About this picture:
an = the probability of getting exactly
n heads when you flip a coin 2n times.
Fun exercises:
1. Prove that
2. Show that
3. Let
. Prove that
Show
that
.
.
.
. Set
.
. Lastly, show
Using this picture on the left:
One can show the following is true:
.
Curious to learn more:
See the paper I wrote inspired by Don
Knuth’s Bowdoin visit in Sp.2011
http://www.bowdoin.edu/~ambirika/
CoinFlipProblem.pdf
MATH 258 Syllabus – Fall 2012
(A guide to Prof aBa’s Combinatorics Class)
Instructor:
Prof. Aba Mbirika
Searles 102, Phone number 207-725-3131
Email: ambirika@bowdoin.edu
Webpage: www.bowdoin.edu/~ambirika
Textbook: Applied Combinatorics, Fifth Edition, by Alan Tucker
5th EDITION
(not 6th
Edition)
Course Information:
Lectures: Tuesdays and Thrursdays 1:00pm – 2:25pm in Searles 113
Office Hours: Mondays, Tuesdays, Wednesday and Thursdays 2:30pm – 3:30pm
Course Website: Go to http://blackboard.bowdoin.edu and click on the “Courses” tab.
Final Exam: Wednesday Dec 12th, 2012, at 2:00pm (Room TBD)
Course Content: We will try to cover most of the first eight chapters of Tucker’s book.
Occasionally we might skip sections of the text OR cover material outside of the first eight
chapters. I will occasionally introduce accessible material that is CURRENT RESEARCH in
combinatorial mathematics.
Homework and Take-home Quizzes: For the most part, this is a course in problem
solving. There is a theoretical portion too that will take advantage of your proof-writing
abilities and previous skills learned in Math 200. To learn the material, you need to actually
solve problems. Doing the assigned homework and take-home quizzes is the BEST way to learn
the material. BOTTOMLINE: Doing the exercises is the most important part of the course!
Many of the problems on the exams will be based on problems you will see in HW/Quizzes.
What? Zero credit for the CORRECT answer? Some combinatorics problems are
“obvious” in the sense that one can do the problem in one’s head. This is not the case of more
process-oriented homework/exam exercises. You should SHOW your work or EXPLAIN your
reasoning. An answer (even correct) alone is worth little to no credit. You have been warned!
Objectives and Goals of the Course:
Develop Logical Reasoning in a Problem-Solving Framework. One goal is to develop a
disciplined logical analysis of “word” problems. Such reasoning is the foundation for building
simple mathematical models of problems – models implicit in counting expressions built of
sums and products of binomial coefficients, in generating function models, in recurrence
relation models, or in graph models. A good way to learn this logical thinking is by working
many exercises. Students will use this reasoning often (consciously or unconsciously) in
computer science, operations research, and probability and statistics. However, a logical mind
will serve a person well in ANY field. So, one may view a part of this course as IQ-enrichment.
Secondary Goal. To introduce students to the basic concepts and tools of enumeration and
graph theory: counting methods such as generating functions, recurrence relations, and the
inclusion-exclusion formula, and basic types of graphs, such as planar graphs and trees, and the
uses of graphs in applied mathematics fields. Recurrence relations develop recursive reasoning
(oh so important in computer science), the inclusion-exclusion formula is a grand exercise in
applied set theory, Polya’s formula is a practical introduction to group theory. Finally, it is our
goal to achieve these learning objectives in as appealing a manner as possible, with real-world
problems, card problems, and logical puzzles.
You will be encouraged (without free choice) to collaborate with one another in collaborative
assignments. We hope you will do so too when not “encouraged”. Before coming to class
please read the sections that I will cover. Occasional 5 minute s’prise quizzes might pop up if I
feel people are not looking at the material before class. We will be INTERACTIVE in the class
(translate that as “I will ask you what YOU would do on a certain problem or what YOU think
might work”). MAIN OBJECTIVE: Have fun, enjoy the course, and learn that combinatorics can
be a lot of fun.
SOME RULES / REGULATIONS / REMINDERS
HOMEWORK PRESENTATION: Homework papers must be STAPLED, have NO FRAYED
EDGES (ie. don’t just rip it out your spiral notebook, pleaseeee), the individual cover sheet
(found on blackboard) must be the FIRST page of every assignment. A similar coversheet (also
downloadable on Blackboard) is used for collaborative assignments. Please do not deviate from
these rules. We’ll start with a two-strikes-you’re out foundation. Neatness and presentation
help the grading AND the learning (especially when you can clearly see what you have written).
