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    % Draw the faded diagonal line
    \draw[dashed, gray] (#1) -- +(#2,#2);
    
    % Add numbers within each box in the grid

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\title{Canon Permutation Posets}

%% define your authors in the usual way
%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
%% for more than 3 names, use the Oxford comma: One Author, Two Author, Red Author, \and Blue Author
\author[Matthias Beck \and Danai Deligeorgaki]{\texorpdfstring{Matthias
Beck\thanks{\href{mailto:mattbeck@sfsu.edu}{mattbeck@sfsu.edu}.}\addressmark{1}, Danai Deligeorgaki\thanks{\href{mailto:danaide@kth.se}{danaide@kth.se}. D.\ D.\ acknowledges support from the Wallenberg Autonomous Systems and
Software Program (WASP), funded by the Knut and Alice Wallenberg Foundation. The full results
mentioned in this extended abstract can be found at {\tt arxiv:2410.03245}.}\addressmark{2}}{Matthias
Beck, Danai Deligeorgaki}}

%% then use \addressmark to match authors to institutions here
\address{\addressmark{1}Department of Mathematics,
         San Francisco State University,
         San Francisco, CA 94132,
         U.S.A. \\ \addressmark{2}Institutionen f\"or Matematik, KTH, SE10044, Stockholm, Sweden}

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%% put your English abstract here, or comment this out if you don't have one yet
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\abstract{A permutation of the multiset $\{1^m,2^m,\dots,n^m\}$ is a {\em canon permutation} if
the subsequence formed by the $j$th copy of each element of $[n]:=\{1,2,\dots,n\}$ is identical for
all $j\in[m]$. Canon permutations were introduced by Elizalde and are motivated by pattern-avoiding 
concepts, such as (quasi-)Stirling permutations. He proved that the descent polynomial of canon
permutations exhibits a surprising product structure; as a further consequence, it is palindromic.
Our goal is to understand canon permutations from the viewpoint of Stanley's $(P,\omega)$-partitions,
along the way generalizing Elizalde's definition and results. We start with a labeled poset $P$ and
extend it in a natural way to canon labelings of the product poset $P \times [n]$. The resulting descent
polynomial has a product structure which arises naturally from the theory of $(P,\omega)$-partitions. When
$P$ is graded, this theory also implies palindromicity. We include results on weak descent
polynomials, an amphibian construction between canon permutations and multiset permutations, as well as
$\gamma$-positivity and interpretations of descent polynomials of canon permutations.}

%% put your French abstract here, or comment this out if you don't have one
\resume{Una permutación del multiconjunto $\{1^m,2^m,\dots,n^m\}$ es una {\em permutación {canon}} si la subsecuencia formada por la $j$-ésima copia de cada elemento de $[n]:=\{1,2,\dots,n\}$ es idéntica para todo $j\in[m]$. Las permutaciones canon fueron introducidas por Elizalde y están motivadas por conceptos que evitan patrones, como por ejemplo permutaciones (cuasi-)Stirling. 
Elizalde demostró que el polinomio de descensos de cualquier permutación canon exhibe una sorprendente estructura de producto; por lo tanto, es palindrómico.
Nuestro objetivo es entender las permutaciones canon desde el punto de vista de las $(P,\omega)$-particiones de Stanley, generalizando, a medida que avanzamos, la definición y los resultados de Elizalde. Partiendo de un poset etiquetado $P$, lo extendemos de manera natural a {\em etiquetamientos canon} del poset producto $P \times [n]$. El polinomio de descensos resultante tiene una estructura de producto que surge naturalmente de la teoría de $(P,\omega)$-particiones. Cuando $P$ es graduado, esta teoría también implica palindromicidad. Incluimos resultados sobre polinomios de descensos débiles, una construcción que generaliza permutaciones canon y permutaciones de multiconjuntos, así como la $\gamma$-positividad e interpretaciones de polinomios de descensos de permutaciones canon.}

%% put your keywords here, or comment this out if you don't have them yet
\keywords{labeled poset, canon permutation,  Eulerian polynomial, Narayana number, gamma positivity.}

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\begin{document}

\maketitle
%% note that you DO NOT have to put your abstract here -- it is generated by \maketitle and the \abstract and \resume commands above
\section{Introduction}

A permutation $\pi$ of the multiset $\{1^m,2^m,\dots,n^m\}$ is a {\em canon permutation} if
the subsequence formed by the $j$th copy of each element of $[n]:=\{1,2,
\dots,n\}$ is identical for all $j\in[m]$. For instance, $223143213144$ is a canon
permutation of $\{1^3,2^3,3^3,4^3\}$, with the sequence formed by the $j$th copy being $2314$.
Canon permutations were introduced by Elizalde~\cite{sergi} and are motivated by
pattern-avoiding permutations of the multiset $\{1,1,2,2,\dots,n,n\}$, such as Stirling
\cite{GESSEL197824} and quasi-Stirling \cite{Archer2018PatternRQ} permutations. 
For $m = 2$, canon permutations are sometimes called \emph{nonnesting permutations}; they
are precisely the permutations of the multiset $\{1,1,2,2,\dots,n,n\}$ avoiding the patterns
$1221$ and $2112$.
Following the notation
in~\cite{sergi}, we denote by $\cC_n^{m,\sigma}$ the set of canon permutations whose $j$th copy of each entry forms the permutation $\sigma\in S_{n}$ and by $\cC_n^{m}$ the set of all canon permutations for given $m,n$. Here, $S_{n}$ denotes the set of permutations of $[n]$.

