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\title{Some combinatorial results on smooth permutations}

\author{Shoni Gilboa\addressmark{1}\thanks{\href{mailto:shoni@openu.ac.il}{shoni@openu.ac.il}.} \and Erez Lapid\addressmark{2}\thanks{\href{mailto:erez.lapid@weizmann.ac.il}{erez.lapid@weizmann.ac.il}.}}

\address{\addressmark{1}Department of Mathematics and Computer Science, The Open University of Israel, Raanana, Israel \\ \addressmark{2}Department of Mathematics, Weizmann Institute of Science, Rehovot, Israel}

\received{\today}

\abstract{We show that any smooth permutation $w$ is characterized by the set $\tbl(w)$ of transpositions and $3$-cycles that
are $\le w$ in the Bruhat order and that $w$ is the product (in a certain order) of the transpositions
in $\tbl(w)$. We also characterize the image of the map $w\mapsto\tbl(w)$.
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.}

\keywords{Bruhat order, smooth permutations, pattern avoidance, Covexillary permutations}

\begin{document}

\maketitle

\section{Introduction}
This is an extended abstract to the paper \cite{1912.04725}, which contains all the proofs.

Fix an integer $n\ge1$ and an $n$-dimensional vector space $V$ over $\C$.
Consider the (complete) \emph{flag variety} $\Fl_n$ consisting of all flags
\[
0= V_0\subset V_1\subset\dots\subset V_n=V,\ \ \dim V_i=i,\ i=0,\dots,n.
\]
This is a homogeneous space under the action of the general linear group $\GL(V)$.
It can be identified with $\GL_n(\C)/B_n(\C)$ where $B_n$ is the Borel subgroup of upper triangular matrices.
By \emph{Gauss elimination} (which is a special case of the \emph{Bruhat decomposition}), the orbits of $B_n(\C)$ on $\Fl_n$ are naturally
indexed by the symmetric group $S_n$ (the Weyl group of $\GL_n$). The \emph{Schubert cell} $Y_w$ pertaining to $w\in S_n$ is by definition
the orbit of the permutation matrix of $w$. The \emph{Schubert variety} $X_w$ is by definition the closure of $Y_w$.
For instance, for the identity permutation $e$, $Y_e=X_e$ is a singleton consisting of the standard flag
(whose stabilizer is $B_n$), while for the longest permutation $w_0$, $Y_{w_0}$ is the open cell
defined by the non-vanishing of all minors in the bottom left corners and $X_{w_0}=\Fl_n$.

It is well known that many geometric properties of Schubert varieties can be described combinatorially.
For instance, the \emph{Bruhat order} given by
\[
x\le w\iff Y_x\subseteq X_w
\]
admits the following simple combinatorial description:
\[
x\le w\iff r_x(i,j)\ge r_w(i,j)\text{ for all }i,j\in [n]:=\{1,\dots,n\},
\]
where for any permutation $y\in S_n$,
\[
r_y(i,j)=\#\big(y([i])\cap [j]\big),\ i,j\in [n].
\]
(We refer to \cite{MR2133266} for standard facts about the Bruhat order.)
A more striking result due to Lakshmibai--Sandhya is that the Schubert variety $X_w$ is \emph{smooth}
if and only if $w$ is $3412$ and $4231$ avoiding \cite{MR1051089}. (In this case we say that $w$ is smooth.)
This beautiful result opened the door to far-reaching relations between
the geometry of Schubert varieties and combinatorics (in particular, pattern avoidance).
We refer the reader to \cite{MR3644818} for a recent survey.

Smooth permutations admit other (even earlier) combinatorial characterizations. For instance, by analyzing the
tangent space of $X_w$ at $Y_e$, Lakshmibai--Seshadri \cite{MR752799} proved that
\[
w\text{ is smooth }\iff \#\{i<j:T_{i,j}\le w\}=\ell(w):=\{i<j:w(i)>w(j)\},
\]
where $T_{i,j}\in S_n$ denotes the transposition $i\leftrightarrow j$.
Another characterizing property is that the \emph{Kazhdan--Lusztig polynomial} $P_{e,w}$ is $1$ \cite{MR788771}.
This property is important in representation theory because of the celebrated \emph{Kazhdan--Lusztig conjecture} \cite{MR560412}
(proved independently by Bernstein--Beilinson and Brylinski--Kashiwara).
For a more recent surprising occurrence of smooth permutations in representation theory see \cite{MR3866895}.
We refer the reader to \cite{MR1782635} for more information about singularities
of Schubert varieties, excluding however more recent exciting developments in Kazhdan--Lusztig theory.

