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\newcommand\NN{\mathbb{N}}
\newcommand\QQ{\mathbb{Q}}
\newcommand{\CC}{\mathbb{C}}
\newcommand{\ZZ}{\mathbb{Z}}
\newcommand{\RR}{\mathbb{R}}
\newcommand\LL{\mathcal{L}}
\newcommand\LLT{\operatorname{LLT}}
\newcommand\area{\operatorname{area}}
\newcommand\PTab{\mathcal{T}}
\newcommand\att{\operatorname{att}}
\newcommand\SSS{\operatorname{S}}
\newcommand\SYT{\operatorname{SYT}}
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\newcommand\asc{\operatorname{asc}}
\newcommand\inv{\operatorname{inv}}
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\newcommand\I{\operatorname{I}}
\newcommand\comaj{\operatorname{comaj}}
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\newcommand\AAA{\mathcal{A}}
\newcommand\BBB{\mathcal{B}}
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\newcommand\Dyck{\operatorname{Dyck}}
\newcommand\Hess{\operatorname{Hess}}
\renewcommand\vec[1]{\mathbf{#1}}
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%% define your title in the usual way
\title[\texorpdfstring{$\alpha$}{alpha}-chromatic symmetric functions]{\texorpdfstring{$\alpha$}{alpha}-chromatic symmetric functions}

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\author[Jim Haglund, Jaeseong Oh and Meesue Yoo]{\texorpdfstring{Jim Haglund\thanks{\href{mailto:jhaglund@math.upenn.edu}{jhaglund@math.upenn.edu}}\addressmark{1}, 
Jaeseong Oh\thanks{\href{mailto:jsoh@kias.re.kr}{jsoh@kias.re.kr} Jaeseong Oh was supported by Korea Institute for Advanced Study (HP083401).}\addressmark{2}, 
\and Meesue Yoo\thanks{\href{mailto:meesueyoo@chungbuk.ac.kr}{meesueyoo@chungbuk.ac.kr} Meesue Yoo was supported by NRF grant RS-2024-00344076.}\addressmark{3} }{Jim Haglund, 
Jaeseong Oh, 
and Meesue Yoo }}


%% then use \addressmark to match authors to institutions here
\address{
\addressmark{1}Department of Mathematics, University of Pennsylvania, Philadelphia, PA 19104, USA \\ 
\addressmark{2}June E Huh Center for Mathematical Challenges, Korea Institute for Advanced Study, Seoul 02455, South Korea \\
\addressmark{3}Department of Mathematics, Chungbuk National University, Cheongju 28644,
South Korea
}

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\received{\today}

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\abstract{
In this extended abstract, we introduce the \emph{$\alpha$-chromatic symmetric functions} $\chi^{(\alpha)}_\pi[X;q]$, extending Shareshian and Wachs' chromatic symmetric functions with an additional real parameter $\alpha$. We present positive combinatorial formulas with explicit interpretations. Notably, we show an explicit monomial expansion in terms of the $\alpha$-binomial basis and an expansion into certain chromatic symmetric functions in terms of the $\alpha$-falling factorial basis. 
We also exhibit various Schur positivity results.
}

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\keywords{$\alpha$-chromatic symmetric functions, Schur positivity, $\alpha$-binomial coefficients, rook polynomials, hit polynomials}

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\begin{document}

\maketitle
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\section{Introduction}

In his seminal paper \cite{Sta95}, Stanley introduced the \emph{chromatic symmetric function} $\chi_G[X]$ as a symmetric function generalization of the chromatic polynomial for graphs. Subsequently, Shareshian and Wachs \cite{ShWa16} refined this by defining the \emph{chromatic (quasi)symmetric function} $\chi_G[X;q]$ with an additional parameter $q$. Notably, Shareshian and Wachs showed that $\chi_{G(\pi)}[X;q]$ is symmetric if the graphs $G(\pi)$ are associated with Dyck paths $\pi$. 

In this extended abstract, we consider even further generalization of Shareshian and Wachs' chromatic symmetric functions. 
We introduce a new parameter $\alpha$ to define the \emph{$\alpha$-chromatic symmetric function} $\chi^{(\alpha)}_\pi[X;q]$ for a Dyck path $\pi$, for $\alpha \in \mathbb R$, as 
\begin{equation*}
    \chi^{(\alpha)}_\pi[X;q] := 
\chi_\pi\left[\frac{q^\alpha-1}{q-1}X;q\right].
\end{equation*}
Notably, for a Dyck path $\pi$ of size $n$ (referred to as an $n$-Dyck path from now on), the coefficients of the $\alpha$-chromatic symmetric function $\chi^{(\alpha)}_\pi[X;q]$ lie within the $\mathbb{C}(q)$-span of $\{[\alpha]_q, [\alpha]_q^2, \dots, [\alpha]_q^n\}$, where $[\alpha]_q := \dfrac{q^\alpha-1}{q-1}$. We can express each coefficient in terms of two alternative bases:
\begin{equation}\label{eq: two bases}
    \left\{\scriptstyle{\begin{bmatrix}\alpha+k\\n\end{bmatrix}_q}\right\}_{k=0,1,\dots,n-1}\qquad\text{and}\qquad \left\{[\alpha]_q^{\underline{k}}\right\}_{k=1,2,\dots,n}.
\end{equation}
Here, \(
  [\alpha]_q^{\underline{k}} := [\alpha]_q[\alpha-1]_q\cdots[\alpha-k+1]_q
\)
is the $q$-falling factorial \and $\scriptstyle{\begin{bmatrix}\alpha+k\\n\end{bmatrix}_q} = \dfrac{[\alpha+k]_q^{\underline{n}}}{[n]_q!}$ denotes the $q$-binomial coefficient.