As usual, we call $j$ a \emph{descent} of $\pi$ if $\pi(j+1) < \pi(j)$ and denote by
$\des(\pi)$ the number of descents of $\pi$. We further denote by $\text{wdes}(\pi)$ the
number of \emph{weak descents} of $\pi$, i.e., the number of positions $j$ such that
$\pi(j+1) \leq  \pi(j)$. Elizalde's work %, and the previous work he extended,
centered on
understanding the distribution of descents of canon permutations by studying the
\emph{descent polynomial}
\[C^{m}_n(x):=\sum_{\pi \in \cC_n^{m}}x^{\des(\pi)} . \]

Let $A_n(x)$ be an Eulerian polynomial (i.e., the descent polynomial of
permutations on $[n]$) and $N_n(x)$ a Narayana polynomial, enumerating high peaks in
Dyck paths (we give a precise definition later).
Both of these polynomials are \emph{palindromic}, that is, their coefficient
sequences are symmetric.
Elizalde found that the descent polynomial $C^2_n(x)$ has the following
surprisingly simple structure~\cite[Theorem~2.1]{sergi}.

\begin{theorem}[Elizalde]\label{thm: sergi}
For $n \geq 1$, $C^2_n(x) = A_n(x) \, N_n(x)$.
In particular, $C^2_n(x)$ is palindromic.
\end{theorem}

As Elizalde noted, the palindromicity of $C^2_n(x)$ is a priori unexpected.
Elizalde proved a bivariate generalization of \Cref{thm: sergi}, involving also the number of the so-called \textit{plateaus} of permutations. 
Another consequence of the bivariate extension of \Cref{thm: sergi} is that the number of
canon permutations with $r$ weak descents is the same as that for $2n-r$ descents. (We will see an explanation of this palindromicity in \Cref{prop:weakdes}.)
In a follow-up paper~\cite{sergi2}, Elizalde extended \Cref{thm: sergi} from $m = 2$
to the general case. Here the Narayana polynomial is generalized so that the role of Dyck
paths is replaced by standard Young tableaux of rectangular shape. Elizalde's proofs are
bijective, and the resulting descent polynomial $C^m_n(x)$ is, again surprisingly,
palindromic, due to the (older) fact that the generalized Narayana polynomials are
palindromic~\cite{sulanke}.

Our goal is to understand \Cref{thm: sergi} and its extensions from the viewpoint
of Stanley's $(P,\omega)$-partitions~\cite{stanleythesis}, along the way generalizing
Elizalde's definition and results. We describe our \emph{ansatz} next.
Consider a poset $P$ with $m$ elements 
and a labeling $\omega$ (i.e., a bijection $\omega: P \to [m]$). We extend this labeling, via a
given $\sigma \in S_n$, to the {\em canon labeling} $\omega \times \sigma$ of the poset $P \times [n]$
(here we think of $[n]$ as a poset---a chain) defined by
\[
  (\omega \times \sigma)(p, j) := \omega(p) + (\sigma(j) - 1)m \, .
\]
% For example, in \Cref{fig: labeled poset}, the permutation $\sigma$ is the identity permutation and $n=2$. 
A \emph{linear extension} of $P \times [n]$ is defined, as usual, as an
order-preserving bijection $\pi : [mn] \to P \times [n]$; writing $\pi$ as a word in
terms of the canon labeling $\omega \times \sigma$ of $P \times [n]$, it is a short step to view $\pi$ as a
permutation of the multiset $\{1^m,2^m,\dots,n^m\}$. 
Let $\cC_n^{ P, \omega \times \sigma }$ consist of all such multiset permutations, and define
\[
  \cC_n^{ P, \omega } := \left\{ \cC_n^{ P, \omega \times \sigma } : \, \sigma \in S_n \right\} 
\]
which we call the set of all \emph{canon permutations} of the labeled poset $(P,\omega)$.
The classical canon permutations  $\cC_n^m$ stem from the case that $P=[m]$ is a chain and $\omega$ is
the identity (i.e., $[m]$ is \emph{naturally labeled}).
For instance, starting with the $2$-element chain poset, $P=[2]$, and the identity permutation of $[4]$,  $\text{id}\in S_4$, the canon labeling $\omega \times \sigma$ of the product poset $[2]\times [4]$ is shown in \Cref{fig: labeled poset} (left). In this example, $\cC_4^{ P, \omega }$ is identified with the set $\cC_4^2$ of canon permutations of $\{1,1,2,2,3,3,4,4\}$.


Finally, we define the \emph{canon polynomial} 
\[C^{P,\omega}_n(x):=\sum_{\pi \in \mathcal{C}_n^{P,\omega}}x^{\des(\pi)} . \]
Again, we note the special case $C_n^m(x) = C_n^{ [m], \id } (x)$.
Indeed, in all but one of our applications, we will use a natural labeling for~$\omega$.
The one exception (at least in this current work) is captured by
the \emph{weak descent polynomial} $ C_n^{ P, \upsilon } (x)$, where the labeling $\upsilon$ is reverse
natural. In the case $P=[m]$, this corresponds to the polynomial
enumerating weak descents in canon permutations of $\{1^m,2^m, \dots, n^m\}$. We will see
that it is a translate of $C_n^{P,\omega}(x)$ for the posets that we study
(\Cref{prop:weakdes} below). 
Our generalization of \Cref{thm: sergi} and its extensions is as follows.

\begin{theorem}\label{thm:main}
Let $P$ be a poset with a natural labeling $\omega$. Then $C_n^{P,\omega}(x) = A_n(x) \, h^*_{P \times [n]}(x)$.
Furthermore, if $P$ is graded then $C_n^{P,\omega}(x)$ is palindromic.
\end{theorem}

Here $P$ being \emph{graded} means that all maximal chains have the same length, 
and 
{\parfillskip=0pt \par} \noindent
$h^*_{P \times [n]}(x)$ is the numerator of the rational generating function of the
order polynomial of $P \times [n]$, which is in a sense the descent polynomial of $P
\times [n]$; we will give detailed definitions in \Cref{sec:ordpollabelposets}.
In fact, \Cref{thm:main} is a special case of a general result (\Cref{thm:
multiplication} below).

