# On the M\"obius Function and Topology of General Pattern Posets

## TL;DR

This paper introduces a unified framework for pattern posets, analyzing their M"obius function and topology, and applies these findings to derive new insights and alternative proofs for known results in the field.

## Contribution

It provides a general definition of pattern posets, linking their M"obius function to embeddings, and explores topological properties like Cohen-Macaulayness and shellability.

## Key findings

- M"obius function linked to the number of embeddings
- Topological properties preserved under certain fibrations
- Alternative proofs for known results like Bj"orner's subword order

## Abstract

We introduce a formal definition of a pattern poset which encompasses several previously studied posets in the literature. Using this definition we present some general results on the M\"obius function and topology of such pattern posets. We prove our results using a poset fibration based on the embeddings of the poset, where embeddings are representations of occurrences. We show that the M\"obius function of these posets is intrinsically linked to the number of embeddings, and in particular to so called normal embeddings. We present results on when topological properties such as Cohen-Macaulayness and shellability are preserved by this fibration. Furthermore, we apply these results to some pattern posets and derive alternative proofs of existing results, such as Bj\"orner's results on subword order.