  We consider a one-dimensional variant of a recently introduced settlement
planning problem in which houses can be built on finite portions of the
rectangular integer lattice subject to certain requirements on the amount of
insolation they receive. In our model, each house occupies a unit square on a
$1 \times n$ strip, with the restriction that at least one of the neighboring
squares must be free. We are interested mostly in situations in which no
further building is possible, i.e. in maximal configurations of houses in the
strip. We reinterpret the problem as a problem of restricted packing of
vertices in a path graph and then apply the transfer matrix method in order to
compute the bivariate generating functions for the sequences enumerating all
maximal configurations of a given length with respect to the number of houses.
This allows us to determine the asymptotic behavior of the enumerating
sequences and to compute some interesting statistics. Along the way, we
establish close connections between our maximal configurations and several
other types of combinatorial objects, including restricted permutations and
walks on certain small oriented graphs. In all cases we provide combinatorial
proofs. We then generalize our results in several directions by considering
multi-story houses, by varying the insolation restrictions, and, finally, by
considering strips of width 2 and 3. At the end we comment on several possible
directions of future research.
