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%
%\usepackage{lipsum}

\usepackage{ytableau}
\newcommand{\BB}{\mathsf{B}}
\newcommand{\PP}{\mathsf{P}}
\newcommand{\bbb}{\mathsf{b}}
\newcommand{\inc}{\ensuremath{\mathrm{Inc}}}
\newcommand{\Frame}{\ensuremath{\mathsf{Frame}}}

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\title[Proof of the Cameron and Fon-Der-Flaass periodicity conjecture]{Proof of the Cameron and Fon-Der-Flaass periodicity conjecture}

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%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
\author[Rebecca Patrias and Oliver Pechenik]{Rebecca Patrias\thanks{\href{mailto:patr0028@stthomas.edu}{patr0028@stthomas.edu}. Oliver Pechenik was partially supported by a Mathematical Sciences Postdoctoral Research Fellowship (\#1703696) from the National Science Foundation.}\addressmark{1} \and Oliver Pechenik\addressmark{2}}

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\address{\addressmark{1}Department of Mathematics, University of St.\ Thomas, St.\ Paul, MN 55105, USA\\ \addressmark{2}Department of Combinatorics \& Optimization, University of Waterloo, ON  N2L 3G1, Canada}

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\abstract{One of the oldest outstanding problems in dynamical algebraic combinatorics is the following conjecture of P.~Cameron and D.~Fon-Der-Flaass (1995). Consider a plane partition $P$ in an $a \times b \times c$ box ${\sf B}$. Let $\Psi(P)$ denote the smallest plane partition containing the minimal elements of ${\sf B} - P$. Then if $p= a+b+c-1$ is prime, Cameron and Fon-Der-Flaass conjectured that the cardinality of the $\Psi$-orbit of $P$ is always a multiple of $p$.

This conjecture was established for $p \gg 0$ by Cameron and Fon-Der-Flaass (1995) and for slightly smaller values of $p$ in work of K.~Dilks, J.~Striker, and the second author (2017). 
Our main theorem specializes to prove this conjecture in full generality. }

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\resume{L'un des probl\`emes les plus anciens de la combinatoire alg\'ebrique dynamique est la conjecture suivante de P.~Cameron et D.~Fon-Der-Flaass (1995). Nous consid\'erons une partition de plans $P$ dans une $a \times b \times c$ bo\^{i}te ${\sf B}$. Soit $\Psi(P)$ la plus petite partition de plans contenant les \'el\'ements minimaux de ${\sf B} - P$. Alors, si $p = a + b + c-1$ est premier, Cameron et Fon-Der-Flaass ont suppos\'e que la cardinalit\'e de l'orbite $\Psi$ de $P$ est toujours un multiple de $p$.

Cette conjecture a \'et\'e \'etablie pour $p \gg 0$ par Cameron et Fon-Der-Flaass (1995) et pour des valeurs l\'eg\`erement plus petites de $p$ dans les travaux de K.~Dilks, J.~Striker, et du deuxi\`eme auteur (2017).
Notre th\'eor\`eme principal se sp\'ecialise pour prouver cette conjecture en toute g\'en\'eralit\'e.}

%% put your keywords here, or comment this out if you don't have them yet
\keywords{dynamical algebraic combinatorics, rowmotion, plane partition, increasing tableau, $K$-promotion}

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\begin{document}

\maketitle
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\section{Introduction}\label{sec:intro}

The relatively young field of \emph{dynamical algebraic combinatorics} studies dynamical properties of actions on various fundamental objects of algebraic combinatorics. For example, alternating sign matrices, plane partitions, root systems, and Young tableaux all carry combinatorially natural cyclic group actions. In dynamical algebraic combinatorics, one is interested in establishing features of the resulting orbit structures, such as  \emph{cyclic sieving phenomena} \cite{Reiner.Stanton.White}, \emph{homomesies} \cite{Propp.Roby}, \emph{periodicities}, and \emph{resonance phenomena} \cite{Dilks.Pechenik.Striker}. For an excellent survey of the area, see \cite{Striker:notices}.

