Some Results on Superpatterns for Preferential Arrangements

TL;DR

This paper investigates superpatterns for preferential arrangements, establishing structural equivalences with permutations and exploring probabilistic properties related to minimal superpatterns, contributing to combinatorial pattern containment theory.

Contribution

It proves that superpatterns for all preferential arrangements are equivalent to superpatterns for all permutations and explores probabilistic aspects of minimal superpatterns.

Findings

Superpatterns for all preferential arrangements are equivalent to superpatterns for all permutations.

Probabilistic results on the existence of small superpatterns are provided.

Connections to unresolved conjectures in permutation containment are discussed.

Abstract

A {\it superpattern} is a string of characters of length $n$ that contains as a subsequence, and in a sense that depends on the context, all the smaller strings of length $k$ in a certain class. We prove structural and probabilistic results on superpatterns for {\em preferential arrangements}, including (i) a theorem that demonstrates that a string is a superpattern for all preferential arrangements if and only if it is a superpattern for all permutations; and (ii) a result that is reminiscent of a still unresolved conjecture of Alon on the smallest permutation on $[n]$ that contains all $k$-permutations with high probability.