Simple permutations, alternating permutations, and the superpatterns that contain them

Let Ck be a subset of the set of permutations of length k. We say a Ck-superpattern is a permutation which contains all elements of Ck as a pattern. In this paper, we find bounds on the lengths of superpatterns containing all simple permutations of length k and superpatterns containing all alternating permutations of length k.