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%% define your title in the usual way
\title{Rowmotion on $321$-avoiding permutations}

%% define your authors in the usual way
%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
\author{Ben Adenbaum\thanks{\href{mailto:benjamin.m.adenbaum.gr@dartmouth.edu}{benjamin.m.adenbaum.gr@dartmouth.edu}}\addressmark{1} \and Sergi Elizalde\thanks{\href{mailto:sergi.elizalde@dartmouth.edu}{sergi.elizalde@dartmouth.edu}}\addressmark{1}}

%% then use \addressmark to match authors to institutions here
\address{\addressmark{1}Department of Mathematics, Dartmouth College, Hanover, NH 03755, USA}

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\received{November 14 2022}

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%\revised{}

%% put your English abstract here, or comment this out if you don't have one yet
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\abstract{
We give a natural definition of rowmotion for $321$-avoiding permutations, by translating, through bijections involving Dyck paths and the Lalanne--Kreweras involution, the analogous notion for antichains of the positive root poset of type $A$. We prove that some permutation statistics, such as the number of fixed points, are homomesic under rowmotion, meaning that they have a constant average over its orbits. 
Finally, we show that the Armstrong--Stump--Thomas equivariant bijection between antichains in types $A$ and $B$ and non-crossing matchings can be described more naturally in terms of the Robinson--Schensted--Knuth correspondence on permutations. }


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\keywords{rowmotion, homomesy, pattern avoidance}

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\begin{document}

\maketitle
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\section{Introduction}
Let $\S_n$ denote the set of permutations of $\{1,2,\dots,n\}$.
We say that $\pi\in\S_n$ is {\rm $321$-avoiding} if there do not exist $i<j<k$ such that $\pi(i)>\pi(j)>\pi(k)$. 
Let $\S_n(321)$ denote the set of $321$-avoiding permutations in $\S_n$.
 We can represent $\pi \in \S_n(321)$ as an $n\times n$ array with crosses in squares $(i,\pi(i))$ for $1\le i\le n$; we call this the {\em array} of $\pi$. Rows and columns are indexed using cartesian coordinates, so that $(i,j)$ denotes the cell in the $i$th column from the left and $j$th row from the bottom.
 We say that $(i,\pi(i))$ is a {\em fixed point} (respectively {\em excedance}, {\em weak excedance}, {\em deficiency}, {\em weak deficiency}) if
$\pi(i)=i$ (respectively $\pi(i)>i$, $\pi(i)\ge i$, $\pi(i)<i$, $\pi(i)\le i$). 

Let $\P$ be a finite poset, and let 
$\cA(\P)$ denote the set of antichains of $\P$.
{\em Antichain rowmotion} is the map $\rA:\cA(\P)\to \cA(\P)$ defined as follows: for $A\in\cA(\P)$, let $\rA(A)$ be the minimal elements of the complement of the order ideal generated by $A$. See \cref{rowmotion} for more details and definitions. 

Historically, rowmotion was first described for general posets by Brouwer and Schrijver~\cite{BS74}, and then again by Cameron and Fon-der-Flaas~\cite{CF95} as a composition of certain involutions called toggles. The name of rowmotion comes from the work of Striker and Williams~\cite{SW12} where, for certain posets, rowmotion is described as a composition of toggles along the rows, and related to another operation called promotion.

Restricting our attention to the poset of positive roots for the type $A$ root system, we will show that antichains of this poset are in bijection with Dyck paths and with $321$-avoiding permutations. This will allow us to define a natural rowmotion operation on $\S_n(321)$.

When studying rowmotion, it is common to look for statistics that exhibit a property called {\em homomesy} \cite{PR15}. Given a set $S$ and a bijection $\tau:S\to S$ so that each orbit of the action of $\tau$ on $S$ has finite order, we say that a statistic on $S$ is {\em homomesic} under this action if its average on each orbit is constant. 
More specifically, the statistic is said to be {\em $c$-mesic} if its average over each orbit is $c$. We prove that several statistics on $321$-avoiding permutations, including the number of fixed points, are homomesic under rowmotion.

Furthermore, we use the viewpoint of $321$-avoiding permutations to shed new light into a celebrated bijection 
of Armstrong, Stump and Thomas \cite{AST13} between antichains in root posets of finite Weyl groups (also known as nonnesting partitions) and noncrossing partitions.  We will show that, in the case of types $A$ and $B$, the Armstrong--Stump--Thomas (AST) bijection 
has a simple interpretation in terms of the Robinson--Schensted--Knuth (RSK) correspondence applied to $321$-avoiding permutations.

\section{Background}\label{background}

  In this section we review some notions about Dyck paths, noncrossing matchings, and the RSK correspondence, in particular as it applies to $321$-avoiding permutations. We also provide a basic overview of rowmotion.

\subsection{Dyck paths}\label{sec:Dyck}

Let $\D_n$ be the set of words over $\{\uu,\dd\}$ consisting of $n$ $\uu$s and $n$ $\dd$s, and satisfying that every prefix contains at least as many $\uu$s as $\dd$s. Elements of $\D_n$ are called Dyck paths, and they will be drawn in three different ways as lattice paths in $\mathbb{Z}^2$ starting at the origin. Replacing $\uu$ and $\dd$ with $(0,1)$ and $(1,0)$ (respectively, $(1,0)$ and $(0,1)$), we obtain paths that stay weakly above (respectively, below) the diagonal $y=x$. We denote these by $\Dul_n$ (respectively, $\Dlr_n$). The sets $\Dul_n$ and $\Dlr_n$ are in bijection with each other, by simply reflecting along the diagonal. The third way to draw Dyck paths that we will use is when $\uu$ and $\dd$ are replaced with $(1,1)$ and $(1,-1)$, respectively. 
In all cases, a pair of consecutive steps $\uu\dd$ is called a {\em peak}, and a pair $\dd\uu$ is called a {\em valley}. 
 Interpreting $\uu$ and $\dd$ steps of $D\in\D_n$ as opening and closing parentheses, respectively, and matching them in the usual way, a pair of matched steps will be called a \emph{tunnel}, following \cite{EP1}.
 
