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\documentclass[finalversion]{FPSAC2018}
\articlenumber{34}
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\newtheorem{thm}{Theorem}[section]
\newtheorem{lem}[thm]{Lemma}

\usepackage{lipsum}

\usepackage{tikz,amssymb,amsmath,listings,amsthm,graphicx,enumitem,hyperref}
\usetikzlibrary{patterns,decorations.pathreplacing}

\definecolor{mypink}{RGB}{255,192,203}
\definecolor{myblue}{RGB}{160,203,251}

\newcommand{\rank}{\mathsf{rank}}
\newcommand{\Id}{\mathsf{Id}}
\newcommand{\degree}{\mathsf{degree}}
\newcommand{\sort}{\mathsf{sort}}
\newcommand{\compact}{\mathsf{compact}}
\newcommand{\xpara}{\mathsf{xpara}}
\newcommand{\ypara}{\mathsf{ypara}}
\newcommand{\configuration}[2]{\binom{#1}{#2}}
\newcommand{\topplingconfiguration}[1]{\Delta^{(#1)}}
\newcommand{\topplingequiv}{\equiv_\Delta}
\newcommand{\park}{\mathsf{park}}
\newcommand{\diracconfiguration}[1]{\epsilon^{(#1)}}
\newcommand{\topplingandpermutingequiv}[1]{\equiv_{\Delta,#1}}
\newcommand{\Ta}{\mathsf{T_a}}
\newcommand{\Tb}{\mathsf{T_b}}
\newcommand{\prank}{\mathsf{prank}}
\newcommand{\pdegree}{\mathsf{pdegree}}


\newcommand{\area}{\mathsf{area}}
\newcommand{\dinv}{\mathsf{dinv}}
\newcommand{\bounce}{\mathsf{bounce}}
\newcommand{\pbounce}{\mathsf{pbounce}}
\newcommand{\pmaj}{\mathsf{pmaj}}

\newcommand{\Val}{\mathsf{Val}}
\newcommand{\Rise}{\mathsf{Rise}}
\newcommand{\Peak}{\mathsf{Peak}}
\newcommand{\DVal}{\mathsf{DVal}}
\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{\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}}} 
\newcommand{\SQN}{\mathsf{SQ^{N}}} 
\newcommand{\LSQE}{\mathsf{LSQ^{E}}} 
\newcommand{\LSQN}{\mathsf{LSQ^{N}}} 

%% define your title in the usual way
\title{Decorated Dyck paths and the Delta conjecture}

%% define your authors in the usual way
%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
\author{Michele D'Adderio\thanks{\href{mailto:mdadderi@ulb.ac.be}{mdadderi@ulb.ac.be}}\addressmark{1} \and Anna Vanden Wyngaerd \thanks{\href{mailto:anvdwyng@ulb.ac.be}{anvdwyng@ulb.ac.be}}\addressmark{1}}

%% then use \addressmark to match authors to institutions here
\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{}

%% put your English abstract here, or comment this out if you don't have one yet
%% 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 discuss the combinatorics of the decorated Dyck paths appearing in the Delta conjecture framework in (Haglund et al 2015) and (Zabrocki 2016), by introducing two new statistics, $\mathsf{bounce}$ and $\mathsf{bounce}'$. We then provide plethystic formulae for their $q,t$-enumerators, by proving in this way a decorated version of Haglund's $q,t$-Schr\"{o}der theorem, answering a question in (Haglund et al 2015). In particular we provide both an algebraic and a combinatorial explanation of a symmetry conjecture in (Haglund et al 2015) and (Zabrocki 2016). 

This is an extended abstract of (D'Adderio, Vanden Wyngaerd 2017).}

%% put your French abstract here, or comment this out if you don't have one
%\resume{\lipsum[2]}

%% put your keywords here, or comment this out if you don't have them yet
\keywords{Macdonald Polynomials, Delta conjecture, decorated Dyck paths}

%% you can include your bibliography however you want, but using an external .bib file is STRONGLY RECOMMENDED and will make the editor's life much easier
%% regardless of how you do it, please use numerical citations, ie. [xx, yy] in the text

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\usepackage[backend=bibtex]{biblatex}
\addbibresource{34-DAdderio-VandenWyngaerd.bib}
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\begin{document}

\maketitle
%% note that you DO NOT have to put your abstract here -- it is generated by \maketitle and the \abstract and \resume commands above

\section{Introduction}

Motivated by the problem of proving the Schur positivity of the modified Macdonald polynomials, Garsia and Haiman introduced the $\mathfrak{S}_n$-module of diagonal harmonics, and they conjectured that its Frobenius characteristic is $\nabla e_n$, where the operator $\nabla$ is just a special case of a whole family of so called delta operators $\Delta_f$, where $f$ is a symmetric function.

In \cite{Haglund-Remmel-Wilson-2015} a combinatorial formula for $\nabla e_n$ has been conjectured, in terms of labelled Dyck paths (i.e. parking functions). This formula has been known as the \emph{shuffle conjecture}, and it has been recently proved by Carlsson and Mellit in \cite{Carlsson-Mellit-ShuffleConj-2015}, where they actually proved a refinement of this formula, known as the \emph{compositional shuffle conjecture}, proposed in \cite{Haglund-Morse-Zabrocki-2012}.

