Compact enumeration formulas for generalized partitions

Efficient counting of (various types of somewhat regular) objects may belong to one of the most fundamental skills society expects from mathematicians. By efficient counting we refer to a procedure that is less time consuming than the naive approach, where we would simply make a list of all objects and then count them. In this project we have solved various enumeration problems in the presumably most efficient manner, and, along the way, invented new methods for accomplishing such results, which is probably the more important contribution in the long run. The counting procedure is indisputable very efficient if we manage to derive a simple formula for the number of objects that involves, say, only the basic mathematical operations. For a generic enumeration problem this is however not possible. The project was centered around combinatorial objects such as plane partitions, alternating sign matrices and their relatives. Counting problems related to such objects ever now and then admit the nicest type of enumeration formulaclosed product formulashowever the proofs have been notorious for their difficulty as well as for the lack of insight they provide with regard to when to expect a simple formula. This was the main motivation for the research undertaken in this project. In the past eight years, we have proved a variety of new explicit enumeration formulas in this eld. Notably we have been able to settle the last conjecture on the enumeration of symmetry classes of alternating sign matrices involving a closed product formula, which was discovered by Robbins in the 1980s upon a suggestion by Stanley. This conjecture was probably the most important open problem (depending on ones taste of course) in this eld involving a closed product formula before the beginning of the project. Excitingly, we actually found a whole new zoo of subclasses of alternating sign matrices that are also counted by simple product formulas. These are actually formulas that had appeared before, e.g., for the number of all alternating sign matrices, or for the number of certain subclasses of plane partitions. This circumstance is far from being obvious. This contributes to another important and extremely challenging classical topic, namely the search for explicit explanations of relations between the different types of objects that are counted by the same formula. Such problems are naturally approached by identifying as many equidistributed statistics as possible on the two types of objects. Previously, we had to rely very much on our intuition (or patience and ability to write computer programs) to guess such statistics, but as another outcome of the project we have developed methods that have the potential to help to construct such parameters in a more systematically way.