Welcome to the Math 28 Web Page

Math 28

Introduction to Combinatorics

Instructor: Prof. Peter Winkler (peter.winkler at dartmouth.edu)

Abstract | Classes | Staff | Textbooks | Grading | News and current assignment | Past assignments | Exams | Honor Code


Final Exam is 8am Tuesday, March 11, in Kemeny 007(!).

Sections of the text covered in the course (roughly): 2.1, 2.2, 2.3, 2.6; Chap 3; 4.5; 5.1, 5.2, 5.6; 6.1, 6.3, 6.4, 6.5; 7.1, 7.2, 7.4; 8.1; Chap 9; 10.2, part of 10.3 and 10.4; 11.1, 11.2, 11.3, 11.5, 11.7; 13.3.


Combinatorics began as the study of counting, e.g., how many ways are there to order an n-element set? With the ascent of the digital computer, combinatorics has taken on vastly more applications and has become a mature and indeed extremely active area of mathematical research. We will learn basic combinatorial techniques and we will apply them to solve problems in design and optimization. The course will take us in many directions including algorithms, probability, graphs, and games; but most importantly it will help us reason mathematically with discrete objects.

Prerequisites: Math 8, placement into Math 11, or permission of the instructor. Dist: QDS.

Here is a (tentative) rough weekly syllabus.

1. Basic Counting

2. Permutations and Combinations

3. Inclusion-Exclusion

4. Recurrence Relations

5. Generating Functions

6. Matchings

7. Basic Graph Theory

8. Networks

9. Designs


Room: Kemeny Hall 108
Lectures: "12" slot, in particular: Monday, Wednesday and Friday 12:30-1:35.
X-hour: Tuesdays (same room) 1:00 pm--1:50pm. Will be used only when so announced in class.


Peter Winkler -- Kemeny Hall 231 / Tel. 6-3468
Office Hours: M 1:45-2:45; W 9:30-10:30; F 1:45-2:45.


Richard Brualdi, Introductory Combinatorics, 5th Edition, Prentice Hall 2010.


Your grade will be based on homework (20%), class participation (10%), two in-class exams (15% each) and final exam (40%).


There will be two in-class hour exams, the first on Monday Jan 27, the second on Monday Feb 17.
Let me know immediately if you might not be able to make it to an exam.
The final exam will be in the regular slot for MWF12 courses, namely 8am on Tuesday, March 11.


Homework will be assigned at each class period, due at the beginning of the next class.


Due Wednesday Jan 8: Read Chapter 1 "for fun" and do Exercise 1.8.3 (that is, Ex. 3 p. 20). (For those of you who don't have the text yet: you are asked to determine whether you can start at one corner of a chessboard and, moving at each turn to an orthogonally adjacent square, visit each square exactly once and end at the diagonally opposite corner.)

Due Friday Jan 10: Read 2.1 (and further, if you like). Do Exercises 2, 4 and 7 of 2.7 (p. 60).

Due Monday Jan 13: Finish reading Chapter 2. Do Exercises 20 and 26 of 2.7 (p. 60).

Due Wednesday Jan 15: Begin reading Chapter 3. Do Exercises 5 and 11 of 3.4 (p. 83).

Due Friday Jan 17: Finish reading Chapter 3. Do Exercises 17 and 23 of 3.4 (pp. 84-85).

Due Tuesday Jan 21: Do Exercises 20 and 27 of 3.4 (pp. 84-85). Note hints on p. 584.

Due Wednesday Jan 22: No written assignment; read about binary relations in Chapter 4.

Due Friday Jan 24: Do #36 and #38, Exercises 4.6 at end of Chapter 4. Note that the text's definition of "antisymmetric" is slightly different from mine; use the text's.

Due Friday Jan 31: Read 5.1-5.3, and 5.6. Do #7 and #30, Exercises 5.7. Use a chain partition (not Sperner's Theorem) to do #30.

Due Monday Feb 3: (b) List the linear extensions of the "N" poset in which a is less than b, c is less than b, and c is less than d. (b) Compute the probability that a is less than d in a random linear extension of this poset.

Due Wednesday Feb 5: Read 6.1-6.3, do #2 and #11 of Exercises 6.7.

Due Monday Feb 10: Exercises 6.7, #15 and #24.

Due Tuesday Feb 11: Exercises 7.7, #13.

Due Wednesday Feb 12: Exercises 7.7, #15.

Due Friday Feb 14: Exercises 7.7, #4 and #11.

Due Friday Feb 21: Exercises 9.4, #1 and #6. Read Sections 9.1 and 9.2.

Due Monday Feb 24: Exercises 9.4, #9 and #12.

Due Wednesday Feb 26: Read Section 9.3, and do Exercises 9.4, #16 and #17.

Due Friday Feb 28: Exercises 9.4, #25. NOTE: there are a couple of typos in this exercise: the lower-right "4" in the matrix should be a "5", and on the last line, the columns should of correspond to a,b,c,d,e,f.

Due Monday Mar 3: Find 12 subsets of size 3 of the set V = {1,2,...,9} which have the property that every pair of elements of V belong to exactly one of your subsets. (In other words, find a BIBD with v=9, k=3 and lambda = 1.)

Due Wednesday Mar 5: Count the number of trees on the vertex set V = {1,2,3,4}.

Due Friday Mar 7: Draw the tree whose Pruefer code is 10,3,2,11,9,4,3,10,3,10.

Honor Code

Students are encouraged to work together to do homework problems. What is important is a student's eventual understanding of homework problems, and not how that is achieved. The honor principle applies to homework in the following way. What a student turns in as a written homework solution is to be his or her own understanding of how to do the problem. Students must state what sources they have consulted, with whom they have collaborated, and from whom they have received help. Students are discouraged from using solutions to problems that may be posted on the web, and as just stated, must reference them if they use them. The solutions you submit must be written by you alone. Any copying (electronic or otherwise) of another person's solutions, in whole or in part, is a violation of the Honor Code.

If you have any questions as to whether some action would be acceptable under the Academic Honor Code, please speak to me, and I will be glad to help clarify things. It is always easier to ask beforehand than to have trouble later!


I encourage any students with disabilities, including "invisible" disabilities such as chronic diseases and learning disabilities, to discuss appropriate accommodations with me, which might help you with this class, either after class or during office hours. Dartmouth College has an active program to help students with disabilities, and I am happy to do whatever I can to help out, as appropriate.

The Student Disabilities Center is located at 318 Wilson Hall, ext. 6-9900, http://www.dartmouth.edu/~accessibility, if you have any questions. Any student with a documented disability requiring academic adjustments or accommodations is requested to speak with me by the end of the second week of the term. All discussions will remain confidential, although the Academic Skills Center may be consulted to verify the documentation of the disability and advise on an appropriate response to the need. It is important, however, that you talk to me soon, so that I can make whatever arrangements might be needed in a timely fashion.