The number of polygons on a lattice

In this paper an n-stepped self-avoiding walk is defined to be an ordered sequence of n + 1 mutually distinct points, each with (positive, negative, or zero) integer coordinates in d -dimensional Euclidean space (where d is fixed and d ≥ 2), such that any two successive points in the sequence are neighbours , i.e. are unit distance apart. If further the first and last points of such a sequence are neighbours, the sequence is called an ( n + 1)- sided self-avoiding polygon . Clearly, under this definition a polygon must have an even number of sides. Let f ( n ) and g ( n ) denote the numbers of n -stepped self-avoiding walks and of n -sided self-avoiding polygons having a prescribed first point. In a previous paper (3), I proved that there exists a connective constant K such that Here I shall prove the truth of the long-standing conjecture that I shall also show that (2) is a particular case of an expression for the number of n -stepped self-avoiding walks with prescribed end-points, a distance o ( n ) apart, this being another old and popular conjecture.