Excluded Volume Effect in Polymer Chains. I

The number of allowed configurations of a polymer chain is considered on the basis of a random walk on an arbitrary ``regular'' lattice. Upper and lower bounds for the number of nonoverlapping configurations in various lattices have been derived by means of a recursion formula method. The probability density function and its moments for the ``head-to-tail'' distance for short-range nonoverlapping chains are shown to the calculable by the use of a generating function. The order of the logarithm of the number of nonsuperposable ring polymer chains has been shown to be directly proportional to the number of segments composing the ring.