Self-avoiding walk enumeration via the lace expansion

We introduce a new method for the enumeration of self-avoiding walks based on the lace expansion. We also introduce an algorithmic improvement, called the two-step method, for self-avoiding walk enumeration problems. We obtain significant extensions of existing series on the cubic and hypercubic lattices in all dimensions d >= 3: we enumerate 32-step self-avoiding polygons in d = 3, 26-step self-avoiding polygons in d = 4, 30-step self-avoiding walks in d = 3, and 24-step self-avoiding walks and polygons in all dimensions d >= 4. We analyze these series to obtain estimates for the connective constant and various critical exponents and amplitudes in dimensions 3 <= d <= 8. We also provide major extensions of 1/d expansions for the connective constant and for two critical amplitudes.