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\articlenumber{65}
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% Tikz package and libraries
\usepackage{tikz}
\usepackage{tikz-cd}
%\usepackage{tikzducks}
\usetikzlibrary{arrows,backgrounds,chains,decorations.pathreplacing,patterns,shapes}


%% Generic commands macros

% Functions
\newcommand{\area}{\mathsf{area}}
\newcommand{\dinv}{\mathsf{dinv}}
\newcommand{\inv}{\mathsf{inv}}
\newcommand{\bounce}{\mathsf{bounce}}
\newcommand{\pbounce}{\mathsf{pbounce}}
\newcommand{\pmaj}{\mathsf{pmaj}}
\newcommand{\maj}{\mathsf{maj}}
\newcommand{\ides}{\mathsf{ides}}

\newcommand{\Val}{\mathsf{Val}}
\newcommand{\Rise}{\mathsf{Rise}}
\newcommand{\Peak}{\mathsf{Peak}}
\newcommand{\ZVal}{\mathsf{ZVal}}
\newcommand{\DRise}{\mathsf{DRise}}
\newcommand{\DPeak}{\mathsf{DPeak}}

\newcommand{\DP}{\mathsf{DP}} % Dyck paths
\newcommand{\D}{\mathsf{D}} % Dyck paths (new)
\newcommand{\LD}{\mathsf{LD}} % Labelled Dyck paths
\newcommand{\DD}{\mathsf{DD}} % Decorated Dyck paths
\newcommand{\LDD}{\mathsf{LDD}} % Labelled decorated Dyck paths
\newcommand{\PDD}{\mathsf{PDD}} % parking decorated Dyck paths
\newcommand{\DDd}{\mathsf{DDd}} % Decorated Dyck paths dinv set
\newcommand{\DDb}{\mathsf{DDb}} % Decorated Dyck paths bounce set
\newcommand{\DDp}{\mathsf{DDp}} % Decorated Dyck paths pbounce set
\newcommand{\SQE}{\mathsf{SQ^{E}}} % square paths ending east
\newcommand{\SQN}{\mathsf{SQ^{N}}} % square paths ending north
\newcommand{\PLSQE}{\mathsf{PLSQ^{E}}} 
\newcommand{\PLSQN}{\mathsf{PLSQ^{N}}} 
\newcommand{\PF}{\mathsf{PF}} % Parking functions
\newcommand{\PLD}{\mathsf{PLD}} % Partially labelled Dyck paths
\newcommand{\PLDd}{\mathsf{PLDd}} % Partially labelled Dyck paths
\newcommand{\PLDb}{\mathsf{PLDb}} % Partially labelled Dyck paths

\newcommand{\PP}{\mathsf{PP}} % Parallelogram polyominoes
\newcommand{\RP}{\mathsf{RP}} % Reduced parallelogram polyominoes
\newcommand{\LPP}{\mathsf{LPP}} % Labelled parallelogram polyominoes

\newcommand{\gendelta}{\mathsf{PLD}_{\underline x,q,t}(m,n)^{\ast k}} % the combinatorial def of generalized delta conjecture.
\newcommand{\gensquare}{\mathsf{PLSQ^E}_{\underline x,q,t}(m,n)^{\ast k}} % the combinatorial def of the generalized square conjecture. 
\newcommand{\OP}{\mathsf{OP}} % ordered set partitions 
\newcommand{\OPi}{\mathsf{OPi} } % ordered set partitions with inv
\newcommand{\OPd}{\mathsf{OPd} } % ordered set partitions with dinv
\newcommand{\OPm}{\mathsf{OPm} } % ordered set partitions with maj
\newcommand{\SOP}{\mathsf{SOP}} % standard ordered set partitions 
\newcommand{\SOPi}{\mathsf{SOPi} } % standard ordered set partitions with inv
\newcommand{\SOPd}{\mathsf{SOPd} } % standard ordered set partitions with dinv
\newcommand{\SOPm}{\mathsf{SOPm} } % standard ordered set partitions with maj


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\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
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\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}


