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%\def\B{\mathbf{B}}}
\def\B{B}
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%\def\SS{\mathbf{S}}
\def\Ss{\SS_{m,n}^{(k)}}
\def\Bb{\B_{m,n}^{(k)}}
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%\parindent=0pt
\begin{document} 

\title[Schr\"oder Combinatorics]{\bleu{A note on: Rectangular Schr\"oder Parking Functions Combinatorics}}
\author[J.-C.~Aval]{Jean-Christophe Aval}
\address{LaBRI, CNRS - Universit\'e de Bordeaux,
351 cours de la Lib\'eration, 33405 Talence, France}
\author[F.~Bergeron]{Fran\c{c}ois Bergeron}
\address{D\'epartement de Math\'ematiques, UQAM,  C.P. 8888, Succ. Centre-Ville, 
 Montr\'eal,  H3C 3P8, Canada.}
%\date{\today. This work was supported by NSERC-Canada}
\thanks{This work was supported by NSERC-Canada}

\begin{abstract}
We study Schr\"oder paths drawn in an $(m,n)$ rectangle,
for any positive integers $m$ and $n$.
We get explicit enumeration formulas, closely linked to
those for the corresponding $(m,n)$-Dyck paths.
Moreover, we study a Schr\"oder version of $(m,n)$-parking functions,
and associated $(q,t)$-analogs.
\end{abstract} 
  \maketitle

% \parskip=0pt

%%%%%%%%%%%%%%%%%%%%%%%%%%   
{ \setcounter{tocdepth}{1}\parskip=0pt\footnotesize \tableofcontents}
%\parskip=8pt  

\thispagestyle{myheadings}
\font\rms=cmr8 
\font\its=cmti8 
\font\bfs=cmbx8

\markright{\its S\'eminaire Lotharingien de
Combinatoire \bfs 79 \rms (2018), Article~B79a\hfill}
\def\thepage{}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}

Many objects are enumerated by Schr\"oder numbers:
       $$1, 2, 6, 22, 90, 394, 1806, \dots\qquad  \hbox{(sequence A006318 in \cite{oeis})},$$ 
but the most classical are probably paths from $(0, 0)$ to $(n, n)$ with 
steps either $(1, 0)$, $(0, 1)$, or $(1, 1)$,  never going below the diagonal (although they may touch it).
These are generalizations of Dyck paths
(for which only the first two types of steps are allowed), 
well-known to be enumerated by the Catalan numbers.
The aim of this work is to investigate properties of an analogous notion 
for the $m\times n$-rectangle, 
rather than for the $n\times n$-square, the latter corresponding to the classical case. Our $(m,n)$-Schr\"oder paths
(defined in Section~\ref{sec:enum}) have the same kind of steps,
but we change the endpoint to $(m,n)$, and use the diagonal of the rectangle to crop the set of allowed paths. 
Partial results are already known, in the case where $m$ and $n$ are coprime,
in particular when $m=rn+1$ (see~\cite{song}).
We first obtain (see Corollary~\ref{prop:bizley_schroder}) a symmetric function enumeration formula for the general case (no coprimality condition).  This result, as well as the methods used to obtain it, closely parallels the case of Dyck paths
studied in~\cite{AB14}. Next, we consider the notion of Schr\"oder parking functions, i.e., labeled 
rectangular Schr\"oder paths, and investigate associated $q$-analogs and $(q,t)$-analogs
of our enumeration formulas; with the parameter $q$ accounting for the area
between the path and the diagonal. 

%First page headline in AmS-LaTeX for S\'eminaire Lotharingien de Combinatoire
%--restoring the headers and pagenumbering
\pagenumbering{arabic}
\addtocounter{page}{1}
\markboth{\SMALL J.-C. AVAL AND F. BERGERON}{\SMALL SCHR\"ODER COMBINATORICS}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Schr\"oder path enumeration}
\label{sec:enum}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection*{Schr\"oder paths} 
Our \defn{$(m,n)$-Schr\"oder  paths} are sequences 
   $$(0,0)=(x_0,y_0),(x_1,y_1),\ldots, (x_N,y_N)=(m,n)$$
with $(x_k,y_k)$ in $\N\times\N$, and such that 
\begin{itemize}\itemsep=4pt
   \item $(x_{k+1},y_{k+1})=(x_k,y_k)+(s,t)$, with $(s,t)\in \{(0,1),(1,1),(1,0)\}$,
   \item  such that $my_k-nx_k\geq 0$, for all $k$.
 \end{itemize}
Depending on $(s,t)$ being equal to $(0,1)$, $(1,1)$, or $(1,0)$ in the first condition, we say that we have an \defn{up}, \defn{diagonal}, or \defn{right} step. The inequality of the second condition ensures that the paths does not go below the diagonal. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection*{Schr\"oder polynomials}
Recall that the enumerating polynomials for $(n,n)$-Schr\"oder paths are given by the formula
   \begin{equation}\label{formule}
         \bleu{\SS_n(y)}:=\bleu{\sum _{k=0}^{n} \frac {1}{n-k+1}\binom{n}{k}\binom{2n-k}{n}\,y^k},
            \end{equation}
 with $\SS_0=1$. These ``Schr\"oder polynomials'' have a long and interesting history. The coefficient of $y^k$, denoted by $\SS_n^{(k)}$, enumerates Schr\"oder paths having $k$ 
%Christian: already defined
%``diagonal steps'' (relevant definitions are recalled below). 
diagonal steps.
Hence, the case $k=0$ corresponds to the usual notion of Dyck paths,   well-known to be counted by the Catalan numbers 
    $${\SS_n^{(0)} = \frac{1}{n+1}\binom{2n}{n}}.$$ 
Small values of these polynomials are
\begin{align*}
      {\SS_1(y)} &={1+y},\\
      {\SS_2(y)} &={2+ 3\, y+y^2},\\
      {\SS_3(y)} &={5+10\, y+6\, y^2 +{y}^{3} },\\
      {\SS_4(y)} &={14+35\,y+30\,{y}^{2}+10\,{y}^{3}+{y}^{4}},\\
       {\SS_5(y)} &={42+126\,y+140\,{y}^{2}+70\,{y}^{3}+15\,{y}^{4}+{y}^{5}}.
    \end{align*}
We consider here an $(m,n)$-rectangular analog of these polynomials, for any pair $(m,n)$ of positive integers. Just as for Dyck paths, the case when $m$ and $n$ are coprime is somewhat simpler. To make this more apparent notation-wise, we often write our pairs of integers in the form $(m,n)=(ac,bc)$, with $(a,b)$ coprime. Hence $c$ is the greatest common divisor of $m$ and $n$. In particular, the $(a,b)$-case corresponds to the coprime situation. We recall that the number of $(a,b)$-Dyck paths is simply given by the formula
    $${C_{a,b}=\frac{1}{a+b}\binom{a+b}{a}}.$$
The general case of $(m,n)$-Dyck path enumeration, $C_{m,n}$ giving the number of such paths, is probably best 
%Christian:
%coined
presented in generating series format as 
\begin{equation}\label{bizley}
\bleu{\sum_{d\geq 0}C_{ad,bd}z^d = 
          \exp\!\left(\sum_{j>1}\frac{1}{a}\binom{ja+jb}{ja}\,\frac{z^j}{j}\right)}.
  \end{equation}
The proof of this formula, going back to 1954, is due to Bizley~\cite{bizley}, who attributes it to Grossman~\cite{grossman}. 