Several bijections between $321$-avoiding permutations and Dyck paths are known. For $\pi \in \S_n(321)$, let $E_p(\pi)\in\Dul_n$ be the path whose peaks occur at the weak excedances of $\pi$, let $E_v(\pi)\in\Dul_n $ be the path whose valleys occur at the excedances of $\pi$, and let $D_v(\pi)\in\Dlr_n$ be the path whose valleys occur at the weak deficiencies of $\pi$. The bijection that maps $E_p(\pi)$ to $D_v(\pi)$ is known as the {\em Lalanne--Kreweras involution} on Dyck paths \cite{KRE70,LAL92}, which we denote by 
$\LK=D_v\circ E_p^{-1}$; see the examples in \cref{fig:rS_example_paths}.

 Let $\NC_n$ denote the set of {\em noncrossing matchings} of $\{1,2,\dots,2n\}$, i.e., perfect matchings with the property that there do not exist $i<j<k<\ell$ such that $i$ is matched with $k$ and $j$ is matched with $\ell$. We will draw the points $1,2,\dots,2n$ around a circle in clockwise order, with a line segment connecting each pair of matched points. There is a straightforward bijection between Dyck paths and noncrossing matchings.

\begin{definition}\label{def:Match}
Let $\Match:\D_n\to\NC_n$ be the bijection defined as follows.
Given $D\in\D_n$, the points $i$ and $j$ are matched in $\Match(D)$ if the steps of $D$ in positions $i$ and $j$ form a tunnel.
 \end{definition}
Define promotion of Dyck paths to be the following map $\Pro:\D_n\to \D_n$. Given $D\in\D_n$, consider its first-return decomposition $D=\uu A\dd B$ where $A,B$ are Dyck paths, and let $\Pro(D)=A\uu B\dd$. The reason for this name is that, as observed by White \cite[Sec.\ 8]{RHO10}, applying this operation to a path $D\in\D_n$ is equivalent to applying promotion on standard Young tableaux of shape $(n,n)$.


 \subsection{RSK and 321-avoiding permutations}\label{sec:RSK321Basics}
 The RSK correspondence is a bijection between permutations and pairs of standard Young tableaux of the same shape. We refer the reader to~\cite[Sec.\ 7.11]{EC2} for definitions. Given a permutation $\pi\in\S_n$, we denote its image by $\RSK(\pi)=(P,Q)$, where $P$ and $Q$ are standard Young tableaux of the same shape. In, particular, $\pi$ is $321$-avoiding if and only if $P$ and $Q$ have at most two rows. This property is used to define the following map from $\S_n(321)$ to $\D_n$.
\begin{definition}\label{def:RSKD}
Let $\pi\in \S_n(321)$, and suppose that $\RSK(\pi)=(P,Q)$. Define a Dyck path $\RSKD(\pi)$ as follows. For $1\le i\le n$, let the $i$th step be a $\uu$ if $i$ is in the top row of $P$, and a $\dd$ otherwise; let the $(2n+1-i)$th step be a $\dd$ if $i$ is in the top row of $Q$, and a $\uu$ otherwise.
\end{definition}


 \subsection{Rowmotion}\label{rowmotion}

Let $\P$ be a finite poset. An {\em antichain} of $\P$ is a subset of pairwise incomparable elements. 
An {\em order ideal} of $\P$ is a subset $I$ with the property that if $x\in I$ and $y\le x$, then $y\in I$.
Similarly, an {\em order filter} of $\P$ is a subset $F$ with the property that if $x\in F$ and $x\le y$, then $y\in F$. Let
$\cA(P)$, $\I(\P)$ and $\F(\P)$ denote the sets of antichains, order ideals and order filters of $\P$, respectively.
The complementation map $\Theta$, defined on subsets $S$ of $\P$ by $\Theta(S) = \P\setminus S$, restricts to a bijection between $\I(P)$ and $\F(P)$.

Following~\cite{HJ20}, define the up-transfer map $\Delta:\I(\P)\to \cA(\P)$ by letting $\Delta(I)$ be the set of maximal elements of $I\in\I(P)$, and the down-transfer map $\nabla: \F(\P)\to \cA(\P)$ by letting $\nabla(F)$ be the set of minimal elements of $F\in\F(P)$. The inverses of these maps are given by 
\begin{align*}\Delta^{-1}(A) &= \{x\in \P : x\le y\text{ for some } y\in A\},\\
\nabla^{-1}(A) & = \{x\in \P : x\ge y\text{ for some } y\in A\},\end{align*} 
for $A\in\cA(P)$. 
Antichain rowmotion is the map $\rA:\cA(\P)\to \cA(\P)$ defined as the composition $\rA = \nabla\circ\Theta\circ\Delta^{-1}$.

The poset of positive roots in type $A$, which we denote by $\A^{n-1}$, can be described as the set of intervals $\{[i,j]: 1\le i\le j\le n-1\}$ ordered by inclusion. It is a ranked poset, with rank function given by $\rk([i,j])=j-i$. 
The set $\cA(\A^{n-1})$ is in bijection with $\D_n$. One such bijection consists of mapping each path $P$ to the antichain $\ant(P)$ whose elements are at the valleys of $P$. This bijection allows us to define rowmotion on Dyck paths as $\rD=\ant^{-1}\circ \rA\circ \ant$. A second bijection between $\cA(\A^{n-1})$ and $\D_n$ consists of mapping the antichain $A$ to the path $\Path(A)$ whose peaks are at the elements of the antichain. Note that $\rA =\ant\circ\Path$ and $\rD = \Path\circ \ant$.

Panyushev \cite{PAN04,PAN09} considered the map on $\cA(\A^{n})$ defined by mapping an antichain given by 
$\{[i_1,j_1],\dots [i_k, j_k]\}$ to $\{[i'_1,j'_1],\dots [i'_{n-k},j'_{n-k}]\}$
with $\{i'_1,\dots, i'_{n-k}\}= \{1,2,\dots,n\}\setminus \{j_1,\dots, j_k\}$ and $\{j'_1,\dots j'_{n-k}\}= \{1,2,\dots,n\}\setminus \{i_1,\dots, i_k\}$. As noted in~\cite{HJ20}, this map equals the composition $\ant \circ \LK\circ \ant^{-1}$, so we will denote it by $\LKA$. Hopkins and Joseph show in \cite[Thm.\ 3.5]{HJ20} that a certain map known as \emph{antichain rowvacuation}, denoted in general by $\Rvac_\cA$, coincides with $\LKA$ in the case of the type $A$ root poset.

