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\title{On $e$-positivity and $e$-unimodality of chromatic quasisymmetric functions}

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%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
\author{Soojin
Cho\thanks{\href{mailto:chosj@ajou.ac.kr}{chosj@ajou.ac.kr}. This work is
supported by the Basic Science Research Program through the National
Research Foundation of Korea (NRF) funded by the Ministry of Education
(NRF-2015R1D1A1A01057476).}\addressmark{1}
\and JiSun Huh\thanks{\href{mailto:hyunyjia@ajou.ac.kr}{hyunyjia@ajou.ac.kr}}\addressmark{1}}


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\address{\addressmark{1}Department of Mathematics, Ajou University, Suwon 16499 Republic of Korea}
%\\ \addressmark{2}Department of Mathematics, Ajou University, Suwon 16499 Republic of Korea}

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\abstract{The $e$-positivity conjecture and the $e$-unimodality conjecture of chromatic quasisymmetric functions are proved for some classes of natural unit interval orders. Recently, J. Shareshian and M. Wachs introduced chromatic \emph{quasisymmetric} functions as a refinement of Stanley's chromatic symmetric functions and conjectured the $e$-positivity and the $e$-unimodality of these functions. Our work resolves the Stanley's conjecture on chromatic symmetric functions of $(3+1)$-free posets for two classes of natural unit interval orders.}

%% put your French abstract here, or comment this out if you don't have one
\resume{La conjecture d'$e$-positivit\'e ainsi que celle d'$e$-unimodalit\'e sur les fonctions chromatiques 
quasi-sym\'etriques ont \'et\'e d\'emontr\'ees pour quelques classes d'ordres naturels sur l'ensemble des intervalles unitaires. 
R\'ecemment, J. Shareshian et M. Wachs ont introduit les fonctions chromatiques 
quasi-sym\'etriques comme un raffinement des fonctions chromatiques sym\'e\-tri\-ques, et ont conjectur\'e qu'elles sont $e$-positives et $e$-unimodales. L'$e$-positivit\'e d'une fonction chromatique quasi-sym\'etrique implique celle de la fonction chromatique sym\'etrique correspondante, et ce travail r\'esout donc la conjecture de Stanley pour les fonctions chromatiques sym\'etriques des ensembles ordonn\'es sans $(3+1)$ pour deux ordres naturels sur les intervalles unitaires.}

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\keywords{chromatic quasisymmetric function, $e$-positivity, $e$-unimodality,  $(3+1)$-free poset, natural unit interval order}

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\begin{document}
\nocite{DW,Gasha,GS,Guay,Hai,HP,SW,S1,S2,S3,SS} 

\maketitle
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\section{Introduction}

In 1995, R. Stanley \cite{S1} introduced the {\it chromatic symmetric function} $X_G(\bold{x})$ associated with any simple graph $G$, which generalizes the chromatic polynomial $\chi_G(n)$ of $G$. One of the long standing and well known conjecture due to Stanley on chromatic symmetric functions states that a chromatic symmetric function of any $(3+1)$-free poset is a linear sum of elementary symmetric function basis $\{e_\la\}$ with \emph{nonnegative} coefficients. Recently, Shareshian and Wachs \cite{SW} introduced a chromatic \emph{quasi}symmetric refinement $X_G({\bf x}, t)$ of  chromatic symmetric function $X_G({\bf x})$ for a graph $G$. They conjectured the $e$-positivity and the $e$-unimodality of the chromatic quasisymmetric functions of natural unit interval orders: That is, if $X_G({\bf x}, t)=\sum_{j=0}^m a_j({\bf x}) t^j$ for  the incomparability graph $G$ of a natural unit interval order, then $a_j({\bf x})$, $0\leq j\leq m$, is a nonnegative linear sum of $e_\la$'s, which is a refinement of the Stanley's conjecture. Moreover, $e$-unimodality conjecture states that $a_{j+1}({\bf x})-a_j({\bf x})$ is a sum of $e_\la$'s with nonnegative coefficients for $0\leq j<\frac{m-1}{2}$.

In this paper, we give combinatorial proofs of the conjectures on
$e$-positivity and $e$-unimodality of chromatic quasisymmetric functions
for certain classes of natural unit interval orders. We use the Schur
function expansion of chromatic quasisymmetric functions in terms of
Gasharov's $P$-tableaux (\cite{Gasha}), that was done by Shareshian and
Wachs in \cite{SW}. We use the Jacobi--Trudi expansion of Schur functions into
elementary symmetric functions to write chromatic quasisymmetric
functions as a sum of elementary symmetric functions with coefficients of
signed sum of positive $t$ polynomials. We then define a sign reversing
involution to cancel out negative terms and obtain only positive terms.
We use an inductive argument on $P$-tableaux to find explicit
$e$-expansion formulae of $X_G({\bf x}, t)$ and this shows the
$e$-unimodality. We remark that one class of natural unit interval orders
we consider (in Section~\ref{firstmain}) was considered by
Stanley--Stembridge and recently by Harada--Precup  and  the $e$-positivity of the corresponding chromatic symmetric functions was proved (Remark 4.4 in \cite{SS}, \cite{HP}). 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Preliminaries}\label{pre}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Chromatic quasisymmetric functions and the positivity conjecture.}
We set $\mathbb{P}=\{1,2,\dots\}$ and $[n]=\{1,2,\dots,n\}$. For $n\in \mathbb{P}$, a \emph{partition} $\la=(\la_1, \dots,\la_{\ell})$ of $n$ is a sequence of positive integers such that $\la_i\geq \la_{i+1}$ for all $i$ and $\sum_i \la_i =n$. For a partition $\la$, the \emph{conjugate} of $\la$ is the partition $\la'=(\la'_1, \dots,\la'_{\la_1})$ with $\la'_j=|\{ i\,|\,\la_i\geq j\}|$. For $n\in \mathbb{P}$, the \emph{$n^{\mbox{\tiny{th}}}$ elementary symmetric function} $e_n$ is defined as 
$e_n=\sum_{i_1<\cdots<i_n} x_{i_1}\cdots x_{i_n}\,,$ and the \emph{$\mathbb{Q}$-algebra $\Lambda_{\mathbb{Q}}$ of symmetric functions} is the subalgebra of $\mathbb{Q}[[x_1, x_2, \dots]]$ generated by the $e_n$'s;
$\Lambda_{\mathbb{Q}}=\Lambda=\mathbb{Q}[e_1, e_2, \dots]\,.$ Then
$\Lambda=\bigoplus_{n=0}^\infty \Lambda^n$, where $\Lambda^n$ is the
subspace of symmetric functions of degree $n$, and $\{e_\la \,|\,
\la\vdash n\}$ is a basis of $\Lambda^n$, where $e_\la=e_{\la_1}\cdots
e_{\la_\ell}$. The set of \emph{Schur functions} $\{s_\la\,|\,\la\vdash
n\}$ forms another  basis of $\Lambda^n$, where
$s_\la=\det[e_{\la'_i-i+j}]$ is a determinant of a $\la_1\times \la_1$
matrix, that is called \emph{Jacobi--Trudi identity}. A {\it proper coloring} of a simple graph $G=(V,E)$ is a function $\kappa : V \rightarrow \mathbb{P}$
satisfying $\kappa(u)\neq \kappa(v)$ for any $u, v\in V$ such that $\{u,v\}\in E$. For a proper coloring $\kappa$, ${\rm asc}(\kappa)=\{\{i,j\}\in E~|~i<j~\text{and}~\kappa(i)<\kappa(j)\}.$
 
