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)
02/27/2019 update: 5 bonus problems are added
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.
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.
Distributive lattices.
Birkhoff's fundamental theorem for finite distributive lattices.

Wed 03/13/2019:
Up and down operators.
Differential posets. Fibonacci lattice.
Unimodality of Gaussian coefficients.

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

Mon 03/18/2019:

Wed 03/20/2019:

Fri 03/22/2019:
Mon 03/25/2019  Fri 03/29/2019 no classes (Spring Break)

Mon 04/01/2019:

Wed 04/03/2019:

Fri 04/05/2019:

Mon 04/08/2019:

Wed 04/10/2019:

Fri 04/12/2019:
Mon 04/15/2019: no classes (Patriots' Day)

Wed 04/17/2019:

Fri 04/19/2019:

Mon 04/22/2019:

Wed 04/24/2019:

Fri 04/26/2019:

Mon 04/29/2019:

Wed 05/01/2019:

Fri 05/03/2019:

Mon 05/06/2019:

Wed 05/08/2019:

Fri 05/10/2019:

Mon 05/13/2019:

Wed 05/15/2019:
Last updated: February 27, 2019 