\section{A rowmotion operation on 321-avoiding permutations}\label{defs}
To define rowmotion on $321$-avoiding permutations, first consider the following bijection to antichains of $\A^{n-1}$.
\begin{definition}\label{def:Exc}
Let $\Exc:\S_n(321)\to \cA(\A^{n-1})$ be the bijection where, for $\pi\in\S_n(321)$, we define
$$
\Exc(\pi)=\{[i,\pi(i)-1]\,:\,(i,\pi(i)) \text{ is an excedance of } \pi\}.
$$
\end{definition}
See the left column of \cref{fig:rS_example_paths} for an example
of $\Exc:\S_9(321)\to \cA(\A^{8})$, which maps the permutation $241358967$ to the antichain $\{[1,1],[2,3],[6,7],[7,8]\}$.

To see that $\Exc$ is a bijection, note that it can be written as $\Exc=\Path^{-1}\circ E_p=\ant\circ E_v$. The map $\Exc$ provides the following natural way to translate rowmotion into an operation on $\S_n(321)$.

\begin{definition}\label{def:rowmotion}
Rowmotion on $321$-avoiding permutations is the map $\rS:\S_n(321)\to \S_n(321)$ defined by $\rS=\Exc^{-1}\circ\rA\circ \Exc$.
\end{definition}

An equivalent description of $\rS$ can be given in terms of the bijections to Dyck paths, as the composition
\begin{equation}\label{eq:rS_paths}
    \rS=E_v^{-1}\circ E_p.
\end{equation}
\cref{fig:rS_example_paths} gives examples of $\rS$ computed in both ways.
Note that if $\pi\in\S_n(321)$ has upper and lower paths given by $P=E_p(\pi)$ and $Q=D_v(\pi)$, then $\sigma=\rS(\pi)$ has upper and lower paths given by $\rD(P)=E_p(\sigma)$ and $\rD^{-1}(Q)=D_v(\sigma)$. Equivalently, at the level of antichains, we have $\Exc(\sigma)=\rA(\Exc(\pi))$ and $\Exc(\sigma^{-1})=\rA^{-1}(\Exc(\pi^{-1}))$, noting that
$\Exc(\pi^{-1})$ is the antichain formed by the deficiencies of $\pi$. This follows from the fact that 
\begin{equation}\label{eq:Pan}\LKA\circ\rA=\rA^{-1}\circ\LKA,\end{equation}
which was proved by Panyushev \cite[Thm.\ 3.5]{PAN09}.
\begin{figure}[h]
\centering
\begin{tikzpicture}[scale=0.5]
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 \perm{{2,4,1,3,5,8,9,6,7}}
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  \crosscolor{1}{2}{orange}\crosscolor{2}{4}{orange}\crosscolor{6}{8}{orange}\crosscolor{7}{9}{orange}
  \begin{scope}[shift={(.5,1.5)}]
\typeAtilted{7}
\an{1}{1}\an{2}{3}\an{6}{7}\an{7}{8}
\end{scope}
 \draw[red,ultra thick] (0,0) \N\N\E\N\N\E\E\E\N\E\N\N\N\E\N\E\E\E;
 \draw[blue,ultra thick] (0,0) \E\E\N\E\N\N\E\E\E\E\N\N\N\E\N\E\N\N;
 \permc{{3,1,2,5,6,9,4,7,8}}
\draw[->] (9.5,4.5)-- node[above] {$\rS$} (10.5,4.5);
  \draw[->] (4.5,-.5)-- node[right] {$\Exc$} (4.5,-1.5);
  \draw[->] (9.5,4.5-9)-- node[above] {$\rA$} (10.5,4.5-9);
  
\begin{scope}[shift={(11,0)}]
 \draw (0,0) grid (9,9);
 \draw[dotted] (0,0)--(9,9);
 \perm{{3,1,2,5,6,9,4,7,8}}
  \draw (4.5,10) node {$\ora{3}12\ora{569}478$};
  \crosscolor{1}{3}{orange}\crosscolor{4}{5}{orange}\crosscolor{5}{6}{orange}\crosscolor{6}{9}{orange}
 \draw[red,ultra thick] (0,0) \N\N\N\E\E\E\N\N\E\N\E\N\N\N\E\E\E\E;
 \draw[blue,ultra thick] (0,0) \E\N\E\N\E\E\E\E\N\N\E\N\N\N\E\N\E\N;
 \permc{{1,2,4,6,7,3,5,8,9}}
 \draw[->] (9.5,4.5)-- node[above] {$\rS$} (10.5,4.5);
   \draw[->] (9.5,4.5-9)-- node[above] {$\rA$} (10.5,4.5-9);
  \draw[->] (4.5,-.5)-- node[right] {$\Exc$} (4.5,-1.5);
\end{scope}

\begin{scope}[shift={(22,0)}]
 \draw (0,0) grid (9,9);
 \draw[dotted] (0,0)--(9,9);
 \perm{{1,2,4,6,7,3,5,8,9}}
  \draw (4.5,10) node {$12\ora{467}3589$};
   \crosscolor{3}{4}{orange}\crosscolor{4}{6}{orange}\crosscolor{5}{7}{orange}
 \draw[red,ultra thick] (0,0) \N\E\N\E\N\N\E\N\N\E\N\E\E\E\N\E\N\E;
 \draw[blue,ultra thick] (0,0) \E\E\E\E\E\N\N\N\E\N\N\E\E\E\N\N\N\N;
 \permc{{2,3,5,7,1,4,8,9,6}}
 % \draw[->] (9.5,4.5)-- node[above] {$\rS$} (10.5,4.5);
   \draw[->] (4.5,-.5)-- node[right] {$\Exc$} (4.5,-1.5);
     % \draw[->] (9.5,4.5-9)-- node[above] {$\rA$} (10.5,4.5-9);
\end{scope}

