Doron Zeilberger proposes to continue to develop methodologies for harnessing the great potential of Computer Algebra to do research in Combinatorics and related areas, and design experiments for (rigorous) computer-assisted and computer-generated research. In particular he hopes to develop a new algorithmic theory to be named `Symbolic Moment Calculus', that would produce algorithms for computing, symbolically and automatically, (statistical) moments of interesting combinatorial quantities. He also plans to develop a general theory of recurrence equations that would include both the `Dynamic Programming' recurrences featuring the maximum, ubiquitous in Computational Biology, and the mex operation, that occurs in the theory of Combinatorial Games. He also proposes to continue his efforts in `Artificial Combinatorics', and develop algorithms for the discovery and rigorous proof of enumeration schemes for counting permutation classes, and for automatically deducing generating functions. Another line of research concerns automatic determinant evaluations that has potential applications in applied mathematics and science.
This research should be symbiotic, as it is expected that both the concrete results and the underlying methodologies, would help computer algebra developers to improve and enhance their systems. It is also hoped that this research will contribute to the budding field of Experimental Mathematics, in that it will help develop a research methodology for conducting computer experiments that output rigorous (and interesting!) mathematical theorems (and proofs), rather than just verifying and formulating conjectures.
This research is in the field of Combinatorics, whose usefulness to science and technology is well-known. In particular, computer science is largely based on combinatorics, as is electronic communication and the World Wide Web.