We want to remark that the expansions of \(  \chi^{(\alpha)}_\pi[X;q]\) in terms of \( \alpha \)-binomial bases have
close connections to the original chromatic symmetric functions and the Schur expansion of the unicellular LLT
polynomials, namely, the coefficients of the \( \alpha \)-chromatic symmetric functions expressed in terms of the \( q \)-falling factorial basis serve as an interpolation between the chromatic symmetric functions and the unicellular LLT polynomials. For more details, refer to \cite{HOY}.  

This extended abstract presents various positive combinatorial formulas for the $\alpha$-chromatic symmetric functions in these bases or their specialization at $q=1$. 



\section{Backgrounds}

\subsection{Combinatorics}
Throughout this paper, we set $[n]=\{ 1,2,\dots, n\}$.
We shall mainly follow the notations and definitions of \cite{Mac1995, Stanley2} for symmetric functions.
For a partition $\lambda$ of $n$, we let  $m_\lambda[X], ~e_\lambda[X],~h_\lambda[X],~p_\lambda[X],~s_\lambda[X]$ denote 
the \emph{monomial}, \emph{elementary}, \emph{complete homogeneous}, \emph{power sum} and \emph{Schur symmetric functions}, respectively. 
Let \( \Lambda_F \) denote the \( F \)-algebra of symmetric functions with coefficients in \( F=\mathbb{Q}(q,t) \).
The elements in \( \Lambda_F \) may be viewed as formal power series over \( \mathbb{Q}(q,t) \) in infinitely many
variables \( X=(x_1, x_2,\dots) \) with finite degrees that are invariant under permutations of variables.
In this extended abstract, we often replace \( t\) by \( q^\alpha \), for some \( \alpha \).

We briefly review plethystic notation. 
Let \( E=E(t_1, t_2, \dots) \) be a formal Laurent series with rational coefficients in \( t_1, t_2,\dots \). The
\emph{plethystic substitution} \( p_k [E] \) is defined by replacing each \( t_i \) in \( E \) by \( t_i ^k \), i.e.,
\( p_k[E] = E\left(t_1 ^k, t_2 ^k, \dots\right)\) and
$\displaystyle p_\lambda [E] = \prod_{i=1}^{\ell(\lambda)} p_{\lambda_i}[E]$. 
 The plethystic substitution has the following properties: 
(1).  If \( X=x_1+x_2+\cdots \), then
$\displaystyle p_k[X]=p_k (x_1,x_2,\dots)$; 
(2). For a real parameter $z$, $p_k [zX]=z^k p_k[X]$; 
(3). $p_k[(1-t)X]=\sum_{i}x_i ^k (1-t^k)=(1-t^k)p_k[X]$; 
(4). $\displaystyle p_k \left[ \frac{X}{1-q}\right]=\sum_{i}\frac{x_i ^k}{1-xq^k}=\frac{1}{1-q^k}p_k[X]$.

Then for an arbitrary symmetric function \( f \), we define \( f[E] \) by \( \sum_{\lambda}c_\lambda p_\lambda [E] \)
if \( f=\sum_{\lambda}c_\lambda p_\lambda \). Due to the above property (1), it easily follows that for any \( f\in \Lambda_F \), \( f[X]=f(x_1,x_2,\dots) \). For this reason, it is a common convention
in plethystic expression that \( X \) stands for \( x_1+x_2+\cdots \). In a similar vein, for $X=x_1+x_2+\cdots, Y=y_1+y_2+\cdots$, we have $f[X+Y]=f(x_1,x_2,\dots,y_1,y_2,\dots)$ and $f[XY]=f(x_1y_1,x_1y_2,\dots,x_2y_1,x_2y_2,\dots,)$.
Refer to  \cite{Hag2008, Mac1995} for a more comprehensive account. 


Given a Dyck path $\pi$, there is a natural way to associate a Hessenberg function $h(\pi)$, a natural unit interval order $P(\pi)$, and a graph $G(\pi)$ for a Dyck path $\pi$. We outline those correspondences. 

 A \emph{Hessenberg function} is a nondecreasing function $h:[ n] \rightarrow  [n]$ such that $h(i)\ge i$ for all $1\le i \le n$. For a Dyck path $\pi$, if we let $h(\pi)$ be a function whose value at $i$ is the $i$-th column height of $\pi$, then $h(\pi)$ is a Hessenberg function. Therefore, a Dyck path determines a Hessenberg function and vice versa.

The \emph{natural unit interval order} is a poset arising from an arrangement of unit intervals. There is a well-known correspondence between Dyck paths and natural unit interval orders and we will take it as a definition of natural unit interval order.  For a Dyck path $\pi$, $P=P(\pi)$ is a poset on $[n]$ whose order relation is defined by \( i<_{P(\pi)} j \) if $i<n$ and $h(\pi)_i +1\le j\le n$.
In this case, we also denote the order relation by $i<_\pi j$ for \( i<_{P({\bf m})} j \). In addition, we say $(i,j)$ is an \emph{attacking pair} of $\pi$ if $i\nless_\pi j$. 

Given a poset $P$, the \emph{incomparability graph}  $\inc(P)$ is a graph whose vertex set is the elements of $P$
and edges are connecting pairs of incomparable elements in $P$. Notice that for a natural unit interval order $P(\pi)$ associated with a Dyck path $\pi$, the corresponding incomparability graph is encoded in the cells between the Dyck path and the main diagonal. We denote this graph by $G(\pi)$. 