One of the most studied actions in dynamical algebraic combinatorics is called \emph{rowmotion}. Rowmotion can be defined as an action on the order ideals of any finite poset $\PP$. Interesting dynamical properties appear when $\PP$ is chosen to be a poset of significance in algebraic combinatorics. While much of the dynamical algebraic combinatorics literature dates from the past 15 or so years, rowmotion has older roots; it first appeared in 1974 through independent work of P.~Duchet \cite{Duchet} (in a special case) and of A.~Brouwer and A.~Schrijver \cite{Brouwer.Schrijver} (in full generality).

One of the oldest open problems in dynamical algebraic combinatorics has been a 1995 conjecture of P.~Cameron and D.~Fon-Der-Flaass \cite{Cameron.Fonderflaass} on the periodicity of rowmotion for plane partitions. This paper is an extended abstract of \cite{Patrias.Pechenik}, whose main goal is to prove their conjecture, which we now recall.

Fix positive integers $a,b,c \in \mathbb{Z}^+$ and consider plane partitions sitting inside a rectangular $a \times b \times c$ box. We identify this box with the poset $\BB_{a,b,c} = {\bf a} \times {\bf b} \times {\bf c}$ that is the product of three chains, and identify plane partitions in this box with order ideals of the poset $\BB_{a,b,c}$. 

We write $J(\PP)$ for the set of all order ideals of a poset $\PP$. Given $I \in J(\PP)$, define $\Psi(I)$ to be the order ideal generated by the minimal elements of the complementary order filter $\PP - I$. Following \cite{Striker.Williams}, we refer to the operator $\Psi$ as \emph{rowmotion}. It is straightforward to see that the action of $\Psi$ is reversible, so it permutes the elements of $J(\PP)$ and partitions them into disjoint orbits. For a general poset $\PP$, these orbits tend to be large and without discernible structure. However, for special posets $\PP$, intricate structure has been discovered; see, e.g., \cite{Armstrong.Stump.Thomas,Brouwer.Schrijver,Cameron.Fonderflaass,Mandel.Pechenik,Panyushev,Propp.Roby,Rush.Shi,Striker.Williams,Vorland} for various such results.

Cameron and Fon-Der-Flaass \cite{Cameron.Fonderflaass} made the following periodicity conjecture for rowmotion on the poset $\BB_{a,b,c}$.

\begin{conjecture}[{\cite{Cameron.Fonderflaass}}]\label{conj:main}
	Suppose $p=a+b+c-1$ is prime. Then the cardinality of every $\Psi$-orbit of $J(\BB_{a,b,c})$ is a multiple of $p$.
\end{conjecture}

\begin{remark}
	Conjecture~\ref{conj:main} proposes a special kind of \emph{resonance} in the sense of \cite{Dilks.Pechenik.Striker}. That is, while the $\Psi$-orbit cardinalities remain unknown, they all ``resonate with the frequency $p$,'' being all of the form $hp$ for some positive integers $h$. It would be very interesting to understand the values $h$ that appear. Experimentally, there appears to be a strong bias toward odd values of $h$. We currently have no explanation for this phenomenon, nor do we have good upper bounds on the values $h$.
\end{remark}

Our main result is the following, which implies Conjecture~\ref{conj:main}. 

\begin{thm}\label{thm:main}
	Let $k$ be the cardinality of any $\Psi$-orbit of $J(\BB_{a,b,c})$. Then 
	\[
	\gcd(k,a+b+c-1) > 1.
	\]
\end{thm}

Previous work had succeeded in establishing Conjecture~\ref{conj:main} only for very small and very large values of $c$.
The case $c=1$ was established earlier by Brouwer and Schrijver \cite{Brouwer.Schrijver} and the case $c=2$ by Cameron and Fon-Der-Flaass \cite{Cameron.Fonderflaass}. (Indeed, in these ``small $c$'' cases the size of every $\Psi$-orbit is exactly $p$.) Cameron and Fon-Der-Flaass \cite{Cameron.Fonderflaass} also established the ``large $c$'' case 
$
c > ab - a - b + 1.
$
This bound was later improved to 
\[
c > \frac{2ab - 2}{3} -a - b + 2
\] 
in \cite[Theorem~4.13]{Dilks.Pechenik.Striker}.

