1234-avoiding permutations and Dyck paths
  We define a map $\nu$ between the symmetric group $S_n$ and the set of pairs of Dyck paths of semilength $n$. We show that the map $\nu$ is injective when restricted to the set of 1234-avoiding permutations and characterize the image of this map. 
Marilena Barnabei, Flavio Bonetti, and Matteo Silimbani
math.CO