Wang]Shiyun Wang
Shiyun Wang: University of Southern California, Los Angeles, CA 90089-2532, USA
The e-positivity of the chromatic symmetric functions and the inverse Kostka matrix

We expand the chromatic symmetric functions for Dyck paths of bounce number three in the elementary symmetric function basis using a combinatorial interpretation of the inverse of the Kostka matrix studied in Eg̃eciog̃lu-Remmel (1990). We prove that certain coefficients in this expansion are positive. We establish the $e$-positivity of an extended class of chromatic symmetric functions for Dyck paths of bounce number three beyond the "hook-shape" case of Cho-Huh (2019).

§ INTRODUCTION

Chromatic symmetric functions were first introduced by Stanley <cit.> and have grown to be a fruitful area of research in relation to enumerative combinatorics, algebraic geometry and representation theory. They can form numerous bases for the algebra of symmetric functions and have close relationship with Hessenberg varieties as well. Chromatic symmetric functions are an algebraic tool to study graphs.

Let $G$ be a finite graph with $V(G)=\{v_1, v_2, \cdots, v_n\}$ and $\mathbb{P}=\{1,2,3,\cdots\}$. The chromatic symmetric function $X_G(\mathbf{x})$ associated with $G$ is defined as
\[X_G(\mathbf{x})=X_G(x_1,x_2,\cdots)=\sum_{\kappa}x_{\kappa(v_1)}x_{\kappa(v_2)}\cdots x_{\kappa(v_n)},\]
where the sum ranges over all proper colorings $\kappa: V \rightarrow \mathbb{P} $. Stanley <cit.> considered the expansion of $X_G(\mathbf{x})$ into various symmetric function bases. For example, the expansion of $X_G(\mathbf{x})$ in terms of the monomial symmetric functions $m_\lambda$ can be considered as a generating function for stable partitions into independent sets of the vertices of $G$. The coefficients of $X_G(\mathbf{x})$ in the expansion of power sum symmetric functions $p_\lambda$ are related to the Mobius function of the lattice of contractions of $G$. If the chromatic symmetric function is expanded into the elementary symmetric function $e_\lambda$ as $X_G(\mathbf{x})=\sum_\lambda c_\lambda e_\lambda$, then the summation of the coefficients $\sum_{\lambda\vdash V, l(\lambda)=j}c_\lambda$ is the number of acyclic orientations of $G$ with $j$ sinks. Furthermore, it is natural to think about the combinatorial interpretation of the coefficients of $e_\lambda$. Thus the famous Stanley-Stembridge conjecture is the following.

Let $P$ be a finite poset and $\inc(P)$ be its incomparability graph, that is, the graph $G$ with vertices the elements of $P$ where two vertices are connected by an edge if and only if the elements are incomparable in $P$. We say $P$ is $(\bf{a}+\bf{b})$-free if $P$ does not contain an induced subposet isomorphic to a disjoint union of an $\bf{a}$-element chain and a $\bf{b}$-element chain.

If $P$ is $(\bf{3}+\bf{1})$-free, then $X_{\inc(P)}$ is $e$-positive.

Some critical results related to this conjecture have been shown. Gasharov<cit.> proved the incomparability graphs of $(\bf{3}+\bf{1})$-free posets are Schur-positive. In particular, $X_G(\mathbf{x})$ can be expanded as $X_G(\mathbf{x})=\sum_\lambda c_\lambda s_\lambda$, where $c_\lambda$ is the number of $P$-tableaux of shape $\lambda$. Guay-Paquet<cit.> reduced the conditions in Conjecture <ref>, so that it is enough to prove the $e$-positivity of the incomparability graph of a natural unit interval order, that is, posets which are both $(\bf{3}+\bf{1})$-free and $(\bf{2}+\bf{2})$-free.

Shareshian-Wachs<cit.> introduced a quasisymmetric refinement of Stanley's chromatic symmetric function with a variable $q$ counting ascents in the colorings. For $G$ the incomparability graph of a natural unit interval order this quasisymmetric function $X_G(\mathbf{x},q)$ turns out to be symmetric, and they generalized the expansion of $X_G(\mathbf{x})$ in terms of various symmetric function bases. For example, The coefficients $c_\lambda(q)$ of $X_G(\mathbf{x},q)$ in the expansion in terms of the $e_\lambda$ were generalized as $\sum_{\lambda\vdash |V|, l(\lambda)=j}c_\lambda(q)=\sum_{o\in O(G,j)}q^{\text{asc}(o)}$, where $O(G,j)$ is the set of acyclic orientations of $G$ with $j$ sinks and $\text{asc}(o)$ is the number of directed edges $(a,b)$ of $o$ such that $a<b$. The Schur-basis expansion also has its quasisymmetric version in <cit.>, which can be reduced to the special case in <cit.> with $q=1$. Furthermore, Shareshian-Wachs obtained the following version of the Stanley-Stembridge conjecture.

Let $G$ be the incomparability graph of a natural unit interval order, then $X_G(\mathbf{x},q)$ is an $e$-positive polynomial in $q$.

These long-time open conjectures were widely studied and proved for some special cases, such as cycle or complete graphs <cit.>, lollipop graphs <cit.>, melting lollipop graphs <cit.>, graphs generated by abelian Dyck paths <cit.> and some certain classes of non-abelian ones<cit.>. More precisely, since the  unit interval order of $n$ are in bijection with Dyck paths on an $n\times n $ board, Cho-Huh<cit.> proved Conjecture <ref> for Dyck paths with bounce number two and a "hook-shape" case with bounce number three. Cho-Hong<cit.> showed the Conjecture <ref> is true when the corresponding Dyck paths have bounce number $\leq 3$ and $q=1$.

In this paper, we start with introducing basic definitions and facts that we need about chromatic symmetric functions. We develop a new way in Section <ref> to expand $X_G(\mathbf{x},q)$ by using the inverse Kostka numbers<cit.> and Conjecture <ref>. From there we will construct sign-reversing involutions in Theorem <ref> and Theorem <ref> to show that some coefficients of the new expansion satisfy the $e$-positivity property with generalized $q$, that is, the inversions of $P$-tableaux are preserved. In Section <ref>, we will prove Conjecture <ref> for Dyck paths of bounce number three beyond the "hook-shape" case shown in <cit.>. Denote the coefficient of $e_\lambda$ in the expansion of $X_G(\mathbf{x},q)$ in $e$-basis by $[e_\lambda]X_G(\mathbf{x},q)$. We state our main results as follows:

* (Theorem <ref> and Theorem <ref>) Let $X_G(\mathbf{x},q)$ be the chromatic symmetric function for Dyck path $\mathbf{d}$ of bounce number three. Then we have $[e_{(n-2l,l,l)}]X_G(\mathbf{x},q)$ and $[e_{(n-2l-1,l+1,l)}]X_G(\mathbf{x},q)$ are positive.

* (Theorem <ref>) Let $X_G(\mathbf{x},q)$ be the chromatic symmetric function for Dyck path $\mathbf{d}=(d_1, d_2, n-1,\cdots,n-1,n,\cdots,n)$ of bounce number three. Then we have $X_G(\textbf{x},q)$ is $e$-positive.

§ BACKGROUND

In this section we introduce definitions and background on chromatic symmetric functions and inverse Kostka matrix needed for our results.

Following Shareshian-Wachs<cit.>, the chromatic quasisymmetric function is defined as
\[X_G(\mathbf{x}, q)=\sum_\kappa q^{\text{asc}(\kappa)}\mathbf{x}_\kappa,\]

where the sum ranges over all proper colorings $\kappa: V \rightarrow \mathbb{P}$ and $\text{asc}(\kappa):=|\{(v_i,v_j)\in E:i<j\,\, \text{and}\,\, \kappa(i)<\kappa(j)\}|$.

In <cit.>, $P$-tableaux were defined to expand $X_G(\mathbf{x})$ in terms of the Schur basis. Let $a_{i,j}$ be the entry of a Young diagram at $i^{th}$ row and $j^{th}$ column. A $P$-tableaux of shape $\lambda$ is a filling of a Young diagram of shape $\lambda$ with elements of the poset $P$ satisfying the following.

* Each element in $P$ appears once in the diagram.

* $a_{i,j}\prec a_{i,j+1}$.

* $a_{i+1,j}\nprec a_{i,j}$.

If $X_G(\mathbf{x})=\sum_{\lambda}c_\lambda s_\lambda$, then $c_\lambda$ is the number of $P$-tableaux of shape $\lambda$.

This expansion has a quasisymmetric version with the inversions of a $P$-tableaux considered. Let $\PT$ be the set of all P-tableaux. Then for $T\in \PT$, an inversion of T is a pair of entries $i$ and $j$ of $T$ such that $(i,j)\in E, i>j$ and $i$ appears above $j$ in $T$. Let $\iv(T)$ be the number of inversions of $T$ and $\sh(T)$ be the shape of $T$.

Let $G$ be the incomparability graph of a natural unit interval order $P$. Then
\[X_G(\mathbf{x},q)=\sum_{T\in \PT}q^{\iv(T)}s_{\sh(T)}\]

We can rewrite the expansion as the following lemma.

\[X_G(\textbf{x},q)=\sum_{\lambda}B_{\lambda}(q)s_{\lambda},\]

where $B_{\lambda}=\displaystyle\sum_{\substack{T\in \PT\\ \sh(T)=\lambda}}q^{\iv(T)}$.

Shareshian-Wachs<cit.> provided a useful characterization of natural unit interval orders which are descried in the introduction as posets both $(3+1)$-free and $(2+2)$-free.

Let $\mathbf{d}:=(d_1,d_2,\cdots,d_{n-1})$ be a sequence of positive integers satisfying $d_1\leq d_2\leq\cdots\leq d_{n-1}\leq n$ and $d_i\geq i$ for all $i$. $\mathbf{d}$ is also called a Dyck path on an $(n\times n)$ board, taking a staircase walk from the bottom left corner $(0,0)$ to the top right corner $(n,n)$ that lies above the diagonal. It is connecting $(0,0),(0,d_1),(1,d_1),(1,d_2),(2,d_2),(2,d_3),(3,d_3),\cdots,(n-2,d_{n-1}),(n-1,d_{n-1}),(n-1,n),(n,n)$ with either horizontal or vertical steps.

Let $P(\mathbf{d})$ be the poset on $[n]$ with the order relation given by $i\prec j$ if $i<n$ and $j\in \{d_i+1,d_i+2,\cdots,n\}$. i.e., the poset relations $i\prec j$ in $P(\mathbf{d})$ are given by all cells $(i,j)$ that lie above the Dyck path $\mathbf{d}$. Denote the partition of the Young diagram consisting of all cells above $\mathbf{d}$ by $\tau$.

Let $P$ be a poset on $[n]$, then $P$ is a natural unit interval order iff $P\simeq P(\mathbf{d})$ for some $\mathbf{d}=(d_1,d_2,\cdots,d_{n-1})$.

Given a natural unit interval order $P$ on $[n]$, we can obtain its corresponding Dyck path $\mathbf{d}=(d_1,d_2,\cdots,d_{n-1})$ on an $(n\times n)$ board and the Young diagram $\tau$ determined by $\mathbf{d}$. Let $G$ be the incomparability graph of $P$ and denote the chromatic symmetric function of $G$ by $X_\tau(\textbf{x},q)$. Let $\mathbf{m}:=(m_0,m_1,\cdots,m_l)$ be a sequence of positive integers satisfying $d_1=m_0<m_1<\cdots<m_{l-1}<m_l=n$, and for all $i\in [1,l-1]$, $m_i$ is determined recursively by $m_{i-1}$ such that $m_i=d_{m_{i-1}+1}$. Let $b(\mathbf{d})$ be the bounce path connecting $(0,0),(0,m_0),(m_0,m_0),(m_0,m_1),(m_1,m_1),(m_1,m_2),(m_2,m_2),\cdots,(n,n)$. Denote the bounce number, that is, the number of the bounce path hitting the diagonal including the end point $(n,n)$ by $|\mathbf{m}|$. See an example in Figure <ref>.

Figure <ref> illustrates a case when bounce number $|\mathbf{m}|=3$. We let $\{1,2,\cdots,m_0\}$ be the set $S_1$, $\{m_0+1,\cdots,m_1\}$ be the set $S_2$, and $\{m_1+1,\cdots,m_2\}$ be the set $S_3$. Denote the order of $S_1,S_2,S_3$ by $c,b,a$ respectively.

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(-1.6,-1.6) – (1.6,1.6);
[red,very thick] (-1.6,-1.6)–(-1.6,0)–(-1.2,0)–(-1.2,+.8)–(+0.4,+.8)–(0.4,1.2)–(.8,1.2)–(.8,1.6)–(1.6,1.6);
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(-1.8,1.6)–(-1.8,.8) node[midway,left]$a$;
(-1.8,.8)–(-1.8,0) node[midway,left]$b$;
(-1.6,-0.2)–(0,-0.2) node[midway,below]$c$;

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A natural unit interval order $P=P(\mathbf{d})$ for Dyck path $\mathbf{d}=(4,6,6,6,6,7,8,8)$ with the corresponding partition $\tau=(6511)$. $\mathbf{m}=(4,6,8)$. The bounce path is $(0,0),(0,4),(4,4),(4,6),(6,6),(6,8),(8,8)$ and the bounce number $|\mathbf{m}|=3$. $G=\inc(P)$.

Following <cit.>, A rim hook with length $l$ of a partition $\lambda\vdash n$ is a sequence of $l$ connected cells in the Young diagram satisfying the following:

* Any two adjacent cells have a common edge.

* It starts from a cell on the southeast boundary in the diagram and travels along its northeast rim.

* We can obtain a legal Young diagram with size $n-l$ by removing the rim hook.

A special rim hook $H$ is a rim hook with at least one cell in the first column. Define the sign of $H$ as $\sign(H)=(-1)^{\heigt(H)-1}$, where $\heigt(H)$ is the height of the rim hook.

