Profile classes and partial well-order for permutations

TL;DR

This paper characterizes when profile classes of permutations, defined via 0/+-1 matrices, are partially well-ordered, linking this property to the graph being a forest and introducing exotic antichains.

Contribution

It provides a complete characterization of the partial well-order property for profile classes of permutations based on associated graphs.

Findings

Profile class of M is pwo iff related graph is a forest.

Constructs exotic fundamental antichains lacking periodicity.

Links permutation classes to graph-theoretic properties.

Abstract

It is known that the set of permutations, under the pattern containment ordering, is not a partial well-order. Characterizing the partially well-ordered closed sets (equivalently: down sets or ideals) in this poset remains a wide-open problem. Given a 0/+-1 matrix M, we define a closed set of permutations called the profile class of M. These sets are generalizations of sets considered by Atkinson, Murphy, and Ruskuc. We show that the profile class of M is partially well-ordered if and only if a related graph is a forest. Related to the antichains we construct to prove one of the directions of this result, we construct exotic fundamental antichains, which lack the periodicity exhibited by all previously known fundamental antichains of permutations.