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\definecolor{darkblue}{rgb}{0,0,0.7} % darkblue color
\definecolor{green}{RGB}{57,181,74} % green color
\newcommand{\darkblue}{\color{darkblue}} % darkblue command
\newcommand{\green}{\color{green}} % green command
\newcommand{\red}{\color{red}} % red command
\newcommand{\defn}[1]{\emph{\darkblue #1}} % emphasis of a definition

% Numbers
\newcommand{\NN}{\mathbb{N}}
\newcommand{\QQ}{\mathbb{Q}}
\newcommand{\CC}{\mathbb{C}}
\newcommand{\ZZ}{\mathbb{Z}}
\newcommand{\RR}{\mathbb{R}}
\newcommand{\PP}{\mathbb{P}}
\newcommand{\KK}{\mathbb{K}}

\newcommand{\arm}{a}
\DeclareMathOperator{\leg}{\ell}
\DeclareMathOperator{\area}{area}
\DeclareMathOperator{\ssim}{sim}
\DeclareMathOperator{\ddef}{def}
\DeclareMathOperator{\dist}{dist}

\newcommand{\SSYT}{\mathrm{SSYT}}
\newcommand{\SYT}{\mathrm{SYT}}

\newcommand{\vlmin}{v_\lambda^{-}}
\newcommand{\vlmax}{v_\lambda^{+}}
\newcommand{\vlminc}{v^{-} (c, \lambda)}
\newcommand{\vlmaxc}{v^{+} (c, \lambda)}
\newcommand{\vmminc}{v^{-} (c, \mu)}
\newcommand{\vmmaxc}{v^{+} (c, \mu)}

\newcommand{\areap}[2]{\area_{#1} (#2)}
\newcommand{\simp}[2]{\ssim_{#1} (#2)}
\newcommand{\defp}[2]{\ddef_{#1} (#2)}

\newcommand{\arealm}{\areap{\lambda}{\mu}}
\newcommand{\simlm}{\simp{\lambda}{\mu}}
\newcommand{\Alqt}{A_{\lambda} (q,t)}
\newcommand{\defthm}{\defp{\theta}{\mu}}
\newcommand{\deflm}{\defp{\lambda}{\mu}}
\newcommand{\simthm}{\simp{\theta}{\mu}}
\newcommand{\Athqt}{A_{\theta} (q,t)}

\usepackage{tikz}



\newtheorem{theorem}{Theorem}
\newtheorem{definition}{Definition}[section]
\newtheorem{proposition}[definition]{Proposition}
\newtheorem{lemma}[definition]{Lemma}
\newtheorem{conjecture}[theorem]{Conjecture}


%% define your title in the usual way
\title{Deficit and $(q,t)$-symmetry in triangular Dyck paths}

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%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
\author{Loïc Le Mogne\thanks{\href{mailto:loic.le-mogne@universite-paris-saclay.fr}{loic.le-mogne@universite-paris-saclay.fr}.}\addressmark{1} \and Viviane Pons\thanks{\href{mailto:viviane.pons@lisn.upsaclay.fr}{viviane.pons@lisn.upsaclay.fr}. Supported by Project PAGCAP ANR-21-CE48-0020.}\addressmark{1}}

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\address{\addressmark{1}CNRS, IRL CRM Montréal, Canada -- Université Paris-Saclay, CNRS,
Laboratoire Interdisciplinaire des Sciences du Numérique, Orsay, France}

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\received{\today}

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\abstract{We study the $(q,t)$-enumeration of triangular Dyck paths considered by Bergeron and Mazin. To do so, we introduce the notion of \emph{triangular} and \emph{sim-sym} tableaux and the \emph{deficit} statistic which is a new interpretation of the \emph{dinv}. We use it to obtain new results and proofs on triangular $2$-partitions and an interesting conjecture for a certain lattice interval $(q,t,r)$-enumeration.}

%% put your French abstract here, or comment this out if you don't have one
\resume{Nous étudions la symétrie $(q,t)$ dans l'énumération des chemins de Dyck triangulaires décrits par Bergeron et Mazin. Nous introduisons les notions de \emph{tableau triangulaire} et \emph{sim-sym} ainsi que la statistique du \emph{deficit} qui ré-interprète le \emph{dinv}. Nous utilisons cela pour obtenir de nouveaux résultats et nouvelles démonstrations pour les $2$-partages ainsi qu'une conjecture sur un comptage trivarié $(q,t,r)$ des intervalles de certains treillis. }

%% put your keywords here, or comment this out if you don't have them yet
\keywords{triangular partitions, $(q,t)$-Catalan, Dyck paths}

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\begin{document}

\maketitle
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\section{Introduction}

In~\cite{Berg1}, Bergeron and Mazin study a certain family of partitions that they call \defn{triangular partitions}, motivated by the results of~\cite{BHMPS}. A triangular partition is the maximal partition lying under a given line. The sub-partitions of a triangular partition are called the triangular Dyck paths, generalizing classical and rational Dyck paths. The $(q,t)$-enumeration of paths, such as the $(q,t)$-Catalan numbers~\cite{Hag}, has raised interesting combinatorial questions in recent years, related in particular to representation theory. The Negut formula (see for instance~\cite{Ber22, GHSR20, GN15}) gives a $(q,t)$-enumeration of the sub-partitions of any given partition but the signification of each $(q,t)$ monomial is not clear and the coefficients are not always positive. In a recent work generalizing the \emph{shuffle theorem}~\cite{BHMPS}, the authors found out a combinatorial interpretation when the partition is triangular, using two statistics generalizing the classical \emph{area} and \emph{dinv} of a Dyck path. Indeed, in this case the $(q,t)$ polynomials appear as the coefficients of some symmetric functions and are by nature symmetric themselves. More generally, Conjecture 7.1.1 from~\cite{BHMPS} states that the Negut formula gives positive coefficients if the partition lies under a certain convex curve. As expressed in~\cite{Ber22}, Bergeron further conjectures that this condition is not only sufficient but necessary. Moreover, the $(q,t)$-enumeration appears to be \emph{Schur positive} in the triangular case~\cite[Conjecture 1]{Berg1}.

In terms of combinatorics, this raises two (very) difficult questions: finding an elementary, combinatorial proof of the $(q,t)$-symmetry for  the enumeration of triangular Dyck paths and proving that they are Schur positive. This includes the classical and rational Dyck path cases, which are special cases of triangular Dyck paths. This is the problem that motivates this extended abstract. Our main results are stated in \cref{thm:alqt2,thm:qtrsym} where we prove the conjecture in the (limited) case of $2$-partitions and extend the symmetry to a certain interval enumeration.  Moreover, we introduce new combinatorial tools such as the \emph{triangular} and \emph{sim-sym} tableaux as well as the \emph{deficit} statistic which explore the combinatorial interpretation of the \emph{dinv} statistic. This led us to state \cref{conj:lattice} in relation to interval enumeration in the generalized Tamari lattices of~\cite{PRV15}.

The paper is organized as follows. \cref{sec:background} recalls the basic notions related to partitions and the definition and main properties of triangular partitions from~\cite{Berg1}. In \cref{sec:triangular}, we define and explore some new combinatorial tools on triangular partitions, namely the \emph{triangular tableau}, the \emph{deficit} statistic and the \emph{sim-sym} tableaux. The main result of this section is \cref{prop:deficit} where we prove that the \emph{deficit} statistic actually gives the \emph{dinv}. \cref{sec:2parts} uses these tools to obtain explicit results on $2$-partitions: \cref{thm:alqt2,thm:simsym}. Finally, the lattice enumerations are succinctly presented in \cref{sec:lattice} in order for us to state \cref{conj:lattice} and \cref{thm:qtrsym}.

