Enumerations in statistical mechanics and combinatorics

Mathematicians require proofs, mathematical physicists are content to know that a result is exact, and physicists require only a numerical approximation adequate for their purposes. This caricature is becoming increasingly inaccurate as computers become capable of refining approximations to the extent that we can make exact conjectures. Once you have a conjecture that you believe is exact, providing a proof is generally easier. Furthermore, computer programs that can provide proofs, or exact conjectures, or even conjectured analytic information, are becoming more widespread and increasingly powerful. The article focuses primarily on the methods that the author has developed that provide the raw material for such conjectures, i.e., the enumerative techniques that produce a generating function´s early terms. Using polyomino enumeration and the self-avoiding walk problem as examples, the author shows how to produce enough terms of the generating function to enable soundly based conjectures about that function´s analytic properties