Instructor: Alexander Postnikov
Time: Monday, Wednesday, Friday 1:002:00 pm
Place:
Room 4237.
Canvas: For announcements and problem set submissions:
https://canvas.mit.edu/courses/24334
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,
random walks, convex polytopes, and other topics.
Course Level: Advanced Undergraduate.
Keywords:
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,
Young diagrams, Young's lattice, Gaussian qbinomial coefficients,
standard Young tableaux (STY),
RobinsonSchenstedKnuth (RSK) correspondence,
partitions, Euler's pentagonal theorem, Jacobi triple product,
noncrossing paths, LindstromGesselViennot lemma,
spanning trees, parking functions, Prufer codes,
matrixtree theorem, electrical networks, random walks on graphs,
graph colorings, chromatic polynomial, Mobius function,
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.
Grading: Based on several Problems Sets.
Problem Sets:
Lectures:
We include links to related lecture notes from previous years.
The notes may contain some additional material not covered in the lectures.

Mon, Feb 5. The Catalan numbers, Dyck paths,
Bertrand's ballot problem, reflection method.
Notes

Wed, Feb 7. The Catalan numbers (cont'd): recurrence relation,
generating function, generalized binomial coefficients,
binary trees.
Notes

Fri, Feb 9. The Catalan numbers (cont'd): combinatorial interpretations
(binary trees, plane trees, triangulations of polygons,
noncrossing and nonnesting matchings, etc).
Notes

Mon, Feb 12.
Integer partitions. Young diagrams and standard Young tableaux.
The hook length formula.
Notes
(pages 17)
 Wed, Feb 14.
Probabilistic "hook walk" proof of the hook length formula.
Notes
(pages 817)
 Fri, Feb 16.
Set partitions. The Bell numbers and the Stirling numbers of the second kind.
Rook placements.
Notes
(pages 111)
 Tue, Feb 20. Exponential generating functions. The exponential formula.
Notes (pages 17)
 Wed, Feb 21. Applications of the exponential formula.
Notes (pages 815)
 Fri, Feb 23.
Cycles in permutations. The Stirling numbers of the first kind.
The Stirling numbers (of both kinds) as coefficients in
change of bases matrices.
Recurrence relations for the Stirling numbers and their ``Pascallike''
triangles.
Notes (pages 1621)
and these
Notes (pages 915)
 Mon, Feb 26.
The Eulerian numbers and the Eulerian triangle.
Notes (page 16)
 Wed, Feb 28.
Statistics on permutations: inversions, descents, cycles, major index,
records, exceedances.
Notes
(pages 18)
 Fri, March 1.
Sperner's Theorem. Posets.
Notes
(pages 18).
 Mon, March 4.
Rank unimodality and Sperner property of posets.
Symmetric chain decompositions (SCD). Products of posets.
Notes (pages 915)
 Wed March 6.
Dilworth's, Mirsky's, and Greene's theorems for posets.
Posets associated to permutations. Increasing and decreasing subsequences
in permutations.
Notes
(pages 14)
 Fri March 8.
Ramsey's and ErdosSzekeres theorems. Graphs vs posets vs permutations.
Young's lattice.
Notes (page 7)
 Mon, March 11.
The RobinsonSchensted(Knuth) correspondence (RSK).
Schensted's insertion algorithm.
Increasing and descreasing subsequences in permutations.
Notes
(pages 17)
 Wed March 13. Symmetry of RSK. 321avoiding permutations.
Walks on Young's lattice.
Notes
(pages 89)
 Fri March 15.
Oscillating tableaux and rook placements.
Up and down operators on Young's lattice.
Notes
(pages 1015)
 Mon, March 18.
Differential posets. Hopping particles and antiparticles.
The YoungFibonacci lattice.
Notes
(pages 123)
 Wed, March 20. qanalogs.
Notes
(pages 15)
 Fri, March 22. Gaussian qbinomial coefficients and Grassmannians
over finite fields.
Notes
(pages 614)
March 2529: Spring break.
 Mon, Apr 1. Theory of partitions.
Euler's theorem on odd and distinct partitions.
Notes
(pages 17)
 Wed, Apr 3. Theory of partitions (cont'd).
Euler's pentagonal number theorem. Franklin's proof of pentagonal
theorem. Involution principle.
Notes
(pages 815)
 Fri, Apr 5.
Partition theory (cont'd).
Jacobi's triple product formula.
Notes
(pages 19)
 Mon, Apr 8.
Counting labelled trees.
Cayley's formula.
Notes
(pages 110)
 Wed, Apr 10.
Two bijective proofs of Cayley's formula:
Prufer's coding and
EgeciogluRemmel bijection.
Notes
(pages 1014) and
Notes
(pages 17)
 Fri, Apr 12.
Spanning trees in graphs.
Adjacency matrix. Incidence matrix. Laplacian matrix.
Kirchhoff's Matrix Tree Theorem.
Notes
(pages 815)
 Wed, Apr 17.
Proof of the Matrix Tree Theorem. Oriented incidence matrix.
CauchyBinet formula.
Notes
(pages 110).
 Fri, Apr 19.
Spectral graph theory. Products of graphs.
Spanning trees in the hypercube graph.
Notes
(pages 19)
 Mon, Apr 22.
Reciprocity formula for spanning trees.
Notes
(pages 14)

Wed, Apr 24.
Weighted and directed versions of matrix tree theorem. Counting arborescences.
Notes
(pages 515)
and Notes
(pages 15).
 Fri, Apr, 26. Electrical networks. Effective resistance and spanning
trees.
Notes
 Mon, Apr 29.
Eulerian cycles in digraphs. B.E.S.T. Theorem.
Notes
(pages 314)
 Wed, May 1.
Random walks on graphs. Probability = potential.
Notes
(pages 611).
 Fri, May 3.
Parking functions.
Notes
(pages 110)
 Mon, May 6. Tree inversion polynomials and parking functions. Alternating
permutations.
Notes
 Wed, May 8. The chip firing game and the abelian sandpile model.
Notes
 Fri, May 10. The abelian sandpile model (cont'd). The sandpile
group of a graph.
Notes
 Mon, May 13. Domino tilings and perfect matchings.
Kasteleyn's and Temperley's theorems.
Notes
Additional reading materials:
(The students are not required to buy these books.)
[AC]
Algebraic Combinatorics: Walks, Trees, Tableaux, and More
by R. P. Stanley, Springer.
Book info
including Text of
version of 2013 (without exercises), available at Richard Stanley's webpage.
[EC1]
Enumerative Combinatorics Vol 1 by R. P. Stanley,
Cambridge University Press.
Book info
from Richard Stanley's webpage.
[18.212'2019]
18.212 Lecture Notes from 2019 (the notes are taken by Andrew Lin).
Last updated: May 13, 2024 