\begin{defn}[\cite{SW}] 
For a simple graph $G=(V,E)$ which has a vertex set $V\subset\mathbb{P}$, the \emph{chromatic quasisymmetric function} of $G$ is a sum over all proper colorings of $G$:
\begin{center}
$\displaystyle{X_G(\bold{x},t)=\sum_{\kappa}t^{{\rm asc}( \kappa )}\bold{x}_{\kappa}.}$
\end{center}
\end{defn}

Note that chromatic quasisymmetric function $X_G(\bold{x},t)$ is a refinement of Stanley's \emph{chromatic symmetric function} $X_G(\bold{x})$ introduced in \cite{S1};
$X_G(\bold{x},t)|_{t=1}=X_G(\bold{x}).$


The {\it incomparability graph} $\text{inc}(P)$ of a poset $P$ is a graph which has as vertices the elements of $P$, with edges connecting pairs of incomparable elements.  \emph{Natural unit interval orders} are the posets we are interested in. 

\begin{defn}[\cite{SW}]
Let $\bold{m}:=(m_1, m_2, \dots, m_{n-1})$ be a list of integers satisfying $i\leq m_i \leq m_{i+1} \leq n $ for all $i$. The corresponding \emph{natural unit interval order} $P(\bold{m})$ is the poset on $[n]$ with the order relation given by $i<_{P(\bold{m})} j$ if $i<n$ and $j\in \{m_i +1, m_i +2, \dots, n\}$.
\end{defn}

Note that Catalan number $C_n=\frac{1}{n+1}\binom{2n}{n}$ counts the natural unit interval orders with $n$ elements. Shareshian and Wachs showed that if $G$ is the incomparability graph of a natural unit interval order then $X_G(\bold{x},t)$ is a polynomial with very nice properties. They also made a conjecture on the $e$-positivity and the $e$-unimodality of $X_G(\bold{x},t)$. Remember that a symmetric function $f({\bf{x}})\in\Lambda^n $ is $b$-{\it positive} if the expansion of $f(\bf{x})$ in the basis $\{b_{\la}\}$ has nonnegative coefficients when $\{b_\la\,|\,\la\vdash n\}$ is a basis of $\Lambda^n$.

\begin{thm}[Theorem 4.5 and Corollary 4.6 in \cite{SW}]\label{thm:SW}
If $G$ is the incomparability graph of a natural unit interval order then the coefficients of $t^i$ in $X_G(\bold{x},t)$ are symmetric functions and form a palindromic sequence in the sense that $X_G(\bold{x},t)=t^{|E|}X_G(\bold{x},t^{-1})$.
\end{thm}


\begin{conj}[\cite{SW}]\label{conj:SW}
If $G$ is the incomparability graph of a natural unit interval order, then $X_G(\bold{x},t)$ is $e$-positive and $e$-unimodal. That is, if $X_G(\bold{x},t)=\sum_{i=0}^{m}a_i(\bold{x})t^i$ then $a_{i}(\bold{x})$ is $e$-positive for all $i$, and $a_{i+1}(\bold{x})-a_{i}(\bold{x})$ is $e$-positive whenever $0\leq i < \frac{m-1}{2}$.
\end{conj}


A finite poset $P$ is called $(r+s)$-{\it free} if $P$ does not contain an induced subposet isomorphic to the direct sum of an $r$ element chain and an $s$ element chain. Due to the result by Guay-Paquet \cite{Guay}, Conjecture~\ref{conj:SW} specializes to the famous $e$-positivity conjecture on the chromatic symmetric functions of Stanley and Stembridge:

\begin{conj}[\cite{S1,SS}]\label{Con1} If a poset $P$ is $(3+1)$-free, then $X_{{\rm inc}(P)}(\bold{x})$ is $e$-positive. 
\end{conj}