Pen users – PLEASE USE TAPE WITE-OUT instead of SCRIBBLING OUT a mess of ink. Graders and
instructors abhor large continents of scribble (we call it scratch work, not turn-in homework).
FINAL EXAM: Wednesday Dec 12th, 2012, at 2:00pm (Room TBD). Take home due date TBD.
CLASS CONDUCT: Here are the simple 3 rules:
(1) Try to come to class on time. However if you are running very late, don’t hesitate to
come. Better late than absent! Attendance is closely monitored as you’ll notice.
(2) TURN OFF THE CELL PHONES. Pretend you are on an airplane just before take-off.
(3) Love thy neighbor and all that groovy stuff.
Grades
Midterm 1
Midterm 2
Final Exam
Homework
Quizzes and Graded Group Work
15%
20%
25%
25%
15%
You are expected to participate in class. This “ungraded category” can influence
your final grade if you lie between B+ and A- for instance at semester’s end.
We will have semi-weekly problem sets in addition to sporadic collaborative
problem sets (done in groups in class) and take-home quizzes. Occasionally the
in-class group work may be collected for either grading/ feedback.
Each midterm and final may have both an in-class component and a take-home
component.
Important Point: As you can see in the tentative schedule below, some topics are
skipped in our readings, however, there may be problems assigned from those
sections (eg., I may assign interesting problems from 7.5 as by then, you will be
masters of generating functions for instance making that section an easy selflearner). So do not hold the schedule below as ALL you need to know!
Tentative Class Schedule and Readings
Week 0
Thurs 8/30
Week 1
Tues 9/4
Thurs 9/6
Tues 9/11
Thurs 9/13
Week 2
Week 3
Week 4
Tues 9/18
Thurs 9/20
Tues 9/25
Thurs 9/27
Overview of combinatorics, and
Current combinatorial research problems
1.1: Graph Models
1.2: Isomorphism, 1.3: Edge Counting
1.4: Planar graphs, 2.1: Euler cycles
Common Good Day Sat 9/15
Go out there U-bears and volunteer!
2.2: Hamilton circuits, 2.3: Graph coloring
2.4: Coloring Theorems, 3.1: Tree properties
Family Weekend
Fri 9/28 to Sun 9/30
Introduce me to your family if they come!
Week 5
Tues 10/2
Thurs 10/4
Midterm 1 In-class portion
Midterm 1 Take-home portion due
Fall Vacation
Fri 10/5 to Wedn 10/10
Have fun! And be safe!
Week 6
Tues 10/9
Thurs 10/11
No class as fall vacation is underway.
Prof aBa away. William Silver might fill in?
Week 7
Week 8
Week 9
Week 10
Week 11
Week 12
Tues 10/16
Thurs 10/18
Tues 10/23
Thurs 10/25
Tues 10/30
Thurs 11/1
Tues 11/6
Thurs 11/8
Tues 11/13
Thurs 11/15
Tues 11/20
5.1: Two basic counting principles,
5.2: Simple arrangements and selections
Homecoming Weekend
Fri 10/19 to Sun 10/21
5.3: Arrangements and selections with
repetition, 5.4: Distributions
5.5: Binomial identities, 6.1: Generating
function models
6.2: Calculating coefficients of gen. functions
6.3: Partitions OR 6.4: Exp. gen. functions
Midterm 2 In-class portion
Midterm 2 Take-home portion due
Thanksgiving Break
Wedn 11/21 to Mon 11/26
Gobble Gobble!
Week 13
Week 14
Tues 11/27
Thurs 11/29
Tues 12/4
Thurs 12/6
Week 15
Wed 12/12 2pm
8.1: Counting with Venn diagram, 8.2:
Inclusion-exclusion, and maybe 8.3: Rook
polynomials
Fun New Topic? Class outside in the snow?
Last day of classes is Fri 12/7
Reading Period from Sat 12/8 to Tues 12/11
Final Exam (room TBD)
And Take-home due date TBD
Dates I will be away:
Wednesday October 10th thru Sunday October 14th, 2012 giving a math talk in Seattle. So I will
be absent one day of class (Thur 10/11). There is a high probability that William Silver (CS
instructor) will come on this day to speak about counting legal Sudoku boards. This will be an
excellent introduction to combinatorial counting techniques (which is our next area of study
when I get back from Seattle). Otherwise, there will be some collaborative group assignment to
be done in class that day.