Some combinatorial results on smooth permutations

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.

Keywords

Bruhat order, smooth permutations, pattern avoidance, covexillary permutations

2010 Mathematics Subject Classification

05A05

Full Text (PDF format)

Received 11 December 2019

Accepted 7 July 2020

Published 16 July 2021