Automating Combinatorics

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.