A generalization of the shuffle conjecture, known as the \emph{Delta conjecture}, has been proposed in \cite{Haglund-Remmel-Wilson-2015}, in terms of decorated labelled Dyck paths. Some special cases and consequences of this conjecture have been proved: see \cite{Haglund-Remmel-Wilson-2015} for a survey of these partial results. We should mention here also the work in \cite{Romero-Deltaq1-2017}, where the conjecture at $q=1$ is proved, the work in \cite{Garsia-Haglund-Remmel-Yoo-2017}, where the conjecture at $q=0$ is proved, as they do not appear in \cite{Haglund-Remmel-Wilson-2015}. Finally, in \cite{DAdderio-Iraci-polyominoes-2017}, relying on results in the present paper, the ``$h_jh_{n-j}$'' case of the Delta conjecture is proved. To this date, the general problem remains open.

\medskip

In \cite{Zabrocki-4Catalan-2016}, Zabrocki established one of the consequences of the Delta conjecture, the so called \emph{$4$-variable Catalan conjecture}. In fact, this conjecture, stated in \cite{Haglund-Remmel-Wilson-2015}, makes other predictions that have not been explained in \cite{Zabrocki-4Catalan-2016}. One of them is the symmetry 
\begin{equation} \label{eq:intro_HRW_symmetry}
\langle\Delta_{h_{\ell}}\nabla e_{n-\ell},s_{k+1,1^{n-\ell-k-1}}\rangle=\langle\Delta_{h_{k}}\nabla e_{n-k},s_{\ell+1,1^{n-k-\ell-1}}\rangle\qquad \text{for }n>k+\ell.
\end{equation}

In \cite{Haglund-Remmel-Wilson-2015} and \cite{Wilson-PhD-2015}, the authors asked for a proof of the decorated version of the famous $q,t$-Schr\"{o}der theorem of Haglund \cite{Haglund-Schroeder-2004}. This is a combinatorial interpretation of the formula
\begin{equation} \label{eq:intro_qtSchroederLHS}
\langle \Delta_{e_{a+b-k-1}}'e_{a+b},e_a h_{b}\rangle
\end{equation}
as a sum of weights $q^{\mathsf{dinv}(D)}t^{\mathsf{area}(D)}$ of decorated Dyck paths, where dinv and area are certain functions from the space of decorated Dyck paths to the natural numbers. 

\medskip

In the present article we provide a solution of the last two problems. In particular we give both an algebraic and a combinatorial explanation of the symmetry in \eqref{eq:intro_HRW_symmetry}, and we prove three combinatorial interpretations of \eqref{eq:intro_qtSchroederLHS}.

In fact, we introduce two more statistics on decorated Dyck paths, $\mathsf{bounce}$ and $\mathsf{pbounce}$, which both extend Haglund's bounce statistic on Dyck paths. We prove that the zeta map of Haglund (also known as sweep map) switches the bistatistics $(\mathsf{dinv},\mathsf{area})$ and $(\mathsf{area},\mathsf{bounce})$ and the numbers of decorated rises and decorated peaks. Also, we provide a recursion for the $q,t$-enumerators of these two bistatistics, which matches a recursion in \cite{Wilson-PhD-2015}.

We also prove a recursion for the $q,t$-enumerator of the bistatistic $(\mathsf{area},\mathsf{pbounce})$, and we define a bijection $\psi$ on decorated Dyck paths that preserves the bistatistic $(\mathsf{area},\mathsf{pbounce})$ but switches the numbers of decorated rises and decorated peaks: this will eventually explain combinatorially the symmetry in \eqref{eq:intro_HRW_symmetry}.

This is an extended abstract of \cite{DAdderio-VandenWyngaerd-2017}: we refer to it for some definitions and full proofs.

\section{Basics of symmetric functions and Macdonald polynomials}

Given $f\in K[x_1,\dots,x_n]$ for some field $K$ and $\sigma\in \mathfrak S_n$, we set \[\sigma\cdot f(x_1,\dots, x_n)=f(x_{\sigma(1)}, \dots, x_{\sigma(n)}). \] We say that $f$ is a \emph{symmetric function} if for all $\sigma \in \mathfrak S_n $ we have $\sigma\cdot f=f$. 

Let $\Lambda$ be the ring of symmetric functions in $x_1,x_2,\dots$. As we are working with Macdonald symmetric functions involving two parameters $q$ and $t$, we will consider this polynomial ring over the field $\mathbb{Q}(q,t)$. This ring has a grading $\Lambda=\bigoplus_{n\geq 0}\Lambda^{(n)}$ where $\Lambda^n$ is the sub vector space of $\Lambda$ consisting of symmetric functions of degree $n$. The standard basis of symmetric functions, indexed by partitions are 
\begin{center}
\begin{tabular}{lcc}
	the complete homogeneous &$h_n=\sum_{i_1\leq...\leq i_n}x_{i_1}...x_{i_n}$ &$h_\lambda=h_{\lambda_1}... h_{\lambda_n}$\\
	the elementary & $e_n=\sum_{i_1<...<i_n}x_{i_1}...x_{i_n}$ &$e_\lambda=e_{\lambda_1}... e_{\lambda_n}$ \\
	the power symmetric &$p_n=\sum_{i}x_{i}^n$ &$p_\lambda=p_{\lambda_1}... p_{\lambda_n}$\\ 
	The Schur &$s_\lambda = \sum_{T\in SSYT(\lambda)}x^T$.
\end{tabular}
\end{center}
In this last definition $SSYT(\lambda)$ are the semi-standard Young tableaux of shape $\lambda$ and for such a tableau $T$ we set $x^T=x_1^{t_1}x_2^{t_2}\dots $ if the number $i$ appears $t_i$ times in $T$. We will use implicitly the usual convention that $e_0=h_0=1$ and $e_k=h_k=0$ for $k<0$.