\section{Background}
\label{sec:background}

\subsection{Partitions and Young tableaux}

%We start by fixing the terminology used in the paper and recall some very classical definitions.

\begin{definition}
A \defn{partition}\footnote{This is actually an integer partition. As there is no ambiguity in the paper, we simply call them partitions.} $\lambda$ of size $n$ is a tuple $\lambda_1 \geq \lambda_2 \geq \dots \lambda_k > 0$ such that \linebreak $|\lambda | := \sum_i^k \lambda_i = n$. For convenience, we consider partitions ending with an infinite number of $0$-parts, $\lambda_j := 0$ for $j > k$.  A \defn{$k$-partition} is a partition $\lambda = (\lambda_1, \dots, \lambda_{k'})$ with $k' \leq k$. We say that a partition $\mu$ is a \defn{sub-partition} of another partition $\lambda$ if $\mu_i \leq \lambda_i$ for all $i$.
\end{definition}

As usual, we represent partitions by their \defn{Ferrers diagram} (French style) such that each value $\lambda_i$, from bottom to top, corresponds to a line of $i$ square cells as shown on \cref{fig:young}. 

% définition racourcie
\begin{definition}
A \defn{standard Young tableau} on a partition $\lambda$ is an integer filling of the Ferrers diagram of $\lambda$ (we place a number in each cell) such that the numbers are strictly increasing along each row and each column. We identify the cells of $\lambda$ by their coordinates in the Ferrers diagram (the cell $(0,0)$ is the bottom left-most cell). We consider a tableau $\theta$ as a function from the cells of $\lambda$ to $\{1, \dots, |\lambda|\}$ and write $\theta(c)$ the value of the cell $c$ in $\theta$.
\end{definition}

% Definition with Semi standard
%\begin{definition}
%The (French) \defn{Young diagram} of a partition is a square diagram such that the number of squares on the $i$-th line from bottom to top is equal to $\lambda_i$. A \defn{Young tableau} on a partition $\lambda$ is an integer filling of the Young diagram of $\lambda$. A Young tableau is said to be \defn{semi-standard} if the numbers are strictly increasing along each row and weakly increasing along each column. It is said to be \defn{standard} if it is also strictly increasing along the columns and filled with the numbers $1$ to $n$ where $n$ is the size of $\lambda$. We denote by $\SSYT(\lambda)$ the set of semi-standard Young tableau over $\lambda$, and by $\SYT(\lambda)$ the set of standard Young tableaux of $\lambda$. Finally, for a Young tableau $\theta$, we write $\theta_{i}$ for the number of $i$ appearing in the tableau. 
%\end{definition}

See \cref{fig:young} for an example. \defn{Schur functions} form a famous basis of symmetric functions indexed by partitions. Any symmetric polynomial can thus be expressed as a sum of Schur functions. A symmetric polynomial is said to be \defn{Schur positive} if all the coefficients in its expansion are positive integers. For the precise definition and properties of Schur functions, we refer the reader to~\cite{Mcdo}. 


%We represent partitions through their (French style) \defn{Young diagram} as shown on \cref{fig:young}, \emph{i.e.}, 



%\textbf{A partition} $\lambda$ is a tuple  $\lambda= (\lambda_{1},\lambda_{2}, ... , \lambda_{k})$ such that $\lambda_{1} \geq \lambda_{2} \geq ... \geq \lambda_{k}\geq 0$, for $k$ an integer. We say that $\lambda$ is a \textbf{2-partition} if $k=2$. Furthermore, we also have $\lambda= (\lambda_{1},\lambda_{2}, ... , \lambda_{k},0)$ (i.e. we can add or erase as many zeroes as we like), meaning that, for example, $\mu = (n)$ is a 2-partition. However, we often omits the zeros. The \textbf{conjugate} of $\lambda$ is the partition $\lambda' = (\#\{\lambda_{k} | \lambda_{k} \geq 1 \},\#\{\lambda_{k} | \lambda_{k} \geq 2 \}, ..., \#\{\lambda_{k} | \lambda_{k} \geq n \}, ....)$. Finally, we define size$(\lambda) = \sum \lambda_{i}$.


%A partition can be represented as a \textbf{Ferrers Diagram}, as shown in figure \ref{Ferrer}, where the number of squares on the $i$-th line is equal to $\lambda_{i}$. We will, most of the time, visualize a partition with this diagram.

\begin{figure}[ht]
    \centering
    \begin{tabular}{cc}
    \scalebox{.7}{\input{figures/partition.tex}}
    &
    \scalebox{.7}{\input{figures/Young_tableau.tex}}
    \end{tabular}
    
    \caption{The Ferrers diagram of $(4,3,1)$ and an example of standard Young tableau.}
    \label{fig:young}
\end{figure}


%\begin{definition}
%A \defn{Young tableau} on a partition $\lambda$ is an integer filling of the Young diagram of $\lambda$. A Young tableau is said to be \defn{semi-standard} if the numbers are strictly increasing along each row and weakly increasing along each column. It is said to be \defn{standard} if it is also strictly increasing along the columns and filled with the numbers $1$ to $n$ where $n$ is the size of $\lambda$. We denote by $\SSYT(\lambda)$ the set of semi-standard Young tableau over $\lambda$, and by $\SYT(\lambda)$ the set of standard Young tableaux of $\lambda$. Finally, for a Young tableau $\theta$, we write $\theta_{i}$ for the number of $i$ appearing in the tableau. 
%\end{definition}



%\begin{figure}[ht]
%    \centering
%    \input{figures/tableaux.tex}
%    \caption{Here are 3 Young tableaux for the partition $(3,3)$, the first one is not semi-standard, the second is semi-standard but not standard and the last is standard.}
%    \label{tableau}
%\end{figure}

%\defn{Schur functions} are infinite series defined on an infinite alphabet $x_1, x_2, \dots$. They form a famous basis of symmetric functions~\cite{Mcdo}. We will use the following definition.
%
%%We define \textbf{Schur functions} as follows. For a partition $\lambda$, the Schur function $s_{\lambda}$ in $n$ variables is 
%
%\begin{equation}
%s_{\lambda} := \sum_{\theta \in \SSYT(\lambda)}\prod_{i \in \NN} x_{i}^{\theta_{i}} 
%\end{equation}
%
%In practice, Schur functions can be developed on finite number of variables by setting all other variables to~$0$. For example, we have 
%
%\begin{align}
%s_{3,1}(q,t) &= q^{3}t + q^{2}t^{2} + qt^{3} \\
%s_{2,1}(q,t,r) &= q^{2}t+qt^{2}+qtr+ qr^{2}+rt^{2}+r^{2}t \\
%s_{1,1,1}(q,t,r) &= qtr
%\end{align}


%These functions are symmetric which is to say that they are invariant under all permutations of their variables. Even more importantly, they form a basis of the set of symmetric functions (\cite{Mcdo}), meaning that any symmetric function on $n$ variables can be described as a finite sum of Schur functions on $n$ variables. 


\subsection{Triangular partitions}
\label{subsec:triangular}

%A \textbf{sub-partition} $\tau$ of $\lambda$ is a partition such that $\forall i, \tau_{i} \leq \lambda_{i}$.

%As defined in \cite{Berg1}, a partition $\lambda$ is said to be \textbf{triangular} if there exist two real numbers $r$ and $s$ such that

\begin{definition}[From~\cite{Berg1}]
A \defn{triangular partition} $\lambda$ is a partition such that there exist two real numbers $r$ and $s$ with $\lambda_{j} = \lfloor r - \frac{jr}{s} \rfloor$ for all $j \leq s$, and $\lambda_j = 0$ otherwise.
\end{definition}

We say that the line joining $(r,0)$ and $(0,s)$ is an~\defn{$r$-$s$-line}. Then $\lambda$ is the greatest partition lying under the $r$-$s$-line. We also say that this line \defn{cuts off} $\lambda$ . See an example on \cref{fig:triang}.

