MATH 630-600. Enumerative Combinatorics

Fall, 2008, MWF 1:50- 2:40p.m., HELD 122


    Instructor: Dr. Catherine Yan

    Office: Milner 220

    Telephone: (979) 862-4476

    E-mail: cyan@math.tamu.edu,

    Class: MWF 1:50--2:40, HELD 122

    Office hours: Monday, 12:00p.m.--1:30 p.m. or by appointment

    Test Schedule:

    Midterm: October 17, 2008, take-home exam, due on Monday, October 20

    Final Exam: Take-home. December 9, 2008,Tuesday. Due on December 10, 2pm.

    Project: due on December 1st, 2008.

    Level : Graduate

    Prerequisites : Math 302 (Discrete Mathematics), or Math 431 (Structures and methods in Combinatorics), or equivalent will be sufficient.

    Course Description : This course will be an introduction at graduate level to the fundamental ideas and results of combinatorics. The course moves quickly but does not assume prior study in combinatorics. It is intended for graduate students from mathematics or related areas wanting a good one-semester background in fundamental and applicable discrete mathematics. It also provides solids preparation for advanced combinatorics and for various courses in computer sciences. The course will cover structures and methods of combinatorics, including enumerative techniques, sieve methods, partially ordered sets, and generating functions. One emphasis of the course will be bijective proofs illustrating the frequent, and often surprising, interrelations between these structures. The prerequisite for this course is an undergraduate discrete mathematics course or permission of the instructor.

    Text: : Enumerative Combinatorics, Volumn I, by Richard P. Stanley, published by Cambridge University Press, 1997.

    Grading: Tne plan is to cover the first three chapters of Stanley's textbook, on basic counting, sieve methods, and partially ordered set, together with a chapter on exponential generating functions. Extensive homework assignments will be given bi-weekly. As exercise is an important part of combinatorics, anyone who doesn't hand in homework fails the course automatically. No late home work will be accepted except for university-approved excuses.

    The grade will be determined by four parts: homework assignment, a take-home middle term exam, an in-class final exam, and a project. The weights of each of these are as follows

    Midterm 25%
    Project 25%
    Homework 25%
    Final 25%
    The stardard cutoffs for letter grades will be used:
    A 90/100; B 80/100; C 70/100; D 60/100.

    Homework Assignment

    Homework 1 Due on Sep. 10th, in class.
    Homework 2 Due on Sep. 24th, in class.
    Homework 3 Due on Oct. 8th, in class.
    Homework 4 Due on Oct. 22th, in class.
    Homework 5 Due on Nov. 5th, in class.
    Homework 6 Due on Nov. 19th, in class.
    Homework 7 Due on Dec. 3rd, in My office, Milner 220.

    Description of Project

    See the pdf file here, posted on September 3, 2008.

    Lecture Schedule

    (updated regularly)
    § 1. Baisc Counting

    Week 1 .
    A. How to count, formal power series
    B. Limit of formal power series, permutation and combination of set
    C. Composition, permutation and combination of multiset.

    Week 2. Permutation statistics.
    A. Cycle structures
    B. Stirling number of the first kind. Inversions.
    C. Inversions and descents

    Week 3.
    A. Tree representations for permutations
    B. q-analog, and Gaussian coefficients
    (School canceled on Friday.)

    Week 4
    A. Inversions for multiset and lattice path couting.
    B. Integer partitions
    C. Twelvefold Way and Set partitions

    § 2.Sieve Methods

    Week 5
    A. Stirling number of the second kind.
    B. Difference operators, and basic principle of inclusion-exclusion.
    C. The algebraic formula for PIE

    Week 6.
    A. Descents of permutations
    B. Permutations with restricted positions
    C. Ferrer boards

    Week 7.
    A. Reflection principles and Ballot problem
    B. Ballot with many candidates
    C. Non-crossing lattice paths.

    § 3. Partially Ordered Sets

    Week 8.
    A. Basic concept of posets
    B. New Poset from Old
    Take-home Midterm Exam on Friday, October 17, 2008. (No class on Friday.)

    Week 9
    A. Lattice and Distributive law.
    B. Modularity and Semi-modularity.
    Class on Friday, October 24 is cancelled.

    Week 10
    A. Dilworth's Chain Decomposition Theorem, Sperner's theorem
    B. Symmetric chain decomposition in Boolean lattice, the incidence algebra
    C. zeta function and Mobius function.

    Week 11.
    A. Mobius inversion formula, B_n and D_n. Mobius algebra.
    B. Mobius algebra and Mobius functions in \Pi_n and L_n(q).
    C. Zeta polynomials

    § 4. Exponential Generating functions

    Week 12.
    A. Binomial posets and generating functions
    B. Permutation Enumeration
    C. Exponential generating functions

    Week 13.
    A. Application of exponential Formula
    B. Rooted labeled trees
    C. Unlabeled plane trees.

    Week 14
    A. Lagrange inversion: the theorem
    Thanksgiving Holidays
    B. Lagrange inversion: examples

    Dec 9 FINAL Exam

    MAKE-UP POLICY: Make-ups for missed quizzes and exams will only be allowed for a university approved excuse in writing. Wherever possible, students should inform the instructor before an exam or quiz is missed. Consistent with University Student Rules , students are required to notify an instructor by the end of the next working day after missing an exam or quiz. Otherwise, they forfeit their rights to a make-up.

    POLICY FOR ABSENCES: Attendance on a regular basis is expected. While there are occasionally good reasons for you to choose to miss class, my experience has been that there is a strong correlation between attendance (as long as you are awake and listening) and performance in the course.

    For absence related to injure or illness, students who are absent from class three or more days should provide instructors with confirmation from a medical provider for an excused absence.

    SCHOLASTIC DISHONESTY: Copying work done by others, either in-cl ass or out of class, is an act of scholastic dishonesty and will be prosecuted to the full extent allowed by University policy. Collaboration on assignments, either in-class or out-of-class, is forbidden unless permission to do so is granted by your instructor. For more information on university policies regarding scholastic dishonesty, see University Student Rules .

    COPYRIGHT POLICY: All printed materials disseminated in class or on the web are protected by Copyright laws. One xerox copy (or download from the web) is allowed for personal use. Multiple copies or sale of any of these materials is strictly prohibited.