This award supports the research of Professor Doron Zeilberger in the areas of algebraic combinatorics, the theory of special functions, and computer algebra. One direction of this project is a continuation of his research program on the use of computer algebra for computerized proofs of identities in combinatorics and the theory of special functions. Other goals of this project include the development of a general theory of permutation statistics, the study of holonomic aspects of combinatorial statistical mechanics, and the extension of methods developed in the principal investigator's recent proof of the alternating sign matrix conjecture to other, still open, problems in this area. Because both special functions and combinatorics are so prevalent in the natural sciences, this research has many potential applications to both science and technology, as well as to pure mathematics. This research is in the general area of combinatorics. Combinatorics attempts to find efficient methods to study how discrete collections of objects can be arranged. The behavior of discrete systems is extremely important to modern communications. For example, the design of large networks, such as those occurring in telephone systems, and the design of algorithms in computer science deal with discrete sets of objects, and this makes use of combinatorial research.
