University of Connecticut College of Liberal Arts and Sciences
Department of Mathematics
Math 3250 Spring 2013 (Roby) Tom Roby's Math 3250 Home Page (Spring 2013)

Questions or Comments?

Class Information

COORDINATES: Classes meet Tuesdays and Thursdays 12:30–1:45 in MSB 215. The registrar calls this Section 001.


TEXT: Miklos Bona: A Walk through Combinatorics, 3rd Ed. Errata for this edition are available here.

WEB RESOURCES: The homepage for this course is I may list some others resources below.

SOFTWARE: For generating data to understand properties of integers, or to solve equations, computer software can be a valuable tool. Doing some computations by hand is generally good for learning, but having software that can do bigger computations or check your work is very useful. One free source on the web is WolframAlpha. For a full-fledged programming environment, check out the free open-source computer algebra system called Sage, which has a number of commands that do specialized computations in combinatorics.

GRADING: Your grade will be based on two midterm exams, one final exam, and homework.

The breakdown of points is:

HW Midterms Final
20% 25% each 30%

MIDTERM EXAMS: Will cover all the material up to that point in the term. They are currently scheduled for THURSDAY 28 FEBRUARY 2013 and THURSDAY 25 APRIL 2013 during our usual class meeting. Please let me know immediately if you have a conflict with those dates. There are no makeup exams.

HOMEWORK: Homework will be assigned most weeks, and should be attempted by the following TUESDAY, when I will be happy to answer questions or provide hints. It will generally be due Thursdays at the start of class. Since I may discuss the homework problems in class the day they are due, late assignments will be accepted only under the most extreme circumstances. (Please let me know as soon as possible if you find yourself with a situation that might qualify.) The lowest written homework score will be dropped in any event.

You may find some homework problems to be challenging, even frustrating, leading you to spend lots of time and effort working on them. This is a natural part of doing mathematics, faced by everyone from school children to top researchers. I encourage you to work with other people in person or electronically. It's OK to get significant help from any resource, but in the end, please write your own solution in your own words. Copying someone else's work without credit is plagiarism and will be dealt with according to university policy. It also is a poor learning strategy.

ACADEMIC INTEGRITY: Please make sure you are familiar with and abide by The Student Code governing Academic Integrity in Undergraduate Education and Research. For quizzes and exams you may not discuss the material with anyone other than the instructor or offical proctor, and no calculators, phones, slide rules or other devices designed to aid communication or computation may be used.

ACKNOWLEDGEMENTS: Many people have influenced my approach to this material, particularly Richard Stanley, Gian-Carlo Rota, and many of my research collaborators.

CONTENT: Combinatorics is related to the word "combinations", and much of the field deals with counting problems ("enumeration"). More broadly, combinatorics considers discrete structures, such as graphs, (as opposed to the continuous structures considers in calculus and analysis) and questions that go beyond simply counting the number of possibilities. With the advent of computers in the second half of the last century, such questions have become increasingly important in applications as well as more managable to solve. Specifically, we plan to go through the following topics from the text:

  • Pigeon-Hole Principle;
  • Permutations & Combinations;
  • Binomial Theorem;
  • Partitions:
  • Sieve methods (Principle of Inclusion-Exclusion);
  • Generating functions;
  • Graph theory;
  • Trees;
  • Matchings;
  • and Partially-ordered sets ("Posets").

ACCESSIBILITY & DISABILITY ISSUES: Please contact me and UConn's Center for Students with Disabilities as soon as possible if you have any accessibility issues, have a (documented) disability and wish to discuss academic accommodations, or if you would need assistance in the event of an emergency.

LEARNING: The only way to learn mathematics is by doing it! Complete each assignment to the best of your ability, and get help when you are confused. Come to class prepared with questions. Don't hesitate to seek help from other students. Sometimes the point of view of someone who has just figured something out can be the most helpful.

Especially for learning this subject, it is important to do lots of problems—many more than I could reasonably assign you to hand in. The text has a number of problems ("Exercises") with solutions for you to try to test your understanding.

READING INCENTIVE: When you prepare for class, you may fill up to one side of an 8.5 x 11-inch sheet of paper with any notes from the assigned reading you wish. I will collect these, and return them to you to use on the midterms or final as crib sheets. Make sure you put your name on these! Since this is an incentive for you to come to class prepared, I will only accept notes on previously covered sections if you had an excused absence.

