Topics:
A general introduction to algebraic combinatorics with a particular
emphasis on methods of enumeration in ordered and geometric
structures
(partially ordered sets, simplicial complexes and polytopes).
Possible topics include (but are not limited to)
- Permutations and partitions,
- Partially ordered sets (posets) and lattices,
- Möbius inversion (inclusion-exclusion over posets),
- Posets as topological objects,
- Geometric lattices, including their relationship to hyperplane
arrangements,
- Generating functions of combinatorial objects as Hilbert functions
of algebraic objects, and the influence of geometric/topological
properties on both.
We will assume only a basic knowledge of linear algebra and ring theory (say at the level of Math 433-4) and will develop the necessary ideas from commutative algebra and topology as they are needed.
About 60% of what we will do can be found in Stanley's book Enumerative Combinatorics, Vol. I (Cambridge, 2012).
Homework: If you want a letter grade (as opposed to S/U or Audit), then you will have to do 6 of the available hw problems. They can be done anytime before classes end. Problems will be available approximately once every two weeks.
Hw3 Here is a complete proof of the Dehn-Sommerville relations for the toric h-vector of semi-Eulerian posets. Included are three additional problems.