Syllabus

Course Meeting Times

Lectures: 3 sessions / week, 1 hour / session

Prerequisites

18.701 Algebra I or 18.703 Modern Algebra

Course Description and Topics

This course is the applications of algebra to combinatorics and vise versa. The following topics are discussed:

  • Catalan numbers, Dyck paths, triangulations, noncrossing set partitions
  • Symmetric group, statistics on permutations, inversions and major index
  • Partially ordered sets and lattices, Sperner's and Dilworth's theorems
  • q-binomial coefficients, Gaussian coefficients, Young diagrams
  • Young's lattice, tableaux, Schensted's correspondence, RSK
  • Partitions, Euler's pentagonal theorem, Jacobi triple product
  • Noncrossing paths, Lindstrom lemma (aka Gessel-Viennot method)
  • Spanning trees, parking functions, Prufer codes
  • Matrix-tree theorem, electrical networks, random walks on graphs
  • Graph colorings, chromatic polynomial, Mobius function
  • Lattice paths, continued fractions
  • Enumeration under group action, Burnside's lemma, Polya theory
  • Transportation and Birkhoff polytopes, cyclic polytopes, permutohedra
  • Domino tilings, matching enumeration, Pfaffians, Ising model

Recommended Text

The course more or less covers the textbook:

Buy at Amazon Stanley, Richard P. Algebraic Combinatorics: Walks, Trees, Tableaux, and More. Springer, 2018. ISBN: 9783319771724. Online 2013 version.

Grading

Grading is based on three problems sets.