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\title{Conjectured Statistics for the Higher $q,t$-Catalan Sequences}

\author{Nicholas A. Loehr\thanks{Work supported by an NSF Graduate Research 
Fellowship and an NSF Postdoctoral Research Fellowship}\\
\small Department of Mathematics\\
\small University of Pennsylvania \\
\small Philadelphia, PA 19104 \\
\small \texttt{nloehr@math.upenn.edu}\\}

\date{\small
Submitted: Oct 22, 2002; Accepted: Jan 24, 2005; Published: Feb 14, 2005\\
\small Mathematics Subject Classifications: 05A10, 05E05, 05E10, 20C30, 11B65}

\begin{document}
\maketitle

\begin{abstract}
This article describes conjectured combinatorial interpretations
for the higher $q,t$-Catalan sequences introduced by Garsia and Haiman,
which arise in the theory of symmetric functions and Macdonald polynomials.
We define new combinatorial statistics generalizing those proposed by Haglund 
and Haiman for the original $q,t$-Catalan sequence. We prove explicit
summation formulas, bijections, and recursions involving the new statistics.
We show that specializations of the combinatorial sequences
obtained by setting $t=1$ or $q=1$ or $t=1/q$ agree with the corresponding
specializations of the Garsia-Haiman sequences.
A third statistic occurs naturally in the combinatorial setting, 
leading to the introduction of $q,t,r$-Catalan sequences. 
Similar combinatorial results are proved for these trivariate sequences.
\end{abstract}

\section{Introduction}
\label{sec:intro}
In \cite{origpaper}, Garsia and Haiman introduced a $q,t$-analogue of the 
Catalan numbers, which they called the $q,t$-Catalan sequence.
In the same paper, they introduced a whole family of ``higher'' $q,t$-Catalan
sequences, one for each positive integer $m$.
We begin by describing several equivalent characterizations of the
original $q,t$-Catalan sequence.
We then discuss analogous characterizations of the
higher $q,t$-Catalan sequences.

In the rest of the paper, we present some conjectured combinatorial
interpretations for the higher $q,t$-Catalan sequences.
We prove some combinatorial formulas, recursions, and bijections
and introduce a three-variable version of the Catalan sequences.
We also show that certain specializations of our combinatorial
sequences agree with the corresponding specializations of the
higher $q,t$-Catalan sequences.

\subsection{The Original $q,t$-Catalan Sequence}
\label{subsec:origcat}
To give Garsia and Haiman's original definition of the $q,t$-Catalan sequence,
we first need to review some standard terminology associated with integer
partitions.  A \emph{partition} is a sequence
$\lambda=(\lambda_1\geq\lambda_2\geq\cdots\geq\lambda_k)$
of weakly decreasing positive integers, called the \emph{parts} of
$\lambda$.  The integer $N=\lambda_1+\lambda_2+\cdots+\lambda_k$
is called the \emph{area of $\lambda$} and denoted $|\lambda|$.
In this case, $\lambda$ is said to be a \emph{partition of $N$},
and we write $\lambda\vdash N$.
The number of parts $k$ is called the \emph{length of $\lambda$}
and denoted $\ell(\lambda)$.  We often depict a partition $\lambda$
by its \emph{Ferrers diagram}. This diagram consists of $k$ left-justified
rows of boxes (called \emph{cells}), where the $i$'th row from the top
has exactly $\lambda_i$ boxes.  Figure \ref{ptndef} shows the Ferrers diagram
of $\lambda=(8,7,5,4,4,2,1,1)$, which is a partition of $32$ having eight parts.

\begin{figure}[ht]
\begin{center}
\epsfig{file=ptndef.eps}
\caption{Diagram of a partition.}
\label{ptndef}
\end{center}
\end{figure}

%Given a partition $\lambda$, the \emph{transpose} $\lambda'$ of
%$\lambda$ is the partition obtained by interchanging the rows
%and columns of $\lambda$.  For example, the transpose
%of the partition in Figure \ref{ptndef} is
%$\lambda'=(8,6,5,5,3,2,2,1)$.

Let $\lambda$ be a partition of $N$. Let $c$ be one of the $N$ cells
in the diagram of $\lambda$.  We make the following definitions.
\begin{enumerate}
\item The \emph{arm of $c$},
denoted $a(c)$, is the number of cells strictly right
of $c$ in the diagram of $\lambda$. 
\item The \emph{coarm of $c$},
denoted $a'(c)$, is the number of cells strictly left
of $c$ in the diagram of $\lambda$. 
\item The \emph{leg of $c$},
denoted $l(c)$, is the number of cells strictly below
$c$ in the diagram of $\lambda$. 
\item The \emph{coleg of $c$},
denoted $l'(c)$, is the number of cells strictly above
$c$ in the diagram of $\lambda$. 
\end{enumerate}
For example, the cell labelled $c$
in Figure \ref{ptndef} has $a(c)=4$, $a'(c)=2$, $l(c)=3$, and $l'(c)=1$.

We define the \emph{dominance partial ordering} on partitions of $N$
as follows. If $\lambda$ and $\mu$ are partitions of $N$, we write
$\lambda\geq\mu$ to mean that
\[ \lambda_1+\cdots+\lambda_i\geq \mu_1+\cdots+\mu_i
\mbox{ for all $i\geq 1$. } \]

Fix a positive integer $n$ and a partition $\mu$ of $n$.
Let $\mu'$ denote the transpose of $\mu$, obtained by interchanging
the rows and columns of $\mu$.  Define the following abbreviations:
\begin{eqnarray*}
 h_{\mu}(q,t) &=& \prod_{c\in\mu} (q^{a(c)}-t^{l(c)+1}) \\
 h'_{\mu}(q,t) &=& \prod_{c\in\mu} (t^{l(c)}-q^{a(c)+1}) \\
  n(\mu) &=& \sum_{c\in\mu} l(c) \\
  n(\mu') &=& \sum_{c\in\mu'} l(c)=\sum_{c\in\mu} a(c) \\
  B_{\mu}(q,t) &=& \sum_{c\in\mu} q^{a'(c)}t^{l'(c)} \\
 \Pi_{\mu}(q,t) &=& \prod_{c\in\mu,c\neq(0,0)} (1-q^{a'(c)}t^{l'(c)}) 
\end{eqnarray*}
In all but the last formula above, the sums and products range
over all cells in the diagram of $\mu$. In the product defining
$\Pi_{\mu}(q,t)$, the northwest corner cell of $\mu$ is omitted
from the product.  This is the cell $c$ with $a'(c)=l'(c)=0$;
if we did not omit this cell, then $\Pi_{\mu}(q,t)$ would be zero.

Finally, we define the \emph{original $q,t$-Catalan sequence}
to be the following sequence of rational functions in the variables
$q$ and $t$:
\begin{equation}\label{origcat}
 OC_n(q,t) = \sum_{\mu\vdash n} \frac{t^{2n(\mu)}q^{2n(\mu')}
(1-t)(1-q)\Pi_{\mu}(q,t) B_{\mu}(q,t)}{h_{\mu}(q,t) h'_{\mu}(q,t)}
\ \ \mbox{($n=1,2,3,\ldots$).}
\end{equation} 
It turns out that, for all $n$, $OC_n(q,t)$ is a polynomial in $q$
and $t$ with nonnegative integer coefficients.  But this fact
is very difficult to prove.  See Theorem 1 below.

\subsection{Symmetric Function Version of the $q,t$-Catalan Sequence}
\label{subsec:sfcat}
This section assumes familiarity with basic symmetric function theory,
including Macdonald polynomials. We begin by briefly recalling
the definition of the modified Macdonald polynomials and the nabla
operator.

Let $\Lambda$ denote the ring of symmetric functions in the variables
$x_1,\ldots,x_n,\ldots$ with coefficients in the field $K=\Q(q,t)$.
Let $\alpha$ denote the unique automorphism of the ring $\Lambda$ that
interchanges $q$ and $t$. Let $\phi$ denote the unique
$K$-algebra endomorphism of $\Lambda$ that sends
the power-sum symmetric function $p_k$ to $(1-q^k)p_k$.
Let $\geq$ denote the usual dominance partial ordering on partitions.
Then the \emph{modified Macdonald basis} is the
unique basis $\tilde{H}_{\mu}$ of $\Lambda$ (indexed by
partitions $\mu$) such that:
\begin{itemize}
\item[(1)] $\phi(\tilde{H}_{\mu})=
\sum_{\lambda\geq\mu} c_{\lambda,\mu} s_{\lambda}$
for certain scalars $c_{\lambda,\mu}\in K$.
\item[(2)] $\alpha(\tilde{H}_{\mu})=\tilde{H}_{\mu'}$.
\item[(3)] $\tilde{H}_{\mu}|_{s_{(n)}}=1$.
\end{itemize}
The \emph{nabla operator} is the unique linear operator
on $\Lambda$ defined on the basis $\tilde{H}_{\mu}$ by the formula
\[ \nabla(\tilde{H}_{\mu})=q^{n(\mu')}t^{n(\mu)}\tilde{H}_{\mu}. \]
(The nabla operator was introduced by F. Bergeron and A. Garsia 
in \cite{nablaref1}.  See also \cite{nablaref2} or \cite{nablaref3}
for more information about nabla).  

Now, we define the \emph{symmetric
function version of the $q,t$-Catalan sequence} by the formula
\begin{equation}\label{symmcat}
 SC_n(q,t) = \left.\nabla (e_n)\right|_{s_{1^n}}\ \ \mbox{($n=1,2,3,\ldots$)},
\end{equation}
where $e_n$ is an elementary symmetric function, $s_{1^n}$ is a 
Schur function, and the vertical bar indicates extraction
of a coefficient. In more detail,
to calculate $SC_n(q,t)$, start with the elementary symmetric function
$e_n$ (regarded as an element of the $K$-vector space 
$\Lambda$), and perform the following steps:
\begin{enumerate}
\item Find the unique expansion of the vector $e_n$ as a linear combination
of the modified Macdonald basis elements $\tilde{H}_{\mu}$.
% (see \cite{origpaper} or \cite{modmac2} for definitions of this basis). 
The scalars appearing in this expansion are elements of $K=\Q(q,t)$.
\item Apply the nabla operator to this expansion by multiplying the coefficient
of $\tilde{H}_{\mu}$ by $q^{n(\mu')} t^{n(\mu)}$, for every $\mu$.
\item Express the resulting vector as a linear combination of
the Schur function basis $s_{\mu}$.
\item Extract the coefficient of $s_{1^n}$ in this new expansion.
This coefficient (an element of $\Q(q,t)$) is $SC_n(q,t)$.
\end{enumerate}

\subsection{The Representation-Theoretical $q,t$-Catalan Sequence}
\label{subsec:repcat}
This section assumes familiarity with representation theory
of the symmetric groups. Let $R_n=\C[x_1,\ldots,x_n,y_1,\ldots,y_n]$
be a polynomial ring over $\C$ in two independent sets of $n$ variables.
Let the symmetric group $S_n$ act on the variables by
\[ \sigma(x_i)=x_{\sigma(i)} \mbox{ and }
   \sigma(y_i)=y_{\sigma(i)} \mbox{ for $\sigma\in S_n$}. \]
Extending this action by linearity and multiplicativity, 
we obtain an action of $S_n$ on $R_n$ which is called
the \emph{diagonal action}. This action turns the vector space $R_n$
into an $S_n$-module. We define a submodule $DH_n$ of $R_n$, called
the space of \emph{diagonal harmonics}, as follows.  
A polynomial $f\in R_n$ belongs to $DH_n$ iff $f$ simultaneously solves
the partial differential equations
\[ \sum_{i=1}^n \frac{\partial^h}{\partial x_i^h}
                \frac{\partial^k}{\partial y_i^k} f = 0, \]
for all integers $h,k$ with $1\leq h+k\leq n$.

Let $R_{h,k}$ consist of polynomials in $DH_n$ that
are homogeneous of degree $h$ in the $x_i$'s, and homogeneous
of degree $k$ in the $y_i$'s, together with the zero polynomial.
Then each $R_{h,k}$ is a finite-dimensional submodule of $DH_n$, and we have
\[ DH_n=\bigoplus_{h\geq 0} \bigoplus_{k\geq 0} R_{h,k}. \]
Thus, $DH_n$ is a bigraded $S_n$-module.

Suppose we decompose each $R_{h,k}$ into a direct sum of 
irreducible modules (which correspond to the irreducible characters 
of $S_n$). Let $a_{h,k}(n)$ be the number of occurrences of the
module corresponding to the sign character $\chi_{1^n}$
in $R_{h,k}$. Then we define the \emph{representation-theoretical
$q,t$-Catalan sequence} by
\[ RC_n(q,t)=\sum_{h\geq 0}\sum_{k\geq 0} a_{h,k}(n) q^h t^k
\ \ \mbox{($n=1,2,3,\ldots$)}. \]
Thus, $RC_n(q,t)$ is the generating function for occurrences
of the sign character in $DH_n$. By the symmetry of $x_i$ and
$y_i$ in the definition, we see that $RC_n(q,t)=RC_n(t,q)$.

\subsection{The Two Combinatorial $q,t$-Catalan Sequences}
\label{subsec:combcat}
We next present a combinatorial construction due to Haglund,
and a related construction found later by Haiman, which
interpret the $q,t$-Catalan sequence as a weighted sum of Dyck paths.

A \emph{Dyck path of height $n$} is a path in the $xy$-plane from $(0,0)$
to $(n,n)$ consisting of $n$ north steps and $n$ east steps (each of length
one), such that the path never goes strictly below the diagonal 
line $y=x$. See Figure \ref{dyck1} for an example.
Let ${\cal D}_n$ denote the collection of Dyck paths of height $n$.
For $D\in{\cal D}_n$, let $\area(D)$ be the number of complete lattice
squares (or \emph{cells}) 
between the path $D$ and the main diagonal.  

For $0\leq i< n$,
define $\gamma_i(D)$ to be the number of cells between the path and
the main diagonal in the $i$'th row of the picture, where we let
the bottom row be row zero.
Thus, $\area(D)=\sum_{i=0}^{n-1} \gamma_i(D)$.  
Following Haiman, we set 
\begin{equation}\label{haiman}
 \dinv(D)=\sum_{i<j} \left[\chi(\gamma_i(D)=\gamma_j(D))
+\chi(\gamma_i(D)=\gamma_j(D)+1)\right]. 
\end{equation}
Here and below, we set $\chi(A)=1$ if $A$ is a true statement,
$\chi(A)=0$ if $A$ is a false statement.

\begin{figure}[ht]
\begin{center}
\epsfig{file=dyck1.eps}
\caption{A Dyck path.}
\label{dyck1}
\end{center}
\end{figure}

Define \emph{Haiman's combinatorial $q,t$-Catalan sequence} to be
\[ HC_n(q,t)=\sum_{D\in{\cal D}_n} q^{\dinv(D)}t^{\area(D)}
\ \ \mbox{($n=1,2,3,\ldots$)}. \]

Next, following Haglund (see \cite{haglundstat}), 
we define a ``bounce'' statistic for each Dyck 
path $D$.  Given $D$, we define a \emph{bounce path} derived
from $D$ as follows.  The bounce path begins at $(n,n)$ and moves
to $(0,0)$ via an alternating sequence of horizontal and vertical moves.
Starting at $(n,n)$, the bounce path proceeds due west until it reaches
the north step of the Dyck path going from height $n-1$ to height $n$.  From
there, the bounce path goes due south until it
reaches the main diagonal line $y=x$.  This process continues recursively:
When the bounce path has reached the point $(i,i)$ on the main
diagonal ($i>0$), the bounce path goes due west until it hits
the Dyck path, then due south until it hits the main diagonal.
The bounce path terminates when it reaches $(0,0)$.
See Figure \ref{dyck2} for an example. 

Suppose the bounce path derived from $D$ hits the main diagonal at the points
\[ (n,n),\ \ (i_1,i_1),\ \ (i_2,i_2),\ \ \ldots,\ \ (i_s,i_s),\ \ (0,0). \]
Then Haglund's bounce statistic is defined by
\[ \bounce(D)=\sum_{k=1}^s i_k. \]

\begin{figure}[ht]
\begin{center}
\epsfig{file=dyck2.eps}
\caption{A Dyck path with its derived bounce path.}
\label{dyck2}
\end{center}
\end{figure}

We define \emph{Haglund's combinatorial $q,t$-Catalan sequence} by
\[ C_n(q,t)=\sum_{D\in{\cal D}_n} q^{\area(D)}t^{\bounce(D)}
\ \ \mbox{($n=1,2,3,\ldots$)}. \]

\subsection{Equivalence of the $q,t$-Catalan Sequences}
\label{subsec:equivcat}
The five $q,t$-Catalan sequences discussed in the preceding sections have
quite different definitions. In spite of this, we have the following theorem.

\noindent\textbf{Theorem 1.} For every positive integer $n$,
\[ OC_n(q,t)=SC_n(q,t)=RC_n(q,t)=HC_n(q,t)=C_n(q,t). \]
In particular, $OC_n(q,t)$ is a polynomial in $q$ and $t$ with
nonnegative integer coefficients for all $n$.

This theorem was proved in various papers of Garsia, Haiman, and Haglund.
In \cite{origpaper}, Garsia and Haiman proved that $SC_n(q,t)=OC_n(q,t)$ using
symmetric function identities.  Haglund discovered the combinatorial
sequence $C_n(q,t)$ (see \cite{haglundstat}), 
and Haiman proposed his version $HC_n(q,t)$
shortly thereafter. Haiman and Haglund easily proved that $HC_n(q,t)=C_n(q,t)$ 
by showing that both satisfy the same recursion. We discuss this
recursion later (\S \ref{sec:recursions}). 
Similarly, Garsia and Haglund proved in \cite{nablaproof,PNAS-GH}
that $C_n(q,t)=SC_n(q,t)$ by showing that both sequences satisfied the same
recursion. This proof is much more difficult and requires substantial
machinery from symmetric function theory. 
Finally, Haiman proved that $RC_n(q,t)=SC_n(q,t)$ using 
sophisticated algebraic geometric methods (see \cite{bighaimantheorem}).

A consequence of Theorem 1 is that $C_n(q,t)=C_n(t,q)$ for all $n$,
since this symmetry property holds for $RC_n$.  (It is also
easily deduced from the formula for $OC_n$, by replacing the summation
index $\mu$ by the conjugate of $\mu$ and simplifying.) An open question
is to give a combinatorial proof that $C_n(q,t)=C_n(t,q)$. Later,
we give bijections proving the weaker result that $C_n(q,1)=C_n(1,q)=
HC_n(q,1)=HC_n(1,q)$.  This says that the new statistics of Haiman
and Haglund have the same univariate distribution as the area statistic
on Dyck paths.

\subsection{The Higher $q,t$-Catalan Sequences}
\label{subsec:highercat}
We now discuss various descriptions of the higher $q,t$-Catalan
sequences, also introduced by Garsia and Haiman in \cite{origpaper}.
Fix a positive integer $m$.  The \emph{original higher $q,t$-Catalan
sequence of order $m$} is defined by
\begin{equation}\label{origmcat}
 OC_n^{(m)}(q,t) = \sum_{\mu\vdash n} \frac{t^{(m+1)n(\mu)}q^{(m+1)n(\mu')}
(1-t)(1-q)\Pi_{\mu}(q,t) B_{\mu}(q,t)}{h_{\mu}(q,t) h'_{\mu}(q,t)}
\ \ \mbox{($n=1,2,3,\ldots$).}
\end{equation} 
This formula is the same as (\ref{origcat}), except that
the factors $t^{2n(\mu)}q^{2n(\mu')}$ in $OC_n(q,t)$ have been
replaced by $t^{(m+1)n(\mu)}q^{(m+1)n(\mu')}$. Clearly,
$OC_n^{(1)}(q,t)=OC_n(q,t)$.

Next, the \emph{symmetric function version} of the higher
$q,t$-Catalan sequence of order $m$ is defined by
\begin{equation}\label{symmmcat}
 SC_n^{(m)}(q,t) 
= \left.\nabla^m(e_n)\right|_{s_{1^n}}\ \ \mbox{($n=1,2,3,\ldots$)},
\end{equation}
where $\nabla^m$ means apply the nabla operator $m$ times in succession.
To calculate $SC_n^{(m)}(q,t)$ for a particular $m$ and $n$,
one should express $e_n$ as a linear combination of the  modified Macdonald
basis elements $\tilde{H}_{\mu}$, multiply the coefficient of each
$\tilde{H}_{\mu}$ by $t^{mn(\mu)}q^{mn(\mu')}$, express the result
in terms of the Schur basis $\{s_{\mu}\}$, and extract the coefficient
of $s_{1^n}$.
Garsia and Haiman proved in \cite{origpaper}
that $OC_n^{(m)}(q,t)=SC_n^{(m)}(q,t)$
using symmetric function identities.

