Lectures: Tuesday, Thursday 1:002:30 Room 2102 
Office hours: TR 45pm, Room 2389 
Course advertisement: PostScript
PDF
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
Past Lectures (with suggested reading):

Lecture 1 (02/05/02): Catalan numbers

[EC2] pp. 219229, 256265.
This can be found on the web at
"Exercises on Catalan and related numbers"
(PostSctipt
PDF).
See also "Catalan addendum"
(PostScript
PDF).

Lecture 2 (02/07/02): Pattern avoidance in permutations,
Young tableux, Schensted correspondence, longest increasing
subsequences

[TAC] Section 8 "A glimpse of Young tableaux".

C. Schensted "Longest increasing and decreasing subsequences"
Canadian Journal of Mathemetics 13 (1961), 179191.
(A copy is available upon request.)

Lecture 3 (02/12/02): qBinomial coefficients

[TAC] Section 6 "Young diagrams and qbinomial coefficients".

[CC] Section 24 "Gaussian numbers and qanalogues"

[EC1] Section 1.3 "Permutation statistics"

Lecture 4 (02/14/02): Labelled trees, Prufer's codes, increasing trees

[CC] Section 2 "Trees"

[EC1] Section 1.3 "Permutation statistics"

Lecture 5 (02/21/02): Symmetric group, statistics on permutations

[EC1] Section 1.3 "Permutation statistics"

Lecture 6 (02/26/02): Posets, lattices, distributive lattices, Young's lattice

[EC1] Sections 3.1  3.4

Lecture 7 (02/28/02):
Up and Down operators, unimodality of Gaussian coefficients.

[TAC] Section 6 "Young diagrams and qbinomial coefficients"
and Section 8 "A glimpse of Young tableux".

Lecture 8 (03/05/02): Sperner's and Dilworth's theorems

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

[TAC] Section 4 "The Sperner property"

Lecture 9 (03/07/02): De Bruijn sequences

[CC] Section 8 "De Bruijn sequences"

Lecture 10 (03/12/02):
Partitions: Euler's pentagonal theorem, Jacobi triple product

[CC] Section 15 "Partitions"

Lecture 11 (03/14/02):
Lindstrom lemma (GesselViennot method). Exponential formula
 [EC2] Section 5.1 "Exponential formula"

Lecture 12 (03/19/02): Review of Problem Sets 1 and 2

Lecture 13 (03/21/02): Plane partitions, rombus tilings of hexagon,
pseudoline arrangements
 [EC2] Section 7.21 "Plane partitions with bounded part size"

Lecture 14 (04/02/02): Spanning trees, greedy algorithm, MatrixTree theorem
 [CC] Section 2 "Trees"
 [CC] Section 34 "Electrical networks and squared matrices"

Lecture 15 (04/04/02): MatrixTree theorem (cont'd)
 [CC] Section 34 "Electrical networks and squared matrices"

Lecture 16 (04/09/02): Review of Problem Set 3.

Lecture 17 (04/11/02): Weighted lattice paths and continued fractions
 I.P.Goulden, D.M.Jackson: Combinatorial Enumeration,
John Wiley & Sons, 1983
Section 5 "Combinatorics of Paths"

Lecture 18 (04/18/02): Electrical networks
 [TAC] Section 11.3 "Electrical networks"
 [CC] Section 34 "Electrical networks and squared squares"

Lecture 19 (04/23/02): Chromatic polynomial, acyclic orientations,
Mobius function
 [CC] Section 25 "Lattices and Mobius inversion"

Lecture 20 (04/25/02): Permutohedra, Newton polytopes, zonotopes

Lecture 21 (04/30/02): Birkhoff polytope and Hall's marriage theorem.
 [CC] Section 5 "Systems of distinct representatives

Lecture 22 (05/02/02): Domino tilings of rectangles

Lecture 23 (05/07/02): TBA

Lecture 24 (05/09/02): Guest lecture by James Propp

Lecture 25 (05/14/02): TBA
Recommended Texts:
(The students are not required to have these books.)
[TAC]
Topics in Algebraic Combinatorics,
Course notes by R. P. Stanley for Mathematics 192
(Algebraic Combinatorics), Harvard University, Fall 2000.
PostScript
PDF
[CC]
A course in Combinatorics by J. H. van Lint and R. M. Wilson,
Cambridge University Press, 1992 (reprinted 1994, 1996).
[EC1]
Enumerative Combinatorics, Vol 1 by R. P. Stanley,
Wadsworth and Brooks/Cole,
Pacific Grove, CA, 1986; second printing, Cambridge University Press, 1996.
[EC2]
Enumerative Combinatorics, Vol 2 by R. P. Stanley,
Cambridge University Press, 1999.
Information on [EC1] and
[EC2].
Problem Sets
Tentative List of Topics:
 Catalan Numbers, Triangulations, Catalan Paths, Noncrossing Partitions
 Walks in Graphs, Random Walks on the nCube, Spectra of Graphs
 Symmetric Group, Statistics on Permutations,
Inversions and Major Index
 Partially Ordered Sets and Lattices, Sperner's Theorem
 qBinomial Coefficients, Gaussian coefficients, and Young Diagrams
 Young's Lattice, Tableaux, and Schensted's Correspondence
 Trees, Parking Functions, and Prufer Codes
 MatrixTree Theorem, Electrical Networks
 Enumeration under Group Action, Burnside's Lemma, Polya Theory
 Transportation and Birkhoff Polytopes, Cyclic Polytopes, Permutohedra
 Matching Enumeration, Pfaffians, Ising Model, Domino Tilings
Grading Bonuses:

There will be several bonus problems.

You can get a grading bonus if you invent a new interesting integer sequence
and submit it to the
OnLine
Encyclopedia of Integer Sequences. You can get up to 5% of
the final grade for each new entry in the Encyclopedia.
Last updated: February 8, 2002 