Some combinatorial results on smooth permutations

Shoni Gilboa, Erez Lapid

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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.

Original languageEnglish
Article number81
JournalSeminaire Lotharingien de Combinatoire
Issue number84
StatePublished - 2020

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  • Bruhat order
  • Covexillary permutations
  • pattern avoidance
  • smooth permutations


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