A special rim hook tabloid $T$ of shape $\mu$ and type $\lambda=(\lambda_1,\cdots,\lambda_l)$ is a tiling of the Young diagram of shape $\mu$  with special rim hooks of sizes $\{\lambda_1,\cdots,\lambda_l\}$. Define the sign of $T$ as $\sign(T)=\prod_{H}\sign(H)$.

Let $T$ be a special rim hook tabloid of shape $\mu=(4333)$ and type $\lambda=(5422)$. Then we can obtain two such tabloids as in Figure <ref>.

[very thick,blue] (0.5,-3.5)–(2.5,-3.5)–(2.5,-2.5);
[very thick,blue]
[very thick,blue]
[very thick,blue]

[very thick,blue] (0.5,-3.5)–(2.5,-3.5)–(2.5,-1.5);
[very thick,blue]
[very thick,blue]
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Special rim hook tabloid $T_1$ (left) and $T_2$ (right)

Hence $\sign(T_1)=1\cdot(-1)\cdot1\cdot(-1)=1$ and $\sign(T_2)=1\cdot1\cdot1\cdot1=1$.

Let $K_{\lambda,\mu}$ be the Kostka numbers enumerating SSYT of shape $\lambda$ and type $\mu$. Then we have $s_{\lambda}=\sum_{\nu}K_{\lambda,\nu}m_{\nu}$. Since $\langle h_{\mu}, m_{\nu}\rangle =\delta_{\mu\nu}$ for the symmetric functions scalar product $\langle \, , \,\rangle$, we can obtain that $h_{\mu}=\sum_{\lambda}K_{\lambda,\mu}s_{\lambda}$. Eg̃eciog̃lu-Remmel<cit.> introduced a combinatorial interpretation of the inverse of the Kostka matrix. Since both $s$ and $h$ form bases for the symmetric functions, we can invert the Kostka matrix as $s_{\mu}=\sum_{\lambda}K\inv_{\lambda,\mu}h_{\lambda}$, where $K\inv$ are the inverse Kostka numbers. Applying the involution $\omega$ to both sides we can also get $s_{\mu}=\sum_{\lambda}K\inv_{\lambda,\mu'}e_{\lambda}$.

\[K\inv_{\lambda, \mu}=\sum_{T}\sign(T),\]

where the summation is over all special rim hook tabloids of type $\lambda$ and shape $\mu$.

As in Example <ref>, $K\inv_{\lambda, \mu}=\sign(T_1)+\sign(T_2)=2$.

§ THE CHROMATIC SYMMETRIC FUNCTIONS FOR BOUNCE NUMBER THREE

§.§ Expansion of the chromatic symmetric functions in terms of the inverse Kostka number

In an $(n\times n)$ board, let $\mathbf{d}$ be a Dyck path with the bounce number $|\mathbf{m}|$. Let $P$ be the associated unit interval order and $G=\inc(P)$. By Lemma <ref> and the expansion of Schur function in terms of the inverse Kostka numbers, we can write the chromatic symmetric function as follows.

\begin{equation} \label{inv kostka expand}
\begin{split}
&=\sum_{\mu}B_{\mu}(q)s_{\mu} \\
&=\sum_{\mu}B_{\mu}(q)(\sum_{\lambda}K\inv_{\lambda,\mu'}e_{\lambda}) \\
\end{split}
\end{equation}

In Equation (<ref>), the coefficient $B_\mu(q)$ is nonzero only if $\mu_1\leq$ the bounce number $|\mathbf{m}|$. This is because for $B_{\mu}(q)=\sum_{T\in \PT,\sh(T)=\mu}q^{\iv(T)}$, a $P$-tableaux $T$ of shape $\mu$ exists only if $\mu_1 \leq$ the longest chain of relations in the poset $P$. In a longest chain of $P$, we can have at most one entry chosen from $\{1,\cdots,m_0\}$, and at most one entry chosen from $\{m_0+1,\cdots,m_1\}$ and so forth. Thus the length of the chain can be at most $|\mathbf{m}|$.

We consider the chromatic symmetric functions determined by Dyck paths with bounce number $|\mathbf{m}|=3$. Following the notations in Figure <ref>, we have $|S_1|=c, |S_2|=b, |S_3|=a$ and let $a+b=k$.

Given a poset $P$ on $[n]$ with the associated Dyck path $|\mathbf{m}|=3$, if a $P$-tableaux $T$ with $\sh(T)=3^l2^j1^{n-3l-2j}$ exists, i.e., $B_{3^l2^j1^{n-3l-2j}}\neq 0$, then $2l+j\leq k$.

This is immediate from Remark <ref>. There are in total $2l+j$ entries in the second and third column of $T$, which can at most exhaust all elements of $P$ in $S_2$ and $S_3$.

Given a Dyck path with $|\mathbf{m}|=3$, by Remark <ref>, we can clear up all zero terms and reduce Equation (<ref>) by finding all possible shapes of $\mu$'s, that is, $\mu=2^i1^{n-2i}$ for $i\in [0,k]$, or $\mu=3^l2^j1^{n-3l-2j}$ for $l\in [1,\min\{a,b,c\}], j\in [0, k-2l]$.

Following Remark <ref> and Theorem $\ref{inv Kostka}$, we can compute the inverse Kostka number for each possible $s_\mu$ as follows:

* For $s_{1^n}: K\inv_{\lambda,(n)}=\begin{cases}1 &\text{if}\,\, \lambda=(n)\\0 &\text{otherwise}\end{cases}$

* For $s_{2^i1^{n-2i}}: K\inv_{\lambda,(n-i,i)}=$
1 & \text{for}\,\, \lambda=(n-i,i) \\
-1 & \text{for}\,\, \lambda=(n-i+1,i-1)\\
0 & \text{otherwise}
\end{cases}$

* For $s_{3^l2^j1^{n-3l-2j}}:K\inv_{\lambda,(n-2l-j,l+j,l)}=$
1 & \text{for}\,\,\lambda=(n-2l-j,l+j,l) \\
-1 & \text{for}\,\,\lambda=(n-2l-j+1,l+j-1,l) \\
-1 & \text{for}\,\, \lambda=(n-2l-j,l+j+1,l-1) \\
1 & \text{for}\,\, \lambda=(n-2l-j+2,l+j-1,l-1) \\
0 & \text{otherwise}
\end{cases}$

We can combine the above cases and derive the formula for computing $K\inv$ by restricting to only valid $\lambda$. Namely, $K\inv_{\lambda,\mu'}\neq 0$ provided $\lambda_i\geq 0$ for each $i$. Thus we have for $\mu=3^l2^j1^{n-3l-2j}$ with $l\in [0,\min\{a,b,c\}], j\in [0, k-2l]$,

\begin{equation}
\begin{cases}
1 & \text{for}\,\,\lambda=(n-2l-j,l+j,l) \\
-1 & \text{for}\,\,\lambda=(n-2l-j+1,l+j-1,l) \\
-1 & \text{for}\,\, \lambda=(n-2l-j,l+j+1,l-1) \\
1 & \text{for}\,\, \lambda=(n-2l-j+2,l+j-1,l-1) \\
0 & \text{otherwise}
\end{cases}
\end{equation}

Now we can rewrite Equation (<ref>) as

\begin{align*} \numberthis \label{expansion}
& X_\tau(\textbf{x},q) \\
=& \sum_{l=0}^{\min\{a,b,c\}}\sum_{j=0}^{k-2l}B_{3^l2^j1^{n-3l-2j}}s_{3^l2^j1^{n-3l-2j}} \\
=& \sum_{l=0}^{\min\{a,b,c\}}\sum_{j=0}^{k-2l}B_{3^l2^j1^{n-3l-2j}}\big(\sum_\lambda K\inv_{\lambda,(n-2l-j,l+j,l)}e_\lambda \big) \\
=& \sum_{l=0}^{\min\{a,b,c\}}\sum_{j=0}^{k-2l}B_{3^l2^j1^{n-3l-2j}}\big(e_{(n-2l-j,l+j,l)}-e_{(n-2l-j+1,l+j-1,l)}-e_{(n-2l-j,l+j+1,l-1)}+e_{(n-2l-j+2,l+j-1,l-1)}\big) \\
=& \sum_{l=0}^{\min\{a,b,c\}}\Bigg[\sum_{j=0}^{k-2l}B_{3^l2^j1^{n-3l-2j}}e_{(n-2l-j,l+j,l)}-\sum_{j=0}^{k-2l}B_{3^l2^j1^{n-3l-2j}}e_{(n-2l-j+1,l+j-1,l)}\Bigg] \\
& +\sum_{l=-1}^{\min\{a,b,c\}-1}\Bigg[\sum_{j=0}^{k-2l}B_{3^{l+1}2^j1^{n-3l-2j-3}}e_{(n-2l-j,l+j,l)}-\sum_{j=0}^{k-2l}B_{3^{l+1}2^j1^{n-3l-2j-3}}e_{(n-2l-j-2,l+j+2,l)}\Bigg] \\
=& \sum_{l=0}^{\min\{a,b,c\}}\Bigg[\sum_{j=0}^{k-2l-1}\big(B_{3^l2^j1^{n-3l-2j}}-B_{3^l2^{j+1}1^{n-3l-2j-2}}\big)e_{(n-2l-j,l+j,l)} \\
& +B_{3^l2^{k-2l}1^{n-2k+l}}e_{(n-k,k-l,l)}-B_{3^l1^{n-3l}}e_{(n-2l+1,l,l-1)}\Bigg] \\
& +\sum_{l=-1}^{\min\{a,b,c\}-1}\Bigg[\sum_{j=2}^{k-2l-2}\big(B_{3^{l+1}2^j1^{n-3l-2j-3}}-B_{3^{l+1}2^{j-2}1^{n-3l-2j+1}}\big)e_{(n-2l-j,l+j,l)} \\
& +B_{3^{l+1}1^{n-3l-3}}e_{(n-2l,l,l)}+B_{3^{l+1}2^11^{n-3l-5}}e_{(n-2l-1,l+1,l)} \\
& -B_{3^{l+1}2^{k-2l-3}1^{n+l-2k+3}}e_{(n-k+1,k-l-1,l)}-B_{3^{l+1}2^{k-2l-2}1^{n+l-2k+1}}e_{(n-k,k-l,l)}\Bigg] \\
=& \sum_{l=0}^{\min\{a,b,c\}}\Bigg[\sum_{j=2}^{k-2l-2}\big(B_{3^l2^j1^{n-3l-2j}}-B_{3^l2^{j+1}1^{n-3l-2j-2}}+B_{3^{l+1}2^j1^{n-3l-2j-3}}-B_{3^{l+1}2^{j-2}1^{n-3l-2j+1}}\big)e_{(n-2l-j,l+j,l)} \\
& +\big(B_{3^l1^{n-3l}}-B_{3^l2^11^{n-3l-2}}+B_{3^{l+1}1^{n-3l-3}}\big)e_{(n-2l,l,l)} \\
& +\big(B_{3^l2^11^{n-3l-2}}-B_{3^l2^21^{n-3l-4}}+B_{3^{l+1}2^11^{n-3l-5}}\big)e_{(n-2l-1,l+1,l)} \\
& +\big(B_{3^l2^{k-2l-1}1^{n+l-2k+2}}-B_{3^l2^{k-2l}1^{n+l-2k}}-B_{3^{l+1}2^{k-2l-3}1^{n+l-2k+3}}\big)e_{(n-k+1,k-l-1,l)} \\
& +\big(B_{3^l2^{k-2l}1^{n+l-2k}}-B_{3^{l+1}2^{k-2l-2}1^{n+l-2k+1}}\big)e_{(n-k,k-l,l)} \\
& -B_{3^l1^{n-3l}}e_{(n-2l+1,l,l-1)} \Bigg] \\
=& \sum_{l=0}^{\min\{a,b,c\}}\Bigg[\sum_{j=2}^{k-2l}\big(B_{3^l2^j1^{n-3l-2j}}-B_{3^l2^{j+1}1^{n-3l-2j-2}}+B_{3^{l+1}2^j1^{n-3l-2j-3}}-B_{3^{l+1}2^{j-2}1^{n-3l-2j+1}}\big)e_{(n-2l-j,l+j,l)} \\
& +\big(B_{3^l1^{n-3l}}-B_{3^l2^11^{n-3l-2}}+B_{3^{l+1}1^{n-3l-3}}\big)e_{(n-2l,l,l)} \\
& +\big(B_{3^l2^11^{n-3l-2}}-B_{3^l2^21^{n-3l-4}}+B_{3^{l+1}2^11^{n-3l-5}}-B_{3^{l+1}1^{n-3l-3}}\big)e_{(n-2l-1,l+1,l)}\Bigg]
\end{align*}

Note that in the last equality of Equation <ref>, $B_{3^l2^{j+1}1^{n-3l-2j-2}}=0$ when $j=k-2l$, and $B_{3^{l+1}2^j1^{n-3l-2j-3}}=0$ when $j=k-2l$ and $j=k-2l-1$. This is because the corresponding P-tableaux does not exist by Lemma <ref>.

§.§ Proof of $e$-positivity of chromatic symmetric functions with $|\mathbf{m}|=3$

We will prove Conjecture <ref> for some coefficients when the bounce number is three. In particular, for each term of the elementary functions in the expansion of Equation <ref>, we construct injective functions such that each $P$-tableaux with a negative sign is mapped to one with a positive sign.

Given a poset $P$ on $[n]$ with the associated Dyck path $|\mathbf{m}|=3$, let $L$ be a vertical consecutive sequence of entries in the first column of a $P$-tableaux $T$. Denote the entry at the $i$th row in $L$ by $a_{i,1}$. For every $i$, there are at most two indices $s$ such that $s>i$ and $a_{i,1},a_{i+1,1}\cdots,a_{s-1,1}\prec a_{s,1}$.