%We clearly have that the conjugate of a triangular partition is triangular.



\begin{figure}[ht]
    \centering
    \begin{tabular}{cc}
    \input{figures/triangular_partition.tex}
    &
    \input{figures/non_triangular.tex}
    \end{tabular}
    \caption{Left: the triangular partition $(4,3,1)$ with one line that cuts it off and its slope vector. Right: a non-triangular partition $(4,4)$.}
    \label{fig:triang}
\end{figure}

When there exists an $r$-$s$-line cutting off $\lambda$ with $r$ and $s$ integer values, $\lambda$ is said to be a \defn{rectangular} partition. It is further said to be \defn{rational} when $r$ and $s$ are coprime. Later on, it will be important to characterize the \defn{slope} of the line. In accordance with~\cite{Berg1}, this can be done through a single parameter $v$ which characterizes a \emph{slope vector}.

\begin{definition}
The \defn{slope vector} of an $r$-$s$-line is the vector orthogonal to the line and of coordinates $(v,1-v)$. For simplicity, we write ``the slope $v$''. We say that $\lambda$ admits a slope $v$ if it is cut off by an $r$-$s$-line of slope vector $(v, 1-v)$.
\end{definition}

A triangular partition admits an infinite number of slope vectors which corresponds to certain open interval for the value of $v$. Bergeron and Mazin~\cite{Berg1} give a simple way to compute the slope interval using the hooks of the partition. For each cell $c$ of $\lambda$, let $\arm(c)$ be the \defn{arm} of the cell, \emph{i.e.} the number of cells lying to the right of $c$ in $\lambda$, and $\leg(c)$ be the \defn{leg} of $c$, \emph{i.e.} the number of cells lying above $c$ in $\lambda$. Then 
\begin{align}
\vlminc &:= \frac{\leg(c)}{\arm(c) + \leg(c) + 1} &
\vlmaxc &:= \frac{\leg(c) + 1}{\arm(c) + \leg(c) + 1}.
\end{align}
We say that $]\vlminc, ~\vlmaxc[$ is the \defn{admissible interval} of the cell $c$. The slope interval of $\lambda$ is the intersection of all admissible intervals $]\vlmin, \vlmax[$ where $\vlmin := \max_{c \in \lambda}(\vlminc)$ and  $\vlmax := \min_{c \in \lambda}(\vlmaxc)$. In particular, $\lambda$ is a triangular partition if and only if $\vlmin < \vlmax$~\cite[Lemma 1.2]{Berg1}. %Finally, we say that a triangular partition $\mu$ is \defn{similar} to $\lambda$ if they share a slope vector. In practice, this means that $\mu$ and $\lambda$ can be cut off by two parallel lines.  

%A \textbf{slope vector} for a triangular partition $\lambda$ is a positive coordinate vector which is orthogonal to one of the lines that cuts off $\lambda$, normalized so that its coordinates sum up to 1. As such, they can be identified with only their first coordinate.\\
%The set of slope vectors of a triangular partition is an open interval, with $t_{\lambda}$ the mean of the inf and the sup of said interval.


\subsection{Area, sim and $(q,t)$-enumeration}

In this paper, we are interested in the $(q,t)$-enumeration of sub-partitions of a given triangular partition $\lambda$. Following the terminology of~\cite{Berg1}, we call such objects \defn{triangular Dyck paths}.

\begin{definition}
A \defn{triangular Dyck path} is a tuple $(\lambda, \mu)$ such that $\mu$ is a sub-partition of $\lambda$ and $\lambda$ is a triangular partition. The \defn{area} of a triangular Dyck path is the number of cells belonging to $\lambda$ but not to $\mu$, given by $\arealm:= \sum_{i \in \NN} \lambda_i - \mu_i$.
\end{definition}

%In other words, a triangular Dyck path is a skew-partition $\lambda \setminus \mu$ where $\lambda$ is triangular. 
Note that we do not require that $\mu$ is also triangular. See \cref{fig:Area} for an example. This name is justified in the sense that when $\lambda$ is the staircase partition, \emph{i.e.} \linebreak $\lambda = (n-1, n-2, \dots, 1)$, then the triangular Dyck paths $(\lambda, \mu)$ correspond to actual Dyck paths of length $n$. Similarly, when $\lambda$ is a rational partition, then its sub-partitions are rational Dyck paths. 

%The \textbf{area} of a sub-partition $\tau$ of $\lambda$ is area$_{\lambda}(\tau) = \sum_{i \in \mathbb{N}} \lambda_{i} - \tau_{i} $. It is the number of cells of $\tau$ \textbackslash$ \lambda$ (The skew Ferrers diagram of $\tau$ without the diagram of $\lambda$). In figure \ref{Area}, $\tau$ \textbackslash$ \lambda$ is the set of red cells.

\begin{figure}[ht]
    \centering
    \begin{tabular}{ccc}
    \input{figures/area.tex} &
    \input{figures/hook.tex} &
    \input{figures/sim.tex}
    \end{tabular}
    \caption{A triangular Dyck path $(\lambda, \mu)$. The left picture shows $\mu$ in blue and the area in red with $\arealm = 6$. The middle picture illustrates the hook of a given cell in $\mu$. The right picture shows all similar cells in green with $\simlm = 13$.}
    
    %\caption{A triangular Dyck path $(\lambda, \mu)$. On the left, the sub-partition $\mu$ is in blue. We have $\arealm = 6$, the number of red cells. On the right, we have the similar cell of $\mu$ in green.}
    %\caption{A triangular partition (green and red cells) with one of its sub-partition, in green. The area is the number of red cells (here, it is 15).}
    \label{fig:Area}
\end{figure}


\begin{definition}[From~\cite{Berg1}, Section 4.1]
Let $(\lambda, \mu)$ be a triangular Dyck path. A cell $c$ of $\mu$ is said to be \defn{similar} if
\begin{equation}
\vmminc \leq \frac{\vlmin + \vlmax}{2} < \vmmaxc.
\end{equation}
In other words, the \defn{mean slope} of the slope interval of $\lambda$ belongs to the semi-closed admissible interval of the cell $c$ in $\mu$. Then $\simlm$ is the number of similar cells in the triangular Dyck path $(\lambda, \mu)$.
\end{definition}

For example on \cref{fig:Area}, the mean slope of $\lambda$, $\frac{45}{112}$ belongs to the admissible interval of the selected cell given by $\vmminc = \frac{1}{3}$ and $\vmmaxc = \frac{1}{2}$. It is important to understand that the similarity of a cell depends on both partitions of the triangular Dyck path. Indeed, the mean slope is computed on $\lambda$ but the admissible cell interval is computed on $\mu$. If $(\lambda, \mu)$ is a triangular Dyck path such that all the cells of $\mu$ are similar, then in particular $\mu$ is triangular: a slight perturbation of the mean slope of $\lambda$ will always cut off $\mu$ (see~\cite[Lemma 4.1]{Berg1}). 
%The converse is not true: we might have a sub-partition $\mu$ of $\lambda$ which is similar to $\lambda$ but not cut-off by the mean-slope. 
When all cells of $\mu$ are similar cells, we say that $(\lambda, \mu)$ is a \defn{mean-similar} triangular Dyck path.

%It is shown in~\cite{Berg1} that $\tau$ is similar to $\lambda$ if and only if all the cells of $\tau$ in the triangular skew-partition $(\lambda, \tau)$ are similar cells. WRONG??

The \emph{sim} statistic already appears in~\cite{BHMPS} and is a generalization of the famous \emph{dinv} statistic that can be found in~\cite{Hag} for the classical $(q,t)$-Catalan enumeration. Indeed, when $\lambda$ is the staircase partition, then the \emph{sim} of $(\lambda, \mu)$ is actually the \emph{dinv} of the corresponding Dyck path.

%For a cell $c$ of coordinates $(i,j)$ in a partition $\lambda$, we write $a_{\lambda}(c) = \lambda_{j} - i$ for the \textbf{arm length}. This the number of cells in $\lambda$ to the right of c. Similarly, we write $l_{\lambda}(c) = \lambda'_{i} - j$ for the \textbf{leg length}, the number of cells in $\lambda$ above c. Finally $\epsilon_{\lambda}(c) = 1 + a_{\lambda}(c) + l_{\lambda}(c)$ is the \textbf{hook length} of $c$ in $\lambda$. These notions are illustrated on figure \ref{SimC}.