\title[Delta conjectures]{Delta Conjectures}

\author{Michele D'Adderio\thanks{\href{mailto:mdadderi@ulb.ac.be}{mdadderi@ulb.ac.be}}\addressmark{1}, Alessandro Iraci\thanks{\href{mailto: airaci@ulb.ac.be}{airaci@ulb.ac.be}}\addressmark{1}, \and Anna Vanden Wyngaerd\thanks{\href{mailto:anvdwyng@ulb.ac.be}{anvdwyng@ulb.ac.be}}\addressmark{1}}

\address{\addressmark{1}Universit\'e Libre de Bruxelles (ULB), D\'epartement de Math\'ematique,  Boulevard du Triomphe, B-1050 Bruxelles, Belgium}

%% put the date of submission here
\received{\today}

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%\revised{}

%% please don't use custom commands in your abstract / resume, as these will be displayed online
%% likewise for citations -- please don't use \cite, and instead write out your citation as something like (author year)
\abstract{We state the so called generalized Delta conjecture of Haglund, Remmel and Wilson, and we survey the state of the art about this problem. We do the same with our related generalized Delta square conjecture. Both surveys include some of our recent results.}

%% put your French abstract here, or comment this out if you don't have one
%\resume{ Ceci est un résumé }

\keywords{Delta conjecture, Delta square conjecture, Macdonald polynomials}


%\usepackage[backend=bibtex]{biblatex}
%\addbibresource{Biblebib.bib}

\begin{document}

\maketitle

\section{Introduction}
In \cite{Haglund-Remmel-Wilson-2018}, Haglund, Remmel and Wilson conjectured a combinatorial formula for $\Delta_{e_{n-k-1}}'e_n$ in terms of decorated labelled Dyck paths, which they called \emph{Delta conjecture}, after the so called delta operators $\Delta_f'$ introduced by Bergeron, Garsia, Haiman and Tesler \cite{Bergeron-Garsia-Haiman-Tesler-Positivity-1999} for any symmetric function $f$. In fact in the same article \cite{Haglund-Remmel-Wilson-2018} the authors conjectured a combinatorial formula for the more general $\Delta_{h_m}\Delta_{e_{n-k-1}}'e_n$ in terms of decorated partially labelled Dyck paths, which we call \emph{generalized Delta conjecture}.

The special case $m=k=0$ gives precisely the \emph{Shuffle conjecture} in \cite{HHLRU-2005}, now a theorem of Carlsson and Mellit \cite{Carlsson-Mellit-ShuffleConj-2018}. The latter turns out to be a combinatorial formula for the Frobenius characteristic of the $\mathfrak{S}_n$-module of diagonal harmonics studied by Garsia and Haiman in relation to the famous \emph{$n!$ conjecture}, now $n!$ theorem of Haiman \cite{Haiman-nfactorial-2001}.


In \cite{Loehr-Warrington-square-2007} Loehr and Warrington conjectured a combinatorial formula for $\Delta_{e_{n}}\omega(p_n)=\nabla \omega(p_n)$ in terms of labelled square paths (ending east), called \emph{square conjecture}. The special case $\<\cdot ,e_n\>$ of this conjecture, known as \emph{$q,t$-square}, has been proved earlier by Can and Loehr in \cite{Can-Loehr-2006}. Recently the full square conjecture has been proved by Sergel in \cite{Leven-2016} after the breakthrough of Carlsson and Mellit in \cite{Carlsson-Mellit-ShuffleConj-2018}.

In \cite{DAdderio-Iraci-VandenWyngaerd-DeltaSquare-2019}, we conjectured a combinatorial formula for $\frac{[n-k]_t}{[n]_t}\Delta_{h_m}\Delta_{e_{n-k}}\omega(p_n)$ in terms of \emph{decorated partially labelled square paths} that we call \emph{generalized Delta square conjecture}. In analogy with the Delta conjecture in \cite{Haglund-Remmel-Wilson-2018}, we call simply \emph{Delta square conjecture} the special case $m=0$. Our conjecture extends the square conjecture of Loehr and Warrington \cite{Loehr-Warrington-square-2007} (now a theorem \cite{Leven-2016}), i.e. it reduces to that one for $m=k=0$. Moreover, it extends the generalized Delta conjecture in the sense that on decorated partially labelled Dyck paths gives the same combinatorial statistics. Our conjecture answers a question in \cite{Haglund-Remmel-Wilson-2018}.

