On partitions avoiding right crossings
  Recently, Chen et al. derived the generating function for partitions avoiding right nestings and posed the problem of finding the generating function for partitions avoiding right crossings. In this paper, we derive the generating function for partitions avoiding right crossings via an intermediate structure of partial matchings avoiding 2-right crossings and right nestings. We show that there is a bijection between partial matchings avoiding 2-right crossing and right nestings and partitions avoiding right crossings. 
Sherry H.F. Yan and Yuexiao Xu
math.CO