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Séminaire Lotharingien de Combinatoire, 84B.81 (2020), 10 pp.

# Shoni Gilboa and Erez Lapid

# Some Combinatorial Results on Smooth Permutations

**Abstract.**
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* -> **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.

Received: November 20, 2019.
Accepted: February 20, 2020.
Final version: April 30, 2020.

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