Following the notations defined above Figure <ref>, we have the fact that $S_1\prec S_3$. Now suppose that there exist two indices $s$ satisfying the chain relations described in the lemma, denote them by $s_1$ and $s_2$. Without loss of generality, we can assume $s_1<s_2$. Then we have $a_{i,1},a_{i+1,1}\cdots,a_{s_1-1,1}\prec a_{s_1,1}$ and $a_{i,1},\cdots,a_{s_1,1},\cdots,a_{s_2-1,1}\prec a_{s_2,1}$. Since $|\mathbf{m}|=3$, so the length of a poset chain is at most three. It is necessary that $a_{i,1},\cdots,a_{s_1-1,1}\in S_1, a_{s_1,1}\in S_2$, and $a_{s_2,1}\in S_3$. We have $a_{i,1}\prec a_{s_1,1}\prec a_{s_2,1}$, thus no element is greater than $a_{s_2,1}$ in $P$. The result follows.

Let $X_G(\mathbf{x},q)$ be the chromatic symmetric function of a Dyck path with a bounce number three. In the expansion of $X_G(\mathbf{x},q)$ in terms of the elementary symmetric functions, $[e_{(n-2l,l,l)}]X_G(\mathbf{x},q)$ for $l\in[0,\min\{a,b,c\}]$ is positive.

We obtain the expansion of $X_G(\mathbf{x},q)$ in Equation <ref>. We prove the $e$-positivity for the term $e_{(n-2l,l,l)}$ by showing that $(B_{3^l1^{n-3l}}-B_{3^l2^11^{n-3l-2}}+B_{3^{l+1}1^{n-3l-3}})$ is a polynomial in $q$ with positive coefficients. In particular, we will build an injective map from the set of $P$-tableaux $\{T^-\in\PT|\sh(T^-)=3^l2^11^{n-3l-2}\}$ to the set of $P$-tableaux $\{T^+\in\PT|\sh(T^+)=3^l1^{n-3l}\,\, \text{or}\,\, 3^{l+1}1^{n-3l-3}\}$.

Let $a_{i,j}$ be the entry at the $i$th row and $j$th column of a P-tableaux $T^-$ of shape $\sh(T^-)=3^l2^11^{n-3l-2}$. For $s\in[l+2,n-3l-2]$, if there exists an entry $a_{s,1}$ in the first column such that $a_{l+1,1},a_{l+2,1},\cdots,a_{s-1,1}\prec a_{s,1}\prec a_{l+1,2}$, then we move $a_{s,1}$ to the second position of row $l+1$ and move $a_{l+1,2}$ after $a_{s,1}$ in the same row. See Figure <ref> (left). Note that in this case we are guaranteed to obtain a valid $P$-tableaux since such $a_{s,1}$ is unique by Lemma <ref> and $\{a_{s-1,1}\}\in S_1$, $\{a_{l,2},a_{s,1}\}\in S_2$ and $\{a_{l,3},a_{l+1,2}\}\in S_3$. Otherwise, find the smallest $s$ such that $a_{s,1}\nprec a_{l+1,2}$ and insert $a_{l+1,2}$ above $a_{s,1}$. See Figure <ref> (right). If no such $a_{s,1}$ exists, equivalently, $\{a_{l+1,1},\cdots,a_{n-3l-2,1}\}\prec a_{l+1,2}$, then we simply move $a_{l+1,2}$ to the bottom of the first column. In both cases, the relative positions of any two cells having no poset relations all remain unchanged, which preserves the inversions of all $P$-tableaux. Moreover, from the way we construct this map, it is not hard to see that it is injective.

mathmode, boxsize=2em

a_1,1     a_1,2     a_1,3

a_l,1     a_l,2     a_l,3

a_l+1,1     a_l+1,2

a_1,1     a_1,2     a_1,3

a_l,1     a_l,2     a_l,3

a_l+1,1     a_s,1     a_l+1,2

a_1,1     a_1,2     a_1,3

a_l,1     a_l,2     a_l,3

a_l+1,1     a_l+1,2

a_1,1     a_1,2     a_1,3

a_l,1     a_l,2     a_l,3

The injective map for coefficients of $e(n-2l,l,l)$

Let $X_G(\mathbf{x},q)$ be the chromatic symmetric function of a Dyck path with a bounce number three. In the expansion of $X_G(\mathbf{x},q)$ in terms of the elementary symmetric functions, $[e_{(n-2l-1,l+1,l)}]X_G(\mathbf{x},q)$ for $l\in[0,\min\{a,b,c\}]$ is positive.

Again by Equation <ref>, we want to show that $(B_{3^l2^11^{n-3l-2}}-B_{3^l2^21^{n-3l-4}}+B_{3^{l+1}2^11^{n-3l-5}}-B_{3^{l+1}1^{n-3l-3}})$ is a polynomial in $q$ with positive coefficients.

Define two maps

\[f: \Big\{T^-\in\PT|\sh(T^-)=3^{l+1}1^{n-3l-3}\Big\}\mapsto \Big\{T^+\in\PT|\sh(T^+)=3^l2^11^{n-3l-2}\Big\}\]

\[g: \Big\{T^+\in\PT|\sh(T^+)=3^{l+1}2^11^{n-3l-5}\Big\}\mapsto \Big\{T^-\in\PT|\sh(T^-)=3^l2^21^{n-3l-4}\Big\}\]

For map $f$, given any P-tableaux $T^-$ of shape $\sh(T^-)=3^{l+1}1^{n-3l-3}$, we find the smallest $s$ for $s\in[l+2,n-3l-3]$ such that $a_{s,1}\nprec a_{l+1,2}$, then insert $a_{l+1,2}$ above $a_{s,1}$ and move $a_{l+1,1}$ left by one cell. See Figure <ref>. If $a_{s,1}$ exists, then it is unique by Lemma <ref> since $a_{l+1,2}\in S_2$. If there does not exist such $a_{s,1}$, then we simply move $a_{l+1,2}$ to the bottom of the first column. The image of the map $f$ gives us legal $P$-tableaux $T^+$ of shape $\sh(T^+)=3^l2^11^{n-3l-2}$ since $a_{l+1,1}\prec a_{l+1,3}$ and for any row of length three, we always have entries of the first column in $S_1$, entries of the second column in $S_2$, entries of the third column in $S_3$. Moreover, moving $a_{l+1,2}$ and $a_{l+1,3}$ will not affect any inversions in $T^-$.

mathmode, boxsize=2em

a_1,1     a_1,2     a_1,3

a_l,1     a_l,2     a_l,3

a_l+1,1     a_l+1,2     a_l+1,3

a_1,1     a_1,2     a_1,3

a_l,1     a_l,2     a_l,3

a_l+1,1    a_l+1,3

The injective map $f$

For map $g$, given any P-tableaux $T^+$ of shape $\sh(T^+)=3^{l+1}2^11^{n-3l-5}$, we discuss two cases. In case 1, if $a_{l+2,2}\prec a_{l+1,3}$, we have $a_{l+1,1}, a_{l+2,1}\in S_1$ and $a_{l+1,2},a_{l+2,2}\in S_2$ and $a_{l+1,3}\in S_3$. Then we bump $a_{l+2,2}$ with $a_{l+1,3}$, and find the smallest $s$ for $s\in[l+3,n-3l-5]$ such that $a_{s,1}\nprec a_{l+2,2}$ and insert $a_{l+2,2}$ above $a_{s,1}$. See Figure <ref> (left). If $a_{s,1}$ exists, then it is unique by Lemma <ref> since $a_{l+1,2}\in S_2$ (Same reasoning for all sub-cases discussed below). If there does not exist such $a_{s,1}$, then we move $a_{l+2,2}$ to the bottom of the first column. In case 2, if $a_{l+2,2}\nprec a_{l+1,3}$, we have three possibilities. For case $2(a)$, if $a_{l+1,2}\nprec a_{l+2,2}$ and $a_{l+2,1}\nprec a_{l+1,2}$, find the smallest $s$ for $s\in[l+3,n-3l-5]$ such that $a_{s,1}\nprec a_{l+1,2}$ and insert $a_{l+1,2}$ above $a_{s,1}$. Move $a_{l+1,3}$ left by one cell. See Figure <ref> (right). We can carefully check inversions in this mapping. $a_{l+1,2}<a_{l+2,2}$ since $a_{l+1,2}\prec a_{l+1,3}$ while $a_{l+2,2}\nprec a_{l+1,3}$. Similarly, $a_{l+2,1}<a_{l+1,2}$ since $a_{l+2,1}\prec a_{l+2,2}$ while $a_{l+1,2}\nprec a_{l+2,2}$. Thus the odd inversion $(a_{l+1,2},a_{l+2,1})$ is cancelled while a new inversion $(a_{l+2,2},a_{l+1,2})$ is created, resulting in an unchanged total number of inversions; For case $2(b)$, if $a_{l+1,2}\nprec a_{l+2,2}$ and $a_{l+2,1}\prec a_{l+1,2}$, we firstly have the same mapping as described in case $2(a)$, and then switch $a_{l+1,3}$ and $a_{l+2,2}$ on top of it. See Figure <ref> (left). Note that $a_{l+2,2}<a_{l+1,3}$ since $a_{l+1,2}\prec a_{l+1,3}$ but $a_{l+1,2}\nprec a_{l+2,2}$. Reasoning on inversions is also similar except that $a_{l+1,2}$ and $a_{l+2,1}$ is not an inversion anymore. Switching $a_{l+1,3}$ and $a_{l+2,2}$ will help to decease one inversion, offsetting the new inversion $(a_{l+2,2},a_{l+1,2})$ created; For case $2(c)$, if $a_{l+1,2}\prec a_{l+2,2}$, then find the smallest $s$ for $s\in[l+2,n-3l-5]$ such that $a_{s,1}\nprec a_{l+1,2}$ and insert $a_{l+1,2}$ above $a_{s,1}$. Move $a_{l+1,3}$ left by one cell. See Figure <ref> (right). All inversions remain unvaried in this case.

As the ways that maps $f$ and $g$ are constructed, they are both injective.

mathmode, boxsize=2em

a_1,1     a_1,2     a_1,3

a_l,1     a_l,2     a_l,3

a_l+1,1     a_l+1,2     a_l+1,3

a_l+2,1     a_l+2,2

a_1,1     a_1,2     a_1,3

a_l,1     a_l,2     a_l,3

a_l+1,1    a_l+1,2

a_l+2,1     a_l+1,3

a_1,1     a_1,2     a_1,3

a_l,1     a_l,2     a_l,3

a_l+1,1     a_l+1,2     a_l+1,3

a_l+2,1     a_l+2,2

a_1,1     a_1,2     a_1,3

a_l,1     a_l,2     a_l,3

a_l+1,1     a_l+1,3

a_l+2,1     a_l+2,2

The injective map $g$ for case 1 (left) and case $2(a)$ (right)

mathmode, boxsize=2em

a_1,1     a_1,2     a_1,3

a_l,1     a_l,2     a_l,3

a_l+1,1     a_l+1,2     a_l+1,3

a_l+2,1     a_l+2,2

a_1,1     a_1,2     a_1,3

a_l,1     a_l,2     a_l,3

a_l+1,1    a_l+2,2

a_l+2,1     a_l+1,3

a_1,1     a_1,2     a_1,3

a_l,1     a_l,2     a_l,3

a_l+1,1     a_l+1,2     a_l+1,3

a_l+2,1     a_l+2,2

a_1,1     a_1,2     a_1,3

a_l,1     a_l,2     a_l,3

a_l+1,1     a_l+1,3

[⋯]    a_l+2,2

The injective map $g$ for case $2(b)$ (left) and case $2(c)$ (right)

Now we let

\[A=\Big\{\{T^+\in\PT|\sh(T^+)=3^l2^11^{n-3l-2}\}-\im(f)\Big\}\]

\[B=\Big\{\{T^-\in\PT|\sh(T^-)=3^l2^21^{n-3l-4}\}-\im(g)\Big\}\]

We can finish the proof of the $e$-positivity of the coefficients of $e_{(n-2l-1,l+1,l)}$ by finding an injection from $B$ to $A$. We start by characterizing the sets $A$ and $B$ more precisely. $A$ is the set of $P$-tableaux $T^+$ of shape $3^l2^11^{n-3l-2}$ such that in the first column, for any $r\in [l+2,n-3l-2]$, there does not exist a vertical sequence of consecutive entries $a_{l+1,1},\cdots,a_{{r-1},1}$ such that $a_{l+1,1},\cdots,a_{{r-1},1}\prec a_{r,1}\prec a_{l+1,2}$. $B$ is the set of $P$-tableaux $T^-$ of shape $3^l2^21^{n-3l-4}$ satisfying one of the following:

* (The complement of case $1$) If $a_{l+1,2}\prec a_{l+2,2}$, then for any $r\in [l+3,n-3l-4]$ in the first column, there does not exist a vertical sequence of consecutive entries $\{a_{l+2,1},\cdots,a_{{r-1},1}\}$ such that $a_{l+2,1},\cdots,a_{{r-1},1}\prec a_{r,1}\prec a_{l+2,2}$.

* (The complement of case $2(a)$) If $a_{l+1,2}\nprec a_{l+2,2}$, then for any $r\in [l+3,n-3l-4]$ in the first column, there does not exist a vertical sequence of consecutive entries $\{a_{l+3,1},\cdots,a_{{r-1},1}\}$ such that $a_{l+3,1},\cdots,a_{{r-1},1}\prec a_{r,1}\prec a_{l+1,2}, a_{r,1}<a_{l+2,2}$ and $a_{l+2,1}\nprec a_{r,1}$.

* (The complement of case $2(b)$) If $a_{l+1,2}\nprec a_{l+2,2}$, then for any $r\in [l+3,n-3l-4]$ in the first column, there does not exist a vertical sequence of consecutive entries $\{a_{l+2,1},\cdots,a_{{r-1},1}\}$ such that $a_{l+2,1},\cdots,a_{{r-1},1}\prec a_{r,1}\prec a_{l+2,2}$ and $a_{r,1}<a_{l+1,2}$.

