Pairs of Noncrossing Free Dyck Paths and Noncrossing Partitions
  Using the bijection between partitions and vacillating tableaux, we establish a correspondence between pairs of noncrossing free Dyck paths of length $2n$ and noncrossing partitions of $[2n+1]$ with $n+1$ blocks. In terms of the number of up steps at odd positions, we find a characterization of Dyck paths constructed from pairs of noncrossing free Dyck paths by using the Labelle merging algorithm. 
William Y.C. Chen, Sabrina X.M. Pang, Ellen X.Y. Qu, and Richard P.   Stanley