%We can now define the \textbf{sim} statistic of $\tau \subseteq \lambda$ as describe in \cite{Berg1} by $$\mathrm{sim}_{\lambda}(\tau) = \# \{ c \in \tau | l_{\tau}(c)/\epsilon_{\tau}(c) \leq t_{\lambda} < (l_{\tau}(c)+1)/\epsilon_{\tau}(c)\}.$$ It is the number of cells whose hook, formed with the leg and arm illustrated in figure \ref{SimC}, share a common slope with the initial partition. One such a cell is called \textbf{similar}, or $\lambda$-similar, hence why this statistic is called sim. It is important to understand that the similarity of a cell depends on both the partition and the sub-partition we consider.

%It can be proven that the sim statistic is equal to the dinv statistic on partition where it is defined. For further information on the dinv statistic, see \cite{Hag}.

%\begin{figure}[ht]
%    \centering
%    \begin{tabular}{cc}
%    
%    &
%    TODO : the same with all sim cells
%    \end{tabular}
%        
%    \caption{Representation of the hook of the green cell, with the arm in yellow and leg in blue. In this case, the hook length is 3+2+1=6.}
%    \label{fig:SimC}
%\end{figure}


We can now define the $(q,t)$ polynomials associated to a triangular partition $\lambda$ as:

\begin{equation}
\label{eq:alqt}
\Alqt = \sum_{\mu \subseteq \lambda}~q^{\arealm}~ t^{\simlm}.
\end{equation}

These polynomials have been shown to be symmetric in $q$ and $t$ via an algebraic proof~\cite{BHMPS}. However, the question of finding a combinatorial proof is still open even for the classical case. For example, for the partition $\lambda = (3,2)$, we have the following:

\begin{equation}
A_{3,2}(q,t) = q^{5} + q^{4} t + q^{3} t^{2} + q^{2} t^{3} + q t^{4} + t^{5} + q^{3} t + q^{2} t^{2} + q t^{3}
\end{equation}

%\begin{center}
%\begin{tabular}{ |c c|c c| } 
%Sub-partition & Monomial & Sub-partition & Monomial\\
% \hline
% $(3,2):$ & $q^{0}t^{5}$ & $(3,1):$ & $q^{1}t^{4}$ \\ 
% $(3):$ & $q^{2}t^{2}$ & $(2,2):$ & $q^{1}t^{3}$ \\ 
% $(2,1):$ & $q^{2}t^{3}$ & $(2):$ & $q^{3}t^{2}$ \\ 
% $(1,1):$ & $q^{3}t^{1}$ & $(1):$ & $q^{4}t^{1}$ \\
% $():$ & $q^{5}t^{0}$ & & \\
% \hline
%\end{tabular}
%\end{center}

As the $\Alqt$ polynomials are symmetric, they can be expressed in terms of Schur functions with the following conjecture.

\begin{conjecture}[Conjecture 1 of~\cite{Berg1}]
\label{conj:schur}
The polynomials $\Alqt$ are Schur positive.
%\begin{equation}
%\Alqt = \sum_\mu c_\mu^\lambda s_\mu(q,t)
%\end{equation}
%with $c_\mu^\lambda \in \NN$, summing over partitions $\mu$ with $ |\mu |\leq | \lambda |$.

\end{conjecture}

%In particular, the symmetric functions described previously can be expressed with Schur functions on 2 variables. They can also be expanded in more variables (the process is described in \cite{Berg1}). Furthermore, the previously described symmetric functions are conjectured to be Schur-positive: meaning that the coefficient before each Schur functions in the sum is a positive integer.

See~\cite{Berg1} for many examples of computation, here are $A_{3,2,1}(q,t)$ and $A_{3,2}(q,t)$:
\begin{align}
\label{eq:qtschur_a321}
A_{3,2,1}(q,t) &= s_{6}(q,t) + s_{4,1}(q,t) + s_{3,1}(q,t) ,
& A_{3,2}(q,t) &= s_{5}(q,t) + s_{3,1}(q,t).
\end{align}

%\begin{align}
%A_{3,2,1}(q,t) &= s_{6}(q,t) + s_{4,1}(q,t) + s_{3,1}(q,t) 
%A_{3,2}(q,t) &= s_{5} + s_{3,1} \\
%A_{2,2,1}(q,t) &= s_{5} + s_{3,1}
%\end{align}


%\subsection{(Semi) Standard Young Tableau and Schur functions}

\section{Combinatorial interpretation of similar cells}
\label{sec:triangular}

In this paper, we explore some combinatorial tools related to the aforementioned conjecture and obtain a proof for the case of $2$-partitions. The first step is to define a certain Young tableau associated to each triangular partition.


\subsection{The triangular Young tableau}

\begin{proposition}
Let $\lambda$ be a triangular partition. Then for all $0 \leq k \leq |\lambda |$, there exists a unique mean-similar triangular Dyck path $(\lambda, \mu)$ of area $k$. Besides, if $(\lambda, \alpha)$ and $(\lambda, \mu)$ are two mean-similar triangular Dyck paths with $|\alpha| \leq |\mu |$, then $\alpha \subseteq \mu$.
\end{proposition}

In other words, if $(\lambda, \mu)$ is such that all cells of $\mu$ are similar cells, then there is a unique way to remove a cell from $\mu$ such that all remaining cells stay similar. This is a consequence of~\cite[Lemma~4.1]{Berg1}: the mean-slope of $\lambda$ can always be slightly shifted to obtain an irrational slope. By \emph{moving} this irrational slope towards the origin, you will ``touch'' the cells of $\lambda$ one by one, which gives the removal order on the cells of the sub-partitions in order to stay mean-similar, see \cref{fig:TYT} for an example. This leads to the following definition.


%For $\lambda$ a triangular partition and $\tau \subset \lambda$, we say that $\tau$ is \textbf{similar} to $\lambda$ if and only if $\mathrm{sim}_{\lambda}(\tau) = \mathrm{size}(\tau)$, meaning that each cells of $\tau$ is $\lambda$-similar. Be careful, this definition of similar isn't the same as that in \cite{Berg1}. 

%We get the following lemma, heavily inspired by lemma 4.1 in \cite{Berg1}, with a slight precision on the possible irrationality of $\epsilon$:

%\begin{lemma}
%There exists $\epsilon>0$ irrational such that $t = t_{\lambda}+\epsilon$ is a slope vector of all the sub-partition of $\lambda$ similar to $\lambda$, %$\lambda$ included, and all sub-partition of $\lambda$ with $t$ as a slope vector are similar to $\lambda$.
%\end{lemma}

%With this lemma and the properties of a ray (definition and properties given in \cite{Berg1}), we have:

%\begin{proposition}
%For $\lambda$ a triangular partition of size $n$ and $\tau \subset \lambda$ of size $m<n$, $\tau$ similar to $\lambda$, there exist a unique $\alpha \subset \lambda$ of size $m+1$ simlar to $\lambda$ such that $\tau \subset \alpha$. This means that we get $\alpha$ by adding a cell in $\tau$, and that this cell is the only one that can be added and that keeps the sub-partition similar to $\lambda$.
%\end{proposition}
%
%\begin{proof}
% Let $\lambda$ and $\tau$ be such partitions. There is a line of slope $t$ that cuts off $\tau$ without touching any point of $\mathbb{N}^{2}$. We can slide this line to the right until it touches a point of $\mathbb{N}^{2}$ for the first time. Because $t$ is irrational, there is at most one point of $\mathbb{N}^{2}$ that touches a line of slope $t$. In this case, the cell in the bottom left of the point is the new cells to add
%\end{proof}
%As the empty partition is a sub-partition of any partition and is similar to any triangular partition, we deduce that there is exactly one subpartition of a triangular partition similar to it for any given size, smaller than the size of the original, and that they are increasing in the inclusion order.
%
%As such, we get that, in each partition $\tau$ similar to $\lambda$, there is a unique cell that can be removed while keeping the sub-partition similar to $\lambda$.

