Poincar\'e polynomial for fully commutative elements in the symmetric group

Let $W^c(A_n)$ be the set of fully commutative elements of the Coxeter group $W(A_n)$. Let $$a_n(q)=\sum _{w\in W^c(A_n)}{q}^{l(w)}\mathrm{.}$$ We compute $a_n(q)$.