Partitions and partial matchings avoiding neighbor patterns

Authors:
William Y. C. Chen;Neil J. Y. Fan;Alina F. Y. Zhao
Affiliations:
Center for Combinatorics, LPMC-TJKLC, Nankai University, Tianjin 300071, PR China;Center for Combinatorics, LPMC-TJKLC, Nankai University, Tianjin 300071, PR China;School of Mathematical Science, Nanjing Normal University, Nanjing 210046, PR China
Venue:
European Journal of Combinatorics
Year:
2012

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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 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.