Our purpose is to give another way of looking at smooth permutations combinatorially.
Our main result is the characterization of smooth permutations in terms of their \emph{$2$-$3$-table}.
By definition, the $2$-$3$-table of a permutation $w$ is the set of transpositions and the $3$-cycles that are $\le w$.
The $2$-$3$-table of a smooth permutation satisfies some simple combinatorial properties and
conversely, any set of transpositions and $3$-cycles satisfying these conditions arises from a smooth permutation
(\cref{thm:bijection}).
Moreover, we can recover a smooth permutation from its $2$-$3$-table by taking the product of the transposition $T_{i,j}\le w$
in a suitable \emph{compatible order}, governed by the additional data in the $2$-$3$-table. In fact, the set of compatible orders (with respect
to the $2$-$3$-table) has a structure of a connected graph, in a way reminiscent of the graph of reduced decomposition of $w$
under Coxeter moves (\cref{thm: comporder}).

The result is in accordance with known enumerative results of smooth permutations (e.g., \cite{MR1626487, MR2376109, MR3645580}).
It also gives a bijection between smooth permutations and Dyck paths with additional data (\cref{thm:sigma_fg_bijection}).
Another interesting consequence is yet another combinatorial characterization of smooth permutations (\cref{thm: bruintH}).
This characterization is of a rather different nature than the above-mentioned.
Finally, an intriguing relation between covexillary permutations and smooth ones is given (\cref{thm: idemp}).

The $2$-$3$-table of permutation is closely related to the notion of \emph{essential set} conceived by Fulton
in his study of degeneracy loci \cite{MR1154177}. This notion was further studied combinatorially by Eriksson--Linusson \cite{MR1412437}.
In the case of smooth permutations the situation is particularly simple.

%We are very grateful to the anonymous referee for making several important suggestions. In particular, the possible relation to the essential set was suggested to us by her.

\section{The \texorpdfstring{$2$-$3$}{23}-table of a permutation}
Fix an integer $n\ge1$.
Consider the symmetric group $S_n$ of all the permutations of the set $[n]=\{1,2,\ldots,n\}$
with the Bruhat order $\le$.
Let $\Trans=\{T_{i,j}:1\le i<j\le n\}\subset S_n$ be the set of transpositions.
For every permutation $w\in S_n$ define the \emph{2-table} of $w$ to be
\[
\tbl_\Trans(w)=\{x\in\Trans:x\le w\}.
\]
For every $w\in S_n$ we have $\ell(w)\le\# \tbl_\Trans(w)$ where
\[
\ell(w)=\#\{i<j:w(i)>w(j)\}
\]
is the number of inversions of $w$ \cite{MR752799}.
If $\ell(w)=\# \tbl_\Trans(w)$, then $w$ is called \emph{smooth},
a terminology that is justified by the fact that this condition also characterizes the smoothness
of the Schubert variety $X_w$ pertaining to $w$ [ibid.].
Another well-known combinatorial characterization of the smoothness of $w$ is that
$w$ is $4231$ and $3412$ avoiding \cite{MR1051089}.
We refer to \cite{MR1782635} and the references therein for more information about singularities of Schubert varieties.

Distinct smooth permutations may have the same $2$-table (for example, for $n=3$,
$\tbl_\Trans((231))=\{T_{1,2},T_{2,3}\}=\tbl_\Trans((312))$).
However, we show that smooth permutations are distinguishable from each other at the `next level'.
More precisely, let $\Special\subset S_n$ be the set of permutations consisting of a single cycle of length $2$ or $3$.
Denote the $3$-cycle permutation $i\mapsto j\mapsto k\mapsto i$ with $i<j<k$ by $R_{i,j,k}$, so that
\[
\Special=\Trans\cup\{R_{i,j,k},R_{i,j,k}^{-1}:i<j<k\}.
\]
We define the \emph{$2$-$3$-table} of a permutation $w\in S_n$ to be
\[
\tbl(w)=\{x\in\Special:x\le w\}.
\]
Clearly, $\tbl(w)$ is downward closed and it is easy to see that
if $R_{i,j,l},R_{i,k,l}^{-1}\in\tbl(w)$ with $i<j,k<l$, then $T_{i,l}\in\tbl(w)$.