When referring to a Dyck path $\pi$, we implicitly encompass the associated objects discussed above. That is, $\pi$ will represent not only the Dyck path itself but also its corresponding Hessenberg function $h(\pi)$, the natural unit interval order $P(\pi)$, and the associated incomparability graph $G(\pi)$. 


\subsection{Chromatic symmetric functions and unicellular LLT polynomials}


We define the \emph{inversion} statistic as 
$\inv_{\pi} (w)= |\{ (i, j) ~:~ i<j,\, i\nless_{\pi} j,\,\text{ and } w(i) > w(j)\}|$, for a word $w\in \mathbb{Z}_{>0}^n$ of length $n$.

Given a Dyck path $\pi$, the \emph{chromatic symmetric function} of $\pi$ is defined by 
\begin{equation}\label{def:chromsym}
  \chi_{\pi} [X;q]=\sum_{\substack{w\in \mathbb{Z}_{>0}^n\\ \text{proper}}}q^{\inv_{\pi}(w)}x^w,
\end{equation}
where the sum is over `proper' colorings of $G(\pi)$, that is, the sum is over the words of length $n$ such that $w(i)\ne w(j)$ for $i\nless_{\pi} j$.  
In definition (\ref{def:chromsym}) for $\chi_{\pi}$, if we remove the proper condition for $w$, then it becomes the \emph{unicellular LLT polynomial} indexed by $\pi$, namely 
\begin{equation}\label{def:unicellularLLT}
\LLT_{\pi}[X;q]=\sum_{w\in \mathbb{Z}_{>0}^n}q^{\inv_{\pi}(w)}x^w.
\end{equation}

Carlsson and Mellit \cite{CM2018} found an explicit relationship between unicellular LLT polynomials and chromatic symmetric functions via a plethystic substitution.
\begin{prop}\cite{CM2018} \label{prop: plethystic relation between X and LLT Carlsson Mellit} For a Dyck path $\pi$, we have
\begin{equation}\label{eqn:CMrelation}
 (q-1)^n \,  \chi_\pi[X;q] = \LLT_{\pi}[(q-1)X;q].
\end{equation}
\end{prop}

\section{\texorpdfstring{$\alpha$}{alpha}-chromatic symmetric functions}

Throughout this paper, we let $Q_\alpha=\frac{q^\alpha-1}{q-1}$.
\begin{defn}\label{def:alpha_chromsym}
Given a Dyck path $\pi$ and $\alpha\in \mathbb{R}$, we define the \emph{$\alpha$-chromatic symmetric function} 
\begin{equation}\label{eqn:alpha_chromsymdef}
   \chi^{(\alpha)}_\pi[X;q] := \chi_\pi\left[Q_\alpha X;q\right] =\frac{ \LLT_{\pi}[(q^\alpha-1)X;q]}{(q-1)^n}.
\end{equation}
\end{defn}


Given a Dyck path $\pi$, for $1\le i \le n$, let  $b_i(\pi)$ and $a_i(\pi)$ denote the number of cells below $\pi$ and strictly above the diagonal $y=x$, 
in the $i$th column (row, respectively), from the right (top, respectively).  Then, by applying the \emph{superization technique} (cf. \cite{Hag2008}) on the 
right most side of \eqref{eqn:alpha_chromsymdef},
we can prove the following proposition.
\begin{prop}\label{prop: q-chromatic}
For an $n$-Dyck path $\pi$ and a real parameter $\alpha$,
\begin{equation}\label{QQ}
 \chi _{\pi}^{ (\alpha)}[1 ;q]= \chi_{\pi}[Q_\alpha;q]= q^{\area(\pi)}\prod_{i=1}^n [\alpha -a_i(\pi)]_q 
= q^{\area(\pi)}\prod_{i=1}^n [\alpha -b_i(\pi)]_q.
\end{equation}
\end{prop}

\begin{remark}
The products occurring in the middle and the right hand side terms of \eqref{QQ} are of the form so called \emph{rook product}. Note that the close connection between the chromatic 
quasisymmetric functions and  rook theory has already been revealed in various works (cf. \cite{AN21, CMP23}). 
We also observe many interesting combinatorial properties of rook theory related to the $\alpha$-chromatic symmetric functions. In particular, we obtain a new solution to Garsia and Remmel's problem of finding a combinatorial description for their $q$-hit numbers for any Ferrers board. See \cite{HOY} for the details.
\end{remark}

\begin{remark}
Recently, Hikita \cite{Hikita} announced a proof of the Stanley--Stembridge conjecture, and his approach uses probabilistic argument observing that 
\begin{equation}\label{eqn:sumisone}
\sum_{\lambda\vdash n}q^{|\mathsf{e}|-|\mathsf{e}_{\lambda}|}\frac{c_{\lambda, \pi}(q)}{\prod_i [\lambda_i]_q !}=1
\end{equation}
where $\chi_{\pi}(X;q)=\sum_{\lambda\vdash n}c_{\lambda,\pi}(q) e_{\lambda}[X]$ is the $e$-expansion of the chromatic quasisymmetric function indexed by a Dyck path $\pi$,
$|\mathsf{e}|$ is the number of cells outside of $\pi$ and $|\mathsf{e}_\lambda|=\sum_{i<j}\lambda_i \lambda_j$.
We can easily derive \eqref{eqn:sumisone} from \eqref{QQ}, by applying the principal specialization of the elementary symmetric functions and  by letting $\alpha\rightarrow \infty$.
\end{remark}

\subsection{Monomial expansion into \texorpdfstring{$\alpha$}{alpha}-binomial bases}\label{subsec: monomial expansion}

