 Noncrossing partition

A partition of cyclically ordered points into blocks whose connections can be drawn inside the circle without crossings between blocks. Equivalently, no two distinct blocks contain alternating points $a,b,c,d$ with $a,c$ in one block and $b,d$ in the other. The number on $n$ points is the $n$th Catalan number.

A **noncrossing partition** is a specific type of partition of a set that has a particular property related to the arrangement of its elements. To understand noncrossing partitions, let's first clarify what a partition is and what we mean by "noncrossing." ### Partition A partition of a set is a way of dividing that set into disjoint subsets, such that every element of the original set belongs to exactly one of these subsets.