Classification of maximizers of the embedding-time functional

Classify, for all sufficiently large finite \(k\), the permutations maximizing \(\beta(\pi)\) over \(\Pi_k\), and determine whether these extremizers admit a simple description, possibly through their associated tree representations.

Background

The maximum $\beta_k^+=\max_{\pi\in\Pi_k}\beta(\pi)$ has an effective dynamic program, so maximizers can be constructed for each fixed $k$. However, the authors do not know whether the resulting permutations form a recognizable family. They note that the known constructions are neither close to monotone nor approximately uniformly random and may instead be characterized by recursive tree structure.