Our proof of Theorem~\ref{thm:main} and Conjecture~\ref{conj:main} is uniform and does not rely on any of these previous partial results. Nonetheless, we are heavily indebted to previous work that was not available when Cameron and Fon-Der-Flaass first made their conjecture. Explicitly, our proof calls upon some of the main results of \cite{Dilks.Pechenik.Striker} and \cite{Pechenik:frames}. In a deeper sense, our proof builds on technology and theorems developed previously in various other papers \cite{Pechenik:CSP, Striker.Williams, Thomas.Yong:K, Thomas.Yong:Plancherel}, as well. 

More specifically, we  use the results of \cite{Dilks.Pechenik.Striker} to translate Theorem~\ref{thm:main} into an equivalent statement about the combinatorics of \emph{$K$-promotion} on \emph{increasing tableaux}. $K$-promotion was first studied in \cite{Pechenik:CSP} as an outgrowth of the combinatorics of $K$-theoretic Schubert calculus for Grassmannians introduced in \cite{Thomas.Yong:K}, and has since been studied in several purely combinatorial contexts. We then prove this translated conjecture, relying on the main theorem of \cite{Pechenik:frames}.



\section{Main results}\label{sec:reformulation}

Our first step in proving Theorem~\ref{thm:main} is to use the results of \cite{Dilks.Pechenik.Striker} to translate it into an equivalent statement regarding different combinatorics. First, we recall the definitions of increasing tableaux and the $K$-promotion operator on them.

We write $a \times b$ to denote the grid of boxes with $a$ rows and $b$ columns.  Equivalently, this is the Young diagram of the partition with $a$ parts all of size $b$.
Index the boxes of $a \times b$ as in a matrix, so the box $(1,2)$ is the box in the second column from the left in the top row. For a box $\bbb$ in $a \times b$, we write $\bbb^\rightarrow$ for the box immediately right of $\bbb$, $\bbb^\downarrow$ for the box immediately below $\bbb$, etc. A \emph{short ribbon} in $a \times b$ is an edge-connected subset of boxes with at most two in any row or column.

An \emph{increasing tableau} of shape $a \times b$ is a filling $T$ of the boxes of $a \times b$ with positive integers, so that rows strictly increase from left to right and columns strictly increase from top to bottom. That is, for every box $\bbb$, we have $T(\bbb) < T(\bbb^\rightarrow)$ and $T(\bbb) < T(\bbb^\downarrow)$, provided that $\bbb^\rightarrow$ (resp.\ $\bbb^\downarrow$) exists. 
We write $\inc(a \times b)$ for the set of all increasing tableaux of shape $a \times b$ and write $\inc^q(a \times b)$ for the finite subset with entries at most $q$. Note that in an increasing tableau, if we look at the set of boxes containing either $i$ or $i+1$, the edge-connected components of this set are all short ribbons.
 

\begin{example}
An increasing tableau of shape $3 \times 6$ is
\[
T = \ytableaushort{12356{10}, 24589{11},679{10}{13}{17}} \in \inc^{17}(3 \times 6).
\]
Note that not every number from $1$ to $17$ need appear. Note also that, for example, the boxes labeled $4$ and $5$ make up two short ribbons, while the boxes labeled $1$ and $2$ make up a single short ribbon.
\end{example}

Following \cite{Buch.Samuel}, we say that $T \in \inc(a \times b)$ is \emph{minimal} if we have 
\begin{itemize}
	\item $T(1,1) = 1$,
	\item $T(\bbb^\rightarrow) = T(\bbb) + 1$ (for all $\bbb$ not in the rightmost column), and
	\item $T(\bbb^\downarrow) = T(\bbb) + 1$ (for all $\bbb$ not in the bottom row).
\end{itemize}
Note that there is a unique minimal tableau $M_{a \times b}$ of each shape $a \times b$, and that $M_{a \times b}$ is the unique element of $\inc^{a+b-1}(a \times b)$. Moreover, $\inc^q(a \times b)$ is empty if $q < a+b-1$. 