A possible representation-theoretical version of the higher $q,t$-Catalan
sequences is given in \cite{origpaper}; we will not discuss it here.

A problem mentioned but not solved in \cite{origpaper}
is to give a combinatorial 
interpretation for the sequences $OC_n^{(m)}(q,t)$.  That paper does
give a simple interpretation for $OC_n^{(m)}(q,1)$, which we now
describe.  Given positive integers $m$ and $n$, let us define
an \emph{$m$-Dyck path of height $n$} to be a path in the $xy$-plane
from $(0,0)$ to $(mn,n)$ consisting of $n$ north steps and $mn$ east  
steps (each of length one), such that the path never goes strictly
below the slanted line $x=my$. See Figure \ref{dyck3} for an example
with $m=3$ and $n=8$. % Got this from NB3, p.158
Let ${\cal D}_{n}^{(m)}$ denote the collection of $m$-Dyck paths of
height $n$.  For $D\in {\cal D}_{n}^{(m)}$, let
$\area(D)$ be the number of complete lattice squares strictly
between the path $D$ and the line $x=my$.
For instance, $\area(D)=23$ for the path $D$ shown in Figure \ref{dyck3}.

We then have (see \cite{origpaper})
\[ OC_n^{(m)}(q,1)=OC_n^{(m)}(1,q)=\sum_{D\in {\cal D}_{n}^{(m)}} 
q^{\area(D)}. \]

\begin{figure}[ht]
\begin{center}
\epsfig{file=dyck3.eps}
\caption{A 3-Dyck path of height 8.}
\label{dyck3}
\end{center}
\end{figure}

\section{Conjectured Combinatorial Interpretations
for the Higher $q,t$-Catalan Sequences}
\label{sec:combconj}
Fix a positive integer $m$.
We next describe two statistics defined on $m$-Dyck paths that each have
the same distribution as the area statistic. The first statistic generalizes
Haiman's statistic for Dyck paths; the second statistic generalizes
Haglund's bounce statistic. We conjecture that either statistic,
when paired with area and summed over $m$-Dyck paths of height $n$,
will give a generating function that equals $OC_n^{(m)}(q,t)$.

\subsection{A Version of Haiman's Statistic for $m$-Dyck Paths}
\label{subsec:mhmstat}
The statistic discussed here was derived from a statistic communicated
to the author by M. Haiman \cite{haiman-email}.
Let $D\in{\cal D}_{n}^{(m)}$ be an $m$-Dyck path of height $n$. As in
\S \ref{subsec:combcat}, 
we define $\gamma_i(D)$ to be the number of cells in the
$i$'th row that are completely contained in the region between the path $D$ 
and the diagonal $x=my$, for $0\leq i < n$. Here, the lowest row is 
row zero.  Note that $\area(D)=\sum_{i=0}^{n-1} \gamma_i(D)$.
Next, define a statistic $h(D)$ by
\begin{equation}\label{mhmdef}
 h(D) = \sum_{0\leq i<j<n} 
\sum_{k=0}^{m-1}
 \chi\left( \gamma_i(D)-\gamma_j(D)+k \in \{0,1,\ldots, m\} \right). 
\end{equation}
See Figure \ref{mhmfig} for an example.

\begin{figure}[ht]
\begin{center}
\epsfig{file=mhmdef.eps}
\caption{Defining the generalized Haiman statistic for a 2-path.}
\label{mhmfig}
\end{center}
\end{figure}

It is easy to see that $h(D)$ reduces to the statistic $\dinv(D)$
from \S \ref{subsec:combcat} when $m=1$.  Here is another formula for $h(D)$ 
which will be useful later. Define a function $\mysc_m:\Z\rightarrow\Z$ by 
\[ \mysc_m(p)=\left\{\begin{array}{ll}
m+1-p & \mbox{ if $1\leq p\leq m$;} \\
m+p   & \mbox{ if $-m\leq p \leq 0$;} \\
0     & \mbox{ for all other $p$.} \end{array}\right. \]
Note that, given the value of a particular difference 
$\gamma_i(D)-\gamma_j(D)$ for a fixed $i$ and $j$, we can
evaluate the inner sum
$\sum_{k=0}^{m-1} \chi ( \gamma_i(D)-\gamma_j(D)+k \in \{0,1,\ldots, m\} )$
in (\ref{mhmdef}). By checking the various cases, one sees that
the value of this sum is exactly $\mysc_m(\gamma_i(D)-\gamma_j(D))$.
For instance, if $\gamma_i(D)-\gamma_j(D)$ is $0$ or $1$, then
we get a contribution for each of the $m$ values of $k$,  in agreement
with the fact that $\mysc_m(0)=\mysc_m(1)=m$. 
Similarly, if $\gamma_i(D)-\gamma_j(D)$
is $-(m-1)$, then only the summand with $k=m-1$ will cause a
contribution, in agreement with the fact that $\mysc_m(-(m-1))=1$.
The remaining cases are checked similarly. We conclude that
\begin{equation}\label{scoredef}
 h(D)=\sum_{0\leq i<j<n} \mysc_m(\gamma_i(D)-\gamma_j(D)). 
\end{equation}

We now define the \emph{first conjectured combinatorial version}
of the higher $q,t$-Catalan sequence of order $m$ by
\[ HC_n^{(m)}(q,t)=\sum_{D\in{\cal D}_{n}^{(m)}} 
q^{h(D)}t^{\area(D)} \ \ \mbox{($n=1,2,3,\ldots$)}. \]
In \S \ref{subsec:bijections}, 
we will prove that $HC_n^{(m)}(q,1)=HC_n^{(m)}(1,q)$.
This says that the statistic $h$ has the same univariate distribution
as the area statistic.

\subsection{A Bounce Statistic for $m$-Dyck paths}
\label{subsec:mbouncestat}
We now discuss how to define a bounce statistic for $m$-Dyck paths
that generalizes Haglund's statistic on ordinary Dyck paths.
To define this statistic, we must first define the
\emph{bounce path} derived from a given $m$-Dyck path $D$.

In \S \ref{subsec:combcat}, 
we obtained the bounce path by starting at $(n,n)$
and moving southwest towards $(0,0)$ according to certain rules
(see Figure \ref{dyck2}).
It is clear that, for ordinary Dyck paths, we could have obtained
a similar statistic with the same distribution by starting at $(0,0)$
and moving northeast. In the case of $m$-Dyck paths, it is more convenient
to start the bouncing at $(0,0)$.

Fix an integer $m\geq 2$.
As before, the bounce path will consist of a sequence of alternating
\emph{vertical moves} and \emph{horizontal moves}. We begin
at $(0,0)$ with a vertical move, and eventually end at $(mn,n)$
after a horizontal move.  Let $v_0,v_1,\ldots$ denote the lengths
of the successive vertical moves in the bounce path, and let $h_0,h_1,\ldots$
denote the lengths of the successive horizontal moves. These lengths
are calculated as follows. (Refer to Figures \ref{mbouncefig} and 
\ref{bounce3} for examples.)

\begin{figure}[ht]
\begin{center}
\epsfig{file=mbouncedef.eps}
\caption{Defining the bounce statistic for a 2-path.}
\label{mbouncefig}
\end{center}
\end{figure}

To find $v_0$, move due north from $(0,0)$ until
you reach an east step of the $m$-Dyck path; the distance traveled is 
$v_0$.  Next, move due east $v_0$ units (so $h_0=v_0$).
Next, move north from the current position until you reach an
east step of the $m$-Dyck path; let $v_1$ be the distance traveled.
Next, move due east $v_0+v_1$ units (so $h_1=v_0+v_1$).
In general, for $i<m$, we move north $v_i$ units from our current
position until we are blocked by the $m$-Dyck path, and
then move east $h_i=v_0+v_1+\cdots+v_i$ units.

For $i\geq m$, the rules change. At stage $i$, we still move north $v_i$
units until we are blocked by the path.  But we then move east 
$h_i=v_i+v_{i-1}+v_{i-2}+v_{i-(m-1)}$ units. In other words,
the length of the next horizontal move is the sum of the $m$ preceding
vertical moves.

If we define $v_i=0$ and $h_i=0$ for all negative indices $i$, we can state
a single rule that works for all the bounces. Start at $(0,0)$. Assuming 
inductively that $v_{j}$ and $h_{j}$ have been determined for all $j<i$
(where $i\geq 0$), move north from the current position until you are blocked 
by the $m$-Dyck path; define the distance traveled to be $v_{i}$.
Then move east $h_{i}=v_i+v_{i-1}+\cdots+v_{i-(m-1)}$ units.
We continue bouncing until we reach $(mn,n)$.
(In fact, it suffices to stop once we reach the top rim of the figure,
which is the horizontal line $y=n$.)
Finally, we define the \emph{bounce statistic} $b(D)$ to be
\begin{equation}\label{mbouncedef}
 b(D)=\sum_{k\geq 0} k\cdot v_k(D), 
\end{equation}
a weighted sum of the lengths of the vertical segments
in the bounce path derived from $D$.
For example, in Figure \ref{mbouncefig}, we have
\[ b(D')=0\cdot 2 + 1\cdot 3+ 2\cdot 1 + 3\cdot 2 + 4\cdot 1+ 5\cdot 3 = 30. \]

When $m=1$, the new rule just says that $h_i=v_i$ for all $i$.
In other words, we move north until we hit the Dyck path, and then
move east the same distance, bringing us back to the main diagonal $y=x$.
Thus, we obtain Haglund's bounce path (modified to start at $(0,0)$,
of course). To see that $b(D)$ agrees with the earlier statistic
$\bounce(D)$, we first give an alternate formula for $b(D)$.
Let $s$ be the number of vertical moves needed to reach the top rim.
Then $v_0+v_1+\cdots+v_{s-1}=n$, where $n$ is the height of $D$.
We claim that
\begin{equation}\label{bounce2}
 b(D)=\sum_{k=0}^{s-1} (n-v_0-v_1-\cdots-v_k).
\end{equation}
To see this, just replace $n$ by $v_0+\cdots+v_{s-1}$ in (\ref{bounce2})
and simplify the resulting sum. We get
\[ b(D)=(v_1+v_2+v_3+\cdots+v_{s-1})+(v_2+v_3+\cdots+v_{s-1})
   +(v_3+\cdots+v_{s-1})+\cdots = \sum_{k\geq 0} k\cdot v_k, \]
which is formula (\ref{mbouncedef}).
When $m=1$, the numbers $i_k$ in the definition of
$\bounce(D)$ (see \S \ref{subsec:combcat}) are exactly the quantities
$n-v_0-v_1-\cdots-v_k$. (Here we must reflect the shape in Figure 
\ref{dyck2} so that
the bounce path starts at $(0,0)$.)  This shows that the new 
statistic does generalize the original one.

Note that, for $m>1$, the bounce path does not necessarily
return to the diagonal $x=my$ after each horizontal move.
Consequently, it may occur that we cannot move north at all after
making a particular horizontal move.  This situation occurs for the bounce path
shown in Figure \ref{bounce3}, 
which is derived from the $3$-path shown in Figure \ref{dyck3}.
In this case, we define the next $v_i$ to be zero,
and compute the next $h_i=v_i+v_{i-1}+\cdots+v_{i-(m-1)}$ just as before.
In other words, vertical moves of length zero can occur, and are
treated the same as nonzero vertical moves when computing the $h_i$'s
and the $b$ statistic.

\begin{figure}[ht]
\begin{center}
\epsfig{file=bounce3.eps}
\caption{A bounce path with vertical moves of length zero.}
\label{bounce3}
\end{center}
\end{figure}

The possibility now arises that the bounce path could get ``stuck''
in the middle of the figure. To see why,
suppose that $m$ consecutive vertical moves $v_i,\ldots,v_{i+m-1}$
in the bounce path had length zero. Then the next horizontal move
$h_{i+m-1}$ would be zero also. As a result, our position in the
figure at stage $i+m$ is exactly the same as the position at
the beginning of stage $i+m-1$, since $v_{i+m-1}=h_{i+m-1}=0$.
{}From the bouncing rules, it follows that $v_{i+m}=0$ also.
But then $v_j=h_j=0$ for all $j\geq i+m$, so that the bouncing
path is stuck at the current position forever.

We now argue that the situation described in the last paragraph
will never occur. Since the $m$-Dyck path must start with a north 
step, we have $v_0>0$, and so we do not get stuck at $(0,0)$.
The evolving bounce path will continue to make progress eastward with
each horizontal step, unless $h_i=0$ for some $i\geq 0$.
Note that $h_i=0$ iff $v_i+v_{i-1}+\cdots+v_{i-(m-1)}=0$.
Fix such an $i$, and consider the situation just after
making the vertical move of length $v_{i-1}$ and the horizontal
move of length $h_{i-1}$.  Let $(x_0,y_0)$ denote the position of the
bounce path at this instant.  Then $y_0=v_0+v_1+\cdots+v_{i-1}$ is the total
vertical distance moved so far. Since $v_{i-1}=\cdots=v_{i-(m-1)}=0$,
we have $y_0=v_0+\cdots+v_{i-m}$.  On the other hand, the total horizontal
distance moved so far is $x_0=h_0+h_1+\cdots+h_{i-1}$. From
the definition of the $h_j$'s and the fact that
$v_{i-1}=\cdots=v_{i-(m-1)}=0$, it follows that
$x_0=mv_0+mv_1+\cdots+mv_{i-m}$. In more detail, note that the last nonzero 
$v_j$, namely $v_{i-m}$, contributes to the $m$ horizontal moves
$h_{i-m},\ldots,h_{i-1}$. Similarly, for $j<i-m$, $v_j$ has contributed
to $m$ horizontal moves that have already occurred at the end of 
stage $i-1$. Since $v_j=0$ for $i-m<j\leq i-1$, the stated formula for $x_0$
accounts for all the horizontal motion so far. Comparing the formulas for
$x_0$ and $y_0$ gives
$x_0=my_0$, so that the bounce path has returned to the bounding
diagonal $x=my$. If $y_0=n$, the bounce path has reached its destination.
If $y_0<n$, the $m$-Dyck path continues above height $y_0$.
But now $v_i>0$ is forced; otherwise, the $m$-Dyck path must have gone
east from $(my_0,y_0)$, violating the requirement of always staying
weakly above the line $x=my$. This argument is illustrated by the path
in Figure \ref{bounce3}.

Thus, the bounce path does not get stuck. The argument at the end
of the last paragraph can be modified to show that the bounce path (like
the $m$-Dyck path itself) never goes below the line $x=my$.
For, after moving $v_0+\cdots+v_{i-1}$ steps vertically at some time,
we will have gone at most $mv_0+\cdots+mv_{i-1}$ steps horizontally.
Therefore, our position is on or above the line $x=my$.

Now that we know the bounce path is always well-defined, we
can define the \emph{second conjectured combinatorial version}
of the higher $q,t$-Catalan sequence of order $m$ by
\[ C_n^{(m)}(q,t)=\sum_{D\in{\cal D}_{n}^{(m)}} 
q^{\area(D)}t^{b(D)} \ \ \mbox{($n=1,2,3,\ldots$)}. \]
In \S \ref{subsec:bijections}, we will give a bijective proof
that $HC_n^{(m)}(q,t)=C_n^{(m)}(q,t)$.
Setting $t=1$ or $q=1$ here shows that both new statistics
($h$ and $b$) have the same distribution on $m$-Dyck paths of height $n$
as the area.

\smallskip
\noindent\textbf{Conjecture:} For all $m$ and $n$, we have
\[ OC_n^{(m)}(q,t)=HC_n^{(m)}(q,t)=C_n^{(m)}(q,t). \]
A possible approach to proving this conjecture will be indicated
in \S \ref{sec:recursions}.

\subsection{A Formula for $C_n^{(m)}(q,t)$}
\label{subsec:sumformula}
In this section, we give an explicit algebraic formula
(\ref{sumformula}) for $C_n^{(m)}(q,t)$ by analyzing bounce paths.
This formula, while messy, is obviously a polynomial in $q$ and $t$
with nonnegative integer coefficients, unlike the formula defining
$OC_n^{(m)}(q,t)$. A disadvantage of the new formula is that
the (conjectured) symmetry $C_n^{(m)}(q,t)=C_n^{(m)}(t,q)$
is not evident from inspection of the formula.

Before stating the formula, we briefly review $q$-binomial
coefficients.  Let $q$ be an indeterminate. Set
$[n]_q=1+q+q^2+\cdots+q^{n-1}$ for each positive integer $n$.
Set $[0]_q!=1$ and $[n]_q!=\prod_{i=1}^n [i]_q$ for  $n>0$.
Finally, set $\tqbin{n}{k}=\frac{[n]_q!}{[k]_q! [n-k]_q!}$
for $0\leq k\leq n$, and set $\tqbin{n}{k}=0$ for other values of $k$.
When we replace $q$ by $1$, the expressions $[n]_q$, $[n]_q!$,
and $\tqbin{n}{k}$ evaluate to the numbers $n$, $n!$,  and $\tbinom{n}{k}$,
respectively. Note also that $\tqbin{n}{k}=\tqbin{n}{n-k}$. We will often
write $\tqbin{a+b}{a,b}$ to denote
$\tqbin{a+b}{a}=\tqbin{a+b}{b}$ (multinomial coefficient notation).

We shall use the following well-known combinatorial interpretations
of the $q$-binomial coefficient $\tqbin{n}{k}$. Let $R_{a,b}$ denote
a rectangle of height $a$ and width $b$. We write $\lambda\subset R_{a,b}$
for a partition $\lambda$ if the Ferrers diagram of $\lambda$ fits
inside this rectangle.  Then 
\begin{equation}\label{qrect}
 \dqbin{a+b}{a,b}=\sum_{\lambda\subset R_{a,b}} q^{|\lambda|}
      =\sum_{\lambda\subset R_{a,b}} q^{ab-|\lambda|}.
\end{equation} 
(The second equality follows from the first by rotating the rectangle 
$180^{\circ}$ and considering the area cells inside the rectangle but 
outside $\lambda$.) We prefer the notation 
 $\tqbin{a+b}{a,b}$ 
because the bottom row displays both dimensions of the containing rectangle.

Here are two useful ways to rephrase (\ref{qrect}).
Let ${\cal P}_{a,b}$ denote the collection of all paths that
proceed from the lower-left corner of $R_{a,b}$ to the upper-right
corner by taking $a$ north steps and $b$ east steps of length one.
(There is no other restriction on the paths.) If $P$ is such a path,
let $\area(P)$ be the number of cells in the rectangle lying below
the path $P$. Then
\[ \dqbin{a+b}{a,b}=\sum_{P\in{\cal P}_{a,b}} q^{\area(P)}
  =\sum_{P\in{\cal P}_{a,b}} q^{ab-\area(P)}. \]
Similarly, let $R(0^a 1^b)$ denote the collection of all rearrangements
of $a$ zeroes and $b$ ones. If $w=(w_1 w_2\ldots w_{a+b})\in R(0^a 1^b)$, 
define the \emph{inversions of $w$} by
$\inv(w)=\sum_{i<j} \chi(w_i>w_j)$ and the \emph{coinversions of $w$} by
$\coinv(w)=\sum_{i<j} \chi(w_i<w_j)$. Then
\begin{equation}\label{coinv}
 \dqbin{a+b}{a,b}=\sum_{w\in R(0^a 1^b)} q^{\inv(w)}
                  =\sum_{w\in R(0^a 1^b)} q^{\coinv(w)}. 
\end{equation}
This follows by representing $w$ as a path $P\in{\cal P}_{a,b}$,
which is obtained by replacing each zero in $w$ by a north step
and each one in $w$ by an east step. Then the area above (resp., below)
the path in $R_{a,b}$ is easily seen to be $\inv(w)$ (resp., $\coinv(w)$).

We are now ready to state the summation formula for $C_n^{(m)}(q,t)$.
Let ${\cal V}_{n}^{(m)}$ denote the set of all
sequences $v=(v_0,v_1,v_2,\ldots,v_s)$ such that: each $v_i$ is
a nonnegative integer; $v_0>0$; $v_s>0$; $v_0+v_1+v_2+\cdots+v_s = n$; and
there is never a string of $m$ or more consecutive zeroes in $v$.
As usual, let $v_i=0$ for all negative $i$.