In the present article we sketch the state of the art about the generalized Delta conjecture and the generalized Delta square conjecture. In this survey we highlights some results from our more recent works \cite{DAdderio-Iraci-VandenWyngaerd-DeltaSquare-2019,DAdderio-Iraci-VandenWyngaerd-Delta-t0-2020,DAdderio-Iraci-VandenWyngaerd-GenDeltaSchroeder-2019}.


\section{Combinatorial definitions}
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 \cref{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 \cref{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 paths 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 \cref{fig: labelled square} and \cref{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 \cref{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 \cref{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}

\section{Symmetric functions}
In this short section we limit ourself to recall the definitions needed to state the Delta conjectures, in particular the definition of the Delta operators. For the missing notation we refer to \cite[Section~1]{DAdderio-VandenWyngaerd-2017} or \cite{Haglund-Book-2008}.

We denote by $\Lambda$ the algebra over the field $\mathbb{Q}(q,t)$ of symmetric functions in the variables $x_1,x_2,\dots$. We denote by $e_n$, $h_n$ and $p_n$ the \emph{elementary}, \emph{complete homogeneous} and \emph{power symmetric function} of degree $n$, respectively. We denote by $\omega$ the involution of $\Lambda$ defined by $\omega(e_n):=h_n$ for all $n$.

Also, for any partition $\mu$, we denote by $s_{\mu}\in \Lambda$ the corresponding \emph{Schur function}. It is well-known that the symmetric functions $\{s_{\mu}\}_{\mu}$ form a basis of $\Lambda$. The \emph{Hall scalar product} on $\Lambda$, denoted $\<\, ,\, \>$, can be defined by stating that the Schur functions are an orthonormal basis.

Let $\widetilde{H}_{\mu}\in \Lambda$ denote the \emph{(modified) Macdonald polynomial} indexed by the partition $\mu$. As the polynomials $\{\widetilde{H}_{\mu}\}_{\mu}$ form a basis of $\Lambda$, given a symmetric function $f\in \Lambda$, we can define the \emph{Delta operators} $\Delta_f$ and $\Delta_f'$ on $\Lambda$ by setting
%
\begin{align}
\Delta_f \widetilde{H}_{\mu} \coloneqq f[B_{\mu}(q,t)] \widetilde{H}_{\mu} \quad \text{ and } \quad 
\Delta_f' \widetilde{H}_{\mu}  \coloneqq f[B_{\mu}(q,t)-1] \widetilde{H}_{\mu}, \quad \text{ for all } \mu,
\end{align}
where $B_{\mu}(q,t)=\sum_{c\in \mu}q^{a_\mu'(c)}t^{l_\mu'(c)}$ ($a_\mu'(c)$ and $l_\mu'(c)$ are the coarm and coleg of $c$ in $\mu$, respectively) and the square brackets denote the \emph{plethystic} substitution: if $X = x_1 + x_2 + \dots$ is a sum of monomials, then $f[X] \coloneqq f(x_1, x_2, \dots)$. 

For example, if $\mu=(3,2)$ and $f=e_4$, then $B_{(3,2)}(q,t)=1+q+q^2+t+qt$, so that $f[B_\mu(q,t)]=e_4(1,q,q^2,t,qt)=q^4 t^2+q^4 t+q^3 t^2+q^3 t+q^2 t^2$ and $f[B_\mu(q,t)-1]=e_4(q,q^2,t,qt)=q^4t^2$.

%and both $B_\mu$ and $B_\mu - 1$ are, in fact, sum of monomials.

Finally, we recall here the notation for the \emph{$t$-analogue} of $n\in \mathbb{N}$:
\[[n]_t\coloneqq 1+t+t^2+\cdots +t^{n-1}. \]

 

\section{The generalized Delta conjecture}
\subsection{Statement}

%
%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-2018}.
\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}).
  



\subsection{State of the art}

The case $m=k=0$, i.e. the Shuffle conjecture, has been proved by Carlsson and Mellit in \cite{Carlsson-Mellit-ShuffleConj-2018}. Several special cases have also been proved in the original work of Haglund, Remmel and Wilson \cite{Haglund-Remmel-Wilson-2018}.

