Enumerating Up-Side Self-Avoiding Walks on Integer Lattices

A self-avoiding walk (saw) is a path on a lattice that does not pass through the same point twice. Though mathematicians have studied saws for over fifty years, the number of n-step saws is unknown. This paper examines a special case of this problem, finding the number of n-step "up-side'' saws (ussaws), saws restricted to moving up and sideways. It presents formulas for the number of n-step ussaws on various lattices, found using generating functions with decomposition and recursive methods.Mathematic

Self-Avoiding Walks and Fibonacci Numbers

By combinatorial arguments, we prove that the number of self-avoiding walks on the strip {0, 1} × Z is 8Fn − 4 when n is odd and is 8Fn − n when n is even. Also, when backwards moves are prohibited, we derive simple expressions for the number of length n self-avoiding walks on {0, 1} × Z, Z × Z, the triangular lattice, and the cubic lattice