\smallskip
\noindent\textbf{Theorem.}  With ${\cal V}_{n}^{(m)}$ defined as above, we have:
\begin{equation}\label{sumformula}
 C_n^{(m)}(q,t) =
\sum_{v\in {\cal V}_{n}^{(m)}} 
t^{\sum_{i\geq 0} i v_i} q^{m\sum_{i\geq 0}
% \frac{1}{2} v_i(v_i-1)}
\tbinom{v_i}{2}}
 \prod_{i\geq 1} q^{v_i\sum_{j=1}^m (m-j)v_{i-j}}
\dqbin{v_i+v_{i-1}+\cdots+v_{i-m}-1}
     {v_i,v_{i-1}+\cdots+v_{i-m}-1}. 
\end{equation}
Equivalently, we may sum over all compositions $v$ of $n$ with
zero parts allowed, if we identify compositions that differ only
in trailing zeroes. The same formula holds for $HC_n^{(m)}(q,t)$,
hence $C_n^{(m)}(q,t)=HC_n^{(m)}(q,t)$.

\smallskip
\noindent\textbf{Remark:} When $m=1$, this formula reduces to
a formula for $C_n(q,t)$ given by Haglund in \cite{haglundstat}.

\smallskip
\noindent\textbf{Proof, Part 1:}
Let $D\in{\cal D}_{n}^{(m)}$ be a typical object counted by $C_n^{(m)}(q,t)$.
We can classify $D$ based on the sequence $v(D)=(v_0,v_1,\ldots,v_s)$
of vertical moves in the bounce path derived from $D$.  Call this sequence
the \emph{bounce composition} of $D$. By the discussion
in the preceding section, the vector $v=v(D)$ belongs to ${\cal V}_{n}^{(m)}$.
To prove the formula for $C_n^{(m)}(q,t)$, it suffices to show that
\[ \begin{split}\lefteqn{\sum_{D:\ v(D)=v } \!\! q^{\area(D)}t^{b(D)} =}
\\ &
t^{\sum_{i\geq 0} i v_i} 
q^{m\sum_{i\geq 0}
% \frac{1}{2} v_i(v_i-1)} 
\tbinom{v_i}{2}}
 \prod_{i=1}^s q^{v_i\sum_{j=1}^m (m-j)v_{i-j}}
\dqbin{v_i+v_{i-1}+\cdots+v_{i-m}-1}
     {v_i,v_{i-1}+\cdots+v_{i-m}-1} \end{split} \]
for each $v=(v_0,\ldots,v_s)\in {\cal V}_{n}^{(m)}$.
By our conventions for $q$-binomial coefficients, the right side
of this expression is zero if any $m$ consecutive $v_i$'s are zero
(in particular, this occurs if $v_0=0$). Thus, it does no harm 
in (\ref{sumformula}) to
sum over all compositions $v$ of $n$ with zero parts allowed,
not just the compositions $v$ belonging to ${\cal V}_{n}^{(m)}$.

Now, fix $v\in {\cal V}_{n}^{(m)}$ and consider only the $m$-Dyck paths
of height $n$ having bounce composition $v$.
By definition of the bounce statistic, every such path $D$ will have 
the same $t$-weight, namely
\[ t^{b(D)}=t^{\sum_{i\geq 0} i v_i}. \]

To analyze the $q$-weights, note that we can construct all $m$-Dyck
paths of height $n$ having bounce composition $v$ as follows.
\begin{enumerate}
\item Starting with an empty diagram, draw the bounce path with
vertical segments $v_0,\ldots,v_s$.  There is exactly one way to do this,
since the horizontal moves $h_i$ are completely determined by the
vertical moves.
\item Having drawn the bounce path, there are now $s$ empty rectangular
areas just northwest of the ``left-turns'' in the bounce path.
See Figure \ref{brectangles} for an example.  
Label these rectangles $R_1,\ldots, R_s$, as shown.
By definition of the bounce path, rectangle $R_i$ has height
$v_i$ and width $h_{i-1}=v_{i-1}+\cdots+v_{i-m}$ for each $i$.
To complete the $m$-Dyck path, draw a path in each rectangle $R_i$
from the southwest corner to the northeast corner, \emph{where each path
begins with at least one east step}. The first east step in $R_i$ 
must be present, by definition of $v_{i-1}$. 
\end{enumerate}

\begin{figure}[ht]
\begin{center}
\epsfig{file=brectangles.eps}
\caption{Rectangles above the bounce path.}
\label{brectangles}
\end{center}
\end{figure}

We can rephrase the second step as follows. Let $R_i'$ denote the
rectangle of height $v_i$ and width $h_i=v_{i-1}+\cdots+v_{i-m}-1$
obtained by ignoring the leftmost column of $R_i$. Then we can
uniquely construct the path $D$ by filling each shortened rectangle $R_i'$
with an \emph{arbitrary}
path going from the southwest corner to the northeast corner.

The generating function for the number of ways to perform this second step, 
where the exponent of $q$ records the total area above the bounce path, is
\[ \prod_{i=1}^s \dqbin{v_i+v_{i-1}+\cdots+v_{i-m}-1}
     {v_i,v_{i-1}+\cdots+v_{i-m}-1} \]
by the preceding discussion of $q$-binomial coefficients.

We still need to multiply by a power of $q$ that records
the area under the bounce path,
which is independent of the choices in the second step.
We claim that this area is
\[ {m\sum_{i=0}^s \frac{1}{2} v_i(v_i-1)} +
 \sum_{i=1}^s \left(v_i\sum_{j=1}^m (m-j)v_{i-j}\right), \]
which will complete the proof.

\begin{figure}[ht]
\begin{center}
\epsfig{file=dissect.eps}
\caption{Dissecting the area below the bounce path.}
\label{dissect}
\end{center}
\end{figure}

To establish the claim, dissect the area below the bounce path
as shown in Figure \ref{dissect}.  There are $s+1$ triangular pieces $T_i$,
where the $i$'th triangle contains $0+m+2m+\cdots+(v_i-1)m
=m\frac{v_i(v_i-1)}{2}$ complete cells. In Figure \ref{dissect}, for instance, 
where $v_1=3$, we have shaded the $0+2+4=6$ cells in $T_1$ that contribute 
to the area statistic.  The total area coming from the triangles is
\[ m\sum_{i=0}^s \frac{1}{2} v_i(v_i-1). \]

There are also $s$ rectangular slabs $S_i$ (for
$1\leq i\leq s$).  The height of slab $S_i$ is $v_i$.
What is the width of $S_i$? To answer this question, fix $i$, 
let $(a,c)$ be the coordinates of the southeast corner of 
$S_i$, and let $(b,c)$  be the coordinates of the southwest corner
of $S_i$.  First note that $c=v_0+v_1+\cdots+v_{i-1}$, the sum
of the vertical steps prior to step $i$. Therefore, 
\[ a=mc=m(v_0+\cdots+v_{i-1})=mv_{i-1}+mv_{i-2}+\cdots+mv_{i-m}+mv_{i-m-1}
  +\cdots \]
since the southeast corner of $S_i$ lies on the line $x=my$.
Next, $b=h_0+h_1+\cdots+h_{i-1}$, the sum of the horizontal steps
prior to step $i$.  Recall that each $h_j$ is the sum of the $m$
preceding $v_i$'s (starting with $i=j$). Substituting into the expression for
$b$ gives $b=1v_{i-1}+2v_{i-2}+\cdots+mv_{i-m}+mv_{i-m-1}+mv_{i-m-2}+\cdots$.
We conclude that the width of $S_i$ is
\[ a-b= (m-1)v_{i-1}+(m-2)v_{i-2}+\cdots+(m-m)v_{i-m}+0+0+\cdots. \]
Finally, the area of $S_i$ is the height times the width, which is
\[ v_i(a-b)=v_i\sum_{j=1}^m (m-j)v_{i-j}. \]
Adding over all $i$ gives the term
\[ \sum_{i=1}^s \left(v_i\sum_{j=1}^m (m-j)v_{i-j}\right), \]
completing the proof of the claim and the first part of the theorem.

\subsection{Proving the Formula for $HC_n^{(m)}(q,t)$}
\label{subsec:HCformula}
To finish the proof of the theorem,
we now give a counting argument to show that
$HC_n^{(m)}(q,t)$ is also given by the formula (\ref{sumformula}).
This will show that $HC_n^{(m)}(q,t)=C_n^{(m)}(q,t)$.
In the next section, we combine the two different proofs of
this formula to obtain a bijective proof of the identity
$HC_n^{(m)}(q,t)=C_n^{(m)}(q,t)$.

Recall that an $m$-Dyck path $D$ can be represented
by a vector $$\gamma(D)=(\gamma_0(D),\ldots,\gamma_{n-1}(D)),$$
where $\gamma_i(D)$ is the number of area cells between the path
and the diagonal in the $i$'th row from the bottom. Clearly,
the path $D$ is uniquely recoverable from the vector $\gamma$.
Also, a vector $\gamma=(\gamma_0,\ldots,\gamma_{n-1})$ represents
an element $D\in{\cal D}_{n}^{(m)}$ iff the following three conditions hold:
\begin{enumerate}
\item $\gamma_0=0$.
\item $\gamma_i\geq 0$ for all $i$.
\item $\gamma_{i+1}\leq \gamma_i+m$ for all $i<n-1$.
\end{enumerate}
The first condition reflects the fact that the lowest row cannot have
any area cells.  The second condition ensures that the path $D$
never goes below the diagonal $x=my$. The third condition follows
since the path is not allowed to take any west steps.

Let ${\cal G}_n^{(m)}$ denote the set of all $n$-long vectors $\gamma$
satisfying these three conditions. Then the preceding remarks show that
\[ HC_n^{(m)}(q,t)=\sum_{\gamma\in {\cal G}_n^{(m)}} 
q^{h(\gamma)} t^{\sum_{i\geq 0}
\gamma_i}, \]
where $\sum_{i\geq 0} \gamma_i$ is the area of the path $D$ corresponding
to $\gamma$,
and where we set
\[ h(\gamma)=\sum_{0\leq i<j<n}\sum_{k=0}^{m-1} \chi(\gamma_i-\gamma_j+k\in
\{0,1,\ldots,m\}), \]
so that $h(\gamma)$ is the $h$-statistic of the path $D$.

Given a vector $\gamma\in {\cal G}_n^{(m)}$, let $v_i(\gamma)$
be the number of times $i$ occurs in the sequence 
$(\gamma_0,\ldots,\gamma_{n-1})$ for each $i\geq 0$. 
Let $v(\gamma)=(v_0(\gamma),v_1(\gamma),\ldots,v_s(\gamma))$
where $s$ is the largest entry appearing in $\gamma$.
We call $v(\gamma)$ the \emph{composition of $\gamma$}.
{}From the definitions of ${\cal G}_n^{(m)}$ and $v(\gamma)$, we see that
$v_0>0$, $v_s>0$, $v_0+\cdots+v_s=n$, and there
is never a string of $m$ consecutive zeroes in $v$ (lest
$\gamma_{i+1}>\gamma_i+m$ for some $i$). In other words, $v$ belongs to
${\cal V}_{n}^{(m)}$.

We now classify the objects $\gamma$ in ${\cal G}_n^{(m)}$ based on 
their composition.  To prove the summation formula for
$HC_n^{(m)}(q,t)$, it suffices to show that
\begin{equation}\label{gammaclaim}
\begin{split}
\lefteqn{ \sum_{\gamma:\ v(\gamma)=v } q^{h(\gamma)}t^{\sum_{i\geq 0}\gamma_i} 
=  } \\ &
t^{\sum_{i\geq 0} i v_i} 
q^{m\sum_{i\geq 0} \frac{1}{2} v_i(v_i-1)} 
 \prod_{i=1}^s q^{v_i\sum_{j=1}^m (m-j)v_{i-j}}
\dqbin{v_i+v_{i-1}+\cdots+v_{i-m}-1}
     {v_i,v_{i-1}+\cdots+v_{i-m}-1} 
\end{split}
\end{equation}
for each $v=(v_0,\ldots,v_s)\in {\cal V}_{n}^{(m)}$.
It is clear that the powers of $t$ on each side of this equation agree,
since $v_i$ is the number of occurrences of the value $i$ 
in the summation $\sum_{i\geq 0} \gamma_i$.

Before considering the powers of $q$,
note that we can uniquely construct all vectors $\gamma\in {\cal G}_n^{(m)}$
having composition $v$ as follows.
\begin{enumerate}
\item Initially, let $\gamma$ be a string of $v_0$ zeroes.
\item Next, insert $v_1$ ones in the gaps to the right of these
zeroes. There can be any number of ones in each gap, but no $1$
may appear to the left of the leftmost zero.
\item Continue by inserting $v_2$ twos into  valid locations,
then $v_3$ threes, etc. The general step is to insert
$v_i$ copies of the symbol $i$ into valid locations in the current string.
Here, a ``valid'' location is one such that inserting $i$ in
that location will not cause a violation of the three
conditions in the definition of ${\cal G}_n^{(m)}$.
\end{enumerate}
How many ways are there to perform the $i$'th step of this insertion
process, for $i>0$? To answer this, note that a new symbol $i>0$
can only be placed in a gap immediately to the right of the existing
symbols $i-1, i-2, \ldots, i-m$ in the current string.
There are $v_{i-1}+v_{i-2}+\cdots+v_{i-m}$ such symbols, and hence the
same number of gaps. Since multiple copies of $i$ can be placed in
each gap, the number of ways to insert the
$v_i$ new copies of the symbol $i$ is $\dbinom{v_i+v_{i-1}+\cdots+v_{i-m}-1}
{v_i,v_{i-1}+\cdots+v_{i-m}-1}$. (To see this, represent a particular way of 
inserting the new $i$'s by a string of $v_i$ ``stars'' representing the $i$'s
and $v_{i-1}+\cdots+v_{i-m}-1$ ``bars'' that separate the
$v_{i-1}+\cdots+v_{i-m}$ available gaps.)
Multiplying these expressions as $i$ ranges from $1$ to $s$, we
see that formula (\ref{gammaclaim}) is correct when $q=1$. 

It remains to see that the power of $q$ is correct as well.
We prove this by induction on the largest symbol $s$ appearing in $\gamma$.
If $s=0$, then $v=(n)$, and $\gamma$ must consist of a string
of $n$ zeroes. From the definition, we see that $h(\gamma)=mn(n-1)/2$.
This is the same as the  power of $q$ on the right side of
(\ref{gammaclaim}), since $v_0=n$ and $v_i=0$ for $i>0$.

Now assume that $s>0$. Fix $v=(v_0,\ldots,v_s)\in {\cal V}_{n}^{(m)}$.
Let $v'=(v_0,\ldots,v_{s-1})$, which is an element of 
${\cal V}_{n-v_s}^{(m)}$ (ignore trailing zeroes in $v'$ if necessary).
Our induction hypothesis says that
\[ \sum_{\delta: v(\delta)=v'} q^{h(\delta)} =
   q^{m\sum_{i=0}^{s-1} v_i(v_i-1)/2}
\prod_{i=1}^{s-1} q^{v_i\sum_{j=1}^m (m-j)v_{i-j}}
\dqbin{v_i+v_{i-1}+\cdots+v_{i-m}-1}
      {v_i,v_{i-1}+\cdots+v_{i-m}-1}; \]
note that any trailing zeroes in $v'$ just contribute extra
factors of $1$ to the right side, which are harmless.
We want to establish the analogous formula for
\[ \sum_{\gamma: v(\gamma)=v} q^{h(\gamma)}. \]
For this purpose, recast the construction given in the $q=1$ case as follows.
We can uniquely produce every $\gamma$ with $v(\gamma)=v$ by: first, choosing
a $\delta$ with $v(\delta)=v'$; and second, choosing a way to insert
$v_s$ copies of $s$ into $\delta$ in valid locations.
The generating function for the number of ways to choose $\delta$,
where the power of $q$ records $h(\delta)$, is by assumption
\[   q^{m\sum_{i=0}^{s-1} v_i(v_i-1)/2}
\prod_{i=1}^{s-1} q^{v_i\sum_{j=1}^m (m-j)v_{i-j}}
\dqbin{v_i+v_{i-1}+\cdots+v_{i-m}-1}
      {v_i,v_{i-1}+\cdots+v_{i-m}-1}. \]
To complete the proof, we need to show that the increase in 
the $h$-statistic caused by the second choice (namely,
$h(\gamma)-h(\delta)$) has generating function
\begin{equation}\label{addings}
 q^{mv_s(v_s-1)/2}q^{v_s\sum_{k=1}^m (m-k)v_{s-k}}
  \dqbin{v_s+v_{s-1}+\cdots+v_{s-m}-1}
        {v_s,v_{s-1}+\cdots+v_{s-m}-1};
\end{equation}
then the desired result will follow from the product rule for
generating functions (\cite{benderbook}, Ch. 10).

We encode the choice of how to insert the $v_s$ copies of $s$
into $\delta$ as a word $$w\in R(0^{v_s} 1^{v_{s-1}+\cdots
+v_{s-m}-1}).$$ To find $w$, read the symbols in the completed vector $\gamma$
from left to right. Write down a zero in $w$ every time an $s$ 
occurs in $\gamma$; write down a one in $w$ every time one of the symbols
$s-1,\ldots,s-m$ occurs in $\gamma$; ignore all other symbols in $\gamma$.
By the conditions on $\gamma$, the first symbol in $w$ must be a one
(since some symbol in $\{s-1,\ldots,s-m\}$ must appear just before
the leftmost $s$ in $\gamma$). Erase this initial $1$ to obtain the 
word $w$.

We will prove that
\begin{equation}\label{hdiff}
 h(\gamma)-h(\delta)= mv_s(v_s-1)/2+v_s\sum_{k=1}^m (m-k)v_{s-k}
     +  \coinv(w);
\end{equation} 
if this equation holds, then (\ref{addings}) immediately follows
from it because of (\ref{coinv}).

The proof of (\ref{hdiff}) proceeds by induction on the value of $\coinv(w)$.
Suppose $\coinv(w)=0$ first. This happens iff all $v_s$ copies of $s$
were inserted into $\delta$ immediately following the last occurrence
of any symbol in the set $\{s-1,\ldots,s-m\}$. How do these $v_s$
newly inserted symbols affect the $h$-statistic? To answer this, we must
compute the sum (see (\ref{scoredef}))
\[ \sum_{i<j} \mysc_m(\gamma_i-\gamma_j) \]
over all pairs $(i,j)$ such that $\gamma_i=s$ or $\gamma_j=s$.

First, consider the pairs $(i,j)$ for which $i<j$ and $\gamma_i=s$
and $\gamma_j=s$. There are $\tbinom{v_s}{2}$ such pairs, and
each contributes $\mysc_m(s-s)=\mysc_m(0)=m$ to the $h$-statistic.
This gives the term $mv_s(v_s-1)/2$ in (\ref{hdiff}).

Second, consider the pairs $(i,j)$ for which $i<j$ and
$\gamma_i=s$ and $\gamma_j\neq s$.  Since all the copies of $s$
in $\gamma$ occur in a contiguous group following all instances
of the symbols $s-1,\ldots, s-m$, and since $s$ is the largest symbol
appearing in $\gamma$, $j>i$ implies that $\gamma_j<s-m$.
Then $\mysc_m(\gamma_i-\gamma_j)=0$, since $\gamma_i-\gamma_j>m$.
So these pairs contribute nothing to the $h$-statistic.

Third, consider the pairs $(i,j)$ for which $i<j$ and
$\gamma_i\neq s$ and $\gamma_j=s$. Since $s$ is the largest symbol,
we have $\gamma_i<s$.  Write $\gamma_i=s-k$ for some $k>0$,
and consider various subcases.  Suppose $k\in\{1,2,\ldots,m\}$.
Then $\mysc_m(\gamma_i-\gamma_j)=\mysc_m(-k)=m-k$. For how many pairs
$(i,j)$ does it happen that $i<j$, $\gamma_i=s-k$, and $\gamma_j=s$?
There are $v_s$ choices for the index $j$ and $v_{s-k}$ choices
for the index $i$; the condition $i<j$ holds automatically, since all 
occurrences of $s$ occur to the right of all occurrences of $s-k$. Thus, we
get a total contribution to the $h$-statistic of $(m-k)v_s(v_{s-k})$
for this $k$. Adding over all $k$, we obtain the term
\[ v_s\sum_{k=1}^m (m-k)v_{s-k} \]
appearing in (\ref{hdiff}). On the other hand, if $k>m$,
then $\mysc_m(\gamma_i-\gamma_j)=\mysc_m(-k)=0$, so there is no
contribution to the $h$-statistic.

