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\title[Colored multiset Eulerian polynomials]{Colored multiset Eulerian polynomials}

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\author[Danai Deligeorgaki, Bin Han, and Liam Solus]{ \texorpdfstring{Danai Deligeorgaki\thanks{\href{mailto:danaide@kth.se}{danaide@kth.se}. Partially supported by the Wallenberg Autonomous Systems and Software Program (WASP) funded by the Knut and Alice Wallenberg Foundation.}\addressmark{1},
Bin Han\thanks{\href{mailto:bin.han@gu.se}{bin.han@gu.se}. Supported by the The Swedish Research Council (Vetenskapsr{\.a}det), Starting Grant No.~2019-05195 as a postdoctoral fellow at KTH, during which most of the work was done.}\addressmark{2}, \and Liam Solus\thanks{\href{mailto:solus@kth.se}{solus@kth.se} 
Partially supported by the Wallenberg Autonomous Systems and Software Program (WASP) funded by the Knut and Alice Wallenberg Foundation, the Digital Futures
Lab at KTH, the G\"oran Gustafsson Stiftelse Prize for Young Researchers, and Starting Grant No.~2019-05195 from The Swedish Research Council (Vetenskapsr{\.a}det).}\addressmark{1}}{Danai Deligeorgaki, Bin Han, and Liam Solus} }

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\address{
\addressmark{1} Institutionen f\"or Matematik, KTH, SE10044, Stockholm, Sweden.\\
\addressmark{2} School of Public Health and Community Medicine, Sahlgrenska academy, University of Gothenburg, 41390 Gothenburg, Sweden.}



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\abstract{
This extended abstract is a summary of a paper that studies the colored multiset Eulerian polynomials.
These polynomials are a common generalization of MacMahon's multiset Eulerian polynomials and the colored Eulerian polynomials, both of which are known to satisfy well-studied distributional properties including real-rootedness, log-concavity and unimodality. 
The symmetric colored multiset Eulerian polynomials are characterized and used to prove sufficient conditions for a colored multiset Eulerian polynomial to be self-interlacing. 
The latter property implies the aforementioned distributional properties as well as others, including the alternatingly increasing property and bi-$\gamma$-positivity. 
To derive these results, multivariate generalizations of an identity due to MacMahon are deduced.  
The results are applied to a pair of questions, both previously studied in several special cases, that are seen to admit more general answers when framed in the context of colored multiset Eulerian polynomials. The first question pertains to $s$-Eulerian polynomials, and the second to interpretations of $\gamma$-coefficients. }

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\keywords{Eulerian polynomials; colored permutations;  $h^*$-vectors; interlacing polynomials; $\r$-positivity; weakly increasing trees}

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\begin{document}

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%\subjclass[2010]{05A05, 05A15, 05A19, 15B36}

%\tableofcontents



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}\label{sec1: intro}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
For a positive integer $n$, we let $[n]:=\{1,2,...,n\}$ and $[n]_0:=\{0,1,...,n\}$. 
We denote by $M_{\textbf{m}}=\{1^{m_1},2^{m_2},...,n^{m_n}\}$, where $\textbf{m}:=(m_1,m_2,...,m_n)\in \Z_{>0}^n$, the multiset of cardinality $m:=m_1+\cdots+m_n$ containing $m_k$ copies of $k$, for $1\leq k \leq n$. 
The permutations of the multiset $M_{\textbf{m}}$, i.e., words $\pi=\pi_1\pi_2\ldots \pi_{m}$ in which $k$ appears exactly $m_k$ times, are called \emph{multiset permutations}, the set of which we denote $\S_{M_\textbf{m}}$. 
For $\textbf{r}:=(r_1,...,r_n)\in \Z_{>0}^n$, 
we denote by $\S_{M_\textbf{m}^{\textbf{r}}}$ the set of \emph{colored multiset permutations} of $M_\textbf{m}$, with color-vector $\textbf{r}:=(r_1,...,r_n)$; that is, words of the form $\pi^{\textbf{c}}=\pi_1^{c_1}\pi_2^{c_2}\ldots \pi_{m}^{c_{m}}$, where $k$ appears exactly $m_k$ times  and $1\leq c_k \leq r_{\pi_k}$,
for each $1\leq k \leq m$. 
When $\textbf{r}= \mathbf{1} := (1,...,1)$, the set $\S_{M_\textbf{m}^{\textbf{r}}}$ coincides with the multiset permutations $\S_{M_\textbf{m}}$. 
When $\mathbf{m} = \mathbf{1}$,  $\S_{M_\mathbf{1}^{\mathbf{1}}}$ is the permutations of $[n]$, $\S_{M_\mathbf{1}^{2\mathbf{1}}}$ is the {\em signed permutations} of $[n]$ and more generally $\S_{M_\mathbf{1}^{r\mathbf{1}}}$ for $r\geq 1$ is the \emph{$r$-colored permutations} of $[n]$, all of which appear frequently in algebraic combinatorics. 

Of particular interest are properties of descent statistics defined on  $\S_{M_\mathbf{1}^{r\mathbf{1}}}$ and $\S_{M_\mathbf{m}^{\mathbf{1}}}$, which admit a common generalization via a descent statistic on $\S_{M_\mathbf{m}^\mathbf{r}}$.
Let $k^{c_i}$ represent the multiset element $k$ with color $c_i\in[r_k]$.
We impose the \emph{color ordering} \cite{STEINGRIMSSON1994187} 
\[
1^1<2^1<\cdots<n^1<(n+1)^1<1^2<2^2<\cdots<n^2<1^3<\cdots<n^{\text{max}\{r_1,...,r_n\}}, 
\]
on the elements in the ground set $M_\mathbf{m}^{\mathbf{r}}$ of $\S_{M_\mathbf{m}^\mathbf{r}}$,
where we include the extra term $(n+1)^1$ by setting $\pi_{m+1}^{c_{m+1}}:=(n+1)^1$ in every colored permutation $\pi_1^{c_1}\pi_2^{c_2}\cdots \pi_{m}^{c_{m}}\in \S_{M_\textbf{m}^{\textbf{r}}}$. 
We say that $j\in [m]$ is a \emph{descent} (respectively, \emph{ascent}) of $\pi^{\textbf{c}}=\pi_1^{c_1}\pi_2^{c_2}\cdots \pi_{m}^{c_{m}}\pi_{m+1}^{c_{m+1}}$ if $\pi_j^{c_j}>\pi_{j+1}^{c_{j+1}}$ (respectively, $\pi_j^{c_j}<\pi_{j+1}^{c_{j+1}}$) according to the color ordering. 
We denote by $\DES(\pi^{\textbf{c}})$ the set of descents of a colored multiset permutation $\pi^{\textbf{c}}$, and we let $\des(\pi^{\textbf{c}}) = |\DES(\pi^{\textbf{c}})|$.

%We call the {\em descent polynomial} of $\S_{M_{\textbf{m}}}^{\textbf{r}}$ the polynomial $\sum_{\pi\in  \S_{M_\textbf{m}^{\mathbf{r}}}}x^{\text{des}(\pi)}$.


Early in the 20th century, MacMahon proved the following identity for the multiset permutations $\S_{M_{\mathbf{m}}^{\mathbf{1}}}$,

\begin{equation}
\label{macmahon's formula}\frac{\sum\limits_{\pi\in  \S_{M_\textbf{m}^\mathbf{1}}}x^{\text{des}(\pi)}}{(1-x)^{m+1}}= \sum_{t\geq 0}
 {\binom{t+m_1}{ m_1}}{\binom{t+ m_2}{ m_2}}\cdots 
 {\binom{t+ m_{n}}{ m_{n}}}x^t.
 \end{equation}
(see, for instance, \cite[Volume 2, Chapter IV, page 211]{Macm}.)
When $\textbf{m} = \mathbf{1}$, MacMahon's formula recovers a well-known identity for the {\em $n$-th Eulerian polynomial}
\[
A_n = \sum_{\pi\in\S_n}x^{\des(\pi)},
\]
which is ubiquitous in enumerative and algebraic combinatorics.  
The Eulerian polynomials $A_n$ enjoy a variety of sought-after distributional properties including symmetry, unimodality, log-concavity, real-rootedness as well as $\gamma$-positivity \cite{PB}. 
When the Eulerian polynomials arise as special cases of larger families of generating polynomials, such as the \emph{$P$-Eulerian polynomials} \cite[Chapter 3]{stanley2011enumerative} or the \emph{$s$-Eulerian polynomials} \cite{savage2016mathematics}, many of these desirable distributional properties are often seen to extend to the larger family. 
In this paper, we consider the {\em colored multiset Eulerian polynomial} 
\[
A_{{M_\textbf{m}^{\textbf{r}}}} = \sum_{\pi^{\mathbf{c}} \in \S_{M_\textbf{m}^{\textbf{r}}}}x^{\des(\pi)}.
\]
These polynomials are a common generalization of both the \emph{$r$-colored Eulerian polynomials} $A_{M_{\mathbf{1}}^{r\mathbf{1}}}$ and MacMahon's \emph{multiset Eulerian polynomials} $A_{M_{\textbf{m}}^{\mathbf{1}}}$. 
The distributional properties of both $r$-colored Eulerian polynomials and MacMahon's multiset Eulerian polynomials have been studied extensively. 
For instance, the colored Eulerian polynomials were shown to be real-rooted, log-concave and unimodal \cite{savagevisontai}, and they are symmetric if and only if $r \in\{1, 2\}$.  
More recently, they were shown to possess a strong distributional property investigated by Br\"and\'en and Solus \cite{PS}; namely, they are interlaced by their own reciprocal.  
This implies additional distributional properties including { bi-$\gamma$-positivity}, and real-rootedness, unimodality and log-concavity of their {symmetric decomposition}.  


