A technique for random walk enumeration : counting the elementary paths on a rectangular grid

A new method is presented for the construction and analysis of non self-intersecting random walks which: (a) makes economic use of digital computer time, both in the exact enumeration of walks and in applying Monte-Carlo methods; and (b) appears to provide a sympathetic environment for analytical investigations such as obtaining the upper bound for the number of paths on a grid. An algorithm for the enumeration of elementary paths on a rectangular grid is presented and discussed; a program based on this algorithm is exhibited. This program was written for a square, n x n, grid and results are given for n = 2, 3, 4, and 5.