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:
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.