We will make extensive use of the \emph{plethystic notation} (see \cite{Loehr-Remmel-plethystic-2011} for a nice exposition). With this notation we will be able to add and subtract alphabets, which will be represented as sums of monomials $X = x_1 + x_2 + x_3+\cdots $. Then, given a symmetric function $f$, and thinking of it as an element of $\Lambda$, we denote by $f[X]$ the expression $f$ with $p_k$ replaced by $x_{1}^{k}+x_{2}^{k}+x_{3}^{k}+\cdots$, for all $k$. 

The \emph{Kostka coefficients} $K_{\lambda, \mu}$ the coefficients appearing in \[h_\mu=\sum_{\lambda}K_{\lambda,\mu}s_\lambda. \]


For a partition $\mu\vdash n$, we denote by
\begin{equation}
\widetilde{H}_{\mu}:=\widetilde{H}_{\mu}[X]=\widetilde{H}_{\mu}[X;q,t]=\sum_{\lambda\vdash n}\widetilde{K}_{\lambda \mu}(q,t)s_{\lambda}
\end{equation}
the \emph{(modified) Macdonald polynomials}, where 
\begin{equation}
\widetilde{K}_{\lambda \mu}:=\widetilde{K}_{\lambda \mu}(q,t)=K_{\lambda \mu}(q,1/t)t^{n(\mu)}\quad \text{ with }\quad n(\mu)=\sum_{i\geq 1}\mu_i(i-1)
\end{equation}
are the \emph{(modified) Kostka coefficients} (see \cite[Chapter~2]{Haglund-Book-2008} for more details). 

The set $\{\widetilde{H}_{\mu}[X;q,t]\}_{\mu}$ is a basis of the ring of
symmetric functions $\Lambda$ with coefficients in $\mathbb{Q}(q,t)$.
This is a modification of the basis introduced by Macdonald
\cite{Macdonald-Book-1995}, and they are the Frobenius characteristic of
the so called Garsia--Haiman modules (see \cite{Garsia-Haiman-2003}).

If we identify the partition $\mu$ with its Ferrers diagram, i.e. with the collection of cells $\{(i,j)\mid 1\leq i\leq \mu_i, 1\leq j\leq \ell(\mu)\}$, then for each cell $c\in \mu$ we refer to the \emph{arm}, \emph{leg}, \emph{co-arm} and \emph{co-leg} (denoted respectively as $a_\mu(c), l_\mu(c), a_\mu(c)', l_\mu(c)'$) as the number of cells in $\mu$ that are strictly to the right, above, to the left and below $c$ in $\mu$, respectively. We define for every partition $\mu$
\begin{align}
B_{\mu} & :=B_{\mu}(q,t)=\sum_{c\in \mu}q^{a_{\mu}'(c)}t^{l_{\mu}'(c)}\\
w_{\mu} & :=w_{\mu}(q,t)=\prod_{c\in \mu}(q^{a_{\mu}(c)}-t^{l_{\mu}(c)+1})(t^{l_{\mu}(c)}-q^{a_{\mu}(c)+1}).
\end{align}


\section{Combinatorics of decorated Dyck paths}

\subsection{Basic definitions}

We will use the notation $\D(n)$ to denote the set of Dyck paths of size $n$, i.e. the set of lattice paths starting from $(0,0)$ and ending at $(n,n)$, using only north and east steps and staying weakly above the diagonal $x=y$, also called the \emph{main diagonal}. Each Dyck path $D\in \D(n)$ can be described uniquely by a sequence $w_1(D)w_2(D)\cdots w_n(D)$ of $n$ integers called the \emph{area word} of $D$, where $w_i(D)$ is the number of whole squares on the $i$-th row between $D$ and the main diagonal. A sequence $w_1w_2\cdots w_n$ of $n$ nonnegative integers is an area word if and only if $w_1=0$ and $w_{i+1}\leq w_i+1$ for $i=1,...,n-1$.  

The \emph{rises} of a Dyck path $D$ are the indices \[ \Rise(D) := \{2\leq i \leq n\mid a_{i}(D)>a_{i-1}(D)\},\] or the vertical steps that are directly preceded by another vertical step.

The \emph{peaks} of a Dyck path $D$ are the indices \[ \Peak(D) := \{1\leq i\leq n \mid a_{i+1}(D) \leq a_i(D)\}, \] or the vertical steps that are followed by a horizontal step.