The three cases just considered are exhaustive, so we conclude
that (\ref{hdiff}) is true when $\coinv(w)$ is zero.

For the inductive step, consider what happens when we replace
two consecutive symbols $10$ in $w$ by $01$, thus increasing
$\coinv(w)$ by one.  Let $w'$ be the new word after the replacement,
and let $\gamma'$ be the associated vector obtained by inserting
$s$'s into $\delta$ according to the encoding $w'$. We may assume,
by induction, that (\ref{hdiff}) is correct for $\gamma$ and $w$.
Passing from $w$ to $w'$ increases the right side of (\ref{hdiff})
by one. Hence, (\ref{hdiff}) will be correct for $\gamma'$ and $w'$,
provided that $h(\gamma')=h(\gamma)+1$.   To obtain $\gamma'$ from $\gamma$,
look at the symbols in $\gamma$ corresponding to the replaced string
$10$ in $w$. The symbol corresponding to the $0$ is an $s$.
This $s$ is immediately preceded in $\gamma$ by a symbol
in $\{s-1,\ldots,s-m\}$ which corresponds to the $1$, by the conditions
on $\gamma$ and the fact that $s>0$. Say $s-k$ immediately precedes
this $s$. The effect of replacing $10$ by $01$ in $w$ is to move
the $s$ leftwards, past its predecessor $s-k$, and re-insert it in the next
valid position in $\gamma$. This valid position occurs immediately
to the right of the next occurrence of a symbol in $\{s,s-1,s-2,\ldots,s-m\}$
left of the symbol $s-k$.  
Pictorially, we have:
\[ \mbox{original $\gamma$} = \ldots\ (s-j)\ z_1\ z_2\ \ldots\ z_{\ell}\ 
(s-k)\ s\ \ldots \]
where $0\leq j\leq m$, $1\leq k\leq m$, $\ell\geq 0$,
and every $z_i<s-m$. After moving $s$ left, we have
\[ \mbox{ new $\gamma'$} = \ldots\ (s-j)\ s\ z_1\ z_2\ \ldots\ z_{\ell}\ 
(s-k)\ \ldots. \]
Note that the symbol $s-j$ must exist, lest $\gamma'_0=s>0$.

Now, let us examine the effect of this motion on the $h$-statistic.
When we move the $s$ left past its predecessor $s-k$ in $\gamma$, we get
a net change in the $h$-statistic of
\[ \mysc_m(s-[s-k]) - \mysc_m([s-k]-s)
  =\mysc_m(k)-\mysc_m(-k) = +1, \]
since $1\leq k\leq m$ (see (\ref{scoredef})).
As before, since $|s-z_i|>m$, moving the $s$ past each $z_i$ will
not affect the $h$-statistic at all. Thus, the total change
in the $h$-statistic is $+1$,  as desired.

We can obtain an arbitrary encoding word $w$ from the word 
$11\ldots 100\ldots 0$ with no coinversions by doing a finite
sequence of interchanges of the type just described.
Thus, the validity of (\ref{hdiff}) for all words $w$ follows
by induction on the number of such interchanges required
(this number is exactly $\coinv(w)$, of course).
This completes the proof of the theorem.

\subsection{A Bijection Proving that $HC_n^{(m)}(q,t)=C_n^{(m)}(q,t)$}
\label{subsec:bijections}
The two proofs just given to show that formula (\ref{sumformula})
holds for $C_n^{(m)}(q,t)$ and $HC_n^{(m)}(q,t)$ were completely
combinatorial.  Hence, we can combine these proofs to get a bijective proof
that $HC_n^{(m)}(q,t)=C_n^{(m)}(q,t)$.  Fix $m$ and $n$.
We describe a bijection $\phi:{\cal D}_{n}^{(m)}\rightarrow{\cal D}_{n}^{(m)}$
such that
\[ h(D) = \area(\phi(D)) \mbox{ and }
  \area(D) = b(\phi(D)) \mbox{ for $D\in{\cal D}_{n}^{(m)}$} \]
and a bijection $\psi=\phi^{-1}:{\cal D}_{n}^{(m)}\rightarrow{\cal D}_{n}^{(m)}$
such that
\[ b(D) = \area(\psi(D)) \mbox{ and }
  \area(D) = h(\psi(D)) \mbox{ for $D\in{\cal D}_{n}^{(m)}$.} \]
These bijections will show that the three statistics $\area$, $h$, and $b$
all have the same univariate distribution on ${\cal D}_{n}^{(m)}$.

\smallskip\noindent\textbf{Description of $\phi$.}
Let $D$ be an $m$-Dyck path of height $n$. To find the path $\phi(D)$:
\begin{itemize}
\item
Represent $D$ by the vector of row lengths $\gamma(D)=(\gamma_0(D),\ldots,
\gamma_{n-1}(D))$, where $\gamma_i(D)$ is the number of area cells
in the $i$'th row from the bottom.
\item Define $v=(v_0,\ldots,v_s)$ by letting $v_j$ be the number
of occurrences of the value $j$ in the vector $\gamma(D)$.
\item Starting with an empty triangle, draw a bounce path from $(0,0)$
with successive vertical segments $v_0,\ldots,v_s$ and
horizontal segments $h_0,h_1,\ldots$, where $h_i=v_i+v_{i-1}+\cdots+
v_{i-(m-1)}$ for each $i$.
\item For $1\leq i\leq s$, form a word $w_i$ from $\gamma(D)$ as follows.
Initially, $w_i$ is empty. Read $\gamma$ from left to right.
Write down a zero every time the symbol $i$ is seen in $\gamma$.
Write down a one every time a symbol in $\{i-1,\ldots,i-m\}$ is
seen in $\gamma$. Ignore all other symbols in $\gamma$.
At the end, erase the first symbol in $w_i$ (which is necessarily a $1$).
\item Let $R_1,\ldots,R_s$ be the empty rectangles above the bounce
path. Let $R_1',\ldots,R_s'$ be these rectangles with the leftmost
columns deleted (as in \S \ref{subsec:sumformula}). 
For $1\leq i\leq s$, use the
word $w_i$ to fill in the part of the path lying in $R_i'$, from
the southwest corner to the northeast corner,
by taking a north step for each zero in $w_i$, and an east step
for each one in $w_i$. Call the completed path $\phi(D)$.
\end{itemize}
The two preceding proofs have already shown that $\phi$ has
the desired effect on the various statistics.

\smallskip\noindent\textbf{Example.} 
Let $D$ be the $2$-Dyck path of height 12 depicted in Figure \ref{mhmfig}.
We have
\[ \gamma(D)=(0,0,1,3,5,1,2,3,5,5,4,1);\ \ 
 \area(D)=30;\ \ h(D)=41. \]
Doing frequency counts on the entries of $\gamma$, we compute
\[ v=(v_0,v_1,v_2,v_3,v_4,v_5)=(2,3,1,2,1,3). \]
Given $v$, we can draw the bounce path shown in Figure \ref{brectangles} 
with $5$ empty rectangles above it. Now, we compute the words $w_i$:
\[ w_1=1000;\ \ w_2=11101;\ \ w_3=01101;\ \ w_4=110;\ \ w_5=01001. \]
Using these words to fill in the partial paths, we obtain
the path $D'$ in Figure \ref{mbouncefig}, which has
$b(D)=30$ and $\area(D)=41$.

Here is a mild simplification of the bijection.
Leave the first $1$ at the beginning of each $w_i$ instead of erasing it.
Then the $w_i$ tell us how to construct the partial paths in the full
rectangles $R_i$ (rather than the shortened rectangles $R_i'$).
Every such partial path begins with an east step, as required
by the bouncing rules.

\smallskip\noindent\textbf{Description of $\psi$.}
Let $D$ be an $m$-Dyck path of height $n$. To find the path $\psi(D)$:
\begin{itemize}
\item Draw the bounce path derived from $D$ according to the bouncing
rules (see \S \ref{subsec:mbouncestat}). Let $v=(v_0,\ldots,v_s)$ be the 
lengths of the vertical moves in this bounce path. 
\item Let $R_1,\ldots,R_s$ be the rectangular regions above the bounce 
path. These regions contain partial paths going from the southwest
corner to the northeast corner. For $1\leq i\leq s$, find the word
$w_i$ by traversing the partial path in $R_i$ and writing 
a one for each east step and a zero for each north step.
Note that every $w_i$ has first symbol one.
\item Build up $\gamma$ as follows. Start with a string of $v_0$
zeroes. For $i=1,2,\ldots,s$, insert $v_i$ copies of $i$ into
the current string $\gamma$ according to $w_i$. More explicitly, read $w_i$
left to right. When a $1$ is encountered, scan $\gamma$ from
left to right for the next occurrence of a symbol in $\{i-1,\ldots,i-m\}$.
When a $0$ is encountered, place an $i$ in the gap immediately
to the right of the current symbol in $\gamma$. Continue until
all symbols $i$ have been inserted.
\item Use $\gamma$ to draw the picture of a new $m$-Dyck path $D'$
of height $n$, by placing $\gamma_i$ area cells in the $i$'th
row of the figure. Since $\gamma\in {\cal G}_n^{(m)}$, the resulting picture
will be a valid path.
\end{itemize}

\smallskip\noindent\textbf{Example.}
Let $D$ be the $3$-Dyck path of height $8$ shown in Figure \ref{bounce3}.
{}From the bounce path drawn in that figure, we find that
\[ v=(v_0,\ldots,v_9)=(1,1,1,1,2,0,0,1,1). \]
Examining the rectangles above the bounce path
(several of which happen to be empty or have height zero),
we get the words $w_i$:
\[ w_1=10;\ w_2=110;\ w_3=1110;\ w_4=10011;\ 
 w_5=1111;\ w_6=111;\ w_7=110;\ w_8=10. \]
Now, build up the vector $\gamma$ as follows:
\begin{itemize}
\item Initially, $\gamma=0$ (since $v_0=1$).
\item Use $w_1=10$ to insert one $1$ into $\gamma$ to get
   $\gamma=01$.
\item Use $w_2=110$ to insert one $2$ into $\gamma$ to get
   $\gamma=012$.
\item Use $w_3=1110$ to insert one $3$ into $\gamma$ to get
   $\gamma=0123$.
\item Use $w_4=10011$ to insert two $4$'s into $\gamma$ to get
   $\gamma=014423$.
\item Use $w_5=1111$ to insert zero $5$'s into $\gamma$ to get
   $\gamma=014423$.
\item Use $w_6=111$ to insert zero $6$'s into $\gamma$ to get
   $\gamma=014423$.
\item Use $w_7=110$ to insert one $7$ into $\gamma$ to get
   $\gamma=0144723$.
\item Use $w_8=10$ to insert one $8$ into $\gamma$ to get
   $\gamma=01447823$.
\end{itemize}
Thus, the image path $D'$ is the unique $3$-Dyck path of height
8 such that $\gamma(D')=(0,1,4,4,7,8,2,3)$. $D'$ is pictured
in Figure \ref{psiimage}.

\begin{figure}[ht]
\begin{center}
\epsfig{file=psiimage.eps}
\caption{The image $\psi(D)$ for the path $D$ from Figure 7.}
\label{psiimage}
\end{center}
\end{figure}

As this example indicates, the presence of vertical moves of length zero
does not alter the validity of the preceding proofs and bijections.

\smallskip\noindent\textbf{Remark.}
The main difficulty involved in the combinatorial investigation
of the original $q,t$-Catalan sequence $OC_n(q,t)$ was discovering
the two statistics \emph{dinv} and \emph{bounce} defined
in \S \ref{subsec:combcat}. The area statistic, on the other hand,
is quite natural to consider once one notices that $OC_n(1,1)$
counts the number of Dyck paths of height $n$. Similar comments
apply to the higher $q,t$-Catalan sequences.

Having introduced the bijections $\phi$ and $\psi=\phi^{-1}$,
we can consider the problem of finding these statistics in a new light.
It is natural to count Dyck paths (or $m$-Dyck paths) by constructing
the associated $\gamma$-sequences through successive insertion
of zeroes, ones, twos, etc., as done in \S \ref{subsec:HCformula}.
The map $\phi$ arises by representing the insertion choices
geometrically as paths inside rectangles and positioning these rectangles 
in a nice way (as in Figure \ref{brectangles}). The remarkable coincidence
is that the resulting picture is another $m$-Dyck path.

We may thus regard the area statistic and the map $\phi$
as the ``most fundamental'' concepts. Then the two new statistics
$h$ and $b$ can be ``guessed'' by simply looking at what happens
to the area statistic when we apply $\phi$ (or $\phi^{-1}$)!
We find that $\phi$ sends area to the bounce statistic $b$,
and $\phi^{-1}$ sends area to the generalized Haiman statistic $h$.
%We get two different statistics that can be paired with area
%because $\phi$ is \emph{not} an involution.

This suggests a possible approach to other problems
in which there are two variables with the same univariate
distribution, but a combinatorial interpretation is only known for one
of the variables. Finding a combinatorial interpretation for the
Kostka-Macdonald coefficients (see \cite{macbook})
provides an example of such a problem.
There, the $q$-statistic is known (the so-called ``cocharge statistic''
on tableaux), but the $t$-statistic has not been discovered.  
For other examples of this technique of ``guessing'' new statistics,
consult \cite{mythesis}.

\section{Recursions for $C_n^{(m)}(q,t)$}
\label{sec:recursions}
In this section, we prove several recursions for $C_n^{(m)}(q,t)$
and related sequences (see (\ref{messyErec}) and (\ref{nicerec})). 
Of course, the same recursions hold for $HC_n^{(m)}(q,t)$. 
These recursions are more convenient 
for some purposes than the summation formula given 
in \S \ref{subsec:sumformula}.  As an example, we use the
recursion to prove a formula for $C_n^{(m)}(q,1/q)$
which shows that $C_n^{(m)}(q,1/q)=OC_n^{(m)}(q,1/q)$
(see (\ref{monster1}) and (\ref{eq:OCmn-specz})).

We begin by describing Haglund's recursion for $C_n(q,t)$ 
(see \cite{haglundstat}). This recursion is a key
ingredient in the long proof that $SC_n(q,t)=C_n(q,t)$.
We will just give the idea of the proof here; full details 
may be found in \cite{nablaproof,PNAS-GH}.

\subsection{Haglund's Recursion for $C_n(q,t)$}
\label{subsec:haglundrec}
Fix $n$. Let ${\cal F}_{n,s}$ denote the set of
Dyck paths of height $n$ that terminate in exactly $s$ east steps.
For such a path, the length of the first bounce step will be
$s$ (see Figure \ref{haglundpic} below).  Define
\[ F_{n,s}(q,t)=\sum_{D\in{\cal F}_{n,s}} q^{\area(D)}
  t^{\bounce(D)}. \]
These generating functions are related to $C_n(q,t)$ by the identities
\[ C_n(q,t)=\sum_{s=1}^n F_{n,s}(q,t)  \]
\[  t^n C_n(q,t)= F_{n+1,1}(q,t). \]
The first identity follows by classifying Dyck paths of height $n$
by the number $s$ of east steps in the topmost row.
To prove the second identity, augment the diagram of a Dyck path 
of height $n$ by adding a new top row with no area cells. The result is
a Dyck path of height $n+1$ terminating in one east step
preceded by one north step. All elements of ${\cal F}_{n+1,1}$
arise uniquely in this way. The bounce path derived
from this augmented Dyck path starts with a bounce of size $1$ contributing
$n$ to the bounce statistic, and afterwards bounces in the same
way that the original bounce path did. See Figure \ref{haglundtoprow},
and compare to Figure \ref{dyck2}.

\begin{figure}[ht]
\begin{center}
\epsfig{file=haglundtoprow.eps}
\caption{Adding an empty top row to a Dyck path.}
\label{haglundtoprow}
\end{center}
\end{figure}

\smallskip\noindent\textbf{Theorem (Haglund, \cite{haglundstat}).}
The generating functions $F_{n,s}$ satisfy the recursion
\begin{equation}\label{haglundrecursion}
 F_{n,s}(q,t)=t^{n-s}q^{s(s-1)/2}\sum_{r=1}^{n-s}
   \dqbin{r+s-1}{r,s-1} F_{n-s,r}(q,t) \mbox{ for $1\leq s<n$} 
\end{equation}
with initial condition $F_{n,n}(q,t)=q^{n(n-1)/2}$.

\smallskip\noindent\textbf{Remark.} Note that the initial condition
and recursion uniquely determine the polynomials $F_{n,s}(q,t)$
and allow these polynomials to be computed rapidly.

\smallskip\noindent\textbf{Proof.}
Consider the initial condition first. If $D\in{\cal F}_{n,n}$,
then $D$ is a Dyck path of height $n$ terminating in exactly $n$
east steps in the top row. This can only happen if $D$ is the
path consisting of $n$ north steps followed by $n$ east steps.
Then $\area(D)=n(n-1)/2$ (since $\gamma(D)=(0,1,\ldots,n-1)$)
and $\bounce(D)=0$ (since the only bounce hits the diagonal
at $(0,0)$). So, $F_{n,n}(q,t)=q^{n(n-1)/2}t^0$ as claimed.

The recursion for $F_{n,s}$ follows by ``removing the first bounce''
from a Dyck path to obtain a smaller Dyck path of height $n-s$.
More precisely, let $D\in{\cal F}_{n,s}$. Then $D$ ends in $s$
east steps, so the derived bounce path starts with a bounce of size $s$
ending at $(n-s,n-s)$. See Figure \ref{haglundpic}.
If we ignore the top $s$ rows of the figure, we see a smaller
Dyck path $D'$ of height $n-s$. Observe that the derived bounce path 
of $D'$ is just the bounce path of $D$ with the first bounce removed.

\begin{figure}[ht]
\begin{center}
\epsfig{file=haglundpic.eps}
\caption{Proving the recursion by removing the first bounce.}
\label{haglundpic}
\end{center}
\end{figure}

We can uniquely construct a path $D\in{\cal F}_{n,s}$ as follows.
Choose a number $r\in\{1,2,\ldots,n-s\}$. Given $r$, build $D$
by making a sequence of choices. First, choose a path $D'\in {\cal F}_{n-s,r}$.
The generating function for this choice is $F_{n-s,r}(q,t)$.
Second, draw a vertical and horizontal segment to create a triangle
with vertices $(n-s,n-s)$ and $(n-s,n)$ and $(n,n)$.  This triangle
adds $s(s-1)/2$ area cells to the path being constructed, giving
a factor $q^{s(s-1)/2}$.  Also, the path $D$ will have a new bounce
going from $(n,n)$ to $(n-s,n-s)$, so we get a contribution of
$t^{n-s}$ as well.
Third, draw a subpath ending with a north step in the rectangular region
above the top row of $D'$ and left of the triangle just drawn.
This subpath does not change the bounce statistic (since it ends in a north 
step), but the area increases by the number of cells beneath the
subpath in its rectangle. The generating
function for this choice is thus $\tqbin{r+s-1}{r,s-1}$.
The recursion follows immediately from the sum and product
rules for generating functionons (\cite{benderbook}, Ch. 10).

\smallskip\noindent\textbf{Proving that $C_n(q,t)=SC_n(q,t)$.}\\
In \cite{nablaproof,PNAS-GH}, Garsia and Haglund used the recursion
(\ref{haglundrecursion}) to prove that $C_n(q,t)=SC_n(q,t)$ for all $n$.
More specifically, they defined
\[ Q_{n,s}(q,t)=\left.t^{n-s}q^{s(s-1)/2} \nabla(e_{n-s}[X(1+q+\cdots+q^{s-1})])
\right|_{s_{1^{n-s}}}. \]
Here, $X$ is a formal infinite alphabet $X=x_1+x_2+\cdots$,
and the square brackets denote plethystic substitution;
in particular $e_n[X]=e_n$.  Garsia and Haglund showed that
\[ Q_{n,s}(q,t)=t^{n-s} q^{s(s-1)/2} \sum_{r=1}^{n-s}
 \dqbin{r+s-1}{r,s-1} Q_{n-s,r}(q,t) \mbox{ and }
Q_{n,n}(q,t)=q^{n(n-1)/2}. \]
In other words, $Q_{n,s}$ satisfies the same recursion and initial 
condition that $F_{n,s}$ does. By uniqueness, $Q_{n,s}=F_{n,s}$
for all $n$ and $s$. In particular,
\[ C_n(q,t)=F_{n+1,1}(q,t)/t^n=Q_{n+1,1}(q,t)/t^n=\left.\nabla(e_n[X])
\right|_{s_{1^n}}=SC_n(q,t). \]

\subsection{A recursion based on removing the first bounce}
\label{subsec:firstbounce}
Our goal here is to modify the idea in the proof of Haglund's
recursion to get a recursion for $C_n^{(m)}(q,t)$.
The main difficulty is that the bounce path depends
on the prior bouncing history when $m>1$, so that we cannot
simply remove the first bounce and restart ``from scratch.''
Consequently, we must add more subscripts that keep track
of the lengths of the first $m$ vertical moves in the bounce path.

