Nonnesting partitions

A set partition of an $n$ element set into blocks $B_1, B_2, ..., B_k$ is nonnesting if for any 4 elements $1 \leq a < b < c < d \leq n$ satisfying $a, d \in B_i \text{ and }b, c \in B_j$, we have $i = j$.

They are counted by Catalan numbers.

A000108 on the OEIS

Source

Mamede: A bijection between noncrossing and nonnesting partitions of types A and B