Given an $n$-Dyck path \( \pi \), we define an ordering \( \prec_\pi \) for \emph{bi-letters} $(i,j)\in \mathbb{Z}_{> 0}\times [n]$ as follows:
\[
  (i,j) \prec_\pi (i',j') ~\text{if and only if}~
  \begin{cases}
    &  i < i' ~\text{ or }~\\
    & i = i'~ \text{ and } ~ j<_\pi j'.
    \end{cases}
\]
Here, we consider the second coordinate $j\in[n]$ as elements in the poset $P(\pi)$ corresponding to the Dyck path $\pi$. For $\lambda \vdash n$, let $m_i(\lambda)$ denote the number of times $i$ occurs as a part in $\lambda$, and let $M(\lambda)$ be the set of all words 
of {\it weight} $\lambda$, which means all words where $i$ occurs $\lambda_i$ times for $i\ge 1$. For a bi-word \(\displaystyle (w, \sigma) \in M(\lambda) \times \mathfrak{S}_n\), we say that \( i\in [n-1] \)
is an \emph{ascent} of \( (w, \sigma) \) with respect to \( \pi \) if
\[ \displaystyle (w(i), \sigma(i)) \prec_\pi (w(i+1), \sigma(i+1)) .\]
We define \( \asc_\pi (w, \sigma) \) as the number of ascents of \( (w, \sigma) \) with respect to \( \pi \). 
We let $\comaj_\pi (w,\sigma)$ be the sum of positions where the ascents of $(w,\sigma)$ occur. Finally, for a bi-word $(w,\sigma)\in M(\lambda)\times\mathfrak{S}_n$, we define a statistic $\stat_\pi$ by
\[
    \stat_\pi(w,\sigma)={\inv_\pi(w) +\sum_{i=1}^{\ell(\lambda)}\inv_{\pi[w^{-1}(i)]}(\sigma[w^{-1}(i)]) + \binom{n}{2}-nk+\comaj_\pi(w,\sigma)},
\]
where $\pi[A]$ is the induced subgraph $\pi$ on $A$. Using these notions, we state a monomial expansion of $\chi^{(\alpha)}_\pi[X;q]$ into the basis $\left\{\scriptstyle{\begin{bmatrix}\alpha+k\\n\end{bmatrix}_q}\right\}_{0\le k \le n-1}$. 



\begin{thm}\label{thm: alpha+k choose n basis monomial expansion} Given a Dyck path $\pi$, we have
\[
  \chi_\pi^{(\alpha)}[X;q] = \sum_{\lambda\vdash n} \sum_{k=0}^{n-1}\sum_{\substack{(w, \sigma)\in M(\lambda) \times \mathfrak{S}_n\\ \asc_\pi(w,\sigma)=k}} q^{\stat_\pi(w,\sigma)}
  \scriptstyle{\begin{bmatrix}\alpha+k\\n\end{bmatrix}_q} m_\lambda [X].
\]  
\end{thm}

\begin{proof} We briefly sketch the proof.

We consider an ordering $ 1<\overline{1}<2<\overline{2}<\cdots < n <\overline{n}$
on the set of signed alphabets $\mathcal{A}_{+}\cup \mathcal{A}_-.$
By applying the superization technique to the LLT polynomials, we get 
\begin{equation*}\label{Eq: LLT superization}
    \LLT_\pi[tX-Y;q] = \sum_{w\in\mathbb{Z}_{>0}^n}q^{\inv_\pi(w)}\prod_{i=1}^n (t-q^{N_\pi(i)})x^w,
\end{equation*}
where \( N_\pi(i) \) is the number of $j$'s such that $i<j$, $i\nless_\pi j$ and $w(i)=w(j)$.
On the left-hand side, by replacing $t$ by $q^\alpha$ and dividing by \( (q-1)^n \), we get the $\alpha$-chromatic symmetric function:
\begin{align}
\chi_{\pi}^{(\alpha)} [X;q]&=\frac{\LLT_{\pi}[(q^\alpha-1)X;q]}{(q-1)^n}\nonumber\\
&=\sum_{\lambda\vdash n}\sum_{w\in M(\lambda)}
 q^{{\inv_\pi}(w)}\prod_{i=1}^{\ell(\lambda)}\left( q^{{\area}(\pi[w^{-1}(i)])}
 \prod_{j=1}^{\lambda_i}[\alpha-a_j(\pi[w^{-1}(i)])]_q\right)m_\lambda [X].\label{eq: LLT m}
\end{align}

For a given positive integer $\alpha$, a word (coloring) $c\in \mathbb{Z}_{>0}^n$, and another word (decoration) $d\in {\{0,1,\dots, \alpha-1\}}^n$, we say that the bi-word $(c,d)$ is an \emph{$\alpha$-decorated proper coloring} of $\pi$ of weight $\lambda$ if the weight of $c$ is $\lambda$, and for any cell $(i,j)$ with $i<j$ and $i\nless_\pi j$, we have $(c(i),d(i))\neq (c(j),d(j))$. Let $\Colo^{(\alpha)}_{\pi,\lambda}$ denote the set of $\alpha$-decorated proper colorings of weight $\lambda$. By \Cref{prop: q-chromatic} we have
\begin{equation*}
  \prod_{j=1}^n [\alpha-a_j(\pi)]_q =q^{-|\area(\pi)|} \sum_{\substack{\sigma\in \{0,1,\dots,\alpha-1\}^n \\ \text{ proper coloring of $\pi$}}} q^{\inv_\pi(\sigma)+|\sigma|}.
\end{equation*}
By utilizing \eqref{eq: LLT m} and the identity above, we get the following monomial expansion
\begin{equation}\label{eqn:monoex_1}
\chi^{(\alpha)}_\pi [X;q] = \sum_{\lambda\vdash n }\sum_{(c,d)\in \Colo^{(\alpha)}_{\pi,\lambda}}  q^{\inv_\pi (c)+|d|}\prod_{i=1}^{\ell(\lambda)}
q^{\inv_{\pi[c^{-1}(i)]}(d^{-1}(i))} m_\lambda [X].
\end{equation}