In \Cref{sec:narayana}, we give a poset model whose descent polynomial is the Narayana
polynomial. The underlying posets turn out to be structured such that all labeled versions of them, in the
sense of $(P,\omega)$-partitions, have similar descent polynomials, from which \Cref{thm:
sergi} follows in a few short steps, as we outline in \Cref{sec:ordpollabelposets}.
The philosophy of our \emph{ansatz} is that we consider the poset from
\Cref{sec:narayana}, whose descent polynomial is the Narayana polynomial,
and then sum $(P,\omega)$-descent polynomials over certain labelings $\omega$ of this fixed
poset, which gives rise to the product structure exhibited in \Cref{thm: sergi}.
Our proof generalizes immediately to the canon polynomial $C^m_n(x)$ for general~$m$.
The palindromicity of canon polynomials follows organically from the structure of the
involved posets;
indeed, we give three direct explanations for this palindromicity (\Cref{thm: multiplication}, \Cref{prop:oneposet}, and \Cref{them: subposet} below).

In \Cref{sec:amphibians}, we extend our results to 
subposets of $P\times [n]$ with some relations missing between different
copies of $P$, i.e., those of the form $(p,j) \prec (p,j+1)$. 
These subposets give intermediary descent polynomials situated between those for canon permutations
and multiset permutations. We show that palindromicity extends to this class.

A distributional property that is stronger than palindromicity (and unimodality \cite{pettersurvey}) is $\gamma$-positivity: a palindromic polynomial of degree $d$ is {\em $\gamma$-positive} if its coefficients are non-negative when expressed in the \emph{$\gamma$-basis}
$ \{x^i(x + 1)^{d - 2i} : 0 \leq i \leq \lfloor d/2\rfloor\}$ (see, for instance, \cite{athanasiadis2017gamma}).
In \Cref{sec: canon positivity}, we show that our viewpoint implies that the descent polynomials of canon 
permutations are $\gamma$-positive and give a combinatorial interpretation for their coefficients,
based on a result of Br\"and\'en. We conjecture the intermediate class of polynomials
constructed in \Cref{sec:amphibians} to have $\gamma$-positive descent polynomials and ask
for a $\gamma$-coefficient interpretation, which would recover $\gamma$-positivity exhibited by both canon permutations and multiset permutations.

% ===========================

\section{Narayana Polynomials as Descent Polynomials}\label{sec:narayana}

We start, as a warm-up of sorts, by realizing the Narayana polynomials as descent
polynomials of the posets $[2] \times [n]$.
As usual, the set $D_n$ of \emph{Dyck paths} consists of all lattice paths from $(0, 0)$ to
$(n, n)$ with steps $e := (1, 0)$ and $n := (0, 1)$ that do not go above the diagonal $y = x$. 
A \emph{peak} in a Dyck path is an occurrence of two adjacent
steps $en$. 
% A peak is called a \emph{low peak} if these steps touch the diagonal and a \emph{high peak} otherwise. 
A peak is called a \emph{high peak} if these steps do not touch the diagonal.
We denote the number of high peaks of $D \in D_n$ by $\text{hpea}(D)$. For example, the Dyck path from $(0,0)$ to $(4,4)$ appearing in \Cref{fig: labeled poset} has three peaks out of which one is a high peak. We refer to the polynomial 
 $N_n(x):=\sum_{D \in D_n} x^{\text{\rm hpea}(D)}$ as a {\em Narayana polynomial}. 
(Our convention is slightly nontraditional, as the \emph{Narayana numbers} are the coefficients of $x
\, N_n(x)$.)

\begin{figure}[ht]
    \centering
\begin{tikzpicture}[ main/.style = {}] 
\node[main] (1) {$1$}; 
\node[main] (2) [above of=1] {$2$};
\node[main] (3) [right of=2] {$3$};
\node[main] (4) [above of=3] {$4$};
\node[main] (5) [right of=4] {$5$};
\node[main] (6) [above of=5] {$6$};
\node[main] (7) [right of=6] {$7$};
\node[main] (8) [above of=7] {$8$};
\draw (2) -- (1);
\draw (3) -- (4);
\draw (5) -- (6);
\draw (7) -- (8);
\draw (3) -- (1);
\draw (2) -- (4);
\draw (5) -- (7);
\draw (6) -- (8);
\draw (3) -- (5);
\draw (6) -- (4);
\end{tikzpicture} 
\hspace{2.5 cm}
\begin{tikzpicture}[scale=0.9]

  \dyckpath{0,0}{4}{0,1,0,0,1,1,0,1}{dyck4}{ }

\end{tikzpicture}
    \caption{The poset $[2]
\times [4]$ with a natural labeling (left). The Dyck path corresponding to the linear extension $12354678$ of $[2]
\times [4]$ (right).}
    \label{fig: labeled poset}
\end{figure} 

The following can be proved bijectively.
 \begin{theorem}\label{thm: narayana bijection}
The linear extensions of the poset $[2]
\times [n]$ are in bijection with the Dyck paths in $D_n$. Furthermore, the descents of a
linear extension of $[2] \times [n]$, labeled via $\id \times \id$, are in bijection with the high peaks of the corresponding Dyck path. 
 \end{theorem}
For example, the linear extension $12354678\in \mathcal{L}([2]\times [4], \id\times \id)$ corresponds to the Dyck path shown in \Cref{fig: labeled poset}.

\begin{corollary}\label{cor:h*asnarayana} 
Let $\cL([2] \times [n])$ denote the set of linear extensions of $[2] \times [n]$. Then
     \[\sum_{\pi \in \cL([2]\times [n])} x^{\des(\pi)}=N_n(x)\, . \]
 \end{corollary}

