Discrete Morse theory and the consecutive pattern poset

TL;DR

This paper applies discrete Morse theory to analyze the structure of the consecutive pattern poset, providing new proofs for its Mobius function and homotopy type, and exploring relationships with factor order.

Contribution

It offers a novel proof of the Mobius function and homotopy type of the consecutive pattern poset using discrete Morse theory, and compares it with factor order.

Findings

Derived the homotopy type of the consecutive pattern poset.

Provided an alternative proof for the Mobius function.

Explored the relationship between the pattern poset and factor order.

Abstract

We use discrete Morse theory to provide another proof of Bernini, Ferrari, and Steingrimson's formula for the Mobius function of the consecutive pattern poset. In addition, we are able to determine the homotopy type of this poset. Earlier, Bjorner determined the Mobius function and homotopy type of factor order and the results are remarkably similar to those in the pattern case. In his thesis, Willenbring used discrete Morse theory to give an illuminating proof of Bjorner's result. Since our proof parallels Willenbring's, we also consider the relationship between the two posets.