  We show that any smooth permutation $\sigma\in S_n$ is characterized by the
set ${\mathbf{C}}(\sigma)$ of transpositions and $3$-cycles in the Bruhat
interval $(S_n)_{\leq\sigma}$, and that $\sigma$ is the product (in a certain
order) of the transpositions in ${\mathbf{C}}(\sigma)$. We also characterize
the image of the map $\sigma\mapsto{\mathbf{C}}(\sigma)$. As an application, we
show that $\sigma$ is smooth if and only if the intersection of
$(S_n)_{\leq\sigma}$ with every conjugate of a parabolic subgroup of $S_n$
admits a maximum. This also gives another approach for enumerating smooth
permutations and subclasses thereof. Finally, we relate covexillary
permutations to smooth ones and rephrase the results in terms of the
(co)essential set in the sense of Fulton.