\begin{figure}
\begin{center}
\begin{tikzpicture}[scale=.7]
\ligneaire{0}{1}{1};
\ligneaire{0}{2}{2};
\ligneaire{0}{3}{4};
\ligneaire{2}{4}{3};
\ligneaire{2}{5}{4};
\ligneaire{2}{6}{6};
\ligneaire{3}{7}{6};
\ligneaire{7}{8}{3};
\grille{12}{9};
\nord{0}{0};
\nord{0}{1};
\nord{0}{2};
\nordest{0}{3};
\est{1}{4};
\nord{2}{4};
\nord{2}{5};
\nordest{2}{6};
\nordest{3}{7};
\est{4}{8};
\est{5}{8};
\est{6}{8};
\nord{7}{8};
\est{7}{9};
\est{8}{9};
\est{9}{9};
\est{10}{9};
\est{11}{9};
\end{tikzpicture}
\begin{picture}(0,11)(11.5,.8)\setlength{\unitlength}{6.9mm}
\put(8.5,9){$3$}
\put(8.5,8){$6$}
\put(8.5,7){$6$}
\put(8.5,6){$4$}
\put(8.5,5){$3$}
\put(8.5,4){$4$}
\put(8.5,3){$2$}
\put(8.5,2){$1$}
\put(8.5,1){$0$}
\put(7,10){\underline{\bleu{Row areas}}}
\put(-5,9){$0$}
\put(-5,8){$1$}
\put(-5,7){$2$}
\put(-5,6){$3$}
\put(-5,5){$4$}
\put(-5,4){$5$}
\put(-5,3){$6$}
\put(-5,2){$7$}
\put(-5,1){$8$}
\put(-6,10){\underline{\bleu{Row labels}}}
\end{picture}
\end{center}
\vskip-10pt
\caption{The $(12,9)$-Schr\"oder path $000\overline{0}22\overline{2}\overline{3}7$.}\label{fig2}
\end{figure}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection*{Sequence encoding}
It will be practical to encode our paths as sequences $\alpha=a_0a_1 \cdots a_{n-1}$ of barred or unbarred integers,  with one $a_i$ for each up or diagonal step 
%Christian:
%on
of the path, reading them from top to bottom. For up steps we set $a_i=k$, whereas diagonal steps are encoded $a_i=\overline{k}$ (barred steps). In both cases, $k$ is equal to to the number of entire ``cells'' that lie to the left of the unique up (or diagonal) step at height $n-i$. The $a_i$ are the \defn{parts} of $\alpha$. 
In this encoding, $\alpha=a_0a_1 \cdots a_{n-1}$ corresponds to an $(m,n)$-Schr\"oder path if and only if
\begin{itemize}\itemsep=4pt
   \item[(1)] $\bleu{a_0\leq a_1\leq \ldots \leq a_{n-1}}$, (with the order $0<\overline{0}<\dots< k<\overline{k}$),
   \item[(2)]  if $\bleu{a_i=\overline{k}}$, then necessarily $\bleu{a_i<a_{i+1}}$, and
   \item[(3)] for each $i$, we have  $\bleu{a_i\leq \overline{\lfloor i\,m/n\rfloor}}$.
\end{itemize}
Each unbarred $k$, between $0$ and $m$, occurs with some \defn{multiplicity}\footnote{possibly equal to 0.} $n_k$ in a path $\alpha$. Removing $0$-multiplicities, we obtain the  \defn{(multiplicity) composition} $\gamma(\alpha)$ of the sequence $\alpha$, reading these multiplicities in increasing values of $k$. For example, 
   $$ \gamma(0\overline{0}11\overline{1}244\overline{4}) = (1,2,1,2).$$
Clearly $\gamma(\alpha)$ is a composition of $n-k$, 
%Christian: There is a notation clash:
%    Here, k is the number of diagonal steps (also earlier, in fact),
%    whereas just before it was used as a variable in the definition
%    of sequence alpha.
where $k$ stands for the number of diagonal steps in $\alpha$. The parts of $\gamma(\alpha)$ may be understood as the lengths of \defn{risers} (also called vertical runs) in the path. These are maximal sequences of consecutive up-steps.