To prove \Cref{thm: alpha+k choose n basis monomial expansion}, it suffices to verify it for $n$ distinct values of $\alpha$. To that end, we fix an integer $\alpha$ and prove the identity using the bijection below.

\begin{defn}\label{def: the map Phi}

Consider an integer $0\le \alpha \le n-1$. Let $\scriptstyle{\begin{pmatrix}[\alpha+k]\\ n\end{pmatrix}}$ denote the multiset of size $n$ whose elements are in $\{ 0, 1, \dots, \alpha+k-n\}$, namely  
\[
\binom{[\alpha+k]}{n} = \{ \tau=(\tau(1), \tau(2),\dots, \tau(n)) ~ | ~ 0\le \tau(i)\le \alpha+k-n,\,  \tau(1)\le \cdots \le \tau(n)\}.
\]
Note that the size of this set is equal to $\binom{\alpha+k}{n}$. 
We define 
\[ 
    \Phi:\bigcup_{k=0}^{n-1} \left\{(w,\sigma)\in M(\lambda)\times \mathfrak{S}_n:\asc(w,\sigma)=k\right\} \times \binom{[\alpha+k]}{n} \rightarrow \Colo^{(\alpha)}_{\pi,\lambda}
\]
 by $ \Phi(w,\sigma,\tau)_{\sigma(i)} = (w(i), \tau(i)+(i-1)-\asc_\pi^{<i}(w,\sigma))$.
Here, $\asc_\pi^{<i}(w,\sigma)$ counts the ascents occurring in the first $i-1$ positions.
\end{defn}
Then the fact that the map $\Phi$ is a bijection gives a way to collect $\begin{pmatrix}\alpha+k \\n\end{pmatrix}$ terms 
in the monomial expansion of \Cref{thm: alpha+k choose n basis monomial expansion}.
See \cite{HOY} for a fuller account.
\end{proof}



\subsection{The XY technique and  \texorpdfstring{$(\alpha)^{\underline{k}}$}{alphak} expansion}\label{Sec:XYTechnique}
Now we consider the  $(\alpha)^{\underline{k}}$ basis expansion.

Given two sets of variables  $X,Y$, let $XY = \{x_iy_j:i,j\ge 1\}$ and $\pi$ be a Dyck path.  
\begin{thm}
\label{XYThm}
\begin{align}
\label{XYresult}
\chi _{\pi}[XY;q] = \sum_{\lambda \vdash n } m_{\lambda}[X]
\sum_{w \in M(\lambda)} 
q^{\inv_{\pi}(w)} \chi _{\beta (\pi,w)} (Y;q),
\end{align}
where the $n$-Dyck path $\beta(\pi,w)$ corresponds to the graph $G_{\beta}$ obtained by starting with $G_{\pi}$ colored by $w$, and then
removing all edges which connect  two vertices $a,b$ with different colors 
(i.e. vertices satisfying $w(a)\ne w(b)$).
\end{thm}

\begin{proof} Define a total order on ordered pairs of positive integers $(a,b)$ as follows:
\begin{align}
\label{order}
\text{if $a<c$ then $(a,b)<(c,d)$ for all $b,d$ and also
$(b,a)<(b,c)$ for all $b$}.
\end{align}
By definition,
\begin{align}
\label{DefXY}
\chi _{\pi} [XY;q] = 
\sum _{\substack{C \in (\mathbb Z ^2)^n \\ C \text{ proper }} }
q^{\inv_{\pi}(C)} 
\prod_{i\ge 1} x_{w_{i}}y_{z_{i}},
\end{align}
where $C_i=(w_i,z_i)$ with both $w$ and $z$ in $\mathbb Z_{+}^n$, and we use (\ref{order}) to determine inequalities among the $C_i$'s.  Now for any  $i <j$ where $(j,i) \in G_{\pi}$,  if $w_{i}\ne w_{j}$, then
whether or not $C_i$ and $C_j$ contribute to $\inv_{\pi}(C)$ is completely determined by the values of $w_{i}$ and $w_{j}$.  On the other hand, if 
$w_{i}=w_{j}$, then
whether $C_i$ and $C_j$ contribute to $\inv_{\pi}(C)$ is completely determined by the values of $z_{i}$ and $z_{j}$, and in this case we also must have 
 $z_{i}\ne z_{j}$, otherwise the coloring $C$ of $G_{\pi}$ would not be proper.  It follows that 
 \begin{align}
 \inv_{\pi}(C) = \inv_{\pi}(w_{1},w_{2},\ldots ,w_{n}) + 
 \inv_{\beta (\pi,w)}(z_{1},z_{2},\ldots ,z_{n}),
 \end{align}
 where the first term on the right counts the inversions amongst $C_i,C_j$ which have unequal first coordinates, and 
  $ \inv_{\beta (\pi,w)}$ counts the inversions amongst $C_i,C_j$ which have equal first coordinates.  Thus the contribution to $\chi _{\pi}[XY;q]$ from all
  colorings $C$ whose first coordinate $w$ has weight $\lambda$ on the right-hand-side of (\ref{DefXY}) factors into a term of weight $\lambda$ in the $X$ variables, times a sum of terms in the $Y$ variables, which are over proper colorings of $G_{\beta (\pi,w)}$.    We leave it to the reader to show that $G_{\beta(\pi,w)}$ is a unit interval order when 
  $G_{\pi}$ is.  
\end{proof}



