On the topology of the permutation pattern poset

TL;DR

This paper investigates the topological properties of the permutation pattern poset, revealing that most intervals are disconnected and not shellable, but identifying classes of shellable intervals, especially in layered permutations.

Contribution

It provides the first explicit results on the topology of intervals in the permutation pattern poset, characterizing shellability and disconnectedness in layered and generalized subword order permutations.

Findings

Most intervals in the permutation pattern poset are disconnected and not shellable.

All intervals of layered permutations without large disconnected subintervals are shellable.

Characterization of disconnected intervals in layered permutations.

Abstract

The set of all permutations, ordered by pattern containment, forms a poset. This paper presents the first explicit major results on the topology of intervals in this poset. We show that almost all (open) intervals in this poset have a disconnected subinterval and are thus not shellable. Nevertheless, there seem to be large classes of intervals that are shellable and thus have the homotopy type of a wedge of spheres. We prove this to be the case for all intervals of layered permutations that have no disconnected subintervals of rank 3 or more. We also characterize in a simple way those intervals of layered permutations that are disconnected. These results carry over to the poset of generalized subword order when the ordering on the underlying alphabet is a rooted forest. We conjecture that the same applies to intervals of separable permutations, that is, that such an interval is shellable if and only if it has no disconnected subinterval of rank 3 or more. We also present a simplified version of the recursive formula for the M\"obius function of decomposable permutations given by Burstein et al.