 \begin{scope}[shift={(0,-8)},scale=1.3]
 \typeA{7}
\an{1}{1}\an{2}{3}\an{6}{7}\an{7}{8}
  \end{scope} 

 \begin{scope}[shift={(11,-8)},scale=1.3]
 \typeA{7}
\an{1}{2}\an{4}{4}\an{5}{5}\an{6}{8}
  \end{scope}
  
 \begin{scope}[shift={(22,-8)},scale=1.3]
 \typeA{7}
\an{3}{3}\an{4}{5}\an{5}{6}
  \end{scope}
\end{tikzpicture}
\caption{Two applications of rowmotion starting at $\pi=241358967\in\S_9(321)$, 
computed using \cref{def:rowmotion}, or alternatively the composition~\eqref{eq:rS_paths}. In the upper left diagram, the crosses represent $\pi$, the red path is $E_p(\pi)$, the blue path is $D_v(\pi)=\LK(E_p(\pi))$, and the dots represent $\rS(\pi)=E_v^{-1}(E_p(\pi))=312569478$.
}
\label{fig:rS_example_paths}
\end{figure}

A diagram of our bijections for permutations, paths and antichains, as well as their interactions, appears in \cref{fig:bijections}. 
We can translate the Lalanne--Kreweras involution for Dyck paths into an involution on $\S_n(321)$ by defining 
\begin{equation}\label{def:LKS}\LKS= E_p^{-1}\circ\LK\circ E_p.\end{equation}
Using that $\LKA=\ant \circ \LK\circ \ant^{-1}$, the maps $\LKS$ and $\LKA$ are related by
\begin{equation}\label{eq:LKS_LKA}\LKS= E_p^{-1}\circ\ant^{-1}\circ\LKA\circ\ant\circ E_p.\end{equation}

\begin{figure}[htb]
\centering
\begin{tikzpicture}[scale=1.7]
\draw node (S0) at (0,0) {$\S_n(321)$};
\draw node (S1) at (2,0) {$\S_n(321)$};
\draw node (D0) at (.7,1) {$\D_n$};
\draw node (D1) at (2.7,1) {$\D_n$};
\draw node (A0) at (1.4,2) {$\cA(\A^{n-1})$};
\draw node (A1) at (3.4,2) {$\cA(\A^{n-1})$};

\draw[gray] node (S0L) at (0,-3) {$\S_n(321)$};
\draw[gray] node (S1L) at (2,-3) {$\S_n(321)$};
\draw[gray] node (D0L) at (.7,-2) {$\D_n$};
\draw[gray] node (D1L) at (2.7,-2) {$\D_n$};
\draw[gray] node (A0L) at (1.4,-1) {$\cA(\A^{n-1})$};
\draw[gray] node (A1L) at (3.4,-1) {$\cA(\A^{n-1})$};
\draw[<->,dotted,thick,green!50!black] (S0L) to [bend right=20] (S1);

\draw[<->,dashed,purple] (D0) --  (D0L);
\draw[<->,dashed,purple] (D1) -- node[right]{$\LK$} (D1L);
\draw[<->,dashed,purple] (S0) --  (S0L);
\draw[<->,dashed,purple] (S1) -- node[right]{$\LKS$} (S1L);
\draw[<->,dashed,purple] (A0) --  (A0L);
\draw[<->,dashed,purple] (A1) -- node[right]{$\LKA$} (A1L);
\draw[->] (S0)-- node[below left=-2mm]{$D_v$} (D0L);

\draw[->] (D0) -- node[above right]{$\rD$} (D1);
\draw[->] (S0) -- node[above]{$\rS$} (S1);
\draw[->] (A0) -- node[above]{$\rA$} (A1);
\draw[->] (S0) -- node[left]{$E_p$} (D0);
\draw[->] (S1) -- node[right]{$E_p$} (D1);
\draw[->] (S1) -- node[left]{$E_v$} (D0);
\draw[->] (D0) -- node[left]{$\ant$} (A0);
\draw[->] (D1) -- node[left]{$\ant$} (A1);
\draw[->,brown] (S1) -- node[below right]{$\Exc$} (A0);
\draw[->] (A0) -- node[right]{$\Path$} (D1);

\draw[->,gray] (D1L) -- node[above]{$\rD$} (D0L);
\draw[->,gray] (S1L) -- node[above]{$\rS$} (S0L);
\draw[->,gray] (A1L) -- node[above]{$\rA$} (A0L);
\draw[->,gray] (S0L) -- node[left]{$E_p$} (D0L);
\draw[->,gray] (S1L) -- node[right]{$E_p$} (D1L);
\draw[->,gray] (D0L) -- node[left]{$\ant$} (A0L);
\draw[->,gray] (D1L) -- node[left]{$\ant$} (A1L);
\end{tikzpicture}

\caption{Diagram of the bijections $\rS$, $\rD$, $\rA$, $\ant$, $\Path$, $E_p$, $E_v$, $D_v$, $\LK_S$, $\LK_A$, and $\LK_D$. The vertical dashed arrows are various versions of the Lalanne--Kreweras involution, and the dotted curved arrow is the map sending a permutation to its inverse.}
\label{fig:bijections}
\end{figure}

The map $\LKS$ is closely related to the operation that sends each permutation to its inverse, as the next lemma shows.

\begin{lemma}\label{LK_of_Perm}
	If $\pi\in \S_n(321)$ then $\pi^{-1} = \rho_{\S}(\LKS(\pi))=\LKS(\rho_{\S}^{-1}(\pi))$.
 \end{lemma}

Finally, let us mention that rowmotion on $321$-avoiding permutations provides a more direct proof of a result of Hopkins and Joseph~\cite[Thm.\ 6.2]{HJ20} that the number of antichains of $\A^{n-1}$ that are fixed by the involution $\LKA\circ\rA$ is $\binom{n}{\lfloor\frac{n}{2}\rfloor}$,
by instead reducing it to a classical result of Simion and Schmidt~\cite{SS85} on the enumeration of $321$-avoiding involutions.



%%%%%%%%%%%%%

\section{Statistics and homomesies}\label{homomesies}
In this section we  show that certain statistics on $321$-avoiding permutations exhibit homomesy under the action of $\rS$. 

