Bounds on Superpatterns Containing All Layered Permutations

In the study of pattern containment, a k-superpattern is a permutation which contains all k! permutations of length k as a pattern. One may also consider restricted superpatterns, i.e. a permutation which contains, as a pattern, every element in some subclass of the set of permutations of length k. Here, we find lower and upper bounds on a superpattern which contains all layered k-permutations. Also, we exhibit a connection between the sum of depths of null-balanced binary trees on k vertices