Math 731 — Fall 2001 Enumerative Combinatorics

A general introduction to algebraic combinatorics with a particular emphasis on methods of enumeration in ordered and geometric structures (partially ordered sets, complexes and polytopes). Topics to be included (as time permits) are:

• General techniques of generating functions,
   
• Partially ordered sets (posets) and lattices,
   
• Möbius inversion (inclusion-exclusion over posets),
   
• Posets as topological objects,
   
• Generating functions of combinatorial objects as Hilbert functions of algebraic objects, and the influence of topological properties on both.
   
• An introduction to quasisymmetric functions and their relevance to all of the above.

I 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 as they are needed. Much (but not all) of what we do can be found in Stanley's book Enumerative Combinatorics, Vol. I (Cambridge, 1997), Chapters 1 and 3.