The colored Eulerian polynomials as well as the (uncolored) multiset Eulerian polynomial for $\textbf{m} = 2\mathbf{1}$ were further shown to be {$s$-Eulerian polynomials} by Savage and Visontai \cite{savagevisontai}.  
Savage and Visontai conjectured that the same is true for the Type-B analog $A_{M_{2\mathbf{1}}^{2\mathbf{1}}}$ \cite[Conjecture 3.25]{savagevisontai} in an effort to more fully describe the combinatorial generating polynomials that are \emph{$s$-Eulerian}; i.e., equal to some $s$-Eulerian polynomial. 
To settle this conjecture, Lin \cite[Theorem 6]{LZC} generalized MacMahon's formula to signed multiset permutations, showing that
\begin{align}
 %\frac{\sum\limits_{\pi^{\mathbf{c}}\in  \S_{M_\textbf{m}^{2\mathbf{1}}}}x^{\text{des}(\pi)}}{(1-x)^{m+1}} & = \sum_{t\geq 0}\prod_{j=1}^{n} \bigg(\sum_{{{i=0}}}^{ m_j }{\binom{t+m_j-i}{ m_j-i}} {\binom{t+ i-1}{ i}} \bigg)x^t, \label{lin's formula}\\
  \frac{\sum\limits_{\pi^{\mathbf{c}}\in  \S_{M_\textbf{m}^{2\mathbf{1}}}}x^{\text{des}(\pi)}}{(1-x)^{m+1}} &= \sum_{t\geq 0}\prod_{j=1}^{n}
 {\binom{2t+m_j}{ m_j}}
 x^t. \label{eqn:lin2}
% \end{split}
\end{align}

% In fact, several multivariate analogs have been studied for the formulas appearing in equations \eqref{macmahon's formula}, \eqref{lin's formula} \cite{LZC,tielker} and \textcolor{red} {$https://irma.math.unistra.fr/~guoniu/papers/p56lectnotes2.pdf$ and MacMahon \cite{Macm} again}.

In this extended abstract, we study how the various distributional properties of the Eulerian polynomials and colored Eulerian polynomials extend to the larger family of colored multiset Eulerian polynomials. 
A characterization of $A_{M_{\textbf{m}}^{\textbf{r}}}$ that are symmetric with respect to degree $m$ (the cardinality of $M_{\textbf{m}}$) is provided. 
This characterization is applied to show that all $A_{M_{\textbf{m}}^{\textbf{r}}}$ of degree $m$ are interlaced by their own reciprocal.  
This condition implies that these polynomials satisfy a wealth of desirable distributional properties, including real-rootedness, log-concavity, unimodality, the alternatingly increasing property and bi-$\gamma$-positivity. 
To prove these results, an identity of multivariate generating functions, generalizing MacMahon's formula, is proven and utilized. 
Connections to prior work on $s$-Eulerian polynomials and combinatorial interpretations of $\gamma$-coefficients of certain polynomials are drawn. 
Some open questions are also proposed.
% In particular, sufficient conditions are given for real-rootedness, from which log-concavity and unimodality also follow.  
% A necessary and sufficient condition is derived for the polynomial $A_{{M_\textbf{m}^{r}}}$ to be symmetric with respect to degree $m$. 
% This condition is in turn used to generalize known results for colored Eulerian polynomials to the multiset setting; namely, a sufficient condition for the polynomial $A_{\S_{M_\textbf{m}^{\textbf{r}}}}$ to interlace its own reciprocial is derived. 
% As a direct result, these polynomials have a real-rooted, log-concave and unimodal symmetric decomposition and are therefore bi-$\gamma$-positive. 
% It is additionally shown when the known connections between colored multiset Eulerian polynomials and $s$-Eulerian polynomials can be generalized.  
% Generalizations of prior results on combinatorial interpretations of the $\gamma$-coefficients are also given. 

\section{Preliminaries}
\label{sec: preliminaries}
The main results of this paper describe the distributional properties of the colored multiset Eulerian polynomials.  
We now recall the definitions of the distributional properties of interest and their basic properties.
For a more detailed discussion of these properties and their significance in combinatorics, we recommend the survey article \cite{PB}. 

A polynomial $p(x)=p_0+p_1x+\cdots+p_dx^d$ of degree $d$ is \emph{unimodal} if its coefficients satisfy $p_0\leq p_1\leq \cdots \leq p_s\geq \cdots \geq p_d$ for some $s\in \{0,...,d\}$. %; $s$ is then a {\em mode} of $p(x)$. 
It is \emph{log-concave} if $p_i^2 \geq p_{i-1}p_{i+1}$ for all $i = 1,\ldots, d$. 
It is well-known that a log-concave polynomial with no internal zeros is also unimodal. 

The polynomial $p$ is called {\em symmetric} with respect to degree $n$ if $p_s=p_{n-s}$ for all $s=0,...,n$. 
The linear space of polynomials that are symmetric with respect to degree $n$ can be expressed in the basis
\[
\{x^i(x + 1)^{n - 2i} : 0 \leq i \leq \lfloor n/2\rfloor,\}
\]
commonly referred to as the \emph{$\gamma$-basis}.  
If a polynomial has nonnegative coefficients in the $\gamma$-basis it is called \emph{$\gamma$-positive} (or \emph{$\gamma$-nonnegative}). 
If $p$ is $\gamma$-positive then it is also unimodal. 

The polynomial $p$ is said to be {\em real-rooted} if $p$ is a constant polynomial or all its zeros are real numbers. 
When the coefficients of $p$ are nonnegative, it follows that $p$ is log-concave.  
Moreover, when $p$ is real-rooted and symmetric, $p$ is also $\gamma$-positive.  
Hence, a proof that $p$ with only positive coefficients is real-rooted and symmetric is a proof that $p$ satisfies all of the above distributional properties. 

The following distributional property has recently received much attention in the context of algebraic combinatorics and discrete geometry:
The polynomial $p$ is {\em alternatingly increasing} if  $p_0\leq p_d \leq p_1 \leq p_{d-1}\leq \cdots \leq p_{\lfloor \frac{d+1}{2}\rfloor}$. 
When $p$ is alternatingly increasing, it is also unimodal. 
The property has received much attention in algebraic combinatorics since the following alternative characterization makes proving alternatingly increasing particularly useful when studying the unimodality of Hilbert polynomials of graded rings expressed in the multinomial basis. 

Given $n\geq d$, it can be shown that there exist unique polynomials $a,b\in\mathbb{R}[x]$ such that
(1) $p = a + xb$, (2) $\deg(a) \leq n$, (3) $\deg(b)\leq n-1$, (4) $a$ is symmetric with respect to $n$, and (5) $b$ is symmetric with respect to $n-1$.
The pair $(a,b)$ is called the \emph{symmetric decomposition} of $p$ with respect to $n$, or the \emph{$\mathcal{I}_n$-decomposition} of $p$. 
A basic observation is that $p$ is alternatingly increasing if and only if both $a$ and $b$ have only nonnegative coefficients and are unimodal.
It is therefore of interest to investigate the distributional properties of the polynomials in a symmetric decomposition. 
We say that $p$ has a real-rooted, unimodal, or log-concave symmetric decomposition whenever both $a$ and $b$ fulfill the specified condition. 
When $a$ and $b$ are both $\gamma$-positive $p$ is said to be \emph{bi-$\gamma$-positive}. 

Note that it is possible for a polynomial that is not real-rooted to have a real-rooted symmetric decomposition with respect to its degree.  
For instance, the polynomial
$
1 + 5x + 17x^2 + 15x^3 + 2x^4
$
has no real zeros but $\mathcal{I}_4$-decomposition $((1+x)^4, A_3(x))$. 
In \cite{PS}, it was shown that a sufficient condition for a symmetric decomposition to be real-rooted is that it is \emph{interlaced} by its own reciprocal.  
Given two real-rooted polynomials $p$ and $q$ with respective zeros $\alpha_1\geq \alpha_2\geq \cdots$ and $\beta_1\geq \beta_2\geq \cdots$, we say that $p$ is \emph{interlaced} by $q$, denoted $q\prec p$ if
\[
\alpha_1\geq \beta_1\geq \alpha_2\geq \beta_2\geq \cdots.
\]
In \cite{PS}, it is shown that a polynomial $p$ of degree $d$ with $\mathcal{I}_d$-decomposition $(a,b)$ having only nonnegative coefficients is interlaced by its own reciprocal $\mathcal{I}_d(p) = x^dp(1/x)$ if and only if $b\prec a$.
In this case, it follows that the symmetric decomposition of $p$ is real-rooted, log-concave, and unimodal, but we also recover that $p$ is alternatingly increasing, real-rooted, log-concave, unimodal and bi-$\gamma$-positive.  
Hence, it is of interest to determine when a given combinatorial generating polynomial interlaces its own reciprocal.  
This property was shown to hold, for example, for colored Eulerian polynomials and then applied to settle several conjectures in algebraic combinatorics \cite{PS}.  
Below, we describe when this condition generalizes to the class of colored multiset Eulerian~polynomials.