\subsection{The number of fixed points and the statistics $h_i$ and $\ell_i$}
The first statistic that we consider is the number of fixed points of a permutation $\pi$, denoted by $\fp(\pi)= |\{i : \pi(i) = i\}|$.
%The proof of the following result is omitted from this extended abstract.
Let $\exc(\pi)=|\{i:\pi(i)>i\}|$ and $\wexc(\pi)=|\{i:\pi(i)\ge i\}|$ denote the number of excedances and weak excedances of $\pi$, respectively.

\begin{theorem}\label{thm:fp}
The statistic $\fp$ is 1-mesic under the action of $\rS$ on $\S_n(321)$.
\end{theorem}

\begin{proof}
Let $\pi\in\S_n(321)$, and let $\mathcal{O}$ be the orbit of $\pi$ under $\rS$. Then
$$\fp(\pi)=\wexc(\pi)-\exc(\pi)=n-\exc(\pi^{-1})-\exc(\pi)=n-\card{\Exc(\pi)}-\card{\Exc(\pi^{-1})}.$$
Summing over the orbit, $$\frac{1}{|\mathcal{O}|}\sum_{\pi\in \mathcal{O}}\fp(\pi) = n-\frac{1}{|\mathcal{O}|}\sum_{\pi\in \mathcal{O}}\card{\Exc(\pi)}-\frac{1}{|\mathcal{O}|}\sum_{\pi\in \mathcal{O}}\card{\Exc(\pi^{-1})}.$$
As noted above equation~\eqref{eq:Pan}, applying $\rS$ to $\pi$ corresponds to applying $\rA$ to $\Exc(\pi)$ and $\rA^{-1}$ to $\Exc(\pi^{-1})$. It then follows by \cref{LK_of_Perm} that the sets $\{\Exc(\pi) : \pi\in \mathcal{O}\}$ and $\{\Exc(\pi^{-1}): \pi\in \mathcal{O}\}$ are complete orbits under $\rA$.
It is known~\cite{AST13} that the antichain cardinality statistic is $\frac{n-1}{2}$-mesic under the action of $\rA$ on $\cA(\A^{n-1})$.
Thus $\frac{1}{|\mathcal{O}|}\sum_{\pi\in \mathcal{O}}\card{\Exc(\pi)}=\frac{1}{|\mathcal{O}|}\sum_{\pi\in \mathcal{O}}\card{\Exc(\pi^{-1})}=\frac{n-1}{2}$, and we conclude that $\fp$ is $1$-mesic. 
\end{proof}

See \cref{fig:orbit} for a computation of $\fp(\pi)$ over a rowmotion orbit for $n=4$. As \cref{thm:fp} asserts, the average of this statistic is 1 over each orbit.

Next we consider two families of statistics on $321$-avoiding permutations, and show that they are also homomesic under rowmotion. 
The first family are the statistics $h_i$ introduced by  Hopkins and Joseph \cite{HJ20}. For $1\le i\le n-1$, they define $h_i$ on antichains $A\in\cA(\A^{n-1})$ as
$$h_i(A)=\sum_{j=1}^i \one_{[j,i]}(A)+\sum_{j=i}^{n-1}\one_{[i,j]}(A),$$ 
where $\one_{[i,j]}(A)$ is the indicator function that equals 1 if $[i,j]\in A$ and $0$ otherwise. 
For $\pi \in \S_n(321)$, we now define $h_i(\pi)$ by $h_i(\Exc(\pi))$. In terms of the array of $\pi\in\S_n(321)$, $h_i(\pi)$ counts the number of crosses of the form $(j,i+1)$ with $1\le j\le i$, or $(i,j)$ with $i+1\le j\le n$. 
%See the left of \cref{fig:hi_li} for a visualization.

Hopkins and Joseph prove in \cite[Thm. 4.3]{HJ20}
that the statistics $h_i$ on antichains are $1$-mesic under $\rA$. This result can be translated in terms of $321$-avoiding permutations as follows.

\begin{theorem}[\cite{HJ20}]\label{thm:hi} 
For $1\le i\le n-1$, the statistic $h_i$ is $1$-mesic under the action of $\rS$ on $\S_n(321)$.
\end{theorem}


\def\backgr{
\fill[brown!35!white] (0,1) rectangle (2,2);
\fill[brown!35!white] (1,2) rectangle (2,4);
\fill[pattern=north east lines,pattern color=violet] (0,0) rectangle (1,1);
\fill[pattern=north east lines,pattern color=violet] (1,1) rectangle (2,2);
\fill[pattern=north east lines,pattern color=violet] (2,2) rectangle (3,3);
\fill[pattern=north east lines,pattern color=violet] (3,3) rectangle (4,4);
 \draw (0,0) grid (4,4);
 \draw[dotted] (0,0)--(4,4);
}

