catalan number pdfs

http://mathcircle.berkeley.edu/BMC6/pdf0607/catalan.pdf
http://www-math.mit.edu/~rstan/ec/catadd.pdf

How many paths from one corner to the diagonally opposite corner in an n x n matrix do not cross the diagonal?

A sequence of n numbers, 1 thru n, is processed as follows: we either drop the next number into a stack, or we pop the topmost number of the stack. In 2n operations, all numbers would have been processed and the stack would also be empty. The output is the sequence of elements that were popped. How many output sequences are possible?