Let m_n be the number ways to form "mountain ranges" with n upstrokes and n downstrokes that all stay above th?

Given the other question, this is only a matter of establishing, for each n, a bijection between the set of "moutain ranges" and the set of parenthesis sequences with n pairs.

Given a "range", transform it into a parenthesis sequence by associating "/" with "(" and "\" with ")".

I'll leave the detailed verification that this defines a bijection to you.