CLASSTIME: We will sometimes spend classtime doing things in groups, presenting mathematics to one another, or having interactive discussions. There will not be time to "cover" all material in a lecture format so you will need to read and learn some topics on your own from handouts, web sources, or otherwise.

SCHEDULE: The following is a the start of a tentative schedule that I will update throughout the semester. If you have a religious observance that conflicts with your participation in the course, please meet with me within the first two weeks of the term to discuss any appropriate accommodations.

Date Section Topics HW, etc.
1/22T § 1.1–2 Pigeon-Hole Principle HW #1: Ch. 1: #22, 26, 31; Ch. 2: #18, 29, 33
1/24R § 2.1–2 More PHP & Mathematical Induction Notes on Chapter 1 due;
1/29T § 3.1–2 Permutations & Words HW #2: Ch. 3: #27, 33, 34, 35, 37, 38, 41, 49
1/31R § 3.3 Choice problems HW #1 due.
2/5T § 4.1 Binomial Coefficients & Theorem HW #3: Ch. 4: #29, 39, 53 [no comp.]; Ch. 5: #23, 35; (E1)
2/7R § 4.2–3 Binomial& Multinomial Theorems HW #2 due.
2/12T § 5.1,3 Compositions and partitions of integers HW #4: Ch. 5: #21, 24, 26, 28; (E2)
2/14R § 5.2–3 Integer partition identities & Set Partitions HW #3 due.
2/19T § 5.2 Stirling numbers & Twelvefold Way HW #5: Ch. 5: #34 (only k=1); Ch. 6: #27,31,43,49,(E4),(E5)
2/21R § 6.1–2 Permutations HW #4 due.
2/26T Ch. 1–6 Catchup & Review Day Do Practice Midterm 1 by today
3/5T § 6.1–2 More Permutation Cycles HW #6: Ch. 7: #16,27,36; Ch. 8: #23,25,27,28,29,35,(E9)
3/7R § 6.1–2 More Stirling Numbers;
3/12T § 7.1–2 Sieve Methods, Derangements HW #5 due
3/14R § 7.1–2 More Sieve Methods
3/26 T § 8.1 Generating Functions Midterm Exam 1 Rewrites due (+(E3))
3/28R § 8.1–2 More Generating Functions
4/2T § 8.2 Generating Functions for partitions HW #6 due
4/4R §8.1 Catalan Numbers Over 200 intepretations available here and here
4/9T § 8.2–3 Still more generatingfunctionology HW #7: Ch8: #32,34,37; (E 7–11)
4/11R § 8.3 Exponential Formula
4/16T Applications of GF to partitions, etc.
4/18R § 9.1–2 Graph Theory & Eulerian Cycles HW #7 due
4/23T Ch. 6–8 Catchup & Review Day HW #8: Ch. 9: #24,30,34,41, (E12), (E13)
4/25R § 10.1–2 Trees
4/30T § 10.3–4 Counting labeled trees
5/2R Ch. 1–10 Matrix-tree Thm HW #8 due

Web Resources

  • The Wikipedia page on combinatorics has lots of information and links to longer articles on various subfields.

Anonymous Feedback

Use this form to send me anonymous feedback or to answer the question: How can I improve your learning in this class?  I will respond to any constructive suggestions or comments in the space below the form. 

Feedback & Responses

  1. Professor, I am curious about Tuesday's [26 February's] class. In terms of studying for the exam, will there be new material on Tuesday that we will also be responsible for on Thursday? I feel that review would be very helpful at this point.

    Thanks for mentioning this. I generally never test on material that was first mentioned the previous class. Tuesday will be a catchup and review day, and any new material will not be tested before the second midterm. Please bring any questions you have about the practice midterm (whose solutions are now posted) or about anything we've covered in the course so far.

  2. Hello professor, I have tried all the problems on our practice exam during the weekend, fortunately I have worked most of them out, BUT I really don't think I can finish them in 1 hour and 15 minutes! I remember I spent at least 2 hours on it. So could you please make our in-class midterm shorter than this one? I'll be really appreciated. Thank you.

    Practice midterms are sometimes longer than the actual exam just to give a better sense of the range of possible topics. I definitely plan to make the actual midterm shorter or otherwise more manageable (e.g., give a choice of problems).


    Here are a few extra HW problems. (This file may be updated during the semester.)

    Here are some other handouts:

    Back to my home page.

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