Course description
Philosophically a sequel to MAT315/515 Combinatorial Enumeration, but do not worry if you have not taken that course — all the basic objects are recalled here. The syllabus on AURIS is followed more or less, but there will be a lot of digressions.
We start with basic combinatorial objects that recur throughout, then move to tableau combinatorics: in particular a proof of the RSK correspondence and of the hook-length formula, two gems of modern combinatorics. The remainder of the course covers symmetric functions in considerable detail.
Grading
- Problem sets 60% — assigned regularly through the term
- End-semester exam 40% — 180 minutes, written
Grades of A and A− are at the instructor's discretion. Solving all the assigned problems and taking part in class will be reflected positively.
Lectures · 23
- 01 Wed 06 Jan Introduction to the course
- 02 Wed 15 Jan Monomials
- 03 Tue 21 Jan Young diagrams
- 04 Wed 22 Jan Euler's pentagonal number theorem
- 05 Tue 28 Jan Young's lattice, \(q\)-binomial coefficients
- 06 Wed 29 Jan \(q\)-Catalan numbers, skew partitions, compositions
- 07 Tue 04 Feb Words
- 08 Tue 11 Feb Hook-length formula
- 09 Wed 12 Feb Jeu de taquin
- 10 Tue 18 Feb Reverse insertion
- 11 Wed 19 Feb Symmetric functions
- 12 Tue 25 Feb Antisymmetric functions
- 13 Wed 26 Feb Fomin's growth diagrams
- 14 Tue 11 Mar Pieri rules
- 15 Wed 12 Mar Jacobi–Trudi identities
- 16 Tue 18 Mar Knuth equivalence
- 17 Wed 19 Mar Littlewood–Richardson rule
- 18 Tue 25 Mar Student presentations: equivalence of the definitions of Schur functions; determinant evaluations
- 19 Wed 26 Mar Student presentations: Murnaghan–Nakayama rule
- 20 Tue 01 Apr Student presentations: plane partitions (RSK and generating functions)
- 21 Wed 02 Apr Student presentations: plane partitions (non-intersecting lattice paths)
- 22 Tue 08 Apr Other classes of symmetric functions
- 23 Wed 09 Apr Guest lecture by Dr Bishal Deb
Submissions, late work and collaboration
Submissions
- Due at 11:59 pm on the due date.
- Typed in LaTeX, submitted as PDF or printed.
- Begin each solution on a new page.
- State your sources at the top of each problem, even if you worked independently.
Late policy
Late submissions are penalised 20% per day. No extensions, except for approved medical leave covering the day before and the day of the deadline.
Collaboration and sources
- Work on the problems independently first, then collaborate — meaningful collaboration is encouraged.
- You must write up your own solutions.
- At the top of each problem write "Collaborators and sources:" followed by everyone and everything you consulted, or "none". Failure to acknowledge costs 20% of that problem.
- Fine: looking up a standard theorem, formula or technique; using Wolfram Alpha, Mathematica or Python for a calculation.
- Not fine: looking up the problem itself online or in the literature, or using AI to solve the problem. Once you have solved it, seeking other solutions is welcome.
Intentional violations may be treated as academic misconduct.