\begin{cor}\label{Cor: Schur positivity alpha N}
Given an $n$-Dyck path $\pi$, let
\begin{align}
\chi_{\pi}^{(\alpha)} [X;q] = \sum _{\lambda\vdash n} 
C_{\pi,\lambda}(\alpha) s_\lambda [X].
\end{align}
Then if $\alpha \in \mathbb N$, $C_{\pi,\lambda}(\alpha) \in \mathbb N [q]$ and furthermore has a combinatorial interpretation.
\end{cor}
\begin{proof}   Let $X=Q_\alpha$ in (\ref{XYresult}).  Clearly  $m_{\lambda}\left[Q_\alpha \right]\in \mathbb N[q]$ since $\alpha \in \mathbb N$.
Since all the Schur 
coefficients of the $\chi _{\beta(\pi,w)}(Y;q)$ are in $\mathbb N [q]$ and count weighted $P$-tableaux by the result of Shareshian and Wachs, 
everything in (\ref{XYresult}) is positive and has a combinatorial interpretation.
\end{proof}



A \emph{set partition} of $[n]$ is a collection of nonempty subsets of $[n]$ such that each element in $[n]$ is included in exactly one subset. Each subset in a set partition is called a \emph{part}. The set of set partitions of $[n]$ into $k$ parts is denoted by $\SSS(n,k)$. 



For an $n$-Dyck path $\pi$ and a set partition $S$ of $[n]$, we assign a Dyck path $\beta(\pi,S)$ as follows. First, order the parts of $S$ as $S=\{S^{(1)},\dots,S^{(k)}\}$ and let $w_S$ be the word obtained by replacing elements in $S^{(i)}$ with $i$. Now define
\[
    \beta(\pi,S) := \beta(\pi,w_S).
\]
Using this, we present the expansion of $\alpha$-chromatic symmetric functions into $(\alpha)^{\underline{k}}$ basis.



\begin{thm}\label{thm: falling factorial basis} For an $n$-Dyck path $\pi$, the $\alpha$-chromatic symmetric function $\chi^{(\alpha)}_\pi[X;1]$ at $q=1$ in terms of the falling factorial basis $\left\{(\alpha)^{\underline{k}}\right\}_{1\le k \le n}$ is
\begin{equation}\label{Eq: falling factorial expansion}
\chi_{\pi}^{(\alpha)}[X;1] = \sum_{k=1}^n (\alpha)^{\underline{k}}\sum_{S\in \SSS(n,k)} \chi_{\beta(\pi,S)}[X;1].
\end{equation}
In particular, the $\alpha$-chromatic symmetric function \( \chi_{\pi}^{(\alpha)} [X;1] \) is positively expanded in terms of the basis \( \left\{ (\alpha)^{\underline{k}}s_{\lambda} \right\}_{\substack{1\le k \le n, \\ \lambda\vdash n}}\).

\end{thm}



\begin{proof}[Proof of \Cref{thm: falling factorial basis}]
Let $w$ be a word of length $n$. Define $S(w)$ to be the set partition such that $i$ and $j$ belong to the same part of $S(w)$ if and only if $w(i)=w(j)$. For example, $S(31321)=\{\{1,3\},\{2,5\},\{4\}\}$. 
By letting $X=Q_\alpha$ and $Y=X$ in (\ref{XYresult}), we have
\[
    \chi_{\pi}^{(\alpha)}[X;q] = \sum_{\lambda \vdash n } m_{\lambda}\left[Q_\alpha \right] 
    \sum_{w \in M(\lambda)} 
    q^{{\inv}_{\pi}(w)} \chi _{\beta (\pi,w)} (Y;q),
\]
We let $q=1$ to obtain 
\begin{equation}\label{eq: alpha chromsym at q=1 first equation}
    \chi^{(\alpha)}_\pi[X;1] = \sum_{\lambda\vdash n}\sum_{w\in M(\lambda)} \binom{\alpha}{\ell(\lambda)}\binom{\ell(\lambda)}{m_1(\lambda),m_2(\lambda),\cdots,m_{\ell(\lambda)}(\lambda)} \chi_{\beta(\pi,w)}[X;1],    
\end{equation}
where $m_i(\lambda)$ denotes the multiplicity of $i$ in the partition $\lambda$.

For a set partition $S$ of $n$, one can associate a partition $\lambda(S)$ by reordering the sizes of each part of $S$ to a non-increasing sequence. Let $\SSS(n,\lambda)$ be the set of set partitions $S$ with $\lambda(S)=\lambda$. 

