Some combinatorial results on smooth permutations

We show that any smooth permutation w is characterized by the set C(w) of transpositions and 3-cycles that are ≤ w in the Bruhat order and that w is the product (in a certain order) of the transpositions in C(w). We also characterize the image of the map w 7! C(w). This map is closely related to the essential set (in the sense of Fulton) and gives another approach for enumerating smooth permutations and subclasses thereof. As an application, we obtain a result about the intersection of the Bruhat interval defined by a smooth permutation with a conjugate of a parabolic subgroup of the symmetric group. Finally, we relate covexillary permutations to smooth ones.