Instructor: Alexander Postnikov
Time: Monday, Wednesday, Friday 2:00-3:00 pm
Place: MIT, Room 2-131
Office hours: By appointment
Graders: Daniil Kluev
and Milan Haiman
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.
Catalan numbers, Dyck paths, triangulations, non-crossing 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 q-binomial coefficients,
standard Young tableaux (STY),
Robinson-Schensted-Knuth (RSK) correspondence,
partitions, Euler's pentagonal theorem, Jacobi triple product,
non-crossing paths, Lindstrom-Gessel-Viennot lemma,
spanning trees, parking functions, Prufer codes,
matrix-tree theorem, electrical networks, random walks on graphs,
graph colorings, chromatic polynomial, Mobius function,
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.
(We include links to the lecture notes from last year.)
Mon, Dec 31. The Catalan numbers: Dyck paths, recurrence relation,
and exact formula.
Wed, Feb 2. The Catalan numbers (cont'd):
reflection method and cyclic shifts.
Fri, Feb 4. The Catalan numbers (cont'd): combinatorial interpretations
(binary trees, plane trees, triangulations of polygons,
non-crossing and non-nesting matchings, etc).
Pattern avoidance for permutations.
Mon, Feb 7.
Integer partitions. Young diagrams and standard Young tableaux.
The hook length formula.
- Wed, Feb 9.
Probabilistic "hook walk" proof of the hook length formula.
- Fri, Feb 11.
Set partitions. The Bell numbers and the Stirling
numbers of the second kind. Rook placements.
Non-crossing and non-nesting set partitions. Peaks and valleys.
(pages >= 5)
Mon, Feb 14.
Exponential generating functions. The exponential formula.
The Bell numbers and the Stirling numbers of the second kind.
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,
Fri, Feb 18.
Statistics on permutations: inversions, descents, cycles, major index,
records, exceedances. The Eulerian numbers.
The Eulerian triangle.
(pages 2-8, 16)
Mon, Feb 21. Presidents' Day - holiday.
Tue, Feb 22. (Monday schedule of classes to be held.)
Posets. Chains and antichains.
The Boolean lattice. Symmetric chain decompositions (SCD).
Wed, Feb 23.
Dilworth's, Mirsky's, and Greene's theorems.
Increasing and decreasing subsequences in permutations.
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.
Wed, Mar 2.
Linear extensions of posets and saturated chains in distributive lattices.
q-analogs: q-factorials and Gaussian q-binomial coefficients.
Fri, Mar 4. q-binomial coefficients and q-Pascal's triangle.
Young diagrams inside a rectangle.
Mon, Mar 7. Grassmannians over finite fields
and Gaussian elimination. Young diagrams
as reduced row-echelon forms of matrices.
Permutations of multisets and their inversions.
Wed, Mar 9.
Up and down operators. Differential posets.
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.
Wed, Mar 30. Partition theory (cont'd).
Franklin's proof of Euler's Pentagonal Theorem.
Jacobi triple product formula. Euler's and Gauss' identities.
(pages 11-15) and
- Fri, Apr 1. Proof of Jacobi's triple product (cont'd).
q-binomial coefficients again.
Cayley's formula for the number of labelled trees.
(page 15) and
Mon, Apr 4. Three proofs of Cayley's formula:
Algebraic proof of Renyi, bijection of Egecioglu-Remmel,
(pages 6-14) and
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 Cauchy-Binet formula.
(pages 11-15) and
- 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 1-9)
- Wed, Apr 13.
Extensions of the MTT: weighted version and
directed version. Arborescences and cofactors
of the Laplacian matrix.
(pages 5-14) and
Fri, Apr 15.
Electrical networks. Kirchhoff's and Ohm's laws.
The Kirchhoff's matrix. Resistances of networks via spanning trees.
Mon, Apr 18. Patriots' Day - holiday.
Wed, Apr 20.
Electrical networks (cont'd).
Inverse boundary problem. Y-Delta transform.
Random walks on graphs and electrical networks.
Fri, Apr 22.
Parking functions. Labelled Dyck paths.
Mon, Apr 25.
Chip firing game and the Abelian sandpile model.
Wed, Apr 27.
The Abelian sandpile model (cont'd).
Avalanche operators, recurrent configurations, the sandpile group.
Fri. Apr 29.
Inversions in trees. Tree inversion polynomials and parking functions.
Recurrence relations for the inversion polynomials.
Mon, May 9. The last lecture.
(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.)
Algebraic Combinatorics: Walks, Trees, Tableaux, and More
by R. P. Stanley, Springer:
Enumerative Combinatorics Vol 1 by R. P. Stanley,
Cambridge University Press.
Last updated: April 29, 2022