Fix $m>1$. Define ${\cal F}_{n;v_0,v_1,\ldots,v_{m-1}}$ to
be the collection of $m$-Dyck paths of height $n$ whose
derived bounce paths start with vertical moves of lengths
$v_0,v_1,\ldots,v_{m-1}$, in that order. Define $F_{n;v_0,\ldots,v_{m-1}}(q,t)$
to be the sum of $q^{\area(D)}t^{b(D)}$ over all paths
$D\in{\cal F}_{n;v_0,\ldots,v_{m-1}}$. (An empty sum is defined to
be zero.) We make the following observations about these definitions.
\begin{itemize}
\item If ${\cal F}_{n;v_0,v_1,\ldots,v_{m-1}}$ is a nonempty collection
of paths, then we must have $v_0>0$, $v_i\geq 0$ for $i>0$,
and $v_0+\cdots+v_{m-1}\leq n$.
\item If $v_0=n$ and $v_i=0$ for $i>0$, then
${\cal F}_{n;n,0,\ldots,0}$ consists of the single path $D$ that
goes north $n$ steps and then east $mn$ steps. Hence,
$F_{n;n,0,\ldots,0}(q,t)=q^{mn(n-1)/2}t^0$.
\item Consider the collection ${\cal F}_{n+1;1,0,\ldots,0}$.
A path $D$ in this collection starts by going north one unit
and then east $m$ units (since $v_1=\cdots=v_{m-1}=0$).
At this point, $D$ has returned to the diagonal $x=my$.
If we look at the rest of the path beyond this point, we get
an arbitrary $m$-Dyck path $D'$ of height $n$. Also, the bounce
path for $D'$ is the same as the latter part of the bounce path
for $D$ (starting with $v_m$). Note that the prior history in $D$
is immaterial, since $v_{m-1}=\cdots=v_1=0$.  See Figure \ref{extrabottomrow}.
We conclude that
\[ t^{mn}C_n^{(m)}(q,t)=F_{n+1;1,0,\ldots,0}(q,t). \]
The extra factor of $t^{mn}$ accounts for the contribution
of the first $m$ bounces to $b(D)$, which is not present in $b(D')$.

\begin{figure}[ht]
\begin{center}
\epsfig{file=extrabottomrow.eps}
\caption{Removing a trivial bottom row of an $m$-Dyck path.}
\label{extrabottomrow}
\end{center}
\end{figure}

\item There is a version of the formula (\ref{sumformula}) for
$F_{n;v_0,\ldots,v_{m-1}}(q,t)$. Specifically,
\begin{eqnarray*}
\lefteqn{F_{n;v_0,\ldots,v_{m-1}}(q,t)=}
\\ & & \sum_{(v_m,v_{m+1},\ldots)}
t^{\sum_{i\geq 0} i v_i} q^{m\sum_{i\geq 0}
% \frac{1}{2} v_i(v_i-1)}
\tbinom{v_i}{2}}
 \prod_{i\geq 1} q^{v_i\sum_{j=1}^{m} (m-j)v_{i-j}}
\dqbin{v_i+v_{i-1}+\cdots+v_{i-m}-1}
     {v_i,v_{i-1}+\cdots+v_{i-m}-1}. 
\end{eqnarray*}
This equation follows immediately from the combinatorial interpretation
of the summation index $v=(v_0,v_1\ldots)$ appearing in (\ref{sumformula})
as the lengths of the vertical segments in the bounce path.
Since $v_0,\ldots,v_{m-1}$ are fixed in advance, we need only
sum over the remaining segments $v_m,v_{m+1},\ldots$.
\end{itemize}

To state the new recursion, it is convenient to introduce a modified
version of the generating functions $F_{n;v_0,\ldots,v_{m-1}}(q,t)$.
Intuitively, we need to remove the influence of $v_0$ on the 
future bouncing history to obtain a recursion.  Assume that $v_0>0$ first.
Define ${\cal E}_{n;v_0,\ldots,v_{m-1}}$ to be the collection
of all $m$-Dyck paths $D$ of height $n$ with the following properties.
First, the bounce path derived from $D$ starts with vertical moves
of lengths $v_0,\ldots,v_{m-1}$. Second, the first $m-1$ rectangles
$R_1,\ldots,R_{m-1}$ above the bounce path of $D$ (see Figure \ref{brectangles})
are all empty. This means that the subpath in each rectangle goes
all the way east before turning north, so that there are no
area cells in the rectangle. Then define
\[ E_{n;v_0,\ldots,v_{m-1}}(q,t)=
   \sum_{D\in{\cal E}_{n;v_0,\ldots,v_{m-1}}} q^{\area(D)}t^{b(D)}. \]

By filling the empty rectangles $R_1,\ldots,R_{m-1}$ in an object
$D\in {\cal E}_{n;v_0,\ldots,v_{m-1}}$ according to the bouncing rules,
we deduce that
\begin{equation}\label{EvsF}
 F_{n;v_0,\ldots,v_{m-1}}(q,t)=
   E_{n;v_0,\ldots,v_{m-1}}(q,t)\prod_{i=1}^{m-1}
  \dqbin{v_i+v_{i-1}+\cdots+v_{i-m}-1}
        {v_i,v_{i-1}+\cdots+v_{i-m}-1} \mbox{ when $v_0>0$.} 
\end{equation}
This relation gives an exact formula for
$E_{n;v_0,\ldots,v_{m-1}}(q,t)$ when $v_0>0$:
\begin{equation}\label{messyE1}
\begin{split}
\lefteqn{E_{n;v_0,\ldots,v_{m-1}}(q,t)=}
\\ & \sum_{(v_m,v_{m+1},\ldots)}
t^{\sum_{i\geq 0} i v_i} q^{m\sum_{i\geq 0}
 \frac{1}{2} v_i(v_i-1)}
 \prod_{i\geq 1} q^{v_i\sum_{j=1}^{m\wedge i} (m-j)v_{i-j}}
 \prod_{i\geq m}
\dqbin{v_i+v_{i-1}+\cdots+v_{i-m}-1}
     {v_i,v_{i-1}+\cdots+v_{i-m}-1}. 
\end{split}
\end{equation}
Here, we have written $m\wedge i$ to denote the minimum of $m$ and $i$.
Note that the validity of equation (\ref{messyE1}) does not depend
on the earlier convention that $v_i=0$ for all negative $i$.
Now, if $v_0=0$, we simply \emph{define} $E_{n;v_0,\ldots,v_{m-1}}(q,t)$
by formula (\ref{messyE1}). 

It follows from (\ref{EvsF}) that 
$E^{(m)}_{n+1;1,0,\ldots,0}(q,t)=F^{(m)}_{n+1;1,0,\ldots,0}(q,t)$.
Therefore,
\begin{equation}\label{CfromE}
 C^{(m)}_n(q,t)=t^{-mn}E^{(m)}_{n+1;1,0,\ldots,0}(q,t). 
\end{equation}

We can obtain a recursion for $E_{n;v_0,\ldots,v_{m-1}}$ by breaking up
the summation in (\ref{messyE1})
based on the value of $v_m$.  Consider a fixed choice
of $v_m$ in the range $\{0,1,\ldots,n-v_0-\cdots-v_{m-1}\}$.
Write down (\ref{messyE1}) with $n$ replaced by $n-v_0$ and 
$v_k$ replaced by $v_{k+1}$ for all $k\geq 0$:
\begin{equation}
E_{n-v_0;v_1,\ldots,v_m}(q,t) = \sum_{(v_{m+1},v_{m+2},\ldots)} \!\!\!\!
t^{\sum_{i\geq 0} i v_{i+1}} q^{\pow_1}
\prod_{i\geq m} \dqbin{v_{i+1}+v_{i}+\cdots+v_{i+1-m}-1}
     {v_{i+1},v_i+\cdots+v_{i+1-m}-1},
\end{equation}
\[ \pow_1= {m\sum_{i\geq 0} \binom{v_{i+1}}{2}}+
 \sum_{i\geq 1} {v_{i+1}\sum_{j=1}^{m\wedge i} (m-j)v_{i+1-j}}. \]
Replace $i$ by $i-1$ in this formula to get
\begin{equation}
\label{messyE2}
E_{n-v_0;v_1,\ldots,v_m}(q,t) = 
\sum_{(v_{m+1},v_{m+2},\ldots)} \!\!\!\!
t^{\sum_{i\geq 1} (i-1) v_i} q^{\pow_2}
\prod_{i>m} \dqbin{v_{i}+v_{i-1}+\cdots+v_{i-m}-1}
     {v_{i},v_{i-1}+\cdots+v_{i-m}-1},
\end{equation}
\[ \pow_2={m\sum_{i\geq 1} \binom{v_{i}}{2}}+
 \sum_{i\geq 2} {v_i\sum_{j=1}^{m\wedge (i-1)} (m-j)v_{i-j}}. \]
In the original formula for $E_{n;v_0,\ldots,v_{m-1}}$, we can
sum over $v_m$ first and then sum over the remaining $v_j$'s.
The resulting formula is:
\begin{equation}\label{messyE3}
\begin{split}
\lefteqn{E_{n;v_0,\ldots,v_{m-1}}(q,t)=} \\ &
\sum_{v_m=0}^{n-v_0-\cdots-v_{m-1}} \!\!\!\! % negative space!
\sum_{(v_{m+1},v_{m+2},\ldots)}
t^{\sum_{i\geq 0} i v_i} q^{\pow_3}
 \prod_{i\geq m} \dqbin{v_i+v_{i-1}+\cdots+v_{i-m}-1}
     {v_i,v_{i-1}+\cdots+v_{i-m}-1}, 
\end{split}
\end{equation}
\[ \pow_3= {m\sum_{i\geq 0} \binom{v_i}{2}}
 +\sum_{i\geq 1} {v_i\sum_{j=1}^{m\wedge i} (m-j)v_{i-j}}. \]
To go from the formula in (\ref{messyE2}) to the corresponding
summand in (\ref{messyE3}), we need to multiply the former by the expression
\[ t^{0v_0+v_1+v_2+v_3+\cdots} q^{m\binom{v_0}{2}}
 q^{v_1v_0(m-1)}\prod_{i=2}^m q^{v_i(m-i)v_0}
\dqbin{v_{m}+\cdots+v_0-1}{v_{m},v_{m-1}+\cdots+v_0-1}. \]
Doing this multiplication and adding over all choices of $v_m$,
we obtain the recursion
\begin{equation}\label{messyErec}
\begin{split}
\lefteqn{ E_{n;v_0,\ldots,v_{m-1}}(q,t)= }
\\& t^{n-v_0}q^{m\binom{v_0}{2}}
\prod_{i=1}^{m-1} q^{v_0v_i(m-i)}
\sum_{v_m=0}^{n-v_0-\cdots-v_{m-1}}
  \dqbin{v_m+\cdots+v_0-1}{v_m,v_{m-1}+\cdots+v_0-1}
  E_{n-v_0;v_1,\ldots,v_{m-1},v_m}(q,t).  
\end{split}\end{equation}
The initial conditions are
\[ E_{n;n,0,\ldots,0}(q,t)=q^{mn(n-1)/2}t^0 \]
\[ E_{n;0,0,\ldots,0}(q,t)=0. \]
Observe that we recover Haglund's original recursion when $m=1$.

It is hoped that (\ref{messyErec}) could be used to prove
the conjecture $C_n^{(m)}(q,t)=SC_n^{(m)}(q,t)$.
One difficulty is finding the analogues of
$E_{n;v_0,\ldots,v_{m-1}}$ in the symmetric function setting.
Computer experiments suggest that
\[ E_{n;v_0,0,\ldots,0}(q,t)=q^{mv_0(v_0-1)/2} t^{(m-1)(n-v_0)}
\left.\nabla^m(e_{n-v_0}[X(1+q+q^2+\cdots
+q^{v_0-1})])\right|_{s_{1^{n-v_0}}}, \]
However, we have not found a conjectured formula for the general 
$E_{n;v_0,\ldots,v_{m-1}}$ in terms of the nabla operator.

It is clear that we could perform a similar manipulation
of (\ref{sumformula}) to obtain a recursion based on removing
the last nontrivial vertical bounce $v_s$. The inductive
proof in \S \ref{subsec:HCformula} that (\ref{sumformula}) equals 
$HC_n^{(m)}(q,t)$ was based on this idea.  There is a slight added
complication because one must know $s$, not just $v_s$,
to determine the effect of removing the last bounce on $b(D)$.
On the other hand, $v_s$ only affects the dimensions of one nontrivial
rectangle in Figure \ref{brectangles}.

\subsection{Application: A Formula for the Specialization
$C^{(m)}_n(q,1/q)$}
\label{subsec:specz}
We now use the recursion of the preceding subsection
to derive an exact formula for the specialization
\[ E^{(m)}_{n;v_0,\ldots,v_{m-1}}(q,1/q). \]
In particular, using this formula together with (\ref{CfromE}),
we prove that
\[ q^{mn(n-1)/2}C^{(m)}_n(q,1/q)=\frac{1}{[mn+1]_q}\dqbin{mn+n}{mn,n}. \]
Garsia and Haiman proved the same formula for $OC^{(m)}_n(q,1/q)$
in \cite{origpaper}. It follows that
\[ C^{(m)}_n(q,1/q)=OC^{(m)}_n(q,1/q). \]

\medskip\noindent\textbf{The Formula for the $E$'s.}
Fix $m$, $N$, and $v=(v_0,\ldots,v_{m-1})$. Our formula
for $E^{(m)}_{N;v}(q,1/q)$ will involve various intermediate
quantities $A$, $B$, etc., depending on $N$, $m$, and $v$.
If the dependence on the variables needs to be made explicit,
we will write $A(N,m,v)$, $B(N,m,v)$, etc.

The basic formula is
\begin{equation}\label{monster1}
 E^{(m)}_{N;v}(q,1/q)=A_0-B_1-B_2-\cdots-B_m, 
\end{equation}
where $A_0$ and each $B_j$ is a certain $q$-binomial coefficient
multiplied by a certain power of $q$. Specifically, define
\begin{eqnarray*}
A = A(N,m,v_0,\ldots,v_{m-1}) &=& \dqbin{
(m+1)N-1-\sum_{k=0}^{m-1} (m-k)v_k}{N-\sum_{k=0}^{m-1}v_k} \\
B = B(N,m,v_0,\ldots,v_{m-1}) &=& \dqbin{
(m+1)N-1-\sum_{k=0}^{m-1} (m-k)v_k}{N-1-\sum_{k=0}^{m-1}v_k} \\
P_0=P_0(N,m,v_0,\ldots,v_{m-1}) &=& -\frac{m}{2}(N^2+N)
   +\left[m\left(\sum_{k=0}^{m-1} v_k\right)-(m-1)\right]N \\ &&
+\sum_{k=0}^{m-2} (m-1-k)v_k + \sum_{0\leq j<k\leq m-1} (j-k)v_j v_k \\
P_j=P_j(N,m,v_0,\ldots,v_{m-1}) &=& 
 v_{m-1}+(j-1)N \\ & & -\sum_{\ell=0}^{m-2}\min(j-1,m-2-\ell)v_{\ell}
\ \ \ (1\leq j\leq m) 
\end{eqnarray*}
Finally, define
\begin{eqnarray*}
A_0=A_0(N,m,v_0,\ldots,v_{m-1}) &=& Aq^{P_0} \\
B_j=B_j(N,m,v_0,\ldots,v_{m-1}) &=& Bq^{P_0+P_j} 
\ \ \ (1\leq j\leq m) 
\end{eqnarray*}

\medskip\noindent\textbf{Examples.} (1) Let $m=1$ and $v_0=w$. Then
\[ E^{(1)}_{N;w}(q,1/q)=q^{-(N^2+N)/2+wN}\left\{
\dqbin{2N-w-1}{N-w,N-1}-q^w \dqbin{2N-w-1}{N-w-1,N}\right\}. \]
This is equivalent to a formula for $F_{n,s}(q,1/q)$ proved
by Haglund in \cite{haglundstat}.
\\ (2) Let $m=2$, $v_0=w$, $v_1=x$. Then
\begin{eqnarray*}
 E^{(2)}_{N;w,x}(q,1/q) &=& q^{\pow_2}\left\{\dqbin{3N-2w-x-1}
{N-w-x,2N-w-1}\right. \\ & &\left.
 -(q^x+q^{N+x})\dqbin{3N-2w-x-1}{N-w-x-1,2N-w}\right\}, 
 \\ \pow_2&=&-(N^2+N)+(2w+2x-1)N-w(x-1). 
\end{eqnarray*}
\\ (3) Let $m=3$, $v_0=w$, $v_1=x$, $v_2=y$. Then
\begin{eqnarray*}
 E^{(3)}_{N;w,x,y}(q,1/q) &=& q^{\pow_3}\left\{\dqbin{4N-3w-2x-y-1}
{N-w-x-y,3N-2w-x-1} \right.\\ & &\left. -(q^y+q^{y+N-w}+q^{y+2N-w})
\dqbin{4N-3w-2x-y-1}{N-w-x-y-1,3N-2w-x}\right\},  \\
 \pow_3 &=& -3(N^2+N)/2+(3w+3x+3y-2)N+(y-1)(-2w-x)-wx. 
\end{eqnarray*}
\\ (4) Let $m=5$ and $(v_0,v_1,v_2,v_3,v_4)=(v,w,x,y,z)$. Then
\[ \begin{array}{ccl}
 P_1 &=& z \\
 P_2 &=& z+N-v-w-x \\
 P_3 &=& z+2N-2v-2w-x \\
 P_4 &=& z+3N-3v-2w-x \\
 P_5 &=& z+4N-3v-2w-x.
\end{array} \]

\medskip\noindent\textbf{Proof of the Formula.}
To prove (\ref{monster1}), we need to check that
the right side satisfies the same initial conditions
and recursion that the specialization $E^{(m)}_{N;v}(q,1/q)$
satisfies.  This check requires an inordinate amount of
tedious manipulations of powers of $q$. Therefore, we only give
an outline of the proof here, omitting routine algebraic
manipulations. We refer the interested reader to the
author's thesis \cite{mythesis} for more details.

\medskip\emph{Step 1.} We begin by establishing the following identity,
valid for nonnegative integers $C$, $D$, and $E$:
\begin{equation}\label{comb-lemma1}
 \sum_{i=0}^{D-E} \dqbin{C+i}{C,i}\dqbin{D-i}{E,D-i-E}
q^{(E+1)i}=\dqbin{C+D+1}{D-E,C+1+E}.
\end{equation}
Recall from \S \ref{subsec:sumformula} that
\[ \dqbin{a+b}{a,b}=\sum_{P\in{\cal P}_{a,b}} q^{\area(P)}
  =\sum_{P\in{\cal P}_{a,b}} q^{ab-\area(P)}, \]
where ${\cal P}_{a,b}$ is the set of lattice paths
contained in an $a\times b$ rectangle. Using this fact,
we can prove (\ref{comb-lemma1}) by drawing a picture.
See Figure \ref{fig:lemma1-pic}. 

\begin{figure}[ht]
\begin{center}
\epsfig{file=lemma1-pic.eps}
\caption{Picture used to prove (\ref{comb-lemma1}).}
\label{fig:lemma1-pic}
\end{center}
\end{figure}

We classify paths $P$ contained in a rectangle of height $C+E+1$ and width $D-E$
based on what happens in row $(C+1)$ from the top.
This row contains exactly one vertical step $s$ of $P$;
let $i$ denote the distance of this vertical step from the left edge.
Evidently, $0\leq i\leq D-E$. Given $i$, we can uniquely construct
such a path $P$ as follows. First, choose a subpath $P_1$ 
in the rectangle $R_1$ northwest
of $s$, which has height $C$ and width $i$. Second,
choose a subpath $P_2$ in the rectangle $R_2$ southeast of $s$, which has 
height $E$ and width $D-E-i$.  Then $P$ is the concatenation
of $P_1$ and the vertical step $s$ and $P_2$.

