Bounds on Superpatterns Containing all Layered Permutations

Abstract

In the study of pattern containment, a $$k$$k-superpattern is a permutation which contains all $$k!$$k! permutations of length $$k$$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 $$k$$k. Here, we find lower and upper bounds on a superpattern which contains all layered $$k$$k-permutations. Also, we exhibit a connection between the sum of depths of null-balanced binary trees on $$k$$k vertices, as defined in (Proceedings of American Conference on Applied Mathematics, Cambridge, MA, pp 377---381 2012).