Since $e_\la$ is $s$-positive, Conjecture~\ref{conj:SW} implies the
$s$-positivity of $X_G(\bold{x},t)$, that was proved using the notion of
$P$-tableaux by Shareshian--Wachs and Gasharov ($t=1$):

\begin{defn}[Gasharov \cite{Gasha}]
Given a poset $P$ with $n$ elements and a partition $\la$ of $n$, a \emph{$P$-tableau of shape} $\la$ is a filling $T=[a_{i,j}]$ of a Young diagram of shape $\la$ in English notation 

\begin{center}
\begin{ytableau}      
a_{1,1}&a_{1,2}&\cdots\cr
a_{2,1}&a_{2,2}&\cdots\cr
\vdots&\vdots \cr
\end{ytableau}\\
\end{center}
with all elements of $P$ so that $a_{i,j}<_Pa_{i,j+1}$ and $a_{i+1,j}\not <_P a_{i,j}$ for all $i$ and $j$.
\end{defn}

\begin{defn}[\cite{SW}] For a finite poset $P$ on a subset of $\mathbb P$ and a $P$-tableau $T$, let $G$ be the incomparability graph of $P$. An edge $\{i,j\}\in E(G)$ is a \emph{$G$-inversion of $T$} if $i<j$ and $i$ appears below $j$ in $T$. We let ${\rm inv}_G(T)$ be the number of $G$-inversions of $T$.
\end{defn}


The following theorem is the $s$-positivity result for the chromatic quasisymmetric functions of natural unit interval orders, which specializes to Gasharov's $s$-positivity result for chromatic symmetric functions of $(3+1)$-free posets.

\begin{thm}[\cite{SW},\cite{Gasha}]\label{Gasha Thm} Let $G$ be the incomparability graph of a natural unit interval order $P$. If we let $\la(T)$ be the shape of $T$, then $X_G(\bold{x},t)=\sum_{T} t^{{\rm inv}_G(T)}s_{\la(T)},$
where the sum is over all $P$-tableaux and therefore, $X_G(\bold{x},t)$ is $s$-positive.
\end{thm}

We state some related results that are useful for our arguments. 

\begin{lem}[\cite{S3}]\label{lem:basic} If  $A(t)$  and $B(t)$ are unimodal and palindromic polynomials with nonnegative coefficients and centers of symmetry $m_A$, $m_B$ respectively, then  $A(t)B(t)$ is unimodal and palindromic with nonnegative coefficients and center of symmetry $m_A+m_B$.
\end{lem}

From Lemma \ref{lem:basic}, if $G$ is the incomparability graph of a natural unit interval order and $X_G(\bold{x},t)=\sum_{i=0}^{m} a_i(\bold{x})\, t^i= \sum_{\la\vdash n} C_\la(t)\, e_\la(\bold{x})$, then
\begin{enumerate}
\item[$\bullet$] $X_G(\bold{x},t)$ is palindromic in $t$ with center of symmetry $\frac{m}{2}$ if and only if $C_\la(t)$ is a palindromic polynomial with center of symmetry $\frac{m}{2}$,
\item[$\bullet$] $X_G(\bold{x},t)$ is $e$-positive if and only if $C_\la(t)$ is a polynomial with nonnegative coefficients for all $\la$,
\item[$\bullet$] $X_G(\bold{x},t)$ is $e$-unimodal with center of symmetry $\frac{m}{2}$ if every $C_\la(t)$ is unimodal with the same center of symmetry $\frac{m}{2}$. 
\end{enumerate}

\begin{thm}[\cite{SW}]\label{en}
For $G={\rm inc}(P)$ of a natural unit interval order $P$ on $[n]$, if we let $X_G(\bold{x},t)= \sum C_\la(t)\, e_\la(\bold{x})$, then $C_{(n)}(t)=[n]_t \prod_{i=2}^{n} [b_i]_t$ and is therefore positive and unimodal with center of symmetry $\frac{|E(G)|}{2}$. Here, $[n]_t=1+t+\cdots+t^{n-1}$ and
$b_i=|\{ \{j,i\}\in E(G)~|~j<i\}|.$  
\end{thm}

We summarize known results on $e$-positivity and $e$-unimodality of $X_{{\rm inc}(P)}(\bold{x}, t)$ for natural unit interval orders $P=P(\bold{m})$ on $[n]$ with $\bold{m}=(m_1,m_2,\dots,m_{n-1})$.
\begin{enumerate}
\item For $\bold{m}=(2,3,\dots,n)$, Stanley obtained the formula for the generating function and Haiman \cite{Hai} showed that $X_{{\rm inc}(P)}(\bold{x},t)$ is $e$-positive and $e$-unimodal.
\item For $\bold{m}=(m_1, m_2,n,\dots,n)$, Shareshian and Wachs \cite{SW} obtained an $e$-positive and $e$-unimodal explicit formula of $X_{{\rm inc}(P)}(\bold{x},t)$. 
\item For a composition $\alpha=(\alpha_1,\alpha_2,\dots, \alpha_k)$ of $n+k-1$, let $m_1=\cdots=m_{\alpha_1-1}=\alpha_1$ and $m_i=\sum_{j=1}^{\ell}(\alpha_j-1)+1$ if $\sum_{j=1}^{\ell-1}(\alpha_j-1)+1 \leq i \leq \sum_{j=1}^{\ell}(\alpha_j-1)$ for $\ell\geq2$. The graph ${\rm inc}(P)$ is a {\it $K_{\alpha}$-chain}.
Gebhard and Sagan \cite{GS} proved that $X_{{\rm inc}(P)}(\bold{x})$ is $e$-positive.
\item For $\bold{m}=(2,3,\dots, m+1,n,\dots,n)$, the graph ${\rm inc}(P)$ is a {\it lollipop graph} $L_{m,n-m}$. Dahlberg and van Willigenburg \cite{DW} computed an explicit $e$-positive formula for $X_{L_{m,n-m}}(\bold{x})$. 
\item For $\bold{m}=(r, m_2, m_3 \dots,m_r, n,\dots,n)$, the graph ${\rm inc}(P)$ is a complement graph of a bipartite graph. Stanley and Stembridge \cite{SS} proved that $X_{{\rm inc}(P)}(\bold{x})$ is $e$-positive. 
\end{enumerate}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Basic setup}\label{nui}
We often use Catalan paths from $(0,0)$ to $(n,n)$ to represent $P=P(m_1, m_2, \dots, m_{n-1})$: The corresponding Catalan path of $P$ has the $i$th horizontal step on the line $y=m_i$ for all $i\leq n-1$ and the last horizontal step on the line $y=n$. Figure \ref{Fig1} shows an example. 

