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 ∈ S k , there is a subsequence of σ that is order-isomorphic to π. A simple counting argument shows that σ can be a ksuperpattern only if n ≥ (1/e 2 + o(1))k 2 , 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.