We say that a downward closed subset $A$ of $\Special$ is \emph{\adm} if it satisfies the following two conditions.
\begin{itemize}
\item If $R_{i,j,l},R_{i,k,l}^{-1}\in A$ with $i<j,k<l$, then $T_{i,l}\in A$.
\item Whenever $T_{i,j},T_{j,k}\in A$, $i<j<k$, at least one of $R_{i,j,k}$ and $R_{i,j,k}^{-1}$ belongs to $A$.
\end{itemize}

Our main result is the following.
\begin{theorem}\label{thm:bijection}
The map $w\mapsto\tbl(w)$ defines a bijection between the smooth permutations of $S_n$
and the {\adm} sets. The inverse bijection $A\mapsto\pi(A)$ is given by
\begin{equation*}
\pi(A)=\max\{x\in S_n:\tbl(x)=A\}=\max\{x\in S_n:\tbl_\Trans(x)=A_\Trans,\,\tbl(x)\subseteq A\},
\end{equation*}
where $\max$ denotes the greatest element with respect to the Bruhat order.
\end{theorem}

\section{Compatible orders}
We give an alternative, more constructive definition of $\pi(A)$ for an {\adm} set $A\subseteq\Special$.
We say that a total order $\prec$ on $A_\Trans=A\cap\Trans$ is \emph{compatible} (with $A$) if whenever $T_{i,j},T_{j,k}\in A$, $i<j<k$, the following hold:
\begin{enumerate}
\item If $T_{i,k}\in A$, then either $T_{i,j}\prec T_{i,k}\prec T_{j,k}$ or $T_{j,k}\prec T_{i,k}\prec T_{i,j}$.
\item If $T_{i,k}\notin A$, then $R_{i,j,k}\in A\iff T_{i,j}\prec T_{j,k}$.
\end{enumerate}


Note that the first condition also occurs in the notion of \emph{reflection order} (cf. \cite{MR1248893}, \cite[Section 5.2]{MR2133266})
except that we do not consider a total order on the whole of $\Trans$.

\begin{theorem} \label{thm: comporder}
Let $A$ be an {\adm} subset of $\Special$. Then, a compatible order on $A_\Trans$ always exists
and $\pi(A)$ is equal to the product of the elements of $A_\Trans$ taken with respect to a compatible order $\prec$.
(In particular, the product depends only on $A$.)
Consequently, every smooth permutation may be written as the product, in an appropriate order,
of the transpositions in its $2$-table (each appearing exactly once).
\end{theorem}

More precisely, we define a graph $\grph_A$ whose vertices are the compatible orders on $A_\Trans$
and whose edges connect two compatible orders that can be obtained from one another by one of the following elementary operations.
\begin{enumerate}
\item Interchanging the order of two adjacent commuting transpositions, or
\item Switching the order of consecutive $T_{i,j}$, $T_{i,k}$, $T_{j,k}$ to $T_{j,k}$, $T_{i,k}$, $T_{i,j}$,
or vice versa.
\end{enumerate}
These operations do not change the product of the elements of $A_\Trans$, taken in the respective orders.
We show that $\grph_A$ is connected (and in particular, non-empty).
In other words, every two compatible orders are obtained from one another
by a sequence of elementary operations. The situation is reminiscent of the case of reduced decompositions
of a permutation $w$, which form the vertices of a connected graph $G(w)$ whose edges are given by basic Coxeter relations.
In fact, for $A=\Special$ itself, there is a natural isomorphism between $\grph_A$ and $G(w_0)$ where
$w_0$ is the longest permutation \cite{MR911771}.
However, for a general smooth permutation $w$, the number of compatible orders on $\tbl_\Trans(w)$
with respect to $\tbl(w)$ does not agree with the number of reduced decompositions of $w$, which is given by a well-known formula of Stanley \cite{MR782057}.