\begin{figure}[h]
\centering {\includegraphics[height=3cm]{Fig1} }
\caption{The corresponding Catalan path of $P=P(3,3,4,6,6)$ and $\text{inc}(P)$. 
}\label{Fig1}
\end{figure}


If a graph $G$ is a disjoint union of subgraphs $G_1, \dots, G_\ell$, then  
$X_{G}(\bold{x},t)=\prod_{i=1}^\ell X_{G_{i}}(\bold{x},t).$ Moreover, it is easy to see that $G={\rm inc}(P)$ for $P=P(m_1,\dots, m_{n-1})$ is connected if and only if $m_i>i$ for all $i$ or equivalently, $C$ meets the line $y=x$ only at $(0,0)$ and $(n,n)$.
Hence, we may restrict our attention to natural unit interval orders $P(m_1,\dots, m_{n-1})$ with $m_i\neq i$ for all $i=1,\dots,n-1$. Note that there are $C_{n-1}$ such Catalan paths of length $n$. Given a Catalan path $C$ of length $n$, define the {\it bounce path} of $C$ as follows: Starting at $(0,0)$, and travel north until you encounter the beginning of an east step. Then turn east and travel straight until you hit the main diagonal $y=x$. Then turn north until you again encounter the beginning of an east step of $C$, then turn east and travel to the diagonal, etc. Continue in this way until you arrive at $(n,n)$. The \emph{bounce number} of $C$ is the number of connected regions that the bounce path of $C$ and the line $y=x$ enclose; that is one less than the number of times the bounce path of $C$ meets the line $y=x$.

The following useful lemmas can be proved using Theorems~\ref{Gasha Thm} and \ref{thm:SW}, respectively.

\begin{lem} \label{chain}
Let $C$ be the corresponding Catalan path of a natural unit interval order $P$ and $r$ be the bounce number of $C$. If we let $X_{{\rm inc}(P)}(\bold{x},t)=\sum_{\la}B_{\la}(t)s_{\la},$ then $B_{\la}(t)=0$ for partitions $\la=(\la_1, \dots, \la_k)$ with $\la_1>r$. 
\end{lem} 

\begin{lem} \label{reflection}
Given a natural unit interval order $P$ of size $n$, let $C$ be the corresponding Catalan path of $P$ and $G={\rm inc}(P)$. If $\widetilde{C}$ is the reflection of the Catalan path $C$ about the line $y=n-x$, then $X_{G}(\bold{x},t)=X_{\widetilde{G}}(\bold{x},t),$ where $\widetilde{P}$ is the corresponding natural unit interval order of the Catalan path $\widetilde{C}$ and $\widetilde{G}={\rm inc}(\widetilde{P})$. 
\end{lem}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Two $e$-positive classes of natural unit interval orders} \label{main}

In this section we provide two classes of natural unit interval orders $P=P(\bold{m})$ satisfying that $X_{{\rm inc}(P)}(\bold{x},t)$ is $e$-positive.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{The first $e$-positive class} \label{firstmain}

We consider natural unit interval orders $P=P(r,m_2, \dots, m_{r}, n, \dots,n)$ on $[n]$ for  $r<n$. Let $G={\rm inc}(P)$. Figure \ref{Fig3} shows a corresponding Catalan path (solid line) of $P$. The dotted line indicates the bounce path, and the bounce number of $P$'s we consider is \emph{two}.

\begin{figure}[h]
\centering {\includegraphics[height=3.4cm]{Fig3} }
\caption{Catalan path, bounce path, and $G={\rm inc}(P)$ of $P(4,4,5,6,9,9,9,9)$}\label{Fig3}
\end{figure}

If we let $X_{G}(\bold{x},t)=\sum_{\la \vdash n}B_{\la}(t)s_{\la},$ then by Lemma \ref{chain}, $B_{\la}(t)\neq 0$ only for $\la$'s whose conjugate has only two parts. By Theorem \ref{Gasha Thm}, $B_{\la}(t)=\sum_{T\in\mathcal{T}_{P,\la}}t^{\rm{inv}_{G}(T)}$, where $\mathcal{T}_{P,\la}$ is the set of $P$-tableaux of shape $\la$. Let $k$ be the largest positive integer satisfying $B_{2^k1^{n-2k}}(t)\neq 0$.\\ 
Due to Jacobi-Trudi identity, we have
$s_{1^n}=e_n$ and $s_{2^{\ell}1^{n-2\ell}}=e_{(n-\ell,\ell)}-e_{(n-\ell+1,\ell-1)}$ for $\ell \geq 2$.
Hence, we can derive the $e$-basis expansion of $X_{G}(\bold{x},t)$ as follows.
\begin{eqnarray*}
X_{G}(\bold{x},t)&=&B_{1^n}(t)s_{1^n}+\sum_{\ell=1}^{k}B_{2^{\ell}1^{n-2\ell}}(t)s_{2^{\ell}1^{n-2\ell}}\\
 &=& \sum_{\ell=0}^{k-1}(B_{2^\ell1^{n-2\ell}}(t)-B_{2^{\ell+1}1^{n-2\ell-2}}(t))e_{(n-\ell,\ell)}+B_{2^k1^{n-2k}}(t)e_{(n-k,k)}
\end{eqnarray*}

By Lemma \ref{lem:basic}, if we show that $B_{2^{\ell}1^{n-2\ell}}(t)-B_{2^{\ell+1}1^{n-2\ell-2}}(t)$ is a polynomial in $t$ with nonnegative coefficients for each $\ell=0,1,\dots,k-1$, then $X_{G}(\bold{x},t)$ is $e$-positive.

