Spring 2023

Tuesdays and Thursdays, 10:10am - 11:25am
328 Milbank Hall (Barnard)

Dave Bayer
Office Hours

Directory of Classes | Spring 2023 Mathematics | MATH BC2006
Student Services Online


This is an introductory course in combinatorics, with a focus on counting problems. As there are many methods that have been developed in calculus and analysis for measuring continuous quantities, there are also many methods that have been developed in combinatorics for counting finite sets. We will survey these methods, with a focus on problem solving.


Course grades will be determined by three take-home exams. The first two exams will be challenging. The third exam will be primarily review, and will serve as our final:

There will be no class on exam days.


Our textbooks are available freely online to Columbia University affiliates, mostly through SpringerLink.

These books overlap in content, with varying styles and levels. As we study each topic, please work with the book(s) that you prefer.

How to choose? This is personal. For me, books that are too low level think that talking too much makes math easier. That isn’t my experience.

On the other hand, books that are too high level can be hard to read. The sweet spot for me is a book that is considered high level because of its depth, but is extraordinarily clear with no wasted words. Go in with purpose, knowing what you’re looking for. This is why one learns faster in grad school than as an undergraduate; one reads with a shopping list. From this perspective, Aigner is a gem:

The following are good introductory expositions:

The following are also of interest, but don’t match our syllabus as closely:

For the curious, here is the definitive graduate text on enumeration:

The On-Line Encyclopedia of Integer Sequences

This is a fantastic resource. Has anyone else seen before the sequence of integers that you’re studying? This is “Google” for counting problems.

Previous semesters

A complete set of video lectures from Spring 2021 are available to students registered in this course:

For convenience, here are all class notes and exam solutions from Spring 2021 and Spring 2022, combined into single files:

Here are our exams and solutions:

One can find old exam solutions, and other useful resources, in my past course web pages:

I have also lead undergraduate seminars on this topic, in several recent years:


This calendar gives our schedule of classes and exams. There will be no class on exam days:

Follow the Week 1, Week 2… links for class summaries, further reading, and blackboard photographs.

Monday Tuesday Wednesday Thursday Friday
16 Jan 17   Week 1    18 19 20
23 Jan 24   Week 2    25 26 27
30 Jan 31   Week 3    1 Feb 2 3
6 Feb 7   Week 4    8 9 10
13 Feb 14   Week 5    15 16   Exam 1    17
20 Feb 21   Week 6    22 23 24
27 Feb 28   Week 7    1 Mar 2 3
6 Mar 7   Week 8    8 9 10
13 Mar 14 15 16 17
20 Mar 21   Week 9    22 23 24
27 Mar 28   Week 10    29 30 31
3 Apr 4   Week 11    5 6   Exam 2    7
10 Apr 11   Week 12    12 13 14
17 Apr 18   Week 13    19 20 21
24 Apr 25   Week 14    26 27   Exam 3    28
1 May 2 3 4 5

source links validate