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\newtheorem{MMrefine}{Theorem}
\newtheorem{BKrefine}[MMrefine]{Theorem}
\newtheorem{MM}[MMrefine]{Theorem}
\newtheorem{BK}[MMrefine]{Theorem}
\newtheorem{MMrefineLem1}[MMrefine]{Proposition}
\newtheorem{MMrefineLem2}[MMrefine]{Proposition}
\newtheorem{BKrefineLem1}[MMrefine]{Proposition}
\newtheorem{BKrefineLem2}[MMrefine]{Proposition}
\newtheorem{NLPnonrestr1}[MMrefine]{Theorem}
\newtheorem{NLPrestr1}[MMrefine]{Theorem}
\newtheorem{NLPnonrestr2}[MMrefine]{Theorem}
\newtheorem{NLPrestr2}[MMrefine]{Theorem}
\newtheorem{NLPrestr3}[MMrefine]{Theorem}
\newtheorem{Det}[MMrefine]{Lemma}
\newtheorem{Gust}[MMrefine]{Lemma}
\newtheorem{Schursquare}[MMrefine]{Theorem}
\newtheorem{qGauss}[MMrefine]{Theorem}
%\newtheorem{}[MMrefine]{Theorem}
%\newtheorem{}[MMrefine]{Theorem}
%\newtheorem{}[MMrefine]{Theorem}


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\title{THE MAJOR COUNTING OF NONINTERSECTING LATTICE PATHS AND 
GENERATING FUNCTIONS FOR TABLEAUX\\
Summary\\
\normalsize(The full-length article will appear in Mem.\ Amer.\
Math.\ Soc.)}
\author{C.~Krattenthaler\\
Institut f\"ur Mathematik der Universit\"at Wien,\\
Strudlhofgasse 4, A-1090 Wien, Austria.\\
e-mail: KRATT@PAP.UNIVIE.AC.AT
}
%\tracingmacros=2 \tracingcommands=2
\date{}

\begin{document}

\maketitle
\begin{abstract}A theory of counting nonintersecting lattice paths by the
major index and generalizations of it is developed. We obtain determinantal
expressions for the corresponding generating functions for families of
nonintersecting lattice paths with given starting points and given
final points, where the starting points lie on a line parallel to
$x+y=0$. In some cases these determinants can be evaluated to result
into simple products. As applications we compute the generating
function for tableaux with $p$ odd rows, with at most $c$ columns,
and with parts between $1$ and $n$.
Besides, we compute the generating function for the same kind of
tableaux which in addition have only odd parts. We thus also obtain a
closed form for the generating function for symmetric plane
partitions with at most $n$ rows, with parts between $1$ and $c$, and
with $p$ odd entries on the main diagonal. In each case the
result is a simple product. By summing with respect to $p$ we provide
new proofs of the Bender--Knuth and MacMahon \hbox{(ex-)}Conjectures,
which were first proved
by Andrews, Gordon, and Macdonald. The link between nonintersecting
lattice paths and tableaux is given by variations of the Knuth
correspondence.
\end{abstract}

\section{Summary of results and sketch of proofs}
We announce the proof of the following refinements of the MacMahon\break
\hbox{(ex-)}Conjecture and the Bender--Knuth
\hbox{(ex-)}Conjecture. (All the
definitions can be found in the Appendix.)

\begin{MMrefine}[Refinement of the MacMahon (ex-)Conjecture]\label{MMrefine}
The generating\break function for
tableaux with $p$ odd rows (i.e.\  exactly $p$ rows have
odd length), with at most $c$ columns,
and with only odd parts which lie between $1$ and $2n-1$, is given by
\alphaeqn
\begin{eqnarray} 
\lefteqn{q^{p^2} \frac {[2r+2p]_{q^2}\,[r]_{q^2}}
{[2r+p]_{q^2}\,[r+p]_{q^2}}\bmatrix n\\p\endbmatrix_{q^2}
\frac {\bmatrix n+2r\\n\endbmatrix_{q^2}\,} {\bmatrix
n+2r+p\\n\endbmatrix_{q^2}}}\hskip2cm\nonumber\\
&&\times\prod _{i=1} ^{n}\frac {[r+i]_{q^2}} {[i]_{q^2}}
\prod _{1\le i<j\le n} ^{}\frac {[2r+i+j]_{q^2}} {[i+j]_{q^2}}\quad \quad 
\mbox {if }c=2r \label{MMrefineA}
\end{eqnarray}
and
\begin{equation}
q^{p^2}\bmatrix n\\p\endbmatrix_{q^2}
\prod _{i=1} ^{n}\frac {[r+i]_{q^2}} {[i]_{q^2}}
\prod _{1\le i<j\le n} ^{}\frac {[2r+i+j]_{q^2}} {[i+j]_{q^2}}\quad \quad 
\mbox {if }c=2r+1.\label{MMrefineB}
\end{equation}
\reseteqn
An equivalent formulation is: The generating function for symmetric plane
partitions with at most $n$ rows, with parts between $1$
and $c$, and with exactly $p$ odd entries on the main diagonal, is
given by the expressions in (\ref{MMrefineA}) respectively
(\ref{MMrefineB}).