For a fixed $\ell\in\{0,1,\dots,k-1\}$, let $\mathcal{T'}_{P,2^{\ell}1^{n-2\ell}}$ be the subset of $\mathcal{T}_{P,2^{\ell}1^{n-2\ell}}$, each of whose elements $T=[a_{i,j}]$ has some $s \geq \ell+2$ satisfying $a_{i,1}<_Pa_{s,1}$ for all $\ell+1 \leq i \leq s-1$. 

Now we define a map $\psi_{\ell}:\mathcal{T}_{P,2^{\ell+1}1^{n-2\ell-2}} \rightarrow \mathcal{T'}_{P,2^{\ell}1^{n-2\ell}}$. Given $T=[b_{i,j}] \in \mathcal{T}_{P,2^{\ell+1}1^{n-2\ell-2}}$, we let $s$ be the smallest $i>\ell+1$ such that $b_{i,1}\not<_Pb_{\ell+1,2}$; otherwise $s=n-\ell$, then let $\psi_{\ell}(T)$ be the tableau obtained by inserting $b_{\ell+1,2}$ right below $b_{s-1,1}$. Then $\psi_{\ell}(T)$ is a $P$-tableau in $\mathcal{T'}_{P,2^{\ell}1^{n-2\ell}}$. We can show that $\psi_{\ell}$ is a weight preserving bijection:\\
\noindent We now turn to the construction of the inverse map of $\psi_{\ell}$.
Given $T=[a_{i,j}]\in\mathcal{T'}_{P,2^{\ell}1^{n-2\ell}}$, if we move $a_{s,1}$ to the right side of ${a_{\ell+1,1}}$, then we have the $P$-tableau $\psi_{\ell}^{-1}(T)$ in $\mathcal{T'}_{P,2^{\ell+1}1^{n-2\ell-2}}$.

Let $\widebar{\mathcal{T'}}_{P,2^{\ell}1^{n-2\ell}}$ be the set $\mathcal{T}_{P,2^{\ell}1^{n-2\ell}}-\mathcal{T'}_{P,2^{\ell}1^{n-2\ell}}$. Then
\begin{center}
$\displaystyle{B_{2^{\ell}1^{n-2\ell}}(t)-B_{2^{\ell+1}1^{n-2\ell-2}}(t)=\sum_{T\in\widebar{\mathcal{T'}}_{P,2^{\ell}1^{n-2\ell}}}t^{{\rm inv}_{G}(T)},}$
\end{center}
and this gives a combinatorial interpretation of the coefficient of $e_{(n-\ell,\ell)}$ in $X_{G}(\bold{x},t)$. 

We can now give our main result.

\begin{thm}\label{mainthm} Let $P=P(r,m_2,\dots, m_{n-1})$ be a natural unit interval order and let $G$ be the incomparability graph of $P$. If $2\leq r \leq n-1$ and $m_{r+1}=n$, then
\begin{center}
 $\displaystyle{X_{G}(\bold{x},t)=\sum_{\ell=0}^{k}C_{(n-\ell,\ell)}(t)e_{(n-\ell,\ell)},}$ 
 \end{center}
where $k$ is a positive integer and 
\begin{center}
 $\displaystyle{C_{(n-\ell,\ell)}(t)=\sum_{T\in\widebar{\mathcal{T'}}_{P,2^{\ell}1^{n-2\ell}}}t^{{\rm inv}_G(T)}.}$
 \end{center}
Here, $\widebar{\mathcal{T'}}_{P,2^{\ell}1^{n-2\ell}}$ is the set of elements $T=[a_{i,j}]\in\mathcal{T}_{P,2^{\ell}1^{n-2\ell}}$ having no $s\geq\ell+2$ such that $a_{i,1}<_P a_{s,1}$ for all $i\in \{\ell+1, \ell+2, \dots, s-1\}$. Consequently $X_{G}(\bold{x},t)$ is $e$-positive.
\end{thm}


\begin{example} Let $P=P(2,3,4)$ be a natural unit interval order. There are $14$ $P$-tableaux:
\begin{center}
$T_1$=\begin{ytableau}      
1&3\cr
2&4\cr
\end{ytableau} 
$~T_2$=\begin{ytableau}      
2&4\cr
1&3\cr
\end{ytableau} 
$~T_3$=\begin{ytableau}      
1&4\cr
2\cr
3\cr
\end{ytableau} 
$~T_4$=\begin{ytableau}      
1&4\cr
3\cr
2\cr
\end{ytableau} 
$~T_5$=\begin{ytableau}      
1&3\cr
2\cr
4\cr
\end{ytableau}
$~T_6$=\begin{ytableau}      
2&4\cr
1\cr
3\cr
\end{ytableau} \\[2ex]

