2 7 M ay 1 99 8 ENUMERATION SCHEMES AND , MORE IMPORTANTLY , THEIR AUTOMATIC GENERATION

It is way too soon to teach our computers how to become full-fledged humans. It is even premature to teach them how to become mathematicians, it is even unwise, at present, to teach them how to become combinatorialists. But the time is ripe to teach them how to become experts in a suitably defined and narrowly focused subarea of combinatorics. In this article, I will describe my efforts to teach my beloved computer, Shalosh B. Ekhad, how to be an enumerator of Wilf classes. Just like the Wright Brothers' first flight, and Len Adelman's first DNA computer (and perhaps like the first Fleischmann-Pons Cold Fusion experiment), the actual accomplishments are very meager. But I do hope to demonstrate feasibility, and jump-start a research area that is not quite part of AI, and not quite algorithmic combinatorics. It is something brand-new, let's call it Artificial Combinatorics (AC). The closest analogy is Deep Blue, but instead of playing Chess, Shalosh will play a game called 'Wilf-class enumeration'. It is still very far from beating the Kasparovs of the area (to name a few), but I am sure that the ultimate version will. With all due respect to Wilf classes and Enumeration, and even to Combinatorics, the main point of this article is not to enhance our understanding of Wilf classes, but to illustrate how much (if not all) of mathematical research will be conducted in a few years. It goes as follow. Have a (as of now, human) mathematician get a brilliant idea. Teach that idea to a computer, and let the computer 'do research' using that idea. Accompanying Software In a sense, this article is a user's manual for the Maple package WILF, available from my homepage http://www.math.temple.edu/~ zeilberg (click on Maple packages and programs, and then download WILF). The empirical version of the rigorous WILF is HERB, also available there. Enumeration Schemes Suppose that we have to find a 'formula' (in the sense of Wilf[Wi], i.e. a polynomial-time algorithm) for computing a n := |A n |, where A n is an infinite family of finite sets, parameterized by n. Usually A n is a natural subset of a larger set B n , and is defined as the set of members of B n that satisfy a certain set of conditions C n. For example if A n is the set of permutations on {1, 2,. .. , n}, …