* (The complement of case $2(c)$) If $a_{l+1,2}\nprec a_{l+2,2}$, then for any $r\in [l+3,n-3l-4]$ in the first column, there does not exist a vertical sequence of consecutive entries $\{a_{l+1,1},\cdots,a_{{r-1},1}\}$ such that $a_{l+1,1},\cdots,a_{{r-1},1}\prec a_{r,1}\prec a_{l+2,2}$ and $a_{r,1}\prec a_{l+1,2}$. For $r=l+2$, there does not exist an $a_{r,1}$ such that $a_{l+1,1}\prec a_{r,1}\prec a_{l+2,2}$, $a_{r,1}\prec a_{l+1,2}$ and $a_{l++3,1}\prec a_{l+2,2}$.

Given a $P$-tableaux $T$ with $\sh(T)=3^l2^21^{n-3l-4}$, if there exists some $r$ for $r\in[l+3,n-3l-4]$ such that $a_{l+2,1},\cdots,a_{r-1,1}\prec a_{r,1}\prec a_{l+2,2}$, then $T\notin B$.

Let $T$ be a $P$-tableaux with $\sh(T)=3^l2^21^{n-3l-4}$. Suppose that there exists some $r$ for $r\in[l+3,n-3l-4]$ such that $a_{l+2,1},\cdots,a_{r-1,1}\prec a_{r,1}\prec a_{l+2,2}$. If $a_{l+1,2}\prec a_{l+2,2}$, then $T\in \im(g)$ (case $1$); If $a_{l+1,2}\nprec a_{l+2,2}$ and $a_{r,1}\nprec a_{l+1,2}$, then $T\in \im(g)$ (case $2(b)$); If $a_{l+1,2}\nprec a_{l+2,2}$ and $a_{r,1}\prec a_{l+1,2}$, then $T\in \im(g)$ (case $2(c)$). Thus, $T\notin B$.

Let $\phi: B\mapsto A$ be the map from the set $B$ to set $A$, where $\phi$ is defined in the following way. Given a $P$-tableaux $T^-\in B$ with $\sh(T)=3^l2^21^{n-3l-4}$, find the smallest $s$ for $s\in[l+3,n-3l-4]$ such that $a_{s,1}\nprec a_{l+2,2}$ and insert $a_{l+2,2}$ above $a_{s,1}$. See Figure <ref>. If such $a_{s,1}$ does not exist, we simply move $a_{l+2,2}$ to the bottom of the first column. The resulting tableaux $T^+$ with $\sh(T)=3^l2^11^{n-3l-2}$ lies in $A$ by the characterization of set $A$. By Lemma <ref>, since $T^-\in B$, there does not exist an entry $a_{r,1}$ for $r\in[l+3,n-3l-4]$ such that $a_{l+2,1},\cdots,a_{r-1,1}\prec a_{r,1}\prec a_{l+2,2}$, thus we are guaranteed to find an unique $a_{s,1}$ in the mapping by Lemma <ref> and $\phi$ is an injection. Moreover, all inversions are stable in the mapping.

mathmode, boxsize=2em

a_1,1     a_1,2     a_1,3

a_l,1     a_l,2     a_l,3

a_l+1,1     a_l+1,2

a_l+2,1     a_l+2,2

a_1,1     a_1,2     a_1,3

a_l,1     a_l,2     a_l,3

a_l+1,1    a_l+1,2

The injective map $\phi$

§ $E$-POSITIVITY FOR SOME CLASSES OF BOUNCE NUMBER THREE

Using Equation <ref>, we show that a class of chromatic symmetric functions for Dyck paths of bounce number three (and their transposes) are $e$-positive.

Given a natural unit interval order $P(\bf{d})$ on $[n]$, for which the associated Dyck path is $\mathbf{d}=(d_1, d_2, n-1,\cdots,n-1,n,\cdots,n)$ and $|\bf{m}|=3$. Let $\tau=(\tau_1, \cdots,\tau_{a+b})$ with $\tau_i\leq 2$ for $i\geq 2$ be the Young diagram determined by $\bf{d}$, then the chromatic symmetric function $X_\tau(\textbf{x},q)$ is $e$-positive.

Figure <ref> illustrates two examples of the unit interval orders descried in Theorem <ref>. Lemma <ref> below explains why we display these two examples and they are essential for classifying the sets $S_1,S_2,S_3$ and conducting our cases discussion in later proof.

[every node/.style=minimum size=.3cm-, outer sep=0pt]
[step=0.3cm] (-1.5001,-1.5001) grid (1.5,1.5);
(-1.5,-1.5) – (1.5,1.5);
[red,very thick] (-1.5,-1.5)–(-1.5,-.3)–(-1.2,-.3)–(-1.2,0)–(-.9,0)–(-.9,1.2)–(.6,1.2)–(.6,1.5)–(1.5,1.5);
[green,ultra thick, dashed] (-1.5,-1.5)–(-1.5,-.3)–(-.3,-.3)–(-.3,1.2)–(1.2,1.2)–(1.2,1.5)–(1.5,1.5);
[fill=yellow!30] at (-1.35,1.35) ;
[fill=yellow!30] at (-1.05,1.35) ;
[fill=yellow!30] at (-.75,1.35) ;
[fill=yellow!30] at (-0.45,1.35) ;
[fill=yellow!30] at (-0.15,1.35) ;
[fill=yellow!30] at (0.15,1.35) ;
[fill=yellow!30] at (0.45,1.35) ;
[fill=yellow!30] at (-1.35,1.05) ;
[fill=yellow!30] at (-1.05,1.05) ;
[fill=yellow!30] at (-1.35,.75) ;
[fill=yellow!30] at (-1.05,.75) ;
[fill=yellow!30] at (-1.35,.45) ;
[fill=yellow!30] at (-1.05,.45) ;
[fill=yellow!30] at (-1.35,.15) ;
[fill=yellow!30] at (-1.05,.15) ;
[fill=yellow!30] at (-1.35,-.15) ;

[every node/.style=minimum size=.3cm-, outer sep=0pt]
[step=0.3cm] (-1.5001,-1.5001) grid (1.5,1.5);
(-1.5,-1.5) – (1.5,1.5);
[red,very thick] (-1.5,-1.5)–(-1.5,-1.2)–(-1.2,-1.2)–(-1.2,-.3)–(-.9,-.3)–(-.9,1.2)–(-.3,1.2)–(-.3,1.5)–(1.5,1.5);
[green, ultra thick, dashed] (-1.5,-1.5)–(-1.5,-1.2)–(-1.2,-1.2)–(-1.2,-.3)–(-.3,-.3)–(-.3,1.5)–(1.5,1.5);
[fill=yellow!30] at (-1.35,1.35) ;
[fill=yellow!30] at (-1.05,1.35) ;
[fill=yellow!30] at (-.75,1.35) ;
[fill=yellow!30] at (-0.45,1.35) ;
[fill=yellow!30] at (-1.35,1.05) ;
[fill=yellow!30] at (-1.05,1.05) ;
[fill=yellow!30] at (-1.35,.75) ;
[fill=yellow!30] at (-1.05,.75) ;
[fill=yellow!30] at (-1.35,.45) ;
[fill=yellow!30] at (-1.35,.15) ;
[fill=yellow!30] at (-1.35,-.15) ;
[fill=yellow!30] at (-1.05,.45) ;
[fill=yellow!30] at (-1.05,.15) ;
[fill=yellow!30] at (-1.05,-.15) ;
[fill=yellow!30] at (-1.35,-.45) ;
[fill=yellow!30] at (-1.35,-.75) ;
[fill=yellow!30] at (-1.35,-1.05) ;

Examples of the class of natural unit interval orders in Theorem <ref>

Following the context of Theorem <ref> and the notations from Figure <ref>, if $S_3\neq \{n\}$, then we must have $m_0=1$ for $\mathbf{m}=(m_0,m_1,n)$.

Let $\mathbf{m}=(m_0,m_1,n)$. If $S_3\neq \{n\}$, then the bounce path must touch the diagonal for the first time at $m_0=1$. Consequently, $S_1=\{1\}$ and $2\prec S_3$.

Since each part of $\tau$ has length $\leq 2$ except the first row, then a valid $P$-tableaux with longest chain of length three can have at most one 3-chain and one 2-chains. A valid $P$-tableaux with longest chain of length two can have at most three 2-chains. Thus in Equation <ref>, all terms vanished except when $l=0$ for $j\in [0,3]$ and $l=1$ for $j\in[0,1]$, which leads to the following expansion.

\begin{align*} \numberthis \label{expansionepos1}
& X_\tau(\textbf{x},q) \\
=& \sum_{l=0}^{\min\{a,b,c\}}\Bigg[\sum_{j=2}^{k-2l}\big(B_{3^l2^j1^{n-3l-2j}}-B_{3^l2^{j+1}1^{n-3l-2j-2}}+B_{3^{l+1}2^j1^{n-3l-2j-3}}-B_{3^{l+1}2^{j-2}1^{n-3l-2j+1}}\big)e_{(n-2l-j,l+j,l)} \\
& +\big(B_{3^l1^{n-3l}}-B_{3^l2^11^{n-3l-2}}+B_{3^{l+1}1^{n-3l-3}}\big)e_{(n-2l,l,l)} \\
& +\big(B_{3^l2^11^{n-3l-2}}-B_{3^l2^21^{n-3l-4}}+B_{3^{l+1}2^11^{n-3l-5}}-B_{3^{l+1}1^{n-3l-3}}\big)e_{(n-2l-1,l+1,l)}\Bigg] \\
=&\sum_{l=0,1}\Bigg[\big(B_{3^l1^{n-3l}}-B_{3^l2^11^{n-3l-2}}+B_{3^{l+1}1^{n-3l-3}}\big)e_{(n-2l,l,l)} \\
&+\big(B_{3^l2^11^{n-3l-2}}-B_{3^l2^21^{n-3l-4}}+B_{3^{l+1}2^11^{n-3l-5}}-B_{3^{l+1}1^{n-3l-3}}\big)e_{(n-2l-1,l+1,l)}\Bigg] \\
&+\big(B_{2^21^{n-4}}-B_{2^31^{n-6}}-B_{3^11^{n-3}}\big)e_{(n-2,2)} \\
\end{align*}

In the context of Theorem <ref> combining Lemma <ref>, for a $P$-tableaux $T$ with $\sh(T)=3^12^11^{n-5}$, let $a_{i,j}$ be the entry of $T$ at the $i$th row and $j$th column. There are only two possibilities. First, if $a_{1,3}=n$, then we must have $a_{1,1},a_{2,1}\in \{1,2\}$. Second, if $a_{1,3}\neq n$, then we have $S_1=\{1\}$, $a_{1,1}=1, a_{1,2}=2$ and $a_{2,2}=n$.

We prove the following lemmas to show that $X_\tau(\textbf{x},q)$ is $e$-positive. Remark <ref> is very useful to help us obtain the next lemma.

In Equation <ref>, the coefficient $(B_{2^31^{n-6}}-B_{3^12^11^{n-5}}\big)$ of $e_{(n-3,3)}$ is a polynomial in $q$ with positive coefficients.

Using similar methods in Theorem <ref> and Theorem <ref>, we will build an injective map from the set of $P$-tableaux $\{T^-\in\PT|\sh(T^-)=3^12^11^{n-5}\}$ to the set of $P$-tableaux $\{T^+\in\PT|\sh(T^+)=2^31^{n-6}\}$. Let $a_{i,j}$ be the entry at the $i$th row and $j$th column of a P-tableaux $T^-$ with $\sh(T^-)=3^12^11^{n-5}$. We consider four cases:

* Case 1: $a_{2,2}\prec a_{1,3}$ and $a_{3,1}\prec a_{1,3}$.

$a_{3,1}\prec a_{1,3}$ provides that $a_{1,3}=n$. Thus we have the first case in Remark <ref>. To obtain an injective map, we move $a_{1,3}=n$ as a new entry $a_{3,2}$ to obtain a P-tableaux $T^+$ of shape $\sh(T^+)=2^31^{n-6}$ with inversion relations preserved. See Figure <ref>.

mathmode, boxsize=2em

a_1,1     a_1,2     n

a_2,1     a_2,2

a_1,1     a_1,2

a_2,1     a_2,2

a_3,1     n

The injective map for coefficients of $e(n-3,3)$: Case 1

* Case 2: $a_{2,2}\nprec a_{1,3}$ and $a_{3,1}\prec a_{1,3}$.

For the same reason as in case 1, we have $a_{1,3}=n$. In addition, $a_{3,1}\prec a_{1,3}$ and $a_{1,2}\prec a_{1,3}$ while $a_{2,2}\nprec a_{1,3}$ implies that $a_{3,1}<a_{2,2}$ and $a_{1,2}<a_{2,2}$. Then we have three sub-cases:

If $a_{2,1}\nprec a_{3,1}$, then we swap $a_{2,1}$ and $a_{3,1}$, replace $a_{2,2}$ with $n$ and move $a_{2,2}$ as a new entry $a_{3,2}$. In this sub-case, $(a_{3,1},a_{2,1})$ is a new inversion while the original inversion $(a_{2,2},a_{3,1})$ is removed, resulting in an unvaried total number of inversions. See Figure <ref>.

mathmode, boxsize=2em

a_1,1     a_1,2     n

a_2,1     a_2,2

a_1,1     a_1,2

a_3,1     n

a_2,1     a_2,2

The injective map for coefficients of $e(n-3,3)$: Case 2a

If $a_{2,1}\prec a_{3,1}$ and $a_{2,1}<a_{1,2}$, it implies that $a_{1,2}<a_{3,1}$. Note that $a_{1,2}\nprec a_{3,1}$ since otherwise $a_{1,2}\in S_2,a_{3,1}\in S_3$, which cannot happen in the setting of Theorem <ref>. Thus we use the map shown in Figure <ref> (left). Note that the number of inversions is unchanged since the original inversions $(a_{1,2},a_{2,1})$ and $(n,a_{2,2})$ are replaced with new inversions $(a_{2,2},a_{1,2})$ and $(a_{3,1},a_{1,2})$.

