# Dyck Path w/ Descents of Length 2

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.

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?
• @user7638746 Happy to help. You should click the $\color{green}\checkmark$ next to my answer if you found it solved your problem. Feb 22 '19 at 2:25