\begin{figure}[htb]
    \centering
\begin{tikzpicture}[scale=0.5]
\backgr
 \perm{{1,4,2,3}}
 \draw[red,ultra thick] (0,0) \N\E\N\N\N\E\E\E;
 \draw[blue,ultra thick] (0,0) \E\E\N\N\E\N\E\N;
 \permc{{2,1,3,4}}
 \draw (5,2) node {$\mapsto$};
   \draw[brown] (3,-.5) node {1};
 \draw (2,-.5) node {1};
 \draw[violet] (1,-.5) node {1};
% \draw (3.5,-.5) node {2};
\begin{scope}[shift={(6,0)}]
\backgr
 \perm{{2,1,3,4}}
 \draw[red,ultra thick] (0,0) \N\N\E\E\N\E\N\E;
 \draw[blue,ultra thick] (0,0) \E\N\E\E\E\N\N\N;
 \permc{{1,3,4,2}}
 \draw (5,2) node {$\mapsto$};
  \draw[brown] (3,-.5) node {1};
 \draw (2,-.5) node {0};
 \draw[violet] (1,-.5) node {2};
 %\draw (3.5,-.5) node {1};
 \end{scope}
 \begin{scope}[shift={(12,0)}]
\backgr
 \perm{{1,3,4,2}}
 \draw[red,ultra thick] (0,0) \N\E\N\N\E\N\E\E;
 \draw[blue,ultra thick] (0,0) \E\E\E\N\N\E\N\N;
 \permc{{2,4,1,3}}
 \draw (5,2) node {$\mapsto$};
 \draw[brown] (3,-.5) node {1};
 \draw (2,-.5) node {2};
 \draw[violet] (1,-.5) node {1};
 \end{scope}
  \begin{scope}[shift={(18,0)}]
\backgr
 \perm{{2,4,1,3}}
 \draw[red,ultra thick] (0,0) \N\N\E\N\N\E\E\E;
 \draw[blue,ultra thick] (0,0) \E\E\N\E\N\N\E\N;
 \permc{{3,1,2,4}}
 \draw (5,2) node {$\mapsto$};
 \draw[brown] (3,-.5) node {2};
 \draw (2,-.5) node {1};
 \draw[violet] (1,-.5) node {0};
 \end{scope}
\begin{scope}[shift = {(0,-6)}]
\backgr
 \perm{{3,1,2,4}}
 \draw[red,ultra thick] (0,0) \N\N\N\E\E\E\N\E;
 \draw[blue,ultra thick] (0,0) \E\N\E\N\E\E\N\N;
 \permc{{1,2,4,3}}
 \draw (5,2) node {$\mapsto$};
  \draw[brown] (3,-.5) node {0};
 \draw (2,-.5) node {1};
 \draw[violet] (1,-.5) node {1};
 \end{scope}
\begin{scope}[shift={(6,-6)}]
\backgr
 \perm{{1,2,4,3}}
 \draw[red,ultra thick] (0,0) \N\E\N\E\N\N\E\E;
 \draw[blue,ultra thick] (0,0) \E\E\E\N\N\N\E\N;
 \permc{{2,3,1,4}}
 \draw (5,2) node {$\mapsto$};
  \draw[brown] (3,-.5) node {1};
 \draw (2,-.5) node {0};
 \draw[violet] (1,-.5) node {2};
 \end{scope}
 \begin{scope}[shift={(12,-6)}]
\backgr
 \perm{{2,3,1,4}}
 \draw[red,ultra thick] (0,0) \N\N\E\N\E\E\N\E;
 \draw[blue,ultra thick] (0,0) \E\E\N\E\E\N\N\N;
 \permc{{3,1,4,2}}
 \draw (5,2) node {$\mapsto$};
  \draw[brown] (3,-.5) node {2};
 \draw (2,-.5) node {2};
 \draw[violet] (1,-.5) node {1};
 \end{scope}
  \begin{scope}[shift={(18,-6)}]
\backgr
 \perm{{3,1,4,2}}
 \draw[red,ultra thick] (0,0) \N\N\N\E\E\N\E\E;
 \draw[blue,ultra thick] (0,0) \E\N\E\E\N\E\N\N;
 \permc{{1,4,2,3}}
\draw (5,2) node {$\mapsto$};
 \draw[brown] (3,-.5) node {0};
 \draw (2,-.5) node {1};
 \draw[violet] (1,-.5) node {0};
 \end{scope}
\end{tikzpicture}
    \caption{The rowmotion orbit containing 1423. The numbers below each diagram are the values  $\textcolor{violet}{\fp(\pi)}$, $h_2(\pi)$ and $\textcolor{brown}{\ell_2(\pi)}$, from left to right.}
    \label{fig:orbit}
\end{figure}

Next we define a new family of permutation statistics, that we denote by $\ell_i$ for $1\le i \le n$. For $\pi\in\S_n(321)$, let $\ell_i(\pi)$ be the number of crosses in the array of $\pi$ of the form $(j,i)$ with $1\le j\le i$, plus the number of crosses of the form $(i,j)$ with $i< j\le n$.
It turns out that the statistics $\ell_i$ are homomesic as well, see \cref{fig:orbit} for an example.

\begin{theorem}\label{LiHom}
		For $1\le i\le n$, the statistic $\ell_i$ is $1$-mesic under the action of $\rho_{\S}$ on $\S_n(321)$. 
	\end{theorem}
 
\subsection{The sign statistic}
In this subsection we describe how rowmotion interacts with the sign of a $321$-avoiding permutation.
\begin{theorem}\label{SignRowmotionTheorem}
For all $\pi\in\S_n(321)$,
\[\sgn(\rS(\pi))=\sgn(\LKS(\pi))=\begin{cases}
\sgn(\pi) & \text{if $n$ is odd,}\\
-\sgn(\pi) & \text{if $n$ is even.}
 \end{cases}\]
\end{theorem} 

It is a classical result of Simion and Schmidt \cite[Prop.\ 2]{SS85} 
that, when $n$ is even, the set $\S_n(321)$ contains the same number of odd and even permutations. A bijective proof of this fact was given by Reifegerste~\cite{RE05}. \cref{SignRowmotionTheorem} gives two new bijections, $\rS$ and $\LKS$, between the subsets of odd and even permutations in $\S_n(321)$. Furthermore, $\LKS$ has the additional property of being a sign-reversing involution on $\S_n(321)$. 

Finally, we note the following two immediate consequences of \cref{SignRowmotionTheorem}.
\begin{corollary}
For even $n$, the statistic $\sgn$ on $\S_n(321)$ is $0$-mesic under the action of $\rS$.
\end{corollary}

\begin{corollary}\label{NoLKFixed}
For even $n$, the map $\LK$ on $\D_n$ has no fixed points.
\end{corollary}
\Cref{NoLKFixed} is equivalent to \cite[Thm.\ 4.6]{PAN04} for even~$n$. 
 	
\section{The Armstrong--Stump--Thomas bijection as a map on permutations}\label{AST}

In~\cite{AST13}, Armstrong, Stump and Thomas constructed a bijection between antichains in root posets of finite Weyl groups and noncrossing matchings, having the property that it translates rowmotion on antichains into rotation of noncrossing matchings. In type $A$, we can interpret antichains as $321$-avoiding permutations via the bijection $\Exc$. It turns out that, with this interpretation, the Armstrong--Stump--Thomas bijection is equivalent to the well-known Robinson--Schensted--Knuth correspondence restricted to $321$-avoiding permutations. This fact extends to the root poset in type~$B$.