A \emph{decorated Dyck path} is a Dyck path where certain peaks are decorated with a symbol $\circ$ (near the point joining the peak to the horizontal step following it) and certain rises are decorated with a symbol $\ast$. By $\DD(n)^{\circ a, \ast b}$ we denote the set of Dyck paths of size $n$ with $a$ decorated peaks and $b$ decorated rises.
 By $\DD(n)^{\circ a, \ast b}$ we denote the set of Dyck paths of size $n$ with $a$ decorated peaks and $b$ decorated rises. See Figure~\ref{fig:example_deco_Dyck} for an example.

\begin{figure}[h!]
\centering
	\begin{tikzpicture}[scale=.6]
	\draw[gray](0,0)grid(5,5);
	\draw[gray](0,0)--(5,5);
	\draw[blue, opacity=.4,ultra thick](0,0)--(0,2)--(1,2)--(1,3)--(3,3)|-(5,5);
	\draw (2.5,4.5) node {$\ast$}
		  (3,5) node[anchor=south east] {$\circ$}
		  (0,2) node[anchor=south east] {$\circ$};
	\end{tikzpicture}
	\caption{Example of an element in $\DD^{\circ 2,\ast 1}(5)$}
	\label{fig:example_deco_Dyck}
\end{figure}

If a \emph{fall} is a horizontal step that is followed by a second horizontal step, then there is a natural way of mapping rises into falls bijectively. Indeed the endpoint of a rise is a point where $D$ vertically crosses a certain diagonal parallel to the main diagonal; since the path must end at the main diagonal it must cross the same diagonal horizontally with a fall at least once: choose the first occurrence and this yields a bijective map (see Figure~\ref{rises vs falls}). Therefore we can decorate rises or falls equivalently. 
\begin{figure}[h]
	\centering
	\begin{tikzpicture}[rotate=0, scale=0.5]
	\draw[gray] (0,0) grid (11,11)(0,0) to (11,11)(-1,1) to (10,12);
	\draw[blue, ultra thick, opacity=0.4] (0,0) to (0,2) to (1,2) to (1,3) to (2,3) to (2,6) to (4,6) to (4,9) to (5,9) to (5,10) to (7,10) |-(11,11);
	\draw[ultra thick] (2,4) to (2,5) (8,11) to (9,11) ;
	\draw[];
	\fill[pattern=north west lines, pattern color=gray] (2,4) rectangle (4,5) (8,11) rectangle (9,9);
	\draw  (1.5,4.5) node {$\ast$};
	\end{tikzpicture} 
	\caption{Correspondence between rises and falls}\label{rises vs falls}
\end{figure}


\subsection{The statistics $\mathsf{area}$, $\mathsf{dinv}$, $\mathsf{bounce}$ and $\pbounce$.} 

We introduce here four statistics on the set $\DD(n)^{\circ a, \ast b}$, all of which are generalisations of the usual statistics on undecorated Dyck paths. The $\area$ and $\dinv$ statistics are the ``unlabelled'' versions of the statistics defined on labelled decorated Dyck paths in \cite{Haglund-Remmel-Wilson-2015} (cf. also \cite{Zabrocki-4Catalan-2016}). The $\mathsf{bounce}$ and $\pbounce$ are new.

\medskip

Take $D\in \DD(n)^{\circ a, \ast b}$, and let $w_1(D)w_2(D)\cdots w_n(D)$ be its area word. 

Let $\mathsf{DRise}(D)\subseteq \Rise(D) $ be the set of indices such that $i\in \mathsf{DRise}(D)$ if the $i$-th vertical step of $D$ is a decorated rise. We define the \emph{area} of $D$ as
\begin{equation} 
\mathsf{area}(D)= \sum_{i \in \{1,\dots,n\}\setminus \DRise(D)} w_i(D).
\end{equation}
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 the vertical steps following a decorated rise.

Equivalently, we could define the area of $D$ as the number of whole squares that lie between the path and the main diagonal, except the ones in the columns containing falls corresponding to decorated rises. For example, the path in Figure~\ref{statdef} has area equal to $6$ (grey in the picture). 

\begin{figure}
\centering
\begin{tikzpicture}[scale=.7]
			\draw (8.5,.5) node {0} (8.5,1.5) node {1} (8.5,2.5) node {2} (8.5,3.5) node {2} (8.5,4.5) node {3} (8.5,5.5) node {1} (8.5,6.5) node {0} (8.5,7.5) node {1};
			\draw (-.5,.5) node {0} (-.5,1.5) node {0} (-.5,2.5) node {0} (-.5,3.5) node {1} (-.5,4.5) node {1} (-.5,5.5) node {2} (-.5,6.5) node {3} (-.5,7.5) node {3};
			\draw[gray](0,0) grid (8,8) (0,0)--(8,8);
			\draw[blue, ultra thick, opacity=0.4](0,0)--(0,3)--(1,3)--(1,5)--(4,5)|-(6,6)|-(8,8);
			\draw 
			(4,6) node[anchor=south east] {$\circ$} 
			(1.5,5.5) node {$\ast$}
			(6.5,8.5) node {$\ast$}
			(6,8) node[anchor=south east] {$\circ$} ;
			\fill[pattern=north west lines, pattern color=gray];
			\draw[ultra thick, dashed, opacity=.7](0,0)|-(3,3)|-(5,5)|-(6,6)|-(8,8);
			\fill[gray, opacity=.4](0,1) rectangle (1,3) (2,3) rectangle (3,5)rectangle (4,4) (4,5) rectangle(5,6);
\end{tikzpicture}
\caption{$D\in \DD^{\circ 2,\ast 2}(8)$, its bounce word (left) and its area word (right).}\label{statdef}
\end{figure}