If $a_{2,1}\prec a_{3,1}$ and $a_{2,1}\prec a_{1,2}$, we simply swap $a_{1,2}$ and $a_{3,1}$ from the last sub-case. Again all inversions remain the same except that $(n,a_{2,2})$ are replaced with $(a_{2,2},a_{1,2})$. See Figure <ref>(right).

mathmode, boxsize=2em

a_1,1     a_1,2     n

a_2,1     a_2,2

a_1,1     a_2,2

a_2,1     a_3,1

a_1,2     n

a_1,1     a_1,2     n

a_2,1     a_2,2

a_1,1     a_2,2

a_2,1     a_1,2

a_3,1     n

The injective map for coefficients of $e(n-3,3)$: Case 2b (left) and case 2c (right)

* Case 3: $a_{2,2}\prec a_{1,3}$ and $a_{3,1}\nprec a_{1,3}$.

$a_{2,2}\prec a_{1,3}$ provides that $a_{1,3}=n$. In addition, $a_{2,2}\prec n$ and $a_{3,1}\nprec n$ implies that $a_{2,2}<a_{3,1}$ and $a_{2,1}\prec a_{3,1}$. We swap $a_{2,2}$ and $a_{3,1}$, and move $n$ as a new entry $a_{3,2}$. Thus we obtain a P-tableaux $T^+$ of shape $\sh(T^+)=2^31^{n-6}$, where $(a_{3,1},a_{2,2})$ contributes to one more inversion offsetting $(n,a_{3,1})$ disappeared caused by moving $n$. See Figure <ref>.

mathmode, boxsize=2em

a_1,1     a_1,2     n

a_2,1     a_2,2

a_1,1     a_1,2

a_2,1     a_3,1

a_2,2     n

The injective map for coefficients of $e(n-3,3)$: Case 3

* Case 4: $a_{2,2}\nprec a_{1,3}$ and $a_{3,1}\nprec a_{1,3}$.

We need to discuss four sub-cases. In the first three sub-cases, $a_{2,1}\prec a_{3,1}$ provides that we have the first possibility in Remark <ref> such that $a_{1,3}=n$. Also note that since $a_{2,2},a_{3,1}\nprec n$ and $a_{1,2}\prec n$, then we have $a_{1,2}<a_{3,1}$ and $a_{1,2}<a_{2,2}$.

If $a_{2,1}\prec a_{3,1}$ and $a_{2,1}\prec a_{1,2}$, then we find an injective map by swapping $a_{1,2}$ and $a_{3,1}$, then swap $a_{3,1}$ and $a_{2,2}$. Lastly move $n$ as a new entry $a_{3,2}$. Two inversions $(n,a_{2,2})$ and $(n,a_{3,1})$ are disappeared by moving $n$. Two new inversions $(a_{2,2},a_{1,2})$ and $(a_{3,1},a_{1,2})$ are added back, keeping the total number of inversions unchanged. See Figure <ref>.

mathmode, boxsize=2em

a_1,1     a_1,2     n

a_2,1     a_2,2

a_1,1     a_2,2

a_2,1     a_3,1

a_1,2     n

The injective map for coefficients of $e(n-3,3)$: Case 4a

If $a_{2,1}\prec a_{3,1}$, $a_{2,1}\nprec a_{1,2}$, and $a_{3,1}>a_{2,2}$, we simply swap $a_{1,2}$ and $a_{3,1}$. Same reasoning can be applied as case 4a, and in addition, the inversion $(a_{1,2},a_{2,1})$ cancelled will be made up with the one $(a_{3,1},a_{2,2})$ created. See Figure <ref> (left).

If $a_{2,1}\prec a_{3,1}$, $a_{2,1}\nprec a_{1,2}$, and $a_{3,1}<a_{2,2}$, we take the map from the last sub-case, and then swap the first row and second row by observing that $a_{1,1}<a_{2,1}$. Then the inversion $(a_{1,2},a_{2,1})$ cancelled will be made up with the one $(a_{2,1},a_{1,1})$ created. See Figure <ref> (right).

mathmode, boxsize=2em

a_1,1     a_1,2     n

a_2,1     a_2,2

a_1,1     a_3,1

a_2,1     a_2,2

a_1,2     n

a_1,1     a_1,2     n

a_2,1     a_2,2

a_2,1     a_2,2

a_1,1     a_3,1

a_1,2     n

The injective map for coefficients of $e(n-3,3)$: Case 4b (left) and 4c (right)

The last sub-case differs from the other three since we need to consider both scenarios in Remark <ref>.

First, if $a_{2,1}\nprec a_{3,1}$, and $a_{1,3}=n$. $a_{1,2}<a_{3,1}$ implies that $a_{2,1}\nprec a_{1,2}$. $a_{2,1}\nprec a_{3,1}$ while $a_{2,1}\prec a_{2,2}$ implies that $a_{3,1}<a_{2,2}$. We can use the same map as in Figure <ref> (right) since all conditions and reasoning are identical excepting the relation between $a_{2,1}$ and $a_{3,1}$, which will not affect the validation of the $P$-tableaux and the total number of inversions.

Second, if $a_{2,1}\nprec a_{3,1}$, and $a_{2,2}=n$. We obtain exactly the same implications as above. If $a_{3,1}<a_{2,2}$, then the map is also the same. If $a_{3,1}\prec a_{2,2}$, then we define the map as Figure <ref>. Inversions remain unvaried except that $(a_{1,3},a_{2,1})$ is replace by $(a_{1,2},a_{2,1})$.

mathmode, boxsize=2em

1     2     a_1,3

a_2,1     n

1     a_2,1

2     a_1,3

a_3,1     n

The injective map for coefficients of $e(n-3,3)$: Case 4d

It is straightforward to check that all images of the injective maps under cases discussed above have no intersection. Note that the image of Figure <ref> does not intersect with the image of Figure <ref>. Since when $S_1=\{1\}$, the $P$-tableaux of shape $3^12^11^{n-5}$ in case 3 does not exist.

In Equation <ref>, the coefficient $\big(B_{2^21^{n-4}}-B_{2^31^{n-6}}-B_{3^11^{n-3}}\big)$ of $e_{(n-2,2)}$ is a polynomial in $q$ with positive coefficients.

We define injections from the sets of $P$-tableaux $\{T^-\in\PT|\sh(T^-)=3^11^{n-3}\}$ and $\{T^-\in\PT|\sh(T^-)=2^31^{n-6}\}$ to the set of $P$-tableaux $\{T^+\in\PT|\sh(T^+)=2^21^{n-4}\}$ respectively, and verify that their images have no intersection.

First, let $T^-$ be a $P$-tableaux with $\sh(T^-)=2^31^{n-6}$ and $a_{i,j}$ be the entry at the $i$th row and $j$th column of $T$. Find the smallest $s$ for $s\in [4,n-3]$ such that $a_{s,1}\nprec a_{3,2}$ and insert $a_{3,2}$ above $a_{s,1}$, otherwise move $a_{3,2}$ to the bottom of the first column. See Figure <ref>. We can carefully check that this map is injective. By Lemma <ref>, there are at most two indices $s$ such that $a_{3,1}, \cdots, a_{s-1,1}\prec a_{s,1}$. We claim that $a_{s,1}$ is unique in this case. Since if there exist two indices $s$ here, denoted by $s_1$ and $s_2 (s_1<s_2)$. Then $a_{3,1}\prec a_{s_1,1}\prec a_{s_2,1}$ is a 3-chain, for which $a_{3,1}\in \{1,2\}$. If $a_{3,2}=n$, then we do not have enough poset relations to fill out another two 2-chains. If $a_{3,2}\neq n$, then $S_1=\{1\}$ by Lemma <ref>, contradicting to the fact that $a_{3,1}=1$ and $1\nprec a_{2,1}$. Now in a $P$-tableaux $T^+$ with $\sh(T^+)=2^21^{n-4}$, we can always find the unique index $s$ (if it exists) such that $a_{3,1}, \cdots, a_{s-1,1}\prec a_{s,1}$ and move $a_{s,1}$ back as the entry $a_{3,2}$, which is the inverse of the map in Figure <ref>.

mathmode, boxsize=2em

a_1,1     a_1,2

a_2,1     a_2,2

a_3,1     a_3,2

a_1,1     a_1,2

a_2,1     a_2,2

The injective map for coefficients of $e(n-2,2)$: $\sh(T^-)=2^31^{n-6}$

Second, For a $P$-tableaux of shape $\sh(T^-)=3^11^{n-3}$, we define the injective map according to following cases. The reason why we consider for these two cases will become clear after we describe the map.

* Case 1: There does not exist an index $s$ for $s\in [3,n-2]$ such that $a_{3,1},a_{4,1}\cdots,a_{s-1,1}\prec a_{s,1}\nprec a_{1,3}$.

If $a_{2,1}\prec a_{1,3}$, we simply move $a_{1,3}$ below $a_{1,2}$. See Figure <ref> (left).

If $a_{2,1}\nprec a_{1,3}$, with knowing that $a_{1,2}\prec a_{1,3}$, we have either $a_{1,2}< a_{2,1}$ or $a_{1,2}\prec a_{2,1}$. When $a_{1,2}< a_{2,1}$, we adjust the map in Figure <ref> (left) by swapping $a_{2,1}$ and $a_{1,2}$. The only inversion changed is $(a_{1,3},a_{2,1})$, replacing by $(a_{2,1},a_{1,2})$. See Figure <ref> (right). When $a_{1,2}\prec a_{2,1}$, we swap $a_{2,1}$ and $a_{1,3}$ from Figure <ref> (right), preserving all original inversions. See Figure 17.

mathmode, boxsize=2em

a_1,1     a_1,2    a_1,3

a_1,1     a_1,2

a_2,1     a_1,3

a_1,1     a_1,2    a_1,3

a_1,1     a_2,1

a_1,2     a_1,3

The injective map for coefficients of $e(n-2,2)$: $\sh(T^-)=3^11^{n-3}$, case 1a (left) and case 1b (right)

mathmode, boxsize=2em

a_1,1     a_1,2    a_1,3

a_1,1     a_1,3

a_1,2     a_2,1

The injective map for coefficients of $e(n-2,2)$: $\sh(T^-)=3^11^{n-3}$, case 1c

Now it is clear that the images under the injection in case 1 do not intersect with the images in Figure <ref>. In the images of case 1, we can not find the index $s$ with the poset chain $a_{3,1},a_{4,1}\cdots,a_{s-1,1}\prec a_{s,1}\nprec a_{1,3}$ to recover the inverse map in Figure <ref>.

* Case 2: There exists an index $s$ for $s\in [3,n-2]$ such that $a_{3,1},\cdots,a_{s-1,1}\prec a_{s,1}\nprec a_{1,3}$.

Actually, we can further specify the possibilities in case 2 by Lemma <ref>. If $\mathbf{m}=(m_0,n-1,n)$,i.e., $S_3=\{n\}$, then we must have $a_{1,3}=n$ and the index $s=4$ such that $a_{3,1}\prec a_{4,1}\nprec n$, and then $a_{3,1}\in \{1,2\}$. Thus when $a_{3,1}\prec a_{1,2}$, we define the map as shown in Figure <ref> (left). All changes of inversions can be tracked with the following poset relations. First, $a_{1,2}\prec n$ and $a_{4,1}\nprec n$ implies $a_{1,2}<a_{4,1}$. Second, $a_{3,1}\prec a_{4,1}$ and $a_{3,1}\nprec a_{2,1}$ implies $a_{2,1}<a_{4,1}$. Lastly, $a_{3,1}\prec a_{1,2}$ and $a_{3,1}\nprec a_{2,1}$ implies that $a_{2,1}<a_{1,2}$. With these relations, one can easily verify that the total number of inversions remains unchanged. When $a_{3,1}\nprec a_{1,2}$, we slightly modify the image of Figure <ref> by swapping $a_{2,1}$ and $a_{1,2}$. Consequently, the inversions $(a_{1,2},a_{3,1})$ and $(a_{2,1},a_{3,1})$ will be replaced with $(a_{4,1},a_{1,2})$ and $(a_{4,1},a_{2,1})$. See Figure <ref> (right).

mathmode, boxsize=2em

a_1,1     a_1,2     n

a_1,1     n

a_3,1     a_4,1

a_1,1     a_1,2    n

a_1,1     n

a_3,1     a_4,1

The injective map for coefficients of $e(n-2,2)$: $\sh(T^-)=3^11^{n-3}$, case 2a (left) and case 2b (right)

If $\mathbf{m}=(1,m_1,n)$,i.e., $S_1=\{1\}$, then we must have $a_{1,1}=1$ and $2\prec S_3$. Thus possible $P$-tableaux are very limited under this condition. If $a_{1,3}=n$, then the index $s=4$ such that $a_{3,1}=2\prec a_{4,1}$. We define the map as shown in Figure <ref>. We can check that number of inversions are preserved with $(a_{2,1},2), (n,a_{4,1})$ reduced and $(a_{4,1},a_{1,2}), (a_{4,1},a_{2,1})$ added. Clearly the images in this map does not intersect with the images of set of $P$-tableaux $\{T^-\in\PT|\sh(T^-)=2^31^{n-6}\}$ since the index $s$ is removed. They also do not intersect with the images of case 1b for set of $P$-tableaux $\{T^-\in\PT|\sh(T^-)=3^11^{n-3}\}$, in which $a_{3,1}\nprec a_{2,1}$. If $a_{1,3}\neq n$, then we have $a_{1,2}=2$ and $a_{s,1}=n$. We define the map as shown in Figure <ref> (case 2d, left) when $a_{2,1}\prec n$ and Figure <ref> (case 2e, right) when $a_{2,1}\nprec n$, preserving the number of inversions of both $P$-tableaux. Similar analysis on the images can apply to these two sub-cases. They do no intersect with the images of set of $P$-tableaux $\{T^-\in\PT|\sh(T^-)=2^31^{n-6}\}$ since the index $s$ is removed. They also have no intersection with the images of case 1c for set of $P$-tableaux $\{T^-\in\PT|\sh(T^-)=3^11^{n-3}\}$, in which $a_{3,1}\nprec a_{2,1}$.

mathmode, boxsize=2em

1     a_1,2     n

1     a_4,1

a_1,2     n

The injective map for coefficients of $e(n-2,2)$: $\sh(T^-)=3^11^{n-3}$, case 2c

mathmode, boxsize=2em

1     2     a_1,3

1     a_1,3

2     n

1     2     a_1,3

1     a_3,1

2     n

The injective map for coefficients of $e(n-2,2)$: $\sh(T^-)=3^11^{n-3}$, case 2d (left) and case 2e (right)

The result follows immediately combining Theorem <ref>, Theorem <ref>, Lemma <ref> and Lemma <ref>.