\subsection{$\AST$ in type \emph{A}}

The Armstrong--Stump--Thomas bijection is described in \cite{AST13} in much more generality, but for our purposes we will be restricting exclusively to the cases of type $A$ and $B$. We follow the description given by Defant and Hopkins~\cite{DH21}. In type $A$, we denote this bijection by $\AST$.

\begin{definition}[$\AST$ in type $A$ \cite{DH21}]\label{def:AST}
Let $\AST: \cA(\A^{n-1})\to \NC_{n}$ be the bijection where the image of $A\in\cA(\A^{n-1})$ is obtained as follows.
For each $1\le i\le n$, the vertex $2n+1-i$ of the matching will be denoted by $\ol{i}$, so that the vertices are $1,2,\dots, n, \ol{n}, \ol{n-1},\dots, \ol{1}$ in clockwise direction. 
For each $i$ from $1$ to $n$, consider two options:
\begin{itemize}
    \item if $[i,j-1]\in A$ for some $j$, match the vertex $j$ with the nearest unmatched vertex in  counterclockwise direction;
    \item otherwise, match the vertex $\ol{i}$ with the nearest unmatched vertex in clockwise direction.
\end{itemize}
\end{definition}
We can now state our main result in this section.
\begin{theorem}\label{ASTConnection} 
Let $\Match$, $\RSKD$, $\Exc$ and $\AST$ be the bijections from \cref{def:Match,def:RSKD,def:Exc,def:AST}, respectively. Then $$\AST = \Match\circ \RSKD \circ \Exc^{-1}.$$
\end{theorem}

See \cref{fig:CommutativeDiagramExample} for examples of these maps. 
It is shown in~\cite{AST13} that the bijection $\AST$ is equivariant in the sense that 
$\AST\circ\rA=\Rot\circ\AST$.
\begin{figure}[h!]
\label{ThmASTExample}
    \centering
    \begin{tikzpicture}
\draw[->] (4.2,2) to [bend left =15] node[above]{$\AST$} (15,2);
\draw (3.5,1.7) node {$\cA(\A^{n-1})$};
\draw (6.7,1.9) node {$\S_n(321)$};
\draw (11,1.5) node {$\D_n$};
\draw (15.5,1.7) node {$\NC_n$};
  \begin{scope}[shift={(2.25,-.5)},scale=.8]
\typeA{3}
\an{1}{2}\an{2}{4}
% \draw[-to] (0, 1) -- (-.5,1);
\draw[-to] (1.5,-1) --node[right]{$\rA$} (1.5,-2);
\end{scope}

  \begin{scope}[shift={(5.5,-1)},scale=.5]
	  \draw (0,0) grid (5,5);
 \draw[dotted] (0,0)--(5,5);
 \perm{{3,5,1,2,4}}
  \draw[red,ultra thick] (0,0) \N\N\N\E\N\N\E\E\E\E;
  \draw[-to] (-0.5, 2.66) --node[above]{$\Exc$} (-1.5,2.66);
\draw[-to] (2.5,-.5) --node[right]{$\rS$} (2.5,-2);
\end{scope}
\begin{scope}[shift={(9,0)},scale=.7]
\coordinate (a) at (1,0);
\coordinate (b) at (.5,.5);
\filldraw[thick] (0,0)\start\up\up\dn\up\dn\dn\up\up\dn\dn;
\draw[-to ] (-1.25, .5) --node[above]{$\RSKD$} (-.25,.5);
\draw[-to ] (5.5, .5) --node[above]{$\Match$} (6.75,.5);
\draw[-to] (2.5,-1.5) --node[right]{$\Pro^{-1}$} (2.5,-3);
\end{scope}
\begin{scope}[shift={(15.5,0)},scale=.8]
\foreach \i in {1,...,10} {
\filldraw (90+18-\i*36:1.4) circle (.1);
\draw (90+18-\i*36:1.7) node[scale=.7] {\i};
}
\match{1}{6}\match{2}{3}\match{4}{5}\match{7}{10}\match{8}{9}
\draw[-to] (0,-2) --node[right]{$\Rot$} (0,-3);
\end{scope}
  \begin{scope}[shift={(2.25,-4.5)},scale=.8]
\typeA{3}
\an{1}{3}
% \draw[-to] (0, 1) -- (-.5,1);
\draw[-to] (1.5,-1) --node[right]{$\rA$} (1.5,-2);
\end{scope}

  \begin{scope}[shift={(5.5,-5)},scale=.5]
	  \draw (0,0) grid (5,5);
 \draw[dotted] (0,0)--(5,5);
 \perm{{4,1,2,3,5}}
  \draw[red,ultra thick] (0,0) \N\N\N\N\E\E\E\E\N\E;
  \draw[-to] (-0.5, 2.66) --node[above]{$\Exc$} (-1.5,2.66);
\draw[-to] (2.5,-.5) --node[right]{$\rS$} (2.5,-2);
\end{scope}
\begin{scope}[shift={(9,-4)},scale=.7]
\coordinate (a) at (1,0);
\coordinate (b) at (.5,.5);
\filldraw[thick] (0,0)\start\up\up\up\dn\up\dn\dn\dn\up\dn;
\draw[-to ] (-1.25, .5) --node[above]{$\RSKD$} (-.25,.5);
\draw[-to ] (5.5, .5) --node[above]{$\Match$} (6.75,.5);
\draw[-to] (2.5,-1.5) --node[right]{$\Pro^{-1}$} (2.5,-3);
\end{scope}
\begin{scope}[shift={(15.5,-4)},scale=.8]
\foreach \i in {1,...,10} {
\filldraw (90+18-\i*36:1.4) circle (.1);
\draw (90+18-\i*36:1.7) node[scale=.7] {\i};
}
\match{1}{8}\match{9}{10}\match{4}{3}\match{7}{2}\match{5}{6}
\draw[-to] (0,-2) --node[right]{$\Rot$} (0,-3);
\end{scope}