% We call the {\em symmetric decomposition} of $p$ the expression $p=a+xb$, where $a$ and $b$ are symmetric polynomials of degree $d$ and less than $d$ respecively (see \cite[Lemma 2.1]{BS} for an explicit description). The polynomial $p$ is called $\r$-positive if $p(x)=\sum_{i=0}^{\lfloor \frac{d}{2} \rfloor}\r_i x^i(1+x)^{d-2i}$ where all coefficients $\r_i$ are nonnegative, and bi-$\r$-positive if both $a$ and $b$ are $\r$-positive.


\section{Generating function identities}

To derive the desired results on the distributional properties of the polynomial $A_{{M_\textbf{m}^{\textbf{r}}}}$, we first give a generalization of MacMahon's formula~\eqref{macmahon's formula} to the colored multiset permutations.  
The required univariate identity can be recovered from a multivariate identity that also specializes to other previously observed generalizations and q-analogues of MacMahon's formula. 
Throughout this section, we fix a positive integer $n$ and the positive integral vectors $\mathbf{m}=(m_1,...,m_n)$ and $\mathbf{r}=(r_1,...,r_n)$. 
We let $\pi^{\textbf{c}}=\pi_1^{c_1}\cdots \pi_m^{c_m} \pi_{m+1}^{c_{m+1}}$ denote a permutation in the set $\mathfrak{S}_{M_\textbf{m}^{\textbf{r}}}$.
% \textcolor{violet}{we need that  $c_{\pi_i}:=l$, $1\leq l \leq r_k$, if $\pi_i=k^l$. Does $z_{\pi_1}^{c_{\pi_1}-1}$ need to be $z_{\pi_1}^{c_{1}-1}$ and so on?}
We then have the following result.

\begin{theorem}\label{thm: product formula multivariate}
 For positive integral vectors $\mathbf{m}=(m_1,...,m_n)$ and $\mathbf{r}=(r_1,...,r_n)$, we have
 
\begin{align}\label{first genera} & 
\sum_{\pi^{\textbf{c}} \in \S_{M_\textbf{m}^{\mathbf{r}}}}\frac{
z_{\pi_1}^{c_{1}-1}\cdots z_{\pi_m}^{c_{m}-1}
\prod\limits_{\substack{i\in[m+1]\setminus\{1\} \\ i-1 \in DES(\pi)}}z_{\pi_i}^{r_{\pi_i}}\cdots z_{\pi_m}^{r_{\pi_m}}z_{m+1}}{\prod\limits_{i\in [m+1]}\big(1-z_{\pi_i}^{r_{\pi_i}}\cdots z_{\pi_m}^{r_{\pi_m}}z_{m+1}\big)}
& =\sum_{t\geq 0} z_{m+1}^t   
\prod_{k\in[n]} 
\begin{bmatrix}r_kt +m_k \\  m_k \end{bmatrix}_{z_{k}}.
\end{align}
\end{theorem}

The proof of \Cref{thm: product formula multivariate} is combinatorial, and uses the notion of barred permutations popularized by Gessel and Stanley \cite{Gessel}, further explored by Lin \cite{LZC} in the study of signed multiset permutations.
While somewhat garish in form, the identity in \Cref{thm: product formula multivariate} specializes to several known identities. 
Setting $z_{m+1}=x$, $z_k=1$ and $\textbf{r} = \mathbb{1}$ recovers~\eqref{macmahon's formula}. 
Similarly, \eqref{eqn:lin2} is recovered by setting $z_{m+1}=x$, $z_k=1$ and $\textbf{r} = 2\mathbb{1}$. \Cref{macmahon's formula,eqn:lin2} also have $q$-analogs that are analogously recovered by an appropriate specialization.
When $\textbf{m} = \mathbb{1}$ and $\mathbf{r} = r\mathbb{1}$ for a positive integer $r$, \eqref{first genera} specializes to a formula of Steingrimsson \cite[Theorem 17]{STEINGRIMSSON1994187} for $z_{m+1}=x$ and $z_k=1$ (and its $q$-analog, \cite[Proposition 8.1]{Biagioli2009EnumeratingWP},                                    for $z_k=q$).
Another interesting evaluation arises for $z_{m+1}=x$ and $z_k=q$ for all $k\in [n]$, for $\mathbf{r} = r\mathbb{1}$, where (a flag version of) a permutation statistic similar to the inversion-sequence statistic dmaj considered in \cite{SAVAGE2012850} appears on the enumerator in \eqref{first genera}.
% \textcolor{blue}{can you be more specific about what this flag statistic is?  Define it?} \textcolor{violet}{I can define it if you think it is interesting but for space (and time) reasons it might make more sense to leave this sentence out.}

For our %purposes in the
study of the distributional properties of $A_{{M_{\textbf{m}}^{\textbf{r}}}}$, the relevant specialization of the identity in \Cref{thm: product formula multivariate} is given by $z_{m+1} = x$, $z_k = 1$, which yields the following. %corollary.

\begin{corollary}
    \label{cor: main id}
    For positive integral vectors $\mathbf{m}=(m_1,...,m_n)$ and $\mathbf{r}=(r_1,...,r_n)$, we have
    \[
    \frac{A_{{M_{\textbf{m}}^{\textbf{r}}}}}{(1 - x)^{m+1}} = \sum_{t\geq0}\left(\prod_{i=1}^n\binom{r_it + m_i}{m_i}\right)x^t.
    \]
\end{corollary}

\Cref{cor: main id} says that the colored multiset Eulerian polynomial $A_{{M_{\textbf{m}}^{\textbf{r}}}}$ is the polynomial $\prod_{i=1}^n\binom{r_ix + m_i}{m_i}$ expressed in the multinomial basis for the vector space of univariate polynomials of degree at most $d$.
This observation will allow us to derive real-rootedness and interlacing results on the $\mathcal{I}_m$-decomposition of $A_{{M_{\textbf{m}}^{\textbf{r}}}}$.

\begin{rem}
    \label{rem: altproof}
We note that \Cref{cor: main id} may alternatively be proven using the identity \cite[Proposition~3.5(5)]{Branden2016LectureH} associated to the $s$-lecture hall order polytopes of Br\"and\'en and Leander. 
Specifically, \Cref{cor: main id} arises as a mild generalization of \cite[Corollary~3.6]{Branden2016LectureH}.
%The proof included here shows that the direct combinatorial proof method of Stanley and Gessel \cite[Second Proof of Theorem~2.1]{Gessel} generalizes to colored multiset permutations.
\end{rem}


% \textcolor{blue}{A few words about why one cares about this identity?}
% \textcolor{violet}{Is it interesting that it is more general compared to the previous identity, in the sense that the variables $z_{k^l}$ are not all identified?}

% %%%%%%START: DANAI's VERSION OF THIS SECTION%%%%%%

% % We will now discuss two multivariate analogs of MacMahon's (and Lin's) formula, for colored multiset permutations. 

% Throughout this section, we consider $M_\textbf{m}^{\textbf{r}}$ to be a multiset with multiplicity vector $\mathbf{m}=(m_1,...,m_n)$ and color vector $\mathbf{r}=(r_1,...,r_n)$ for a positive number $n$. 
% Again, we let $m=m_1+\cdots + m_n$. 
% % We also make use of the following notation: for a positive integer $a$, we have $[a]:=\{1,2,...,a\}$ and $[a]_0:=\{0,1,...,a\}$. 
% Lastly, given a permutation $\pi^{\textbf{c}}=\pi_1^{c_1}\cdots \pi_m^{c_m} \pi_{m+1}^{c_{m+1}}\in \frak{S}_{M_\textbf{m}^{\textbf{r}}}$, for every $i\in [m]$ we let $c_{\pi_i}:=l$ and $r_{\pi_i}:=r_k$ where $\pi_i=k^l$ ($1\leq l \leq r_k$, $k\in [n]$).

% Note that the upcoming theorems, use several different variables: $z_1,z_2,...,z_n, \ z_{m+1}$, $\{z_{{k}^l}$ for $l\in [r_k], \ k\in [n]\}$ and  $\{y_j$ for $j\in \cup_{k\in [n]} [r_k]\}$.

% \begin{theorem}\label{thm: product formula multivariate}
% Fix $z_{{k}^l}:=z_k$ for all $l\in [r_k], \ k\in [n]$. The permutation set $\S_{M_\textbf{m}^{\textbf{r}}}$ satisfies:

% \begin{align}\label{first genera} & 
% \sum_{\pi \in \S_{M_\textbf{m}^{\mathbf{r}}}}\frac{
% z_{\pi_1}^{c_{\pi_1}-1}\cdots z_{\pi_m}^{c_{\pi_m}-1}
% \prod\limits_{\substack{i\in[m+1]\setminus\{1\} \\ i-1 \in DES(\pi)}}z_{\pi_i}^{r_{\pi_i}}\cdots z_{\pi_m}^{r_{\pi_m}}z_{m+1}}{\prod\limits_{i\in [m+1]}\big(1-z_{\pi_i}^{r_{\pi_i}}\cdots z_{\pi_m}^{r_{\pi_m}}z_{m+1}\big)}
% & =\sum_{t\geq 0} z_{m+1}^t   
% \prod_{k\in[n]} 
% \begin{bmatrix}r_kt +m_k \\  m_k \end{bmatrix}_{z_{k}}.
% \end{align}
% \end{theorem}

% \begin{remk} Setting $z_{m+1}=x$, $z_k=1$ and $r_k=1$ for all $k\in [n]$ in \eqref{first genera} recovers \eqref{macmahon's formula}. Similarly, \eqref{lin's formula} is recovered in part for the same variable evaluations, where $r_k=2$ for all $k\in [n]$. In the regular-set case, \eqref{first genera} specializes to a formula of Steingrimsson \cite[Theorem 17]{STEINGRIMSSON1994187} for $z_{m+1}=x$, $z_k=1$, $r_k=r$ and $m_k=1$, for $k\in [n]$.  Another interesting evaluation arises for $z_{m+1}=x$ and $z_k=q$ for all $k\in [n]$, for fixed color ranges $r_1=r_2=\cdots =r_n$, where (a flag version of) a permutation statistic similar to the inversion-sequence statistic dmaj defined in \cite{SAVAGE2012850} appears on the enumerator in \eqref{first genera}.
% \end{remk}


% We now give an alternative generalization of MacMahon's formula \eqref{macmahon's formula}, which recovers the first equality in equation \eqref{lin's formula}.

