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To introduce students to the fundamental ideas of enumerative
combinatorics and basic graph theory. The topics in this course will be
accessible to all students who have completed the calculus sequence and who have
taken or are concurrently enrolled in Math 360 or Computer Science 220.
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To develop the ability to write combinatorial proofs.
Combinatorial proofs vary greatly from algebraic proofs in that equations and
formulas are seldom used in an elegant combinatorial proof. Instead,
combinatorialists appeal to basic counting arguments to prove complicated
formulas. To facilitate the development of proof-writing abilities,
students will often be asked to work in small groups in class to construct the
proof of a theorem or to solve a set of problems and then present their
solutions to the class.
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To encourage students to make conjectures, support those
conjectures with intuition and reasoning, and then try to prove their
conjectures. This goal is intended to introduce students to the process of
research mathematics through which mathematicians make educated conjectures
based on intuition and related knowledge and then attempt to prove those
conjectures using known theorems, lemmas and definitions.
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To help students see that mathematics
is a creative endeavor in which it is okay to be wrong, but it is important to
keep trying! Research mathematics can often be frustrating as conjectures
may repeatedly be proven incorrect, but a good mathematician makes changes and
adjustments to those conjectures and continues to try to prove them or disprove
them. Students will regularly be asked to volunteer conjectures or
solutions in class. The class atmosphere will be a welcoming environment in
which students will feel comfortable being wrong and will learn from each
other's mistakes and misconceptions.
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To challenge students academically by setting
high standards for coursework and by encouraging students to meet those
standards. The standards for the course will be clearly defined and you will be
provided with the necessary tools to succeed in the course.
Topics
Covered in this Course
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During the semester we will cover Chapters 1, 2, and 5-9. The course will
begin with Chapter 5 and the basic counting methods and theorems for
combinations, selections, arrangements and permutations. Students will
learn to write basic combinatorial proofs for a number of binomial
identities. In Chapter 6, students will learn to construct basic and
exponential generating functions to solve combinatorial problems. The
ideas of partitions and Ferrers diagrams will be introduced in this chapter
and students will be expected to write combinatorial proofs of results
involving these ideas. In Chapter 7, students will learn to write
recurrence relations and to solve them using the appropriate generating
functions. Students will solve both linear, homogeneous and inhomogeneous
recurrence relations. In Chapter 8 students will learn to count using Venn
diagrams and utilizing the Principle of Inclusion-Exclusion. Students will
study rook polynomials for normal boards and boards with restricted
positions. In Chapter 9, the ideas of equivalence and symmetry groups will
be introduced and students will learn the famous Burnside’s Theorem.
Students will learn to construct a cycle index and will learn Polya’s
Enumeration Formula for the number of colorings of a set S. In the latter
part of the course, we will cover chapters 1 and 2 which introduce students
to the basic ideas of graph theory, including the ideas of Eulerian and
Hamiltonian circuits and graph coloring theorems. In addition to these
topics from the book, we will also study the Catalan numbers and the related
Catalan bijections. Notes will be proved for these topics.
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Homework assignments will be given out daily and will also be posted here. If you miss a class, be sure to check this web page for any changes to the homework!
 Assignment
1 - Due Thursday, January 18th
- Section 5.1: 1, 3, 5, 6, 7, 8, 12, 14, 22, 23, 27, 28, 29,
Challenge Problem #41
- Section 5.2: 3, 5, 7, 8, 10, 12, 13, 17, 21, 24, 25, 26, 30
Assignment
2 - Due Friday, January 26th
- Section 5.3: 2, 3, 5, 8, 10, 11, 12, 19, 20, 22, 30
Challenge Problem #26
- Section 5.4: 1, 2, 4, 5, 8, 10, 12, 16, 18, 19, 20, 22, 26, 30, 39
Challenge Problem #64
Assignment
3 - Due Friday, February 2nd
- Section 5.5: 3bcd, 8, 9a, 14abcdef, 19 Challenge Problem #16
- Section 6.1: 2ac, 3, 4, 5, 8, 10, 14, 17, 23
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MAKE-UP POLICY
- Late Assignments - Homework is due at the BEGINNING of each class period and will not be accepted late.
Anyone coming to class more than 10 minutes past the start of class is considered late and their homework
will not be accepted. Exceptions will not be made. You may still have your late homework graded for your
own personal benefit, but the score will not count towards your homework score. If you turn in an assignment
to the instructor's box (located in the main office), have a secretary date, time and initial the assignment.
DO NOT put assignments underneath my office door. DO NOT submit any assignment electronically or on disk.
- Missed Examinations - There are NO MAKEUP EXAMS unless arrangements have been made prior to the
examination and only as a result of severe illness (must be documented) or other significant reason documented
and approved by the professor. ALL make-up exams, regardless of reason, will be an oral examination,
administered by the professor in her office and lasting approximately one hour.
PLAGIARISM POLICY
- Cheating or plagiarism on a test or other assignment will result in automatic failure on that assignment
and possible failure in the course. Students suspected of cheating will be referred to the Academic Ethics
Committee and face penalties up to expulsion from the University. It is considered cheating on homework to
copy someone else's solutions. Plagiarism on a written assignment is considered to be taking more than 10%
of another's words, sentences, or written material without properly documenting and identifying the source.
DISABILITY SERVICES-
Any student
with a documented disability (physical, learning or psychological) needing
academic accommodations should contact the Disability Services Office (Main
Campus, Tyler Campus Center 264, x6500) as early in the semester as
possible. All discussions will remain confidential. Please visit
http://www.pepperdine.edu/disabilityservices/ for additional
information.
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