Instructor: Alexander Postnikov
Time: Monday, Wednesday, Friday 2:003:00 pm
Place: MIT, Room 2131
Office hours: By appointment
Graders: Daniil Kluev
and Milan Haiman
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 the lecture notes from last year.)

Mon, Dec 31. The Catalan numbers: Dyck paths, recurrence relation,
and exact formula.
Notes

Wed, Feb 2. The Catalan numbers (cont'd):
reflection method and cyclic shifts.
Notes

Fri, Feb 4. The Catalan numbers (cont'd): combinatorial interpretations
(binary trees, plane trees, triangulations of polygons,
noncrossing and nonnesting matchings, etc).
Pattern avoidance for permutations.
Notes

Mon, Feb 7.
Integer partitions. Young diagrams and standard Young tableaux.
The hook length formula.
Notes
(pages 17)
 Wed, Feb 9.
Probabilistic "hook walk" proof of the hook length formula.
Notes
(pages 817)
 Fri, Feb 11.
Set partitions. The Bell numbers and the Stirling
numbers of the second kind. Rook placements.
Noncrossing and nonnesting set partitions. Peaks and valleys.
Notes
(pages >= 5)

Mon, Feb 14.
Exponential generating functions. The exponential formula.
The Bell numbers and the Stirling numbers of the second kind.
Notes

Wed, Feb 16.
Cycles in permutations.
The Stirling numbers of the first kind. Two Stirling triangles.
The Stirling numbers (of both kinds) as matrix coefficients in
change of bases matrices.
Notes (pages 1,
915)

Fri, Feb 18.
Statistics on permutations: inversions, descents, cycles, major index,
records, exceedances. The Eulerian numbers.
The Eulerian triangle.
Notes
(pages 28, 16)
Mon, Feb 21. Presidents' Day  holiday.

Tue, Feb 22. (Monday schedule of classes to be held.)
Sperner's theorem.
Posets. Chains and antichains.
The Boolean lattice. Symmetric chain decompositions (SCD).
Notes
(pages 114)

Wed, Feb 23.
Dilworth's, Mirsky's, and Greene's theorems.
Increasing and decreasing subsequences in permutations.
ErdosSzekeres theorem.
Notes
(pages 14)
Fri, Feb 25. Snow emergency  no lecture.

Mon, Feb 28.
Lattices. Examples: Boolean and Young's lattices.
Order ideals. Distributive lattices. Birkhoff's fundamental theorem
on finite distributive lattices.
Notes
(pages 16)

Wed, Mar 2.
Linear extensions of posets and saturated chains in distributive lattices.
qanalogs: qfactorials and Gaussian qbinomial coefficients.
Notes
(pages 713)

Fri, Mar 4. qbinomial coefficients and qPascal's triangle.
Young diagrams inside a rectangle.
Notes
(pages 18)

Mon, Mar 7. Grassmannians over finite fields
and Gaussian elimination. Young diagrams
as reduced rowechelon forms of matrices.
Permutations of multisets and their inversions.
Notes
(pages 913)

Wed, Mar 9.
Up and down operators. Differential posets.
Notes

Fri, Mar 11. Guest lecture by Anna Weigandt.

Mon, Mar 14. Guest lecture by Anna Weigandt.

Wed, Mar 18. Guest lecture by Anna Weigandt.

Fri, Mar 19. Students' presentations.
Mon, Mar 21  Fri, Mar 25. Spring break.

Mon, Mar 28.
Partition theory. Generating functions. Partitions with
odd and distinct parts. Euler's Pentagonal Theorem.
Notes
(pages 110)

Wed, Mar 30. Partition theory (cont'd).
Franklin's proof of Euler's Pentagonal Theorem.
Jacobi triple product formula. Euler's and Gauss' identities.
Notes
(pages 1115) and
Notes
(pages 110)
 Fri, Apr 1. Proof of Jacobi's triple product (cont'd).
qbinomial coefficients again.
Cayley's formula for the number of labelled trees.
Notes
(page 15) and
Notes
(pages 16)

Mon, Apr 4. Three proofs of Cayley's formula:
Algebraic proof of Renyi, bijection of EgeciogluRemmel,
Prufer's coding.
Notes
(pages 614) and
Notes
(pages 17)

Wed, Apr 6. Spanning trees in graphs.
Graphical matrices: adjacency matrix, incidence matrix, Laplacian matrix.
Matrix Tree Theorem. Oriented incidence matrix.
Proof of the Matrix Tree Theorem based on the CauchyBinet formula.
Notes
(pages 1115) and
Notes
(pages 18).
 Fri, Apr 8. MTT cont'd.
 Mon, Apr 11.
Eigenvalues of the adjacency matrix vs eigenvalues of the Laplacian.
Products of graphs. Spanning trees of the hypercube.
Notes (pages 19)
 Wed, Apr 13.
Extensions of the MTT: weighted version and
directed version. Arborescences and cofactors
of the Laplacian matrix.
Notes
(pages 514) and
Notes
(pages 15).

Fri, Apr 15.
Electrical networks. Kirchhoff's and Ohm's laws.
The Kirchhoff's matrix. Resistances of networks via spanning trees.
Seriesparallel networks.
Notes
Mon, Apr 18. Patriots' Day  holiday.

Wed, Apr 20.
Electrical networks (cont'd).
Seriesparallel networks.
Inverse boundary problem. YDelta transform.
Random walks on graphs and electrical networks.
Notes
(pages 112)

Fri, Apr 22.
Parking functions. Labelled Dyck paths.
Notes
(pages 110)

Mon, Apr 25.
Chip firing game and the Abelian sandpile model.
Notes

Wed, Apr 27.
The Abelian sandpile model (cont'd).
Avalanche operators, recurrent configurations, the sandpile group.
Notes

Fri. Apr 29.
Inversions in trees. Tree inversion polynomials and parking functions.
Recurrence relations for the inversion polynomials.
Alternating permutations.
Notes
................
Mon, May 9. The last lecture.
Recommended Textbooks:
(The students are not required to buy any of these books. This is an optional
reading. Most of the material of the course is contained in the
union of these books. But we will not follow these books section by section.)
[AC]
Algebraic Combinatorics: Walks, Trees, Tableaux, and More
by R. P. Stanley, Springer:
PDF
[EC1]
Enumerative Combinatorics Vol 1 by R. P. Stanley,
Cambridge University Press.
Last updated: April 29, 2022 