ar X iv : h ep - l at / 9 21 10 62 v 1 2 7 N ov 1 99 2 Algebraic Techniques for Enumerating Self - Avoiding Walks on the Square Lattice

We describe a new algebraic technique for enumerating self-avoiding walks on the rectangular lattice. The computational complexity of enumerating walks of N steps is of order 3 N/4 times a polynomial in N , and so the approach is greatly superior to direct counting techniques. We have enumerated walks of up to 39 steps. As a consequence, we are able to accurately estimate the critical point, critical exponent, and critical amplitude.