Similarly let $\mathsf{DPeak}(D)\subseteq \Peak(D)$ be the set of indices such that $i\in \mathsf{DPeak}(D)$ if the $i$-th vertical step of $D$ is a decorated peak. We define the \emph{dinv} of $D$, denoted $\mathsf{dinv}(D)$, to be the number of pairs $(i,j)\in \{1,...,n\}\times \{1,...,n\}$ with $i<j$ such that either \begin{enumerate}
	\item $w_i(D)=w_j(D)$ and $i\not \in \mathsf{DPeak}(D)$
\end{enumerate}
or, \begin{enumerate}[resume]
	\item  $w_i(D)=w_j(D)+1$ and $j\not \in \mathsf{DPeak}(D)$.
\end{enumerate} We refer to the first and second kind of pairs as \emph{primary} and \emph{secondary dinv}, respectively. For example, the path in Figure \ref{statdef} has dinv equal to 5: 4 primary and 1 secondary. 


The definitions of our two bounce statistics both start from the usual definition of the bounce of undecorated Dyck paths, but they are modified (in two different ways) by the decorations on the peaks. 

\medskip

 The \emph{bounce path} of a decorated Dyck path $D\in \DD(n)^{\circ a,
 \ast b}$ starts in $(0,0)$ and travels north until it encounters the
 beginning of an east step of $D$, then it turns east until it hits the
 main diagonal, then it turns north again, and so on. Thus it continues
 until it arrives at $(n,n)$. We label the vertical steps of the bounce
 path as follows: all the steps before the first ``bounce'', i.e. the
 first time that the bounce path changes direction from north to east,
 are labelled with a $0$; next the vertical steps are labelled with a
 $1$, again until the path ``bounces''; then with a $2$, and so on. The labels of the bounce path from bottom to top create a sequence of integers $b_1(D)b_2(D)\cdots b_n(D)$ which we call the \emph{bounce word}. We define the \emph{bounce} of $D$ as 
\begin{equation}
\mathsf{bounce}(D):=\sum_{i\in \{1,...,n\}\setminus \mathsf{DPeak(D)}}b_i(D) .
\end{equation}
The path in Figure~\ref{statdef} has bounce equal to 5. 


We now describe how to compute our \emph{pbounce} statistic of a decorated Dyck path $D\in \DD(n)^{\circ a, \ast b}$. Delete all the horizontal steps that directly follow decorated peaks. When deleting the horizontal steps, move the rest of the path one square to the left, together with part of the main diagonal so that the number of squares between the path and the main diagonal on each row remains the same.  In this way we end up with a path from $(0,0)$ to $(n-a,n)$ and some line segments that form the new \emph{main diagonal}. This is what is called the corresponding \emph{leaning stack} in \cite{Haglund-Remmel-Wilson-2015}. See Figure~\ref{fig:ex_second_bounce} for an example. 

Now we draw the \emph{pbounce path} in this new object as described before: start from $(0,0)$, bounce at the beginning of horizontal steps of the path and against the main diagonal. Then label the vertical steps as before to get a \emph{pbounce word} $b'_1(D)b_2'(D)\cdots b_n'(D)$. The \emph{pbounce} of $D$ is then defined as
\begin{equation}
\pbounce(D):=\sum_{i=1}^nb'_i(D).
\end{equation}
In the example $D\in \mathcal{D}_{11}^{(2,0)}$ of Figure~\ref{fig:ex_second_bounce} its pbounce word is $00111111122$, so that its pbounce is $\pbounce(D)=11$.

\begin{figure}[h]
	\centering
	\begin{minipage}{.55\textwidth}
		\centering
		\begin{tikzpicture}[scale=0.7]
		\draw[gray] (0,0) grid (11,11);
		\draw[blue, ultra thick, opacity=0.4] (0,0) to (0,2) to (1,2) to (1,3) to (2,3) to (2,6) to (3,6) to (3,9) to (5,9) to (5,10) to (7,10) |-(11,11);
		\draw[thick,densely dotted] (0,0) to (11,11);
		\draw (2, 6) node[anchor=south east] {$\circ$} (5,10)node[anchor=south east] {$\circ$};
		\end{tikzpicture} 
		
