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.
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,
random walks, 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
 Young diagrams, Young's lattice, Gaussian qbinomial coefficients
 standard Young 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:
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
Video
 Fri, Feb 26.
Integer partions vs set partitions. The Bell numbers and the Stirling
numbers of the second kind.
Rook placements.
Noncrossing and nonnesting set partitions. Peaks and valleys.
The Narayana numbers.
Notes
Video

Mon, Mar 1.
Exponential generating functions. The exponential formula.
The Bell numbers and the Stirling numbers of the second kind.
Notes
Video

Wed, Mar 3.
Statistics on permutations: inversions, descents, cycles, major index,
records, exceedances. The Eulerian numbers.
The Stirling numbers of the first kind.
The Stirling numbers as the coefficients in the change of bases matrices.
Three "Pascallike" triangles: The Stirling triangles of both kinds and
the Eulerian triangle.
Notes
Video

Fri, Mar 5.
Sperner's theorem.
Posets. Chains and antichains.
The Boolean lattice. Symmetric chain decompositions (SCD).
Notes
Video

Tue, Mar 9.
Dilworth's, Mirsky's, and Greene's theorems.
Increasing and decreasing subsequences in permutations.
ErdosSzekeres theorem.
Lattices. Examples of lattices: The Boolean lattice, Young's lattice,
and the partition lattice.
Notes
Video

Wed, Mar 10.
Order ideals. Distributive lattices. Birkhoff's fundamental theorem
on finite distributive lattices.
Linear extenstions of posets and saturated chains.
Notes
Video

Fri, Mar 12.
qanalogs. The qfactorials and the Gaussian qbinomial coefficients.
Gaussian elimination and Grassmannians over finite fields.
Permutations of multisets.
Notes
Video

Mon, Mar 15.
The RobinsonSchensted(Knuth) correspondence (RSK).
Increasing and descreasing subsequences in permutations.
321avoiding permutations and Dyck paths.
Up and down operators on Young's lattice.
Notes
Video

Wed, Mar 17.
Differential posets. Up and down operators. The Fibonacci lattice.
Notes
Video

Fri, Mar 19.
Differential posets and rook placements. Oscillating tableaux.
Perfect matchings and involutions.
Notes
Video

Wed, Mar 24. Partition theory. Generating functions. Partitions with
odd and distinct parts. Selfconjugate partitions.
Euler's Pentagonal Theorem.
Notes
Video

Fri, Mar 26. Partition theory (cont'd).
Jacobi triple product. Euler's and Gauss' identities. Gaussian qbinomial
coefficients again.
Notes
Video

Mon, Mar 29. Sylvester's proof of unimodality of Gaussian qbinomial
coefficients.
Notes
Video

Wed, Mar 31. Counting labelled trees. Cayley's formula:
algebraic proof by Renyi and bijective proof by EgeciogluRemmel.
Notes
Video

Fri, Apr 2. Two bijective proofs of Cayley's formula.
Prufer coding and decoding. Spanning trees in graphs.
Spanning trees in the complete bipartite graph K_{m,n}.
Laplacian matrix (a.k.a Kirchhoff matrix).
Kirchhoff's Matrix Tree Theorem.
Notes
Video

Mon, Apr 5.
Proof of the Matrix Tree Theorem based on the CauchyBinet formula.
Examples: Cayley's formula and the number of spanning trees in the dhypercube.
Notes
Video
 Wed, Apr 7.
Eigenvalues of the adjacency matrix vs eigenvalues of the Laplacian.
Products of graphs. Spanning trees of the hypercube.
Notes
Video
 Fri, Apr 9.
Reciprocity formula for spanning trees.
Extensions of the MTT: weighted version and
directed version. Arborescences and cofactors
of the Laplacian matrix.
Notes
Video

Mon, Apr 12.
Two proofs of the directed matrix tree theorem:
(a) by induction, (b) by involution principle.
Increasing trees. Abel's binomial formula.
Notes
Video

Wed, Apr 14.
Electrical networks. Kirchhoff's and Ohm's laws.
The Kirchhoff's matrix. Resistances of networks via spanning trees.
Seriesparallel networks.
Notes
Video

Wed, Apr 15.
Electrical networks (cont'd).
Inverse boundary problem. YDelta transforms.
Random walks on graphs and electrical networks.
Interpretations of the Cayley number n^{n2}: spanning trees,
parking functions, regions of the Shi arrangement, volume
of the permutohedron.
Notes
Video

Wed, Apr 21.
Parking functions. Labelled Dyck paths. Generalized parking functions.
Notes
Video

Fri, Apr 23.
Chip firing game and the Abelian sandpile model.
Notes
Video

Mon, Apr 26.
The Abelian sandpile model (cont'd).
Avalanche operators, recurrent configurations, the sandpile group.
Notes
Video

Wed, Apr 28.
Gparking functions. Directed Eulerian cycles.
BEST Theorem.
Notes
Video

Fri. Apr 30.
Inversions in trees. Tree inversion polynomials and parking functions.
Recurrence relations for the inversion polynomials.
Alternating permutations.
Notes
Video

Mon. May 3.
Alternating permutations (cont'd). Tangent and secant numbers.
The EulerBernoulli triangle.
The bijection between (complete) increasing binary trees and (alternating)
permutations. Motzkin paths.
Notes
Video

Wed. May 5.
Enumeration of weighted Motzkin and Dyck paths.
The FranconViennot bijection.
Continued fractions and Flajolet's fundamental lemma.
Increasing 012trees.
Notes
Video

Mon, May 10.
Lattice paths and continued fractions (cont'd).
Lindstrom's lemma aka the GesselViennot method.
Notes
Video

Wed, May 12.
Counting plane partitions: determinantal formula and MacMahon's formula.
Rhombus tilings, pseudoline arrangements, and perfect matchings.
The arctic circle phenomenon.
Notes
Video
 Fri, May 14.
Colorings of graphs. The chromatic polynomial. Acyclic orientations. Chordal
graphs.
Notes
Video

Mon, May 17.
The Tutte polynomial.
Notes
Video

Wed, May 19.
Domino tilings. Enumeration of perfect matchings. Kasteleyn's theorem.
Permanents and determinants. Domino tilings and spanning trees.
Temperley's theorem.
Notes
Video
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: May 19, 2021 