Department of Mathematical Sciences
Binghamton University

Syllabus of Math 386, Combinatorics

Fall 2011

Textbook

List of material covered

• Chapter 1: Seven is More Than Six. The Pigeon-Hole Principle
  • An advanced introduction to combinatorial thinking.


• Chapter 2: One Step at a Time. The Method of Mathematical Induction
  • Challenging combinatorial problems and just a few basics.


• Chapter 3: There are a Lot of Them. Elementary Counting Problems
  • At last: Introduction to basic counting methods and problems. Combinations (subsets) and permutations (sequences without repetition). Multisets: sets with multiple elements.


• Chapter 4: No Matter How You Slice It. The Binomial Theorem and Related Identities
  • Binomial identities. Combinatorial proofs.


• Chapter 5: Divide and Conquer. Partitions
  • Breaking up a set or a number into several parts.


• Chapter 6: Not So Vicious Cycles. Cycles in Permutations
  • Permutations as functions, operating on the ordered set {1,2,...,n}.


• Chapter 7: You Shall Not Overcount. The Sieve
  • The Principle of Inclusion and Exclusion (P.I.E.): a fundamental counting method.
  • More combinatorial proofs, some of them complicated.


• Chapter 8: A Function is Worth Many Numbers. Generating Functions
  • Power series enter combinatorics. We take a very different point of view from that of calculus.


• Chapter 14: So Hard to Avoid. Subsequence Conditions on Permutations
  • Section 14.1: Permutations as sequences again: counting those without specified subpatterns.


• Chapter 17: As Evenly as Possible. Block Designs and Error Correcting Codes
  • Sections 17.1-3: Designs. Latin squares.
  • Section 17.4: Finite affine and projective planes are geometrical structures with design-like properties.
    The main part of this topic is in my notes on "Affine and Projective Planes and Latin Squares".