Counting all self-avoiding walks on a finite lattice strip of width one and two

Abstract

In this paper, a closed-form expression for counting all SAWs, irrespective of length, but restricted to the finite lattice strip $\{ -a,\ldots ,0,\ldots ,b\}\times \{0,1\}$, shall be obtained in terms of the non-negative integer parameters $a$ and $b$. In addition, the argument used to prove this result will be extended to establish an enumerating formula for counting all SAWs, irrespective of length, but restricted to the half-finite lattice strip of width two $\{ 0,1,\ldots ,n\}\times \{ 0,1,2\}$, in terms of $n$.