The analogy between descents of linear extensions and high peaks of Dyck paths extends to weak
descents and (ordinary) peaks, and the bijection discussed extends similarly.\footnote{We note that \Cref{cor:h*asnarayana} also follows from work of Sulanke \cite{sulanke}.}
% \matt{is there a similar theorem/corollary for Sergi's generalized Narayana numbers and $P_{2,n}$? If that's easy to see, we should include it (but only if...).}
% \danai{There should be but we would need to talk about $d$-dimensional paths, which is technical so I will leave it for later right now.}

\begin{remk}
Viewing the descent polynomial of $[2] \times [n]$ as the Ehrhart $h^*$-polynomial of the order
polytope of $[2] \times [n]$, it is a curious fact that the positive root polytope of type $A_{ n-1
}$ has the same $h^*$-polynomial~\cite[Example~6]{braunsurvey}.
\end{remk}


% ===========================

\section{Order Polynomials of Labeled Posets}\label{sec:ordpollabelposets} % {$(P,\omega)$-partitions}

We now recall some fundamental definitions and results on labelled posets and their order polynomials
and generating functions~\cite{stanleythesis}.

\begin{definition}
Let $P$ be a poset of cardinality $m$ with a given labeling $\omega: P \to [m]$.
A \emph{$(P, \omega)$-partition} is a map $\sigma: P \rightarrow \mathbb{Z}_{ \ge 0 } $
satisfying the following conditions:
 \begin{itemize}
     \item If $s \leq t$ in $P$, then $\sigma(s) \leq \sigma(t)$; in other words, $\sigma$
is order-preserving.\footnote{Stanley defines $(P, \omega)$-partitions in an order-\emph{reversing} fashion; the present definition mirrors that of the usual order polynomial.}
\item If $s < t$ and $\omega(s) > \omega(t)$, then $\sigma(s) < \sigma(t)$.
 \end{itemize}
Let $\Omega_{P,\omega}(j)$ be the number of $(P, \omega)$-partitions $\sigma: P \rightarrow
[j]_0 := \{ 0, 1, \dots, j \}$.
% If $\sum\limits_{t\in P} \sigma(t) = n$, then we say that $\sigma$ is a $(P, \omega)$-partition of $n$.
\end{definition}

%By replacing $\sigma(t)$ with $m + 1 - \sigma(t)$, we see that $\Omega_P (m)$ is also the number of order-reversing maps $P \rightarrow m$, i.e., the number of $P$-partitions $P \rightarrow m$. We can therefore extend the definition to labelled posets by defining the $(P, w)$- order polynomial $\Omega_{P,w}(m)$ for $m \in P$ to be the number of $(P, w)$-partitions $\sigma: P \rightarrow m$. Let $e_{P,w}(s)$ be the number of surjective $(P, w)$-partitions $P \rightarrow s$. Note that for any $w$ we have $e_{P,w}(p) = e(P)$. By first choosing the image $\sigma(P)$ of the $(P, w)$-partition $\sigma : P \rightarrow m$, it is clear that \[\Omega_{P,w}(m) = \sum\limits_{s=1}^p e_{P,w}(s)\binom{m}{s}.\] It follows that $\Omega_{P,w}(m)$ is a polynomial in $m$ of degree $p$ and leading coefficient $\frac{e(P)}{p!}$.

The function $\Omega_{P,\omega}(j)$ turns out to be a polynomial of degree $m$, called the
\emph{order polynomial} of $(P,\omega)$.\footnote{
This definition is found in the literature most frequently for the case where $\omega$ is a natural
(i.e., order-preserving) labeling.
} Subsequently, we may define the {\em $h^*$-polynomial} of a labeled poset $(P,\omega)$, $h^*_{ P,\omega }(x)$, via
\[
\sum_{j\geq 0} \Omega_{P,\omega}(j) \, x^j \ = \ \frac{ h^*_{ P,\omega } (x) }{ (1-x)^{m+1} }  \, .
\]
When $\omega$ is a natural labeling, we denote $h^*_{ P,\omega } (x)$ by $h^*_{ P } (x)$.
For example, we can rephrase \Cref{cor:h*asnarayana} as $h^*_{ [2] \times [n] } (x) = N_n(x)$.

Parallel to the classical case, we may think of a linear extension $\sigma$ of $P$ as a
permutation of $\omega$; we denote the set of all such linear extensions by $\cL(P,\omega)$. %\danai{(direct the reader to \Cref{ex: main theorem} once ready)}
The fundamental property of order polynomials is the following~\cite[Proposition~13.3]{stanleythesis}.
\begin{theorem}[Stanley]\label{thm: 3.15.8}
$\displaystyle h^*_{P,\omega}(x) = \sum\limits_{\sigma \in \cL(P,\omega)}x^{\des(\sigma)} . $
\end{theorem}

Given a poset $P$ with a labeling $\omega$ and a chain $C$ of $P$, a \emph{descent} of $C$ is any occurrence of a cover relation $a < b$ in $C$ with $\omega(a) > \omega(b)$.
We define $\Des(C,\omega)$ to be the set of descents in $C$ and let $\des(C,\omega)$ denote the cardinality of $\Des(C,\omega)$. The following theorem, which was stated in \cite[Theorem~4.1]{stembridge} using the notion of shift equivalence,  provides a sufficient condition for the $h^*$-polynomials corresponding to two different labelings of a given poset to be the same up to a shift.

