Partitions and Partial Matchings Avoiding Neighbor Patterns
William Y. C. Chen, Neil J. Y. Fan and Alina F. Y. Zhao
Abstract: 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 three neighbor patterns (neighbor alignments, left nestings and right nestings) and set partitions avoiding right nestings via an intermediate structure of integer compositions. Such integer compositions are known to be in one-to-one correspondence with self-modified ascent sequences or -avoiding permutations, as shown by Bousquet-Mélou, Claesson, Dukes and Kitaev.
AMS Classification: 05A15, 05A19
Keywords: set partition, partial matching, neighbor alignment, left nesting, right nesting, self-modified ascent sequence