Any $(m,n)$-Schr\"oder 
%Christian:
%paths
path may be obtained by either barring or not the rightmost part of a given size in the analogous word encoding of an $(m,n)$-Dyck path. As we will see this makes the enumeration of Schr\"oder
%Christian: paths
paths easy, once we 
%Christian:
%setup
set up the right tools.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection*{Symmetric function weight} 
As we will come to see  later, it is interesting to consider a weighted enumeration of Schr\"oder paths, with the weight lying in the 
%Christian:
%degree 
graded ring 
   $${\Lambda=\bigoplus_{d\geq 0} \Lambda_d}$$
of symmetric ``functions'' (polynomials in a 
%Christian:
%denumerable
countable set of variables $\mbf{x}=x_1,x_2,x_3,\ldots$). Recall that the degree $d$ homogeneous component $\Lambda_d$ affords as a linear basis the set 
    $$\{e_\mu(\mbf{x})\ |\ \mu\vdash d\}$$
  of elementary symmetric functions, with 
     $$e_\mu(\mbf{x}):=e_{\mu_1}(\mbf{x})e_{\mu_2}(\mbf{x})\cdots e_{\mu_\ell}(\mbf{x}),$$ 
  for $\mu=\mu_1\mu_2\ldots \mu_\ell$ running over the set of partitions of $d$. In turn, each factor $e_k(\mbf{x})$ is characterized by the generating function identity
  \begin{displaymath}
     {\sum_{k\geq 0} e_k(\mbf{x})z^k=\prod_{i\geq 1} (1+x_i\,z)}.
   \end{displaymath}
   with $e_0(\mbf{x}):=1$.
It easily follows that
 \begin{equation}\label{eq:eq5}
     {e_k(\mbf{x}+y) = e_k(\mbf{x})+e_{k-1}(\mbf{x})\,y},
   \end{equation}
 where $\mbf{x}+y$ means that we add a new variable $y$ to those occurring in $\mbf{x}$.
 
 With these notions at hand, we now simply set
   \begin{equation}\label{sym_enumer}
      {\SS_{m,n}(\mbf{x};y):=\sum_{\alpha} \alpha(\mbf{x})\,y^{\diag(\alpha)}},\quad{\rm with}\quad {\alpha(\mbf{x}):=\prod_{k\in\gamma(\alpha)}e_k(\mbf{x})},
      \end{equation}
%Christian:  and 
where the sum is over the set of $(m,n)$-Schr\"oder paths $\alpha$, with $\diag(\alpha)$ denoting the number of diagonal steps in $\alpha$. Likewise, we denote by
  \begin{displaymath}
      {\Ss(\mbf{x}):=\sum_{\diag(\alpha)=k} \alpha(\mbf{x})},
      \end{displaymath}
the symmetric function enumerator of $(m,n)$-Schr\"oder paths with exactly $k$ diagonal steps, so that $\SS_{m,n}(\mbf{x};y)=\sum_k \Ss(\mbf{x})\,y^k$.     
  For example, we have
 \begin{align*}
 {\SS_{{1,1}}(\mbf{x};y)}&={e_{{1}}(\mbf{x})+y},\\
{\SS_{{2,2}}(\mbf{x};y)}&={({e_{{11}}}(\mbf{x})+e_{{2}}(\mbf{x}))+3\,e_{{1}}(\mbf{x})\,y+{y}^{2}},\\
{\SS_{{3,3}}(\mbf{x};y)}&={({e_{{111}}}(\mbf{x})+3\,e_{{21}}(\mbf{x})+e_{{3}}(\mbf{x}))}\\
  &\qquad {+ ( 6\,{e_{{11}}}(\mbf{x})+4\,e_{{2}}(\mbf{x}) ) y+6\,e_{{1}}(\mbf{x}){y}^{2}+{y}^{3}}.
\end{align*}
Observe that, for all $r$ and $n$, we have
\begin{equation}\label{classical}
     {\SS_{rn+1,n}(\mbf{x};y)=\SS_{rn,n}(\mbf{x};y)},
 \end{equation}
 since the last step of $(rn+1,n)$ must necessarily be a right step, and the coprimality of $rn+1$ and $n$ 
 implies that staying 
%Christian:
%below
above the $(rn+1,n)$-diagonal insures as well that we stay 
%Christian:
%below
above the $(rn,n)$-diagonal. Hence we get the same set of paths.

To make some expressions more compact, we shall use ``plethystic notation'', recalling that we have
\begin{equation}\label{plet_en}
    {e_{n}[m\,\mbf{x}] :=\sum_{\nu\vdash n} (-1)^{n-\ell(\nu)}\,m^{\ell(\nu)} \frac{p_{\nu}(\mbf{x})}{z_\nu} },
\end{equation}
with $m$ considered as a ``constant'' in the calculation of plethysm. This means that 
%Christian: I don't understand this. What is p?
%   If it is any symmetric polynomial, then why does (6) seem to contradict
%   this?
%   I guess you mean power sums?
$p[m\mbf{x}]=m\,p[\mbf{x}]$.
Recall that it is usual, for any partition $\nu$ of $n$, to denote its length by $\ell(\nu)$, and to use the notation
    $$z_\nu:=1^{d_1}d_1!2^{d_2}d_2!\dots n^{d_n}d_n!,$$ 
where $d_i$ is the number of copies of the part $i$ in $\nu$. 

\subsection*{Main results}

Recall from~\cite{AB14} that we have  a Bizley-like formula  
 for the symmetric function enumeration 
  \begin{displaymath}
     {\mathcal{C}_{m,n}(\mbf{x}):=\sum_{\diag(\alpha)=0} \alpha(\mbf{x})}
  \end{displaymath}
of $(m,n)$-Dyck paths, namely
\begin{equation}\label{bizley2}
\bleu{\sum_{d\geq 0}\mathcal{C}_{ad,bd}(\mbf{x})\,z^d = 
          \exp\!\left(\sum_{j\geq 1} e_{jb}[ja\,\mbf{x}]   \,\frac{z^j}{aj}\right)},
  \end{equation}
 with $a$ and $j$ considered as constants in the plethysm.
We can exploit this formula to get one for our weighted enumeration of Schr\"oder paths. Indeed, 
\begin{proposition}\label{prop1}
    For all $m$ and $n$, we have
\begin{equation}\label{cat_schrod}
     \bleu{\SS_{m,n}(\mbf{x};y) = \mathcal{C}_{m,n}(\mbf{x}+y)}.
 \end{equation}
\end{proposition}

\begin{proof}[\bf Proof.] 
Let us fix the dimensions $m,n$ of the rectangular box.
Recall that, for any given $k\ge 0$, $\Ss$ stands for the set of $(m,n)$-Schr\"oder paths
with exactly $k$ diagonal steps. We denote by $\Cc$ the set $\SS_{m,n}^{(0)}$ of $(m,n)$-Dyck paths.

