
%
%In the present work we prove the so called Schr\"{o}der case of the generalized Delta conjecture, i.e. the special case $\<\cdot , e_{n-d}h_d\>$. This extends some of the results in \cite{DAdderio-VandenWyngaerd-2017}, which in turn extended the $q,t$-Schr\"{o}der of Haglund \cite{Haglund-Schroeder-2004}.
%
%As in the aforementioned works, to prove our result, we will give a recursion, which generalizes the ones in the literature. The essential tool on the symmetric function side will be the most technical result of \cite{DAdderio-VandenWyngaerd-2017}.
%
%Also, we introduce a statistic \emph{bounce} which will provide another combinatorial interpretation of the same polynomial, and we give a statistic preserving bijection between the two interpretations.
%
%It turns out that our recursion is given by iterating a simpler recursion. We will explain this intermediate recursion by giving another combinatorial interpretation of the same polynomial in terms of doubly decorated parallelogram polyominoes. This result extends some of the results in \cite{DAdderio-Iraci-polyominoes-2017}, which in turn extended the results in \cite{Aval-DAdderio-Dukes-Hicks-LeBorgne-2014}.


%In \cite{Haglund-Remmel-Wilson-2015}, the authors conjectured a combinatorial interpretation for the symmetric function \[ \Delta_{h_m}\Delta'_{e_{n-k-1}}e_n \] in terms of partially labelled decorated Dyck paths, known as the \emph{generalized Delta conjecture}, which reduces to the \emph{Delta conjecture} when $m=0$. We give the necessary definitions.
%
%\begin{definition}
%	A \emph{Dyck path} of size $n$ is a lattice path going from $(0,0)$ to $(n,n)$, using only north and east unit steps and staying weakly above the line $x=y$ (also called the \emph{main diagonal}). The set of Dyck paths of size $n$ will be denoted by $\mathsf{D}(n)$. A \emph{partially labelled Dyck path} is a Dyck path whose vertical steps are labelled with (not necessarily distinct) non-negative integers such that the labels appearing in each column are strictly increasing from bottom to top, and $0$ does not appear in the first column. The set of partially labelled Dyck paths with $m$ zero labels and $n$ nonzero labels is denoted by $\PLD(m,n)$.  
%\end{definition}
%
%Partially labelled Dyck paths differ from labelled Dyck paths only in that $0$ is allowed as a label in the former and not in the latter. 
%
%\begin{definition}\label{def: monomial path}
%	We define for each $D\in \PLD(m,n)$ a monomial in the variables $x_1,x_2,\dots$: we set \[ x^D \coloneqq \prod_{i=1}^n x_{l_i(D)} \] where $l_i(D)$ is the label of the $i$-th vertical step of $D$ (the first being at the bottom). Notice that $x_0$ does not appear, which explains the word \emph{partially}.
%\end{definition}
%
%
%
%\begin{definition}
%	Let $D$ be a (partially labelled) Dyck path of size $n+m$. We define its \emph{area word} to be the string of integers $a(D) = a_1(D) \cdots a_{n+m}(D)$ where $a_i(D)$ is the number of whole squares in the $i$-th row (counting from the bottom) between the path and the main diagonal.
%\end{definition}
%
%\begin{definition} \label{def: rise}
%	The \emph{rises} of a Dyck path $D$ are the indices \[ \Rise(D) \coloneqq \{2\leq i \leq n+m\mid a_{i}(D)>a_{i-1}(D)\},\] or the vertical steps that are directly preceded by another vertical step. Taking a subset $\DRise(D)\subseteq \Rise (D)$ and decorating the corresponding vertical steps with a $\ast$, we obtain a \emph{decorated Dyck path}, and we will refer to these vertical steps as \emph{decorated rises}. 
%\end{definition}
%
%\begin{definition}
%Given a partially labelled Dyck path, we call \emph{zero valleys} its vertical steps with label $0$ (which are necessarily preceded by an horizontal step, that is why we call them valleys).
%\end{definition}
%
%The set of partially labelled decorated Dyck paths with $m$ zero labels, $n$ nonzero labels and $k$ decorated rises is denoted by $\PLD(m,n)^{\ast k}$. See 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 $\PLD(2,6)^{\ast 2}$.}
%  \label{fig:pldExample1}
%\end{figure*}
%
%
%We define two statistics on this set.
%
%\begin{definition} \label{def: area DP}
%	We define the \emph{area} of a (partially labelled) decorated Dyck path $D$ as \[ \area(D) \coloneqq \sum_{i\not \in \DRise(D)} a_i(D). \]
%\end{definition}
%
%For a more visual definition, the area is the number of whole squares that lie between the path and the main diagonal, except for the ones in the rows containing a decorated rise. For example, the decorated Dyck path in Figure~\ref{fig:pldExample1} has area $7$. 
%
%Notice that the area does not depend on the labels. 
%
%\begin{definition} \label{def: dinv DP}
%	Let $D \in \PLD(m,n)$. For $1 \leq i < j \leq n+m$, we say that the pair $(i,j)$ is an \emph{inversion} if
%	\begin{itemize}
%		\item either $a_i(D) = a_j(D)$ and $l_i(D) < l_j(D)$ (\emph{primary inversion}),
%		\item or $a_i(D) = a_j(D) + 1$ and $l_i(D) > l_j(D)$ (\emph{secondary inversion}),
%	\end{itemize}
%	where $l_i(D)$ denotes the label of the vertical step in the $i$-th row.
%
%	Then we define \[\dinv(D)\coloneqq \# \{ 0\leq i < j \leq n+m \mid (i,j) \; \text{is an inversion} \}.\]
%\end{definition}
%
%For example, the decorated Dyck path in Figure~\ref{fig:pldExample1} has $1$ primary inversion (the pair $(2,4)$) and $2$ secondary inversions (the pairs $(2,3)$ and $(5,6)$), so its dinv is $3$. 
%
%Notice that the decorations on the rises do not affect the dinv.

\begin{definition}
	We define a formal series in the variables $\underline x=(x_1,x_2,\dots)$ and coefficients in $\mathbb N [q,t]$ 
	\[
	\gendelta \coloneqq \sum_{D\in \PLD(m,n)^{\ast k}}q^{\dinv(D)} t^{\area(D)}x^D.
	\]
\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}
%Notice that $\gendelta$ is in fact a symmetric function (cf. Remark~\ref{rmk:symmetry}).
  


