Self-avoiding walks on oriented square lattices

An analysis is undertaken of the self-avoiding walk problem on two distinct oriented square lattices. The importance of these two oriented cases to the problem on the unoriented square lattice is pointed out by means of a number of transformations showing the interdependence of various apparently different 'walk' problems on the oriented and unoriented square lattices. The total number of self-avoiding walks, C_N, and their mean-square sizes, (R_N²), are exactly enumerated on a computer up to 28 and 36 steps for the two oriented square lattices respectively. Rigorous upper and lower bounds together with estimates are presented for the connective constant mu ( mu identical to lim_{N to infinity} C_N¹N/) for both the oriented square lattices.