For $\beta\in\Cc$, let us denote by $\gamma(\beta)=(\gamma_1,\dots,\gamma_j)$  the composition giving the lengths of the vertical runs of $\beta$.
In view of \eqref{sym_enumer} and \eqref{eq:eq5}, we may write
\begin{align}
\Cc[\mbf{x}+y]&=\sum_{\beta\in\Cc} \prod_{\gamma_i\in\gamma(\beta)} e_{\gamma_i}[\mbf{x}+y]\nonumber\\
&=\sum_{\beta\in\Cc} \prod_{\gamma_i\in\gamma(\beta)} (e_{\gamma_i}[\mbf{x}]+y\,e_{\gamma_i-1}[\mbf{x}])\nonumber\\
&=\sum_{\beta\in\Cc} \sum_{k\ge 0} y^k \sum_\delta e_\delta[y],\label{eq:eqdelta}
\end{align}
where the last sum is over all possible compositions $\delta$ obtained by reducing exactly $k$
of the parts of $\gamma(\beta)$ by $1$.

Let us now consider the projection 
$$\pi\ :\ \Ss \longrightarrow \Cc$$ 
which replaces any diagonal step by a North step followed by an East step, i.e., an (outer) corner.
For a Dyck path $\beta$, let us denote by $\Ss(\beta):=\pi^{-1}(\beta)$ the inverse image 
of $\beta$ under $\pi$.
It is clear that the sets $\Ss(\beta)$, for $\beta$ running over $\Cc$, are the blocks of a partition of $\Ss$.
We may thus rewrite the inner sum in \eqref{eq:eqdelta} as
$$\sum_\delta e_\delta[y]=\sum_{\alpha\in\Ss(\beta)} e_{\gamma(\alpha)}[\mbf{x}],$$
and get
\begin{align*}\Cc[\mbf{x}+y]&=
\sum_{\beta\in\Cc} \sum_{k\ge 0} y^k \sum_{\alpha\in\Ss(\beta)} e_{\gamma(\alpha)}[\mbf{x}]\\
&= \sum_{k\ge 0} y^k \sum_{\alpha\in\Ss} e_{\gamma(\alpha)}[\mbf{x}]
= \SS_{m,n}(\mbf{x};y)
\end{align*}
as announced.
\end{proof}


From Equation \eqref{bizley2}, we may derive the following reformulation of Proposition~\ref{prop1}.


\begin{corollary}\label{prop:bizley_schroder}
The generating function of rectangular Schr\"oder polynomials is
\begin{equation}\label{eq:bizley-schroder}
\bleu{\sum_{d\geq 0}\SS_{ad,bd}(\mbf{x})\,z^d = 
          \exp\!\left(\sum_{j\geq 1} e_{jb}[ja\,(\mbf{x}+y)]   \,\frac{z^j}{aj}\right)}.
  \end{equation}
\end{corollary}



For example, for any $a$ and $b$ coprime, we get
\begin{align*}
  \bleu{\SS_{a,b}(\mbf{x}+y)}&=\bleu{\frac{1}{a} e_b[a\,(\mbf{x}+y)]},\\[6pt]
  \bleu{\SS_{2a,2b}(\mbf{x}+y)}&=\bleu{\frac{1}{2a} e_{2b}[2a\,(\mbf{x}+y)] }+ \bleu{\frac{1}{2a^2}\,e_{b}[a\,(\mbf{x}+y)]^2},\\[6pt]
  \bleu{\SS_{3a,3b}(\mbf{x}+y)}&=\bleu{\frac{1}{3a} e_{3b}[3a\,(\mbf{x}+y)] } +\bleu{ \frac{1}{2a^2}\,e_{b}[a\,(\mbf{x}+y)] \,e_{2b}[2a\,(\mbf{x}+y)]}\\
        &\qquad \qquad     +  \bleu{  \frac{1}{6a^3}\,e_{b}[a\,(\mbf{x}+y)]^3}.
\end{align*}


