18.212      ALGEBRAIC COMBINATORICS         MIT Spring 2021

Instructor: Alexander Postnikov

Time: Monday, Wednesday, Friday 2:00-3:00 pm EST       on-line

Zoom: https://mit.zoom.us/j/94806842612

All lectures will be in real time on zoom. The zoom passcode is the first 2-digit Catalan number, written as a number. You can also contact the instructor to find the passcode.

Canvas: https://canvas.mit.edu/courses/6771

Description:
Applications of algebra to combinatorics and vise versa. We will discuss enumeration methods, permutations, partitions, partially ordered sets and lattices, Young tableaux, graph theory, matrix tree theorem, electrical networks, convex polytopes, and other topics.
Course Level: advanced undergraduate.

Topics:

• 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, and 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, and Prufer codes
• matrix-tree theorem, electrical networks, random walks on graphs
• graph colorings, chromatic polynomial, Mobius function
• lattice paths and 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
• and more...

Problem Sets: TBA

Lectures:

The lecture notes will appear on this page shortly before the lectures. The registered students can view the recordings of lectures on canvas.

1. Wed, Feb 17. The Catalan numbers: Dyck paths, recurrence relation, and exact formula.
   Notes   Video

2. Fri, Feb 19. The Catalan numbers (cont'd): drunkard's walk, reflection principle, cyclic shifts.
   Notes   Video

3. Mon, Feb 22. The Catalan numbers (cont'd): combinatorial interpretations (plane trees, triangulations of polygons, non-crossing and non-nesting arc diagrams, etc). Stack and queue sorting. Pattern avoidance for permutations.
   Notes   Video

4. Wed, Feb 24. Partitions of integers and Young diagrams. Standard Young tableaux. The hook length formula. Probabilistic "hook walk" proof.
   Notes  

Recommended Textbooks: (The students are not required to buy these books.)

The course will more or less cover the textbook:
[AC]  Algebraic Combinatorics: Walks, Trees, Tableaux, and More by R. P. Stanley, Springer: PDF

Additional reading:

[EC1] Enumerative Combinatorics Vol 1 by R. P. Stanley, Cambridge University Press: PDF

[vLW]  A Course in Combinatorics by J. H. van Lint and R. M. Wilson, Cambridge University Press.

Last updated:   February 22, 2021