  \begin{scope}[shift={(2.25,-8.5)},scale=.8]
\typeA{3}
\an{4}{4}
\end{scope}

  \begin{scope}[shift={(5.5,-9)},scale=.5]
	  \draw (0,0) grid (5,5);
 \draw[dotted] (0,0)--(5,5);
 \perm{{1,2,3,5,4}}
  \draw[red,ultra thick] (0,0) \N\E\N\E\N\E\N\N\E\E;
  \draw[-to] (-0.5, 2.66) --node[above]{$\Exc$} (-1.5,2.66);
\end{scope}
\begin{scope}[shift={(9,-8)},scale=.7]
\coordinate (a) at (1,0);
\coordinate (b) at (.5,.5);
\filldraw[thick] (0,0)\start\up\up\up\up\dn\up\dn\dn\dn\dn;
\draw[-to ] (-1.25, .5) --node[above]{$\RSKD$} (-.25,.5);
\draw[-to ] (5.5, .5) --node[above]{$\Match$} (6.75,.5);
\end{scope}
\begin{scope}[shift={(15.5,-8)},scale=.8]
\foreach \i in {1,...,10} {
\filldraw (90+18-\i*36:1.4) circle (.1);
\draw (90+18-\i*36:1.7) node[scale=.7] {\i};
}
\match{1}{10}\match{9}{2}\match{8}{3}\match{7}{6}\match{4}{5}
\end{scope}
\end{tikzpicture}
    \caption{The three terms of the respective orbits of the elements of $\cA(\A^4)$, $\S_5(321)$, $\D_5$, and $\NC_5$ associated to and beginning with the permutation $35124$ together with the respective actions of $\rA$, $\rS$, $\Pro^{-1}$, and $\Rot$. }
    \label{fig:CommutativeDiagramExample}
\end{figure}

\subsection{$\AST$ in type \emph{B}}

Let $\B^m$ denote the positive root poset of the type $B_m$ root system. The poset $\B^m$ is isomorphic to the quotient of $\A^{2m-1}$ by the relations of $[i,j]\sim [2m-j,2m-i]$ for all $[i,j]\in \A^{2m-1}$.
In type $B$, the Armstrong--Stump--Thomas map \cite{AST13} is the map $\AST_B:\cA(\B^m)\to\NC_{2m}$ defined as follows. For $A\in \cA(\B^m)$, consider the antichain in $\A^{2m-1}$ given by
\begin{equation}\label{eq:hatA}
\hat{A}=\{[i,j]:[i,j]\in A\text{ or }[2m-j,2m-i]\in A\}.
\end{equation}
Now let $\AST_B(A)=\AST(\hat{A})$, where
$\AST$ is the map from \cref{def:AST}. 


It is shown in \cite[Lemma 3.5]{AST13} that the map $\rA^n$ on $\cA(\A^{n-1})$, obtained by applying rowmotion $n$ times, sends an antichain $\{[i_1,j_1],\dots,[i_k,j_k]\}$ to the antichain $\{[n-j_1,n-i_1],[n-j_2,n-i_2],\dots,[n-j_k,n-i_k]\}$. 
The antichains that are invariant under this map are those that are symmetric under vertical reflection. When $n=2m$, these are the antichains in the image of the embedding $A\mapsto \hat{A}$ from $\cA(\B^m)$ into $\cA(\A^{2m-1})$ given by~\eqref{eq:hatA}. Thus the image of $\AST_B$ consists of the matchings in $\NC_{2m}$ that are invariant under applying $\Rot^{2m}$, i.e., under rotation by $180^\circ$. 
Denoting by $\CSNC_n$ the set of centrally symmetric matchings in $\NC_n$, one concludes that $\AST_B$ is a bijection between $\cA(\B^m)$ and $\CSNC_{2m}$.

Our alternative description of $\AST$ in terms of $\RSK$ allows us to give a new proof of this property of $\AST_B$. Unlike the proof in~\cite{AST13}, which relies on the properties of rowmotion, ours uses well-known properties of the $\RSK$ correspondence.

For $\pi\in \S_n$, its \emph{reverse complement} is the permutation $\pi^{rc}$ such that $\pi^{rc}(i)=n+1-\pi(n+1-i)$ for all $i$. The array  of $\pi^{rc}$ is obtained by rotating the array of $\pi$ by $180^\circ$. Thus, the array of $(\pi^{rc})^{-1}$ is obtained by reflecting the array of $\pi$ along the secondary diagonal, i.e., the one passing through the bottom-left and upper-right corners of the array. 
The map $\Exc$ from \cref{def:Exc} restricts to a bijection between permutations $\pi\in\S_n(321)$ such that $\pi=(\pi^{rc})^{-1}$, and antichains in $\cA(\A^{2m-1})$ that are invariant under vertical reflection, which in turn are in bijection with $\cA(\B^m)$.
The behavior of $\RSK$ under these symmetries is well understood.

\begin{theorem}[{\cite[Thm.\ A1.2.10]{EC2}}] \label{thm:RSKrc}
Suppose that $\RSK(\pi) = (P,Q)$. Then $\RSK(\pi^{rc})= (\evac(P),\evac(Q))$, and $\RSK((\pi^{rc})^{-1}) = (\evac(Q),\evac(P))$. 
\end{theorem}

These properties allow us to show the following without use of $\AST$.

\begin{theorem}\label{CentralSymmetry}
For $\pi\in\S_n(321)$, the matchings $\Match(\RSKD((\pi^{rc})^{-1}))$ and $\Match(\RSKD(\pi))$ are $180^\circ$ rotations of each other.
Thus, the map $\Match\circ\RSKD$ restricts to a bijection between $\{\pi\in\S_n(321):\pi=(\pi^{rc})^{-1}\}$ and $\CSNC_n$.
\end{theorem}

Finally, another byproduct of our permutation perspective is that we can enumerate antichains of $\B^n$ which are fixed under the action of $\LKA\circ \rA$, answering a question of Hopkins and Joseph \cite[Remark 6.7]{HJP}. 
\begin{proposition}
\[\#\{A\in \cA(\B^n) : \LKA(\rA(A)) = A\} = 2^n.\]
\end{proposition}

\acknowledgements{We thank Sam Hopkins, Michael Joseph, Tom Roby, Jessica Striker, and Justin Troyka for helpful discussions.}

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