  In this paper, we consider a particular class of Kazhdan-Lusztig cells in the
symmetric group $S_n$, the cells containing involutions associated with
compositions $\lambda$ of $n$. For certain families of compositions we are able
to give an explicit description of the corresponding cells by obtaining reduced
forms for all their elements. This is achieved by first finding a particular
class of diagrams $\mathcal{E}^{(\lambda)}$ which lead to a subset of the cell
from which the remaining elements of the cell are easily obtained. Moreover, we
show that for certain cases of related compositions $\lambda$ and
$\hat{\lambda}$ of $n$ and $n+1$ respectively, the members of
$\mathcal{E}^{(\lambda)}$ and $\mathcal{E}^{(\hat{\lambda})}$ are also related
in an analogous way. This allows us to associate certain cells in $S_n$ with
cells in $S_{n+1}$ in a well-defined way, which is connected to the induction
and restriction of cells.
