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 π 1 π 2 ⋯ π n of [ n ] such that π i + 1 − π i ≤ 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 π 1 π 2 ⋯ π n of [ n ] such that either π i + ℓ − π i ≤ 1 for all i = 1 , 2 , … , n − ℓ , or π i + 1 − π i ≤ m for all i = 1 , 2 , … , n − 1 .