This is an introductory combinatorics class at the undergraduate level. Combinatorics is the study of finite mathematical objects, including their enumeration, structural properties, design, and optimization. Combinatorics plays an increasingly important role in various branches of mathematics and in numerous applications, including computer science, statistics and statistical physics, operations research, bioinformatics, and electrical engineering. The first half of the course will cover the fundamentals of enumerative combinatorics. The second half will cover the fundamentals of graph theory.

Potential topics include: basic counting, binomial and multinomial coefficients, binomial and multinomial theorems, multiplication principle for generating functions, Stirling numbers of the first and second kind, linear recurrences with constant coefficients, discrete calculus, Catalan numbers, ballot sequences, triangulations and rooted trees, inclusion-exclusion principle, partitions, graphs and Betti numbers, planarity, Eulerian walks, pigeonhole principle, Erdős-Szekeres theorem, Ramsey numbers, chromatic numbers and chromatic polynomials, flows in networks, matchings, vertex covers, Menger’s theorem, Kőnig’s theorem, partially ordered sets, Sperner’s theorem, Mirsky’s theorem, Dilworth’s theorem, Hamiltonian cycles, Gray codes, Eulerian walks, and De Bruijn sequences.

Syllabus

The syllabus is available here.

Important Dates

• Quiz: [2026-01-20 Tue]
• Last day to add/drop without “W”: [2026-01-27 Tue]
• Exam 1: [2026-02-10 Tue]
• Exam 2: [2026-03-17 Tue]
• Last day to withdraw from course with “W”: [2026-03-20 Fri]
• Exam 3: [2026-04-16 Thu]

Office Hours (subject to change)

• Tuesdays, 12pm–1pm
• Wednesdays, 1:15pm–2:15pm
• Thursdays, 1:15pm–2:15pm

Textbook

There is no textbook for this course.

References

• [B] A Walk Through Combinatorics by Miklós Bóna, 2017. UM access.
• [GKP] Concrete Mathematics: A Foundation for Computer Science, 2nd Edition by Ronald L. Graham, Donald E. Knuth, and Oren Patashnik, 1994. UM access.
• [L] Combinatorics by Nicholas Loehr, 2017. UM access.

Homework Assignments

1. Homework 1 due [2026-01-15 Thu].
2. Homework 2 due [2026-01-22 Thu].
3. Homework 3 due [2026-01-29 Thu].
4. Homework 4 due [2026-02-05 Thu].
5. Homework 5 due [2026-02-19 Thu].
6. Homework 6 due [2026-02-26 Thu].
7. Homework 7 due [2026-03-12 Thu].

Notes

1. Basic counting
2. Binomial and multinomial coefficients
3. Introduction to generating functions
4. Permutations and permutation statistics
5. Stirling numbers of the first and second kinds
6. Linear recurrences
7. Catalan numbers
8. Inclusion-Exclusion
9. Integer partitions
10. Graphs and invariants
11. Graph planarity

List of Lectures

Date	Topics	Class Notes	Additional Reading	Additional Materials
[2026-01-08 Thu]	Basic counting	1	[B, 3.1--3.2]
[2026-01-13 Tue]	Binomial and multinomial coefficients	2, pages 1--3	[B, 3.3], Stars and bars
[2026-01-15 Thu]	Binomial identities and binomial theorem	2, pages 4--7	[B, 4]
[2026-01-20 Tue]	Generating functions and quiz	3, pages 1--2	[L, 5.1--5.4]
[2026-01-22 Thu]	Generating functions, permutations and permutation statistics	rest of 3 + 4	[B, 6.1]
[2026-01-27 Tue]	Stirling numbers of the first and second kinds	5	[B, 5.2 & 6.1], Wiki	Youtube video
[2026-01-29 Thu]	Linear recurrences with constant coefficients	6, pages 1--4	Wiki
[2026-02-03 Tue]	More linear recurrences, Catalan numbers	rest of 6 + 7, page 1
[2026-02-05 Thu]	More Catalan numbers	7, pages 2--4		Youtube video
[2026-02-10 Tue]	Exam 1
[2026-02-12 Thu]	Binary trees, Inclusion-Exclusion	rest of 7 + 8, pages 1--2	[B, 7.1]
[2026-02-17 Tue]	Inclusion-Exclusion, derangements, more permutations	rest of 8	[B, 7.2]
[2026-02-19 Thu]	Integer partitions and q-binomial coefficients	9	[B, 5.3]
[2026-02-24 Tue]	Intro to graphs	10, pages 1--4	[B, 9.1, 9.4, 10.3]
[2026-02-26 Thu]	Betti numbers and planar graphs		[B, 12]
[2026-03-03 Tue]	Break
[2026-03-05 Thu]	Break

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Date published: Wednesday, January 7, 2026