We present two families of Wilf-equivalences for consecutive and quasi-consecutive vincular patterns. These give new proofs of the classification of consecutive patterns of length $4$ and $5$. We then prove additional equivalences to explicitly classify all quasi-consecutive patterns of length $5$ into 26 Wilf-equivalence classes. Comment: 20 pages. Reformatted for DMTCS overlay journal
Discrete Mathematics & Theoretical Computer Science