\begin{theorem}[Stembridge]\label{thm: main extended}
Let $P$ be a poset with two labelings $\omega$ and $\omega'$ such that for each $j \in P$ there exists $t_j$
with the following conditions:
\begin{itemize}
    \item if $j$ is minimal then $t_j = 0$;
    \item if $j$ covers $i$ then for any maximal chain $C$ containing $i$ and $j$
\[
  t_j - t_i = \begin{cases}
     1 & \text{ if } i\in \Des(C ,\omega) \setminus \Des(C ,\omega') \, , \\
    -1 & \text{ if } i\in \Des(C ,\omega') \setminus \Des(C ,\omega) \, , \\
     0 & \text{ otherwise; }
  \end{cases}
\]
    \item there exists $k$ such that $t_j = k$ for any maximal~$j$.
\end{itemize}
Then $h^*_{P,\omega}(x) = x^k \, h^*_{P,\omega'}(x) \, .$
\end{theorem}

We note that our conditions in \Cref{thm: main extended} imply that every maximal chain $C$ in the order ideal $\langle j \rangle$
satisfies $t_j=\des(C,\omega)-\des(C,\omega')$.
By choosing $\omega'$ to be natural (and
so there are no descents) and $t_j=\des(C,\omega)$ for any choice
of maximal chain $C$ in $\langle j\rangle $ for a given $j\in P$, we obtain the
% the fact that $\des(C,\omega)$ is the same for any choice of $C$ follows by the assumption that all maximal chains have the same number of descents.
following result which appeared in \cite[Corollary~2.4]{branden}, 
phrased there in the language of sign-graded posets.\footnote{The notion that all maximal
chains of $(P,\omega)$ have the same number of descents is slightly different than $(P,\omega)$ being
sign-graded; when $P$ is graded, the two notions coincide.}
% if we take the labeled poset $P$ with structure: $3<2,3<1,1<4,$ then it does satisfy
% our definition but it is not sign-graded. The maximal chains here are $3-2$ and $3-1-4$.

\begin{corollary}\label{cor: main}
Consider a poset $P$ with a labeling $\omega$ such that all maximal chains have the same number $k$ of descents. Then $h^*_{P,\omega}(x) = x^k \, h^*_P(x)$. 
\end{corollary}

Our results in the remainder of this section concern posets with maximal chains with the same number
of descents for the sake of simplicity, but most arguments extend to the situation described in \Cref{thm: main extended}. The next theorem follows from \Cref{cor: main} and the fact that the $h^*$-polynomial of a naturally labeled poset is palindromic if and only if the poset is
graded~\cite[Proposition~19.3]{stanleythesis}.

\begin{theorem}\label{thm: multiplication}
Consider a poset $P$ with a labeling $\omega$ such that all maximal chains have the same number $k$ of descents. Then $C_n^{P,\omega}(x) = x^k \, A_n(x) \, h^*_{P \times [n]}(x)$.
Moreover, if $P$ is graded then $C_n^{P,\omega}(x)$ is palindromic.
\end{theorem}

Starting with $P\times [n]$, we now construct a new poset $P \widecheck\times [n]$ 
of cardinality $(m+1)n$ by
adding $n$ elements with no relation among them but which cover all maximal
elements of $P\times [n]$.
The poset in \Cref{fig: labeled poset extended} gives an example where $P=[2]$ and $n=3$. 
We extend a given labeling $\omega$ of $P$, first to a labeling $\omega \times \id$ of $P \times
[n]$, and then, to a labeling $\omega \widecheck\times \id$ of $P \widecheck\times [n]$, by giving the new elements any labels that are larger
than those in $\omega \times \id$.


 \begin{figure}[ht]

    \centering
%\begin{tikzpicture}[main/.style = {}] \node[main] (1) {$1$}; \node[main] (2) [above of=1] {$2$}; \node[main] (3) [right of=2] {$3$}; \node[main] (4) [above of=3] {$4$}; \node[main] (5) [right of=4] {$5$}; \node[main] (6) [above of=5] {$6$}; \node[main] (7) [right of=6] {$7$}; \node[main] (8) [above of=7] {$8$}; \node[main] (10) [above left of=8] {$10$}; \node[main] (9) [left of=10] {$9$}; \node[main] (11) [above right of=8] {$11$}; \node[main] (12) [right of=11] {$12$}; \node[main] (13) [above right of=10] {\textcolor{red}{$13$}};
%\draw (2) -- (1); \draw (3) -- (4); \draw (5) -- (6); \draw (7) -- (8); \draw (3) -- (1); \draw (2) -- (4); \draw (5) -- (7); \draw (6) -- (8); \draw (3) -- (5); \draw (6) -- (4); \draw (8) -- (9); \draw (8) -- (10); \draw (8) -- (11); \draw (8) -- (12); \draw (13) -- (9); \draw (13) -- (10); \draw (13) -- (11); \draw (13) -- (12);
%\end{tikzpicture} 
\begin{tikzpicture}[main/.style = {}] 
\small
\node[main] (1) {$1$}; 
\node[main] (2) [above of=1] {$2$};
\node[main] (3) [right of=2] {$3$};
\node[main] (4) [above of=3] {$4$};
\node[main] (5) [right of=4] {$5$};
\node[main] (6) [above of=5] {$6$};
\node[main] (8) [above left of=6] {$7$};
\node[main] (10) [above of=6] {$8$};
\node[main] (11) [above right of=6] {$9$};
\small
\draw (2) -- (1);
\draw (3) -- (4);
\draw (5) -- (6);
\draw (3) -- (1);
\draw (2) -- (4);
\draw (3) -- (5);
\draw (6) -- (4);
\draw (6) -- (8);
\draw (6) -- (10);
\draw (6) -- (11);
\end{tikzpicture} 
    \caption{A labeled poset $(P \widecheck\times [n],\omega \widecheck\times \id)$ such that $h^*_{P
\widecheck\times [n],\omega \widecheck\times \id}(x)=C_3^2(x)$.}
    \label{fig: labeled poset extended}
\end{figure}

\begin{proposition}\label{prop:oneposet} 
% For the labeled poset $(P \widecheck\times [n],\omega \widecheck\times \id)$ constructed above,
\ $ C_n^{P,\omega}(x)=h^*_{P \widecheck\times [n],\omega \widecheck\times \id}(x) \, . $
\end{proposition}

This gives a direct proof for the palindromicity of $C_n^{P,\omega}$ for $\omega=\id$ as $P
\widecheck\times [n]$ is graded.