\begin{definition}
\label{def:tyt}
The \defn{triangular Young tableau} of a triangular partition $\lambda$ with $|\lambda| = n$ is the unique standard Young tableau such that for all $0 \leq k \leq n$, the cells $1$ to $k$ form a sub-partition $\mu$ with $(\lambda, \mu)$ a mean-similar triangular Dyck path.
\end{definition}


%With this proposition, we can define the \textbf{Triangular Young tableau} of a triangular partition $\lambda$ as the standard young tableau on $\lambda$ with the number of each cell being the size of the smallest subpartition similar to $\lambda$ that contains said cell.

\begin{figure}[ht]
    \centering
    \begin{tabular}{cccc}
    \input{figures/triangular_tableau.tex}
    &
    \input{figures/triangular_tableau2.tex}
    &
    \input{figures/triangular_tableau3.tex}
    & 
    \input{figures/triangular_tableau4.tex}
    \end{tabular}
    \caption{The triangular Young tableaux of $(4,3,1)$ with its slope construction.}
    \label{fig:TYT}
\end{figure}

\subsection{Deficit of a triangular Dyck path}

\begin{definition}
Let $(\lambda, \mu)$ be a triangular Dyck path and $\theta$ a standard Young tableau of shape~$\lambda$.  We say that there is a \defn{$\theta$-inversion} in $(\lambda, \mu)$ if there exists a cell $c = (x,y)$ in $\mu$ and a cell $c' = (x',y')$ in $\lambda \setminus \mu$ such that $\theta(c) > \theta(c')$. In particular, $c$ cannot be in the same line or column as $c'$. Then we say that the cell $d = (\min(x,x'), \min(y,y'))$, which is at the hook of $c$ and $c'$, is a \defn{$\theta$-deficit cell}. We call \defn{deficit} of the triangular Dyck path $(\lambda, \mu)$ with tableau~$\theta$, written $\defthm$, the number of $\theta$-deficit cells.
\end{definition}

See \cref{fig:DefC} for an example: $1$ is a deficit cell because it is at the hook of $6$ and $4$, and $2$ is a deficit cell because it is at the hook of $5$ and $7$. Note that it can happen that multiple $\theta$-inversions give the same deficit cell. The following proposition gives a more combinatorial approach to the computation of similar cells.

%For a triangular partition $\lambda$ and $\tau \subset \lambda$, we define a \textbf{deficit cell} $c \in \tau$ as a cell such that there is $c' \in \tau, c" \in \lambda$ \textbackslash $\tau$ such that the value of $c'$ in the triangular Young tableau associated to $\lambda$ is greater than the value of $c"$ and either $c'$ is above $c$, $c"$ is to the right of $c$, or $c'$ is to the right and $c"$ is above c.

\begin{figure}[ht]
    \centering
    \input{figures/deficit.tex}
    \caption{As a sub-partition of $(4,3,1)$, the partition $(2,2,1)$ has two deficit cells 1 \& 2.}
    \label{fig:DefC}
\end{figure}

\begin{proposition}
\label{prop:deficit}
Let $\lambda$ be a triangular partition and $\theta$ its triangular Young tableau as in \cref{def:tyt}. Then for any triangular Dyck path $(\lambda, \mu)$, the $\theta$-deficit cells of $(\lambda, \mu)$ are exactly the non-similar cells of $\mu$. In particular, we have $|\lambda| = \arealm + \simlm + \defthm$.
%For $\lambda$ a triangular partition and $\tau \subset \lambda$, the deficit cells of $\tau$ are exactly the cells of $\tau$ that aren't $\lambda$-similar.
\end{proposition}

In the rest of the paper, we write $\deflm := \defthm$ of a triangular Dyck path where $\theta$ is the triangular Young tableau. 


%We defined the \textbf{deficit} of a subpartition $\tau \subseteq \lambda$, denoted by $\mathrm{def}(\tau)$ as the number of deficit cells in $\tau$. We then have $\mathrm{size}(\tau) = \mathrm{sim}(\tau) + \mathrm{def}(\tau)$.This definition allows us to have an another interpretation of sim, but purely combinatorial this time.

For the staircase partition (which gives classical Dyck paths), the triangular tableau is the \defn{top-down} tableau where we label the corners of the partition in decreasing order from top to bottom as shown in \cref{fig:staircase}. The deficit then gives a new interpretation for the classical \emph{dinv} statistic.

\begin{figure}[ht]
    \centering
\begin{tabular}{ccc}
\input{figures/step_partition.tex}
&
\input{figures/dinv_area1.tex}
&
\input{figures/dinv_area2.tex}
\end{tabular}
\caption{The triangular tableau of the staircase partition $(3,2,1)$ and two examples of sub-partitions (Dyck paths). In the left example, we have $\mu = (3,1,1)$ and $\simlm = 4$. In the second example, we have $\mu = (1,1,1)$ and $\simlm = 1$.}
%\caption{The triangular tableau of the staircase partition (3,2,1) and some example Dyck paths. The first is the subpartition (3,1,1), with a deficit of 1, a sim of $size(3,1,1) - 1= 4$. It is identified with the Dyck path on the right, with a dinv of 4.}
\label{fig:staircase}
\end{figure}

\subsection{Sim-sym tableau}

The deficit statistic is defined for any tableau $\theta$ and just happens to coincide with the non-similar cells when $\theta$ is the triangular Young tableau of $\lambda$. It is then natural to extend the definition of the \emph{sim} statistic to any Young tableau.

\begin{definition}
Let $\theta$ be a standard Young tableau of triangular shape $\lambda$. Then for any triangular Dyck paths $(\lambda, \mu)$, we say that a cell of $\mu$ is \defn{$\theta$-similar} if it is not a $\theta$-deficit cell. We write $\simthm$ the number of $\theta$-similar cells. 
\end{definition}

%For a Standard Young tableau T over $\lambda$, we can use the way we defined deficit cells with the triangular Young tableau to define \textbf{Tdef($\tau$)} the number of deficit cells of $\tau \subseteq \lambda$ with this tableau T. We can then define \textbf{Tsim($\tau$)} as Tsim$(\tau) = \mathrm{size}(\tau) - \mathrm{Tdef}(\tau)$.

With that definition, again it is natural to extend the definition of~\eqref{eq:alqt} to this new setting, we define
\begin{equation}
\Athqt = \sum_{\mu \subseteq \lambda} ~q^{\arealm}~ t^{\simthm}
\end{equation}
where $\theta$ is a standard Young tableau of shape $\lambda$. These polynomials are not symmetric in general. On the other hand, the triangular Young tableau is not the only tableau giving the symmetry. This led us to introduce the following objects which we believe are of high interest when studying the $(q,t)$-symmetry.

\begin{definition}
A \defn{sim-sym} (similar-symmetric) tableau  is a Young tableau $\theta$ of triangular shape $\lambda$ such that $\Athqt = \Alqt$.
\end{definition}

%We will call \textbf{Sim-Sym Tableau} any standard Young tableau over a triangular partitition $\lambda$ such that $$A_{\lambda}(q,t) = \sum_{\tau \subseteq \lambda}q^{\mathrm{area}_{\lambda}(\tau)}t^{\mathrm{Tsim}_{\lambda}(\tau)}$$

It is important to note that this equality does not necessarily mean that, for all triangular Dyck paths, one has $\simthm = \simlm$. It only means we have a bijection over the triangular Dyck paths of $\lambda$ which keeps the \emph{area} constant while sending the $\theta$-\emph{sim} statistic to the classical \emph{sim}. \cref{fig:simsym2} gives three examples of sim-sym tableaux for $(4,2,1)$.