Fix a partition $\lambda\vdash n$ and a set partition $S\in\SSS(n,\lambda)$. Note that the number of words $w$ satisfying $S(w)=S$ is given by $m_1(\lambda)! m_2(\lambda)!\cdots m_{\ell(\lambda)}(\lambda)!$. Therefore, we can rewrite \eqref{eq: alpha chromsym at q=1 first equation} as 
\begin{align*}
    \chi^{(\alpha)}_\pi[X;1] &= \sum_{\lambda\vdash n}\sum_{S\in \SSS(\lambda)} \binom{\alpha}{\ell(\lambda)}\binom{\ell(\lambda)}{m_1(\lambda),m_2(\lambda),\cdots}m_1(\lambda)!\cdots m_{\ell(\lambda)}(\lambda)! \chi_{\beta(\pi,S)}[X;1]
    \\&= \sum_{\lambda\vdash n} \sum_{S\in \SSS(\lambda)}(\alpha)^{\underline{\ell(\lambda)}}\chi_{\beta(\pi,S)}[X;1],
\end{align*}
which finishes the proof.
\end{proof}

\begin{remark}
Due to Hikita's proof of Stanley--Stembridge conjecture \cite{Hikita}, \Cref{thm: falling factorial basis} gives 
$e$-positivity of the $\alpha$-chromatic symmetric functions. 
\end{remark}



\subsection{Further results on Schur expansions}




This section proves the Schur positivity of $\alpha$-chromatic symmetric functions into two binomial bases $\left\{\scriptstyle{\begin{bmatrix}\alpha+k\\n\end{bmatrix}_q}\right\}_{0\le k \le n-1}$ and $\left\{\scriptstyle{\begin{bmatrix}\alpha\\k\end{bmatrix}_q}\right\}_{1\le k \le n}$. Notably, we establish a general result for Schur positivity of $s_\lambda[Q_\alpha X]$.

\begin{lem}\label{Lemma: Schur positivity q^a-1/q-1}
For a partition $\lambda$, \(s_\lambda\left[Q_\alpha X\right]\) is Schur positive in terms of the bases \begin{center}$\left\{\scriptstyle{\begin{bmatrix}\alpha+k\\n\end{bmatrix}_q}\right\}_{0\le k \le n-1}$ and $\left\{\scriptstyle{\begin{bmatrix}\alpha\\k\end{bmatrix}_q}\right\}_{1\le k \le n}$. \end{center} 
\end{lem}


As a Corollary, the Schur positivity of $\alpha$-chromatic symmetric function $\chi^{(\alpha)}_\pi[X;q]$ in terms of $\left\{\scriptstyle{\begin{bmatrix}\alpha+k\\n\end{bmatrix}_q}\right\}_{0\le k \le n-1}$ and $\left\{\scriptstyle{\begin{bmatrix}\alpha\\k\end{bmatrix}_q}\right\}_{1\le k \le n}$ follows from the Schur positivity of the usual chromatic symmetric functions \cite{Gas96, ShWa16}. 
\begin{cor}\label{Cor: Schur positivity}
    Consequently, if a symmetric function $f$ is Schur positive, then \(f\left[Q_\alpha X\right]\) also exhibits Schur positivity in the bases $\left\{\scriptstyle{\begin{bmatrix}\alpha+k\\n\end{bmatrix}_q}\right\}_{0\le k \le n-1}$ and $\left\{\scriptstyle{\begin{bmatrix}\alpha\\k\end{bmatrix}_q}\right\}_{1\le k \le n}$. In particular, the Schur coefficients of $\alpha$-chromatic symmetric function $\chi^{(\alpha)}_\pi[X;q]$ of a Dyck path $\pi$ are in $\mathbb{N}[q]$ when $\chi^{(\alpha)}_\pi[X;q]$ is expanded 
     in terms of aforementioned bases.
\end{cor}

It is noteworthy that the positivity in terms of the second basis is \emph{weaker} than the positivity in the $q$-falling factorial basis $[\alpha]^{\underline{k}}_q$. In fact, $s_\lambda\left[Q_\alpha X\right]$ might not be in $\mathbb{N}[q]$  in terms of  the $q$-falling factorial basis $[\alpha]^{\underline{k}}_q$. For example,  $s_{(2)}\left[Q_\alpha X\right]$ at $q=1$ is
    \[
      \left(\frac{1}{2} \alpha^2 - \frac{1}{2} \alpha\right)s_{(2)}[X] + \left(\frac{1}{2}\alpha^2 + \frac{1}{2}\alpha\right)s_{(1,1)}[X]
      = \frac{1}{2}(\alpha)^{\underline{2}}s_{(2)}[X]+\left( \frac{1}{2}(\alpha)^{\underline{2}}+(\alpha)^{\underline{1}} \right)s_{(1,1)}[X],
    \]
    in which the coefficients are rational in terms of the falling factorial basis (even for $q=1$).
Nevertheless, we conjecture that the Schur coefficients of the $\alpha$-chromatic symmetric functions lie in $\mathbb{N}[q]$ in terms of the $q$-falling factorial basis.
\begin{conj}\label{conj: falling factorial Schur positivity} 
    For an $n$-Dyck path $\pi$, the coefficients of the $\alpha$-chromatic symmetric function $\chi_\pi^{(\alpha)}[X;q]$ in  \( \left\{[\alpha]^{\underline{k}}_q s_\lambda [X] \right\}_{1\le k \le n,\,\lambda\vdash n} \) expansion are in $\mathbb{N}[q]$. 
\end{conj}


Lastly, we present a symmetry relation for the Schur expansion of the $\alpha$-chromatic symmetric functions. 

\begin{prop} For an $n$-Dyck path $\pi$, if we let 
\[
    \chi^{(\alpha)}_\pi[X;q] = \sum_{k=0}^{n-1}\sum_{\lambda\vdash n} c^{(\alpha)}_{\pi,\lambda,k} (q) \scriptstyle{\begin{bmatrix}\alpha+k\\n\end{bmatrix}_q} s_\lambda [X],
\]
then 
$$c^{(\alpha)}_{\pi,\lambda,k}(q) = q^{\area(\pi)+\binom{n}{2}}c^{(\alpha)}_{\pi,\lambda',n-1-k}(q^{-1}).$$
\end{prop}

\begin{proof}
This symmetry relation easily follows the following relation of the unicellular LLT polynomials \cite[Proposition 4.1.4]{BHMPS} 
 \[
    \omega(\LLT_\pi [X;q]) = q^{\area(\pi)}\LLT_\pi[X;q^{-1}].
 \]
 For the details, see \cite{HOY}.
\end{proof}