We now recall the definition of \emph{$K$-promotion} on increasing tableaux.  Let $T \in \inc^q(a \times b)$. Consider the short ribbons consisting of the boxes labeled $1$ and $2$. Say a short ribbon is \emph{trivial} if it consists of a single box, and \emph{nontrivial} otherwise. For each trivial short ribbon, we do nothing, while for each nontrivial short ribbon, we swap the labels $1$ and $2$. The result is generally not an increasing tableau, but nonetheless consider the short ribbons in it consisting of the boxes labeled $1$ and $3$ and repeat this process, successively swapping the pairs of labels $(1,4),(1,5), \dots, (1,q)$ in nontrivial short ribbons. Note that, if the box in position $(1,1)$ originally had label $1$, then label $1$ finally appears only in position $(a,b)$. To finish, decrement the label in each box by $1$, and replace any resulting $0$ label by $q$. The result is now an increasing tableau in $\inc^q(a \times b)$, the $K$-promotion of $T$. See Example~\ref{ex:promotion} for an example of this process. We will abuse notation by also denoting the $K$-promotion of $T$ by $\Psi(T)$, as there can be no confusion with rowmotion of plane partitions. We write $\Psi^\bullet(T)$ to denote the $\Psi$-orbit of the increasing tableau $T$.

\begin{remark}
	Increasing tableaux are a special case of the more classically studied \emph{semistandard tableaux} and $K$-promotion shares features with M.-P.~Sch\"utzenberger's \emph{promotion} (see \cite{Schutzenberger}) for semistandard tableaux; however, promotion of semistandard tableaux does not preserve the subset of increasing tableaux and $K$-promotion does not coincide with promotion.
\end{remark}

\begin{example}\label{ex:promotion} Starting with the tableau $T\in \inc^9(4\times 4)$ shown below, we illustrate the process of computing its $K$-promotion $\Psi(T)$. At each step, trivial short ribbons are shown in light gray and nontrivial short ribbons are shown in darker gray. 
\ytableausetup{boxsize=1.4em}
\begin{align*}T&=\begin{ytableau}*(gray) 1 & *(gray) 2 & 4 & 5 \\ *(gray) 2 & 3 & 5 & 6\\ 4 & 5 & 7 & 8\\ 5 & 6 & 8 & 9\end{ytableau}\rightarrow\begin{ytableau}2 & *(gray) 1 & 4 & 5 \\ *(gray) 1 & *(gray) 3 & 5 & 6\\ 4 & 5 & 7 & 8\\ 5 & 6 & 8 & 9\end{ytableau}\rightarrow\begin{ytableau}2 & 3 & *(lightgray) 4 & 5 \\ 3 & *(lightgray) 1 & 5 & 6\\ *(lightgray) 4 & 5 & 7 & 8\\ 5 & 6 & 8 & 9\end{ytableau}\rightarrow\begin{ytableau}2 & 3 &  4 & *(lightgray) 5 \\ 3 & *(gray) 1 & *(gray) 5 & 6\\ 4 & *(gray) 5 & 7 & 8\\ *(lightgray) 5 & 6 & 8 & 9\end{ytableau}\rightarrow\begin{ytableau}2 & 3 &  4 & 5 \\ 3 & 5 & *(gray) 1 & *(gray) 6\\ 4 & *(gray) 1 & 7 & 8\\ 5 & *(gray) 6 & 8 & 9\end{ytableau} \\ &\rightarrow \begin{ytableau}2 & 3 &  4 & 5 \\ 3 & 5 & 6 & *(lightgray) 1\\ 4 & 6 & *(lightgray) 7 & 8\\ 5 & *(lightgray) 1 & 8 & 9\end{ytableau} \rightarrow \begin{ytableau}2 & 3 &  4 & 5 \\ 3 & 5 & 6 & *(gray) 1\\ 4 & 6 & 7 & *(gray) 8\\ 5 & *(gray) 1 & *(gray) 8 & 9\end{ytableau} \rightarrow  \begin{ytableau}2 & 3 &  4 & 5 \\ 3 & 5 & 6 & 8 \\ 4 & 6 & 7 & *(gray) 1\\ 5 & 8 & *(gray) 1 & *(gray) 9\end{ytableau} \rightarrow 
 \begin{ytableau}2 & 3 &  4 & 5 \\ 3 & 5 & 6 & 8 \\ 4 & 6 & 7 & 9\\ 5 & 8 & 9 & 1\end{ytableau} \rightarrow 
  \begin{aligned}[t] &\begin{ytableau}1 & 2 &  3 & 4 \\ 2 & 4 & 5 & 7 \\ 3 & 5 & 6 & 8\\ 4 & 7 & 8 & 9\end{ytableau} \\
  & =\Psi(T) \end{aligned} \end{align*}
\end{example}
\ytableausetup{boxsize=normal}

