Math 68: Algebraic Combinatorics

Fall 2015

Syllabus



Course Description

Algebraic combinatorics is a huge subject encompassing several topics such as symmetric functions, Young tableaux, posets, etc. The unifying theme in this area of mathematics is the interplay between combinatorics and abstract algebra. We employ methods in algebra to solve combinatorial problems, and, conversely, apply combinatorial techniques to problems in algebra.

The objectives of the course are:

  • Students will be able to solve problems in algebraic combinatorics.
  • Students will improve their proof-writing skills.
  • Students will be exposed to many topics in algebraic combinatorics.
    • "Mathematics is not for spectators; in order to gain in understanding, confidence, and enthusiasm one has to participate." M.A. Armstrong
    Course Topics
    Here is a tentative list of topics that will be covered.

  • Counting Principles and Basic Coefficients.
  • Inclusion-Exclusion Principle.
  • Generating Functions.
  • Partially ordered sets. Chains and antichains. Sperner's Theorem.
  • Young Tableaux and RSK algorithm.
  • Group actions on boolean algebras.
  • Young diagrams and q-binomial coefficients.
  • Enumeration under group action, counting orbits using Burnside Lemma. Polya's Enumeration Theorem.
  • Intro to symmetric functions.
    • Grades
    The course grade will be computed as follows:

    Percent of Final Grade
    Homework 40
    Participation 10
    Final Exam 25
    Final Project 25

    Students will be graded on class participation. Of course it is difficult to participate if one does not come to class at all, and so habitual absence will also be reflected in the class participation grade.
    Homework Policy
    Written homework will be assigned daily and will be collected once a week at the beginning of class.
    • All homework assignments will be posted on the course assignment's webpage.
    • Unexcused late and missing papers count zero.
    • Homework is to be written neatly using both sides of 8 1/2 x 11 inch paper. Do not use paper from a spiral notebook unless you can tear off the ragged edge. All papers are to be stapled.
    • Use English. If you can't read your solutions aloud as fluently as if you were reading a textbook, try using nouns and verbs in your write ups!
    Textbook
    Richard Stanley, Topics in Algebraic Combinatorics, the book is available without problems online: Here.
    Some additional material will be taken from these notes:
  • Anna de Mier, Lecture notes for "Enumerative Combinatorics", available here.
    • Honor Principle
    Students are encouraged to work together in the thinking process on the homework problems. What is important is a student's eventual understanding of homework problems, and not how that is achieved. However, students must work alone without any assistance when writing up the solutions for the homework.

    The honor principle on homework: What a student turns in as a 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. It is a violation of the honor code to copy solutions from problems posted on the web or book or any other source. 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. For example, it is a breach of the honor code to read the solutions of someone else in order to write your solution.

    The honor principle on exams: Students may not give or receive assistance of any kind on an exam from any person except for clarification from the professor or someone explicitly designated by the professor to answer questions about the exam.

    If you have any questions as to whether some action would be acceptable under the Academic Honor Code, please speak to me I will be glad to help clarify things. It is always easier to ask beforehand than to have trouble later!
    Disabilities and Religious Observances
    Students with disabilities enrolled in this course and who may need disability-related classroom accommodations are encouraged to make an appointment to see your instructor before the end of the second week of the term. All discussions will remain confidential, although the Student Accessibility Services office may be consulted to discuss appropriate implementation of any accommodation requested.

    Some students may wish to take part in religious observances that occur during this academic term. If you have a religious observance that conflicts with your participation in the course, please meet with your instructor before the end of the second week of the term to discuss appropriate accommodations.

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