Vincular pattern posets and the M\"obius function of the quasi-consecutive pattern poset

TL;DR

This paper introduces vincular pattern posets, focusing on the quasi-consecutive pattern poset, and thoroughly analyzes its M"obius function, especially for intervals where the smaller permutation occurs exactly once in the larger.

Contribution

It defines vincular pattern posets and provides a complete determination of the M"obius function for specific intervals in the quasi-consecutive pattern poset.

Findings

M"obius function explicitly computed for certain intervals

Characterization of intervals where the smaller permutation occurs once

Introduction of vincular pattern posets as a new framework

Abstract

We introduce vincular pattern posets, then we consider in particular the quasi-consecutive pattern poset, which is defined by declaring $\sigma \leq \tau$ whenever the permutation $\tau$ contains an occurrence of the permutation $\sigma$ in which all the entries are adjacent in $\tau$ except at most the first and the second. We investigate the M\"obius function of the quasi-consecutive pattern poset and we completely determine it for those intervals $[\sigma ,\tau ]$ such that $\sigma$ occurs precisely once in $\tau$.