	\end{minipage}%
	\begin{minipage}{.45\textwidth}
		\centering
		\begin{tikzpicture}[scale=0.6]
		\draw[gray] (0,0) grid (9,11);
		\draw[blue, ultra thick, opacity=0.4] (0,0) to (0,2) to (1,2) to (1,3) to (2,3) to (2,6) to (2,9) to (4,9) to (4,10) to (5,10) |-(9,11);
		\draw (2, 6) node[anchor=south east] {$\circ$} (4,10) node[anchor=south east] {$\circ$};
		\draw[ultra thick, dashed,opacity=0.7](0,0)|-(2,2)|-(8,9)|-(9,11);
		\draw[thick,densely dotted] (0,0) to (6,6) (5,6) to (9,10) (8,10) to (9,11);
		\draw (-.5,.5) node {0} (-.5,1.5) node {0}(-.5,2.5) node {1}(-.5,3.5) node {1}(-.5,4.5) node {1}(-.5,5.5) node {1}(-.5,6.5) node {1}(-.5,7.5) node {1}(-.5,8.5) node {1}(-.5,9.5) node {2}(-.5,10.5) node {2} ;
		\end{tikzpicture} 
	\end{minipage}
	\caption{Construction of the pbounce path}
	\label{fig:ex_second_bounce}
\end{figure}



Remark that a peak is a vertical step followed by a horizontal step, one could think of these two steps together as a diagonal step. So $\DD(n)^{\circ a, \ast b}$ can be interpreted as the set of \emph{decorated Schr\"oder paths} with $a$ diagonal steps and $b$ decorated rises. The definitions of $bounce$ and $dinv$ given in \cite[Chapter~4]{Haglund-Book-2008} coincide with our definitions of $\pbounce$  and $\mathsf{dinv}$. When $b=0$ the definition of the $area$ in \cite[Chapter~4]{Haglund-Book-2008} and our $\mathsf{area}$ also coincide.


Let us conclude this section by introducing and motivating some notation. For each of the three statistics that depend on the decorations on the peaks, i.e. $\mathsf{dinv}, \mathsf{bounce}$ and $\pbounce$, there is one specific peak that, when decorated, does not alter this statistic. Often, when dealing with one of these statistics, it will be useful to consider not the whole set of decorated Dyck paths, but rather the set of decorated Dyck paths where this specific peak is not decorated. We will use the following notations:
\begin{center}
	\begin{itemize}
		\item $ \DDd(n)^{\circ a, \ast b} \subseteq \DD(n)^{\circ a, \ast b}$  is the subset where the \emph{rightmost highest peak}, i.e. the peak that lies the most to the right on the diagonal that is the farthest removed from the main diagonal, is never decorated, denoted as such because decorating this peak does not alter the $\dinv$ statistic;
		\item $\DDb(n)^{\circ a, \ast b} \subseteq \DD(n)^{\circ a, \ast b}$ is the subset where the \emph{first peak}, i.e. the peak in the leftmost column, is never decorated, denoted as such because decorating this peak does not alter the $\bounce$ statistic; 
		\item $ \DDp(n)^{\circ a, \ast b} \subseteq \DD(n)^{\circ a, \ast b}$ is the subset where the \emph{last peak}, i.e. the peak in the highest row is never decorated, denoted as such because decorating this peak does not alter the $\pbounce$ statistic. 
	\end{itemize}
\end{center}

\subsection{Haglund's $\zeta$ (sweep) map}

\begin{thm}
		There exists a bijective map $$\zeta: \DDd(n)^{\circ a, \ast b} \rightarrow \DDb(n)^{\circ b, \ast a} $$ such that for all $D\in \DDd(n)^{\circ a, \ast b}$ \begin{align*}
		\mathsf{area}(D)&=\mathsf{bounce}(\zeta(D))\\ \mathsf{dinv}(D)&=\mathsf{area}(\zeta(D)).
	\end{align*}
\end{thm}
The map is essentially a decorated version of Haglund's $\zeta$ map on undecorated Dyck paths (see \cite{Egge-Haglund-Killpatrick-Kremer-2003} or \cite[Theorem~3.15]{Haglund-Book-2008}).


\subsection{The $\psi$ map exchanging peaks and falls} 

\begin{thm}\label{thm:psimap}
	There exists a bijective map $$\psi: \DDp(n)^{\circ a, \ast b} \rightarrow  \DDp(n)^{\circ b, \ast a}$$ such that for all $D\in \DDp(n)^{\circ a, \ast b}$ \begin{align*}
		\mathsf{area}(D)&=\mathsf{area} (\psi(D)) \\ \pbounce(D)&=\pbounce(\psi(D)).
	\end{align*}
\end{thm}

Here we merely describe $\psi$. A detailed proof of its properties can be found in \cite{DAdderio-VandenWyngaerd-2017}. 

Let us first look at a simpler map, $\psi_0$, that transforms one decorated fall into a decorated peak. Call the endpoint of the decorated fall $x$. We travel southward from $x$ until we hit the main diagonal, and then travel west until we hit the endpoint of a north step of the path. Call this point $y$. We delete the decorated east step and add an east step right after the point $y$. Since $y$ was the endpoint of a north step and we added an east step after $y$, we have created a peak, which we now decorate. See Figure~\ref{fig:psizero} for an example.

