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

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: 3-0-9           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
• q-binomial 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 Gessel-Viennot method)
• spanning trees, parking functions, and Prufer codes
• matrix-tree 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...

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 n-cube 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

[vL-W]  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):

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
   • Stanley's Exercises on Catalan and related numbers and Catalan addendum

3. Mon 02/11/2019: Pattern avoidance in permutations, stack- and queue-sortable permutations, Young diagrams and Young tableaux, the hook-length formula
   • [AC] Chapter 8: A glimpse of Young tableaux
   • If you want to learn more details about the links between combinatorics of Young tableaux and representation theory, see The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions by B. Sagan.

4. Wed 02/13/2019: Frobenius-Young identity, Schensted correspondence, longest increasing and decreasing subsequences in permutations
   • [AC] Appendix 1 to Chapter 8: The RSK algorithm
   • C. Schensted: Longest increasing and decreasing subsequences Canadian Journal of Mathemetics 13 (1961), 179-191.
   • D. E. Knuth: Permutations, matrices, and generalized Young tableaux Pacific Journal of Mathematics 34 (1970) 709-727.
   • C. Greene: An extension of Schensted's theorem Advances in Mathematics 14 (1974), 254-265.

5. Fri 02/15/2019: Proof of the hook-length formula based on a random hook walk
   • C. Greene, A. Nijenhuis, H. Wilf: A probabilistic proof of a formula for the number of Young tableaux of a given shape, Advances in Mathematics 31 (1979), no. 1

   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.
   • D. E. Knuth: The Art of Computer Programming, Volume 3: Sorting and Searching, 1973, Section 5.1.4.

7. Wed 02/20/2019: q-factorials and q-binomial coefficients
   • [AC] Chapter 6: Young diagrams and q-binomial coefficients

8. Fri 02/22/2019: Grassmannians over finite fields: Gaussian elimination and row-reduced echelon form
   • [EC1] Section 1.7: Permutations of multisets (see Propositions 1.7.2 and 1.7.3)

9. Mon 02/25/2019: Sets and multisets. Statistics on permutations: inversions, cycles, descents.
   • [EC1] 1.2 Sets and multisets, 1.3 Cycles and inversions, 1.4 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.
    • [EC1] Chapter 3: Partially Ordered Sets: 3.1 Basic Concepts, 3.2 New Posets from Old, 3.3 Lattices

16. Wed 03/13/2019: Distributive lattices. Birkhoff's fundamental theorem for finite distributive lattices.
    • [EC1] 3.4 Distributive Lattices

17. Fri 03/15/2019: Sperner's property. Symmetric chain decompositions. Sperner's and Dilworth's theorems. Greene's theorem.
    • [AC] Chapter 4: The Sperner property

18. Mon 03/18/2019: Greene's theorem vs Schensted correspondence. Up and down operators. Differential posets.
    • [EC1] 3.21 Differential Posets

19. Wed 03/20/2019: Differential posets (cont'd). Fibonacci lattice. Unimodality of Gaussian coefficients.
    • [AC] Chapter 6: Young diagrams and q-binomial coefficients (see Corollary 6.10).

20. Fri 03/22/2019: Proof of unimodality of Gaussian coefficients (cont'd). Theory of partitions. Euler's pentagonal number theorem.
    • [EC1] 1.8 Partition Identities

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

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

22. 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.

23. Fri 04/05/2019: Problem Set 2 is due. Two combinatorial proofs of Cayley's formula.
    • [AC] Chapter 9. Appendix: Three Elegant Combinatorial Proofs
    • O. Egecioglu, J. B. Remmel: Bijections for Cayley trees, spanning trees, and their q-analogues, J. Combinatorial Theory, Ser. A, 42 (1986), 15-30.

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

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

26. 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)

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

28. Fri 04/19/2019: Proof of Matrix Tree Theorem using Cauchy-Binnet formula. Weighted and directed version of Matrix Tree Theorem. In-trees and out-trees.

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

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

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

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

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

34. Fri 05/03/2019:

35. Mon 05/06/2019:

36. Wed 05/08/2019:

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

38. Mon 05/13/2019:

39. Wed 05/15/2019:

Last updated:   May 2, 2019