We limit ourselves to summarize in the following table some of the most general special cases that have been proved more recently.

\begin{center}
\begin{tabular}{cc}
	Conditions & Reference \\
	\hline 
	$m=0$ and $q=0$ & \cite{Garsia-Haglund-Remmel-Yoo-2019} \\
	$m=0$ and $q=1$ & \cite{Romero-Deltaq1-2017} \\
	$m=0$ and $\<\cdot , h_{n-d}h_d\>$ & \cite{DAdderio-Iraci-polyominoes-2017} \\
	$\<\cdot , e_{n-d}h_d\>$ & \cref{ssec:oor_delta} \\
	$t=0$ or $q=0$ & \cref{ssec:oor_delta}
\end{tabular}
\end{center}

We should mention here also the proof of the so called \emph{$4$-variable Catalan conjecture} due to Zabrocki \cite{Zabrocki-4Catalan-2016}, which, combined with the results in \cite{DAdderio-VandenWyngaerd-2017}, gives a ``compositional'' refinement of the case $m=0$ and $\<\cdot , e_{n-d}h_d\>$.

For more results on the Delta conjecture, i.e. the case $m=0$ see \cite{Haglund-Rhoades-Shimozono-Advances,Haglund-Xin-Lecture-Notes,Haglund-Rhoades-Shimozono-Advances,Qiu-Remmel-Sergel-Xin-2017,Remmel-Wilson-2015,Rhoades-2018,Wilson-2017,Wilson-PhD-2015}.
\subsection{Our recent results} \label{ssec:oor_delta}
\begin{definition}
	Given two sequences $(a_1, \dots, a_m)$, $(b_1, \dots, b_n)$ of pairwise distinct elements, their \emph{shuffle} $(a_1, \dots, a_m) \shuffle (b_1, \dots, b_n)$ is the set of sequences $(c_1, \dots, c_{m+n})$ such that
	\begin{itemize}
		\item $\{ c_k \mid 1 \leq k \leq m+n \} = \{ a_i, b_j \mid 1 \leq i \leq m, \; 1 \leq j \leq n \}$,
		\item $c_r = a_i, \; c_s = a_j, \; i < j \implies r < s$,
		\item $c_r = b_i, \; c_s = b_j, \; i < j \implies r < s$,
	\end{itemize}
	i.e. it is the set of sequences obtained by interlacing the elements of the two starting sequences while preserving their relative order.
\end{definition}

\begin{definition}
	Given $\mu \vdash n-d$ and $\nu \vdash d$, a $\mu,\nu$-\emph{shuffle} is a sequence of numbers from $1$ to $n$ in
	\begin{align*}
	(1, \dots, \mu_1) & \shuffle \dots \shuffle (n-\mu_{\ell(\mu)}+1, \dots, n-d) \\
	& \shuffle (n-d+\nu_1, \dots, n-d+1) \shuffle \dots \shuffle (n, \dots, n-\nu_{\ell(\nu)}+1)
	\end{align*}
	i.e. a shuffle of $\ell(\mu)$ increasing sequences of length $\mu_1, \dots, \mu_{\ell(\mu)}$, and $\ell(\nu)$ decreasing sequences of length $\nu_1, \dots, \nu_{\ell(\nu)}$, obtained by picking every time the smallest available positive integers.
%	Given $\mu \vdash n$, a $\mu$-shuffle is a string of numbers from $1$ to $n$ such that the substrings $(1, \dots, \mu_1)$, $(\mu_1+1, \dots, \mu_1+\mu_2)$, $\dots$, $(n-\mu_{\ell(\mu)}+1, \dots, n)$ appear in increasing order. Given $\mu \vdash n-d$ and $\nu \vdash d$, a $\mu,\nu$-shuffle is a string of numbers from $1$ to $n$ such that the substrings $(1, \dots, \mu_1)$,  $\dots$, $(n-d-\mu_{\ell(\mu)}+1, \dots, n-d)$ appear in increasing order, and the substrings $(n-d+\nu_1, \dots, n-d+1)$, $\dots$, $(n, \dots, n-\nu_{\ell(\nu)}+1)$ appear in decreasing order.
\end{definition}

It is well known (cf. \cite[Chapter~6]{Haglund-Book-2008}) that taking the scalar product of the combinatorial side of any of the Delta conjectures with $e_\mu \cdot h_\nu$ corresponds to taking the subsets of paths whose reading word is a $\mu,\nu$-shuffle.