Given a natural unit interval order $P(\bf{d})$ on $[n]$, for which the associated Dyck path is $\mathbf{d}=(d_1, n-2,\cdots,n-2,n-1,\cdots,n-1,n,\cdots,n)$ and $|\bf{m}|=3$. Let $\tau=(\tau_1, \cdots,\tau_{a+b})$ with $\tau_i=1$ for $i\geq 3$ be the Young diagram determined by $\bf{d}$, then the chromatic symmetric function $X_\tau(\textbf{x},q)$ is $e$-positive.

The result follows immediately from Theorem <ref> by the fact that the chromatic symmetric functions of two unit interval orders are equal if one's associated Young diagram is the transpose of the other's. To see this, let $G$ and $G_{tr}$ be the incomparability graphs of these two unit interval orders. Then we can just relabel the vertices $i$ of $G_{tr}$ as $n+1-i$ to see $G\cong G_{tr}$. Then all ascents in each coloring are preserved and $X_G(\textbf{x},q)=X_{G_{tr}}(\textbf{x},q)$. Figure <ref> shows an example of this class of natural unit interval orders.

[every node/.style=minimum size=.3cm-, outer sep=0pt]
[step=0.3cm] (-1.5001,-1.5001) grid (1.5,1.5);
(-1.5,-1.5) – (1.5,1.5);
[red,very thick] (-1.5,-1.5)–(-1.5,-0.3)–(-1.2,-0.3)–(-1.2,.9)–(0.3,.9)–(0.3,1.2)–(.9,1.2)–(.9,1.5)–(1.5,1.5);
[green,ultra thick, dashed] (-1.5,-1.5)–(-1.5,-0.3)–(-0.3,-0.3)–(-0.3,.9)–(.9,.9)–(.9,1.5)–(1.5,1.5);
[fill=yellow!30] at (-1.35,1.35) ;
[fill=yellow!30] at (-1.05,1.35) ;
[fill=yellow!30] at (-.75,1.35) ;
[fill=yellow!30] at (-0.45,1.35) ;
[fill=yellow!30] at (-0.15,1.35) ;
[fill=yellow!30] at (0.15,1.35) ;
[fill=yellow!30] at (0.45,1.35) ;
[fill=yellow!30] at (.75,1.35) ;
[fill=yellow!30] at (-1.35,1.05) ;
[fill=yellow!30] at (-1.05,1.05) ;
[fill=yellow!30] at (-.75,1.05) ;
[fill=yellow!30] at (-0.45,1.05) ;
[fill=yellow!30] at (-0.15,1.05) ;
[fill=yellow!30] at (0.15,1.05) ;
[fill=yellow!30] at (-1.35,.75) ;
[fill=yellow!30] at (-1.35,0.45) ;
[fill=yellow!30] at (-1.35,0.15) ;
[fill=yellow!30] at (-1.35,-0.15) ;

An example of the class of natural unit interval orders in Corollary <ref>

§ DISCUSSIONS

§.§ Proof of $e$-positivity of chromatic symmetric function $X_G(\mathbf{x},q)$ for Dyck paths of bounce number three

We showed that $[e_{(n-2l,l,l)}]X_G(\mathbf{x},q)$ and $[e_{(n-2l-1,l+1,l)}]X_G(\mathbf{x},q)$ for $l\in[0,\min\{a,b,c\}]$ are positive. We have encountered difficulties proving $e$-positivity of the rest of the coefficients. According to Equation <ref>, we are looking for injections from the set of $P$-tableaux $\{T^-\in\PT|\sh(T^-)=3^l2^{j+1}1^{n-3l-2j-2}\,\, \text{or}\,\, 3^{l+1}2^{j-2}1^{n-3l-2j+1}\}$ to the set $P$-tableaux $\{T^+\in\PT|\sh(T^+)=3^l2^j1^{n-3l-2j}\,\, \text{or}\,\, 3^{l+1}2^j1^{n-3l-2j-3}\}$. We essentially need an algorithm allowing us to move cells among multiple columns of a $P$-tableaux which preserves the number of inversions.

§.§ For a chromatic symmetric function $X_G(\mathbf{x},q)$ for Dyck path of an arbitrary bounce number, $[e_{(n-i,i)}]X_G(\mathbf{x},q)$ are positive for every $i$

Using the inverse Kostka numbers, we are actually able to expand a chromatic symmetric function for Dyck path $\mathbf{d}$ of an arbitrary bounce number in $e$ basis. Thus developing an algorithm more general than the one described in Section <ref> would be helpful to prove Conjecture <ref>. In particular, to find $[e_{\lambda}]X_G(\mathbf{x},q)$ for some $\lambda$ is equivalent to find the tiling of all possible Young diagrams with type $\lambda$. It would be interesting to investigate $[e_{(n-i,i)}]X_G(\mathbf{x},q)$ for every $i$, in which case the tiling of a special rim hook tabloid would always consists of two special rim hooks of length $i$ and $n-i$ (see Figure <ref>). When these two special rim hooks have opposite signs, we obtain a negative special rim hook tabloid. Thus we are looking for an injection from a negative special rim hook tabloid to a positive one by adjusting cells of the Young diagram such that the two special rim hooks agree with the signs.

[black] (0,0) – (.3,0) – (.3,2) – (2.5,2) – (2.5,2.3)– (0,2.3) – cycle;
[black] (0.3,0.6) – (.6,.6) – (.6,1.7) – (1.9,1.7) – (1.9,2);
[blue, very thick] (0.15,0.15) – (.15,.75) – (.45,.75) – (.45,1.85) – (1.75,1.85)– (1.75,2.15) – (2.35,2.15);
[blue, very thick] (0.15,0.9) – (.15,2.15) – (1.6,2.15);

A special rim hook tabloid of type $(n-i,i)$

Missing 'biblatex' package
The bibliography requires the 'biblatex' package.

journaltitleJournal of Combinatorial Theory, Series A
titleChromatic Symmetric Functions from the Modular Law

abstractWe use a Dyck path model for unit-interval graphs to study the chromatic quasisymmetric functions introduced by Shareshian and Wachs, as well as unicellular LLT polynomials, revealing some parallel structure and phenomena regarding their e-positivity. The Dyck path model is also extended to circular arc digraphs to obtain larger families of polynomials, giving a new extension of LLT polynomials. Carrying over a lot of the non-circular combinatorics, we prove several statements regarding the e-coefficients of chromatic quasisymmetric functions and LLT polynomials, including a natural combinatorial interpretation for the e-coefficients for the line graph and the cycle graph for both families. We believe that certain e-positivity conjectures hold in all these families above. Furthermore, beyond the chromatic analogy, we study vertical-strip LLT polynomials, which are modified Hall–Littlewood polynomials.
journaltitleDiscrete Mathematics
titleLLT polynomials, chromatic quasisymmetric functions and graphs with cycles
Chromatic quasisymmetric functions,Elementary symmetric functions,LLT polynomials,Orientations,Unit interval graphs,Positivity,Diagonal harmonics

journaltitleSIAM J. Discret. Math.
titleOn e-Positivity and e-Unimodality of Chromatic Quasi-symmetric Functions

journaltitleElectron. J. Comb.
titlePositivity of Chromatic Symmetric Functions Associated with Hessenberg Functions of Bounce Number 3

abstractWe compute an explicit $e$-positive formula for the chromatic symmetric function of a lollipop graph, $L_m,n$. From here we deduce that there exist countably infinite distinct $e$-positive and, hence, Schur positive, bases of the algebra of symmetric functions whose generators are chromatic symmetric functions. Finally, we resolve 6 conjectures on the chromatic symmetric function of a lariat graph, $L_n+3$.
journaltitleSIAM Journal on Discrete Mathematics
titleLollipop and Lariat Symmetric Functions

titleA directed graph generalization of chromatic quasisymmetric functions
Combinatorics (math.CO),FOS: Mathematics,FOS: Mathematics,05E05,05A05

Taylor     Francis

journaltitleLinear and Multilinear Algebra
titleA combinatorial interpretation of the inverse kostka matrix

journaltitleDiscrete Mathematics
titleIncomparability Graphs of (3 + 1)-Free Posets Are s-Positive

titleA Modular Relation for the Chromatic Symmetric Functions of (3+1)-Free Posets

abstractIn this work, we generalize and utilize the linear relations of LLT polynomials introduced by Lee (2017). By using the fact that the chromatic quasisymmetric functions and the unicellular LLT polynomials are related via plethystic substitution and thus they satisfy the same linear relations, we can apply the linear relations to both sets of functions. As a result, in the chromatic quasisymmetric function side, we find a class of e-positive graphs, called melting lollipop graphs, and explicitly prove the e-unimodality. On the unicellular LLT side, we obtain Schur expansion formulas for LLT polynomials corresponding to certain set of graphs, namely, complete graphs, path graphs, lollipop graphs and melting lollipop graphs.
journaltitleDiscrete Mathematics
titleMelting lollipop chromatic quasisymmetric functions and Schur expansion of unicellular LLT polynomials
Chromatic quasisymmetric function,-free poset,Lollipop graph,Natural unit interval order,LLT polynomials,-positivity

journaltitleJournal of Combinatorial Theory, Series A
titleOn Immanants of Jacobi-Trudi Matrices and Permutations with Restricted Position

journaltitleAdvances in Mathematics
titleA Symmetric Function Generalization of the Chromatic Polynomial of a Graph

journaltitleAdvances in Mathematics
titleChromatic Quasisymmetric Functions


# Properties of a Majorana fermion ensemble with exciton-like mass

L. E. S. Machado  Programa de Pós-graduação em Física, Universidade Federal do
Ceará, 60455-760, Fortaleza (CE), Brazil.  F. C. E. Lima
<EMAIL_ADDRESS>Programa de Pós-graduação em Física, Universidade
Federal do Maranhão, Campus Universitário do Bacanga, São Luís (MA),
65080-805, Brazil.  C. A. S. Almeida<EMAIL_ADDRESS>Programa de Pós-
graduação em Física, Universidade Federal do Ceará, 60455-760, Fortaleza (CE),
Brazil.  Departamento do Física, Universidade Federal do Ceará, 60455-760,
Fortaleza, CE, Brazil.  Institute of Cosmology, Department of Physics and
Astronomy, Tufts University, Medford, Massachusetts 02155, USA.

###### Abstract

Abstract: Considering the relativistic scenario, we dedicate our study to the
relativistic quantum description of one-dimensional Majorana fermions. Thus,
we focus on aspects related to exciton-like particles. Seeking to reach our
purpose, one analyzes the relativistic quantum mechanical system characterized
by an effective mass distribution. In this context, we adopt an exciton-like
position-dependent mass without impurity, i.e., without electromagnetic
interactions. From this perspective, one notes results of noteworthy interest
as consequences of the theory adopted. For instance, we highlight that, even
without interaction, exciton-like Majorana fermions manifest theoretically
bound states. Also, we construct a Majorana fermion ensemble with effective
mass immersed in a thermal reservoir. That allows for a thorough investigation
of the thermodynamic properties of the system. Among the thermodynamic
characteristics studied in the canonical ensemble, we focus on the Helmholtz
free energy, mean energy, entropy, and heat capacity. The numerical results
obtained for these thermodynamic properties corroborate the validity of the
Dulong-Petit law for our system.

Keywords: Majorana’s fermions. Effective mass. Excitons.

##  I Introduction

Within the microstructure of solids, a notable discrepancy in mass values is
discernible between dynamic and static systems Ref1 . This differentiation
leads us to the theory of effective mass Dresselhaus . This concept has proven
useful in describing electronic bands in the presence of perturbations, such
as external magnetic fields and impurities in semiconductors Ref2 .
Furthermore, one applies this theory to the quantum description of
superlattices Ref3 , heterostructures Ref4 , and thermoelectric materials Ref5
.

In a broad context, one reinterprets the effective mass as a mass distribution
that varies with the position Lima1 , Lima2 . In this scenario, the mass
distribution delineates the spatial coordinates of charge carriers within the
material’s energy band, thereby engendering a system characterized by
Position-Dependent Mass (PDM) Thomsen . Since the inception of the PDM
concept, this theoretical framework has demonstrated its pertinence by
comprehensively addressing a spectrum of issues across diverse domains of
physics Ref6 , Ref7 , Ref8 .

The PDM formalism was employed by von Roos Ref9 , addressing non-homogeneous
semiconductors. In pursuit of its objectives, von Roos utilizes the definition
of the kinetic energy operator to describe the electrons in semiconductors.
The exact form of this operator has been a subject of discussion due to the
ambiguity of the so-called ordering parameters Thomsen , Ref10 , Ref11 , Ref12
. A particularly intriguing discussion involving the PDM concept emerges in
the non-relativistic regime. In this scenario, there is an ambiguity related
to the symmetrization of the kinetic energy operator Ref10 , Ref11 , Ref12 ,
Gora . Cavalcante et al. Ref13  propose the most suitable symmetrization in
the non-relativistic limit using relativistic quantum mechanics. Subsequently,
works adopting the PDM concept of some systems were considered to discuss
solid-state physics problems Ref14 . Naturally, as reported in references
Ref13 , Ref14 , in the context of relativistic quantum mechanics, the PDM
concept makes it more appropriate to address solid-state physics problems
since this ambiguity does not exist in this scenario.

