Partial permutations avoiding pairs of patterns

We continue the study of pattern avoidance in partial permutations initiated by Claesson, Jelínek, Jelínkova, and Kitaev. We extend previous definitions of shape-Wilf-equivalence and ⋆⋆-Wilf-equivalence to sets of patterns, and determine new shape-Wilf-equivalences and shape-⋆⋆-Wilf-equivalences between pairs of patterns of length 3. Using these results, we find infinite families of Wilf-equivalences and ⋆⋆-Wilf-equivalences between pairs of patterns. We also classify pairs of patterns of length up to 4 up to ⋆⋆-Wilf-equivalence.