Lectures: Tuesday, Thursday 1:002:30
Room 2151 
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: 309
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
 qBinomial Coefficients, Gaussian coefficients, and Young Diagrams
 Partitions, Euler's pentagonal theorem, Jacobi triple product
 Noncrossing paths, Lindstrom lemma (GesselViennot method)

MatrixTree 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
Grading: Based on several Problems Sets.
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.)
[vLW]
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), 179191.
 D. E. Knuth: "Permutations, matrices, and generalized Young
tableaux" Pacific Journal of Mathematics 34 (1970) 709727.

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, 104109.

Lecture 4 (Thur 02/10/05)  qanalogues, qbinomial coefficients, qfactorials

[St] Section 6 "Young diagrams and qbinomial coefficients"

[vLW] Section 24 "Gaussian numbers and qanalogues"

Lecture 5 (Tues 02/15/05)  Symmetric group, statistics on permutations, inversions and major index

[EC1] Section 1.3 "Permutation statistics"

[vLW] 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 qbinomial coefficients", Section 8 "A glimpse of Young tableux".

Lecture 8 (Tues 03/01/05)  Sperner's and Dilworth's theorems

[vLW] Section 6 "Dilworth's theorem and extremal set theory"

[St] Section 4 "The Sperner property"

Lecture 9 (Thur 03/03/05)  De Bruijn sequences

[vLW] Section 8 "De Bruijn sequences"

Lecture 10 (Tues 03/08/05) 
Partitions: Euler's pentagonal theorem, Jacobi triple product

[vLW] Section 15 "Partitions"

Lecture 11 (Thur 03/10/05) 
Lindstrom lemma (GesselViennot 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, MatrixTree theorem, directed MatrixTree theorem
 [vLW] 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"
 [vLW] 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
 [vLW] 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 