Prove the number of Dyck paths with $4n$ steps such that every descent is of length exactly two is equal to the Catalan number $C_n$.
I have drawn out some examples to try and solve the problem and have noticed some patterns, but I can not see any obvious bijection or recurrence.
(Big) Hint
I will think of a Dyck path as a string of $+$s and $-s$ where every prefix has at least as many $+$s as $-s$.
In a Dyck path where every descent has length $2$, the $-$s appear in pairs, and each pair $--$ is preceded by a $+$. Replace each occurrence of $+--$ with a $-$. What properties does the resulting string have? Is this transformation reversible?