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

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 $\tau$.