Adopting the PDM formalism in a relativistic quantum-mechanical context, we
explore the one-dimensional quantum-relativistic theory using the Majorana
representation with PDM. The motivation for this investigation arises from the
following inquiry: is it possible to describe an exciton-like system by
considering the Majorana representation with PDM? To achieve our goal, we
assume the fermionic description of Majorana Majorana . In the Majorana
representation, one describes the spinor from the theory as an entity such
that $\Psi^{c}=\Psi$, i.e., the spinor is real Majorana , Elliot . From this
perspective, the equality between the spinor and its conjugate suggests the
absence of a distinction between particle and antiparticle Majorana , Elliot .

The physical nature of Majorana’s fermions has no experimental comprovation.
However, it is possible to experimentally observe excitations concerning
Majorana’s condition in superconductors interface of the type wave-s Ref15
and strong-topological insulators Ref16 . Furthermore, there has been
discussion about the potential of the so-called Majorana Zero Modes (MZM)
applied to topological quantum computing Ref17 , characterizing them as a
topic of growing interest in condensed matter physics.

The Majorana condition is also applicable in semiconductor theory, where the
study of charge carriers leads to the concept of excitons. Briefly, excitons
are structures formed by the binding of an electron at the conduction band and
a hole at the valence band by Coulomb’s interaction Dresselhaus . Thus, these
excitations carry energy and momentum, presenting a null total charge.
Considering that the exciton study requires effective mass approximation Ref18
, one can adopt the PDM concept in the relativistic theory. In this work, one
studies a one-dimensional exciton-like fermionic system with PDM. We undertake
this investigation to comprehend how the eigenstates of Majorana’s fermions
with PDM behave in the absence of impurities.

This paper is structured as follows: we systematically examine the aspects of
PDM formalism in Sec. II. Subsequently, in Sec. III, an analysis of the
relativistic quantum system within Majorana’s representation employing an
exciton-like PDM is performed. One builds an ensemble of Majorana’s particles
with PDM at a thermal bath to study the thermodynamic properties of the system
in Sec. IV. Finally, in Sec. V, we announce our discoveries.

##  II On the effective mass

The theory of effective mass emerges in the description of graded mixed
semiconductors Vliet . In the derivation of this theory, one presupposes that
the variation in chemical composition occurs slowly enough for configuration
changes to become appreciable on extensive scales compared to the lattice
constant Ref9 , Ref18 , Vliet , Slater . Gora Gora , Gora2 , Williams Gora ,
Gora2 , van Vliet Vliet , Vliet2 , and Marshak Vliet , Vliet2  propose a
theory based on Schrödinger’s equation of a compound semiconductor in
Wannier’s representation Ref18 . Following these discussions, one can
demonstrate that the Hamiltonian of a PDM has various ways in the non-
relativistic regime Ref9 , Ref10 , Ref11 , Ref12 , Gora . However, Morrow
Morrow  highlighted that the non-singularity of the Hamiltonians proposed by
Gora and Williams Gora , Gora2  and van Vliet and Marshak Vliet , Vliet2
persists when the wavelengths of the envelope functions are not short compared
to the distance over which the chemical composition changes appreciably.
Nevertheless, this precisely occurs in the case of excited surface impurity
state wavefunctions that extend over hundreds of lattice sites.

Using Bargmann’s theorem Bargmann , von Roos derives a unified Hamiltonian for
PDM problems Ref9 . However, it was only in 1997 that Cavalcante et al. Ref9
a priori resolved the ambiguity in the non-relativistic Hamiltonian. In Ref.
Ref9 , a relativistic approach is adopted, seeking the non-relativistic limit
for the theory. In this case, the non-relativistic Hamiltonian for the system
with PDM assumes the form proposed by Li and Kuhn Ref12 . Since these
discussions, several works have emerged employing the theories with effective
mass. For instance, one uses the PDM concept to investigate the quantum
information of a solitonic mass distribution Lima1 . Meanwhile, one can use
the PDM in explorations of non-Hermitian theories Lima2 . Thus, motivated by
these applications, let us commence our investigation by studying the equation
of motion for an electron in the energy band of a semiconductor with a wave
vector $k$ associated with the group velocity

$\displaystyle v_{g}=\dfrac{1}{\hbar}\dfrac{dE}{dk},$ (1)

where $E$ corresponds to the system’s dispersion energy. Considering the
action of an electric field $\epsilon$ on this electron over a time interval
$\delta t$, one obtains the work $\delta E$, i.e.,

$\displaystyle\delta E=-e\epsilon v_{g}\delta t.$ (2)

which leads us to

$\displaystyle\delta E=\hbar v_{g}\delta k.$ (3)

Comparing the equations 1 and 2, we arrive at

$\displaystyle\delta k=-e\dfrac{\epsilon}{\hbar}\delta t,$ (4)

and

$\hbar\dfrac{dk}{dt}=-e\epsilon.$ (5)

One can write the Eq. (5) in terms of the external force, namely,

$\displaystyle F=\hbar\dfrac{dk}{dt}.$ (6)

The Eq. (6) represents a crucial relationship determining the external force
exerted on the electron within a crystal. The correlation between energy and
wave vector manifests through the expression $E=\hbar^{2}k^{2}/2m$, where
$\hbar^{2}/2m$ defines the curvature of the energy dispersion $E(k)$.

We employ this approach in describing hole-electron pairs that constitute
excitons in the energy bands. In this case, the holes possess the same
expressions as the electrons for wave vector and energy but with opposite
signs. Thus, differentiating the Eq. (1) concerning time, one obtains

$\displaystyle\dfrac{dv_{g}}{dt}=\dfrac{1}{\hbar^{2}}\dfrac{d^{2}E}{dk^{2}}F.$
(7)

Comparing the Eq. (7) with the definition coming from Newton’s second law, we
conclude that the effective mass is

$\displaystyle m(k)=\hbar^{2}\bigg{(}\dfrac{d^{2}E}{dk^{2}}\bigg{)}^{-1}.$ (8)

###  II.1 The effective mass for the one-dimensional exciton-like coupling

Using the definition $m(k)$, we will adopt the dispersion energy $E(k)$
considering the tight-binding approximation or strong binding. The tight-
binding approximation follows the following premises Dresselhaus :

  1. 1.

The eigenvalues and eigenfunctions of energy are known for an electron in an
isolated atom;

  2. 2.

The atoms of the material remain spaced apart such that each electron finds a
specific atomic site;

  3. 3.

One can approximate the periodic potential from the lattice by a superposition
of interactions.

Based on the perturbation theory emerges the tight-binding theory Foulkes ,
where the system’s Hamiltonian is $H=H_{0}+H^{\prime}$. Here, $H_{0}$ is the
atom’s Hamiltonian at the $n$-th site and

$\displaystyle H^{\prime}=V(r)-U(r-R_{n}).$ (9)

The perturbed Hamiltonian for an electron is the difference between its
periodic potential and the atom’s potential in the surrounding neighborhood.
We can then define an atom’s wave function $\varphi(r-R_{n})$ for localized in
the position $r$, where $R_{n}$ is the lattice vector, both measured
concerning the particle. Thus, one can linearly combine these functions as a
Bloch function, namely,

$\displaystyle\psi_{n}=N^{-1/2}\sum_{n}e^{ik\cdot R_{n}}\varphi(r-R_{n}).$
(10)

Using Bloch’s function (11), one obtains the dispersion energy, i.e.,

$\displaystyle E(k)=-\alpha-2\gamma\cos{ka},$ (11)

where $\alpha$ and $\gamma$ are parameters of the crystal lattice. The
effective mass for this dispersion is

$\displaystyle m(k)=\dfrac{\hbar^{2}}{2\gamma a}\sec(ka).$ (12)

We present the behavior of the effective mass $m(k)$ in Fig. 1(a).

Note that the mass is at the reciprocal space $k$. Nevertheless, we wish to
obtain a corresponding PDM. So, to reach our purpose, Fourier’s transform is
employed, considering the range corresponding to the first Brillouin zone111We
consider the first Brillouin zone because the essential information on the
energy and momentum of the crystal are in this domain. $(-\pi/a\leq
k\leq\pi/a)$. Thus, one obtains the exciton-like PDM, i.e.,

$\displaystyle
m(x)=\frac{1}{2\gamma\sqrt{2\pi}}\int_{-\pi/a}^{\pi/a}\text{sec}(ka)\cdot\text{e}^{-ikx}dk.$
(13)

One can obtain the result of the mass integration (13) using the residue
theorem 222The residue theorem is a method for calculating integrals used in
analytical functions along closed paths that generalizes Cauchy’s formula
Nikiforov .. In this case, the residue theorem leads us to

$\displaystyle m(x)=-\dfrac{\sqrt{2\pi}}{\gamma}\sin{\left(\dfrac{\pi
x}{2}\right)}.$ (14)

For this calculation, we have assumed $a=1$. To obtain a positively defined
PDM to the first Brillouin zone, one considers $x\in[-1,0]$. We display the
behavior correponding PDM in Fig. 1(b).

Figure 1: (a) Behavior of the effective mass $m(k)$. (b) PDM for several
values of $\gamma$.

##  III Majorana’s fermions with exciton-like PDM

Let us focus our study on a specific class of particles, namely Majorana’s
fermions. Investigations and manipulations of Majorana’s fermions in condensed
matter systems are currently a topic of significant theoretical and
experimental interest Alicea1 , Alicea2 . As reported by Alicea Alicea1 ,
Alicea2 , Majorana’s fermions constitute "half" of an usual fermion. To create
a fermion $f$, one requests the superposition of two Majorana’s modes
$\gamma_{1,2}$, where these modes are arbitrarily separated Alicea2 . Thus,
Majorana’s fermions admit bound states. That naturally leads to the question:
when constructing a one-dimensional theory with PDM and without impurities,
preserves the presence of bound states in the prototype? To address this
question, let us adopt the one-dimensional Majorana representation with an
effective mass distribution. In this scenario, one describes Majorana’s
fermions by the expression

$\displaystyle[\gamma^{\mu}(p_{\mu}-eA_{\mu})-m(x)]\psi(x,t)=0\hskip
14.22636pt\text{with}\hskip 14.22636pt\mu=0,1;$ (15)

where $m(x)$ is the PDM, $p_{\mu}$ is the particle’s momentum, and
$A_{\mu}=[V(x),0]$ is the gauge field with $V(x)$ corresponding to the
electric potential. One highlights that by adopting $A_{\mu}=[V(x),0]$, only
electric interactions are considered. Furthermore, the metric of the system is

$\displaystyle g_{\mu\nu}=\begin{pmatrix}+1&0\\\ 0&-1\end{pmatrix}.$ (16)

Adopting the previously mentioned definitions, one writes Dirac’s equation as

$\displaystyle[\gamma^{0}p_{0}+\gamma^{1}p_{x}-\gamma^{0}V(x)-m(x)]\psi(x,t)=0.$
(17)

Now, let us use Majorana’s representation, namely,

$\displaystyle\gamma^{0}=\sigma^{2}=\begin{pmatrix}0&-i\\\ i&0\\\
\end{pmatrix}\quad\text{and}\quad\gamma^{1}=i\sigma^{3}=\begin{pmatrix}i&0\\\
0&-i\\\ \end{pmatrix},$ (18)

with $(\gamma^{0})^{2}=\mathbb{I}$.

For this system, the Hamiltonian operator is

$\displaystyle\hat{H}=\alpha p_{x}+\beta m(x)+V(x),$ (19)

where $\alpha=\gamma^{0}\gamma^{1}$ and $\beta=\gamma^{0}$.

To achieve our purpose, allow us to assume that the wave function $\psi(x,t)$
is

$\displaystyle\psi(x,t)=\begin{pmatrix}\varphi(x)\\\ \chi(x)\\\
\end{pmatrix}\text{e}^{-iEt}.$ (20)

This separation of variables leads us to the system of equations:

$\displaystyle
i\bigg{[}\dfrac{d\chi(x)}{dx}-m(x)\chi(x)\bigg{]}=(E-V)\varphi(x),$ (21)

and

$\displaystyle
i\bigg{[}\dfrac{d\varphi(x)}{dx}+m(x)\varphi(x)\bigg{]}=(E-V)\chi(x).$ (22)

Decoupling the Eqs. (21) and (22), one obtains

$\displaystyle-\varphi^{\prime\prime}(x)+\left[-m^{\prime}(x)-2EV(x)+V^{2}(x)\right]\varphi(x)=[E^{2}-m^{2}(x)]\varphi(x),$
(23)

and

$\displaystyle\chi^{\prime\prime}(x)-\left[-m^{\prime}(x)-2EV(x)+V^{2}(x)\right]\chi(x)=[E^{2}-m^{2}(x)]\chi(x).$
(24)

In this scenario, the prime notation denotes the derivative concerning the
position variable. We are interested in studying the description of fermions
with PDM, such as Majorana’s particles with exciton-like PDM without
impurities. Thus, one should consider $V(x)=0$. This consideration leads us to
the Schrödinger-like equation, i.e.,

$\displaystyle-\varphi^{\prime\prime}(x)+[m^{2}(x)-m^{\prime}(x)]\varphi(x)=E^{2}\varphi(x).$
(25)

Comparing Eq. (25) with the usual Schrödinger theory, it is noted that even
$V(x)=0$, the system will exhibit effective interaction333This interaction is
called effective potential. due to the mass distribution. In this case, the
effective potential is

$\displaystyle V_{\text{eff}}=m^{2}(x)-m^{\prime}(x).$ (26)

Replacing the definition of effective potential in Eq. 26, we conclude that

$\displaystyle
V_{\text{eff}}=\dfrac{2\pi}{\gamma^{2}}\left[\sin^{2}{\left(\dfrac{\pi
x}{2}\right)}+\dfrac{\gamma}{4}\sqrt{2\pi}\cos{\left(\dfrac{\pi
x}{2}\right)}\right].$ (27)

This potential is half of a Pöschl-Teller-like potential Radovanovic , Sun ,
Miranda . We expose this potential in Fig. 2.

Figure 2: The effective potential of the system.