% Given a permutation  $\pi=\pi_1\cdots \pi_m \pi_{m+1}\in \frak{S}_{M_\textbf{m}^{\textbf{r}}}$,
% let $c_j(\pi)$ denote the number of entries in $i\in [m]$ that are $j$-colored for $j\in [r]$. For example, for $\pi=1^11^22^22^11^33^1$ we have $c_1(\pi)=3, c_2(\pi)=2, c_3(\pi)=1$. Recall that we defined $c_{\pi_i}=l$ and $r_{\pi_i}=r_k$ where $\pi_i=k^l$ ($1\leq l \leq r_k$, $k\in [n]$). Finally, let $r:=\text{max}\{r_1,...,r_n\}$.

% \begin{theorem}\label{mcmahon general(2)} The permutation set $\S_{M_\textbf{m}^{\textbf{r}}}$ satisfies:

%     \begin{align}\label{mc mahon generali2}
%  & \sum_{\pi \in \S_{M_\textbf{m}^{\mathbf{r}}}}\frac{\prod\limits_{p\in [r]}y_p^{c_{p}(\pi)} 
% \prod\limits_{\substack{i\in[m+1]\setminus\{1\} \\ i-1 \in DES(\pi)}}z_{\pi_i}^{r_{\pi_i}}\cdots z_{\pi_m}^{r_{\pi_m}}z_{m+1}}{\prod\limits_{i\in [m+1]}\big(1-z_{\pi_i}^{r_{\pi_i}}\cdots z_{\pi_m}^{r_{\pi_m}}z_{m+1}\big)}
% \\ &= \sum_{t\geq 0} z_{m+1}^t 
% \prod_{k\in[n]} \bigg(
% \sum_{\footnotesize{\substack{i_{r_k;k}\in [m_k]_0 \\
%     \cdots\\
%     i_{2;k} \in [m_k-\sum\limits_{l=3}^{r_k}i_{l;k}]_0 }}}
%     \Big(\begin{bmatrix}t +i_{1;k} \\  i_{1;k} \end{bmatrix}_{z_{k^1}^{r_k}} y_1^{i_{1;k}}
% \prod\limits_{l\in [r_k]\setminus\{1\}} \begin{bmatrix}t-1 +i_{l;k} \\  i_{l;k} \end{bmatrix}_{z_{k^l}^{r_k}}  y_l^{i_{l;k}}\Big)\bigg),
% \end{align}
%      where $i_{1;k}=m_k-\sum\limits_{l=2}^{r_k}i_{l;k}$ in the sum, for each $k\in [n]$.
     
% \begin{remk} Setting $z_{m+1}=x$, $y_l=1$, $z_{k^l}=1$ and $r_k=1$ for all $l \in [r_k], k\in [n]$ in the equation in \Cref{mcmahon general(2)} recovers \eqref{macmahon's formula}. Similarly, the first equality in equation \eqref{lin's formula} is recovered in part for the same variable evaluations, where $r_k=2$ for all $k\in [n]$.
% \end{remk}

% \end{theorem}

% The proof of Theorems \ref{thm: product formula multivariate} and \ref{mcmahon general(2)} uses the notion of barred permutations, an idea that dates back to Gessel and Stanley \cite{Gessel}. This notion was extended by Lin \cite{LZC} to signed multiset permutations, who used it to prove multivariate analogs of equation \ref{lin's formula}, partly motivated by results in \cite{eulermah}.

% %%%%%%END: DANAI's VERSION OF THIS SECTION%%%%%%

\section{Distributional properties}


We now consider the distributional properties of the colored multiset Eulerian polynomials $A_{M_\mathbf{m}^{\mathbf{r}}}$ for positive integral vectors $\textbf{m} = (m_1,\ldots, m_n)$ and $\textbf{r} = (r_1,\ldots,r_n)$. 
To do so, we make use of the identity in \Cref{cor: main id}, noting first that this identity has a simple geometric interpretation. 

The \emph{Ehrhart polynomial} of a $d$-dimensional convex lattice polytope $P\subset\mathbb{R}^n$ is $\mathcal{L}(P; t) = |tP \cap\mathbb{Z}^n|$, where $tP = \{tp : p\in P\}$ denotes the \emph{$t$-th dilate} of $P$ for $t\in\mathbb{Z}_{>0}$. 
The \emph{(Ehrhart) $h^\ast$-polynomial} of $P$, denoted $h^\ast(P; x)$, is the Ehrhart polynomial of $P$ when expressed in the multinomial basis for the vector space of polynomials of degree at most $d$. 
It is well-known that the $\mathcal{L}(\Delta_d;t)= \binom{t + d}{d}$ when $\Delta_d$ is the the standard $d$-dimensional simplex \cite[Theorem 2.2.]{BR}.
Since $\mathcal{L}(rP;t) = \mathcal{L}(P; rt)$ for any positive integer $r$ and taking a direct product of polytopes corresponds to multiplying the Ehrhart polynomials, we have that the polynomial $\prod_{i=1}^n\binom{r_it + m_i}{m_i}$ appearing in \Cref{cor: main id} is the Ehrhart polynomial of the product of dilated simplices
\[
P^{\mathbf{r}}_{\mathbf{m}}:=\prod_{j=1}^{ n}r_j\Delta_{m_j}.
\]
It follows that $A_{M_{\textbf{m}}^{\textbf{r}}}(x)$ is the $h^\ast$-polynomial of $P^{\mathbf{r}}_{\mathbf{m}}$. 
\begin{lemma}\label{lem: colored eulerian multiset permutation polynomial}
% The colored multiset Eulerian polynomial of $M_\textbf{m}^{\textbf{r}}$ is equal to the $h^*$-vector of the polytope $P^{\mathbf{r}}_{\mathbf{m}}$ defined above, i.e., we have
  $
  A_{M_{\mathbf{m}}^{\mathbf{r}}}(x) =h^*(P^{\mathbf{r}}_{\mathbf{m}};x).
  $
\end{lemma}
\Cref{lem: colored eulerian multiset permutation polynomial} provides a generalization of the well-known relationship between the Eulerian polynomials $A_n(x) = A_{M_{\mathbb{1}}^{\mathbb{1}}}$ and the $h^*$-vector of the $n$-dimensional cube, which is a product of $n$ one-dimensional simplices. We further note that the corresponding relation for uncolored multisets can arise via the theory of order polytopes for a suitable choice of posets. Lastly, the case $\textbf{m}=(2,...,2)$ and $\textbf{r}=(2,...,2)$ has also been studied (see \cite{Ardila2020TheB}).

Basic results in Ehrhart theory then imply the following.


\begin{lemma}
    \label{lem:degree}
    $A_{M_{\mathbf{m}}^{\mathbf{r}}}$ has degree $m + 1 - \max_{k\in[n]}\left\lceil\frac{m_k + 1}{r_k}\right\rceil$.
\end{lemma}


Results of Hibi and DeNegri \cite{DH97} can then be applied to characterize when $A_{M_{\textbf{m}}^{\textbf{r}}}$ is symmetric with respect to degree $m$.

\begin{theorem}
    \label{thm: gorenstein}
    $A_{M_{\mathbf{m}}^{\mathbf{r}}}$ is symmetric (with respect to its degree) if and only if 
    \begin{equation}
        \label{eqn:gorenstein}
        \max_{k\in[n]}\left\lceil\frac{m_k+1}{r_k}\right\rceil r_j  = m_j +1 \qquad \mbox{for all $j\in[n]$.}
    \end{equation}
\end{theorem}

It follows that
$A_{M_{\mathbf{m}}^{\mathbf{r}}}$ is symmetric with respect to its degree $m$ if and only if $r_j = m_j + 1$ for all $j\in[n]$.%, i.e., $P^{\mathbf{r}}_{\mathbf{m}}$ contains an interior lattice point.


