Math 465: Introduction to Combinatorics

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.

Homework Assignments

  1. Homework 1 due [2026-01-15 Thu].
  2. Homework 2 due [2026-01-22 Thu].

Notes

  1. Basic counting
  2. Binomial and multinomial coefficients
  3. Generating functions

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]  
[2026-01-15 Thu] Binomial identities and binomial theorem 2, pages 4--7 [B, 4]  
[2026-01-20 Tue]        
Date published: Wednesday, January 7, 2026