Assume that the power of $q$ records the area below the path $P$.
This area is the sum of the area below $P_1$ inside $R_1$,
the area below $P_2$ inside $R_2$, and the full area of the southwest
rectangle of height $E+1$ and width $i$. These three pieces
of the area are accounted for by the factors
$\tqbin{C+i}{C,i}$, $\tqbin{D-i}{E,D-i-E}$, and $q^{(E+1)i}$,
respectively. Adding over all choices of $i$, we immediately obtain
(\ref{comb-lemma1}).

\medskip\emph{Step 2.} We prove the identity
\begin{equation}\label{comb-lemma2}
q^{-C}\left\{\dqbin{C+D}{C,D}-\dqbin{C+D}{C-1,D+1}\right\}=
\dqbin{C+D}{C,D}-q^{D-C+1}\dqbin{C+D}{C-1,D+1}. 
\end{equation}
This identity is equivalent to the relation
\[ \dqbin{C+D}{C,D}+q^{D+1}\dqbin{C+D}{C-1,D+1}
  =q^C \dqbin{C+D}{C,D}+\dqbin{C+D}{C-1,D+1}, \]
which can also be proved by drawing a picture. See Figure
\ref{fig:lemma2-pic}.

\begin{figure}[ht]
\begin{center}
\epsfig{file=lemma2-pic.eps}
\caption{Picture used to prove (\ref{comb-lemma2}).}
\label{fig:lemma2-pic}
\end{center}
\end{figure}

Here, the power of $q$ records the area above a path
that goes from northwest to southeast in a rectangle of
height $C$ and width $D+1$. The left side classifies
such paths by their \emph{initial} step at the northwest corner.
If this step is horizontal, the remainder of the path
lies in a rectangle of height $C$ and width $D$,
giving the term $\dqbin{C+D}{C,D}$. If this step is vertical,
the remainder of the path lies in a rectangle of height $C-1$
and width $D+1$, giving the term $\dqbin{C+D}{C-1,D+1}$.
However, we must also multiply by $q^{D+1}$ to account for
the $D+1$ area cells in the top row of the original rectangle.

The right side classifies the paths by their \emph{final} step
at the southeast corner.
If this step is horizontal, the remainder of the path
lies in a rectangle of height $C$ and width $D$,
giving the term $\dqbin{C+D}{C,D}$. 
However, we must also multiply by $q^{C}$ to account for
the $C$ area cells in the rightmost column of the original rectangle.
If the final step is vertical,
the remainder of the path lies in a rectangle of height $C-1$
and width $D+1$, giving the term $\dqbin{C+D}{C-1,D+1}$.

\medskip\noindent\emph{Step 3.} The next step is to check
that the right side of (\ref{monster1}) satisfies the specialized
initial conditions
 \[ E^{(m)}_{N;N,0,\ldots,0}(q,1/q)=q^{mn(n-1)/2} \]
 \[ E^{(m)}_{N;0,0,\ldots,0}(q,1/q)=0. \]
We leave this algebraic manipulation to the reader.

\medskip\noindent\emph{Step 4.} The next step is to
check that the right side of (\ref{monster1}) satisfies the
recursion (\ref{messyErec}) with $t$ specialized to $1/q$.
After setting $t=1/q$ and simplifying, this recursion can be written
\begin{equation}\label{spec-rec}
E^{(m)}_{N;v_0,\ldots,v_{m-1}}(q,1/q)=q^{\pow}\sum_{i=0}^{N-v_0-\ldots-v_{m-1}}
\dqbin{C+i}{C,i} E^{(m)}_{N-v_0;v_1,v_2\ldots,v_{m-1},i}(q,1/q),
\end{equation}
where
\begin{eqnarray*}
 \pow &=& v_0-N+m(v_0^2-v_0)/2+\sum_{k=1}^{m-1} (m-k)v_0v_k  \\
 C &=& v_0+v_1+\cdots+v_{m-1}-1
\end{eqnarray*}
The proof will be finished if we can show this same relation
holds with the $E$'s replaced by the appropriate formulas
from the right side of (\ref{monster1}). Specifically, 
write $A'$ for $A(N,m,v_0,\ldots,v_{m-1})$,
write $P_j'$ for $P_j(N,m,v_0,\ldots,v_{m-1})$, and so forth.
Write $A''$ for $A(N-v_0,m,v_1,\ldots,v_{m-1},i)$,
write $P_j''$ for $P_j(N-v_0,m,v_1,\ldots,v_{m-1},i)$, and so forth.
Then we must show that the quantity
\begin{equation}\label{left-spec}
 A_0'-B_1'-B_2'-\cdots-B_m' 
\end{equation}
is equal to the quantity
\begin{equation}\label{right-spec}
 q^{\pow}\sum_{i=0}^{N-v_0-\ldots-v_{m-1}}
\dqbin{C+i}{C,i}\left(A_0''-B_1''-B_2''-\cdots-B_m''\right). 
\end{equation}
To show this, we write the latter expression as the sum of $m+1$
smaller expressions, namely
\[ q^{\pow}\sum_{i=0}^{N-v_0-\ldots-v_{m-1}}
\dqbin{C+i}{C,i}A_0''  \]
and  (for $1\leq j\leq m$)
\[ q^{\pow}\sum_{i=0}^{N-v_0-\ldots-v_{m-1}}
\dqbin{C+i}{C,i}(-B_j''). \]
Each of these $m+1$ expressions can be evaluated (see below) using the
lemma from Step 1. The resulting sum is \emph{almost} the desired quantity
\[ A_0'-B_1'-B_2'-\cdots-B_m'. \]
More specifically, for $2\leq j\leq m$, the expression involving $-B_j''$
will evaluate to $-B_{j-1}'$. On the other hand, the expression
involving $-B_1''$ will evaluate to $-B_m'$ times an unwanted power of $q$.
Similarly, the expression
involving $A_0''$ will evaluate to $A_0'$ times another unwanted power of $q$.
Finally, the lemma from step 2 will show that these last two terms are
in fact equal to $A_0'-B_m'$ without the unwanted powers of $q$!
This will complete the proof of the formula (\ref{monster1}).

\medskip\noindent\emph{Step 5.} We indicate how to evaluate
the expression
\begin{equation}\label{step5-expr}
 q^{\pow}\sum_{i=0}^{N-v_0-\ldots-v_{m-1}} \dqbin{C+i}{C,i}(A_0'') 
\end{equation}
from Step 4. The final answer will be $q^{-(N-v_0-\cdots-v_{m-1})}A_0'$.

One must first verify the algebraic identity
\begin{equation}\label{P0-relation}
 P_0''+\pow=P_0'-(N-v_0-\cdots-v_{m-1})
+i\left[mN-\sum_{k=0}^{m-1}(m-k)v_k\right].
\end{equation}
Using this identity and expanding the definition of $A_0''$,
the expression (\ref{step5-expr}) can be written
\[ q^{P_0'-(N-v_0-\cdots-v_{m-1})}\sum_{i=0}^{D-E}
\dqbin{C+i}{C,i}\dqbin{D-i}{E,D-E-i} q^{i(E+1)}, \]
where
\begin{eqnarray*}
 D &=& (m+1)N-1-\sum_{k=0}^{m-1} (m+1-k)v_k \\
 E &=& mN-1-\sum_{k=0}^{m-1} (m-k)v_k \\
D-E &=& N-v_0-v_1-\cdots-v_{m-1}
\end{eqnarray*}
Using the identity from Step 1, this new expression becomes
\[ q^{-(N-v_0-\cdots-v_{m-1})}q^{P_0'}\dqbin{C+D+1}{D-E,C+1+E}
  =q^{-(N-v_0-\cdots-v_{m-1})}A_0'. \]

\medskip\noindent\emph{Step 6.} We indicate how to evaluate
the expression
\begin{equation}\label{step6-expr}
 q^{\pow}\sum_{i=0}^{N-v_0-\ldots-v_{m-1}} \dqbin{C+i}{C,i}(-B_j'') 
\end{equation}
from Step 4. The answer will be $-B_{j-1}'$ for $j>1$;
it will be $-B'q^{P_0'-(N-v_0-\cdots-v_{m-1})}$ for $j=1$.

The calculation is similar to the one in Step 5.
Using the definition of $B_j''$ and 
and the identity (\ref{P0-relation}),
we can rewrite (\ref{step6-expr}) as
\begin{equation}\label{step6-2}
 -q^{P_0'-(N-v_0-\cdots-v_{m-1})}\sum_{i=0}^{D-E+1}
\dqbin{C+i}{C,i}\dqbin{D-i}{E,D-E-i} q^{iE}q^{P_j''}, 
\end{equation}
where we now set
\begin{eqnarray*}
 D &=& (m+1)N-1-\sum_{k=0}^{m-1} (m+1-k)v_k \\
 E &=& mN-\sum_{k=0}^{m-1} (m-k)v_k \\
D-E &=& N-v_0-v_1-\cdots-v_{m-1}-1.
\end{eqnarray*}
The summand where $i=D-E+1$ is zero, so we may adjust the upper
limit of the sum to be $i=D-E$ instead.
To continue simplifying, one must first verify the identity
\begin{eqnarray*}
 %P_1'' &=& i \\
 %P_m' &=& v_{m-1}+(m-1)N-\sum_{k=0}^{m-2} (m-2-k)v_k \\
 P_{j-1}' &=& P_j''-i-(N-v_0-\cdots-v_{m-1}) \ \ \ \ (j>1).
\end{eqnarray*}

Assume $j>1$ first. Using the last identity to eliminate $P_j''$, 
the expression (\ref{step6-2}) becomes
\[ -q^{P_0'+P_{j-1}'}\sum_{i=0}^{D-E}
\dqbin{C+i}{C,i}\dqbin{D-i}{E,D-E-i} q^{i(E+1)}.  \]
Using the identity from Step 1, the sum (without the outside
power of $q$) evaluates to $B'$.
Thus, when $j>1$, the expression (\ref{step6-expr}) evaluates
to $-B_{j-1}'$ as claimed.
 
Now assume $j=1$. Since $P_1''=i$, the expression (\ref{step6-2}) becomes
\[ -q^{P_0'-(N-v_0-\cdots-v_{m-1})}\sum_{i=0}^{D-E}
\dqbin{C+i}{C,i}\dqbin{D-i}{E,D-E-i} q^{i(E+1)}. \]
Using the identity from Step 1, this becomes
\[ -q^{P_0'-(N-v_0-\cdots-v_{m-1})}B' \]
as claimed.

\medskip\noindent\emph{Step 7.} Let us recap the preceding calculations.
We have evaluated the expression (\ref{right-spec}),
hoping to obtain the answer
\[ (A_0'-B_m')-B_1'-B_2'-\cdots-B_{m-1}'  \]
from (\ref{left-spec}).
Instead, we obtained the answer
\[ q^{-(N-v_0-\cdots-v_{m-1})}(A'q^{P_0'}-B'q^{P_0'})
-B_1'-B_2'-\cdots-B_{m-1}'.  \]
Now, use the identity from Step 2, setting
\begin{eqnarray*}
 C &=& N-v_0-\cdots-v_{m-1} \\
 D &=& mN-1-\sum_{k=0}^{m-1} (m-1-k)v_k.
\end{eqnarray*}
The result is
\[ q^{-(N-v_0-\cdots-v_{m-1})}(A'-B')=A'-q^{P_m'}B', \]
since one can check that $P_m'=D-C+1$ here.
Multiplying by $q^{P_0'}$, we see that
\[ q^{-(N-v_0-\cdots-v_{m-1})}(A'q^{P_0'}-B'q^{P_0'})=(A_0'-B_m'), \]
so that (\ref{right-spec}) does indeed evaluate to
the desired answer (\ref{left-spec}). This completes the proof.

\bigskip
\noindent\textbf{Proving the Formula for $C^{(m)}_n(q,1/q)$.}
We are now ready to verify that
\begin{equation}\label{eq:OCmn-specz}
 q^{mn(n-1)/2}C^{(m)}_n(q,1/q)=\frac{1}{[mn+1]_q}\dqbin{mn+n}{mn,n}. 
\end{equation}
{}From (\ref{CfromE}) with $t=1/q$, we have
\[ C^{(m)}_n(q,1/q)=q^{mn} E^{(m)}_{n+1;1,0,\ldots,0}(q,1/q). \]
Now, we use the formula just proved for the $E$'s with
$N=n+1$, $v_0=1$, and $v_i=0$ for $i>0$. The reader may verify
that, with these substitutions, we obtain
\[ q^{mn(n-1)/2}C^{(m)}_n(q,1/q)=q^{n-nm}\left\{
\dqbin{mn+n}{mn,n}-\dqbin{mn+n}{n-1,mn+1}\cdot\left(
\sum_{j=0}^{m-1} q^{nj+\chi(j=m-1)}\right)\right\}. \]
The expression in the curly braces can be written
\[ \dqbin{mn+n}{mn,n}\cdot\left(1-\frac{[n]_q\sum_{j=0}^{m-1}
q^{nj+\chi(j=m-1)}}{[mn+1]_q}\right), \]
which in turn simplifies to
\[ \dqbin{mn+n}{mn,n}\cdot\left(\frac{\sum_{k=0}^{mn} q^k
 -\sum_{k=0}^{mn} q^k\chi(k\neq mn-n)}{[mn+1]_q}\right)
  =\frac{1}{[mn+1]_q}\dqbin{mn+n}{mn,n}q^{mn-n}. \]
The leftover power of $q$ is exactly what is needed to
cancel the outside power $q^{n-nm}$. Thus, we obtain the desired result
\[ q^{mn(n-1)/2}C^{(m)}_n(q,1/q)=\frac{1}{[mn+1]_q}\dqbin{mn+n}{mn,n}. \]

\subsection{Recursions for $C_n^{(m)}(q,t)$ based on removing the last row}
\label{subsec:lastrow}
We now present one more recursion (\ref{nicerec}) that is not based directly
on formula (\ref{sumformula}). This recursion is simpler in form than
(\ref{messyErec}) because it has only four terms. However,
one must keep track of several new statistics in this recursion.

We need to introduce some temporary notation.  Let $D$ be an $m$-Dyck
path of height $n$. Let the bounce path of $D$ have successive
vertical moves $(v_0,v_1,\ldots,v_s)$ and horizontal moves
$(h_0,h_1,\ldots)$ as usual. Here, $v_s$ is the last nonzero
vertical move. Define
\begin{eqnarray*}
Q(D) &=& \area(D); \\
T(D) &=& b(D); \\
Y(D) &=& s; \\
Z_i(D) &=& v_{s-i} \mbox{ for $i\geq 0$}; \\
K(D) &=& \mbox{the total number of area cells in the top row of $D$;} \\
W(D) &=& \mbox{the number of area cells in the top row of $D$}
  \\ & & \mbox{left of the last vertical move of the bounce path.} 
\end{eqnarray*}
Thus, $Y(D)$ is one less than the total number of bounces 
needed to reach the top rim;
the statistics $Z_i(D)$ record the history of vertical moves near the
end of the bounce path; and $W(D)$ counts the number of ``extra''
cells in the top row left of the bounce path.
Define ${\cal D}_{n,k}^{(m)}$ to be the collection of paths
$D\in{\cal D}_{n}^{(m)}$ with $K(D)=k$, for $0\leq k\leq m(n-1)$.

For example, the $2$-Dyck path $E$ in Figure \ref{mbouncefig}
has $Q(E)=41$, $T(E)=30$, $Y(E)=5$, $Z_0(E)=3$,
$Z_1(E)=1$, $W(E)=1$, and $K(E)=8$. The $3$-Dyck path $D$ in Figure
\ref{bounce3} has $Q(D)=23$, $T(D)=29$, $Y(D)=8$,
$Z_0(D)=1$, $Z_1(D)=1$, $Z_2(D)=0$, $W(D)=0$, and $K(D)=5$.

Now, define
\begin{equation}\label{sixvardef}
 C_{n,k}(q,t,y,z_0,\ldots,z_{m-1},w) =
\sum_{D\in{\cal D}_{n,k}^{(m)}} q^{Q(D)}t^{T(D)}y^{Y(D)}w^{W(D)}
     \prod_{i=0}^{m-1} z_i^{Z_i(D)}. 
\end{equation}
(We suppress the dependence on $m$ from the notation.)
If $k=m(n-1)$, there is only one path $D_0\in{\cal D}_{n,m(n-1)}^{(m)}$,
which goes north $n$ steps and then east $mn$ steps. Thus, we obtain
the initial condition
\[ C_{n,m(n-1)}(q,t,y,z_0,\ldots,z_{m-1},w) =
    q^{mn(n-1)/2}z_0^n, \]
since, by inspection, $D_0$ has area $mn(n-1)/2$ and a single nontrivial
bounce of height $n$.

Write $\vec{z}$ to denote $(z_0,\ldots,z_{m-1})$. 
We will show combinatorially that, for $0\leq k<m(n-1)$,
\begin{equation}\label{nicerec}
\begin{split}
C_{n,k}(q,t,y,\vec{z},w) =
& z_0q^k C_{n-1,k-m}(q,t,ty,\vec{z},w) \\
&+q^{-1}w^{-1}\left(C_{n,k+1}(q,t,y,\vec{z},w)
                   -C_{n,k+1}(q,t,y,\vec{z},0)\right) \\
&+q^{-1}tyz_0z_1^{-1}w^{-2} C_{n,k+1}(q,t,y,wz_1,wz_2,\ldots,
wz_{m-1},w,0).
\end{split}
\end{equation}
With the initial condition, this recursion uniquely determines
the multivariable generating functions $C_{n,k}$
(by induction on $n$ and backwards induction on $k$).

To prove this recursion, we classify a path $D\in{\cal D}_{n,k}^{(m)}$
based on what happens at the left edge of the top row of $D$.
Exactly one of the following three cases must occur:
\begin{itemize}
\item \textbf{Case 1:} The path $D$ reaches the top row
by taking two consecutive north steps. See Figure \ref{mbouncefig}
for an example.
\item \textbf{Case 2:} The path $D$ reaches the top
row by taking a north step preceded by an east step,
AND this east step did not block the progress of
the next-to-last vertical bounce move. This means
that adding one more area cell to the top row of $D$
would not change the derived bounce path. See Figure \ref{case2pic}
for an example.
\item \textbf{Case 3:} The path $D$ reaches the top
row by taking a north step preceded by an east step,
AND this east step did block the progress of
the next-to-last vertical bounce move. This means
that adding one more area cell to the top row of $D$
would enable the next-to-last bounce to reach the top rim,
so that the total number of bounces would decrease by one.
See Figure \ref{bounce3} for an example.
\end{itemize}

\begin{figure}[ht]
\begin{center}
\epsfig{file=case2pic.eps}
\caption{A path satisfying case 2 in the recursion analysis.}
\label{case2pic}
\end{center}
\end{figure}

The three terms on the right side of (\ref{nicerec}) are the
respective generating functions for the paths in the three cases above.

To see this, first consider paths satisfying Case 1.
We can uniquely construct each such path $D$ by first picking
a path $D'$ of height $n-1$ with $k-m$ area cells in row $n-1$,
and then placing $k$ new area cells in row $n$ to obtain $D$.
See Figures \ref{buildcase1} and \ref{mbouncedef} (where $D=E$). 
The generating function for the choice
of $D'$ is $C_{n-1,k-m}(q,t,y,\vec{z},w)$.
Adding the new row influences the statistics as follows.
The power of $q$ increases by $k$ since we
added $k$ new area cells.  Let $(v_0',\ldots,v_{s'}')$ be the 
vertical moves in the bounce path for $D'$.
It is clear from Figure \ref{buildcase1} that the bounce path of
$D$ will have vertical moves $(v_0,\ldots,v_s)$, where
$s=s'$, $v_i=v_i'$ for $i<s$, and $v_s=v_{s'}'+1$.
Since only the last vertical move changed, all horizontal moves
before reaching the top rim are the same.
Since $v_s=v_{s'}'+1$, the power of $z_0$ should increase by one
when we pass from $D'$ to $D$. Since $v_i=v_i'$ for $i<s$,
the powers of $z_1,z_2,\ldots$ should not change. Similarly,
since $s=s'$, the power of $y$ does not change in the passage
from $D'$ to $D$. The power of $w$ does not change either,
since there are the same number of extra cells left of the
last vertical move after adding the new row.
Finally, we have $b(D)=\sum_{i\geq 0} iv_i=\sum_{i\geq 0}iv_i'+s
=b(D')+s$, since $v_s=v_s'+1$. We can increase the power
of $t$ in the generating function by exactly $s$ if we replace $y$ by $ty$ in 
$C_{n-1,k-m}(q,t,y,\vec{z},w)$. To see this, recall that $Y(D')=s'=s$
and compare to definition (\ref{sixvardef}) [with $D$ there replaced by $D'$].
Putting all this together, we see that the generating function for
paths in Case 1 is precisely $z_0q^k C_{n-1,k-m}(q,t,ty,\vec{z},w)$.

\begin{figure}[ht]
\begin{center}
\epsfig{file=buildcase1.eps}
\caption{Constructing a path in Case 1 by adding a row.}
\label{buildcase1}
\end{center}
\end{figure}

We will treat the next two cases together.
Note that all paths $D$ satisfying Case 2 or 3 can be
uniquely constructed by choosing a path $D'\in {\cal D}_{n,k+1}^{(m)}$
and then \emph{removing} the leftmost area cell in the top row
of $D'$. The generating function for the paths $D'$ is
$C_{n,k+1}(q,t,y,\vec{z},w)$. However, to determine
the effect of the cell removal on the bounce statistic,
we must know whether the removed cell was an ``extra'' cell
or one that was part of the bounce path. This complication
forces the introduction of two separate cases.

