Math 222: Combinatorics

Professor: Jonah Blasiak

Fall 2018

Monday, Wednesday 3:30pm - 4:50pm, One Drexel Plaza 002

Course Description: Select combinatorial topics such as induction, generating functions, inclusion-exclusion, and graph theory. Emphasis on proof writing.
Prerequisites: Math 220.
Office Hours: Korman 275, Tuesday 1pm-2:30pm, Wednesday 1pm-2:30pm.
Problem Session: Korman 241, Tuesday 1pm-3pm.
Textbooks: We will use a combination of the following texts:
  • Discrete Mathematics: Elementary and Beyond, by L. Lovász, J. Pelikán, and K. Vesztergombi (Drexel Library online copy)
  • Introduction to Graph Theory, 2nd Edition, by Douglas B. West
  • Enumerative Combinatorics, vol.1, by R.P.Stanley
  • Grade Breakdown:
  • 30% Homework
  • 30% Midterm
  • 40% Final
  • Grading Policy:
  • A: 80-100%
  • B: 60-80%
  • C: 40-60%
  • D-F: 0-40%
  • Exam Policy: No books or electronic devices are allowed on the midterm or exam. No collaboration is permitted at the midterm or exam. THERE WILL BE NO MAKE-UPS FOR EXAMS.
    The midterm will be in-class on Wednesday, October 24; it will be 80 minutes long.
    Students with special exam-taking requirements or time conflicts should contact me by October 10.
    Homework Policy: You may consult each other and the textbook above. List all people and sources who aided you and whom you aided, and write up the solutions independently, in your own language. It is easy nowadays to find solutions to almost anything online. DO NOT consult such solutions until after turning your homework. Solutions to homeworks will be handed out in class and/or discussed in class. Late homeworks will not be accepted.
    Please visit this site frequently for new information. Updates to the syllabus and reading assignments, homeworks, and practice exams will be posted here as the course progresses.


    Since we are using multiple textbooks, there will be some overlap with the reading assignments. The most important/relevant sources will be listed first.

    Week 1: Sep 24, Sep 26
    Set theory, functions, and the watermelon cutting problem: read the handout Joy of Sets, Section 1.2 and Theorem 1.3.1 of LPV (the textbook Discrete Mathematics: Elementary and Beyond).
    Read the handout Mathematical Hygiene. We will discuss some of these concepts throughout the course as needed.
    Homework 1 due Oct 01

    Week 2: Oct 1, Oct 3
    Pascal's triangle, counting, bijective proofs, inclusion-exclusion: Sections 1.7-1.8, 3.5-3.6, 2.3 of LPV
    Wikipedia article on inclusion-exclusion
    Homework 2 due Oct 10. Note: Homeworks will be due on Wednesdays from now on.

    Week 3: Oct 10
    Inclusion-exclusion, Fibonacci numbers: Sections 6.9, 4.1-4.3 of LPV
    Homework 3 due Oct 17

    Week 4: Oct 15, Oct 17
    Catalan numbers, generating functions: 1.1-1.3, 2.1-2.2 of generatingfunctionology
    Homework 4 due Oct 31 (no homework due October 24 because of the midterm)

    Week 5: Oct 22, Oct 24
    Generating functions continued: Chapter 2 of generatingfunctionology
    Reminder: Midterm Wednesday, October 24. It will be in-class and 80 minutes long. It will cover the material up to and including the October 17 lecture and Homeworks 1, 2, 3 and the first five problems of Homework 4.
    Midterm Practice Problems

    Week 6: Oct 29, Oct 31
    Introduction to graph theory: vertex degrees, trees, paths, cycles: Ch. 1-2 of West or Sections 7.1-7.2, 8.1-8.2, 13.2 of LPV
    Homework 5 due Nov 07

    Week 7: Nov 5, Nov 7
    Introduction to graph theory: vertex degrees, trees, paths, cycles: Ch. 1-2 of West or Sections 7.1-7.2, 8.1-8.2, 13.2 of LPV
    Homework 6 due Nov 14

    Week 8: Nov 12, Nov 14
    Kruskal's algorithm, Hall's matching theorem (following parts of Ch. 2 and 3 of West)
    Homework 7 due Nov 28

    Week 9: Nov 19
    Euler's formula, Platonic solids: Ch. 6.1 of West

    Week 10: Nov 26, Nov 28
    Graph coloring, coloring planar graphs: Ch. 5.1 and 6.3 of West
    Homework 8 due Dec 05

    Week 11: Dec 3, Dec 5
    Fisher's inequality and other linear algebra applications (pages 46-56 of notes by Andras Gyarfas)

    The Final Exam is on Tuesday, December 11, in Academic Building 107, from 3:30pm to 5:30pm. It will cover all the material from class and on the homeworks, with more emphasis on the material from weeks 5-11. The format of the final will be similar to the midterm but 120 minutes instead of 80 minutes.

    Homework Help: Math Resource Center (Korman 247)
    Important University Policies:

    Academic Dishonesty

    Course Drop Policy

    Code of Conduct

    Disability Resources:
    Students requesting accommodations due to a disability at Drexel University need to request a current Accommodations Verification Letter (AVL) in the ClockWork database before accommodations can be made. These requests are received by Disability Resources (DR), who then issues the AVL to the appropriate contacts. For additional information, visit the DR website at, or contact DR for more information by phone at 215.895.1401, or by email at

    Outcomes: Students must understand basic mathematical language including sets and functions, apply mathematical induction, count or enumerate objects using various combinatorial formulas, operate with discrete structures including graphs and permutations, and describe simple algorithms. Students will be comfortable writing short mathematical proofs including proofs by induction, bijective proofs, and proofs in number theory.