Recall the following well-known expansion\footnote{This is essentially an instance of the dual Cauchy formula.} of \pref{plet_en}  in terms of elementary symmetric functions:
  \begin{displaymath} %\label{eq:enmx}
  {e_{n}[m\,\mbf{x}]= \sum_{\nu\vdash n} \binom{m}{\ell(\nu)}\binom{\ell(\nu)}{d_\nu}e_\nu(\mbf{x})},
  \end{displaymath}
where $\binom{\ell(\nu)}{d_\nu}$ stands for the multinomial coefficient
     $$\binom{\ell(\nu)}{d_\nu}:=\binom{\ell(\nu)}{d_1,d_2,\ldots, d_n},$$
 with each $d_i$ equal to the multiplicity of the part $i$ in $\nu$.
Using this formula, we get
\begin{align}
\bleu{\SS_{a,b}(\mbf{x};y)} &=  \frac{1}{a} e_b[a\,(\mbf{x}+y)]\nonumber\\
&= \frac{1}{a} \sum_{k=0}^b e_{b-k}[a\,\mbf{x}]\,e_k[a\,y]\nonumber \\
&= \frac{1}{a} \sum_{k=0}^b \binom{a}{k}\,e_{b-k}[a\,\mbf{x}]\, y^k\nonumber\\
&= \bleu{\frac{1}{a} \sum_k y^k\,\sum_{\nu\vdash b-k} \binom{a}{k}\binom{a}{\ell(\nu)}\,\binom{\ell(\nu)}{d_\nu}e_\nu(\mbf{x})}.\label{cas_coprime}
\end{align}
Moreover, we may write the coefficient of $y^k$ in~\pref{cas_coprime} as the following integer coefficient linear combination of the $e_\nu(\mbf{x})$:  
      \begin{displaymath}%\label{formule_coprime}
     {\SS_{a,b}^{(k)}(\mbf{x})= \sum_{\nu\vdash b-k} \frac{1}{a}\binom{a}{k}\binom{a}{\ell(\nu)}\,\binom{\ell(\nu)}{d_\nu}e_\nu(\mbf{x})}.
   \end{displaymath}
Since $\scalar{e_\mu(\mbf{x})}{\sum_{j\geq 0}e_j(\mbf{x})}=1$ for all partition $\mu$, we immediately get
    $$\SS_{a,b}^{(k)}=\Scalar{\SS_{a,b}^{(k)}(\mbf{x})}{\sum_{j\geq 0}e_j(\mbf{x})},$$
        where $\scalar{-}{-}$ stands for the usual scalar product on symmetric function\footnote{For which $\scalar{p_\mu}{p_\nu}= z_\mu\,\delta_{\mu,\nu}$}.
Otherwise stated, for $a$ and $b$ coprime,
  \begin{displaymath}
     {\SS_{a,b}^{(k)}= \sum_{\nu\vdash b-k} \frac{1}{a}\binom{a}{k}\binom{a}{\ell(\nu)}\,\binom{\ell(\nu)}{d_\nu}}.
   \end{displaymath}
 In particular, in view of~\pref{classical}, this covers the classical case ($m=n$) as well as the generalized 
 version ($m=rn$) of \cite{song}.
 One also deduces from Proposition~\ref{prop1} the following generalization of a result of Haglund~\cite{haglund}.
\begin{proposition} For all $m$ and $n$, we have
    \begin{displaymath}
      \bleu{ \SS_{m,n}^{(k)} = \Scalar{\mathcal{C}_{m,n}(\mbf{x})}{e_{n-k}(\mbf{x})h_k(\mbf{x})} }.
     \end{displaymath}
\end{proposition}
\begin{proof}[\bf Proof.]
We start by recalling the symmetric function identity
\begin{equation}\label{translation}
    {f(\mbf{x}+y)=\sum_{k\geq 0} y^k\, h_k^\perp\,  f(\mbf{x})},
 \end{equation}
 where $h_k^\perp$ stands for the dual of the operator of multiplication by $h_k(\mbf{x})$ with respect to the symmetric function scalar product. 
Equation \eqref{translation} may be checked by computation for the basis element $f=e_\lambda$
for $\lambda$ a partition $\lambda=(\lambda_1,\dots,\lambda_t)$,
by using the classical fact that 
$$h_k^\perp(e_\lambda)=\sum_{|\rho|=k} \prod_{i=1}^t e_{\lambda_i-\rho_i},$$
where the sum is over sequences $\rho$ such that $0\le \rho_i\le 1$.
  
It follows directly from~\pref{cat_schrod} that
 \begin{align*}
     \sum_k   \SS_{m,n}^{(k)}\,y^k
          &=\Scalar{\mathcal{C}_{m,n}(\mbf{x}+y)}{\sum_{k\geq 0}e_k(\mbf{x})},\\
          &=\Scalar{\sum_{k\geq 0} y^k\, h_k^\perp\, \mathcal{C}_{m,n}(\mbf{x})}{\sum_{j\geq 0}e_j(\mbf{x})},\\
          &=\Scalar{\mathcal{C}_{m,n}(\mbf{x})}{\sum_{k\geq 0} y^k h_k(\mbf{x}) \sum_{j\geq 0}e_j(\mbf{x})},\\
          &=\sum_{k\geq 0} \Scalar{\mathcal{C}_{m,n}(\mbf{x})}{ h_k(\mbf{x}) e_{n-k}(\mbf{x})}\,y^k.
 \end{align*}
The last equality comes from the fact that $\mathcal{C}_{m,n}(\mbf{x})$ is homogeneous of degree $n$, hence all terms of the wrong degree vanish in the scalar product. Evidently we get the announced result by comparing 
%Christian:
%same degree powers of $y$
powers of $y$ of the same degree
in both sides of the identity obtained.
\end{proof}    

%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection*{Area enumerator}
 The $i^{\rm th}$ \defn{row area} of a path $\alpha$ in $\Schroder_{n}^{(r)}$ is the integer
    \begin{displaymath}
          {\area_i(\alpha):=\lfloor i\,m/n\rfloor- |a_i|},
      \end{displaymath}
  where  we set $|\overline{k}|:=k$.
 Summing over all indices $i$ between $1$ and $n$, we get the \defn{area} of $\alpha$:
    \begin{displaymath} 
              {\area(\alpha):=\sum_{i=0}^{n-1} \area_i(\alpha)}.
      \end{displaymath}   
This generalizes 
%Christian: later
%to $(m,n)$-Schr\"oder paths 
a notion of area on Schr\"oder paths introduced for the case $m=n$ in~\cite{bonin} (and further studied 
%Christian:
%by
in \cite{barcucci}) to $(m,n)$-Schr\"oder paths. Following the presentation of \cite{haglund}, this may also be understood as the number of 
%Christian: This seems to come from some earlier version where the
%    paths ran below the diagonal. Please check whether my corrections
%    are okay.
``upper'' triangles lying 
%Christian:
%above
below the path and 
%Christian:
%below
above the diagonal line (as illustrated in Figure~\ref{fig2}). These triangles are also called \defn{area triangles}.
 We have the area $q$-enumerator symmetric function 
   \begin{displaymath}
      { \SS_{m,n}(\mbf{x};y,q)}:={\sum_{\alpha} \alpha(\mbf{x})\,q^{\area(\alpha)}y^{\diag(\alpha)}}.
   \end{displaymath}
   Keeping up with our previous 
%Christian:
%notation
notational conventions, we also set
   \begin{equation*}
     { \SS_{m,n}^{(k)}(q)}:={\sum_{\diag(\alpha)=k}  q^{\area(\alpha)}},
%Christian: This does not look good.
%{\rm and}\\ 
\end{equation*}
and
\begin{equation*}
      {\mathcal{C}_{m,n}(\mbf{x};q)}:={\sum_{\diag(\alpha)=0} \alpha(\mbf{x})\,q^{\area(\alpha)}}.
   \end{equation*}

