| Lec # | Topics | Readings |
|---|---|---|
| 1 | Catalan Numbers | [EC1] [EC2] Stanley, R. P. "Exercises on Catalan and Related Numbers." Stanley, R. P. "Catalan Addendum." |
| 2 | Pattern Avoidance in Permutations, Young Tableaux, Schensted Correspondence, Longest Increasing Subsequences | [St] Section 8: "A Glimpse of Young Tableaux." |
| 3 | The Hooklength Formula Random Hook Walks A "Hooklength Formula" for Increasing Trees | Greene, C., A. Nijenhuis, and H. Wilf. "A Probabilistic Proof of a Formula for the Number of Young Tableaux of a given Shape." Adv. in Math. 31, no. 1 (1979): 104-109. |
| 4 | 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." |
| 5 | Symmetric Group, Statistics on Permutations, Inversions and Major Index | [EC1] Section 1.3: "Permutation Statistics." [vL-W] Section 13 |
| 6 | Posets, Lattices, Distributive Lattices, Young's Lattice, Differential Posets | [EC1] Sections 3.1-3.4 |
| 7 | Up and Down Operators, Unimodality of Gaussian Coefficients | [St] Section 6: "Young Diagrams and q-binomial Coefficients." Section 8: "A Glimpse of Young Tableaux." |
| 8 | Sperner's and Dilworth's Theorems | [vL-W] Section 6: "Dilworth's Theorem and Extremal Set Theory." [St] Section 4: "The Sperner Property." |
| 9 | De Bruijn Sequences | [vL-W] Section 8: "De Bruijn Sequences." |
| 10 | Partitions: Euler's Pentagonal Theorem, Jacobi Triple Product | [vL-W] Section 15: "Partitions." |
| 11 | Lindstrom Lemma (Gessel-Viennot Method) Exponential Formula | [EC2] Section 5.1: "Exponential Formula." |
| 12 | Weighted Lattice Paths and Continued Fractions | Goulden, I. P., and D. M. Jackson. "Combinatorics of Paths." Section 5 in Combinatorial Enumeration. New York, NY: John Wiley & Sons, 1983. ISBN: 0471866547. |
| 13 | Review of Problem Set 1 | |
| 14 | Review of Problem Set 2 | |
| 15 | Cayley's Formula, Prufer's Codes, Egecioglu and Remmel's Bijection | |
| 16 | Spanning Trees, Matrix-Tree Theorem, Directed Matrix-Tree Theorem | [vL-W] Section 2: "Trees." Section 34: "Electrical Networks and Squared Matrices." |
| 17 | Electrical Networks | [St] Section 11: "Cyles, Bonds, and Electrical Networks." [vL-W] Section 34: "Electrical Networks and Squared Squares." |
| 18 | Review of Problem Set 3 | |
| 19 | BEST Theorem Permutohedra, Newton Polytopes, Zonotopes | |
| 20 | Domino Tilings of Rectangles | |
| 21 | Birkhoff Polytope and Hall's Marriage Theorem | [vL-W] Section 5: "Systems of Distinct Representatives." |
| 22 | Pfaffians and Matching Enumeration, Ising Model | |
| 23 | Plane Partitions, Rhombus Tilings of Hexagon, Pseudoline Arrangements | [EC2] Section 7.21: "Plane Partitions with Bounded Part Size." |
| 24 | Review of Problem Set 4 | |
| 25 | Eulerian Numbers and Hypersimplices | |
| 26 | What Next? |
Table of Contents
Study Materials
Readings
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Some of the readings are based on the suggested textbooks, which are shown in the table as:
[vL-W] = van Lint, J. H., and R. M. Wilson. A Course in Combinatorics. Cambridge, U.K.: Cambridge University Press, 1992. ISBN: 0521422604. (Reprinted 1994, 1996.)
[St] = Stanley, R. P., Sergey Fomin, B. Bollobas, W. Fulton, A. Katok, F. Kirwan, P. Sarnak, and B. Simon. Course notes on Topics in Algebraic Combinatorics. (PDF) (Courtesy of R. P. Stanley. Used with permission.)
[EC1] [EC2] = Stanley, R. P., Sergey Fomin, B. Bollobas, W. Fulton, A. Katok, F. Kirwan, P. Sarnak, and B. Simon. Enumerative Combinatorics. Vol. 1 and 2. Cambridge, U.K.: Cambridge University Press, 2001. ISBN: 0521789877.