The structure of the consecutive pattern poset

TL;DR

This paper investigates the structure of the consecutive pattern poset, revealing properties like unimodality, Sperner-ness, and shellability, and characterizes the nature of its intervals and their topological features.

Contribution

It provides a comprehensive analysis of the intervals in the consecutive pattern poset, including topological, poset-theoretic, and enumerative properties, with new characterizations and proofs.

Findings

All intervals are rank-unimodal and strongly Sperner.

Most intervals are not shellable and have M"obius function zero.

Characterization of disconnected and shellable intervals.

Abstract

The consecutive pattern poset is the infinite partially ordered set of all permutations where $\sigma\le\tau$ if $\tau$ has a subsequence of adjacent entries in the same relative order as the entries of $\sigma$. We study the structure of the intervals in this poset from topological, poset-theoretic, and enumerative perspectives. In particular, we prove that all intervals are rank-unimodal and strongly Sperner, and we characterize disconnected and shellable intervals. We also show that most intervals are not shellable and have M\"obius function equal to zero.