In order to state the generalized Delta conjecture and the generalized Delta square conjecture, we need to introduce some combinatorial objects.

\begin{definition}
	A \emph{square path ending east} of size $n$ is a lattice paths going from $(0,0)$ to $(n,n)$ consisting of east or north unit steps, always ending with an east step. The set of such paths is denoted by $\SQE(n)$. We call \emph{base diagonal} of a square path the diagonal $y=x+k$ with the smallest value of $k$ that is touched by the path (so that $k\leq 0$). The \emph{shift} of the square path is the non-negative value $-k$. The \emph{breaking point} of the square path is the lowest point in which the path touches the base diagonal. A \emph{Dyck path} is a square path whose breaking point is $(0,0)$.
\end{definition}

For example, the path in Figure~\ref{fig: labelled square} has shift $3$. 

\begin{figure}[ht]
	\centering
	\begin{tikzpicture}[scale = 0.8]
	\draw[step=1.0, gray!60, thin] (0,0) grid (8,8);
	
	\draw[gray!60, thin] (3,0) -- (8,5);
	
	\draw[blue!60, line width=1.6pt] (0,0) -- (0,1) -- (1,1) -- (2,1) -- (3,1) -- (4,1) -- (4,2) -- (5,2) -- (5,3) -- (5,4) -- (6,4) -- (6,5) -- (6,6) -- (6,7) -- (7,7) -- (7,8) -- (8,8);
	
	\node at (5.5,5.5) {$\ast$};
	
	\node at (0.5,0.5) {$2$};
	\draw (0.5,0.5) circle (.4cm); 
	\node at (4.5,1.5) {$0$};
	\draw (4.5,1.5) circle (.4cm); 
	\node at (5.5,2.5) {$2$};
	\draw (5.5,2.5) circle (.4cm); 
	\node at (5.5,3.5) {$4$};
	\draw (5.5,3.5) circle (.4cm); 
	\node at (6.5,4.5) {$0$};
	\draw (6.5,4.5) circle (.4cm); 
	\node at (6.5,5.5) {$1$};
	\draw (6.5,5.5) circle (.4cm); 
	\node at (6.5,6.5) {$3$};
	\draw (6.5,6.5) circle (.4cm); 
	\node at (7.5,7.5) {$1$};
	\draw (7.5,7.5) circle (.4cm); 
	\filldraw (4,1) circle (2pt);
	\end{tikzpicture}
	\caption{Example of an element in $\PLSQE(2,6)^{\ast 1 }$ with reading word $241231$.}
	\label{fig: labelled square}
\end{figure}
	
\begin{definition}
	 A \emph{partially labelled square path ending east} is a square path ending east whose vertical steps are labelled with (not necessarily distinct) non-negative integers such that the labels appearing in each column are strictly increasing bottom to top, there is at least one nonzero label labelling a vertical step starting from the base diagonal, and if the path starts with a vertical step, this first step's label is nonzero. The set of partially labelled square paths ending east with $m$ zero labels and $n$ nonzero labels is denoted by $\PLSQE(m,n)$. 
	 
	 The subset of the Dyck paths, called \emph{partially labelled Dyck paths}, is denoted by $\PLD(m,n)$.  
\end{definition}

\begin{definition}
	Let $P$ be a (partially labelled) square path ending east of size $n+m$. We define its \emph{area word} to be the sequence of integers $a(P) = (a_1(P),a_2(P), \cdots, a_{n+m}(P))$ where the $i$-th vertical step of the path starts from the diagonal $y=x+a_i(P)$. For example the path in Figure~\ref{fig: labelled square} has area word $(0, \, -\!3, \, -\!3, \, -\!2, \, -\!2, \, -\!1, \, 0, \, 0)$.
\end{definition}

\begin{definition}\label{def: monomial path}
	We define for each $P\in \PLSQE(m,n)$ a monomial in the variables $x_1,x_2,\dots$: we set \[ x^P \coloneqq \prod_{i=1}^{n+m} x_{l_i(P)} \] where $l_i(P)$ is the label of the $i$-th vertical step of $P$ (the first being at the bottom), where we conventionally set $x_0 = 1$. The fact that $x_0$ does not appear in the monomial explains the word \emph{partially}.
\end{definition}

\begin{definition}\label{def: rise}
	The \emph{rises} of a square path ending east $P$ are the indices \[ \Rise(P) \coloneqq \{2\leq i \leq n+m\mid a_{i}(P)>a_{i-1}(P)\},\] or the vertical steps that are directly preceded by another vertical step. Taking a subset $\DRise(P)\subseteq \Rise (P)$ and decorating the corresponding vertical steps with a $\ast$, we obtain a \emph{decorated square path}, and we will refer to these vertical steps as \emph{decorated rises}.
\end{definition}

\begin{definition}
	Given a partially labelled square path, we call \emph{zero valleys} its vertical steps with label $0$ (which are necessarily preceded by a horizontal step, hence the name valleys).
\end{definition}

\begin{definition}
	Given a partially labelled square path $D$, we define its \emph{reading word} $\sigma(D)$ as the sequence of nonzero labels, read starting from the base diagonal going bottom left to top right, then moving to the next diagonal again going bottom left to top right, and so on.
\end{definition}