\section{Intersection of Bruhat intervals with conjugates of para\-bolic subgroups}
As an application of \cref{thm:bijection}, consider an arbitrary partition $X$ (i.e., an equivalence relation) of $[n]$
and the subgroup $S_X$ of $S_n$ preserving all subsets of $X$.
The group $S_X$ is isomorphic to the direct product of $S_{\# y}$ over $y\in X$.
However, the product order on $S_X$ (which we denote by $\le_X$) is in general stronger than the one induced from $S_n$.
We say that an element of $S_X$ is \emph{$X$-smooth} if all its coordinates in $S_{\# y}$, $y\in X$ are smooth.
This condition is weaker than smoothness in $S_n$.
For instance, if $X$ is the partition $\{\{1,3\},\{2,4\}\}$ then the permutation $(3412)$ is $X$-smooth but not smooth.

\begin{theorem} \label{thm: bruintH}
$w\in S_n$ is smooth if and only if for every partition $X$ of $[n]$, the set
\[
\{x\in S_X:x\le w\}
\]
admits a maximum $w_X$ with respect to $\leq_X$.
Moreover, in this case $w_X$ is $X$-smooth.
\end{theorem}


\section{Relation to Dyck paths} \label{sec: dyck}
We may also interpret the bijection of \cref{thm:bijection} in terms of more familiar
combinatorial objects, namely Dyck paths.
We may view a Dyck path as a weakly increasing function $f:[n]\to[n]$ such that $f(i)\ge i$ for all $i$.
Suppose that in addition to $f$, we are given a function $g:[n]\rightarrow\{0,1\}$ such that
\begin{enumerate}
\item $g(i)=0$ whenever $f(f(i))=f(i)$.
\item $g(i)=g(i+1)$ whenever $i<n$ and $f(i+1)<f(f(i))$.
\end{enumerate}
In this case we say that $(f,g)$ is a \emph{good pair}.
Write $g^{-1}(0)=\{i_1,\dots,i_k\}$ and  $g^{-1}(1)=\{j_1,\dots,j_l\}$ with $i_1<\dots<i_k$ and $j_1<\dots<j_l$.


For every $1\leq i<j\leq n$, let $\rshft ij\in S_n$ be the cycle permutation $i\to i+1\to\dots\to j\to i$
and let $\lshft ij=\rshft ij^{-1}$.
\begin{theorem}\label{thm:sigma_fg_bijection}
The map
\begin{equation} \label{eq: fgbij}
(f,g)\to w(f,g)=\lshft{j_1}{f(j_1)}\cdots\lshft{j_l}{f(j_l)}\rshft{i_k}{f(i_k)}\cdots\rshft{i_1}{f(i_1)}
\end{equation}
is a bijection between good pairs and the smooth permutations in $S_n$.
The inverse is given by $w\mapsto(f,g)$, where for every $i\in[n]$,
\[
\begin{aligned}
f(i)&=\max\Big(\{i\}\cup\{j>i:T_{i,j}\in\tbl(w)\}\Big)\\
g(i)&=\begin{cases}1&\text{if $i<f(i)$ and }R_{i,f(i),f(i)+1}\in\tbl(w),\\
0&\text{otherwise.}\end{cases}
\end{aligned}
\]
Moreover, the expression on the right-hand side of \eqref{eq: fgbij} is reduced.
Finally,
\begin{align*}
\tbl(w(f,g))=&\{T_{i,j}:i<j\le f(i)\}\cup \{R_{i,j,k},R_{i,j,k}^{-1}:i<j<k\le f(i)\}\\
&\bigcup\ \{R_{i,j,k}:i<j\le f(i)<k\le f(j),\ g(i)=1  \}\\
&\bigcup\ \{R_{i,j,k}^{-1}:i<j\le f(i)<k\le f(j),\ g(i)=0 \}.
\end{align*}
\end{theorem}

\cref{thm:sigma_fg_bijection} is in the spirit of Skandera's factorization of smooth permutation \cite{MR2387587}.
Using \cref{thm:sigma_fg_bijection}, we can recover several known enumerative results concerning smooth permutations \cite{MR1626487, MR2376109, MR1603806, MR3645580, MR1417303}.

