Vincular pattern posets and the Möbius function of the quasi-consecutive pattern poset

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