%It is important to note that this equality doesn't necessarely mean that $\mathrm{Tsim}_{\lambda}(\tau) = sim_{\lambda}(\tau)$, only that we have a bijection over subpartition of $\lambda$ that keeps the $area$ constant and send a sub-partition of Tsim $i$ to a sub-partition of sim $i$.

%\begin{figure}[ht]
%    \centering
%    \input{figures/simsym.tex}
%    \caption{Two examples of sim-sym tableaux for (7,2).}
%    \label{fig:simsym}
%\end{figure}

The natural question that follows would be to characterize the sim-sym tableaux. This is not an easy question. One way to find different sim-sym tableaux is to choose a different slope to sweep down the partition. Indeed, the triangular Young tableau is defined using a slight deformation of the mean-slope (to ensure that it is irrational). But any irrational slope would actually give a tableau, and computational explorations suggest that these are always sim-sym tableaux. Nevertheless, they are not the only examples. For example, for the partition $(4,2,1)$, we find only two slope-tableaux which are the first two tableaux of \cref{fig:simsym2} (the first one is the mean-slope triangular tableau), but there are actually seven sim-sym tableaux. In the rest of the paper, we characterize sim-sym tableaux on $2$-partitions, and explore some interesting conjectures related to the lattice structure of triangular Dyck-paths and sim-sym tableaux.

\begin{figure}[ht]
    \centering
\begin{tabular}{ccc}
\scalebox{0.8}{\input{figures/sim-sym_tableau.tex}}
&
\scalebox{0.8}{\input{figures/sim-sym_tableau2.tex}}
&
\scalebox{0.8}{\input{figures/sim-sym_tableau3.tex}}
\end{tabular}
\caption{Three sim-sym tableaux of $(4,2,1)$: the first two are slope-tableaux but not the last one.}
\label{fig:simsym2}
\end{figure}


\section{Results on $2$-partitions}
\label{sec:2parts}

Using the \emph{deficit} statistic on the triangular tableau, we are able to give a direct proof of \cref{conj:schur} for $2$-partitions.

\subsection{Characterization of triangular $2$-partitions}

We have seen in \cref{subsec:triangular} that triangular partitions have an easy characterization through their slope vectors. This gives the following result.

\begin{proposition}
Let $\lambda=(m,n)$ be a 2-partition, $\lambda$ is triangular if and only if $n \leq \lceil m/2 \rceil$.
\end{proposition}

%\begin{proof}
% If $\lambda = (m,n)$ is triangular, then there is a line that cuts off $\lambda$. This line necessarily pass strictly under the point $(1,3)$, otherwise its 3rd part wouldn't be null. And the same goes for the point $(m+1,1)$. As such this line is strictly under all the points between $(1,3)$ and $(m+1,1)$, in particular strictly under it's middle: $(\frac{m}{2} +1, 2)$. Hence why $n \leq \lceil m/2 \rceil$.
% 
%For $m\geq 1$, $(m, \lceil m/2 \rceil)$ is triangular: we just have to take the line going through $(1,3)$ and $(m+1,1)$ and translate it of an $\epsilon>0$ to the left. For $n < \lceil m/2 \rceil$, we take said line and rotate it around its point of coordinate $(x,1)$ until only $n$ cells lie under the line in the second part (with this rotation, as long as the slope stay negative, the number of cells lying under the line in the first part stay the same).
%\end{proof}

We can also explicitly describe the triangular Young tableau associated with a triangular $2$-partition.

\begin{proposition}
Let $\lambda = (m,n)$ be a triangular partition and $\theta$ its triangular Young tableau. Then the values in $\theta$ on the shortest
row of $\lambda$ (of size $n$) are given in decreasing order by \linebreak $k, k-2, k-4, \dots, k-2(n-1)$ with $k = m+n$ if $n = \lceil m/2 \rceil$, and $k = m+n -1$ if $n < \lceil m/2 \rceil$. As $\lambda$ only has two (increasing) rows, this entirely characterizes $\theta$.
\end{proposition}

In other words, the maximal value of $\theta$ will be placed on either of the two rows depending on the length $n$, then the values will alternate between the two rows until the shortest row is filled. See some examples on \cref{fig:Alt}.

\begin{figure}[ht]
    \centering
    \scalebox{0.7}{\input{figures/triangular_2tableaux.tex}}

    \caption{The triangular tableaux of $(7,2)$, $(3,2)$ and $(4,2)$.}
    \label{fig:Alt}
\end{figure}

%\begin{proposition}
%For $\lambda = (m,n)$ a triangular partition, the triangular Young tableau associated to $\lambda$ is the following:
%\begin{itemize}
%    \item for all $i \leq n$ the i-th cell in the second row is numbered with $m+n -2i$ (case $n = \lceil m/2 \rceil)$
%    \item for all $i \leq n$ the i-th cell in the second row is numbered with $m+n -2i -1$ (case $n < \lceil m/2 \rceil$
%\end{itemize}
%\end{proposition}


\subsection{Explicit $(q,t)$-enumeration and Schur positivity}

We are now able to give a direct proof of the following theorem which was stated in~\cite{Berg1}. This formula was actually obtained by~\cite[Theorem 3.1]{GHSR20} which explores a similar $(q,t)$-enumeration in a different combinatorial context. 


\begin{theorem}[Formula (6.3) in~\cite{Berg1}] 
\label{thm:alqt2}
For $\lambda = (m,n)$ a triangular 2-partition, we have
\begin{equation}
\label{eq:alqt2}
\Alqt = \sum_{0 \leq i \leq \min(n, m-n)} s_{m+n-2i,i}(q,t).
\end{equation}
\end{theorem}

Recall that $s_{k,\ell}(q,t) = \sum_{i=k}^{\ell} q^i t^{k+\ell -i}$. In particular, each monomial $q^i t^j$ appears only once in~\eqref{eq:alqt2}. The proof then is purely combinatorial, building an explicit bijection between monomials $q^i t^j$ and triangular Dyck paths which \emph{area} equals to $i$ and \emph{deficit} equals to $m+n-i-j$. In the general case, monomials might appear with coefficients greater than $1$, which explains why this proof does not generalize easily.

%
%Note that $\min(n,m-n) = n$ except when $m$ is odd and $n = \lceil m/2 \rceil$. This is also the only case when the maximal value of the triangular Young tableau $\theta$ appears on the shortest row. 
%
%The proof of the theorem relies solely on the combinatorics of triangular Dyck paths. It thus gives a combinatorial proof of the $(q,t)$ symmetry and of the Schur positivity for $2$-partitions. The Schur expansion here is particular as each Schur function appears at most once. Moreover, no two Schur function give the same monomial $q^i t^j$ in their expansion. This is not true in general and is one of the reasons this result cannot be easily generalized to other triangular partitions. In this extended abstract, we omit the full proof of the theorem but give the general ideas and the lemmas it relies on.
%
%\begin{lemma}
%A monomial $q^i t^j$ appears in~\eqref{eq:alqt2} if and only if there exists a triangular Dyck path $(\lambda, \mu)$ such that $i = \arealm$ and $j = \simlm$.
%\end{lemma}
%
%For a given monomial, the existence of the corresponding triangular Dyck path is proved by explicit construction: removing the cells of $\lambda$ in a certain order depending on the targeted deficit. The converse is proved explicitly by showing that certain combinations of deficit / area cannot be obtained. 
%
%Now to prove the theorem, we need to show that no two triangular Dyck path share the same \emph{area} and \emph{deficit} statistic (each monomial appear exactly once). This is a consequence of a certain property of the dominance order on $2$-partitions. Recall that the dominance order on two partitions $\alpha$ and $\tau$ is given by $\alpha \preceq \tau$ if and only if, for all $i$, $\alpha_1 + \alpha_2 + \dots \alpha_i \leq \tau_1 + \tau_2 + \dots + \tau_i$. In particular, this is a total order on $2$-partitions with $| \alpha | = |\beta |$.
%
%\begin{lemma}
%For $\lambda = (m,n)$, we consider the set of triangular Dyck paths $(\lambda, \beta_i)$ such that $\areap{\lambda}{\beta_i} = k$ (then $|\beta_i| = m+n-k$), ordered by the dominance order such that $\beta_1 \prec \beta_2 \dots \prec \beta_\ell$. And let $(\lambda, \alpha)$ be the only mean similar Dyck path with $\areap{\lambda}{\alpha} = k$. Then
%\begin{itemize}
%    \item for all $\beta_{i} \prec \beta_{j}\preceq \alpha$, we have $\defp{\lambda}{\beta_i} > \defp{\lambda}{\beta_j}$;
%	\item for all $\alpha \preceq \beta_i \prec \beta_j$, we have $\defp{\lambda}{\beta_i} < \defp{\lambda}{\beta_j}$.
%\end{itemize}
%\end{lemma}
%
%You can see an example of this property on \cref{fig:Dom}. The lemma shows that the \emph{deficit} first strictly decreases until $\alpha$ and then strictly increases. It is left to show that two partitions $\beta_i \prec \alpha \prec \beta_j$ cannot share the same deficit which can be done using the properties of the dominance order and of the triangular tableau.