\section{Relation to covexillary permutations}
Using \cref{thm:bijection}, we can also give an interesting relation between smooth
permutations and \emph{covexillary} ones.
Recall that a permutation is called covexillary if it avoids the pattern $3412$.

\begin{theorem}\label{thm: idemp}
For any covexillary $x\in S_n$, $\tbl(x)$ is {\adm}. 
Therefore, the map $x\mapsto\pi(\tbl(x))$ is an idempotent function from
the set of covexillary permutations onto the subset of smooth permutations. 
Moreover, this map is order preserving and for any covexillary $x\in S_n$, 
\[
\pi(\tbl(x))=\min\{w\in S_n\text{ smooth}:w\geq x\}.
\]
\end{theorem}

\section{Relation to coessential set}

In \cite{MR1154177} Fulton introduced the notion of the \emph{essential set} of a permutation $w\in S_n$.
For our purpose it is more convenient to use the following slight variant:
\[
\Ess(w)=\{(i,j)\in [n-1]\times[n-1]:w(i)\le j<w(i+1)\text{ and }w^{-1}(j)\le i<w^{-1}(j+1)\}.
\]
For any $w\in S_n$ we have
\[
\forall x\in S_n,\ x\le w\iff r_x(i,j)\ge r_w(i,j)\text{ for all }(i,j)\in\Ess(w).
\]
Moreover, the set $\Ess(w)$ is minimal with respect to this property.

In particular, $w$ is defined by the set $\Ess(w)$ and the restriction of $r_w$ to $\Ess(w)$.
The image of the injective map
\[
w\in S_n\mapsto (\Ess(w),r_w\big|_{\Ess(w)})
\]
was described in \cite{MR1412437}, extending Fulton's result in the covexillary case.

We say that $w$ is \emph{defined by inclusion} if $r_w(i,j)=\min(i,j)$
(i.e., if $w([i])\subseteq[j]$ or $[j]\subseteq w([i])$) for all $(i,j)\in\Ess(w)$.
It was proved by Gasharov--Reiner that $w$ is defined by inclusion if and only if $w$
is $4231$, $35142$, $42513$ and $351624$ avoiding \cite{MR1934291}. In particular, $w$ is smooth if and only if
$w$ is covexillary and defined by inclusions.

In general, consider the subset
\[
\Ess^\circ(w)=\{(i,j)\in\Ess(w):w([i])\subseteq[j]\text{ or }[j]\subseteq w([i])\}.
\]
Thus, $w$ is defined by inclusion if and only if $\Ess(w)=\Ess^\circ(w)$, in which case $w$ is determined by
the set $\Ess(w)$. In particular, this is the case if $w$ is smooth.

Note that for any $w\in S_n$, the $2$-$3$-table $\tbl(w)$ is determined by the set $\Ess^\circ(w)$.
More precisely, we have
\[
\begin{gathered}
T_{i,j}\in\tbl(w)\iff \Ess^\circ(w)\cap\big([i,j)\times [i,j)\big)=\emptyset,\\
R_{i,j,k}\in\tbl(w)\iff\Ess^\circ(w)\cap\big([i,j)\times [i,j)\big)=\Ess^\circ(w)\cap\big([j,k)\times [i,k)\big)=\emptyset,\\
R_{i,j,k}^{-1}\in\tbl(w)\iff\Ess^\circ(w)\cap\big([i,j)\times [i,k)\big)=\Ess^\circ(w)\cap\big([j,k)\times [j,k)\big)=\emptyset.
\end{gathered}
\]

We say that a subset $E$ of $[n-1]\times[n-1]$ is \emph{permissible} if for every two distinct points
$(i_1,j_1)$ and $(i_2,j_2)$ in $E$ such that $\min(i_2,j_2)\ge\min(i_1,j_1)$ we have
\[
i_2\ge i_1,\ \ j_2\ge j_1,\ \ \max(i_2,j_2)>\max(i_1,j_1)\text{ and }\min(i_2,j_2)>\min(i_1,j_1).
\]

