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Non-crossing partitions

ZaidAlyafey

Well-known member
MHB Math Helper
Jan 17, 2013
1,667
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

Indicium Physicus
Staff member
Jan 26, 2012
4,193
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.
 

ZaidAlyafey

Well-known member
MHB Math Helper
Jan 17, 2013
1,667
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

Indicium Physicus
Staff member
Jan 26, 2012
4,193
Is the part I quoted not in words? Is it an adequate definition? These are not rhetorical questions, but genuine.
 

ZaidAlyafey

Well-known member
MHB Math Helper
Jan 17, 2013
1,667
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 .