Instructor: Alexander Postnikov
Time: Monday, Wednesday, Friday 2:003:00 pm EST
online
Zoom: https://mit.zoom.us/j/94806842612
All lectures will be in real time on zoom.
The zoom passcode is the first 2digit
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
 qbinomial 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 GesselViennot method)
 spanning trees, parking functions, and Prufer codes
 matrixtree 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...
Grading: Based on several Problems Sets.
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.

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

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

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

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 