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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