Partitions and partial matchings avoiding neighbor patterns

We obtain the generating functions for partial matchings avoiding neighbor alignments and for partial matchings avoiding neighbor alignments and left nestings. We show that there is a bijection between partial matchings avoiding the three neighbor patterns (neighbor alignments, left nestings and right nestings) and set partitions avoiding right nestings via an intermediate structure of upper triangular matrices. Combining our bijection and the bijection given by Dukes and Parviainen between upper triangular matrices and self-modified ascent sequences, we get a bijection between partial matchings avoiding the three neighbor patterns and self-modified ascent sequences.