Often calculations in statistical mechanics lead to rather complex counting problems. A problem of this type, which has been studied for a long time, is the determination of the number of ways to place chains on a regular lattice, respecting the excluded volume constraint, which states that each site of the lattice may be occupied by at most one monomer. In particular, one is interested in calculating this number in the thermodynamic limit, in which the lattice becomes infinite with the fraction of sites which are occupied by monomers kept constant. In lattices of dimension larger than one, the only particular case of this problem which was exactly solved is for dimers (chains with two monomers on first neighbor sites) in the limit of full lattice and for two-dimensional lattices. In this paper we present the solution of the problem on a one-dimensional lattice in two different ways. In particular, we solve the problem using a transfer matrix, which may also be applied in the two-dimensional case, leading to quite precise results. Finally, we obtain and discuss the state equations of the lattice gas formed by chains.
polymers; entropy; transfer matrix