Analyzing the effective potential (27), we find that Majorana’s equation for
the spinor $\varphi(x)$ with exciton-like PDM is

$\displaystyle-\varphi^{{}^{\prime\prime}}(x)+\left[\dfrac{2\pi}{\gamma^{2}}\sin^{2}{\left(\dfrac{\pi
x}{2}\right)}+\dfrac{\pi\sqrt{2\pi}}{2\gamma}\cos{\left(\dfrac{\pi
x}{2}\right)}\right]\varphi(x)=E^{2}\varphi(x),$ (28)

where $x\in[-1,0]$.

Equation (28) only admits numerical solutions in the interval $[-1,0]$444This
range is due to the mass profile corresponding to the first Brillouin zone..
Therefore, it is necessary to employ a numerical method to accurately describe
the analytical solution of the wave function $\varphi(x)$. Thus, we utilize
the finite element method555One uses the finite element method to find
approximate solutions to differential equations Hildebrand . This method
involves discretizing the space corresponding to the interval $[-1,0]$ with
steps of $10^{-2}$ Hildebrand . Thus, we determine the solutions within each
finite element in the range., adopting the Dirichlet condition666The Dirichlet
boundary condition specifies the values which function must take on a specific
part of the boundary in the domain where the differential equation admit
solution Hildebrand .. Considering this approach, one obtains the numerical
system’s solutions. We expose these solutions in Fig. 3.

(a) (b)

(c) (d)

Figure 3: Solução numérica de $\varphi(x)$ para os quatro primeiros auto
estados de energia.

By numerical inspection, we observe that the system admits bound states. That
is an attractive result once all interactions in relativistic theory arise
from mass distribution. Consequently, we can infer that Majorana’s fermion
with exciton-like PDM induces a self-interaction leading to quantizable
eigenstates. One presents the detailed numerical quantization of the first one
hundred energy eigenvalues in Fig. 4.

(a) (b)

(c) (d)

Figure 4: The numerical solution of the energy eigenvalues.

##  IV Thermodynamic properties

From now on, we will study the thermodynamic properties of the system. To
reach our purpose, let us assume, a priori, the construction of a Majorana
fermion ensemble with exciton-like PDM. In this scenario, the partition
function Oliveira  is

$\displaystyle Z=\sum_{n=0}^{\infty}\text{e}^{-\beta E_{n}},$ (29)

where $\beta=1/k_{B}T$. Here, $k_{B}$ is Boltzmann’s constant, and $T$ is the
thermal bath temperature.

Using Eq. (29), we can obtain all the thermodynamic properties of a Majorana
fermion ensemble with exciton-like PDM in a thermal bath. These properties are
Helmholtz free energy, internal energy, entropy, and heat capacity. These
quantities are, respectively,

$\displaystyle\mathcal{F}=-\frac{1}{\beta}\text{ln}Z_{N},\hskip
14.22636ptU=-\frac{\partial}{\partial\beta}\text{ln}Z_{N},\hskip
14.22636ptS=k_{B}\beta^{2}\frac{\partial F}{\partial\beta},\hskip
14.22636pt\text{and}\hskip 14.22636ptC_{v}=-k_{B}\beta^{2}\frac{\partial
U}{\partial\beta}.$ (30)

Here, $Z_{N}$ is the total partition function for the ensemble of
indistinguishable N-fermions in a thermal bath,

Note that we obtain the eigenstate results numerically. Thus, we must adopt a
numerical approach to access the thermodynamic properties. In this form, in
the subsection below, one presents the numerical results of the thermodynamic
properties.

###  IV.1 Numerical results

Investigating the thermodynamic properties, we adopted the numerical solution
shown in Fig. 4. Using the numerical solution for the energy spectrum, i.e.,
the numerical expression for $E_{n}$ [Figure 4], we explored the partition
function associated with the Majorana fermion ensemble with PDM using the
numerical interpolation method with steps of $10^{-2}$. Thus, we elucidated
the behavior of the partition function (29). Using the numerical solution of
the partition function, together with definitions of thermodynamic properties,
we obtain results as displayed in Eq. (30). In this way, one investigates
numerically the thermodynamic properties. In Fig. 5 are the results of the
thermodynamic properties.

(a) (b)

(c) (d)

Figure 5: Thermodynamic properties of Majorana fermions with exciton-like PDM.

In Figs. 5(a), 5(b), 5(c), and 5(d), we numerically plotted all profiles of
thermal quantities as functions of temperature $T$ for different lattice
parameter, i.e., $\gamma$. For one-dimensional Majorana fermions with exciton-
like PDM, one notes that Helmholtz’s function $\mathcal{F}(T)/N$ decreases for
high-temperature values and exhibits higher values when the lattice parameter
is small. In contrast, the mean energy $U(T)/N$ shows an almost linear
behavior. As the lattice parameter increases, the curve of the mean energy
undergoes a slight decrease. Meanwhile, the entropy $S(T)/N$ increases with
$T$ and decreases as the lattice parameter increases. The heat capacity
$C_{V}(T)/N$ tends toward an asymptotic behavior fixed at the unit value when
$T$ increases.

From the analysis of the profiles of the curves of the aforementioned
thermodynamic functions, we extract information about the thermal behavior of
the one-dimensional Majorana fermion ensemble with exciton-like PDM.
Physically, it is noticeable that Helmholtz’s free energy has a critical point
for low temperatures, i.e., $T\simeq 0$. Thus, one notes that the energy
available for performing work decreases rapidly when the temperature
increases. On the other hand, we note that when this ensemble tends to thermal
equilibrium in the reservoir, the system’s mean energy continuously increases.
Furthermore, the change in the lattice’s parameter tends to slow the
thermodynamic behavior until the system reaches equilibrium. Finally, we
observe that regardless of the lattice parameter, all quantities assume higher
values at low temperatures, as predicted in the literature. Lastly, one notes
that the well-known Dulong-Petit law is satisfied once that $C_{v}(T\to
0)/N\simeq k_{B}$.

##  V Final remarks

We investigated a relativistic quantum system to describe Majorana fermions
with a mass distribution. The PDM concept was adopted to reach the purpose.
Furthermore, we have assumed that the quantum mechanical system is without
impurities. In this scenario, one adopts that Majorana fermions exhibit an
exciton-like mass distribution to study the quantum dynamics of the theory.
Besides, we employ statistical mechanics arguments to obtain the Helmholtz
free energy, mean energy, entropy, and specific heat.

Assuming the quantum theory, we investigate Majorana’s fermions with an
exciton-like PDM. In this scenario, one notes that the mass distribution
induces an effective interaction purely due to the particle’s mass, i.e.,
$V_{\text{eff}}=m^{2}(x)-m^{\prime}(x)$. That potential at the first Brillouin
zone results in half of a Pöshl-Teller-like self-interaction, confining
Majorana fermions and generating bound states. In other words, even in the
absence of electromagnetic interaction, i.e., $A_{\mu}=(0,0)$, the system will
exhibit the existence of bound states purely due to the presence of the mass
distribution. Thus, if one-dimensional Majorana fermions possess bound states
with exciton-like mass, it admits bound states even in the case of free
particles.

Furthermore, numerical results demonstrate that when the lattice parameter
$\gamma$ is small, regions of maximum probability are near the boundary from
the first Brillouin zone. Meanwhile, as the atoms become more spaced in the
crystal lattice (i.e., as $\gamma$ increases), the probability regions exhibit
a symmetrical probability distribution in the first Brillouin zone.

Considering a Majorana particle ensemble with exciton-like PDM in a thermal
bath, we noted that the Helmholtz free energy has a maximum capacity to
perform work as $T\to 0$ and decreases as the ensemble temperature increases.
Thus, one concludes that the amount of free energy available for work
decreases rapidly. Moreover, this available free energy is higher when the
lattice parameter is smaller. Meantime, the mean energy continuously increases
and is always higher for smaller $\gamma$ values. Additionally, the thermal
variation of the particle ensemble tends to increase until reaching thermal
equilibrium in the considered system. Indeed, this behavior resonates in the
heat capacity, where reaching thermodynamic equilibrium remains constant,
following the Dulong-Petit law. Finally, one can note that the system’s
entropy adheres to the basic principles of thermodynamics with a monotonically
increasing behavior independent of the lattice parameter. However, the
system’s entropy increases rapidly for small $\gamma$ values. In this regime,
the thermodynamic system will have more available energy.

A prospect of this work is the construction of an excitons ensemble with PDM
trapped in an abrupt heterojunction under the non-relativistic limit. We hope
to undertake this study soon.

##  VI Acknowledgment

The authors express their gratitude to FUNCAP, FAPEMA, CNPq, and CAPES
(Brazilian research agencies) for their invaluable financial support. L. E. S.
M. is supported by FUNCAP. F. C. E. L. is supported by FAPEMA BPD-05892/23. C.
A. S. A. is supported by CNPq 309553/2021-0, CNPq/Produtividade. Furthermore,
C. A. S. A. is grateful to the Print-UFC CAPES program, project number
88887.837980/2023-00, and acknowledges the Department of Physics and Astronomy
at Tufts University for hospitality.

##  VII Conflicts of Interest/Competing Interest

All the authors declared that there is no conflict of interest in this
manuscript.

##  VIII Data Availability

The datasets generated during and/or analysed during the current study are
available from the corresponding author on reasonable request.

## References

  * [1] F. Fedele, P. Suryanarayana and A. Yavari, J. Mech. Phys. Solids 179 (2023) 105393.
  * [2] M. S. Dresselhaus, G. Dresselhaus, S. B. Cronin and A. G. S. Filho, Solid states properties from bulk to nano, vol. 1, first ed., Springer, Heidelberg, 2018.
  * [3] M. G. Burt, J. Phys.: Condens. Matt. 4 (1992) 6651.
  * [4] A. Sasaki, Phys. Rev. B, 30 (1984) 7016.
  * [5] D. Jena, Quantum physics of semiconductor materials and devices, (Oxford University Press, Oxford, 2022).
  * [6] Y. Pei, A. D. LaLonde, H. Wang and G. J. Snyder, Energy Environ. Sci. 5 (2012) 7963.
  * [7] F. C. E. Lima, Ann. Phys. 442 (2022) 168906.
  * [8] F. C. E. Lima, L. N. Monteiro and C. A. S. Almeida, Physica E 150 (2023) 115682.
  * [9] J. Thomsen, G. T. Einevoll, and P. C. Hemmer, Phys. Rev. B 39 (1989) 12783.
  * [10] J. Flores, G. Solovey and G. Salvador, Am. J. Phys. 71 (2003) 721.
  * [11] J. Hadjidemetriou, Advances in Astronomy and Astrophysics 5 (1967) 131.
  * [12] H. A. Bethe, Phys. Rev. Lett. 56 (1986) 1305.
  * [13] O. von Roos, Phys. Rev. B 27 (1982) 7547.
  * [14] D. J. BenDaniel and C. B. Duke, Phys. Rev., 152 (1966) 683.
  * [15] Q. G. Zhu and H. Kroemer, Phys. Rev. B 27 (1983) 3519.
  * [16] T. L. Li and K. J. Kuhn, Phys. Rev. B 47 (1993) 12760.
  * [17] T. Gora and F. Williams, Phys. Rev. 177 (1969) 1179.
  * [18] F. S. A. Cavalcante, R. N. Costa Filho, J. Ribeiro Filho, C. A. S. De Almeida and V. N. Freire, Phys. Rev. B 55 (1997) 1326.
  * [19] R. Renan, M. H. Pacheco and C. A. S. Almeida, J. Phys. A: Math. Gen. 33 (2000) L509.
  * [20] E. Majorana, II Nuovo Cimento 5 (1937) 171.
  * [21] S. R. Elliot and M. Franz, Rev. Mod. Phys. 87 (2015) 137.
  * [22] R. Lutchyn, T. D. Sau and S. D. Sarma, Phys. Rev. Lett. 105 (2010) 077001
  * [23] L. Fu and C. L. Kane, Phys. Rev. Lett. 100 (2008) 096407.
  * [24] C. Nayak, S. H. Simon, M. Freedman and S. D. Sarma, Rev. Mod. Phys. 80 (2008) 1083.
  * [25] G. H. Wannier, Phys. Rev. 52 (1937) 191.
  * [26] C. M. van Vliet and A. H. Marshak, Phys. Rev. B 26 (1982) 6734.
  * [27] J. C. Slater, Phys. Rev. 76 (1949) 1592.
  * [28] T. Gora and F. Williams, in II-VI Semiconducting Compounds, edited by D. G. Thomas (Benjamin, New York, 639, 1967).
  * [29] C. M. van Vliet and A. H. Marshak, in Proceedings of the International Workshop on Solid State Devices, (Delhi, India, November, 1981).
  * [30] R. A. Morrow, Phys. Rev. B 35 (1987) 8074.
  * [31] V. Bargmann, Ann. Math. 59 (1954) 1.
  * [32] W. M. C. Foulkes and R. Haydock, Phys. Rev. B 39 (1989) 12520.
  * [33] A. F. Nikiforov and V. B. Uvarov, Special functions of mathematical physics: A unified introduction with applications, (Springer Basel AG, Moscow, 1988).
  * [34] J. Alicea, Rep. Prog. Phys. 75 (2012) 076501.
  * [35] J. Alicea, Phys. Rev. B 81 (2010) 125318.
  * [36] J. Radovanović, V. Milanović, Z. Ikonić and D. Indjin, Phys. Lett. A 269 (2000) 179.
  * [37] G. -H. Sun, M. A. Aoki and S. -H. Dong, Chin. Phys. B 22 (2013) 050302
  * [38] M. G. Miranda, G. -H. Sun and S. -H. Dong, Int. J. Mod. Phys. E 19 (2010) 123.
  * [39] F. B. Hildebrand, Introduction to Numerical Analysis, (Dover Publications, Inc., 2nd ed., New York, 1956).
  * [40] R. R. S. Oliveira, A. A. A. Filho, F. C. E. Lima, R. V. Maluf and C. A. S. Almeida, Eur. Phys. J. Plus 134 (2019) 495.