Lower bounds for superpatterns and universal sequences

Abstract

A permutation σ∈Sn is said to be k-universal or a k-superpattern if for every π∈Sk, there is a subsequence of σ that is order-isomorphic to π. A simple counting argument shows that σ can be a k-superpattern only if n≥(1/e2+o(1))k2, and Arratia conjectured that this lower bound is best-possible. Disproving Arratia's conjecture, we improve the trivial bound by a small constant factor. We accomplish this by designing an efficient encoding scheme for the patterns that appear in σ. This approach is quite flexible and is applicable to other universality-type problems; for example, we also improve a bound by Engen and Vatter on a problem concerning (k+1)-ary sequences which contain all k-permutations.