general CMU student, and consequently, much of the curriculum sequence is original. Homework and exam problems are particularly difficult, and require creative problem-solving rather than application of learned techniques. To encourage students to truly develop these skills, collaboration is encouraged on homework, and exams (which are non-collaborative) will be open-notes.
The only pre-requisite for the course is one of 21-127 Concepts of Mathematics or 21-128 Mathematical Concepts and Proofs.
In order to encourage students to experiment with the concepts taught in class, homework assignments will be given on alternate weeks. They will be due in class on Fridays, at the beginning of lecture. Each assignment will consist of four to five challenging problems, for which the proof or justification of each answer is more important than actual numerical answer.
Since homework is a learning activity, students are welcome to discuss ideas with each other, although collaboration in the writing stage is not permitted. In other words, please do not look at the actual document that another student is handing in. Also, do not use AI to help with the homework at any stage (whether brainstorming or writing the solution).
Other make-up exams will be given only in the case of a documented medical excuse, a university-sanctioned absence (e.g., participation in a varsity sporting event), or a family emergency. However, this must be requested before the official start time of each exam.
Attendance/Participation. Attendance and participation are not part of the course grade, but are strongly encouraged because the curriculum of this class contains many original components.
Academic Integrity and Collaboration. Collaboration as detailed above is permitted only on homework assignments, and not on exams. In the event of academic dishonesty, a score of zero will be assigned, and a communication will be sent to the academic advisor. This policy is motivated by the goal of maintaining a fair environment for all learners.
Late Work. There is no penalty for late homework, as long as it is submitted within 1 week of its original deadline. Any homework submitted after that date receives zero credit. However, students are strongly encouraged to submit homework on time, because this keeps the class in sync for healthy collaboration on homework, and homework is very helpful for exam preparation.
Re-grade Requests. Re-grade requests must be submitted in writing, within one week of receiving the graded exam or assignment. They will be honored, but please note that it is possible for scores to decrease after a re-grade.
Accommodations for students with disabilities. If you have a disability and require accommodations, please contact Catherine Getchell, Director of Disability Resources, 412-268-6121, getchell@cmu.edu. If you have an accommodations letter from the Disability Resources office, I encourage you to discuss your accommodations and needs with me as early in the semester as possible. I will work with you to ensure that accommodations are provided as appropriate.
Statement on Student Wellness. As a student, you may experience a range of challenges that can interfere with learning, such as strained relationships, increased anxiety, substance use, feeling down, difficulty concentrating and/or lack of motivation. These mental health concerns or stressful events may diminish your academic performance and/or reduce your ability to participate in daily activities. CMU services are available, and treatment does work. You can learn more about confidential mental health services available on campus at: http://www.cmu.edu/counseling/. Support is always available (24/7) from Counseling and Psychological Services: 412-268-2922.
| Week | Topic | Book | Work due Fri |
| Week 1
(Mon Jan 12) |
Basic counting; Induction | Chapter 1 | |
| Week 2
(Mon Jan 19) |
Pigeonhole principle; inclusion-exclusion | Chapter 2 | Homework 1 on Wed |
| Week 3
(Mon Jan 26) |
Homework 2 | ||
| Week 4
(Mon Feb 2) |
Binomial coefficients | Chapter 3 | Exam 1 on Fri |
| Week 5
(Mon Feb 9) |
Linear recurrences and matrices | Chapter 4 | Homework 3 |
| Week 6
(Mon Feb 16) |
Homogeneous linear recurrences and generating functions | Chapter 5 | |
| Week 7
(Mon Feb 23) |
More generating functions | Chapter 6 | Homework 4 |
| Spring Break | |||
| Week 8
(Mon Mar 9) |
Introduction to graphs | Chapter 7 | Exam 2 on Fri |
| Week 9
(Mon Mar 16) |
Trees | Chapter 8 | Homework 5 |
| Week 10
(Mon Mar 23) |
Cycles and spanning trees | Chapter 9 | |
| Week 11
(Mon Mar 30) |
Ramsey theory | Chapter 10 | Homework 6 |
| Week 12
(Mon Apr 6) |
Matchings and bipartite graphs
(No class Fri Apr 10) |
Chapter 11 | |
| Week 13
(Mon Apr 13) |
Chromatic number | Chapter 12 | Exam 3 on Mon |
| Week 14
(Mon Apr 20) |
Planar graphs | Homework 7 | |
|
Final exam
TBA |
Room TBA | ||
| You are visitor number since 11 January 2026. |
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