%The use of min$(n,m-n)$ is only used for 2-partitions such that $m$ is odd and $n = \lceil m/2 \rceil$. One may realize that Schur functions appear with positive coefficients (more precisely 0 or 1 in this case): this is the Schur-positivity previously defined. As such this proposition proves both the symmetry but also the Schur-positivity of the symmetric functions of triangular 2-partitions.








%To prove this, we will need the following lemmas:




%
%We use this lemma to prove that, for each monomial $q^{i}t^{j}$ in $\sum_{0 \leq i \leq \mathrm{min}(n, m-n)} s_{m+n-2i,i}(q,t)$, there is a sub-partition $\mu$ of $\lambda$ such that $i=\rm{area}(\mu)$ and $j=\rm{sim}(\mu)$.
%
%\begin{proof}
% Let $d \in \{0;1;...; \rm{min}(n,m-n)\}$. We just have to prove that we can get each monomial in $s{m+n-2d,d}$ with a sub-partition. The monomial in $s_{m+n-2i,i}$ are $q^{d}t^{m+n-2d},$ $q^{d+1}t^{m+n-2d-1},$ $..., q^{m+n-2d}t^{d}$. The monomial $q^{d}t^{m+n-2d}$ can be obtained with the sub-partition where we remove from the original partition $d$ cells in the part without the cell having the largest number in the triangular Young tableau. The monomial $q^{d+1}t^{m+n-2d-1}$ can be obtained by removing from the original partition $d$ cells in the other part (the one without the cell having the largest number) and $1$ from the part without this cell. The next one by removing $d+1$ cells in the part without the cell having the largest number and $1$ in the other. We can continue to alternate like this until we try to remove more cells in the second part than they actually are. When we are at this point, we always keep $d$ cells in the second part and we remove as many cells as necessary in the first to get to the wanted area, until we get to the partition $(d,d)$, that has an area of $m+n-2d$ and a sim of $d$ like we wanted.
% 
% Be careful, this proof does not work for (2,1), but a simple verification is enough to show that the conjecture works for (2,1).
%\end{proof}
% 
% The two next steps to prove the conjecture are to show that there is no sub-partition $\mu$ such that $q^{\rm{area}(\mu)}t^{\rm{sim}(\mu)}$ is not in  $\sum_{0 \leq i \leq \mathrm{min}(n, m-n)} s_{m+n-2i,i}(q,t)$ and that each monomial appears only once.
% 
% We can prove the first point quickly: 
% \begin{proof}
%  For a sub-partition $\mu$, $m+n - (\rm{area}(\mu) + \rm{sim}(\mu)) = \rm{def}(\mu)$. However, each deficit cells in $\mu$ necessarily has a cell above it, this gives that $\rm{def}(\mu) \leq n$. We also have that $n>m-n$ only in the case where $2n=m+1$, meaning that, in this case, $n-1=m-n$. The cell numbered $1$ in the triangular Young tableau can only be a deficit cell in case where either the first cell of the second part is in $\mu$ and the $m-n-1$-th cell in the first part is not, or both the first cell of the second part and the $m-n-2$-th cell in the first part are not. In both cases, at least one cell of $\lambda$ with a cell above it is not part of $\mu$, meaning that $\mu$ as at most $n-1=m-n$ cells with a cell above them, and as such $\rm{def}(\mu) \leq m-n$. The positivity aspect of $\rm{def}(\mu)$ is obvious.
% \end{proof}
% 
%\begin{lemma}
%For $\lambda = (m,n)$ a triangular partition, let us consider the set of sub-partitions of $\lambda$ of a fixed size $k\leq m+n$ with the dominance order ($\alpha \leq \tau \iff \forall i, \alpha_{i} + \alpha_{i+1} + ... \leq \tau_{i} + \tau_{i+1} + ...$), this order is total on 2-partition. As such, we can interpret it as a chain from the minimum to the maximum. Let us consider $\alpha$ the only sub-partition of size $k$ similar to $\lambda$. Then we have the following properties:
%\begin{itemize}
%    \item $\forall \beta_{1}<\beta_{2}\leq \alpha, \mathrm{def}(\beta_{1}) > \mathrm{def}(\beta_{2})$
%    \item $\forall \alpha \leq \beta_{1}<\beta_{2}, \mathrm{def}(\beta_{1}) < \mathrm{def}(\beta_{2})$
%\end{itemize}
%\end{lemma}
%
%\begin{figure}[ht]
%    \centering
%    \input{figures/2partitions-chain.tex}
%    \caption{Triangular Dyck path of $(6,3)$ with \emph{area} 3 ordered by dominance. The deficit cells are in yellow and the similar cells are in green.}
%    \label{fig:Dom}
%\end{figure}

%With this lemma, we can now prove that no two sub-partitions generate the same monomial, ending the proof of the conjecture:
%
%\begin{proof}
% For two sub-partitions to generate the same monomial, they would have to have the same area and the same sim (and, thus, the same deficit). The previous lemma show that, for that condition to happen, one of the sub-partition, that we denote $\alpha$, would need to be smaller than the partition with the same area similar to $\lambda$ in the dominance order, while the other, denoted $\beta$, would need to be bigger.
% 
% One important property, illustrated in figure \ref{Dom}, is that for a fixed area, the sub-partition smaller than the partition similar to $\lambda$ in the dominance order will have a lack of cells in their second part and an excess in the first, creating the deficit. This is the opposite for those greater than the partition similar to $\lambda$.
% 
% As such, $\alpha$ will have a deficit of either $(n-\alpha_{2}) - (m-\alpha_{1})$, if the cell with the greatest number in the triangular Young tableau is in the first part, or $(n-\alpha_{2}) - (m-\alpha_{1})-1$ otherwise (like in figure \ref{Dom}). In the first case, the deficit has the same parity as area$(\alpha)=(n-\alpha_{2})+(m-\alpha_{1})$, while it has the opposite parity than the area otherwise.
% 
% For $\beta$, its deficit is either $(m-\beta_{1}) - (n-\beta_{2}) -1$, if the cell with the greatest number in the triangular Young tableau is in the first part,  $(m-\beta_{1}) - (n-\beta_{2})$ if it isn't or $\beta_{2}$ in some cases that we will specify later. In the first case, the deficit has a different parity than area$(\beta)$, however area$(\beta)=\rm{area}(\alpha)$, as such def$(\beta)\not=\rm{def}(\alpha)$. The same is true in the second case.
% 
% Now, def$(\beta)=\beta_{2}$ is only in the case where $m-\beta_{1} > n$, and as such area$(\beta)=(m-\beta_{1}) + (n-\beta_{2})> n + (n-\beta_{2}) = 2n - \rm{def}(\beta) $. On the other hand, area$(\alpha)=(m-\alpha_{1}) + (n-\alpha_{2})$ and $m-\alpha_{1} + n+\alpha_{2} = 2n - ((n-\alpha_{2}) - (m-\alpha_{1})) = 2n - \rm{def}(\alpha)$ if the cell with the greatest number in the triangular Young tableau is in the first part, and  $m-\alpha_{1} + n+\alpha_{2} = 2n - (n-\alpha_{2}) - (m-\alpha_{1}) = 2n - \rm{def}(\alpha)-1$ otherwise. Because $(m-\alpha_{1}) + (n-\alpha_{2}) \leq m-\alpha_{1} + n+\alpha_{2}$, in both cases, we get def$(\beta) > \rm{def}(\alpha)$.
%\end{proof}
 