$T_7$=\begin{ytableau}      
1\cr
2\cr
3\cr
4\cr
\end{ytableau}
$\quad T_8$=\begin{ytableau}      
2\cr
1\cr
3\cr
4\cr
\end{ytableau}
$\quad T_9$=\begin{ytableau}      
3\cr
2\cr
1\cr
4\cr
\end{ytableau}
$\quad T_{10}$=\begin{ytableau}      
4\cr
3\cr
2\cr
1\cr
\end{ytableau}
$\quad T_{11}$=\begin{ytableau}      
1\cr
2\cr
4\cr
3\cr
\end{ytableau}
$\quad T_{12}$=\begin{ytableau}      
1\cr
4\cr
3\cr
2\cr
\end{ytableau}
$\quad T_{13}$=\begin{ytableau}      
1\cr
3\cr
2\cr
4\cr
\end{ytableau}
$\quad T_{14}$=\begin{ytableau}      
2\cr
1\cr
4\cr
3\cr
\end{ytableau}\\[2ex]
\end{center}
By definition, $\mathcal{T'}_{P,1^{4}}=\{T_{11}=\psi_{0}(T_3),T_{12}=\psi_{0}(T_4),T_{13}=\psi_{0}(T_5),T_{14}=\psi_{0}(T_6)\}$ and  $\mathcal{T'}_{P,21^{2}}=\{T_5=\psi_{1}(T_1),T_6=\psi_{1}(T_2)\}$. Therefore, $\widebar{\mathcal{T'}}_{P,1^{4}}=\{T_7,T_8,T_9,T_{10}\}$,
$\widebar{\mathcal{T'}}_{P,21^{2}}=\{T_3,T_4\}$, and 
$\widebar{\mathcal{T'}}_{P,2^{2}}=\{T_1,T_2\}$,  by Theorem \ref{mainthm},
\begin{eqnarray*}
X_{G}(\bold{x},t)&=&\left( \sum_{i=1}^{2}t^{{\rm inv}_G(T_i)} \right)e_{(2,2)}+\left( \sum_{i=3}^{4}t^{{\rm inv}_G(T_i)} \right)e_{(3,1)}+\left( \sum_{i=7}^{10}t^{{\rm inv}_G(T_i)} \right)e_{4}\\
&=&(t+t^2)e_{(2,2)}+(t+t^2)e_{(3,1)}+(1+t+t^2+t^3)e_{4}.
\end{eqnarray*}
\end{example}

The next two corollaries follow easily from Theorem \ref{mainthm}. 
Note that Corollary \ref{likecor} is a quasisymmetric version of Remark 4.4 in \cite{SS}.

\begin{cor} \label{likecor}
For $G={\rm inc}(P)$ of a natural unit interval order $P$ on $[n]$, if induced subgraphs of $G$ on $\{1,\dots,r\}$ and $\{r+1, \dots,n\}$ are both complete graphs for some $r$, then $X_{G}(\bold{x},t)$ is $e$-positive. 
\end{cor}

\begin{cor}\label{aa}
Let $P_{n,r}=P(r,r+1,r+2,\dots,n)$ be a natural unit interval order on $[n]$ and let $G_{n,r}$ be the incomparability graph of $P_{n,r}$. If $r \geq \lceil \frac{n}{2} \rceil$, then $X_{G_{n,r}}(\bold{x},t)$ is $e$-positive.
\end{cor}

\begin{rem} Corollary~\ref{aa} is proved for $r=2, n-2, n-1$ in \cite{SW}.

\end{rem}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{The second $e$-positive class} \label{secondmain}

In this section, we consider the natural unit interval orders $P=P(r,n-1,m_3,\dots,m_{n-2},n)$ for some $r>1$. If $m_{r+1}=n$, then $X_{{\rm inc}(P)}(\bold{x},t)$ is $e$-positive by Theorem \ref{mainthm}. Hence, we consider the case when $m_{r+1}\neq n$. Figure \ref{Fig4} shows a corresponding Catalan path (solid line) of $P$ with the bounce path (dotted line). In this case, the bounce number is $3$ for $P$.


\begin{figure}[h]
\centering {\includegraphics[height=3.4cm]{Fig4} }
\caption{ Catalan path, bounce path, and $G={\rm inc}(P)$ of $P(4,8,8,8,8,8,9,9)$}\label{Fig4}
\end{figure}

\noindent Using the similar method in the Section \ref{firstmain}, we have the following theorem.

\begin{thm} \label{secondthm}
Let $G$ be the incomparability graph of a natural unit interval order $P=P(\bold{m})$. If $\bold{m}=(r,m_2,\dots,m_{n-1})$ with $m_2=\cdots=m_{s}=n-1$ and $m_{s+1}$=n for $2\leq r<s \leq n-2$, then
\begin{center}
$\displaystyle{X_{G}(\bold{x},t)=\sum_{\la \in A} C_{\la}(t)e_{\la},}$
\end{center}
for $A=\{(n-2,1,1), (n-2,2), (n-1,1), (n)\}$. If we let 
\begin{eqnarray*}
G_1&=&\{[b_{i,j}]\in \mathcal{T}_{P,2^21^{n-4}}~|~ b_{1,1}=1,~b_{1,2}<_Pb_{2,2}\},\\ G_2&=&\{[b_{i,j}]\in \mathcal{T}_{P,2^21^{n-4}}~|~b_{1,1}=1,~b_{1,1}<_Pb_{2,1},~b_{1,2}\not<_Pb_{2,2} \},\\
F_1&=&\{[b_{i,j}]\in \widebar{\mathcal{T'}}_{P,2^11^{n-2}}~|~ b_{1,1}<_P b_{2,1} <_P b_{1,2} \},\\
F_2&=&\{[b_{i,j}]\in \widebar{\mathcal{T'}}_{P,2^11^{n-2}}~|~ b_{1,1}<_P b_{1,2} <_P b_{2,1} \},\\
F_3&=&\{[b_{i,j}]\in \mathcal{T'}_{P,2^11^{n-2}}~|~b_{1,1}=1,~1\not<_Pb_{2,1}~\}\\
&~&\qquad\qquad\qquad\qquad\qquad\qquad\cup \{[b_{i,j}]\in \mathcal{T'}_{P,2^11^{n-2}}~|~b_{1,1}=1,~ b_{1,2}<_Pn~\},\\
F_4&=&\{[b_{i,j}]\in \mathcal{T'}_{P,2^11^{n-2}}~|~b_{2,1}=1,~b_{2,1}<_P b_{3,1}\not<_P b_{1,2}\},
\end{eqnarray*}  
then, the coefficients of the $e$-basis expansion of $X_{G}(\bold{x},t)$ are
\begin{center}
$\displaystyle{C_{\la}(t)=\sum_{T\in U_{\la}}t^{{\rm inv}_G(T)},}$
\end{center}
where 
\begin{eqnarray*}
U_{(n-2,1,1)}&=& \mathcal{T}_{P,3^11^{n-3}},\\
U_{(n-2,2)}&=&\mathcal{T}_{P,2^21^{n-4}}-G_1-G_2,\\
U_{(n-1,1)}&=&\mathcal{T}_{P,2^11^{n-2}}-F_1-F_2-F_3-F_4,\\
U_{(n)}&=& \widebar{\mathcal{T'}}_{P,1^{n}}.
\end{eqnarray*}
Consequently $X_{G}(\bold{x},t)$ is $e$-positive.
\end{thm}

