Math 222: Combinatorics

Professor: Jonah Blasiak

Fall 2023

Monday, Wednesday, Friday 12:00pm - 12:50pm, Pearlstein Business Center 207
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 241, Monday 2pm-3pm, Tuesday 4:30pm-6:00pm.
Problem Session: Korman 241, Tuesday 3:30pm-5:30pm.
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
  • generatingfunctionology by Herbert Wilf
    • Grade Breakdown:
  • 10% Quiz
  • 20% Midterm
  • 30% Homework
  • 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 quiz, midterm, and exam. No collaboration is permitted at the quiz, midterm, and exam. THERE WILL BE NO MAKE-UPS FOR THE QUIZ, MIDTERM, and EXAM. Students with special exam-taking requirements or time conflicts should contact me by October 11.

    Quiz: There will be one 25 minute quiz in class on Wednesday, October 25. It can be thought of as practice for the midterm.

    Midterm: there will be one 50 minute midterm in class on Wednesday, November 1st.

    Homework Policy: You may consult each other and the textbooks 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 (https://www.math.drexel.edu/~jblasiak/Combinatorics222Fall2023.html) frequently for new information. Updates to the syllabus and reading assignments, homeworks, and practice exams will be posted here as the course progresses.

    Syllabus

    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 27, Sep 29
    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 04

    Week 2: Oct 02, Oct 04, Oct 06
    Pascal's triangle, counting, bijective proofs: Sections 1.7-1.8, 3.5-3.6 of LPV
    Homework 2 due Oct 11.

    Week 3: Oct 11, Oct 13
    Inclusion-exclusion, Fibonacci numbers: Sections 2.3, 6.9, 4.1-4.3 of LPV
    Wikipedia article on inclusion-exclusion
    Homework 3 due Oct 18.

    Week 4: Oct 16, Oct 18, Oct 20
    Fibonacci numbers, generating functions: 1.1-1.3, 2.1-2.2 of generatingfunctionology
    Homework 4 due Oct 25.

    Week 5: Oct 23, Oct 25, Oct 27
    Generating functions continued: 1.1-1.3, 2.1-2.2 of generatingfunctionology
    Quiz: Wednesday, October 25 in class.
    Homework 5 due Nov 08

    Week 6: Oct 30, Nov 01, Nov 03
    Catalan numbers, generating functions: Chapter 2 of generatingfunctionology
    Midterm: Wednesday, November 1st in class.

    Week 7: Nov 06, Nov 08, Nov 10
    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 15

    Week 8: Nov 13, Nov 15, Nov 17
    Trees, Kruskal's algorithm: Ch. 2 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, Dec 01
    Graph coloring, Coloring planar graphs: Ch. 5.1 and 6.3 of West
    Homework 8 due Dec 06

    Week 11: Dec 04, Dec 06, Dec 08
    Hall's matching theorem: Ch. 3 of West

    The Final Exam is on Thursday December 14, in PEARL 207 (our normal classroom), from 10:30am to 12:30pm. It will cover all the material from class and on the homeworks, with more emphasis on the material from weeks 6-11. The format of the final will be similar to the midterm but 120 minutes instead of 50 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 drexel.edu/oed/disabilityResources/overview/, or contact DR for more information by phone at 215.895.1401, or by email at disability@drexel.edu.
    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 and bijective proofs.