% Note that Petter's proof of palindromicity for all sign-graded $\omega$ is weaker than what we are working with (but we could extend his proofs to show that this broader family of labeled posets has palindromic $h^*$-polynomial.)
% ===========================

\section{Amphibians}\label{sec:amphibians}

In this section, we will study a broader family of linear extensions/multiset permutations
whose descent polynomials are also palindromic.

Throughout this section,  we denote by $Q$ a subposet of $P\times [n]$ with some relations missing between different
copies of $P$, i.e., those of the form $(p,j) \prec (p,j+1)$. 
The motivation for studying these subposets comes from the following observation. When $P=[m]$, the linear
extensions of $Q$ can be interpreted (in the same way as before) as a collection of multiset
permutations of $\{1^m,2^m,\dots,n^m\}$, where conditions weaker than those for canon
permutations are imposed. 
% The set of canon permutations was seen to correspond to a union of linear extensions of $|S_n|$ posets of the form $[m]\times [n]$ labeled differently.  If we consider the poset $[m]\bar{\times}[n]$ where some of the relations of the form $(j,i)<(j,i+1), i\in [n], j\in [m]$, have been removed, this corresponds to a set of multipermutations that  partially satisfy the constraints canon permutations do.  If we further assume that $ j\in [m]\setminus\{1\}$, this guarantees that the union $\cup_{\sigma \in S_n} \mathcal{L}([m]\bar{\times}[n],\text{id}\times \sigma)$ is disjoint. 
When $P=[m]$ and we remove all relations of the form $(p,j) \prec (p,j+1)$ for all $j\in [m]\setminus
\{1\}$ (where $1$ can be replaced with any other element of $[m]$), we recover the set of multiset permutations of
$\{1^m,2^m,\dots,n^m\}$ whose subsequence formed by the first copy of each element in $[n]$ is fixed. Similarly to canon permutations, summing over the corresponding descent polynomials for all canon labelings $\id \times \sigma$  of $Q$ for $\sigma \in S_n$ gives the descent polynomial of multiset permutations of $\{1^m,2^m,\dots,n^m\}$. %\danai{importantly, $j$ has to not be the minimum element of $[m]$, i.e., one of the chains needs to remain, to avoid double-counting. So we need to specify this on the $j$'s.} 
%\matt{What about general $P$? Do you mean in this case that $j$ cannot be any of the minimal elements of $P$?} 
%\danai{Not for general $P$. Just the construction that gives us all multiset permutations needs to not be double-counting. Or in general when we consider the canon stuff and take that sum we could be double-counting but I don't think that this is a problem.}
%\matt{now I'm confused about the $j>1$ condition... Could you please state it carefully?}
Since the descent polynomials corresponding to $\{1^m,2^m,\dots,n^m\}$ and $\mathcal{C}_n^m$
are both palindromic, it is natural to ask if palindromicity extends to more subposets $Q$ of
 $P\times [n]$.
\Cref{them: subposet} below confirms this. 
We note that the notion of \emph{palindromicity} here might involve polynomials with zero constant
terms (or even more zero coefficients), and so we state, in each case, the relevant functional
equation.

Let $Q$ be a subposet of $P\times [n]$ with some of the cover relations of the form $(p,j) \prec (p,j+1)$ removed.
We define the \emph{dissonant canon polynomial} 
\[ C^{Q,\omega}(x):=\sum_{\sigma \in S_n} h^*_{Q,{\omega}\times \sigma}(x) \, .\]
%Similarly \danai{would update this definition later}, we define the 
%corresponding polynomial for weak descents
%\[ wC^{[m]\bar{\times}[n]}(x):=\sum_{\sigma \in S_n} h^*_{[m]\bar{\times}[n],\bar{\text{id}}\times \sigma}(x).\]
The name is inspired from the case when $P=[m]$ and $Q$ is a subposet of $[m]\times [n]$ with some of
the edges of the form $(p,j) \prec (p,j+1)$ removed, where $p\in P\setminus\{q\}$ for some fixed
$q\in P$. This last condition fixes one of the subsequences and therefore ensures that $C^{Q,\omega}$ is
counting multiset permutations of $\{1^m, \dots,n^m\}$ without doublecounting.

%Let ${P}\bar{\times}[n]$ be a subposet of $P\times [n]$ on the same ground set that contains all relations coming from the copies of $P$ but misses some of the other edges.  


%\begin{lemma}\label{thm: main extended} Consider a poset $P$ with a labeling $w$ such that all maximal chains have the same number $k$ of descents. Then $h^*_{P\bar{\times} [n],w\times \sigma}(x) = x^kh^*_{P\bar{\times} [n],\text{id}\times \sigma}(x)$ for any subposet $P\bar{\times} [n]$ of $P{\times} [n]$. \end{lemma}

If all maximal chains in $(P,\omega)$ have the same number
$k$ of descents, then any maximal chain in $(Q, \omega\times \sigma)$ will contain between $k$ and
$k+\des(\sigma)$ descents. In particular, from \Cref{thm: main extended} we can deduce the following.
\begin{corollary}
    \label{cor: main extended}
Consider a poset $P$ with a labeling $\omega$ such that all maximal chains in $(P,w)$ have the same number $k$ of descents. 
Then, for any $Q$ as above 
\[h^*_{Q,\omega\times \sigma}(x) = x^k \, h^*_{Q,\id\times \sigma}(x) \, .\]
\end{corollary}
\Cref{cor: main extended} is used to prove the following theorem.

