Density maximizers of layered permutations

TL;DR

This paper investigates the structure of layered permutations that maximize the density of a given pattern, disproving previous conjectures and providing conditions for when such maximizers have a bounded number of layers.

Contribution

It disproves a conjecture about layered permutation maximizers and offers new criteria for bounded-layer maximizers.

Findings

Number of layers tends to infinity for certain layered permutations with large first layer and a second layer of size one.

Disproves the conjecture that layered permutations with no consecutive layers of size one are asymptotically maximized by bounded-layer permutations.

Provides sufficient conditions for layered permutations to have maximizers with a bounded number of layers.

Abstract

A permutation is layered if it contains neither 231 nor 312 as a pattern. It is known that, if $\sigma$ is a layered permutation, then the density of $\sigma$ in a permutation of order $n$ is maximized by a layered permutation. Albert, Atkinson, Handley, Holton and Stromquist [Electron. J. Combin. 9 (2002), R#5] claimed that the density of a layered permutation with layers of sizes $(a,1,b)$ where $a,b\geq2$ is asymptotically maximized by layered permutations with a bounded number of layers, and conjectured that the same holds if a layered permutation has no consecutive layers of size one and its first and last layers are of size at least two. We show that, if $\sigma$ is a layered permutation whose first layer is sufficiently large and second layer is of size one, then the number of layers tends to infinity in every sequence of layered permutations asymptotically maximizing the density of $\sigma$. This disproves the conjecture and the claim of Albert et al. We complement this result by giving sufficient conditions on a layered permutation to have asymptotic or exact maximizers with a bounded number of layers.