By \cite[Theorem~4.4]{Dilks.Pechenik.Striker}, there is a $\Psi$-equivariant bijection between the sets $J(\BB_{a,b,c})$ and $\inc^{a+b+c-1}(a \times b)$. Hence, to prove Theorem~\ref{thm:main} and Conjecture~\ref{conj:main}, it is sufficient to establish the following.

\begin{thm}\label{thm:tab}
Let $q > a+b -1$ and suppose the $\Psi$-orbit of $T \in \inc^q(a \times b)$ has cardinality $k$. Then $
\gcd(k, q) > 1.
$
\end{thm}

\begin{remark}
	The hypothesis $q > a+b-1$ in Theorem~\ref{thm:tab} is necessary merely to exclude the minimal tableau $M_{a \times b}$, corresponding under the $\Psi$-equivariant bijection of \cite{Dilks.Pechenik.Striker} to the empty plane partition in the degenerate $a \times b \times 0$ box. Obviously, these objects have $\Psi$-orbits of size $1$. 
\end{remark}


%\section{Proof of Theorem~\ref{thm:tab}}\label{sec:proof}

The \emph{frame} of the shape $a \times b$ is the set $\Frame(a \times b)$ consisting of those boxes in the first or last column, or first or last row, of $a \times b$. The \emph{frame} $\Frame(U)$ of the tableau $U \in \inc^q(a \times b)$ is the restriction of the filling $U$ to $\Frame(a \times b)$.

\begin{example}
	For $T$ as in Example~\ref{ex:promotion}, the frame consists of the boxes shaded in light gray below.
	\[\begin{ytableau}*(lightgray) 1 & *(lightgray) 2 & *(lightgray) 4 & *(lightgray) 5 \\ *(lightgray) 2 &  3 &   5 &  *(lightgray) 6\\ *(lightgray) 4 &   5 & 7 &  *(lightgray) 8\\ *(lightgray) 5 &  *(lightgray) 6 & *(lightgray)  8 &  *(lightgray) 9\end{ytableau}\]
\end{example}

%Consider the cyclic group $C_k = \langle \psi \rangle$ of order $k$. Define an action of $C_k$ on $\Psi^\bullet(T)$ by $\psi \cdot U = \Psi(U)$ for all $U \in \Psi^\bullet(T)$.
% Since $k$ and $q$ are relatively prime, the group element $\psi^q$ also generates $C_k$. Hence, every $U \in \Psi^\bullet(T)$ is of the form $\Psi^{mq}(T)$ for some positive integer $m \in \mathbb{Z}^+$. 
 
 To prove Theorem~\ref{thm:tab}, we use \cite[Theorem~2]{Pechenik:frames}, stating that $\Frame(U) = \Frame(\Psi^q(U))$ for all tableaux $U \in \inc^q(a \times b)$. With this, we establish the following proposition, completing the proof of Theorem~\ref{thm:tab}.

\begin{proposition}\label{prop:sameframe}
	Suppose $V \in \inc^\ell(a \times b)$ satisfies $\Frame(V) = \Frame(\Psi(V))$. Then $V$ is minimal and $\ell = a+b-1$.
\end{proposition}

