  We prove several Wilf-equivalences for vincular patterns of length 4, some of
which generalize to infinite families of vincular patterns. We also present
functional equations for the generating functions for the number of
permutations of length n avoiding a single pattern for the patterns 124-3,
134-2, 231-4, 241-3, 132-4, and 142-3. This nearly completes the
Wilf-classification of vincular patterns of length 4. As a corollary, these
results imply Wilf-equivalences for certain barred patterns of length 5 with a
single bar.
