MHB What Are Non-Crossing Partitions and How Are They Defined?

[alyafey22]

Define a partition of a set $S$ as a collection of non-empty disjoint subsets $\in S$ whose union covers $S$. The number of them is defined using the Bell numbers.

Can we define ''Non-crossing'' partitions in words . I have seen the visualization of these partitions and the number of them is calculated using the Catalan's numbers.

 

[Ackbach]

In the wiki, it says that a noncrossing partition

is a partition in which no two blocks "cross" each other, i.e., if a and b belong to one block and x and y to another, they are not arranged in the order a x b y. If one draws an arch based at a and b, and another arch based at x and y, then the two arches cross each other if the order is a x b y but not if it is a x y b or a b x y. In the latter two orders the partition { { a, b }, { x, y } } is noncrossing.

There's a nice picture illustrating noncrossing partitions.

 

[alyafey22]

In the wiki, it says that a noncrossing partition There's a nice picture illustrating noncrossing partitions.

Yes, I already saw this . But I am surprised to know that this is the only explanation! I mean it should have a mathematical definition '' can be described by words '' .

 

[Ackbach]

Is the part I quoted not in words? Is it an adequate definition? These are not rhetorical questions, but genuine.

 

[alyafey22]

Is the part I quoted not in words? Is it an adequate definition? These are not rhetorical questions, but genuine.

I actually meant something else .But now I got the general idea , thanks .