\Cref{thm: gorenstein} can in turn be applied to describe when the polynomial $A_{M_{\mathbf{m}}^{\mathbf{r}}}$ is interlaced by its own reciprocal, and hence fulfills the numerous distributional properties described in Section~2. 
To do so, one uses the \emph{subdivision operator}, which is the linear transformation on the space of univariate polynomials given by $\mathcal{E}: \binom{x}{k} \longmapsto x^k$. 
When applied to an Ehrhart polynomial $\mathcal{L}(P;t)$ that can be expressed as $\mathcal{L}(P;t) = \sum_{i = 0}^dc_it^i(1+t)^{d-i}$ for nonnegative $c_i$, the subdivision operator returns a $[-1,0]$-rooted polynomial that is the $h^\ast$-polynomial up to a transformation that preserves $(-\infty,0)$-rootedness \cite{PB15}. 
Hence, $h^\ast(P;x)$ is real-rooted, log-concave and unimodal. 
It can be shown that Ehrhart polynomial $\mathcal{L}(P^{\mathbf{r}}_{\mathbf{m}}; t)$ does indeed have nonnegative coefficients in this so-called \emph{magic basis} \cite{Ferroni2023ExamplesAC} $t^i(1+t)^{d-i}$, $i = 0,\ldots, d$. 
Since the polynomials satisfying $r_j=m_j+1$ for all $j\in [n]$  are symmetric with respect to their degree $m$, they are trivially interlaced by their own reciprocal.  
Applying the results of \cite{PS} allows us to deduce the following more general~result. 
\begin{theorem}
\label{thm: interlacing}
Suppose that $\mathbf{m}$, $\mathbf{r} $ satisfy $r_j\geq m_j+1$ for all $j=1,...,n$. Then
% Suppose that $r_j\geq m_j+1$ for all $j=1,...,n$, for the colored multiset $M_\textbf{m}^{\textbf{r}}$ with $\mathbf{m} = (m_1,m_2,\ldots,m_n)$, $\mathbf{r} = (r_1,r_2,\ldots,r_n)$. Then 
 \[
\mathcal{I}_{m}(A_{M_{\mathbf{m}}^{\mathbf{r}}})\prec A_{M_{\mathbf{m}}^{\mathbf{r}}}.
 \]
\end{theorem}
It follows that if $r_j\geq m_j+1$ for all $j=1,...,n$, then the colored multiset Eulerian polynomial $A_{M_\textbf{m}^{\textbf{r}}}(x)$ is real-rooted, log-concave, unimodal, alternatingly increasing and bi-$\r$-positive with a real-rooted, log-concave and unimodal $\mathcal{I}_m$-decomposition.
This generalizes the observation of Br\"and\'en and Solus \cite[Corollary 3.2]{PS} for colored Eulerian polynomials to the colored multiset Eulerian polynomial setting. 

% For a colored multiset $M_\textbf{m}^{\textbf{r}}$, the descent polynomial
% \[A_{M_\textbf{m}^{\textbf{r}}}(x):=\sum_{\pi\in  \S_{M_\textbf{m}^{\textbf{r}}}}x^{\text{des}(\pi)}\]
% is its corresponding {\rm colored Eulerian multiset permutation polynomial}, or simply called {\rm colored Eulerian multiset polynomial}. 
% In the uncolored case, the choice $\textbf{m}=(1,...,1)$, corresponds to  the classical {\em $n$-th Eulerian polynomial} $A_{n}(x)$, and, more generally, when $\textbf{m}=(p,...,p)$ and $\mathbf{r}=(1,...,1)$, $A_{[n]^\mathbf{r}_\mathbf{m}}(x)$ is known as the {\em $p$-multiset Eulerian polynomial}. Further, the colored polynomials that arise when we restrict to the set $[n]$ (i.e., $\textbf{m}=(1,...,1)$) are known as {\em colored Eulerian polynomials}. We will denoted them by $A_{[n]^{\textbf{r}}}$.

% One of the aims of this paper is to understand the distributional properties of the descent polynomial $A_{M_\textbf{m}^{\textbf{r}}}(x)$. To do this, we will need the following notions. 
% We say that a polynomial $p(x)=p_0+p_1x+\cdots+p_nx^n$ (of degree $d$) is \emph{unimodal} if its coefficients satisfy $p_0\leq p_1\leq \cdots \leq p_s\geq \cdots \geq p_n$ for some $s\in \{0,...,n\}$; $s$ is then a {\em mode} of $p(x)$. The polynomial $p$ is called {\em symmetric} if $p_s=p_{d-s}$ for all $s=0,...,d$, and {\em real-rooted} if all its zeros are real numbers (or $p$ is the zero polynomial). Furthermore, it is called {\em alternatingly increasing} if its coefficients are nonnegative such that $ p_0\leq p_d \leq p_1 \leq p_{d-1}\leq \cdots \leq p_{\lfloor \frac{d+1}{2}\rfloor}$. We call the {\em symmetric decomposition} of $p$ the expression $p=a+xb$, where $a$ and $b$ are symmetric polynomials of degree $d$ and less than $d$ respecively (see \cite[Lemma 2.1]{BS} for an explicit description). The polynomial $p$ is called $\r$-positive if $p(x)=\sum_{i=0}^{\lfloor \frac{d}{2} \rfloor}\r_i x^i(1+x)^{d-2i}$ where all coefficients $\r_i$ are nonnegative, and bi-$\r$-positive if both $a$ and $b$ are $\r$-positive.
% Note that a real-rooted polynomial with nonnegative coefficients is $\r$-positive, and a $\r$-positive polynomial is unimodal; if a polynomial is symmetric and unimodal then it necessarily has a mode in the middle. Another interesting property is that of $p$ being interlaced by its own reciprocal $\mathcal{I}_{d}p\prec p$ (see \cite{PS} for a definition). By \cite[Theorem 2.6]{PS} this property implies that the polynomials $a$ and $b$ in the symmetric decomposition interlace, which implies that they have only real zeros, when $a$ and $b$ have nonnegative coefficients.  
% Since symmetric polynomials with only nonnegative coefficients and only real zeros are $\r$-positive \cite{PB}, the bi-$\r$-positivity follows from this stronger condition. 

% In \cite[Corollary 3.2]{PS}, it was shown that the colored Eulerian polynomial $A_{[n]^{\textbf{r}}}$ is real-rooted and alternatingly increasing. In particular, it was shown that this result is a consequence of a stronger property; namely, that the colored Eulerian polynomial with $r$ colors is interlaced by its own reciprocal.  We extend this result to the colored Eulerian multiset polynomials $A_{M_\textbf{m}^{\textbf{r}}}$ whose color-vector is bounded below as in the following theorem.

% \begin{theorem}
% \label{thm: interlacing}
% Suppose that $r_j\geq m_j+1$ for all $j=1,...,n$, for the colored multiset $M_\textbf{m}^{\textbf{r}}$ with $\mathbf{m} = (m_1,m_2,\ldots,m_n)$, $\mathbf{r} = (r_1,r_2,\ldots,r_n)$. Then 
%  \[\mathcal{I}_{m_1+\cdots+m_n}A_{\mathbf{m}}^{\mathbf{r}}\prec A_{\mathbf{m}}^{\mathbf{r}}.\]
% \end{theorem}

% It follows that if $r_j\geq m_j+1$ for all $j=1,...,n$, then the colored multiset Eulerian polynomial $A_{M_\textbf{m}^{\textbf{r}}}(x)$ for $M = \{1^{m_1},2^{m_2},\ldots, n^{m_n}\}$ and $\mathbf{r} = (r_1,r_2,\ldots,r_n)$ is real-rooted, alternatingly increasing and bi-$\r$-positive. 

% The proof of \Cref{thm: interlacing} relies on the realization of $A_{M_\textbf{m}^{\textbf{r}}}$ as the $h^*$-vector of a lattice polytope, in particular as a product of simplices. 
% For every multiset $M^\mathbf{r}_\mathbf{m}$ with multiplicity vector $\mathbf{m}:=(m_1,...,m_n)$ and color vector $\mathbf{r} = (r_1,r_2,\ldots, r_n)$ of positive integers, consider the direct product of simplices
% \[P^{\mathbf{r}}_{\mathbf{m}}:=\prod_{j=1}^{ n}r_j\Delta_{m_j}\]
% where $\Delta_{m_j}$ denotes the standard $m_j$-dimensional simplex, and $r_j\Delta_{m_j}$ its $r_j$-th dilate.

% \begin{lemma}\label{lem: colored eulerian multiset permutation polynomial}
% The colored multiset Eulerian polynomial of $M_\textbf{m}^{\textbf{r}}$ is equal to the $h^*$-vector of the polytope $P^{\mathbf{r}}_{\mathbf{m}}$ defined above, i.e., we have
%   \[A_{\mathbf{m}}^{\mathbf{r}}(x) =h^*(P^{\mathbf{r}}_{\mathbf{m}};x).\] 
% \end{lemma}

% \Cref{lem: colored eulerian multiset permutation polynomial} provides a generalization of the known relationship between the permutation polynomials $A_{[n]^{\textbf{r}}}$ and the $h^*$-vector of the $n$-dimensional cube, which is a product of $n$ one-dimensional simplices. We further note that the corresponding relation for uncolored multisets can arise via the theory of order polytopes for a suitable choice of posets. Lastly, the case $\textbf{m}=(2,...,2)$ and $\textbf{r}=(2,...,2)$ has also been studied (see \cite{Ardila2020TheB}).




% The connection between colored multiset Eulerian polynomials $A_{M_\textbf{m}^{\textbf{r}} }$ and polytopes established in \Cref{lem: colored eulerian multiset permutation polynomial}, further allows us to characterize when $A_{M_\textbf{m}^{\textbf{r}} }$ is symmetric in the case $r_j\geq  m_j+1$ for every $j=1,...,n$. Note that when $r_j\geq m_j+1$, the polytope $P^{\mathbf{r}}_{\mathbf{m}}$ has full degree, equal to $m_1+...+m_n$.

% \begin{lemma}\label{lem: the symmetric case}
% Let $M_\textbf{m}^{\textbf{r}}$ be a multiset with multiplicity vector $\mathbf{m}=(m_1,...,m_n)$ and colored vector $\mathbf{r}=(r_1,...,r_n)$ such that $r_j\geq m_j+1$, for all $j=1,...,n$. The colored multiset Eulerian polynomial $A_{\mathbf{m}}^{\mathbf{r}}$ is symmetric if and only if 
% $r_j= m_j+1$ for every $j=1,...,n$.
% \end{lemma}


\section{Combinatorial interpretations}

We end with a discussion of some alternative interpretations of certain colored multiset Eulerian polynomials as a motivation for future work connecting these polynomials with desirable distributional properties to other lines of combinatorial investigation.  
% Connections are made to the well-studied $s$-Eulerian polynomials and efforts to provide combinatorial interpretations of $\gamma$-coefficients in combinatorial generating polynomials.