\begin{figure}[h!]
	\centering
	\begin{minipage}{.5 \textwidth}
		\centering
		\begin{tikzpicture}[scale=.7]
		\filldraw (2,4) circle (1.5pt) (0,2) circle (1.5pt);
		\draw (2,4)node[anchor=south west] {$x$} (0,2) node[anchor= north west] {$y$};
		\draw[gray] grid (0,0) grid (5,5) (0,0)--(5,5);
		\draw[blue, opacity=.4,ultra thick](0,0)--(0,3)-|(1,4)--(3,4)|-(5,5);
		\draw(1.5,4.5) node {$\ast$};
		\fill[pattern=north west lines, pattern color=gray](1,2) rectangle (2,4);
		\draw[thick, dashed] (2,4)--(2,2)--(0,2);
		\end{tikzpicture}
	\end{minipage}%
	\begin{minipage}{.5\textwidth}
		\centering
		\begin{tikzpicture}[scale=.7]
		\draw (2,4)node[anchor=south west] {$x$} (0,2) node[anchor= north west] {$y$};
		\draw[gray] grid (0,0) grid (5,5) (0,0)--(5,5);
		\filldraw (2,4) circle (1.5pt) (0,2) circle (1.5pt);
		\draw[thick, dashed] (2,4)--(2,2)--(0,2);
		\draw[blue, opacity=.4,ultra thick](0,0)--(0,2)--(1,2)|-(2,3)|-(4,4)|-(5,5);
		\draw (0,2) node[anchor= south east]{$\circ$};
		\end{tikzpicture}
	\end{minipage}
	\caption{The map $\psi_0$}
	\label{fig:psizero}
\end{figure}
Clearly, $\psi_0$ is invertible. Note that $\psi_0^{-1}$ does not yield the desired result when the decoration is on the highest peak: the added horizontal step is the last of the path, so not a fall. This explains why $\psi$ is defined on $\psi$ on $\DDp(n)^{\circ a, \ast b} $. 

The general map $\psi$ relies on $\psi_0$ as follows: apply $\psi_0$ to all the decorated falls and $\psi_0^{-1}$ to all the decorated peaks, one by one. However, we must be careful, because the order in which we transform the decorations matters, see Figure~\ref{fig:psi_definition} for an example. We use the following order:
\begin{enumerate}
	\item transform \emph{all} the decorated falls, from left to right. We end up with a path all of whose decorations are on peaks. 
	\item Transform \emph{some} of the decorated peaks, starting from the top row, down to the to bottom row. Indeed, decorated peaks that came from decorated falls in step (1) must not be transformed again.
	\end{enumerate}
\begin{figure}[h!]
		\centering
		\begin{minipage}{.5 \textwidth}
			\centering
			\begin{tikzpicture}[scale=.55]
			\draw[gray](0,0) grid (8,8) (0,0)--(8,8);
			\draw[blue, ultra thick, opacity=0.4](0,0)--(0,4)--(2,4)--(2,5)--(4,5)|-(5,6)|-(6,7)|-(8,8);
			\draw (2.5,5.5) node {$\ast$}
			(4,6) node[anchor= south east]{$\circ$}
			(.5,4.5) node {$\ast$} 
			(5,7) node[anchor= south east]{$\circ$}
			(2,5) node[anchor= south east]{$\circ$} ;
			\fill[pattern=north west lines, pattern color=gray](0,1) rectangle (1,4) (2,3) rectangle (3,5);
			\end{tikzpicture}	
		\end{minipage}%
		\begin{minipage}{.5 \textwidth}
			\centering
			\begin{tikzpicture}[scale=.55]
			\draw[gray](0,0) grid (8,8) (0,0)--(8,8);
			\draw[blue, ultra thick, opacity=0.4](0,0)|-(1,1)--(1,3)--(2,3)|-(3,4)|-(4,5)|-(5,6)|-(6,7)|-(8,8);
			\draw (0,1) node[anchor= south east]{$  \circ$} (1,3) node[anchor= south east]{$  \circ$};
			\draw (3,5) node[anchor= south east]{$\circ$}
			(5,7) node[anchor= south east]{$\circ$}
			(4,6) node[anchor= south east]{$\circ$}  ;
			\fill[pattern=north west lines, pattern color=gray](0,1);
			\end{tikzpicture}		
		\end{minipage}
		
		\begin{minipage}{\textwidth}
			\centering
			\begin{tikzpicture}[scale=.55]
			\draw[gray](0,0) grid (8,8) (0,0)--(8,8);
			\draw[blue, ultra thick, opacity=0.4](0,0)|-(1,1)--(1,3)--(2,3)|-(3,4)|-(8,8);
			\draw [ultra thick] 
			(0,1) node[anchor= south east]{$ \circ$}
			 (1,3)node[anchor= south east]{$ \circ$}
			 (4.5,8.5) node {$ {\ast}$} 
			 (5.5,8.5) node {$ {\ast}$} 
			 (6.5,8.5) node {$ {\ast}$} ;
			\draw  ;
			\fill[pattern=north west lines, pattern color=gray] (5,5)|-(6,6)|-(7,7)|-(8,8)--(4,8)--(4,5)--(5,5);
			\end{tikzpicture}	
		\end{minipage}
		\caption{The two steps in the definition of the map $\psi$}
		\label{fig:psi_definition}
\end{figure}

