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
• 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