The following theorem is proved in \cite{DAdderio-Iraci-VandenWyngaerd-GenDeltaSchroeder-2019}. It extends the case $m=0$ proved in \cite{DAdderio-VandenWyngaerd-2017}.

\begin{theorem}
For $m,n,k,d\in \mathbb{N}$ with $n>k\geq 0$, $n\geq d\geq 0$ and $m\geq 0$
\begin{equation}
\<\Delta_{h_m}\Delta_{e_{n-k-1}}'e_n,e_{n-d}h_d\> = \mathop{\sum_{D\in \PLD(m,n)^{\ast k}}}_{\sigma(D)\text{ is a }(n-d),(d)\text{-shuffle}} q^{\dinv(D)} t^{\area(D)}x^D.
\end{equation}
\end{theorem}

The following theorem is proved in \cite{DAdderio-Iraci-VandenWyngaerd-Delta-t0-2020}. It extends the case $m=0$ first proved in \cite{Garsia-Haglund-Remmel-Yoo-2019}, giving a new independent proof of that special case (see also \cite{Haglund-Rhoades-Shimozono-Advances} and \cite{DAdderio-Iraci-VandenWyngaerd-DeltaSquare-2019} for alternative proofs).
\begin{theorem}
	For $m,n,k,d\in \mathbb{N}$ with $n>k\geq 0$, $n\geq d\geq 0$ and $m\geq 0$, we have
	\begin{equation}
	\left.\Delta_{h_m}\Delta_{e_{n-k-1}}'e_n\right|_{t=0} =\PLD_{\underline{x},q,0}(m,n)^{\ast k}\quad \text{ and } \quad \left.\Delta_{h_m}\Delta_{e_{n-k-1}}'e_n\right|_{q=0} =\PLD_{\underline{x},0,t}(m,n)^{\ast k}.
	\end{equation}
\end{theorem}

\section{The generalized Delta square conjecture}
\subsection{Statement}
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-DeltaSquare-2019}.

\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}

\begin{remark} \label{rmk:symmetry}
Observe that $\gensquare$ is a symmetric function. Indeed, consider the expression $\sum_Pq^{\dinv(P)}t^{\area(P)}x^P$ where the sum is taken over all $P\in \PLSQE(m,n)^{\ast k}$ of a fixed \emph{shape}, i.e. a fixed underlying square path with prescribed zero valleys. From this sum we can factor $t^{\area(P)}$, as the area is the same for all such paths $P$, and $q^{a(P)}$, where $a(P)$ is the contribution to the dinv of the $0$ labels and of the negative letters of the area word (the bonus dinv): indeed this contribution does not depend on the nonzero labels, but only on the shape, so it will be the same for all our paths. What we are left with is in fact an LLT polynomial: the argument is essentially the same as in \cite[Section~6.2]{Haglund-Remmel-Wilson-2018}, so we omit it (cf also \cite[Remark~6.5]{Haglund-Book-2008}). As it is well-known that the LLT polynomials are symmetric functions (cf. \cite[Appendix]{Haglund-Haiman-Loehr}), we deduce that also $\gensquare$ is symmetric.
\end{remark}

This conjecture answers a question in \cite[Section~8.2]{Haglund-Remmel-Wilson-2018}.

\begin{remark}
Notice that the case $m=k=0$ of the generalized Delta square conjecture reduces precisely to the \emph{square conjecture} of Loehr and Warrington \cite{Loehr-Warrington-square-2007}, recently proved by Sergel \cite{Leven-2016} after the breakthrough of Carlsson and Mellit \cite{Carlsson-Mellit-ShuffleConj-2018}.
\end{remark}

% We need some symmetric functions identities if we want to include this examplet