% In this section, we describe how the results of Sections~3 and~4 yield generalizations of known properties and interpretations of (colored) Eulerian polynomials. 
% We describe the relationship between the well-studied $s$-Eulerian polynomials and the colored multiset Eulerian polynomials. 
% We then extend prior work pertaining to combinatorial interpretations of the gamma-coefficients of the symmetric decomposition of $A_{M_{\mathbf{m}}^{\mathbf{r}}}(x)$ in specific instances. 

\subsection{\texorpdfstring{$s$}{s}-Eulerian polynomials}
The $s$-Eulerian polynomials $E_n^{s}$ are a large family of combinatorial generating polynomials defined for any sequence of positive integers $s = (s_1,\ldots,s_n)$. 
They are known to be real-rooted and admit numerous connections to classically studied problems, including the enumeration of lecture hall sequences, as well as connections to discrete geometry where they are interpreted as $h^\ast$-polynomials of \emph{$s$-lecture hall simplices} $P_n^s$; namely, $E_n^s = h^\ast(P_n^s;x)$. 
The $s$-Eulerian polynomials are surveyed in \cite{savage2016mathematics}. 
It is shown in \cite{savagevisontai}, for instance, that $E_n^{(r,2r,\ldots,rn)} = A_{M_{\mathbb{1}}^{r\mathbb{1}}}$, the $r$-colored Eulerian polynomial and $A_{M_{2\mathbb{1}}^{\mathbb{1}}} = E_{2n}^{s}$, where $s = (1,1,3,2,5,3,\ldots, 2n-1, n)$. 

%In \cite[Theorem 3.23]{savagevisontai}, it was shown that for $\textbf{m}=(2,2,...,2)$ and $\textbf{r}=(1,1,...,1)$, \begin{equation}\label{1,1,3,2}A_{\textbf{m}}^{\textbf{r}}=h^*(P^{(1,1,3,2,5,3,7,4,...2n-1,n)}).    \end{equation}
%We should also be able to show with the same techniques that for $\textbf{m}=(2,2,...,2,1)$ and $\textbf{r}=(1,1,...,1)$, \begin{equation}\label{1,1,3}A_{\textbf{m}}^{\textbf{r}}=h^*(P^{(1,1,3,2,5,3,7,4,...2n-1)}).\end{equation}
%Note that the signed analog of \eqref{1,1,3,2} and \eqref{1,1,3} was shown in \cite{Chen2013sInversionSA}. It can be generalized to colors in the following way

% In \cite[Corollary 3.2]{PS}, it was shown that for  $\mathbf{r}=r\mathbb{1}, \ r\geq 1$, and $\textbf{m}=m\mathbb{1}$, 
% \[\mathcal{I}_{n} A_{M_\textbf{m}^{\mathbf{r}}}\prec A_{M_\textbf{m}^{\mathbf{r}}}, \] 
% using the equality $A_{M_\textbf{m}^{\mathbf{r}}} = E_{n}^{s}$, where $E_{n}^{s}$ is the $s$-Eulerian polynomial for $s = (1,2,3,\ldots, n)$. 

% In this section, we will focus on the colored Eulerian multiset permutation polynomials $A_{\textbf{m}}^{\mathbf{r}}$ for the multisets $\{1^2,2^2,\ldots, n^2\}$ and $\{1^2,2^2,\ldots, (n-1)^2, n\}$ with color vector $\mathbf{r} = (r,r,\ldots, r)$ for a positive integer $r\geq 1$. 
% In \cite[Theorem 3.23]{savagevisontai}, it is observed that, for $r =1$ and $\textbf{m}=(2,2,...,2)$, $A_\textbf{m}^{\mathbf{1}} = E_{2n}^{s}$, where $E_{2n}^{s}$ is the $s$-Eulerian polynomial for $s = (1,1,3,2,5,3,\ldots, 2n-1, n)$.  
% %\textcolor{blue}{We should also be able to show with the same techniques that,for $r=1$ and $\textbf{m}=(2,...,2,1)$,  $A_\textbf{m}^{\mathbf{1}} = E_{2n-1}^{s}$ where $s = (1,1,3,2,5,\ldots,n-1, 2n-1)$.}


In the following, we fix $s = (1,1,3,2,\ldots, 2n-1, n)$, $p = (1,1,3,\ldots, n-1, 2n-1)$ and $rs = (r,r,3r,2r,\ldots,r(2n-1),rn)$,  $rp = (r,r,3r,\ldots,r(n-1),r(2n-1))$ for $r\geq 1$. 
It is seen that the $s$-lecture hall simplex $P_{2n}^{rs}$ is the $r$-th dilate of $P_{2n}^{s}$; that is, $P_{2n}^{rs} = rP_{2n}^s$. Similarly, $P_{2n-1}^{rp} = rP_{2-1n}^p$.
In \cite[Theorem 14]{SAVAGE2012850}, it is shown that 
\begin{align*}
\sum_{t\geq 0}(t+1)^n\left(\frac{t+2}{2}\right)^nx^t = \frac{E_{2n}^{s}}{(1-x)^{2n+1}} \quad \text{and} \quad \sum_{t\geq 0}(t+1)^{n}\left(\frac{t+2}{2}\right)^{n-1}x^t = \frac{E_{2n-1}^{p}}{(1-x)^{2n}}.
\end{align*}
It follows that the Ehrhart polynomials of $P_{2n}^{rs}$ and $P_{2n-1}^{rp}$ are, respectively, 
\[
\mathcal{L}(P_{2n}^{rs};t) = (rt+1)^n\left(\frac{rt+2}{2}\right)^{n} \qquad \mbox{and} \qquad \mathcal{L}(P_{2n-1}^{rp};t) = (rt+1)^n\left(\frac{rt+2}{2}\right)^{n-1}.
\]
% That is, the Ehrhart polynomials of $P_{2n}^{s}$ and $P_{2n-1}^{p}$ are respectively:
% \begin{align*}
% \ell_{2n}^{s}(t) = (t+1)^n\left(\frac{t+2}{2}\right)^n
% \quad \text{and} \quad  \ell_{2n-1}^{s}(t) = (t+1)^{n}\left(\frac{t+2}{2}\right)^{n-1}.
% \end{align*}
% Since $P_{2n}^{rs}$ is the $r$-th dilate of $P_{2n}^{s}$, it follows that its Ehrhart polynomial is 
% \[
% \ell_{2n}^{rs}(t) = (rt+1)^n\left(\frac{rt+2}{2}\right)^{n}.
% \]
% Similarly, 
% \[
% \ell_{2n-1}^{rp}(t) = (rt+1)^n\left(\frac{rt+2}{2}\right)^{n-1}.
% \]
It then follows from \Cref{first genera} that $A_{M_{2\mathbb{1}}^{r\mathbb{1}}} = E_{2n}^{rs}$ and $A_{M_{(2,\ldots,2,1)}^{r\mathbb{1}}} = E_{2n-1}^{rp}$ for any $r\geq 1$.
Hence we have observed the following.
\begin{proposition}
    \label{prop: s-lecture hall}
    The colored multiset Eulerian polynomial $A_{M_{\mathbf{m}}^{\mathbf{r}}}$ is an $s$-Eulerian polynomial for any $\mathbf{m}\in\{(1,\ldots,1),(2,\ldots,2),(2,\ldots,2,1)\}$ and $\mathbf{r} = r\mathbb{1}$ for any $r\geq 1$. 
\end{proposition}
% It then follows from \Cref{first genera} that for any $r\geq 1$:
% \begin{equation}\label{eq: s-euler}
% A_\mathbf{m}^{\mathbf{r}} = E_{2n}^{rs}, \qquad \text{for $\textbf{m}=(2,2,...,2)$ and $\textbf{r}=(r,...,r)$}
% \end{equation}
%  and
% \begin{equation}\label{eq: p-euler}
% A_\mathbf{m}^{\mathbf{r}} = E_{2n-1}^{rp}, \qquad \text{for $\textbf{m}=(2,...,2,1)$ and $\textbf{r}=(r,...,r)$.}
% \end{equation}

It follows from \Cref{thm: interlacing} that for $r\geq3$ the $s$-Eulerian polynomials captured in \Cref{prop: s-lecture hall} satisfy the strongest distributional property in Section~2; namely, they are interlaced by their own reciprocal and hence satisfy all other distributional properties of interest. 
In the case that $r = 1$, the same is true for $E^{s}_{2n}$ since it is known to be symmetric and real-rooted by prior work \cite{savagevisontai} as well as in \cite[Theorem 6.2]{Ardila2020TheB} where these polynomials appear in the study of bipermutohedra. 
$E^{p}_{2n-1}$ is also known to have a real-rooted symmetric decomposition \cite{weaklyincreasingtrees}.
Similarly, when $r=2$, it is only known that $E^{2s}_{2n}$ and $E^{2p}_{2n-1}$ are bi-$\r$-positive.
This connection prompts the following question.
\begin{question}\label{que: s-eul}
    Characterize the colored multiset Eulerian polynomials that are $s$-Eulerian polynomials.  What are the strongest distributional properties they satisfy?
\end{question}
% It would be interesting to fully characterize when a colored multiset Eulerian polynomial appears as an $s$-Eulerian polynomial for some $s$, as well as to fully describe the strongest distributional properties these polynomials satisfy.  
Such results may give us deeper insights into when an $s$-Eulerian polynomial interlaces its own reciprocal. 
To see that such a characterization may be challenging, we note that there is no $s$-sequence and $n\geq 1$ such that $E_n^s=A_{M_{\mathbf{m}}^{\mathbf{r}}}$ for $\mathbf{r}=(1,1)$ and $\mathbf{m}=(3,3)$.