\begin{cor}
Let $G$ be the incomparability graph of a natural unit interval order on $[n]$. If $G$ has the complete graph on $\{2,3, \dots,n-1\}$ as a subgraph, then $X_{G}(\bold{x},t)$ is $e$-positive. 
\end{cor}

Our proof for the $e$-positivity is to find sign reversing involutions on the signed sets of $P$-tableaux. Our hope is to extend it to cover all natural unit interval orders.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Some explicit formulae} \label{last}

In this section we give $e$-unimodal explicit formulae for some natural unit interval orders in the first $e$-positive class in Section \ref{firstmain}. By Lemma \ref{lem:basic}, if we provide explicit formulae for all coefficients in the $e$-basis expansions of $X_{{\rm inc}(P)}(\bold{x},t)$ as sums of polynomials in $t$ which are unimodal with center of symmetry $\frac{|E({\rm inc}(P))|}{2}$, then $X_{{\rm inc}(P)}(\bold{x},t)$ is $e$-unimodal.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{The first explicit formula} \label{firstform}

In this section we consider natural unit interval orders $P=P(m_1,m_2 \dots, m_{n-1})$ with $m_1=\cdots=m_{s}=r$ and $m_{s+1}=\cdots=m_{n-1}=n$ for any positive integers $s\leq r\leq n-1$. Figure \ref{Fig5} shows a corresponding Catalan path of $P$.

\begin{figure}[h]
\centering {\includegraphics[height=3.2cm]{Fig5} }
\caption{A corresponding Catalan path of $P(r, \dots, r, n, \dots, n)$}\label{Fig5}
\end{figure}

\begin{lem}[Stanley \cite{S2}]\label{lem} 
Let $M=\{1^{c_1}, 2^{c_2}, \dots, n^{c_n}\}$ be a multiset of cardinality $n=\sum_{i=1}^{n}c_i$ and $S(M)$ be the set of permutations on $M$. An {\rm inversion} of $\pi=\pi_1\cdots \pi_n\in S(M)$ is a pair $(\pi_i, \pi_j)$ with $i<j$ and $\pi_i>\pi_j$. If we let $i(\pi)$ be the number of inversions of $\pi$, then 
\begin{center}
$\displaystyle{\sum_{\pi \in S(M)} t^{i(\pi)}=\frac{[n]_t!}{[c_1]_t! \cdots [c_n]_t!}.}$
\end{center}
\end{lem}


By using Theorem \ref{en}, Theorem \ref{mainthm}, and Lemma \ref{lem}, we have an explicit formula $X_{G}(\bold{x},t)$ in the $e$-basis expansion.

\begin{prop} \label{form1}
Let $G={\rm inc}(P)$ of a natural unit interval order $P=P(m_1,m_2 \dots, m_{n-1})$. For $r,~s$ satisfying $1\leq s\leq r \leq n-1$, if $m_1=\cdots=m_s=r$ and $m_{s+1}=n$, then 
\begin{center}
$\displaystyle{X_{G}(\bold{x},t)=\sum_{\ell=0}^{\min\{n-r,s\}}t^{(r-s)\ell} \frac{[n-r]_t! [s]_t! [r-\ell-1]_t! [n-s-\ell-1]_t! [n-2\ell]_t}{[n-r-\ell]_t! [s-\ell]_t! [r-s-1]_t!} e_{(n-\ell,\ell)}.}$
\end{center}
Consequently, $X_{G}(\bold{x},t)$ is $e$-positive and $e$-unimodal with center of symmetry $\frac{\binom{n}{2}-(n-r)s}{2}$.
\end{prop} 

\begin{proof} Let $A_1=\{1,\dots, s \}$ and $A_2=\{r+1, \dots, n\}$. For any $B_1\subset A_1$ and $B_2 \subset A_2$ with $|B_1|=|B_2|=\ell$, let $\widebar{\mathcal{T'}}_P(B_1,B_2)$ be the set of $T=[a_{i,j}]$ in $\widebar{\mathcal{T'}}_{P,2^{\ell}1^{n-2\ell}}$ satisfying 
$\{a_{i,1}~|~1\leq i \leq \ell \}=B_1$ and $\{a_{i,2}~|~1\leq i \leq \ell \}=B_2$, then by Theorem \ref{mainthm},
\begin{center}
$\displaystyle{C_{(n-\ell,\ell)}(t)=\sum_{\substack{B_1\subset A_1\\B_2\subset A_2}}\sum_{T\in \widebar{\mathcal{T'}}_P(B_1,B_2)}t^{{\rm inv}_G(T)}.}$
\end{center}
If we let $S(A)$ be the set of permutations for some set $A$, then $C_{(n-\ell,\ell)}(t)$ is equal to
\begin{center}
$\displaystyle{t^{(r-s)\ell} \left(\sum_{\pi \in S(B_1)}t^{i(\pi)}\right)^2 \left( \sum_{ T^b\in \widebar{\mathcal{T'}}_{R_{\ell},1^{n-2\ell} }}t^{{\rm inv}_{{\rm inc}(R_{\ell})}(T^b)} \right) \left(\sum_{\sigma_1\in S(M_1)}t^{i(\sigma_1)} \right) \left(\sum_{\sigma_2\in S(M_2)} t^{i(\sigma_2)} \right),}$
\end{center}
where  $i(\pi)$ is the number of inversions of $\pi$, $R_{\ell}=P(m_1,\dots, m_{n-2\ell-1})$ with $m_1=\dots =m_{s-\ell}=r-\ell$ and $m_{s-\ell+1}=n$, and $M_1=\{1^{s-\ell}2^{\ell}\}$, $M_2=\{1^{n-r-\ell}2^{\ell}\}$ are multisets.\\
By using Theorem \ref{en} and Lemma \ref{lem}, we can have $X_{G}(\bold{x},t)$ in the $e$-basis expansion.
\end{proof}