  \begin{proposition}\label{cor1} For all $m$ and $n$, we have
   \begin{equation}
      \bleu{\SS_{m,n}(\mbf{x};y,q) = \mathcal{C}_{m,n}(\mbf{x}+y;q)},
   \end{equation}
and
       \begin{equation}\label{haglundmn}
        \bleu{\SS_{m,n}^{(k)}(q) = \Scalar{\mathcal{C}_{m,n}(\mbf{x};q)}{e_{n-k}(\mbf{x})h_k(\mbf{x})}}.
     \end{equation}
  \end{proposition}
\begin{proof}[\bf Proof.] 
This comes easily from the fact that the projection $\pi$ in the proof of Proposition~\ref{prop1}
preserves the area. 
\end{proof} 
           
From a result of~\cite{haiman},
it follows that $\mathcal{C}_{rn+1,n}(\mbf{x};q)=\mathcal{C}_{rn,n}(\mbf{x};q)=\nabla^r(e_n)\big|_{t=1}$, where $\nabla$ is a Macdonald ``eigenoperator'' introduced in~\cite{bergerongarsia}. By this, we mean that its eigenfunctions are the (combinatorial) $q,t$-Macdonald polynomials. Thus, a special instance of~\pref{haglundmn} may be formulated as
 \begin{equation}\label{haglundconj}
     \bleu{\SS_{rn,n}^{(k)}(q)=\Scalar{\nabla^r(e_n)\big|_{t=1}}{ e_{n-k}(\mbf{x})h_k(\mbf{x})}}.
       \end{equation}
 In this way, we get back the case $t=1$ of  Proposition 1 in~\cite{haglund}.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Constant term formula} 
The following constant term formula adds an extra parameter to our story. We conjecture that it corresponds to a $(q,t)$-enumeration of $(m,n)$-Schr\"oder parking 
%Christian:
%function,
functions, with $t$ accounting for a  ``dinv''-statis\-tic, still to be defined. 
\begin{conj}\label{conj:negutS}
We have
\begin{align*}
   & \bleu{\SS_{m,n}(\mbf{x};y,q,t)}=\nonumber\\[6pt]
   &   \quad\bleu{\mathrm{CT}_{z_m,\ldots,z_0}\left(\frac{1}{\mbf{z}_{m,n}}\prod_{i=1}^{m}\frac{z_i(1+y\,z_i)}{z_i-q\,z_{i+1}}\Omega'(\mbf{x};z_i)\prod_{j=i+1}^m\frac{(z_i-z_j)(z_i-qt\,z_j)}{(z_i-qz_j)(z_i-tz_j)}\right)},
\end{align*}
where $\Omega'(\mbf{x};z):=\sum_{k\geq 0} e_k(\mbf{x})\,z^k$, and $\mbf{z}_{m,n}:=\prod_{i=0}^{n-1} z_{\lfloor im/n\rfloor}$. 
\end{conj}
We recall that some care must be used in evaluating multivariate constant term expressions. Indeed,  the order in which successive constant terms are taken does have an impact on the overall result. This is why, in the above formula, the indices appearing after  ``{CT}'' specify that this should be done starting with $z_m$, and then going down to $z_0$.
For example, we have
\begin{align*}
  {\SS_{2,2}(\mbf{x};y,q,t)}&=\rouge{(s_{{2}}+ (q+t)\,s_{{11}})}{+ ( q+t+1)\, s_{{1}}\,y+{y}^{2}},\\
  {\SS_{2,3}(\mbf{x};y,q,t)}&=\rouge{(s_{{21}}+ (q+t)\, s_{{111}})}{+ ( s_{{2}}+ ( q+t+1) s_{{11}} )\, y+s_{{1}}\,{y}^{2}},\\
  {\SS_{2,4}(\mbf{x};y,q,t)}&=\rouge{(s_{{22}}+ (q+t)\, s_{{211}}+ ( {q}^{2}+qt+{t}^{2})\, s_{{1111}})}\\
     &\qquad{+ (  ( q+t+1)\, s_{{21}}+ ( {q}^{2}+qt+{t}^{2}+q+t )\, s_{{111}} )\, y}\\
     &\qquad{+ ( s_{{2}}+ ( q+t )\, s_{{11}} )\, {y}^{2}}.
\end{align*}
We underline that Conjecture~\ref{conj:negutS} is simply the evaluation at $\mbf{x}+y$ of a similar formula conjectured in~\cite{negut} 
 in relation with $(m,n)$-parking functions. More precisely, it is conjectured in the mentioned paper, that
\begin{conj}[\sc Negut]\label{conj:negut}
We have
\begin{multline*}
    \bleu{\mathcal{C}_{m,n}(\mbf{x};q,t)}\\
=\bleu{ \mathrm{CT}_{z_m,\ldots,z_0}\left(\frac{1}{\mbf{z}_{m,n}}\prod_{i=1}^{m}\frac{z_i}{z_i-q\,z_{i+1}}\Omega'(\mbf{x};z_i)\prod_{j=i+1}^m\frac{(z_i-z_j)(z_i-qt\,z_j)}{(z_i-qz_j)(z_i-tz_j)}\right)}.
\end{multline*}
\end{conj}
From this identity it is clear that Conjecture~\ref{conj:negutS} follows, using the equality  $\Omega'(\mbf{x}+y;z_i)=(1+y\,z_i)\,\Omega'(\mbf{x};z_i)$.
One may readily show
 that the specialization at $t=1$ of the right-hand side of Conjecture~\ref{conj:negutS} does indeed give back our previous $\mathcal{C}_{m,n}(\mbf{x}+y;q)=\SS_{m,n}(\mbf{x};y,q)$, since the relevant constant term formula is shown to hold in~\cite{compositionalshuffle}.