%Before proving this proposition, we need a few more definitions. Let $V\in \inc^\ell(a×\times b)$. Following \cite{Dilks.Pechenik.Striker}, we define the \emph{flow path} of $V$ to be the set of pairs $\{\bbb,\bbb'\}$ of adjacent boxes of $V$ such that $\bbb$ and $\bbb'$ are at some point part of the same nontrivial short ribbon during the application of $K$-promotion to $V$. We define the \emph{stream-bed} of $V$ to be the union of the flow path, i.e., the set of all boxes $\bbb$ appearing in any pair $\{\bbb, \bbb' \}$ of the flow path of $V$. (Warning: In \cite{Pechenik:CSP}, the term ``flow path'' was used to refer to what we here call a ``stream-bed.'') Observe that if $\bbb \neq (1,1)$ is in the stream-bed of $V$, then either $\{\bbb^\leftarrow, \bbb \}$ or $\{ \bbb^\uparrow, \bbb \}$ is in the flow path of $V$. Similarly observe that if $\bbb \neq (a, b)$ is in the stream-bed of $V$, then either $\{ \bbb, \bbb^\rightarrow \}$ or $\{ \bbb, \bbb^\downarrow \}$ is in the  flow path of $V$.
%
%\begin{example}\label{ex:flow}
%Let $T$ be as in Example~\ref{ex:promotion}. Then its stream-bed is the union of all the darker gray short ribbons in all the tableaux illustrated there.
%\[\begin{ytableau}*(gray) 1 & *(gray) 2 & 4 & 5 \\ *(gray) 2 & *(gray) 3 &  *(gray) 5 &  *(gray) 6\\ 4 &  *(gray) 5 & 7 &  *(gray) 8\\ 5 &  *(gray) 6 & *(gray)  8 &  *(gray) 9\end{ytableau}\]
%\end{example}
%
%\begin{proof}[Proof of Proposition~\ref{prop:sameframe}]
%We have $V(1,1) = 1$, for otherwise $\Psi(V)(1,1) = V(1,1) - 1$, contradicting $\Frame(V) = \Frame(\Psi(V))$. For any tableau $W \in \inc^\ell(a \times b)$ with $W(1,1) = 1$, we have $\Psi(W)(a,b) = \ell$. Hence, by $\Frame(V) = \Frame(\Psi(V))$, we also have $V(a,b) = \ell$.
%
%	Consider $\bbb \in \Frame(a \times b)$. If $\bbb$ is not in the stream-bed of $V$, then $\Psi(V)(\bbb) = V(\bbb) - 1$, contradicting $\Frame(V) = \Frame(\Psi(V))$. Hence, every box of $\Frame(a \times b)$ must be in the stream-bed of $V$.
%	
%	Consider $\{\bbb, \bbb^\rightarrow\}$ in the top row of $V$. Since $\bbb^\rightarrow$ is in the stream-bed of $V$, the pair $\{\bbb, \bbb^\rightarrow\}$ must be in the flow path of $V$. Hence $\Psi(V)(\bbb) = V(\bbb^\rightarrow) - 1$. But by assumption $\Psi(V)(\bbb) = V(\bbb)$, so we have $V(\bbb^\rightarrow) = V(\bbb) + 1$. Similarly, we have $V(\bbb^\downarrow) = V(\bbb) + 1$ for $\bbb$ in the leftmost column of $V$.
%	
%	Consider $\{\bbb, \bbb^\rightarrow\}$ in the bottom row of $V$. Since $\bbb$ is in the stream-bed of $V$, the pair $\{\bbb, \bbb^\rightarrow\}$ must be in the flow path of $V$. Thus again we have $V(\bbb^\rightarrow) = V(\bbb) + 1$. Similarly, we have $V(\bbb^\downarrow) = V(\bbb) + 1$ for $\bbb$ in the rightmost column of $V$.
%
%Therefore, the entries of $V$ increase consecutively around $\Frame(a \times b)$ from upper-left to lower-right. In particular, the largest entry of $V$ must be $a + b - 1$. But we already determined that this largest entry was $\ell$ in position $(a,b)$. Hence, $\ell = a + b - 1$ and $V$ is the minimal tableau $M_{a \times b}$.
%\end{proof}


\acknowledgements{The authors are grateful to Hugh Thomas for comments on a draft of \cite{Patrias.Pechenik}.}


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