Lecture I. Plane Partitions, Determinants, and Hypergeometric Series
I plan to give a survey of the enumeration theory of plane partitions. I will outline the now classical approach to enumerate plane partitions, via nonintersecting lattice paths, which leads to expressions in the form of determinants or Pfaffians. Finally I will address the main techniques in evaluating these determinants, in which identities for hypergeometric series play an important role.
Lecture II. Beyond Plane Partitions: Some Applications of Orthogonal Polynomials and Multiple Hypergeometric Series
There are two combinatorial objects which are intimately related to plane partitions: tilings and alternating sign matrices. (For the second kind of objects this is not really understood.) On several occasions, the solution of enumeration problems involving these kinds of objects required essentially knowledge from special function theory. In the first part, I will consider two such cases in which orthogonal polynomials feature prominently: the enumeration of rhombic tilings which contain a fixed rhombus, and Zeilberger's solution of the refined alternating sign matrix conjecture. In the second part, I will consider two such cases in which identities for multiple hypergeometric series (a subject which, in my opinion, deserves much more attention) feature prominently: the enumeration of cylindric partitions and of tilings of Aztec rectangles.