This, together with the results and conjectures that appear in~\cite{compositionalshuffle}, opens up many new avenues of exploration. In particular, we may obtain explicit candidates for the $(q,t)$-enumeration of special families of $(m,n)$-Schr\"oder paths (say with return conditions to the diagonal), by the simple device of evaluating 
%Christian: later
%at $\mbf{x}+y$  
analogous symmetric function formulas for $(m,n)$-Dyck paths at $\mbf{x}+y$. Several questions regarding this are explored in~\cite{positivity}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Schr\"oder parking functions} 
An \defn{$(m,n)$-Schr\"oder parking function} is a bijective labeling of the up steps of an $(m,n)$-Schr\"oder path $\alpha$ by the elements of $\{1,2,\ldots,n-\diag(\alpha)\}$. One further imposes the condition that consecutive up steps of same $x$-coordinate have decreasing labels reading them from top to bottom. The path involved in this description is said to be the  \defn{shape} of the parking function. For  $\alpha$ an $(m,n)$-Schr\"oder path, we denote by $\P(\alpha)$ the set of parking functions having shape $\alpha$.  
When $\diag(\alpha)=0$, we get the ``usual'' notion of parking functions of shape $\alpha$ (an $(m,n)$-Dyck path).
The $(m,n)$-Schr\"oder parking functions may be understood as preference functions, with some of the parking places being closed to parking (these correspond to diagonal steps). 
For $f\in\P(\alpha)$, we denote 
%Christian: later
%by $\area_i(f)$ 
the row area for the $i^{\rm th}$ row of the shape of $f$ by $\area_i(f)$. 
Figure~\ref{fig3} gives an example of a parking function of shape $000\overline{0}1\overline{1}223$.


\begin{figure}
\begin{center}
\begin{tikzpicture}[scale=.7]
\foreach \x in {0,...,0}
\filldraw[color=yellow, fill=yellow](0+\x,1)-- (1+\x,1) -- (1+\x,1+1) -- cycle;
\foreach \x in {0,...,1}
\filldraw[color=yellow, fill=yellow](0+\x,2)-- (1+\x,2) -- (1+\x,2+1) -- cycle;
\foreach \x in {0,...,3}
\filldraw[color=yellow, fill=yellow](0+\x,3)-- (1+\x,3) -- (1+\x,3+1) -- cycle;
\foreach \x in {0,...,3}
\filldraw[color=yellow, fill=yellow](1+\x,4)-- (2+\x,4) -- (2+\x,4+1) -- cycle;
\foreach \x in {0,...,3}
\filldraw[color=yellow, fill=yellow](2+\x,5)-- (3+\x,5) -- (3+\x,5+1) -- cycle;
\foreach \x in {0,...,5}
\filldraw[color=yellow, fill=yellow](2+\x,6)-- (3+\x,6) -- (3+\x,6+1) -- cycle;
\foreach \x in {0,...,6}
\filldraw[color=yellow, fill=yellow](2+\x,7)-- (3+\x,7) -- (3+\x,7+1) -- cycle;
\foreach \x in {0,...,6}
\filldraw[color=yellow, fill=yellow](3+\x,8)-- (4+\x,8) -- (4+\x,8+1) -- cycle;
\foreach \x in {0,...,12}
\draw[blue] (\x,0) -- (\x,9);
\foreach \x in {0,...,9}
\draw[blue] (0,\x) -- (12,\x);
\draw[black, thick] (0,0) -- (12,9) ;
\draw[red,ultra thick] (0,0) -- (0,0+1);
\draw[red,ultra thick] (0,1) -- (0,1+1);
\draw[red,ultra thick] (0,2) -- (0,2+1);
\draw[red,ultra thick] (0,3) -- (0+1,3+1);
\draw[red,ultra thick] (1,4) -- (1,4+1);
\draw[red,ultra thick] (1,5) -- (1+1,5);
\draw[red,ultra thick] (2,5) -- (2,5+1);
\draw[red,ultra thick] (2,6) -- (2,6+1);
\draw[red,ultra thick] (2,7) -- (2+1,7+1);
\draw[red,ultra thick] (3,8) -- (3,8+1);
\draw[red,ultra thick] (3,9) -- (3+1,9);
\draw[red,ultra thick] (4,9) -- (4+1,9);
\draw[red,ultra thick] (5,9) -- (5+1,9);
\draw[red,ultra thick] (6,9) -- (6+1,9);
\draw[red,ultra thick] (7,9) -- (7+1,9);
\draw[red,ultra thick] (8,9) -- (8+1,9);
\draw[red,ultra thick] (9,9) -- (9+1,9);
\draw[red,ultra thick] (10,9) -- (10+1,9);
\draw[red,ultra thick] (11,9) -- (11+1,9);
\filldraw[color=blue, fill=white] (.5,.5) circle (.4); \node at (.5,.5) {$1$};
\filldraw[color=blue, fill=white] (.5,1.5) circle (.4); \node at (.5,1.5) {$2$};
\filldraw[color=blue, fill=white] (.5,2.5) circle (.4); \node at (.5,2.5) {$6$};
\filldraw[color=blue, fill=white] (1.5,4.5) circle (.4); \node at (1.5,4.5) {$7$};
\filldraw[color=blue, fill=white] (2.5,5.5) circle (.4); \node at (2.5,5.5) {$3$};
\filldraw[color=blue, fill=white] (2.5,6.5) circle (.4); \node at (2.5,6.5) {$5$};
\filldraw[color=blue, fill=white] (3.5,8.5) circle (.4); \node at (3.5,8.5) {$4$};
\end{tikzpicture}
\end{center}\caption{A Schr\"oder parking function.}
\label{fig3}
\end{figure}