\begin{rem}
    \label{rmk:orderpolytopes}
    In \cite{Branden2016LectureH}, Br\"and\'en and Leander introduced a common generalization of $s$-lecture hall simplices and order polytopes called \emph{$s$-lecture hall order polytopes}. %(see Remark~\ref{rem: s-order symmetry}). 
    If $p = h^\ast(\mathcal{O}(P, s); x)$, we say that $p$ is \emph{$(P, s)$-Eulerian}. 
    \Cref{que: s-eul} asks when $A_{M_{\mathbf{m}}^{\mathbf{r}}}$ is $s$-Eulerian. We saw above that not all colored multiset Eulerian polynomials have this property. On the other hand, $A_{M_{\mathbf{m}}^{\mathbf{r}}}$ is always $(P,s)$-Eulerian. 
\end{rem}



% For $r=1$, $E^{s}_{2n}$
% is known to be symmetric \cite{hoggatt} and real-rooted \cite{simion}, %(as noted in \cite{multiseteulerian})
%  shown also in \cite[Theorem 6.2]{Ardila2020TheB}. On the other hand, the polynomial $E^{p}_{2n-1}$ is not symmetric but is bi-$\r$-positive as shown in \cite{weaklyincreasingtrees} (see \Cref{thm: weaktrees} below).

% The equalities in \eqref{eq: s-euler}, \eqref{eq: p-euler} in the case $r=2$  are discussed in \cite[Theorems 4.1-4.2]{Chen2013sInversionSA}. Moreover, it is shown in \cite[Theorems 12-13]{ma2019eulerian} that both $E^{2s}_{2n}$ and $E^{2p}_{2n-1}$ are bi-$\r$-positive.

% For $r\geq 3$, it follows by \eqref{eq: s-euler}, \eqref{eq: p-euler} that 
% \[\mathcal{I}_{2n}E^{rs}_{2n}\prec E^{rs}_{2n} \qquad \text{and} \qquad \mathcal{I}_{2n-1}E^{rp}_{2n-1}\prec E^{rp}_{2n-1}\] since the conditions in \Cref{thm: interlacing} are satisfied. 
% Furthermore, by \Cref{lem: the symmetric case}, $E^{rs}_{2n}$ is not symmetric for $r\geq 4$ and $E^{rp}_{2n-1}$ is not symmetric for $r\geq 3$. 

% Note that the symmetry of $s$-Eulerian polynomials was also studied in \cite[Theorem 4.4]{} \textcolor{blue}{https://arxiv.org/pdf/1211.0258.pdf}.

% We summarize some of the discussion above in the following corollary.

% \begin{corollary} Consider $rs = (r,r,3r,2r,\ldots,r(2n-1),rn)$,  $rp = (r,r,3r,\ldots,r(n-1),r(2n-1))$ for $r\geq 1$. 
% The polynomial $E^{rs}_{2n}$ is symmetric for $r=1,3$, and bi-$\r$-positive for $r=2,r\geq 4$. The polynomial $E^{rp}_{2n-1}$ is bi-$\r$-positive (and not symmetric for $r\geq 1$.
% \end{corollary}

% \begin{remk} In this section, we discussed the connection between $s$-Eulerian polynomials and descent polynomials of multiset permutations $A_{\mathbf{m}}^{\mathbf{r}}$ with multiplicity vector $\mathbf{m}\in \{(1,1,...,1),\ (2,2,...,2),\ (2,...,2,1)\}$ and color vector $\mathbf{r}=(r,...,r)$ for some $r\geq 1$. However, such connection becomes less apparent when we consider other multisets such as $\mathbf{m}=(3,3,...,3)$.
%     For example, there is no sequence $s=(s_1,...,s_6)$ such that $E_6^s=A_{\mathbf{m}}^{\mathbf{r}}$ for $\mathbf{r}=(1,1)$ and $\mathbf{m}=(3,3)$.
% \end{remk}

\subsection{Interpreting \texorpdfstring{$\gamma$}{gamma}-coefficients}

The results of \Cref{thm: interlacing} and the observations on bi-$\gamma$-positivity in the previous subsection yield a large family of relatively unexplored combinatorial generating polynomials that are $\gamma$-positive.  
A natural endeavour is then to explore combinatorial interpretations of these $\gamma$-coefficients.  
Such investigations may prove fruitful in, for instance, the development of a complete understanding of the $\gamma$-coefficients for $s$-Eulerian polynomials. 
We motivate this investigation with a complete characterization of the $\gamma$-coefficients in the symmetric decompositions of $A_{M_{2\mathbb{1}}^{\mathbb{1}}}$ and $A_{M_{(2,\ldots,2,1)}^{\mathbb{1}}}$.
% Partial results in this direction exist for specific examples in the uncolored setting described in the previous subsection.                                                      

A {\em plane tree} is a rooted tree in which the children of each node are ordered (according to any ordering). 
\begin{definition}[Weakly increasing trees~\cite{weaklyincreasingtrees}]
Let $M=\{1^{m_1},2^{m_2},\cdots,n^{m_n}\}$ be a multiset.
A {\em weakly increasing tree} on $M$ is a plane tree  that: 
\begin{enumerate}[label=(\roman*)]
\item  contains $|M|+1$ nodes, labeled by the elements in $M\cup\{0\}$,
\item  its root is labeled by $0$ and each node receives a  label weakly greater than its parent, 
\item  the labels of the children of each node are weakly increasing from left to right. 
\end{enumerate}   
%Let $\mathcal{T}_{M}$ be the set of weakly increasing trees on the multiset $M$. 
\end{definition}
A leaf or an internal node (other than the root) of a weakly increasing tree is {\em old} if it is the rightmost child of its parent; otherwise, it is {\em young}.
The polynomial $A_{M_{2\mathbb{1}}^{\mathbb{1}}}$ is known to be symmetric, and hence it is its own symmetric decomposition.  
Since it is also real-rooted, it is $\gamma$-positive.  
In \cite{weaklyincreasingtrees} it was shown that its $\gamma$-coefficient $\gamma_i$ counts the number of weakly increasing trees on $M_{2\mathbb{1}}^{\mathbb{1}}$ with $i+1$ leaves and no young leaves. 
The polynomial $A_{M_{(2,\ldots,2,1)}^{\mathbb{1}}}$ is not symmetric with respect to its degree, but it is known to be bi-$\gamma$-positive \cite{weaklyincreasingtrees}. 
The $b$-polynomial in its symmetric decomposition is a scalar multiple of $A_{M_{2\mathbb{1}}^{\mathbb{1}}}$ \cite{weaklyincreasingtrees}, and hence we recover a combinatorial description of the $\gamma$-coefficients by the result above.  
For the $a$-polynomial, one can extend the methods of \cite{weaklyincreasingtrees} to prove the following.
\begin{theorem}\label{mainresult:1}
For $n\geq1$, the $a$-polynomial in the symmetric decomposition of $A_{M_{(2,\ldots,2,1)}^{\mathbb{1}}}$ satisfies 
\begin{equation} \label{m:gam}
a(x)=\sum\limits_{i=0}^{(n-1)}\r_{n,i}x^i(1+x)^{2(n-1)+1-2i},
\end{equation}
where $\r_{n,i}$ is the number of  weakly increasing trees on $\{1^2,2^2,\cdots,(n-1)^2,n,n+1\}$ with $i+1$ leaves and no young leaves.
\end{theorem}



% \section{Interpretation of $\r$-coefficients}



% %Our approach requires the use of  Chen's general bijective algorithm for trees~\cite{Chen1990} and a new decomposition for weakly increasing trees.


% Let us now turn our attention to the (uncolored) multiset Eulerian polynomials that correspond to multisets $M=\{1^{m_1},2^{m_2},\cdots,n^{m_n}\}$ and $M=\{1^{m_1},2^{m_2},\cdots,n^{m_n}, n+1\}$ for some positive integers $m_1,...,m_n$. Throughout this section we restrict to the uncolored case $\mathbf{r}=(1,...,1)$, and therefore omit the vector $\mathbf{r}$ in our notation. The (bi)-$\r$-positivity of these polynomials was discussed in \cite{multiseteulerian}. Below, we state and extend some of these results.

% Let us start by defining weakly increasing trees, which is a generalization of both plane trees and increasing trees. Note that a {\em plane tree} is a rooted tree in which the children of each node are ordered (according to any ordering). 
% \begin{definition}[Weakly increasing trees~\cite{weaklyincreasingtrees}]
% Let $M=\{1^{m_1},2^{m_2},\cdots,n^{m_n}\}$ be a multiset.
% A {\em weakly increasing tree} on $M$ is a plane tree  that: 
% \begin{enumerate}[(i)]
% \item  contains $|M|+1$ nodes, labeled by the elements in $M\cup\{0\}$,
% \item  its root is labeled by $0$ and each node receives a  label weakly greater than its parent, 
% \item  the labels of the children of each node are weakly increasing from left to right. 
% \end{enumerate}   
% %Let $\mathcal{T}_{M}$ be the set of weakly increasing trees on the multiset $M$. 
% \end{definition}
% %An example of a weakly increasing tree appears in \Cref{wit}.