\begin{theorem}\label{them: subposet}
Consider a graded poset $P$ with a labeling $\omega$ such that all maximal chains in $(P,w)$ have the same number $k$ of descents (and $k'$ ascents). 
Then, for any $Q$ as above, the dissonant canon polynomial $C^{Q,\omega}(x)$ is palindromic, 
in the sense that
\[
 x^{mn-1+k-k'} \, C^{Q,\omega}(\tfrac{1}{x})= C^{Q,\omega}({x}) \, .
\]
\end{theorem}

A key point in the proof of \Cref{them: subposet} is the decomposition of the mentioned polynomial into (a total of $\frac{n!}{2}$) palindromic polynomials. A natural question is to identify the corresponding bijections between the permutations captured by the coefficients, which would result in a bijective proof of \Cref{them: subposet}. This would also yield bijections in the subcase of canon permutations, addressing questions raised by Elizalde \cite[Problem~4.1]{sergi}. 
The palindromicity of the dissonant canon polynomial
$C^{Q, \id}(x)$ in \Cref{them: subposet} also gives rise to the following problem. 

\begin{question}\label{que: gampos}
    For which subposets $Q$ of $P{\times}[n]$ is $C^{Q,\id}(x)$ $\gamma$-positive? Can we describe the $\gamma$-coefficients in a unified way?
\end{question}

In the next section (\Cref{cor:gamma-interp-canon}), we will see that the answer to \Cref{que: gampos} is positive for canon permutations, which correspond to the poset $Q=[m]\times [n]$.
%\begin{question} Understand the $\gamma$-coefficients of the $\gamma$-positive $C^{Q,\id}$ in a unified way. \end{question}
The answer is also positive for the poset corresponding to all multiset
permutations, i.e., the poset where all conditions of the form $(p,j)\prec (p,j+1)$ are removed. 
This follows from \cite{branden} by realizing the set of multiset permutations of $\{1^m,2^m,\dots,n^m\}$
as linear extensions of the poset consisting of $n$ $m$-element chains, labeled with the regular canon labeling.
A combinatorial interpretation for those $\gamma$-coefficients is given in \cite{BrandenActions}.
%from the real-rootedness and palindromicity of  and a combinatorial interpretation was given in \cite{weaklyincreasingtrees} in terms of a particular family of trees. It also follows from \cite{branden}
%\danai{(I want to discuss citing something else or not.)}
The two combinatorial interpretations (corresponding to the set of multiset permutations and to canon
permutations) rely on partitioning permutations into classes, but the classes are different
in each case. \Cref{que: gampos} asks for a unified interpretation of the
$\gamma$-coefficients for both corresponding polynomials, as well as other $\gamma$-positive
polynomials that arise for different subposets $Q$. We conjecture that this class will contain all such $Q$, at least for $P=[m]$.

\begin{conj}\label{conj: gammapos}
The dissonant canon polynomial $C^{Q,\id}(x)$ is $\gamma$-positive whenever $P=[m]$. 
\end{conj}

%Proving $\gamma$-positivity is one of the most promising avenues to show unimodality, given  that $C^{Q,\id}$ is not always real-rooted \danai{we might remove this}, even for canon permutations (we give an example in \Cref{sec: canon positivity}).


% ===========================

\section{\texorpdfstring{$\gamma$}{gamma}-positivity of Canon Permutations}\label{sec: canon positivity}

%\danai{(We need to understand if gamma-positivity is known. Perhaps it follows from the correspondence with Standard Young Tableau in Sergi's latest paper. We also need to find the degree of $C_n^{P,w}$; I think it  is $|P|(n-1)+\des(w)$.)}
 %update. Real-rootedness might not hold for the generalized Narayana polynomials. The example I checked is for $m=3$ and $n=4$. The resulting $h^*$-polynomial seems to be $1+ 22x+113x^2+ 190x^3 +113x^4 + 22x^5+x^6$, computed thanks to SAGE.
%I found an article that says that the regular Narayana polynomials are real-rooted (even in types B and D). Therefore, I think that we need both $m,n>2$ to get out of the regular Narayana case, and hence the only polynomial of smaller degree that could be non-real rooted would arise for $m=n=3$. It appears that this polynomial is real-rooted if my computations are correct. So this example provided might be the smallest-degree one. Remember though that they are all $\gamma$-positive since we have a sign-graded poset corresponding to them. This motivates the following text.

\Cref{thm:main} shows that certain distributional properties shared among the polynomials
$A_n(x)$ and $h^*_{P\times [n]}(x)$, such as palindromicity and $\gamma$-positivity, transfer to
the canon polynomial $C_n^{P,\omega}(x)$. 
A direct proof of the $\gamma$-positivity of $C_n^m(x)$ can be derived from Br\"and\'en's work in \cite{branden} using the poset $[m] \widecheck\times [n]$ constructed in \Cref{prop:oneposet} (here $P=[m]$). 
Below, we discuss a combinatorial interpretation of the $\gamma$-coefficients 
of $C_n^m(x)$, a consequence of a group action on permutations due to Foata and Strehl extended to posets by Br\"and\'en~\cite{BrandenActions}.

%The following corollary is a consequence of \Cref{prop:oneposet} and Theorem ~\ref{petter_interpretation}.