\section{Final Remarks}


\subsection{Schur expansion of unicellular LLT polynomials}\label{subsec: Schur expansion for unicellular LLT}
\begin{prop}\label{prop: Schur coefficient of LLT}
    For an $n$-Dyck path $\pi$, let $c^k_{\pi,\lambda}(q)$ and $d^k_{\pi,\lambda}(q)$ be the Schur coefficients of $\alpha$-chromatic symmetric function into the bases $\left\{\scriptstyle{\begin{bmatrix}\alpha+k\\n\end{bmatrix}_q}\right\}_{0\le k \le n-1}$ and $\left\{[\alpha]_q^{\underline{k}}\right\}_{1\le k \le n}$, respectively:
    \[
        \chi_\pi^{(\alpha)}[X;q] = \sum_{\lambda\vdash n} \sum_{k=0}^{n-1} c^k_{\pi,\lambda}(q)\scriptstyle{\begin{bmatrix}\alpha+k\\n\end{bmatrix}_q} s_\lambda [X] = \sum_{\lambda\vdash n} \sum_{k=1}^{n} d^k_{\pi,\lambda}(q)[\alpha]_q^{\underline{k}}s_\lambda [X].
    \]
    Then we have
    \[
        \LLT_\pi[X;q] = \sum_{\lambda\vdash n} \dfrac{\sum_{k=0}^{n-1} c^k_{\pi,\lambda}(q)}{[n]_q!}s_\lambda [X] =\sum_{\lambda\vdash n} q^{-\binom{n}{2}} d^n_{\pi,\lambda}(q)s_\lambda [X] .
    \]
\end{prop}

At $\alpha=1$, we have $[\alpha]^{\underline{k}}_q=0$ for all $k$ except for $k=1$. On the other hand, the $\alpha$-chromatic symmetric function evaluated at $\alpha=1$ coincides with the original chromatic symmetric function. Consequently, in the expansion of $\alpha$-chromatic symmetric functions $\chi^{(\alpha)}_\pi[X;q]$ using the $\left\{[\alpha]_q^{\underline{k}}\right\}$ basis, the coefficient corresponding to $[\alpha]_q^{\underline{1}}$ is the chromatic symmetric function $\chi_\pi[X;q]$. Based on this observation, \Cref{prop: Schur coefficient of LLT} implies that the coefficients of the $\alpha$-chromatic symmetric functions expressed in terms of the $q$-falling factorial basis serve as an interpolation between the chromatic symmetric functions and the unicellular LLT polynomials.


\subsection{Geometric interpretations}\label{subsec: geometric interpretation}

For an $n$-Dyck path $\pi$, let $\Hess(\pi)$ be the (regular semisimple) Hessenberg variety associated to $\pi$ (or the Hessenberg function $h(\pi)$). Tymoczko's dot action on torus equivariant cohomology induces an $\mathfrak{S}_n$ action on the cohomology $H^*(\Hess(\pi))$. Brosnan--Chow \cite{BC18}, and Guay-Paquet \cite{Gua13} independently proved the conjecture of Shareshian--Wachs \cite{ShWa16} that this representation coincides with the chromatic symmetric function:
\[
    \omega \chi_\pi[X;q] = \Frob(H^*(\Hess(\pi));q),
\]
where $\Frob$ is the (graded) Frobenius characteristic map. 

Since $\alpha$-chromatic symmetric functions enjoy various Schur positivity phenomena and symmetry relations, it is tempting to seek a geometric model behind it. For fixed $\alpha\in\mathbb{Z}_{>0}$, it is quite direct to obtain the following geometric interpretation for $\alpha$-chromatic symmetric function.

\begin{prop}
    Let $\pi$ be an $n$-Dyck path and $\alpha\in \mathbb{Z}_{>0}$ be a fixed positive integer. Then we have
    \[
        \omega \chi^{(\alpha)}_\pi[X;q] = \Frob\left(H^*\left(\Hess(\pi)\times (\mathbb{P}^{\alpha-1})^n\right);q\right).
    \]
    Here $\mathbb{P}^{\alpha-1}$ is the complex projective space of dimension $\alpha-1$ where $\mathfrak{S}_n$ acts by permuting coordinates. 
\end{prop}

Unfortunately, if we let $\alpha$ be general (as a parameter), it is unclear how to interpret the $\alpha$-chromatic symmetric functions in terms of geometry.  Such an approach could 
guide us toward achieving the desired Schur positivity in \Cref{conj: falling factorial Schur positivity}.



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\end{document}







\subsubsection{A subsubsection}

\lipsum[3]

\begin{equation}\label{eqn:eq1}
\int_0^\infty \exp(-x) \, \mathrm{d}x = 1.
\end{equation}

\begin{lem}\label{lem:lem1}
\lipsum[11]
\end{lem}

\lipsum[10]

\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{plot}
\caption{A plot of a function.}
\label{fig:plot}
\end{figure}

\subsection{A subsection}

\lipsum[5]

\begin{align*}
1 + 1 + 1 &= 2 + 1 \\ &= 3.
\end{align*}

\lipsum[17]

Here is a citation~\cite{greenwade93} with URL. Here is another
citation~\cite{MR4}, which is the earliest-numbered Math Review with a
functioning DOI: MR0000004. Finally, a reference to
equation~\eqref{eqn:eq1}. And a reference to \Cref{fig:plot}.

