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Alexander Postnikov: 18.212 Algebraic Combinatorics
Instructor: Alexander Postnikov
Time: Monday, Wednesday, Friday 2:003:00 pm
Place:
Room 4145
Web: http://math.mit.edu/18.212/
Office hours: Monday 34 pm or by appointment, Room 2367
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 ...
Units: 309
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:
 Problem Set 1 (due March 04, 2019)

Problem Set 2 (due April 5, 2019)

Problem Set 3 (due May 10, 2019)
Correction: In Problem 5 assume that the ncube graph is a bidirectected graph
with each edge directed both way. An Eulerian cycle should pass each edge once in each direction. As a bonus question try to think about undirected
version of this problem.
Recommended Texts:
(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, 2nd ed, 2018.
Version of 2013 is available as
pdf file
Additional reading:
[EC1] [EC2]
Enumerative Combinatorics, Vol 1 and Vol 2, by R. P. Stanley,
Cambridge University Press, 2011 and 2001.
Volume 1 is available as
pdf file
[vLW]
A
Course in Combinatorics by J. H. van Lint and R. M. Wilson,
Cambridge University Press, 2001.
Lecture Notes by Andrew Lin.
Lectures
(with links to additional reading materials):

Wed 02/06/2019: Catalan numbers: drunkard's walk problem, generating function,
recurrence relation

Fri 02/08/2019:
Catalan numbers (cont'd): formula for C_n, reflection principle, necklaces,
triangulations of polygons, plane binary trees, parenthesizations

Mon 02/11/2019:
Pattern avoidance in permutations, stack and queuesortable permutations,
Young diagrams and Young tableaux, the hooklength formula

Wed 02/13/2019:
FrobeniusYoung identity, Schensted correspondence,
longest increasing and decreasing subsequences in permutations

Fri 02/15/2019: Proof of the hooklength formula based on
a random hook walk
Mon 02/18/2019: no classes (President's Day)

Tues 02/19/2019 (Monday schedule):
Hook walks (cont'd). Linear extensions of posets. Hooklengthtype
formulas for shifted shapes and trees.
 D. E. Knuth: The Art of Computer Programming, Volume 3: Sorting and
Searching, 1973, Section 5.1.4.

Wed 02/20/2019: qfactorials and qbinomial coefficients

Fri 02/22/2019: Grassmannians over finite fields: Gaussian elimination
and rowreduced echelon form

Mon 02/25/2019:
Sets and multisets. Statistics on permutations:
inversions, cycles, descents.

Wed 02/27/2019:
Statistics on permutations (cont'd). Equidistributed statistics.
Major index. Records. Exceedances. Stirling numbers.

Fri 03/01/2019: Stirling numbers (cont'd).
Setpartitions. Rook placements on triangular boards.
Noncrossing and nonnesting setpartitions.
 Mon 03/04/2019:
Problem Set 1 is due.
Eulerian numbers.
Increasing binary trees.
3 Pascallike triangles: Eulerian triangles, Stirling
triangles of 1st and 2nd kind.

Wed 03/06/2019: Discussion of Problem Set 1.
Volunteers can present solutions of some problems.
Try to limit your presentations to 510 min.
 Fri 03/08/2019:
Discussion of Problem Set 1 (cont'd).

Mon 03/11/2019:
Posets and lattices. Boolean lattice. Partition lattice. Young's lattice.

Wed 03/13/2019:
Distributive lattices.
Birkhoff's fundamental theorem for finite distributive lattices.

Fri 03/15/2019:
Sperner's property.
Symmetric chain decompositions.
Sperner's and Dilworth's theorems.
Greene's theorem.

Mon 03/18/2019:
Greene's theorem vs Schensted correspondence.
Up and down operators.
Differential posets.

Wed 03/20/2019:
Differential posets (cont'd). Fibonacci lattice.
Unimodality of Gaussian coefficients.

Fri 03/22/2019:
Proof of unimodality of Gaussian coefficients (cont'd).
Theory of partitions. Euler's pentagonal number theorem.
Mon 03/25/2019  Fri 03/29/2019 no classes (Spring Break)

Mon 04/01/2019: Partition theory (cont'd). Franklin's combinatorial
proof of Euler's pentagonal number theorem.
Jacobi's triple product identity.

Wed 04/03/2019:
Partition theory (cont'd).
Combinatorial proof of Jacobi's triple product identity.
Enumeration of trees. Cayley's formula.
Simple inductive proof of Cayley's formula.

Fri 04/05/2019:
Problem Set 2 is due.
Two combinatorial proofs of Cayley's formula.

Mon 04/08/2019:
Discussion of Problem Set 2.

Wed 04/10/2019:
Discussion of Problem Set 2 (cont'd).

Fri 04/12/2019:
Matrix Tree Theorem.
Spanning trees. Laplacian matrix of a graph.
Reciprocity formula for spanning trees.
Examples: complete graphs, complete bipartite graphs.
Mon 04/15/2019: no classes (Patriots' Day)

Wed 04/17/2019:
Matrix Tree Theorem (cont'd).
Products of graphs. Number of spanning trees
in the hypercube graph.
Oriented incidence matrix.

Fri 04/19/2019: Proof of Matrix Tree Theorem using
CauchyBinnet formula.
Weighted and directed version of Matrix Tree Theorem.
Intrees and outtrees.

Mon 04/22/2019: Proof of Directed Matrix Tree Theorem
based on induction.

Wed 04/24/2019: Proof of Directed Matrix Tree Theorem via
the Involution Principle. Electrical Networks and Kirchhoff's Laws.

Fri 04/26/2019: Electrical networks (cont'd). Kirchhoff's matrix.
Relations of electrical networks with
Matrix Tree Theorem and spanning trees.
Seriesparallel connections. Probabilistic interpretation
of the electrical potential in terms of random walks on graphs.

Mon 04/29/2019: Eulerian cycles in digraphs and B.E.S.T. theorem.

Wed 05/01/2019: Parking functions.
Tree inversion polynomials.

Fri 05/03/2019:

Mon 05/06/2019:

Wed 05/08/2019:

Fri 05/10/2019:
Problem Set 3 is due.

Mon 05/13/2019:

Wed 05/15/2019:
Last updated: May 2, 2019 