It is easy to see that $\Ess^\circ(w)$ is permissible for every covexillary $w\in S_n$.

\begin{theorem} \label{thm: allbij}
We have a commutative diagram of bijections
\[
\begin{tikzcd}
\text{smooth permutations}\arrow[r, leftrightarrow] \arrow[rd, leftrightarrow] \arrow[d, leftrightarrow] & \text{{\adm} sets}\\
\text{good pairs}\arrow[r, leftrightarrow] \arrow[ru, leftrightarrow] & \text{permissible sets} \arrow[u, leftrightarrow]
\end{tikzcd}
\]
that is compatible with those of \cref{thm:bijection,thm:sigma_fg_bijection}.
The good pair corresponding to a permissible set $E$ is given by
\[
\begin{aligned}
f(k)&=\min\Big(\{n\}\cup\{\max(i,j):(i,j)\in E,\ i,j\ge k\}\Big),\\
g(k)&=\begin{cases}1&\text{if }j<f(j)\text{ and }(j,f(j))\in E\text{, where } j=\max f^{-1}(f(k)),\\
0&\text{otherwise.}\end{cases}
\end{aligned}
\]
The permissible set corresponding to a good pair $(f,g)$ is
\[
\begin{aligned}
&\{(i,f(i)):i\in[n-1],\ f(i+1)>f(i)\text{ and }g(i)=1\}\\
\bigcup\ &\{(f(i),i):i\in[n-1],\ f(i+1)>f(i)\text{ and }g(i)=0\}.
\end{aligned}
\]
The {\adm} set corresponding to a permissible set $E$ is
\[
\begin{aligned}
&\{T_{i,j}:E\cap\big([i,j)\times [i,j)\big)=\emptyset\}\\
\bigcup\ &\{R_{i,j,k}:E\cap\big([i,j)\times [i,j)\big)=E\cap \big([j,k)\times [i,k)\big)=\emptyset\}\\
\bigcup\ &\{R_{i,j,k}^{-1}:E\cap\big([i,j)\times [i,k)\big)=E\cap \big([j,k)\times [j,k)\big)=\emptyset\}.
\end{aligned}
\]
The permissible set corresponding to an {\adm} set $A$ is
\[
\begin{aligned}
&\{(i,i):i<n,\ T_{i,i+1}\notin A\}\\
\bigcup\ &\{(i,j):i<j<n,\ T_{i,j},T_{i+1,j+1}\in A,\ T_{i,j+1}\notin A,\ R_{i,j,j+1}\in A\}\\
\bigcup\ &\{(i,j):j<i<n,\ T_{j,i},T_{j+1,i+1}\in A,\ T_{j,i+1}\notin A,\ R_{j,i,i+1}^{-1}\in A\}.
\end{aligned}
\]
The permissible set corresponding to a smooth permutation $w$ is $\Ess^\circ(w)$.
\end{theorem}


Finally, we can relate \cref{thm: idemp,thm: allbij} as follows.
\begin{theorem}
For any covexillary $x\in S_n$ we have $\Ess(\pi(\tbl(x)))=\Ess^\circ(x)$.
\end{theorem}


\section{Odds and ends}
\cref{thm: bruintH} was the original motivation of this work.
It came up in studying a related problem, which is discussed in \cite{MR3880654}.
The result of [ibid.] is relevant for a certain representation-theoretic context.
We hope that the same will be true for \cref{thm: bruintH} and its variants,
although we will not discuss these possible applications here.

Likewise, it would be interesting to find a geometric context for \cref{thm:bijection,thm: bruintH}.

It is natural to ask whether \cref{thm: comporder} admits an analogue for other Weyl groups $W$.
In particular, one may ask whether any smooth element $w$ of $W$ can be written as the product (in a suitable order)
of the reflections that are smaller than or equal to $w$ in the Bruhat order (each reflection occurring exactly once).

\acknowledgements{We are very grateful to the anonymous referee for making several important suggestions.
In particular, the possible relation to the essential set was suggested to us by her.}


\printbibliography

\end{document}