If $w(D')=0$, then $D'$ has no extra area cells in its top row.
The path $D$ constructed from $D'$ therefore belongs to case 3.
Consider the definition (\ref{sixvardef}) with $D$ replaced by $D'$
and $k$ replaced by $k+1$. If we substitute $w=0$ in that definition
(with the usual convention that $0^0=1$), we are left with the
generating function for just those paths $D'$ with $w(D')=0$.
By the sum rule, the generating function for just those paths $D'$
with $w(D')>0$ must be $C_{n,k+1}(q,t,y,\vec{z},w)-C_{n,k+1}(q,t,y,
\vec{z},0)$.

In case 2, we start with a path $D'$ counted by the latter generating
function. For example, $D'$ could be the path $D$ shown in 
Figure \ref{case2pic} with the cell $c$ adjoined. To go from $D'$
to $D$, we remove the cell in position $c$. This clearly decreases
$Q(D')$ and $W(D')$ by $1$, but does not affect the other statistics
that are determined by the bounce path. It immediately follows
that the generating function for the paths $D$ in case 2 is
\[q^{-1}w^{-1}\left(C_{n,k+1}(q,t,y,\vec{z},w)
                   -C_{n,k+1}(q,t,y,\vec{z},0)\right) \]

To get a path $D$ belonging to case 3, on the other hand,
we must have started with a path $D'$ such that $w(D')=0$.
For example, the path $D'$ in Figure \ref{buildcase3}
is used to construct the path $D$ in Figure \ref{bounce3}.

\begin{figure}[ht]
\begin{center}
\epsfig{file=buildcase3.eps}
\caption{Constructing a path in Case 3 by deleting one cell.}
\label{buildcase3}
\end{center}
\end{figure}

The generating function for the choice of $D'$ is
$C_{n,k+1}(q,t,y,\vec{z},0)$. We obtain $D$ from $D'$
by removing the leftmost area cell $c$ in the top row of $D'$.
To see how this affects the statistics, compare
Figure \ref{buildcase3} to Figure \ref{bounce3}.
Clearly, the area $Q(D)=Q(D')-1$ because of the removed cell.
Let $(v_0',\ldots,v_{s'}')$ be the lengths of the vertical moves
in the bounce path for $D'$; let $(v_0,\ldots,v_s)$ be
the lengths of the vertical moves in the bounce path for $D$.
In this case, removing the cell forces the last vertical move
in $D'$ to be shortened by $1$ unit, so that there must be
a new vertical move of length $1$ afterwards in $D$.
Thus, $v_i=v_i'$ for $i<s'$, $v_{s'}=v_{s'}'-1$,
$s=s'+1$, and $v_s=1$. We find that $b(D)-b(D')=(s'+1)\cdot 1-s'\cdot 1=1$,
so that the bounce statistic has increased by $1$.
We also have $Y(D)=Y(D')+1$, $Z_0(D)=1$, 
$Z_1(D)=Z_0(D')-1$, and $Z_i(D)=Z_{i-1}(D')$ for $i\geq 2$.
Finally, we must compute the new value $W(D)$. After the bounce
path for $D$ takes the vertical step of length $v_{s'}=v_{s'}'-1$
(this step is blocked by the east step introduced by the removed cell),
the bounce path moves east  
\[ Z_1(D)+Z_2(D)+\cdots+Z_m(D)
  =Z_0(D')-1+Z_1(D')+\cdots+Z_{m-1}(D') \mbox{ units.} \]
All the area cells above this horizontal move were present in $D'$;
in $D$, all these cells exist except the leftmost cell $c$.
This implies that
\[ W(D)=Z_0(D')+\cdots+Z_{m-1}(D')-2. \]

Consider the last term on the right side of (\ref{nicerec}):
\[ q^{-1}tyz_0z_1^{-1}w^{-2} C_{n,k+1}(q,t,y,wz_1,wz_2,\ldots,wz_{m-1},w,0).\]
By the definition in (\ref{sixvardef}) and the comments above,
\[ C_{n,k+1}(q,t,y,z_0,\ldots,z_{m-1},0)
  =\sum_{D' \mbox{ as in Case 3}}
   q^{Q(D')}t^{T(D')}y^{Y(D')}\prod_{i=0}^{m-1} z_i^{Z_i(D')}. \]
Therefore, making the indicated substitutions for the variables,
\begin{equation}
\begin{split}
\lefteqn{
 q^{-1}tyz_0z_1^{-1}w^{-2} C_{n,k+1}(q,t,y,wz_1,wz_2,\ldots,wz_{m-1},w,0)}
\\ & =\sum_{D'}
  q^{Q(D')-1}t^{T(D')+1}y^{Y(D')+1}z_0^1 z_1^{Z_0(D')-1}
    z_2^{Z_1(D')}\cdots z_{m-1}^{Z_{m-2}(D')}
    w^{\pow} 
\\ & \ \ \ (\pow=Z_0(D')+\cdots+Z_{m-2}(D')+Z_{m-1}(D')-2)
\\ & = \sum_{D}
  q^{Q(D)}t^{T(D)}y^{Y(D)}z_0^{Z_0(D)}\cdots z_{m-1}^{Z_{m-1}(D)}w^{W(D)},
\end{split}
\end{equation}
where the sums extend over the paths $D'$ and $D$ appearing
in the description of case 3 above.
Thus, the third term in (\ref{nicerec}) is the correct generating
function for the paths belonging to case 3. This completes the proof
of the recursion.

\smallskip\noindent\textbf{Remarks.} The given recursion
(\ref{nicerec}) keeps track of the last $m$ vertical bounces
$Z_0(D)$, ... , $Z_{m-1}(D)$. This is necessary to determine
what happens to the other statistics in certain cases.
Though it is not necessary here,
we clearly could add even more variables $z_m,\ldots$ to
keep track of the earlier bounce moves $Z_m(D),\ldots$ if we wished.
Later (\S \ref{sec:trivariate}), we shall consider a
more general recursion in which it becomes necessary to
keep track of $Z_m(D)$.

We remark that a similar recursion can be proved for
a suitable generalization of $HC_n^{(m)}(q,t)$.
We do not give the details of the proof, which are quite messy,
but merely list the appropriate reinterpretations of
the statistics. In this setting, one should take
\begin{eqnarray*}
Q(D) &=& h(D); \\
T(D) &=& \area(D); \\
Y(D) &=& \max_{0\leq i<n} \gamma_i(D); \\
Z_i(D) &=& |\{j:\gamma_j(D)=Y(D)-i\}| \mbox{ for $i\geq 0$}; \\
K(D) &=& h(D)-h(D'),
\mbox{where $D'$ is obtained from $D$ by }
\\ & & \mbox{removing the rightmost value $Y(D)$;} \\
W(D) &=& \mbox{the number of symbols in $\{Y(D)-1,\ldots,Y(D)-m\}$ appearing} 
  \\ & & \mbox{in $\gamma(D)$ after the last occurrence of $Y(D)$.} 
\end{eqnarray*}
This gives an alternate way of proving
that $C_n^{(m)}(q,t)=HC_n^{(m)}(q,t)$.

\section{Trivariate Catalan Sequences}
\label{sec:trivariate}
We now introduce three-variable sequences
$C_n^{(m)}(q,t,r)$ that generalize the higher $q,t$-Catalan sequences.
Our point of departure is the observation that
\[ r^{ab}{\displaystyle\genfrac{[}{]}{0pt}{}{a+b}{a,b}_{q/r}}
  =\sum_{\lambda\subset R_{a,b}} q^{|\lambda|}r^{ab-|\lambda|}
  =\sum_{w\in R(0^a 1^b)} q^{\coinv(w)} r^{\inv(w)}. \]
In other words, given a lattice path from the southwest corner
to the northeast corner of the rectangle $R_{a,b}$,
we can keep track of both the area in $R_{a,b}$ below the path
and the area in $R_{a,b}$ above the path by making the
indicated substitution in the $q$-binomial coefficient.
For convenience, set
\[ \dqrbin{a+b}{a,b}=
 r^{ab}{\displaystyle\genfrac{[}{]}{0pt}{}{a+b}{a,b}_{q/r}}. \]

\begin{figure}[ht]
\begin{center}
\epsfig{file=qrtcells.eps}
\caption{Visualizing the three statistics as counting cells.}
\label{qrtcells}
\end{center}
\end{figure}

We introduce a new statistic $\area'$ on $m$-Dyck paths $D$ of height $n$
as follows. Given $D$, draw the bounce path of $D$ and the associated
rectangles $R_i$ as in Figure \ref{brectangles}. Let $R_i'$
denote the rectangle $R_i$ without its leftmost column.
Define $\area'(D)$ to be the number of complete
cells below the bounce path of $D$ plus the number of cells inside
the rectangles $R_i'$ and \emph{above} the path $D$.
By contrast, $\area(D)$ is the number of complete
cells below the bounce path of $D$ plus the number of 
cells inside the rectangles $R_i'$ and \emph{below} the path $D$.
For each $m$ and $n$, define
\[ C_n^{(m)}(q,t,r)=\sum_{D\in {\cal D}_{n}^{(m)}} 
q^{\area(D)}t^{b(D)}r^{\area'(D)}.
\]
See Figure \ref{qrtcells} for an example. 

In this figure, cells below the bounce path contributing to both $\area$
and $\area'$ are labelled by their weight $qr$. Cells above the
bounce path but below the $m$-Dyck path contribute only to area and
are labelled $q$. Cells inside the rectangles $R_i'$ but above the
$m$-Dyck path are labelled $r$. Finally, Figure \ref{qrtcells}
shows how we can interpret the bounce statistic $b(D)$ as counting
certain cells in the picture as well. Specifically, we label each
cell in the column above a vertical bounce move with $t$.
Equation (\ref{bounce2}) shows that the number of such factors $t$
is exactly $b(D)$.

{}From Figure \ref{qrtcells}, we immediately deduce the symmetry result
\[ C_n^{(m)}(q,t,r)=C_n^{(m)}(r,t,q). \]
For, we can interchange the number of cells labelled $q$ and
the number of cells labelled $r$ by merely rotating the contents
of each shortened rectangle $R_i'$ by $180^{\circ}$. Note that
this rotation will not affect the bounce path, since it does
not affect the leftmost columns of the full rectangles $R_i$.
The image of the path in Figure \ref{qrtcells} under this involution
is shown in Figure \ref{qrtflipped}.

\begin{figure}[ht]
\begin{center}
\epsfig{file=qrtflipped.eps}
\caption{Interchanging $\area$ and $\area'$ by flipping rectangles.}
\label{qrtflipped}
\end{center}
\end{figure}

It is easy to incorporate $\area'$ into formula (\ref{sumformula}).  We have
\begin{equation}\label{rsumformula}
\begin{split}
 \lefteqn{C_n^{(m)}(q,t,r) =} \\ &
\sum_{v\in {\cal V}_{n}^{(m)}} 
t^{\sum_{i\geq 0} i v_i} (qr)^{m\sum_{i\geq 0}
 \frac{1}{2} v_i(v_i-1)}
 \prod_{i\geq 1} (qr)^{v_i\sum_{j=1}^m (m-j)v_{i-j}}
\dqrbin{v_i+v_{i-1}+\cdots+v_{i-m}-1}
     {v_i,v_{i-1}+\cdots+v_{i-m}-1}. 
\end{split}
\end{equation}
The new formula follows by recalling that the factors
$\tqbin{v_i+v_{i-1}+\cdots+v_{i-m}-1}{v_i,v_{i-1}+\cdots+v_{i-m}-1}$
keep track of the area cells below the path in the rectangles
$R_i'$, whereas the remaining powers of $q$ in (\ref{sumformula})
count the cells below the bounce path. Hence, to keep track of $\area'$,
it suffices to replace the latter occurrences of $q$ by $qr$
and to use $q,r$-binomial coefficients in place of $q$-binomial coefficients.

The recursion in \S \ref{subsec:firstbounce} is also easily modified.  
Define
\begin{equation}%\label{messyE1r}
\begin{split}
\lefteqn{E_{n;v_0,\ldots,v_{m-1}}(q,t,r)=}
\\ & \sum_{(v_m,v_{m+1},\ldots)}
t^{\sum_{i\geq 0} i v_i} 
(qr)^{\pow_1} r^{\pow_2}
 \prod_{i\geq m}
\dqrbin{v_i+v_{i-1}+\cdots+v_{i-m}-1}
     {v_i,v_{i-1}+\cdots+v_{i-m}-1},
\end{split}
\end{equation}
where
\[ \pow_1= m\sum_{i\geq 0} \frac{1}{2} v_i(v_i-1)+
 \sum_{i\geq 1} v_i\sum_{j=1}^{m\wedge i} (m-j)v_{i-j}, \]
\[ \pow_2=\sum_{i=1}^{m-1} v_i\left((\sum_{j=0}^{i-1} v_j)-1\right). \]
The extra power $r^{\pow_2}$ accounts for the cells in the
first $m-1$ rectangles $R_i'$, which all contribute to the $r$-statistic.

We have the recursion
% was WRONG --- should be OK now.
\begin{equation}%\label{messyErecr}
\begin{split}
\lefteqn{ E_{n;v_0,\ldots,v_{m-1}}(q,t,r)= }
\\& t^{n-v_0} (qr)^{\pow_3} 
%(qr)^{m\binom{v_0}{2}} goes into \pow_3
%\prod_{i=1}^{m-1} (qr)^{v_0v_i(m-i)} goes into \pow_3
\sum_{v_m=0}^{n-v_0-\cdots-v_{m-1}} r^{\pow_4}
  \dqrbin{v_m+\cdots+v_0-1}{v_m,v_{m-1}+\cdots+v_0-1}
  E_{n-v_0;v_1,\ldots,v_{m-1},v_m}(q,t,r),
\end{split}\end{equation}
where
\[ \pow_3=m\binom{v_0}{2}+\sum_{i=1}^{m-1} v_0v_i(m-i), \]
\[ \pow_4=v_0(v_1+\cdots+v_{m-1})-v_1-v_m(v_{m-1}+\cdots+v_0-1). \]
The initial condition is 
\[ E_{n;n,0,\ldots,0}(q,t,r)=(qr)^{mn(n-1)/2}t^0. \]

The recursion from \S \ref{subsec:lastrow} requires a bit more work.
For an $m$-Dyck path $D$, define $R(D)=\area'(D)$, and set
\begin{equation}
 C_{n,k}(q,r,t,y,z_0,\ldots,z_{m},w) =
\sum_{D\in{\cal D}_{n,k}^{(m)}} q^{Q(D)}r^{R(D)}t^{T(D)}y^{Y(D)}w^{W(D)}
     \prod_{i=0}^{m} z_i^{Z_i(D)}. 
\end{equation}
Observe that this generating function, unlike the original,
keeps track of $Z_m(D)$ as well as $Z_i(D)$ for $i<m$.
We need to make one technical adjustment in the definition
of $Z_m$.  If $D_0$ is the special path that goes north $n$
steps and east $mn$ steps, set $Z_m(D_0)=1$; for all other paths,
define $Z_m(D)$ as in \S \ref{subsec:lastrow}.

With this adjustment, the initial condition is
\[ C_{n,k}(q,r,t,y,\vec{z},w)=(qr)^{mn(n-1)/2} z_0^n z_m^1
\mbox{ when $k=m(n-1)$.} \]
The new recursion, valid for $0\leq k<m(n-1)$, is: 
\begin{equation} % from NB 3, p. 148
\begin{split}
\lefteqn{C_{n,k}(q,r,t,y,\vec{z},w) =} \\
& z_0 q^k r^{k-1} C_{n-1,k-m}(q,r,t,ty,z_0,rz_1,\ldots,rz_m,r^{-2}w) \\
&+q^{-1}w^{-1}r^{+1}\left(C_{n,k+1}(q,r,t,y,\vec{z},w)
                   -C_{n,k+1}(q,r,t,y,\vec{z},0)\right) \\
&+q^{-1}tyz_0z_1^{-1}w^{-2}r^{2} C_{n,k+1}(q,r,t,y,
 r^{-1}wz_1,r^{-2}wz_2,\ldots,r^{-2}wz_m,r^{-1},0).
\end{split}
\end{equation}
To verify this equation, we need only check the correctness of the
powers of $r$ and $z_m$. We look at three cases, as in \S \ref{subsec:lastrow}.
% NB3, p. 160 has this counting argument.
In case 1, we go from $D'\in{\cal D}_{n-1,k-m}^{(m)}$
to $D\in{\cal D}_{n,k}^{(m)}$ by adding a new top row with $k$ area cells.
By definition, $Z_m(D')=Z_m(D)$. [Note that the technical adjustment
made to $Z_m(D_0)$ has no effect here, since $k<m(n-1)$ implies
that $(k-m)<m((n-1)-1)$, hence $D'\neq D_0$ and $D\neq D_0$.]
What happens to $\area'$ when we pass from $D'$ to $D$?
In the new top row, $k-W(D)$ of the $k$ new area cells are below the bounce
path for $D$, hence contribute to $\area'$. The last rectangle
$R_s$ has also gained a new top row, which contains
$h_{s-1}=v_{s-1}+\cdots+v_{s-m}$ cells.  Of these cells, the
one in the leftmost column does not count towards $\area'$, nor do
the $W(D)$ new cells below the path $D$.
These observations explain why we replace $z_1,\ldots,z_m$ 
by $rz_1,\ldots,rz_m$ (leaving $z_0$ alone) and multiply
by $r^{k-1}$ in the term
\[ z_0 q^k r^{k-1} C_{n-1,k-m}(q,r,t,ty,z_0,rz_1,\ldots,rz_m,r^{-2}w). \]
For, the net gain in the power of $r$ is
\[ Z_1(D')+\cdots+Z_m(D')+k-1-2W(D') =v_{s-1}(D)+\cdots+v_{s-m}(D)+
  (k-W(D))-(W(D)+1),\]
as required.

The term from Case 2, namely
\[ q^{-1}w^{-1}r^{+1}\left(C_{n,k+1}(q,r,t,y,\vec{z},w)
                   -C_{n,k+1}(q,r,t,y,\vec{z},0)\right), \]
is the easiest to derive. Recall that we go from $D'$ to $D$
by removing the leftmost ordinary area cell in the top row of $D'$,
which is not below the bounce path of $D'$ or $D$.
But ``removing'' this cell from $D'$ causes the cell to
contribute to $\area'$ instead, since it belonges to one of the
rectangles $R'$ and is now above $D$. Thus, we have an extra
factor $r^{+1}$ in the generating function. As for $z_m$,
note that $D'\neq D_0$ since $W(D')>0=W(D_0)$, and
$D\neq D_0$ since $k<k+1\leq m(n-1)$. Thus, $Z_m(D')=Z_m(D)$.

Finally, consider the term from Case 3, namely
\[ q^{-1}tyz_0z_1^{-1}w^{-2}r^{2} C_{n,k+1}(q,r,t,y,
 r^{-1}wz_1,r^{-2}wz_2,\ldots,r^{-2}wz_m,r^{-1},0). \]
In this case, we go from $D'$ to $D$ by removing the
leftmost ordinary area cell $c$ in the top row of $D'$,
causing a change in the end of the bounce path.
(See Figures \ref{buildcase3} and \ref{bounce3}.)
Specifically, the bounce path of $D$ has
a new terminating vertical move $v_s$ of length 1,
and the previous vertical move $v_{s-1}$ is one less
than the corresponding move $v_{s-1}'$ in $D'$.
Note that the top row of the last rectangle $R_{s-1}^*$ in $D'$
does not belong to the rectangle $R_{s-1}$ in $D$.
Every cell in the top row of $R_{s-1}^*$, except the leftmost one,
contributed to $R(D')$, because $w(D')=0$. The
number of contributing cells is one less than the horizontal
dimension of $R_{s-1}^*$; this dimension is $Z_1(D')+\cdots+Z_m(D')$.
The conclusion is that $R(D)$ drops by
\begin{equation}\label{firstloss}
(Z_1(D')+\cdots+Z_m(D')-1) 
\end{equation}
as a result of the lost row in $R_{s-1}$.

