Supertrees

TL;DR

This paper extends the concept of superpermutations to rooted plane trees, introducing $k$-supertrees for different tree classes and pattern notions, providing bounds and exact sizes for their minimal configurations.

Contribution

It introduces the concept of $k$-supertrees for rooted plane trees and establishes bounds and exact sizes for their minimal configurations under various conditions.

Findings

Derived upper and lower bounds for $k$-supertrees.

Determined exact minimal size for one case.

Connected results to recent permutation pattern research.

Abstract

A $k$-universal permutation, or $k$-superpermutation, is a permutation that contains all permutations of length $k$ as patterns. The problem of finding the minimum length of a $k$-superpermutation has recently received significant attention in the field of permutation patterns. One can ask analogous questions for other classes of objects. In this paper, we study $k$-supertrees. For each $d\geq 2$, we focus on two types of rooted plane trees called $d$-ary plane trees and $[d]$-trees. Motivated by recent developments in the literature, we consider "contiguous" and "noncontiguous" notions of pattern containment for each type of tree. We obtain both upper and lower bounds on the minimum possible size of a $k$-supertree in three cases; in the fourth, we determine the minimum size exactly. One of our lower bounds makes use of a recent result of Albert, Engen, Pantone, and Vatter on $k$-universal layered permutations.