Some combinatorial results on smooth permutations

TL;DR

This paper characterizes smooth permutations in the symmetric group using transpositions and 3-cycles within Bruhat intervals, providing new insights into their structure and enumeration.

Contribution

It introduces a characterization of smooth permutations via their Bruhat interval transpositions and 3-cycles, and relates smoothness to intersections with conjugate parabolic subgroups.

Findings

Smooth permutations are characterized by their Bruhat interval transpositions and 3-cycles.

Smooth permutations can be reconstructed as ordered products of certain transpositions.

The intersection properties with conjugate parabolic subgroups characterize smoothness.

Abstract

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.