Key
[AC] = Stanley, Richard P. Algebraic Combinatorics: Walks, Trees, Tableaux, and More. Springer, 2018. ISBN: 9783319771724. Online 2013 version.
[EC1] = ———. Enumerative Combinatorics. Vol. 1. Cambridge, UK: Cambridge University Press, 1997. ISBN: 9780521553094. Online version.
[EC2] = ———. Enumerative Combinatorics. Vol. 2. Cambridge, UK: Cambridge University Press, 2001. ISBN: 9780521789875.
Lecture sessions with reading assignments are listed below.
| LEC # | TOPICS | READINGS |
|---|---|---|
|
2 |
Catalan numbers (cont.): formula for Cn, reflection principle, necklaces, triangulations of polygons, plane binary trees, parenthesizations. |
Stanley, Richard P. “Exercises on Catalan and Related Numbers” (PDF). (excerpted from [EC2]) Stanley, Richard P. “Catalan Addendum” (PDF). |
|
3 |
Pattern avoidance in permutations. Stack- and queue-sortable permutations. Young diagrams and Young tableaux. Hook-length formula. |
[AC] “Chapter 8: A Glimpse of Young Tableaux” (PDF - 1.2MB). If you want to learn more details about the links between combinatorics of Young tableaux and representation theory, see Sagan, Bruce E. The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions. Springer, 2001. ISBN: 9780387950679. |
|
4 |
Frobenius-Young identity. Schensted correspondence. Longest increasing and decreasing subsequences in permutations. |
[AC] “Appendix 1 to Chapter 8: The RSK Algorithm” (PDF - 1.2MB). Schensted, C. “Longest Increasing and Decreasing Subsequences.” Canadian Journal of Mathemetics 13 (1961), 179-191. Knuth, Donald E. “Permutations, Matrices, and Generalized Young Tableaux” (PDF). Pacific Journal of Mathematics 34 (1970), 709-727. Greene, Curtis. “An Extension of Schensted’s Theorem” (PDF). Advances in Mathematics 14 (1974), 254-265. |
|
5 |
Proof of the hook-length formula based on a random hook walk. |
Greene, Curtis, Albert Nijenhuis, and Herbert Wilf. “A Probabilistic Proof of a Formula for the Number of Young Tableaux of a Given Shape” (PDF). Advances in Mathematics 31 (1979), no. 1. |
|
6 |
Hook walks (cont.). Linear extensions of posets. Hook-length-type formulas for shifted shapes and trees. |
Knuth, Donald E. The Art of Computer Programming, Volume 3: Sorting and Searching. Addison-Wesley Professional, 1998. Section 5.1.4. ISBN: 9780201896855. |
|
7 |
q-factorials and q-binomial coefficients. |
[AC] “Chapter 6: Young Diagrams and q-binomial Coefficients” (PDF - 1.2MB). |
|
8 |
Grassmannians over finite fields: Gaussian elimination and row-reduced echelon form. |
[EC1] “Section 1.7: Permutations of Multisets” (PDF - 4.4MB). (see Propositions 1.7.2 and 1.7.3) |
|
9 |
Sets and multisets. Statistics on permutations: inversions, cycles, descents. |
[EC1] “1.2 Sets and Multisets” (PDF - 4.4MB), “1.3 Cycles and Inversions” (PDF - 4.4MB), and “1.4 Descents” (PDF - 4.4MB). |
|
15 |
Posets and lattices. Boolean lattice. Partition lattice. Young’s lattice. |
[EC1] Chapter 3: Partially Ordered Sets: “3.1 Basic Concepts” (PDF - 4.4MB), “3.2 New Posets from Old” (PDF - 4.4MB), and “3.3 Lattices” (PDF - 4.4MB). |
|
16 |
Distributive lattices. Birkhoff’s fundamental theorem for finite distributive lattices. |
|
|
17 |
Sperner’s property. Symmetric chain decompositions. Sperner’s and Dilworth’s theorems. Greene’s theorem. |
|
|
18 |
Greene’s theorem vs Schensted correspondence. Up and down operators. Differential posets. |
|
|
19 |
Differential posets (cont.). Fibonacci lattice. Unimodality of Gaussian coefficients. |
[AC] “Chapter 6: Young Diagrams and q-binomial Coefficients” (PDF - 1.2MB). (see Corollary 6.10) |
|
20 |
Proof of unimodality of Gaussian coefficients (cont.). Theory of partitions. Euler’s pentagonal number theorem. |
|
|
23 |
Two combinatorial proofs of Cayley’s formula. |
[AC] Chapter 9. “Appendix: Three Elegant Combinatorial Proofs” (PDF - 1.2MB). Egecioglu, Ömer and Jeffrey B. Remmel. “Bijections for Cayley Trees, Spanning Trees, and Their q-analogues.” J. Combinatorial Theory, Ser. A, 42 (1986), 15-30. |
