  We show that the number of independent sets in cocomparability graphs can be
counted in linear time, as can counting cliques in comparability graphs. By
contrast, counting cliques in cocomparabilty graphs and counting independent
sets in comparability graphs are #P-complete. We extend these results to
counting maximal cliques and independent sets. We also consider the
fixed-parameter versions of counting cliques and independent sets of given size
$k$. Finally, we combine the results to show that both counting cliques and
independent sets in permutation graphs are in linear time.