%\begin{example}
%Using \eqref{eq:en_expansion} and \eqref{eq:p_expansion} (see Section~\ref{sec:SF} for definitions and notations about symmetric functions), it is easy to see that for $m=0$ and $k=n-1$ we get
%\begin{equation}
%\frac{[1]_t}{[n]_t} \Delta_{h_0}\Delta_{e_{1}}\omega(p_n) = [n]_qe_n.
%\end{equation}
%We leave to the reader the straightforward verification that indeed
%\begin{equation}
%\mathsf{PLSQ^E}_{\underline x,q,t}(0,n)^{\ast n-1} = [n]_qe_n,
%\end{equation}
%proving in this way our conjecture at $m=0$ and $k=n-1$.
%\end{example}


\subsection{State of the art}
We limit ourselves to summarize in the following table some of the most general special cases that have been proved more recently.

\begin{center}
\begin{tabular}{cc}
	Conditions & Reference \\
	\hline 
	$m=0$ and $k=0$ &  \cite{Leven-2016} \\
	$\<\cdot , e_{n-d}h_d\>$ & \cref{ssec:oor_square}\\
	$q=0$ & \cref{ssec:oor_square}
	\end{tabular}
\end{center}

We would like to mention here the early proof by Can and Loehr \cite{Can-Loehr-2006} of the special case $m=k=0$ and $\<\cdot , e_{n}\>$.
\subsection{Our recent results} \label{ssec:oor_square}
We recall that taking a scalar product with $e_\mu \cdot h_\nu$ corresponds to taking the subsets of paths whose reading word is a $\mu,\nu$-shuffle.

The following theorem is proved in \cite{DAdderio-Iraci-VandenWyngaerd-DeltaSquare-2019}. It extends the case $m=k=d=0$ proved by Can and Loehr in \cite{Can-Loehr-2006}.
\begin{theorem}
For $m,n,k,d\in \mathbb{N}$ with $n>k\geq 0$, $n\geq d\geq 0$ and $m\geq 0$
\begin{equation}
\left\<\frac{[n-k]_t}{[n]_t}\Delta_{h_m}\Delta_{e_{n-k}}\omega(p_n),e_{n-d}h_d\right\> = \mathop{\sum_{P\in \PLSQE(m,n)^{\ast k}}}_{\sigma(P)\text{ is a }(n-d),(d)\text{-shuffle}} q^{\dinv(P)} t^{\area(P)}x^P.
\end{equation}
\end{theorem}

The following theorem is proved in \cite{DAdderio-Iraci-VandenWyngaerd-DeltaSquare-2019}.
\begin{theorem}
	For $m,n,k,d\in \mathbb{N}$ with $n>k\geq 0$, $n\geq d\geq 0$ and $m\geq 0$, we have
	\begin{equation}
	\left.\frac{[n-k]_t}{[n]_t}\Delta_{h_m}\Delta_{e_{n-k}}\omega(p_n)\right|_{q=0} =\PLSQE_{\underline{x},0,t}.
	\end{equation}
\end{theorem}
Observe that the case $t=0$ of the generalized Delta square conjecture remains open.

\section{Relations among Delta conjectures}
In \cite{DAdderio-Iraci-VandenWyngaerd-DeltaSquare-2019} we proved the following results.

\begin{proposition} \label{prop:t0}
	For $n\in \mathbb{N}$, $n >0$,
	\begin{equation}
	\left.\Delta_{h_m} \Delta_{e_{n-1}}'e_n\right|_{t=0} = \mathsf{PLD}_{\underline x,q,0}(m,n)^{\ast 0}.
	\end{equation}
\end{proposition}
\begin{theorem} \label{thm:SFinvoDelta}
	Given $m,n\in \mathbb{N}$, $m\geq 0$ and $n\geq 1$, we have the two identities
	\begin{align} 
	\sum_{s= 0}^{n-1}(-t)^s\Delta_{h_m}\Delta_{e_{n-s-1}}'e_n & = \left.\Delta_{h_m}\Delta_{e_{n-1}}' e_n \right|_{t=0}
	\end{align}
	and
	\begin{align}
	\sum_{s= 0}^{n-1}(-t)^s\frac{[n-s]_t}{[n]_t}\Delta_{h_m}\Delta_{e_{n-s}}\omega(p_n) & = \left.\Delta_{h_m}   \Delta_{e_{n-1}}' e_n\right|_{t=0} \,\, . 
	\end{align}
\end{theorem}

