TITLE: Prove that number of non-cross partitions of a set is Catalan number
QUESTION [5 upvotes]: The title should be clear, how to prove that number of non-cross partitions of a set is Catalan number? 
For example, if we have a set S = {1,2,3,4}, the number of non-cross partitions is 14, because of all the 15 possible partitions S, {{1,3},{2,4}} are excluded.

REPLY [2 votes]: The most straightforward way of proving that the number of noncrossing partitions is given by the Catalan number is introducing a bijection between the set of noncrossing partitions of $[n]$ and the set of balanced parentheses or any other well-known set whose cardinality is characterized by the Catalan number. This notes (http://www.maths.usyd.edu.au/u/kooc/catalan/cat46conv2.pdf) provides an algorithm of finding noncrossing partition of $[n]$ from a sequence of balanced parentheses; checking that the algorithm is a bijection is not hard.
Murasaki diagrams are closely related if you are interested: http://www.maths.usyd.edu.au/u/kooc/catalan/cat58mura.pdf