As we did for paths, we consider the $(m,n)$-Schr\"oder parking function polynomial
    \begin{displaymath}
       {P_{m,n}(y,q)= \sum_k P_{m,n}^{(k)}\,y^k} :={\sum _{\alpha} |\P(\alpha)|  
               \,q^{\area(\alpha)}y^{\diag(\alpha)}}.
    \end{displaymath}
It is easy to derive 
%Christian: the following fact
the following fact from Corollary~\ref{cor1}.
% and formula~\pref{formule_coprime} 
%Christian: bad style
%that
 \begin{corollary} For all $m$ and $n$, we have
   $$\bleu{P_{m,n}(y,q)=\LargeScalar{\mathcal{C}_{m,n}(\mbf{x}+y;q)}{\frac{1}{1-p_1(\mbf{x})}}}.$$
Equivalently, for all $k$, we have
          \begin{displaymath}
        \bleu{P_{m,n}^{(k)}(q)=\Scalar{\mathcal{C}_{m,n}(\mbf{x};q)}{p_1(\mbf{x})^{n-k}h_k(\mbf{x})}},
      \end{displaymath}
 \end{corollary}
 It follows from this, and the observation preceding~\pref{haglundconj}, that
 \begin{displaymath}
     \bleu{P_{rn,n}^{(k)}(q)=\Scalar{\nabla^r(e_n)\big|_{t=1}}{p_1(\mbf{x})^{n-k}h_k(\mbf{x})}}.
     \end{displaymath}
Also, for $a$ and $b$ coprime, we have 
       $\bleu{P_{a,b}^{(k)} =\binom{a}{k} a^{b-k-1}}$,
       which in the case $k=0$ reduces to a formula obtained in \cite{ALW}
       for the number of parking functions in the special case of Dyck paths, and $a$ and $b$ coprime.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{thebibliography}{10}   

\bibitem{ALW}
\auteur{D.~Armstrong, N.~Loehr, and G.~Warrington},
\titreref{Rational Parking Functions and Catalan Numbers},
Ann. Combin. 20 (2015), 21--58.

\bibitem{AB14}
\auteur{J.-C. Aval, and F. Bergeron},
\titreref{Interlaced Rectangular Parking Functions},
\href{http://arxiv.org/abs/1503.03991}{arXiv:1503.03991}.

\bibitem{barcucci}
\auteur{E. Barcucci, A. Del Lungo, E. Pergola, and R. Pinzani},
\titreref{Some Combinatorial Interpretations of $q$-Analogs of Schr\"oder Numbers},
 Ann. Combin. 3 (1999), 171--190.
 
 \bibitem{positivity}
 \auteur{F.~Bergeron},
\titreref{Open Questions for Operators Related to Rectangular Catalan Combinatorics},
J. Combin. 8 (2017), 673--703. 
%\href{http://arxiv.org/abs/1603.04476}{arXiv:1603.04476}.

\bibitem{bergerongarsia}
\auteur{F. Bergeron and A. M. Garsia}, 
\titreref{Science Fiction and Macdonald Polynomials}, 
in: Algebraic Methods and $q$-Special Functions (L. Vinet, R. Floreanini, eds.), CRM Proceedings and Lecture Notes, American Mathematical Society, 1999.

\bibitem{compositionalshuffle}
\auteur{F.~Bergeron, E.~Leven,  A.~M. Garsia, and G.~Xin},
\titreref{Compositional \lowercase{$(km,kn)$}-Shuffle Conjectures},
Intern. Math. Research Notices,
Vol. 2016, 4229--4270.
%\href{http://arxiv.org/abs/1404.4616}{arXiv:1404.4616}, (2014).

\bibitem{bizley}
\auteur{T. L. Bizley},
\titreref{Derivation of a New Formula for the Number of Minimal Lattice Paths from $(0,0)$ to $(km,kn)$ Having Just $t$ Contacts with the Line and a Proof of Grossman's Formula for the Number of Paths which May Touch but Do Not Rise above this Line}, J. Inst. Actuar. 80 (1954), 55--62.


\bibitem{bonin}
\auteur{J. Bonin, L. Shapiro, and R. Simion},
\titreref{Some $q$-Analogues of the Schr\"oder Numbers Arising from Combinatorial Statistics on Lattice Paths},
 J. Statist. Plann. Inference 34 (1993), 35--55.

 \bibitem{haglund}
 \auteur{E. S. Egge, J. Haglund, K. Killpatrick, D. Kremer},
 \titreref{A Schr\"oder Generalization of Haglund's Statistic on Catalan Paths},
Electron. J. Combin. 10 (2003), Article~\#R16, 21~pp.

 \bibitem{grossman}
\auteur{H. D.~Grossman},
\titreref{Fun with Lattice Points: Paths in a Lattice Triangle},
Scripta Math. 16 (1950), 207--212.

\bibitem{haiman}
\auteur{M.~Haiman}, 
\titreref{Combinatorics, Symmetric functions and Hilbert Schemes},  
in: CDM 2002: Current Developments in Mathematics in Honor of  Wilfried Schmid \& George Lusztig, International Press Books, 2003, pp.~39--112.

\bibitem{negut}
\auteur{A.~Negut},
\titreref{The Shuffle Algebra Revisited},
Int. Math. Res. Notices (2014) (22): 6242--6275.
doi: 10.1093/imrn/rnt156
\href{http://arxiv.org/abs/1209.3349}{arXiv:1209.3349}, (2012).

\bibitem{oeis}
\auteur{N.J.A Sloane},
\titreref{The On-Line Encyclopedia of Integer Sequences}, 
{\tt http://oeis.org.}

\bibitem{song}
\auteur{C.~Song},
 \titreref{The Generalized Schr\"oder Theory},
Electron. J. Combin. 12 (2005), Article~\#R53,

\end{thebibliography}







\end{document}








