18.312      Algebraic Combinatorics

Time: Tuesday, Thursday 11:00-12:30      Place: Room 4-159

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, 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, Schensted's Correspondence, RSK
• q-Binomial Coefficients, Gaussian coefficients, and Young Diagrams
• Partitions, Euler's pentagonal theorem, Jacobi triple product
• Noncrossing paths, Lindstrom lemma (aka 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: TBA

Problem Set 1 (due Tuesday 03/05/13)
• Problem Set 2 (due Tuesday 04/09/13)
• Problem Set 3 plus additional problems (due Thursday 05/16/13)

Recommended Texts: (The students are not required to buy these books.)

[AC]  Algebraic Combinatorics by R. P. Stanley, forthcoming book: pdf file

[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. (Click title page to read it online.)

Lectures (with links to additional reading materials):

• Lecture 1 (Tues 02/05/2013) - Catalan numbers
  
  • Stanley's Exercises on Catalan and related numbers and Catalan addendum

• Lecture 2 (Thur 02/07/2013) - Permutation patterns, Young tableux, Schensted correspondence
  
  • C. Schensted, Longest increasing and decreasing subsequences, Canadian Journal of Mathemetics 13 (1961), 179-191: pdf file
  • D. E. Knuth, Permutations, matrices, and generalized Young tableaux, Pacific Journal of Mathematics 34 (1970) 709-727: pdf file

• Lecture 3 (Tues 02/12/2013) - The hooklength formula. Random hook walks.
  
  • 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

• Lecture 4 (Thur 02/14/2013) - q-analogues, q-binomial coefficients, q-factorials
  
  • [EC1] Section 1.7 "Permutations of multisets": pdf file
  • [vL-W] Section 24 "Gaussian numbers and q-analogues": link

  Tues 02/19/2013: no class (Monday schedule)

• Lecture 5 (Thur 02/21/2013) - Permutations, wiring diagrams, statistics on permutations (inversions, cycles, records)
  

• Lecture 6 (Tues 02/26/2013) - Stirling numbers, set-partitions, rook placements on triangular boards, more statistics on permutations (descents, excedances, major index)
  

• Lecture 7 (Thur 02/28/2013) - Posets and lattices. Distributive lattices. FTFDL
  
  • [EC1] Sections 3.1--3.4

• Lecture 8 (Tues 03/05/2013) - Sperner's Theorem. Symmetric chain decompositions (SCD)
  
  • [AC] Chapter 4 "The Sperner property"

• Lecture 9 (Thur 03/07/2013) - Up and down operators. Unimodality of Gaussian coefficients
  
  • [AC] Chapter 8 "A glimpse of Young tableux".

• Lecture 10 (Tues 03/12/2013: discussion of Problem Set 1.

• Lecture 11 (Thur 03/14/2013) - ... more on unimodality. Differential posets. Fibonacci lattice.
  

• Lecture 12 (Tues 03/19/2013) - De Bruijn sequences
  
  • [vL-W] Section 8 "De Bruijn sequences"

• Lecture 13 (Thur 03/21/2013) - Theory of partitions. Partitions with distinct and odd parts. Euler's pentagonal theorem. Jacobi's triple product.
  
  • [vL-W] Section 15 "Partitions"

  Mon 03/25/2013 - Fri 03/29/2013 no classes (Spring Break)

• Lecture 14 (Tues 04/02/2013) -
  

• Lecture 15 (Thur 04/04/2013) -
  

• Lecture 16 (Tues 04/09/2013) -
  

• Lecture 17 (Thur 04/11/2013) -
  

  Tues 04/16/2013: no class (Patriots Day)

• Lecture 18 (Thur 04/18/2013) -
  

  Lecture 19 (Tues 04/23/2013) -
  

• Lecture 20 (Thur 04/25/2013) -
  

• Lecture 21 (Tues 04/30/2013) -
  

• Lecture 22 (Thur 05/02/2013) -
  

• Lecture 23 (Tues 05/07/2013) -
  

• Lecture 24 (Thur 05/09/2013) -
  

• Lecture 25 (Tues 05/14/2013) -
  

• Lecture 26 (Thur 05/16/2013) -
  

Last updated:   February 4, 2013