Math 118

Combinatorics

Last updated June 27, 2016 13:25:41 EDT

## Tentative syllabus

This is a tentative list of topics that I plan to cover, with references to the main sources that I will use.

- Partitions [EC 1.8, BS 2, AE, An, Br, Notes]

- Generating functions for partitions
- Euler's pentagonal number theorem
- Jacobi's triple product identity
- Rogers-Ramanujan identities [For applications to physics, see this article.]
- Bijective proofs of classical partition identities, partitions avoiding multisets of parts
- The involution principle [EC 2.6]
- Examples of the involution principle
- Lecture hall partitions

- Tableaux, plane partitions, tilings, and lattice paths [Br, BS 3-9, EC, Notes]

- Plane partitions, MacMahon's theorem [BS 3]
- Refinements of MacMahon's theorem
- Symmetric, cyclically symmetric, and totally symmetric plane partitions [Br]
- P-partitions [EC 3.15]
- Reverse plane partitions, the Hillman-Grassl bijection, the hook-length formula
- Reduced decompositions, the Edelman-Greene bijection [BS 7]
- Domino tilings of rectangles and Aztec diamonds [BS 8], connections of tilings to plane partitions [BS 9]
- Determinants, the Gessel-Viennot formula [EC 2.7]
- Labeled trees, parking functions
- Inclusion-exclusion [EC 2.1]

** Student presentations **
- Friday, May 10: Kassie and Tim,
* A combinatorial proof of the Rogers-Ramanujan identities. *
- Monday, May 13: Zach,
* Generalized vexillary permutations and Stanley symmetric functions. *
- Monday, May 20: Jenny,
* The enumeration of lecture hall partitions. *
- Wednesday, May 22: Megan,
* The alternating sign matrix conjecture. * Jonathan, * Determinants, nonintersecting paths, and Young tableaux. *
- Friday, May 24: Seth,
* A combinatorial proof of the unimodality of the q-binomial coefficients. *
- Wednesday, May 29: Eva,
* A bijection between k-triangulations and k-tuples of non-intersecting Dyck paths. * David, * Permutohedra and associahedra. *

Last updated June 27, 2016 13:25:41 EDT