The set of partially labelled decorated square pahts ending east with $m$ zero labels, $n$ nonzero labels and $k$ decorated rises is denoted by $\PLSQE(m,n)^{\ast k}$. The subset of the Dyck paths is denoted by $\PLD(m,n)^{\ast k}$. See Figure~\ref{fig: labelled square} and Figure~\ref{fig:pldExample1} for an example.

\begin{figure*}[!ht]
	\centering
	\begin{tikzpicture}[scale = .8]
	
	\draw[step=1.0, gray!60, thin] (0,0) grid (8,8);
	
	\draw[gray!60, thin] (0,0) -- (8,8);
	
	\draw[blue!60, line width=2pt] (0,0) -- (0,1) -- (0,2) -- (1,2) -- (2,2) -- (2,3) -- (2,4) -- (2,5) -- (3,5) -- (4,5) -- (4,6) -- (4,7) -- (4,8) -- (5,8) -- (6,8) -- (7,8) -- (8,8);
	
	\draw (0.5,0.5) circle (0.4 cm) node {$1$};
	\draw (0.5,1.5) circle (0.4 cm) node {$3$};
	\draw (2.5,2.5) circle (0.4 cm) node {$0$};
	\draw (2.5,3.5) circle (0.4 cm) node {$4$};
	\draw (2.5,4.5) circle (0.4 cm) node {$6$};
	\draw (4.5,5.5) circle (0.4 cm) node {$0$};
	\draw (4.5,6.5) circle (0.4 cm) node {$2$};
	\draw (4.5,7.5) circle (0.4 cm) node {$6$};
	
	\node at (1.5,3.5) {$\ast$};
	\node at (3.5,6.5) {$\ast$};
	
	\end{tikzpicture}
	\caption{Example of an element in $\PLD(2,6)^{\ast 2}$ with reading word $134626$.}
	\label{fig:pldExample1}
\end{figure*}

We define two statistics on this set that reduce to the same statistics as defined in \cite{Loehr-Warrington-square-2007} when $m=k=0$. 

\begin{definition}
	\label{def:sqarea}
	Let $P \in \PLSQE(m,n)^{\ast k}$ and $s$ be its shift. Define 
	\[
	\area(P) \coloneqq \sum_{i\not \in \DRise(P)} (a_i(P) + s).
	\] More visually, the area is the number of whole squares between the path and the base diagonal and not contained in rows containing a decorated rise. 
\end{definition}

For example, the path in Figure~\ref{fig: labelled square} has area $11$. 

\begin{definition} \label{def: dinv SQ}
	Let $P \in \PLSQE(m,n)$. For $1 \leq i < j \leq n+m$, we say that the pair $(i,j)$ is an \emph{inversion} if
	\begin{itemize}
		\item either $a_i(P) = a_j(P)$ and $l_i(P) < l_j(P)$ (\emph{primary inversion}),
		\item or $a_i(P) = a_j(P) + 1$ and $l_i(P) > l_j(P)$ (\emph{secondary inversion}),
	\end{itemize}
	where $l_i(P)$ denotes the label of the vertical step in the $i$-th row.

	Then we define 
	\begin{align*}
		\dinv(P) & \coloneqq \# \{ 0\leq i < j \leq n+m \mid (i,j) \; \text{is an inversion}\} \\
		& \quad + \#\{0\leq i\leq m+n \mid a_i(P)<0\text{ and } l_i(P)\neq 0 \}.
	\end{align*} 
	This second term is referred to as \emph{bonus dinv}. 
\end{definition}

For example, the path in Figure~\ref{fig: labelled square} has dinv $6$: $2$ primary inversions, i.e. $(1,7)$ and $(2,3)$, $1$ secondary inversion, i.e. $(1,6)$, and $3$ bonus dinv, coming from the rows $3$, $4$ and $6$. Notice that Dyck paths coincide with the square paths with no bonus dinv.

%\begin{definition}
%	We define two formal series in the variables $\underline x=(x_1,x_2,\dots)$ and coefficients in $\mathbb N [q,t]$
%	\[ \gendelta \coloneqq \sum_{D\in \PLD(m,n)^{\ast k}}q^{\dinv(D)} t^{\area(D)}x^D \]
%	\[ \gensquare \coloneqq \sum_{P\in \PLSQE(m,n)^{\ast k}} q^{\dinv(P)}t^{\area(P)}x^P. \]
%\end{definition}
%
%The following conjecture is stated by Haglund, Remmel and Wilson in \cite{Haglund-Remmel-Wilson-2015}.
%
%\begin{conjecture}[Generalized Delta]
%	For $m,n,k\in \mathbb{N}$, $m\geq 0$ and $n>k\geq 0$,
%	\[ \Delta_{h_{m}} \Delta'_{e_{n-k-1}} e_{n} = \gendelta . \]
%\end{conjecture}
%
%In analogy with the Delta conjecture, we will refer to the case $m=0$ of the following conjecture simply as the \emph{Delta square conjecture}.
%
%We formulated the following conjecture in \cite{DAdderio-Iraci-VandenWyngaerd-Delta-Square}.
%
%\begin{conjecture}[Generalized Delta square]
%	For $m,n,k\in \mathbb{N}$, $m\geq 0$ and $n>k\geq 0$,
%	\[
%	\frac{[n-k]_t}{[n]_t} \Delta_{h_m}\Delta_{e_{n-k}}\omega(p_n) = \gensquare.
%	\]	
%\end{conjecture}