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.
Highlights
1.1 BackgroundLet Sn denote the set of permutations of the set [n] = {1, . . . , n}
An enormous amount of research in the past half-century has focused on pattern containment and pattern avoidance in permutations [7, 30, 31]
Plenty of popular permutation pattern problems possess the following form: What is the minimum length of a permutation that contains all patterns of a certain type?
Summary
Consecutive pattern containment and avoidance are defined by requiring the indices i1, . It is natural to ask about the minimum possible sizes of “universal objects” in these contexts. Several variations on the theme of contiguous and noncontiguous pattern containment in rooted plane trees have appeared in [6,15,18,23,24,25,33,35]. The purpose of the present article is to investigate the minimum possible size of a k-universal tree, or k-supertree, in some of these contexts. Our notions of universal rooted plane trees are new and are inspired by more recent definitions of pattern containment in trees
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