18.312      Algebraic Combinatorics

Course description from the MIT Catalog:
Applications of algebra to combinatorics and conversely. Topics include enumeration methods, partially ordered sets and lattices, matching theory, partitions and tableaux, algebraic graph theory, and combinatorics of polytopes.

Units: 3-0-9           Level: advanced undergraduate

Topics:

• Catalan Numbers, Triangulations, Catalan Paths, Noncrossing Set Partitions
• Symmetric Group, Statistics on Permutations, Inversions and Major Index
• Partially Ordered Sets and Lattices, Sperner's and Dilworth's Theorems
• Young's Lattice, Tableaux, and Schensted's Correspondence
• q-Binomial Coefficients, Gaussian coefficients, and Young Diagrams
• Partitions, Euler's pentagonal theorem, Jacobi triple product
• Noncrossing paths, Lindstrom lemma (Gessel-Viennot method)
• Matrix-Tree Theorem, Electrical Networks, Walks in Graphs
• Lattice Paths and Continued Fractions
• Transportation and Birkhoff Polytopes, Cyclic Polytopes, Permutohedra
• Matching Enumeration, Pfaffians, Ising Model, Domino Tilings

Problem Sets

• Problem Set 1: [PS] [PDF] (due Tuesday 03/01/05)

• Problem Set 2: [PS] [PDF] (due Tuesday 03/15/05)

• Problem Set 3: [PS] [PDF] (due Tuesday 04/05/05)

• Problem Set 4: [PS] [PDF] (due Thursday 05/05/05) several problems added

Recommended Texts: (The students are not required to buy these books.)
[vL-W]  A course in Combinatorics by J. H. van Lint and R. M. Wilson, Cambridge University Press, 1992 (reprinted 1994, 1996).
[St]  Topics in Algebraic Combinatorics, Course notes by R. P. Stanley
[EC1] [EC2]  Enumerative Combinatorics, Vol 1 and Vol 2, by R. P. Stanley, Cambridge University Press, 1996 and 1999.

Lectures (tentative list):

• Lecture 1 (Tues 02/01/05) - Catalan numbers
  
  • Stanley's "Exercises on Catalan and related numbers" ([PS] [PDF]) and "Catalan addendum" ([PS] [PDF])

• Lecture 2 (Thur 02/03/05) - Pattern avoidance in permutations, Young tableux, Schensted correspondence, longest increasing subsequences
  
  • [St] Section 8 "A glimpse of Young tableaux".
    
  • 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.

• Lecture 3 (Tues 02/08/05) - The hooklength formula. Random hook walks. A "hooklength formula" for increasing trees.
  • C. Greene, A. Nijenhuis, H. Wilf: "A probabilistic proof of a formula for the number of Young tableaux of a given shape" Adv. in Math. 31 (1979), no. 1, 104--109.

• Lecture 4 (Thur 02/10/05) - q-analogues, q-binomial coefficients, q-factorials
  • [St] Section 6 "Young diagrams and q-binomial coefficients"
    
  • [vL-W] Section 24 "Gaussian numbers and q-analogues"

• Lecture 5 (Tues 02/15/05) - Symmetric group, statistics on permutations, inversions and major index
  • [EC1] Section 1.3 "Permutation statistics"
  • [vL-W] Section 13

• Lecture 6 (Thur 02/17/05) - Posets, lattices, distributive lattices, Young's lattice, differential posets
  • [EC1] Sections 3.1 -- 3.4

  Tues 02/22/05: no class (Monday schedule)

• Lecture 7 (Thur 02/24/05) - Up and Down operators, unimodality of Gaussian coefficients.
  • [St] Section 6 "Young diagrams and q-binomial coefficients", Section 8 "A glimpse of Young tableux".

• Lecture 8 (Tues 03/01/05) - Sperner's and Dilworth's theorems
  • [vL-W] Section 6 "Dilworth's theorem and extremal set theory"
  • [St] Section 4 "The Sperner property"

• Lecture 9 (Thur 03/03/05) - De Bruijn sequences
  • [vL-W] Section 8 "De Bruijn sequences"

• Lecture 10 (Tues 03/08/05) - Partitions: Euler's pentagonal theorem, Jacobi triple product
  • [vL-W] Section 15 "Partitions"

• Lecture 11 (Thur 03/10/05) - Lindstrom lemma (Gessel-Viennot method). Exponential formula
  • [EC2] Section 5.1 "Exponential formula"

• Lecture 12 (Tues 03/15/05) - Weighted lattice paths and continued fractions
  • I.P.Goulden, D.M.Jackson: Combinatorial Enumeration, John Wiley & Sons, 1983
    Section 5 "Combinatorics of Paths"

• Lecture 13 (Thur 03/17/05) - Review of Problem Set 1

  Mon 03/21/05 - Fri 03/25/05: no classes (Spring Break)

• Lecture 14 (Tues 03/29/05) - Review of Problem Set 2

• Lecture 15 (Thur 03/31/05) - Cayley's formula, Prufer's codes, Egecioglu and Remmel's bijection

• Lecture 16 (Tues 04/05/05) - Spanning trees, Matrix-Tree theorem, directed Matrix-Tree theorem
  • [vL-W] Section 2 "Trees", Section 34 "Electrical networks and squared matrices"

• Lecture 17 (Thur 04/07/05) - Electrical networks
  • [St] Section 11 "Cyles, bonds, and electrical networks"
  • [vL-W] Section 34 "Electrical networks and squared squares"

• Lecture 18 (Tues 04/12/05) - Review of Problem Set 3.

• Lecture 19 (Thur 04/14/05) - BEST Theorem. Permutohedra, Newton polytopes, zonotopes

  Tues 04/19/05: no class (Patriots Day)

• Lecture 20 (Thur 04/21/05) - Domino tilings of rectangles

• Lecture 21 (Tues 04/26/05) - Birkhoff polytope and Hall's marriage theorem
  • [vL-W] Section 5 "Systems of distinct representatives

• Lecture 22 (Thur 04/28/05) - Pfaffians and matching enumeration, Ising Model

• Lecture 23 (Tues 05/03/05) - Plane partitions, rombus tilings of hexagon, pseudoline arrangements
  • [EC2] Section 7.21 "Plane partitions with bounded part size"

• Lecture 24 (Thur 05/05/05) - Review of Problem Set 4.

• Lecture 25 (Tues 05/10/05) - Eulerian numbers and hypersimplices.

• Lecture 26 (Thur 05/12/05) - What next?

Last updated:   February 25, 2005