\section{Decorated $q,t$-Schr\"{o}der}
We provide three combinatorial interpretations of the formula $
\langle \Delta_{e_{a+b-k-1}}'e_{a+b},e_a h_{b}\rangle$. 
\begin{thm} \label{thm:decoqtSchroeder}
For $a,b,k\in \mathbb{N}\cup\{0\}$, $a\geq 1$, $a+b\geq k+1$, we have
\begin{equation*}
\langle \Delta_{e_{a+b-k-1}}'e_{a+b},s_{b+1,1^{a-1}}\rangle= \DDd_{q,t}(a+b)^{\circ b, \ast k}
 = \DDp_{q,t}(a+b)^{\circ b, \ast k}
 = \DDb_{q,t}(a+b)^{\circ k, \ast b}.
\end{equation*}
so that, for $a+b\geq k+1$, we have
\begin{align*} 
\langle \Delta_{e_{a+b-k-1}}'e_{a+b},e_a h_{b}\rangle &= \sum_{D\in \DD(a+b)^{\circ b, \ast k} } q^{\dinv(D)}t^{\area(D)} 
\\  &=\sum_{D\in \DD(a+b)^{\circ b, \ast k} } q^{\area(D)}t^{\bounce(D)}
\\  &=\sum_{D\in \DD(a+b)^{\circ b, \ast k} } q^{\area(D)}t^{\pbounce(D)}.
\end{align*}
\end{thm}


The proof of Theorem~\ref{thm:decoqtSchroeder} relies on recursions that
can be shown to be satisfied by both the combinatorial objects and the
symmetric functions. These recursions are crucial also in the proof of
the main results in \cite{DAdderio-Iraci-polyominoes-2017}, and, together
with them, they provide a complete solution of Problem $8.1$ in \cite{Haglund-Remmel-Wilson-2015}.

For both the statements and the proofs of these recursions, we refer to \cite{DAdderio-VandenWyngaerd-2017}. Here we would like to mention only that the most technical part of these results is the following theorem, that generalizes a result of Haglund.
\begin{thm}
	For $m,k\geq 1$ and $\ell\geq 0$, we have
	\begin{equation} 
	\sum_{\gamma\vdash m}\frac{\widetilde{H}_\gamma[X]}{w_\gamma} h_k[(1-t)B_\gamma]e_\ell[B_\gamma]=\qquad \qquad \qquad \qquad  \qquad \qquad  \qquad \qquad 
	\end{equation}
	\begin{align*} 
	=\sum_{j=0}^{\ell} t^{\ell-j}\sum_{s=0}^{k}q^{\binom{s}{2}} \begin{bmatrix}
	s+j\\
	s
	\end{bmatrix}_q \begin{bmatrix}
	k+j-1\\
	s+j-1
	\end{bmatrix}_qh_{s+j}\left[\frac{X}{1-q}\right] h_{\ell-j}\left[\frac{X}{M}\right] e_{m-s-\ell}\left[\frac{X}{M}\right].
	\end{align*}
\end{thm}

\section{A symmetry result}

In \cite{Haglund-Remmel-Wilson-2015}, the following symmetry is conjectured.
\begin{equation} \label{eq:HRW_symmetry}
\langle\Delta_{h_{\ell}}\nabla e_{n-\ell},s_{k+1,1^{n-\ell-k-1}}\rangle=\langle\Delta_{h_{k}}\nabla e_{n-k},s_{\ell+1,1^{n-k-\ell-1}}\rangle\qquad \text{for }n>k+\ell.
\end{equation}
We provide both an algebraic and combinatorial proof of this. The combinatorial proof relies on the $\psi$-map of Theorem \ref{thm:psimap}, which in fact has the property that it preserves the bistatistic $(\mathsf{area},\mathsf{pbounce})$ (see \cite{DAdderio-VandenWyngaerd-2017} for proofs). The algebraic one uses the following theorem.

\begin{thm} \label{thm:schroeder_identity}
	For all $a,b,k\in \mathbb{N}$, with $a\geq 1$, $b\geq 1$ and $1\leq k\leq a$, we have
	\begin{equation} \label{eq:Schroeder_identity}
	\langle \Delta_{e_a}'\Delta_{e_{a+b-k-1}}'e_{a+b},h_{a+b}\rangle = \langle \Delta_{h_k}\Delta_{e_{a-k}}' e_{a+b-k},e_{a+b-k}\rangle.
	\end{equation}
\end{thm}


\begin{thm} \label{thm:symmetry}
	For $n>k+\ell$, the following expression is symmetric in $k$ and $\ell$:
	\begin{equation}
	\langle \Delta_{h_k}\Delta_{e_{n-k-\ell-1}}' e_{n-k},e_{n-k}\rangle.
	\end{equation}
\end{thm}
\begin{proof}[Algebraic proof of Theorem~\ref{thm:symmetry}]
	Using \eqref{eq:Schroeder_identity}, with $n=a+b$ and $\ell=b-1$, we have
	\begin{equation}
	\langle \Delta_{h_k}\Delta_{e_{n-k-\ell-1}}' e_{n-k},e_{n-k}\rangle=\langle \Delta_{e_{n-\ell-1}}'\Delta_{e_{n-k-1}}'e_{n},h_{n}\rangle ,
	\end{equation}
	which is obviously symmetric in $k$ and $\ell$.
\end{proof}

Notice that the expressions in \eqref{eq:HRW_symmetry} and in Theorem \ref{thm:symmetry} can easily be shown to be the same. 




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