  I consider the $\mathbb{Z}_\lambda,$ $\lambda$ prime free-bosonic permutation
orbifolds as interacting physical string systems at $\hat{c} = 26\lambda $. As
a first step, I introduce twisted tree diagrams which confirm at the
interacting level that the physical spectrum of each twisted sector is
equivalent to that of an ordinary $c=26$ closed string. The untwisted sectors
are surprisingly more difficult to understand, and there are subtleties in the
sewing of the loops, but I am able to propose provisional forms for the full
modular-invariant cosmological constants and one-loop diagrams with insertions.
