number of non-crossing partitions with n vertices is catalan

Statement

The number of non-crossing partitions of $n$ -vertices is $C_{n}$ Catalan numbers.

Here the partition is given by

π = (1, 5, 8, 12 ∣ 2, 3, 4 ∣ 6, 7 ∣ 9 ∣ 10, 11)

Left as an exercise