% \begin{theorem}\cite{weaklyincreasingtrees}\label{thm: weaktrees}
% Let ${k}\geq1$ and denote by  $A_{\{1^k,2^k, \cdots,n^k\}}(x)$, $A_{\{1^k,2^k, \cdots,n^k, n+1\}}(x)$ the descent polynomial corresponding to the multiset $M=\{1^k,2^k, \cdots,n^k\}$, $M=\{1^k,2^k, \cdots,n^k, n+1\}$, respectively. Then $A_{\{1^k,2^k, \cdots,n^k, n+1\}}(x)$ is bi-$\r$-positive with symmetric decomposition 
% \begin{equation}\label{sym:decB}
% A_{\{1^k,2^k, \cdots,n^k, n+1\}}(x)=a_n(x)+xb_n(x),
% \end{equation} 
% where 
% \begin{align*}
%     & a_n(x)= \biggl(1+(kn-k+1)x+x(1-x)\frac{\partial}{\partial x}\biggr)A_{\{1^k,2^k, \cdots,n^k\}}(x),
%     \\ & b_n(x)=(k-1)A_{
% \{1^k,2^k, \cdots,n^k\}}(x).
% \end{align*}  Consequently,  $A_{\{1^k,2^k, \cdots,n^k, n+1\}}(x)$ is alternatingly increasing. 
% \end{theorem}

% \begin{example}
% For the first few values of $n$, fixing $k=2$ gives 
% %
% \[  A_{\{1^2,2^2, \cdots,n^2, n+1\}}(x) \ = \ \begin{cases}
%     1 + 2x, & \text{if $n=1$,} \\
%     1 + 12x + 15x^2+2x^3, & \text{if $n=2$,} \\
%     1 + 46x + 244x^2 + 272x^3+65x^4+2x^5, & \text{if $n=3$,} \\
%     1 + 152x + 2313x^2 + 8290x^3\\ + 8855x^4 +2844x^5+223x^6+2x^7, & 
%                              \text{if $n=4$.} \\
%    % 1 + 242t + 2072t^2 + 3072t^3 + 912t^4 + 32t^5, 
%                                %& \text{if $n=5$}. 
%     \end{cases} \]
% %


% %For instance, for $r=2$, writing $\widetilde{B}^+_n 
% %(t)$ and $\widetilde{B}^-_n (t)$ instead of 
% %$\widetilde{A}^+_{n,2}(t)$ and $\widetilde{A}^-_{n,2}
% %(t)$,



% The corresponding symmetric decomposition consists of
% \[  a_n (x) \ = \ \begin{cases}
%     1 + x, & \text{if $n=1$,} \\
%     1 + 11x + 11x^2+x^3, & \text{if $n=2$,} \\
%     1 + 45x + 224x^2 + 224x^3+45x^4+x^5, & \text{if $n=3$,} \\
%     1 + 151x + 2241x^2 + 7687x^3 + 7687x^4+2241x^5+151x^6+x^7, & 
%                             \text{if $n=4$,} \\
%     %1 + 211t + 1371t^2 + 1371t^3 + 211t^4 + t^5, 
%                            %& \text{if $n=5$} 
%     \end{cases}  \]
% %
% and
% %
% \[  xb_n (x) \ = \ \begin{cases}
%     x, & \text{if $n=1$,} \\
%     x + 4x^2+x^3, & \text{if $n=2$,} \\
%     x + 20x^2 + 48x^3+20x^4+x^5, & \text{if $n=3$,} \\
%     x + 72x^2 + 603x^3 + 1168x^4+603x^5+72x^6+x^7, & 
%                          \text{if $n=4$.} \\
%     %31t + 701t^2 + 1701t^3 + 701t^4 + 31t^5, 
%                          % & \text{if $n=5$}. 
%     \end{cases} \]
% %
%  Note that, in this example, $A_{\{1^2,...,n^2,n+1\}}(x)$ is  the $s$-Eulerian polynomial $E^{2p}_{2n+1}$ for $2p= (2,2,6,\ldots,2n,2(2n+1))$, and $b_n(x)=A_{\{1^2,...,n^2\}}(x)$ is  the $s$-Eulerian polynomial $E^{2s}_{2n}$ for $2s = (2,2,6,4,\ldots,2(2n-1),2n)$.
% \end{example}


% In \cite[Theorem 1.3]{weaklyincreasingtrees}, the authors provided a combinatorial interpretation for the $\r$-coefficients of the descent polynomial $A_{
% \{1^k,2^k, \cdots,n^k\}}(x)$ in terms of weakly increasing trees. This result specifies a combinatorial interpretation of the $\r$-coefficients of $b_n (x)$ in \Cref{thm: weaktrees}. We extend their methods to derive a combinatorial interpretation for the polynomial $a_n(t)$ in \Cref{thm: weaktrees}. The following definition is used.

% \begin{definition}\cite{weaklyincreasingtrees,youngleaves}
%  A leaf or an internal node (other than the root) of a weakly increasing tree is {\em old} if it is the rightmost child of its parent; otherwise, it is {\em young}. \end{definition}

% We now state the main result of this section.

% \begin{theorem}\label{mainresult:1}
% Consider a fixed integer $k\geq1$. For any $n\geq1$, the polynomial $a_n(x)$ in \Cref{thm: weaktrees} satisfies
% \begin{equation} \label{m:gam}
% a_n(x)=\sum\limits_{i=0}^{\lfloor \frac{k(n-1)}{2}\rfloor}\r_{n,i}^{(k)}x^i(1+x)^{k(n-1)+1-2i},
% \end{equation}
% where $\r_{n,i}^{(k)}$ is the number of  weakly increasing trees on $\{1^k,2^k,\cdots,(n-1)^k,n,n+1\}$ with $i+1$ leaves and no young leaves.
% \end{theorem}

% For example, when $k=2$ and $n=2$, we have
% \[
% a_2(x)=1+11x+11x^2+x^3=(x+1)^3+8x(x+1), 
% \]
% and the weakly increasing trees corresponding to the $\gamma$-coefficients $\r_{2,0}^{(2)}=1$ and $\r_{2,1}^{(2)}=8$ appear in \Cref{fig:ex2}.


% \begin{figure}
% \tiny
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% \end{tikzpicture}
% \caption{Weakly increasing trees on $\{1,1,2,3\}$ with $1$ or $2$ leaves and no young leaves. \label{fig:ex2}}
% \end{figure}

% \section{Final remarks}
% \Cref{mainresult:1} combined with \cite[Theorem 1.3]{weaklyincreasingtrees} provides a combinatorial interpretation for the $\r$-coefficients of the $\r$-positive polynomials in
% \Cref{thm: weaktrees}.
%In \cite{weaklyincreasingtrees}, the authors asked for further classes of bi-$\r$-positive polynomials. \Cref{thm: interlacing} provides such a class, prompting the following question.
% ; namely, the polynomials
% $A_{\mathbf{m}}^{\mathbf{r}}$ for vectors $(m_1,\ldots,m_n)$, $\mathbf{r} = (r_1,\ldots,r_n)$ such that $r_j\geq m_j+1$ for all $j\in [n]$ (and at least one inequality holds strictly). 
% This naturally leads to the following question.

\Cref{thm: interlacing} yields a large family of relatively unexplored combinatorial generating polynomials that are bi-$\gamma$-positive.  
A natural endeavour is to explore combinatorial interpretations of these $\gamma$-coefficients.  
Such investigations may prove fruitful in, for instance, the development of a complete understanding of the $\gamma$-coefficients for $s$-Eulerian polynomials when interpreted through the lens of \Cref{prop: s-lecture hall}.
Hence, in this section, we provide results pertaining to the following general question:
\begin{question}
    \label{prob: gamma coeffs}
    Let $(a,b)$ be the $\mathcal{I}_d$-decomposition of a bi-$\gamma$-positive colored multiset Eulerian polynomial $A_{M_{\mathbf{m}}^{\mathbf{r}}}$, where $d$ is the degree of $A_{M_{\mathbf{m}}^{\mathbf{r}}}$.  
    What is a combinatorial interpretation of the coefficients of $a$ and $b$ when expressed in the $\gamma$-basis?
\end{question}

\Cref{thm: interlacing} provides a general family of colored multiset Eulerian polynomials that are bi-$\gamma$-positive for which \Cref{prob: gamma coeffs} may be considered.
This family includes the previously studied colored Eulerian polynomials $A_{M_{\mathbf{1}}^{r\mathbf{1}}}$ for $r\geq 1$.
In \cite[Question 2.27]{athanasiadis2017gamma}, Athanasiadis considers a special case, asking for a solution to \Cref{prob: gamma coeffs} for $ A_{M_{\mathbf{1}}^{r\mathbf{1}}}$ for $r\geq 1$. 
A solution to Athanasiadis's question would extend the well-known interpretation of the $\gamma$-coefficients for the Eulerian polynomial $A_n = A_{M_{\mathbf{1}}^{\mathbf{1}}}$ due to Foata and Strehl \cite{foata1973euler} (see for instance \cite[Section~3.1]{PB}), to all colored Eulerian polynomials. 
Hence, one could more generally solve Athanasiadis's question by solving \Cref{prob: gamma coeffs} for all $A_{M_{\mathbf{m}}^{\mathbf{r}}}$ captured in \Cref{thm: interlacing}.
% In particular, it would be interesting to extend the techniques used in \cite{weaklyincreasingtrees} in the proofs of Theorem 1.3 or 1.5 (see \Cref{thm: weaktrees}) to the colored setting.


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