Note on superpatterns

Abstract

Given a set $P$ of permutations, a $P$-superpattern is a permutation that contains every permutation in $P$ as a pattern. The study of the minimum length of a superpattern has been of interest. For $P$ being the set of all permutations of a given length, bounds on the minimum length have been improved over the years, and the minimum length is conjectured to be asymptotic with $k^2∕e^2$. Similar questions have been considered for the set of layered permutations. We consider superpatterns with respect to packing colored permutations or multiple copies of permutations. Some simple but interesting observations will be presented. We also propose a few questions.