Journal of Combinatorics

Volume 12 (2021)

Number 2

Some combinatorial results on smooth permutations

Pages: 303 – 354



Shoni Gilboa (Department of Mathematics and Computer Science, The Open University of Israel, Raanana, Israel)

Erez Lapid (Department of Mathematics, Weizmann Institute of Science, Rehovot, Israel)


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.


Bruhat order, smooth permutations, pattern avoidance, covexillary permutations

2010 Mathematics Subject Classification


Received 11 December 2019

Accepted 7 July 2020

Published 16 July 2021