  We obtain the entropy of flexible linear chains composed of M monomers placed
on the square lattice using a transfer matrix approach. An excluded volume
interaction is included by considering the chains to be self-and mutually
avoiding, and a fraction rho of the sites are occupied by monomers. We solve
the problem exactly on stripes of increasing width m and then extrapolate our
results to the two-dimensional limit to infinity using finite-size scaling. The
extrapolated results for several finite values of M and in the polymer limit M
to infinity for the cases where all lattice sites are occupied (rho=1) and for
the partially filled case rho<1 are compared with earlier results. These
results are exact for dimers (M=2) and full occupation (\rho=1) and derived
from series expansions, mean-field like approximations, and transfer matrix
calculations for some other cases. For small values of M, as well as for the
polymer limit M to infinity, rather precise estimates of the entropy are
obtained.
