Compact enumeration formulas for generalized partitions

Counting the number of elements in finite sets is surely one of the oldest and most fundamental problems in mathematics. It is in the nature of the subject that only a few enumeration problems have a compact solution in terms of a simple explicit formula. More surprisingly, combinatorialists can still hardly predict when this rather rare event that an enumeration problem has a nice and elegant formula occurs. This project is centered around plane partitions, alternating sign matrices and related objects, the enumeration of which subject to a variety of different constraints lead to formulas that are, on the one hand, of compelling simplicity, but, on the other hand, usually still require highly nontrivial proofs. However, the significance of these objects is also due to their close relations to various other areas such as representation theory of classical groups and statistical mechanics. Using our approach that has successfully been applied to give another, elementary proof of the alternating sign matrix theorem, we will attack a number of refined enumerations of alternating sign matrices and symmetry classes thereof. On the other hand, as enumeration problems on plane partitions and alternating sign matrices can usually be formulated as integer point enumerations in rational convex polytopes, we propose a geometric point of view on these problems and aim to study applications of the theory of vector partition functions to these enumeration problems. Lastly, as the objects under consideration are highly related but so far these relations are often not well explained, we will seek for bijective explanations of these connections.