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In this paper, we consider a particular class of Kazhdan-Lusztig cells in thesymmetric group $S_n$, the cells containing involutions associated withcompositions $\lambda$ of $n$. For certain families of compositions we are ableto give an explicit description of the corresponding cells by obtaining reducedforms for all their elements. This is achieved by first finding a particularclass of diagrams $\mathcal{E}^{(\lambda)}$ which lead to a subset of the cellfrom which the remaining elements of the cell are easily obtained. Moreover, weshow 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 relatedin an analogous way. This allows us to associate certain cells in $S_n$ withcells in $S_{n+1}$ in a well-defined way, which is connected to the inductionand restriction of cells.

On embedding certain Kazhdan--Lusztig cells of $S_n$ into cells of $S_{n+1}$