Following the notation in \cite{BrandenActions}, we consider the map $\rho: [m] \widecheck\times [n]\rightarrow
\mathbb{Z}_2$ with values $\rho(q)=0$ if the maximal chains in the poset ideal $\langle q \rangle$ have
even length and $\rho(q)=1$ if the length is odd.
For a permutation $\pi=\pi_1\cdots \pi_{(m+1)n}\in \mathcal{L}([m] \widecheck\times [n])$, 
let us call $j\in [(m+1)n-1]$ a \emph{$\rho$-descent} of $\pi$ if $\pi_{j+1} < \pi_j$ or $\rho(\pi_{j+1})<\rho(\pi_{j})$. % and an ascent if $\pi(j+1) > \pi(j)$ or $\rho(\pi(j+1))>\rho(\pi(j))$.
% I attach $|P|n+1$ in the beginning of the permutation, and at the end a $0$ only if it is of odd rank.
We say that a $\rho$-descent $j\in [(m+1)n-1]$ of $\pi$ is a \emph{double $\rho$-descent} if $j-1$ and $j$ are both $\rho$-descents in $\pi$, or $j=1$.
The following result can now be derived from \cite[Section~6]{BrandenActions}.
 %We can compute what what $\des(\omega)$ should be for canon permutations. It should be $\lceil \frac{n+m-2}{2} \rceil$. We can also write down an explicit $\omega$ that works for canon permutations.

\begin{corollary}\label{cor:gamma-interp-canon}
Writing
 \[C_n^m(x)=h^*_{[m] \widecheck\times [n]}(x)= \sum_{i=0}^{\lfloor
\frac{m(n-1)}{2}\rfloor}\gamma_{i} \, x^{i}(1+x)^{m(n-1)-2i},\]
the coefficient $\gamma_i$ equals the number of linear extensions $\pi=\pi_1\cdots \pi_{(m+1)n}\in \mathcal{L}([m] \widecheck\times [n])$ such that, for $d=\lfloor \frac{m+n-1}{2}\rfloor$, %\danai{we could note that $m+n-1$ is the length of a maximal chain in $[m] \widecheck\times [n]$} %, for a canonical labeling $\omega$ of $Q$, $\omega(\pi):=\omega(\pi_1)\omega(\pi_2)\cdots \omega(\pi_{|P|n})$ is such that:
\begin{itemize}
\item $\pi$ has exactly $i+d$ $\rho$-descents,%+\lceil\frac{\text{max}\{m,n\}}{2}\rceil$ descents.
\item $\pi$ has no double $\rho$-descents,
\item $\pi_{(m+1)n-1} < \pi_{(m+1)n}$ if $\rho(\pi_{(m+1)n-1})=\rho(\pi_{(m+1)n})=1$.
\end{itemize}
\end{corollary}

The weak-descent polynomial of canon permutations is palindromic and $\gamma$-positive with a combinatorial interpretation specified by \Cref{cor:gamma-interp-canon}, as discussed in the following proposition. (The palindromicity of the weak-descent polynomial of canon permutations for the multiset $\{1,1,2,2,...,n,n\}$ was first observed in   \cite[Corollary~2.3]{sergi}.)


\begin{proposition}\label{prop:weakdes} 
$ \displaystyle \sum\limits_{\sigma \in\mathcal{C}_n^m}x^{\operatorname{wdes}(\sigma)}=
x^{m-1}C^{m}_n(x)=x^{m-1}A_{n}(x) \, h^*_{[m]\times [n]} \, . $
\end{proposition}



% Since this is the case for $P_{m,n}$, we conclude that the descent and weak-descent polynomials are palindromic. However, we will develop a more general theory about palindromic polynomials in the next section.

Let $Q$ be a subposet of $[m]\times [n]$ with some of
the edges of the form $(p,j) \prec (p,j+1)$ removed, where $p\in P\setminus\{q\}$ for some fixed
$q\in P$. 
It follows by similar arguments as in the proof of \Cref{prop:weakdes} that the
dissonant canon polynomial $C^{Q,\id}(x)$ (resp. $C^{Q,\upsilon}(x)$) is the descent (resp. weak-descent) polynomial of the set of permutations of $\{1^m,2^m,...,n^m\}$ that satisfy the constraints imposed by the edges of $Q$ between copies of $[m]$.

%Let $P=[m]$ and let $Q$ be a subposet of $[m]\times [n]$ with some of the edges of the form $(p,j) \prec (p,j+1)$ removed, where $p\in P\setminus\{q\}$ for some fixed $q\in P$. Then $C^{Q,\id}$ equals the descent polynomial of the set of multiset permutations of $\{1^m, \dots,n^m\}$ whose subsequences satisfy certain ordering constraints. %We need an example demontrating that.

\begin{corollary}\label{cor:palindromic subposet}
Let $Q$ and $\upsilon$ be as above.
The dissonant canon polynomial $C^{Q, \id}(x)$ is palindromic in the sense that
\[
 x^{m(n-1)} \, C^{Q,\id}(\tfrac{1}{x})= C^{Q,\id}({x}) \, .
\pagebreak\]

The dissonant canon polynomial $C^{Q, \upsilon}(x)$ corresponding to weak descents of multiset permutations is palindromic, in the sense that 
\[
 x^{m(n+1)-2} \, C^{Q,\omega}(\tfrac{1}{x})= C^{Q,\omega}({x}) \, .
\] 
\end{corollary}

\Cref{cor:palindromic subposet} provides a class of sets of permutations of $\{1^m,2^m,...,n^m\}$ with  palindromic (weak) descent polynomial, which simultaneously generalizes canon permutations and the collection of all permutations of $\{1^m,2^m,...,n^m\}$.

%\begin{proof} It is shown in \Cref{them: subposet} that the polynomial $C^{Q,\id}(x)=\sum\limits_{ \sigma\in S_n}h^*_{Q,\text{id}\times \sigma }(x)$ is palindromic with respect to degree $m(n-1)$, which is the degree of $C^{Q,\id}$.
%Similarly, since $\des(\bar{\text{id}})=m-1$, we have that $wC^{[m]\bar{\times}[n]}(x)=\sum\limits_{ \sigma\in S_n}h^*_{ P\bar{\times}[n],\bar{\text{id}}^\sigma }(x)$ is palindromic with respect to its degree, $mn-1$.
%\end{proof}



%\acknowledgements{\lipsum[20]}

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