The next two corollaries follow easily from Proposition \ref{form1}.

\begin{cor} \label{oneline}
For $r\geq 2$, let $P=P(r,n \dots, n)$ be a natural unit interval order of size $n$  then 
$$X_{{\rm inc}(P)}(\bold{x},t)=[n-2]_t!([n]_t [r-1]_t e_n + t^{r-1}[n-r]_t e_{(n-1,1)}).$$
\end{cor}

\begin{cor}
Let $P=P(m_1,m_2 \dots, m_{n-1})$ be a natural unit interval order and $G$ be the incomparability graph of $P$. If $m_1=\cdots=m_{r-1}=r$ and $m_{r}=n$ for $r\leq n-1$, then 
$$X_{G}(\bold{x},t)=\sum_{\ell=0}^{\min\{n-r,r-1\}}t^{\ell}[n-r]_t! [r-1]_t! [n-2\ell]_t e_{(n-\ell,\ell)}.$$
\end{cor}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{The second explicit formula} \label{secondform}

In this section we consider natural unit interval orders $P=P(m_1,m_2 \dots, m_{n-1})$ with $m_1=r$, $m_2=\cdots=m_{s}=n-1$, and $m_{s+1}=\cdots=m_{n-1}=n$ for some $0<s\leq r\leq n-2$. Figure \ref{Fig6} shows a corresponding Catalan path of $P$ with the bounce path in dotted line.

\begin{figure}[h]
\centering {\includegraphics[height=3.2cm]{Fig6} }
\caption{A corresponding Catalan path of $P(r, n-1,\dots, n-1, n, \dots, n)$}\label{Fig6}
\end{figure}

\begin{prop} \label{form2}
Let $P=P(m_1,m_2 \dots, m_{n-1})$ be a natural unit interval order and $G={\rm inc}(P)$. If $m_1=r$, $m_2=\cdots=m_s=n-1$, and $m_{s+1}=n$ for some $r,s$ with $2 \leq s\leq r \leq n-2$, then 
$$X_{G}(\bold{x},t)=[n-4]_t!(\widebar{C}_{(n)}(t) e_{n}+\widebar{C}_{(n-1,1)}(t) e_{(n-1,1)}+\widebar{C}_{(n-2,2)}(t) e_{(n-2,2)}),$$
where

\begin{eqnarray*}
\widebar{C}_{(n)}(t)&=&[n]_t [n-3]_t [r-1]_t [n-s-1]_t\\
\widebar{C}_{(n-2,2)}(t)&=&t^{n+r-s-3} [2]_t [n-r-1]_t [s-1]_t\\
\widebar{C}_{(n-1,1)}(t)&=& \frac{1}{2} [n-2]_t \{t^{r-1}(1+t^{n-r-s})[n-3]_t+t^{n-s-1} [2]_t [r-2]_t [s-1]_t\\
&~&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad +t^{r-1} [2]_t [n-r-1]_t [n-s-2]_t \} .
\end{eqnarray*}

Moreover, $X_{G}(\bold{x},t)$ is $e$-positive and $e$-unimodal with center of the symmetry $\frac{\binom{n}{2}-(n-r+s-1)}{2}$.
\end{prop} 

\begin{proof}
By Theorem \ref{en}, we already know the formula of $C_{(n)}(t)$ which is $e$-unimodal with center of symmetry $\frac{|E(G)|}{2}=\frac{\binom{n}{2}-(n-r+s-1)}{2}$.\\
Let $\widebar{\mathcal{T'}}_{P,(i,j)}$ be the set of all $P$-tableaux of $\widebar{\mathcal{T'}}_{P,2^{1}1^{n-2}}\cup \widebar{\mathcal{T'}}_{P,2^{2}1^{n-4}}$ such that the first row is occupied by $i$ and $j$. If $G-\{i,j\}$ is the induced subgraph of $G$ with vertex set $V(G)-\{i,j\}$, and $X_{G-\{i,j\}}(\bold{x},t)=D_{(n-2)}(t)e_{(n-2)}+D_{(n-3,1)}(t)e_{(n-3,1)}$, then by Theorem \ref{mainthm},  
$$\sum_{T\in \widebar{\mathcal{T'}}_{P,(i,j)}} t^{{\rm inv}_{G}(T)}=t^{b_i+b_j}(D_{(n-2)}(t)e_{(n-1,1)}+D_{(n-3,1)}(t)e_{(n-2,2)}),$$
where $b_k=|\{ \{j,k\}\in E(G)~|~j<k\}|$. By Corollary \ref{oneline}, we can evaluate $\sum_{T\in \widebar{\mathcal{T'}}_{P,(i,j)}} t^{{\rm inv}_{G}(T)}$ and then have formulae of $C_{(n-2,2)}(t)$ and $C_{(n-1,1)}(t)$. By using Lemma \ref{reflection} and Theorem \ref{thm:SW}, we can show that $C_{(n-1,1)}(t)$ is $e$-unimodal with center of symmetry $\frac{|E(G)|}{2}$.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\acknowledgements{The authors would like to thank Stephanie van Willigenburg for helpful conversations.}

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\end{document}
