An asymptotically tight lower bound for superpatterns with small alphabets

TL;DR

This paper establishes an asymptotically tight lower bound for the shortest superpatterns over small alphabets, resolving longstanding conjectures and improving understanding of pattern containment in combinatorics.

Contribution

It proves the optimality of the shortest superpattern length for small alphabets, confirming a conjecture and refuting a 40-year-old conjecture.

Findings

The shortest $k$-superpattern on $[k+1]$ has length asymptotically $(k^2+k)/2$.

This bound is tight up to lower-order terms.

The results resolve open questions in the theory of superpatterns.

Abstract

A permutation $\sigma \in S_n$ is a $k$-superpattern (or $k$-universal) if it contains each $\tau \in S_k$ as a pattern. This notion of "superpatterns" can be generalized to words on smaller alphabets, and several questions about superpatterns on small alphabets have recently been raised in the survey of Engen and Vatter. One of these questions concerned the length of the shortest $k$-superpattern on $[k+1]$. A construction by Miller gave an upper bound of $(k^2+k)/2$, which we show is optimal up to lower-order terms. This implies a weaker version of a conjecture by Eriksson, Eriksson, Linusson and Wastlund. Our results also refute a 40-year-old conjecture of Gupta.