Partitions of a set satisfying certain set of conditions

In this paper we present a new combinatorial class enumerated by Catalan numbers. More precisely, we establish a bijection between the set of partitions π1π2⋯πn of [n] such that πi+1−πi≤1 for all i=,1,2…,n−1, and the set of Dyck paths of semilength n. Moreover, we find an explicit formula for the generating function for the number of partitions π1π2⋯πn of [n] such that either πi+ℓ−πi≤1 for all i=1,2,…,n−ℓ, or πi+1−πi≤m for all i=1,2,…,n−1.