%The generalization of this proof for other partitions is complex because it relies on the fact that each monomial appears at most once in the symmetric function of a 2-partition. However, except for $\lambda = (3,2,1)$, all partitions with 3 parts or more seem to have some monomial appearing multiple times. 
%
%An interesting remark is that the dominance order is not total when we consider partitions on more than 2 parts (e.g. $(3,3)$ and $(4,1,1)$). The fact that some sub-partitions are not comparable could be one of the reasons some monomials appear multiple times in the symmetric functions associated to partitions with more than 2 parts.\\

\subsection{Sim-sym tableaux on 2-partitions}

The key combinatorial ingredient in the proof of \cref{thm:alqt2} is the triangular tableau and especially the alternation of values between the two rows. This leads to the following theorem.

%This proof allows us to understand what are the necessary conditions to have a sim-sym tableau on 2-partitions. As such, we get the following proposition:

\begin{theorem}[Characterization of sim-sym tableau on 2-partitions]
\label{thm:simsym}
Let $\lambda = (m,n)$ be a triangular partition with $n \neq 2$. Then there are exactly $m-2n+2$ sim-sym tableaux of shape~$\lambda$. The values on the shortest row of each tableau are given in decreasing order by \linebreak $k, k-2, k-4, \dots, k-2(n-1)$ with $3n-2 < k \leq m+n$.

Furthermore, when $n = 2$, we have the aforementioned tableaux as well as the tableau containing values $2$ and $5$ on its shortest row.

%For $\lambda = (m,n)$ a triangular partition, there is exactly $i = m-2n +2$ sim-sym tableaux on $\lambda$. These tableaux are given by:\\for $k\in [0;i-1]$, for all $i \leq n$ the i-th cell in the second row is numbered with $m+n -2i -k$. The partition $\lambda = (3,2)$ is an exception, having a 2nd sim-sym tableau with different form than previously described.
\end{theorem}

\begin{figure}[ht]
    \centering
    \begin{tabular}{c}
        \scalebox{.5}{\input{figures/simsym6-2.tex}}\\
        \scalebox{.5}{\input{figures/simsym6-2-2.tex}}

    \end{tabular}
    
    \caption{The five different sim-sym tableaux on $(6,2)$.}
    \label{fig:symT}
\end{figure}

\section{Lattice and interval enumeration}
\label{sec:lattice}

In some instances, the  polynomials $\Alqt$ of~\eqref{eq:alqt} occur as the expansion in two variables of the characters of a certain $GL_2$-action. Hence, they are Schur positive, as the Schur functions are the characters of the irreducibles in this context. For all triangular partitions $\lambda$, one can define a symmetric function $A_\lambda$ corresponding to the Hilbert series of alternating component of multivariate diagonal harmonic polynomials~\cite{Ber22}. The trivariate case has been studied in particular in the context of the Tamari and $m$-Tamari lattices~\cite{BPR12} with various conjectures emitted in~\cite{PR12}. For example, we have that
\begin{equation}
A_{3,2,1} = s_{6} + s_{4,1} + s_{3,1} + s_{1,1,1}.
\end{equation}
When expanded on two variables, the term $s_{1,1,1}(q,t)$ is $0$ and we get \eqref{eq:qtschur_a321}, the $(q,t)$-enumeration of Dyck paths. For the Tamari and $m$-Tamari cases, the $A_\lambda$ expanded in three variables $q$, $t$, and $r$ are conjectured to enumerate the intervals of the $m$-Tamari lattices with $q$ counting the \emph{distance} of the interval (length of the maximal chain) and $t$ counting the \emph{dinv} of the maximal element. The signification of the $r$ statistic as well as a combinatorial proof that the polynomials are symmetric in $q$, $t$ and $r$, and of their Schur positivity is still open. A wider question would be to find a combinatorial interpretation for the terms appearing in the Schur expansion of the general  polynomials $A_\lambda$.

Considering the three variables expansion for the triangular case, the first question would be to find a lattice structure ``compatible'' with the $(q,t,r)$-enumeration. In~\cite{PRV15}, the authors describe a generalization of the Tamari lattice for all sub-partitions of any given partition~$\lambda$. But this lattice does not always give the proper enumeration. In particular, the triangular tableau does not always define a maximal chain in the lattice, which seems to be an important property. This led us to define the \defn{top-down tableau} by iteratively removing the corners of the partition from top to bottom (see last tableau of \cref{fig:simsym2}) and to state the following conjecture.

\begin{conjecture}
\label{conj:lattice}
If $\lambda$ is a triangular partition such that its top-down tableau $\theta$ is a sim-sym tableau, then
\begin{equation}
A_\lambda(q,t,1) = \sum_{(\lambda, \mu_1) \leq (\lambda, \mu_2)} q^{\dist_{\lambda}(\mu_1,\mu_2)} ~ t^{\simp{\theta}{\mu_2}}
\end{equation}
where the sum runs over intervals in the $\lambda$-Tamari lattice as defined in~\cite{PRV15} and $\dist_{\lambda}(\mu_1,\mu_2)$ is the length of the maximal chain between the triangular Dyck paths $(\lambda, \mu_1)$ and $(\lambda, \mu_2)$.
\end{conjecture} 

In the Tamari and $m$-Tamari cases, the top-down tableau is the triangular tableau and this is the trivariate conjecture found for example in~\cite{PR12}. In the triangular case, there are partitions such as $(4,2,1)$ where the top-down tableau is not the triangular tableau (it is the last one of~\cref{fig:simsym2}), and actually allows us to find the $A_\lambda$ enumeration. However, there are cases such as $(4,3,1)$ where the top-down tableau is not sim-sym and the question of finding a compatible lattice structure is still open. Nevertheless, in the case of triangular $2$-partitions, the top-down tableau is always sim-sym by \cref{thm:simsym}. Besides, in this case the $\lambda$-Tamari lattice has a very neat structure as shown on \cref{fig:PRV}, which allowed us to perform some recursive enumeration, and prove the conjecture.

\begin{theorem}
\label{thm:qtrsym}
For $\lambda$ a triangular $2$-partition and $\theta$ its top-down tableau, then 
\begin{equation}
A_\lambda(q,t,1) = \sum_{(\lambda, \mu_1) \leq (\lambda, \mu_2)} q^{\dist_{\lambda}(\mu_1,\mu_2)} ~t^{\simp{\theta}{\mu_2}}
\end{equation}
summing over intervals of the $\lambda$-Tamari lattice.
\end{theorem}

Note that in the case of $2$-partitions, the Schur expression of $A_\lambda$ is actually the same as for $\Alqt$, and is thus given by \cref{thm:alqt2}.


\begin{figure}[ht]
    \centering
\scalebox{0.6}{\input{figures/PRV_lattice2.tex}}
\caption{The structure of the $\lambda$-lattice on a $2$-partition. Elements share the same $\theta$-\emph{deficit} between two red dotted lines and the same $\theta$-\emph{sim} between two green dashed lines.}
\label{fig:PRV}
\end{figure} 




\acknowledgements{The authors thank Olivier Lafitte, the INSMI, the LACIM and the CRM Montréal for the opportunity to work in Montreal on this project, and all the help they gave on administrative issues. We thank François Bergeron for introducing us to the original problem, and for all the very interesting conversations we had during our time in Montreal. Lastly, all computation has been done using SageMath~\cite{SageMath2022}. }


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