Math 222: Combinatorics

Professor: Jonah Blasiak

Fall 2017

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 4pm-5:30pm, Wednesday 1pm-2:30pm.
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
  • Grading Scheme:
  • 30% Homework
  • 30% Midterm
  • 40% Final
  • Grade Distribution: Grades will curved so that approximately the top 50% of the class receive A's and the bottom 50% receive B's. Lower grades will be reserved for students generally not keeping up with the class, missing homeworks, quizzes, and not coming to class.
    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 October 25; it will be 80 minutes long.
    Students with special exam-taking requirements or time conflicts should contact me by October 11.
    Quiz Policy: Quizzes will be given in class most Wednesdays, and will be similar to some of the homework questions due that day. They will be about 15 minutes long. There will be about 8 quizzes total and the lowest quiz grade will be dropped to compute the quiz grade. Same rules as the exams apply. THERE WILL BE NO MAKE-UPS FOR QUIZZES.
    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.
    Homework Help: Math Resource Center (Korman 247)
    Important University Policies:

    Academic Dishonesty

    Disability Resources

    Course Drop Policy

    Code of Conduct

    Please visit this site frequently for new information. Updates to the syllabus and reading assignments, homeworks, homework solutions, 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 25, Sep 27
    Set theory and functions: 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 04

    Week 2: Oct 2, Oct 4
    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 11

    Week 3: Oct 11
    Inclusion-exclusion, Fibonacci numbers: Sections 6.9, 4.1-4.3 of LPV
    Homework 3 due Oct 18

    Week 4: Oct 16, Oct 18
    Catalan numbers, generating functions: Section 3.1 of LPV
    Homework 4 due Nov 01 (no homework due October 25 because of the midterm)

    Week 5: Oct 23, Oct 25
    Generating functions continued: Chapter 2 of generatingfunctionology

    Week 6: Oct 30, Nov 1
    Introduction to graph theory: vertex degrees, trees, paths, cycles: Sections 7.1-7.2, 8.1-8.2, 13.2 of LPV or Ch. 1-2 of West
    Homework 5 due Nov 08

    Week 7: Nov 6, Nov 8
    Introduction to graph theory: vertex degrees, trees, paths, cycles: Sections 7.1-7.2, 8.1-8.2, 13.2 of LPV or Ch. 1-2 of West
    Homework 6 due Nov 15

    Week 8: Nov 13, Nov 15
    Kruskal's algorithm, Hall's matching theorem (following parts of Ch. 2 and 3 of West)
    Homework 7 due Nov 29

    Week 9: Nov 20
    Euler's formula, Platonic solids: Ch. 6.1 of West

    Week 10: Nov 27, Nov 29
    Graph coloring, coloring planar graphs: Ch. 5.1 and 6.3 of West
    Homework 8 due Dec 06

    Week 11: Dec 4, Dec 6
    Fisher's inequality and other linear algebra applications (pages 46-56 of notes by Andras Gyarfas)

    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.