The last theorem, combined with the Delta conjectures, suggested the following one.

\begin{theorem}
	Given $m,n\in \mathbb{N}$, $m\geq 0$ and $n\geq 1$, we have the two identities
	\begin{align}
	\sum_{s=0}^{n-1}(-t)^s\mathsf{PLSQ^E}_{\underline{x},q,t}(m,n)^{\ast s} & = \mathsf{PLD}_{\underline{x},q,0}(m,n)^{\ast 0} \\
	\sum_{s=0}^{n-1}(-t)^s\mathsf{PLD}_{\underline{x},q,t}(m,n)^{\ast s} & = \mathsf{PLD}_{\underline{x},q,0}(m,n)^{\ast 0}.
	\end{align}
\end{theorem}
\begin{proof}
Fix $m,n\in \mathbb{N}$, $m\geq 0$ and $n>0$. Let 
\begin{equation}
X\coloneqq\bigsqcup_{k=0}^{n-1}\mathsf{PLSQ^E}(m,n)^{\ast k},
\end{equation}
and define a map $\varphi \colon X \to X$ in the following way: if $P\in X$ has no rises, i.e. no two consecutive vertical steps, then $\varphi(P) \coloneqq P$; otherwise, consider the first rise encountered by following the path $P$ starting from its breaking point (notice that this rise will always occur before the north-east corner): if the rise is decorated/undecorated, then $\varphi(P)$ is the path obtained from $P$ by undecorating/decorating that rise. Observe that $\varphi$ is clearly an involution, whose fixed points are the paths $P\in X$ with no rises, i.e. the paths of area $0$ with no decorated rises. Notice also that $\varphi$ restricts to an involution of
\begin{equation}
Y\coloneqq\bigsqcup_{k=0}^{n-1}\mathsf{PLD}(m,n)^{\ast k}\subset X.
\end{equation}

For any $P\in X$ we define a \emph{weight} by setting
\begin{equation}
wt(P)\coloneqq(-t)^{\mathsf{dr}(P)}q^{\dinv(P)}t^{\area(P)}x^P
\end{equation}
where $\mathsf{dr}(P)$ is defined to be the number of decorated rises of $P$.

Observe that
\begin{equation}
\sum_{P \in X}wt(P) = \sum_{s=0}^{n-1}(-t)^s\mathsf{PLSQ^E}_{\underline{x},q,t}(m,n)^{\ast s}
\end{equation}
and
\begin{equation}
\sum_{P \in Y}wt(P) = \sum_{s=0}^{n-1}(-t)^s\mathsf{PLD}_{\underline{x},q,t}(m,n)^{\ast s}.
\end{equation}

Suppose that $P\in X$ is such that $\varphi(P)\neq P$. Notice that the rise occurring in the definition of $\varphi$ is always at distance $1$ from the base diagonal, so undecorating/decorating it when it is decorated/undecorated gives $\mathsf{dr}(\varphi(P))=\mathsf{dr}(P)\mp 1$, but $\area(\varphi(P))=\area(P)\pm 1$. Since the decorations of the rises do not affect the dinv, we deduce
that $wt(\varphi(P))=-wt(P)$. This shows that in the sum $\sum_{P \in X}wt(P)$ all the contributions of the $P$ that are not fixed by $\varphi$ cancel out, leaving the sum over the fixed points of $\varphi$, i.e. over the paths with no rises, as we claimed.

The same argument applies to the sum $\sum_{P \in Y}wt(P)$.
\end{proof}


These last results combined give relations among the Delta conjectures. We get immediately the following curious corollary.
\begin{corollary}
	For fixed $m,n\in \mathbb{N}$, with $m\geq 0$ and $n> 0$, the truth of the generalized Delta (square) conjectures for all values of $k$ in $\{0,1,\dots,n-1\}$ except one imply the truth of the missing case.
\end{corollary}


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