18.212      ALGEBRAIC COMBINATORICS         MIT Spring 2019

Instructor: Alexander Postnikov

Time: Monday, Wednesday, Friday 2:00-3:00 pm

Place: Room 4-145

Web: http://math.mit.edu/18.212/

Office hours: Monday 3-4 pm or by appointment, Room 2-367

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: 3-0-9           Level: advanced undergraduate


Grading: Based on several Problems Sets.

Problem Sets:

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

[vL-W]  A Course in Combinatorics by J. H. van Lint and R. M. Wilson, Cambridge University Press, 2001.

Lectures (with links to additional reading materials):

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

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

  3. Mon 02/11/2019: Pattern avoidance in permutations, stack- and queue-sortable permutations, Young diagrams and Young tableaux, the hook-length formula

  4. Wed 02/13/2019: Frobenius-Young identity, Schensted correspondence, longest increasing and decreasing subsequences in permutations

  5. Fri 02/15/2019: Proof of the hook-length formula based on a random hook walk

    Mon 02/18/2019: no classes (President's Day)

  6. Tues 02/19/2019 (Monday schedule): Hook walks (cont'd). Linear extensions of posets. Hook-length-type formulas for shifted shapes and trees.

  7. Wed 02/20/2019: q-factorials and q-binomial coefficients

  8. Fri 02/22/2019: Grassmannians over finite fields: Gaussian elimination and row-reduced echelon form

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

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

  11. Fri 03/01/2019: Stirling numbers (cont'd). Set-partitions. Rook placements on triangular boards. Non-crossing and non-nesting set-partitions.

  12. Mon 03/04/2019: Problem Set 1 is due. Eulerian numbers. Increasing binary trees. 3 Pascal-like triangles: Eulerian triangles, Stirling triangles of 1st and 2nd kind.

  13. Wed 03/06/2019: Discussion of Problem Set 1. Volunteers can present solutions of some problems. Try to limit your presentations to 5-10 min.

  14. Fri 03/08/2019: Discussion of Problem Set 1 (cont'd).

  15. Mon 03/11/2019: Posets and lattices. Boolean lattice. Partition lattice. Young's lattice. Distributive lattices. Birkhoff's fundamental theorem for finite distributive lattices.

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

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

  18. Mon 03/18/2019:

  19. Wed 03/20/2019:

  20. Fri 03/22/2019:

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

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

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Last updated:   February 27, 2019