On the other hand, consider cells in the top row of $D'$ that
are to the right of the bounce path in $D'$. After removing cell $c$
from $D'$, the new bounce path for $D$ stops at the southwest corner
of $c$, then goes east for a distance
\[ Z_0(D)+\cdots+Z_{m-1}(D)
  =(Z_0(D')-1)+Z_1(D')+\cdots+Z_{m-1}(D'), \]
then goes north one unit.  The cells in the top row above this last east step
used to count towards $\area'(D')$, being below the bounce path of $D'$,
but will no longer count towards $\area'(D)$. In more detail, cell $c$
does not count towards $\area'(D)$ because it is in the leftmost column
of its rectangle. The other cells do not count towards $\area'(D)$
because they count towards ordinary area instead. We conclude that
$R(D)$ drops by an additional
\begin{equation}\label{secondloss}
Z_0(D')+\cdots+Z_{m-1}(D')-1
\end{equation}
as a result of the change
in this part of the bounce path. The total change is
\[ R(D)-R(D')=-(1Z_0(D')+2Z_1(D')+\cdots+2Z_{m-1}(D')+1Z_{m}(D'))+2. \]
This change is modelled algebraically by the additional
occurrences of $r$ in the expression
\[ q^{-1}tyz_0z_1^{-1}w^{-2}r^{2} C_{n,k+1}(q,r,t,y,
 r^{-1}wz_1,r^{-2}wz_2,\ldots,r^{-2}wz_m,r^{-1},0). \]

The argument in the last two paragraphs is correct, unless
$R_{s-1}^*$ has width zero. In this situation, there is no leftmost
column in $R_{s-1}^*$, so we should not have subtracted 1
in (\ref{firstloss}).  But it is easy to see that
this situation occurs iff $D'=D_0$. Then our technical
convention that $Z_m(D_0)=+1$ causes (\ref{firstloss})
to be correct after all, and the validity of (\ref{secondloss})
is not affected either.  Since the new value $Z_m(D)$
comes from the old value $Z_{m-1}(D')$ (not from $Z_m(D')$),
the technical convention for $Z_m(D_0)$ does not affect the correctness
of the values of $Z_m(D)$ calculated using the recursion.
This completes the proof of the new recursion.

Finally, we describe an analogous way of adding a third statistic
to the other combinatorial sequence $HC_n^{(m)}(q,t)$.
We can ``guess'' what this statistic should be by seeing what happens 
to $\area'$ when we apply the bijection $\psi$ from
\S \ref{subsec:bijections}. We are led to the following formula.
For an $m$-Dyck path of height $n$, define
\[ h'(D)=\sum_{0\leq i<j<n}\sum_{k=0}^{m-1}
 \chi\left( \gamma_i(D)-\gamma_j(D)+k \in \{-1,0,1,\ldots, m-1\} \right)
 -\sum_{i=0}^{n-1} \chi(\gamma_i(D)>0). \]
The first sum is similar to the one appearing in $h(D)$.
The second sum that is subtracted may look surprising,
but it arises from the fact that the leftmost column of each
rectangle $R_i$ does \emph{not} count towards $\area'$.
Note that the total number of cells in these columns
is $n-v_0(\phi(D))=\sum_{i=0}^{n-1} \chi(\gamma_i(D)>0)$.

There is a formula analogous to (\ref{scoredef}) for $h'(D)$.  Define
$\mysc_m':\Z\rightarrow\Z$ by
\begin{equation}\label{scoredef2}
 \mysc_m'(p)=\left\{\begin{array}{ll}
   m-p & \mbox{ for $0\leq p\leq m$; } \\
  m+1+p & \mbox{ for $-m\leq p\leq -1$; } \\
    0 & \mbox{ for other $p$. } \end{array}\right. 
\end{equation}
Define $adj':\Z\rightarrow\Z$ by $adj'(p)=-1$ for $p>0$ and
$adj'(p)=0$ for other $p$. Then
\[ h'(D) = \sum_{0\leq i<j<n} \mysc_m'(\gamma_i(D)-\gamma_j(D))
          +\sum_{i=0}^{n-1} adj'(\gamma_i). \]
The proof is the same as the corresponding proof of (\ref{scoredef}).

Now, define 
\[ HC_n^{(m)}(q,t,r)=\sum_{D\in {\cal D}_{n}^{(m)}} 
q^{h(D)}t^{\area(D)}r^{h'(D)}.  \]
We show that $HC_n^{(m)}(q,t,r)$ equals the right side of
(\ref{rsumformula}) by modifying the earlier proof to include $r$.
It will follow that the bijection $\phi$ introduced
in \S \ref{subsec:bijections} maps the ordered triple of
statistics $(h,\area,h')$ to the ordered triple $(\area,b,\area')$
(similarly for $\psi=\phi^{-1}$).

As in \S \ref{subsec:HCformula}, we proceed by induction on
the largest symbol $s$ appearing in $\gamma(D)$.  When $s=0$,
$\gamma$ must consist of $n$ zeroes, and $h'(D)=mn(n-1)/2$.
This is the same as the power of $r$ on the right side
of (\ref{rsumformula}).

For the induction step, it suffices to prove the following formula,
which is the analogue of (\ref{hdiff}) for $h'$:
\begin{equation}\label{hdiff2}
 h'(\gamma)-h'(\delta)=mv_s(v_s-1)/2+v_s\sum_{k=1}^m (m-k)v_{s-k}
     + \inv(w). 
\end{equation}
Here, $\gamma=\gamma(D)$ has largest symbol $s>0$;
$v_i$ is the number of occurrences of $i$ in $\gamma$ for $0\leq i\leq s$;
$\delta$ is obtained from $\gamma$ by erasing all the symbols $s$;
and the word $w$ records how to insert the $v_s$ copies of $s$
into $\delta$ to recover $\gamma$.

We still proceed by induction on $\coinv(w)$.
If $\coinv(w)=0$, all $v_s$ copies of $s$ were inserted into $\delta$
just after the last occurrence of any symbol in the
set $\{s-1,\ldots,s-m\}$. The change $h'(\gamma)-h'(\delta)$ caused
by this insertion is
\[ \sum_{i<j} \mysc_m'(\gamma_i-\gamma_j) - v_s\]
where the sum extends over all pairs $(i,j)$ such that $\gamma_i=s$
or $\gamma_j=s$. We subtract $v_s$ since we introduced $v_s$ new 
positive entries (all equal to $s$) in $\gamma$.

First, consider the pairs $(i,j)$ for which $i<j$ and
$\gamma_i=s=\gamma_j$. There are $\tbinom{v_s}{2}$ such pairs,
and each contributes $\mysc_m'(s-s)=\mysc_m'(0)=m$ to the $h'$-statistic.
This gives the term $mv_s(v_s-1)/2$ in (\ref{hdiff2}).

Second, consider the pairs $(i,j)$ for which $i<j$ and
$\gamma_i=s$ and $\gamma_j\neq s$.  Since all the copies of $s$
in $\gamma$ occur in a contiguous group following all instances
of the symbols $s-1,\ldots, s-m$, and since $s$ is the largest symbol
appearing in $\gamma$, $j>i$ implies that $\gamma_j<s-m$.
Then $\mysc_m'(\gamma_i-\gamma_j)=0$, since $\gamma_i-\gamma_j>m$.
So these pairs contribute nothing to the $h'$-statistic.

Third, consider the pairs $(i,j)$ for which $i<j$ and
$\gamma_i\neq s$ and $\gamma_j=s$. Since $s$ is the largest symbol,
we have $\gamma_i<s$.  Write $\gamma_i=s-k$ for some $k>0$,
and consider various subcases.  Suppose $k\in\{1,2,\ldots,m\}$.
Then $\mysc_m'(\gamma_i-\gamma_j)=\mysc_m'(-k)=m+1-k$. For how many pairs
$(i,j)$ does it happen that $i<j$, $\gamma_i=s-k$, and $\gamma_j=s$?
There are $v_s$ choices for the index $j$ and $v_{s-k}$ choices
for the index $i$; the condition $i<j$ holds automatically, since all 
occurrences of $s$ occur to the right of all occurrences of $s-k$. Thus, we
get a total contribution to the $h'$-statistic of $(m+1-k)v_s(v_{s-k})$
for this $k$. Adding over all $k$, we obtain 
\[ v_s\sum_{k=1}^m (m+1-k)v_{s-k} 
  =v_s\sum_{k=1}^m (m-k)v_{s-k} + \sum_{k=1}^m v_s v_{s-k}. \]
On the other hand, if $k>m$,
then $\mysc_m'(\gamma_i-\gamma_j)=\mysc_m(-k)=0$, so there is no
contribution to the $h'$-statistic.

Finally, recall that $w$ is a rearrangement of $v_s$ zeroes
and $v_{s-1}+\cdots+v_{s-m}-1$ ones. Since $\coinv(w)=0$,
all zeroes in $w$ occur at the end, and hence
\[ \inv(w)=v_s(v_{s-1}+\cdots+v_{s-m}-1)
       =\left(\sum_{k=1}^m v_s v_{s-k}\right) -v_s. \]

Thus, the change $h'(\gamma)-h'(\delta)$ is precisely the expression
on the right side of (\ref{hdiff2}). So we
are done when $\coinv(w)=0$.

To finish the induction step, it suffices to show that
replacing $10$ by $01$ in $w$ decreases $h'$ by one
(since this replacement also decreases $\inv(w)$ by one).
Let $w'$ be the new word after the replacement, with
corresponding vector $\gamma'$ As in \S \ref{subsec:HCformula},
we have
\[ \mbox{original $\gamma$} = \ldots\ (s-j)\ z_1\ z_2\ \ldots\ z_{\ell}\ 
(s-k)\ s\ \ldots \]
where $0\leq j\leq m$, $1\leq k\leq m$, $\ell\geq 0$,
and every $z_i<s-m$. Replacing $10$ by $01$ in $w$ causes the
$s$ to move left, resulting in:
\[ \mbox{ new $\gamma'$} = \ldots\ (s-j)\ s\ z_1\ z_2\ \ldots\ z_{\ell}\ 
(s-k)\ \ldots. \]
Note that the symbol $s-j$ must exist, lest $\gamma'_0=s>0$.

Let us examine the effect of this motion on the $h'$-statistic.
When we move the $s$ left past its predecessor $s-k$ in $\gamma$, we get
a net change in the $h'$-statistic of
\[ \mysc_m'(s-[s-k]) - \mysc_m'([s-k]-s)
  =\mysc_m'(k)-\mysc_m'(-k) = -1, \]
since $1\leq k\leq m$ (see (\ref{scoredef2})).
As before, since $|s-z_i|>m$, moving the $s$ past each $z_i$ will
not affect the $h'$-statistic at all. Thus, the total change
in the $h'$-statistic is $-1$,  as desired.

\section{Open Problems and Recent Developments} 
\label{sec:openprobs}
There are several open problems involving the combinatorial
polynomials $C^{(m)}_n(q,t)$.
The main open problem is to prove that the conjectured
combinatorial interpretation of the higher $q,t$-Catalan
sequences is correct, i.e., that $OC^{(m)}_n(q,t)=C^{(m)}_n(q,t)$.
It may be possible to prove the equivalent
assertion that $SC^{(m)}_n(q,t)=C^{(m)}_n(q,t)$ by proving
that both sides satisfy the same recursion.
A recursion characterizing $C^{(m)}_n(q,t)$ appears in
\S \ref{subsec:firstbounce}. The first difficulty with this approach
is finding the symmetric function formulas that correspond
to the generating functions $E_{n;v_0,v_1,\ldots,v_{m-1}}(q,t)$
when some $v_i$ (besides $v_0$) is nonzero.

Another, purely combinatorial problem is to prove that the 
$q$ and $t$ statistics introduced here for $m$-Dyck paths
are jointly symmetric. This conjecture is only known to be true when $m=1$
by invoking the long proof in \cite{nablaproof,PNAS-GH}. Here, we have only 
proved the weaker statement that the univariate distributions
of the $q$ and $t$ statistics are the same. 

Recall that the higher $q,t$-Catalan sequences
all have (conjectural) interpretations in representation
theory, symmetric function theory, and algebraic geometry.
It would be interesting to find analogous interpretations
for the trivariate $q,t,r$-Catalan sequences.

\medskip\noindent\textbf{Remark:}
Since the initial submission of this article, 
a number of related combinatorial developments have appeared
in the literature. The present author has generalized 
many of the combinatorial constructs presented here 
to paths inside certain trapezoidal shapes \cite{mythesis,trapz}.
Haglund, Haiman, Loehr, and Remmel introduced statistics
on labelled Dyck paths and $m$-Dyck paths, which are conjectured
to give the Hilbert series of certain doubly graded $S_n$-modules
\cite{parkpaper,mpark,pmaj,mythesis}. The same four authors and
A. Ulyanov recently conjectured a combinatorial formula for
the monomial expansion of the Frobenius series of these same modules
\cite{hhlru}. Using ideas from these papers, Haglund conjectured
a combinatorial formula for the monomial expansion of the modified
Macdonald polynomials $\tilde{H}_{\mu}$ \cite{mac-conj}. This conjecture
has been proved by Haglund, Haiman, and the present author 
\cite{mac-proof,PNAS}.
Finding combinatorial statistics for the Kostka-Macdonald coefficients 
(which arise in the Schur expansion of $\tilde{H}_{\mu}$) remains open.  
It seems likely that the new combinatorial formula for Macdonald polynomials
may soon shed additional light on the conjectures in this paper
and in \cite{hhlru}.

\medskip\noindent\textbf{Acknowledgement:}
The author thanks Jeffrey Remmel and James Haglund
for many helpful discussions regarding this problem.

\begin{thebibliography}{99}

\bibitem{benderbook}
E. Bender and S. G. Williamson, \emph{Foundations of Applied Combinatorics.}\\
Electronic version available online at \verb-http://math.ucsd.edu/~ebender-.
%This combinatorics text is available online. It can be downloaded from
%Dr. Bender's web page, \verb-http://math.ucsd.edu/~ebender-.
%Chapter 10 describes the sum and product rules for generating functions
%that are used several times in this paper.

\bibitem{nablaref1}
F. Bergeron and A. Garsia, ``Science Fiction and Macdonald Polynomials,''
\emph{CRM Proceedings and Lecture Notes AMS} VI \textbf{3} (1999), 363---429.  
%a paper with information about the nabla operator.

\bibitem{nablaref2}
F. Bergeron, N. Bergeron, A. Garsia, M. Haiman, and G. Tesler,
``Lattice Diagram Polynomials and Extended Pieri Rules,''
\emph{Adv. in Math.} \textbf{2} (1999), 244---334.
%Another paper with information about the nabla operator.

\bibitem{nablaref3}
F. Bergeron, A. Garsia, M. Haiman, and G. Tesler, ``Identities
and Positivity Conjectures for some remarkable Operators  in
the Theory of Symmetric Functions,'' \emph{Methods and Applications of
Analysis} VII \textbf{3} (1999), 363---420.  
%a third paper with information about the nabla operator.

\bibitem{nablaproof}
A. Garsia and J. Haglund, ``A proof of the $q,t$-Catalan positivity
conjecture,'' \emph{LACIM 2000 Conference on Combinatorics,
Computer Science, and Applications} (Montreal), \emph{Discrete Math.}
 \textbf{256} (2002), 677---717.
%This paper proves the combinatorial interpretation
%for the $q,t$-Catalan sequence is correct (i.e., 
%$C^{(m)}_n(q,t)=SC^{(m)}(q,t)$)
%by showing both satisfy Haglund's recursion.

\bibitem{PNAS-GH}
A. Garsia and J. Haglund, ``A positivity result in the theory of
Macdonald polynomials,'' \emph{Proc. Nat. Acad. Sci.} \textbf{98}
(2001), 4313---4316.

\bibitem{origpaper}
A. Garsia and M. Haiman, ``A remarkable $q,t$-Catalan sequence
and $q$-Lagrange Inversion,'' \emph{J. Algebraic Combinatorics} 
5 (1996), no. 3, 191---244.
%This is the original paper introducing the $q,t$-Catalan sequence
%and higher sequences. The paper proves (with different notation)
%that $OC^{(m)}(q,t)=SC^{(m)}(q,t)$ and conjectures that these equal
%$RC^{(m)}(q,t)$. The combinatorial interpretation of the power of $q$ as area
%under an $m$-Dyck path is given also.

\bibitem{mac-conj} J. Haglund, ``A combinatorial model for the
  Macdonald polynomials,'' \emph{Proc. Nat. Acad. Sci. USA},
  \textbf{101} \#46 (2004), 16127--16131.

\bibitem{haglundstat}
J. Haglund, ``Conjectured Statistics for
the $q,t$-Catalan numbers,'' \emph{Advances in Mathematics} 
\textbf{175} (2003), 319---334.  
%This paper proposes the bounce statistic for the $q,t$-Catalan sequence
%and proves the fundamental recursion satisfied by the
%sequences $F_{n,s}(q,t)$, from which $C_n(q,t)$ can be recovered.

\bibitem{mac-proof} J. Haglund, M. Haiman, and N. Loehr,
``A combinatorial formula for Macdonald polynomials,'' 
\verb-arXiv:math.CO/0409538-, 2004.

\bibitem{PNAS} J. Haglund, M. Haiman, and N. Loehr,
``Combinatorial Theory of Macdonald Polynomials I:
  Proof of Haglund's Formula,'' \emph{Proc. Natl. Acad. Sci. USA},
  in press.

\bibitem{hhlru} 
J. Haglund, M. Haiman, N. Loehr, J. Remmel, and A. Ulyanov, 
``A combinatorial formula for the character of the diagonal coinvariants,'' 
\emph{Duke Math. J.} \textbf{126} \#2 (2005), 195---232.

\bibitem{parkpaper}
J. Haglund and N. Loehr, ``A Conjectured Combinatorial Formula
for the Hilbert Series for Diagonal Harmonics.'' 
\emph{Proceedings of FPSAC 2002 Conference} (Melbourne, Australia),
to appear in \emph{Discrete Mathematics}.

\bibitem{modmac2}
M. Haiman, ``Notes on Macdonald Polynomials and the Geometry
of Hilbert Schemes,'' \emph{Symmetric Functions 2001: Surveys of 
Developments and Perspectives,} Proceedings of the NATO Advaced Study
Institute. Sergey Fomin, ed. Kluwer, Dordrecht (2002) 1---64.
%These lecture notes discuss how methods from algebraic geometry
%and commutative algebra can be used to study Macdonald polynomials.
%Lecture 6 contains one definition of the modified Macdonald basis
%$\tilde{H}_{\mu}$.

\bibitem{haiman-email}
Mark Haiman. Personal communication.

\bibitem{bighaimantheorem}
M. Haiman. ``Vanishing theorems and character formulas for the Hilbert
scheme of points in the plane,''  \emph{Invent. Math.} {\bf 149} 
(2002), 371---407.  
%This paper proves the full character formula
%for the module of diagonal harmonics conjectured
%in \cite{origpaper}, which implies in particular that
%$RC_n(q,t)=SC_n(q,t)$.

\bibitem{mythesis} N. Loehr, \emph{Multivariate Analogues
of Catalan Numbers, Parking Functions, and their Extensions}.
Ph.D. thesis, University of California at San Diego, June 2003.

\bibitem{trapz} N. Loehr, ``Trapezoidal Lattice Paths and
Multivariate Analogues,'' \emph{Adv. in Appl. Math.} \textbf{31} (2003),
597---629.

\bibitem{pmaj} N. Loehr, ``Combinatorics of $q,t$-Parking Functions,''
to appear in \emph{Adv. in Appl. Math.}.

\bibitem{mpark} N. Loehr and J. Remmel, ``Conjectured Combinatorial Models
for the Hilbert Series of Generalized Diagonal Harmonics Modules,'' 
\emph{Electronic Journal of Combinatorics} \textbf{11} (2004), R68. 64 pages.

\bibitem{macbook}
I. G. Macdonald, \emph{Symmetric Functions and Hall Polynomials.}
2nd ed. Oxford University Press, 1995.

\end{thebibliography}

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