The formulation of this result in terms of Schur functions is that
the sum $\sum _{} ^{}s_\lambda(q^{2n-1},\break q^{2n-3},\ldots,q)$, where the sum
is over all partitions $\lambda$ with exactly $p$ odd parts, and where all
the parts do not exceed $c$, is given by the
expressions in (\ref{MMrefineA}) respectively (\ref{MMrefineB}).\quad \quad 
\qed
\end{MMrefine}

\begin{BKrefine}[Refinement of the Bender--Knuth (ex-)Conjecture]
\label{BKrefine}
The generating function for tableaux with $p$
odd rows, with at most $c$ columns, and with parts between $1$ and $n$
is given by 
\alphaeqn
\begin{equation}
q^{\binom {p+1}2}\frac {[2r]} {[2r+p]}\bmatrix n\\p\endbmatrix
\frac { \bmatrix n+2r\\n\endbmatrix} {\bmatrix 
n+2r+p\\n\endbmatrix}\prod _{1\le i\le j\le n} ^{}\frac {[2r+i+j]}
{[i+j]}\quad \quad \mbox {if\/ }c=2r\label{BKrefineA}
\end{equation}
and
\begin{equation}
q^{\binom{p+1}2}\bmatrix n\\p\endbmatrix\prod _{1\le i\le j\le n} ^{}
\frac {[2r+i+j]} {[i+j]}\quad \quad \mbox {if\/
}c=2r+1.\label{BKrefineB}
\end{equation}
\reseteqn

In terms of Schur functions: 
The sum $\sum _{} ^{}s_\lambda(q^n,q^{n-1},\ldots,q)$, where
the sum is over all partitions $\lambda$ with $p$ odd parts, where each
part does not exceed c, equals the expressions (\ref{BKrefineA})
respectively (\ref{BKrefineB}).\quad \quad \qed
\end{BKrefine}

In fact, by summing these expressions with respect to $p$ we obtain
new proofs of the MacMahon and Bender--Knuth (ex-)Conjectures itself.
\begin{MM}[MacMahon (ex-)Conjecture]\label{MM}
The generating function for symmetric plane
partitions with at most $n$ rows and with parts between $1$ and $c$
is equal to
\begin{equation}
\prod _{i=1} ^{n}\frac {[c+2i-1]_q} {[2i-1]_q}
\prod _{1\le i<j\le n} ^{}\frac {[c+i+j-1]_{q^2}} {[i+j-1]_{q^2}}\ .
\label{MMeq}\end{equation}
Equivalently, the generating function for tableaux with at most $c$
columns and with only odd parts which lie between $1$ and $2n-1$ is
also given by (\ref{MMeq}).
\end{MM}
{\bf Proof.} The expressions (\ref{MMrefineA}) are summed by a
special case of the very well-poised $_6\phi_5$-summation (see
\cite[Appendix (II.21)]{Gasp}), the expressions (\ref{MMrefineB}) are
summed by the $q$-binomial theorem (see e.g.\ \cite[(3.3.7)]{Andr1}).
\quad \quad \qed

\begin{BK}[Bender--Knuth (ex-)Conjecture]\label{BK}
The generating function for tabl\-eaux with at
most $c$ columns and with parts between $1$ and $n$ equals
\begin{equation}
\prod _{1\le i\le j\le n} ^{}\frac {[c+i+j-1]} {[i+j-1]}\ .
\label{BKeq}\end{equation}
\end{BK}
{\bf Proof.} The expressions (\ref{BKrefineA}) are summed by the
$q$-Kummer summation (see
\cite[Appendix (II.9)]{Gasp}), the expressions (\ref{BKrefineB}) are
summed by the $q$-binomial theorem (see e.g.\ \cite[(3.3.7)]{Andr1}).
\quad \quad \qed

\medskip
\noindent{\bf Remarks.} Theorem~\ref{MM} was conjectured by MacMahon
\cite[p.~270]{MacM} and only much later proved independently by 
Andrews \cite{Andr2}, Macdonald \cite[Ex.~16 and 17,
pp.~51/52]{Macd}, and Proctor \cite[Proposition~7.3]{Proc1}. 
Theorem~\ref{BK} was conjectured by Bender and Knuth \cite[p.~50]{Bend} and
proved by Andrews \cite{Andr3}, Gordon \cite{Gord}, Macdonald
\cite[Ex.~19, p.~53]{Macd}, and Proctor \cite[Proposition~7.2]{Proc1}.
The $p=0$ cases of Theorems~\ref{MMrefine} and \ref{BKrefine} were previously obtained by 
D\'esarm\'enien \cite[Th\'eor\'eme~1.2]{Desa1}, 
Proctor \cite[Theorem~1, cases (CYH) and (CYI), respectively]{Proc2},
 and Stembridge
\cite[Corollary~4.3~(a,b)]{Stem1}. The $c=2r+1$ special cases of
Theorems~\ref{MMrefine} and \ref{BKrefine} have already been discovered by
D\'esarm\'enien \cite[Th\'eor\'eme~2]{Desa2}.\quad \quad \qed
\medskip

In the sequel we give a brief outline of the proof of 
Theorems~\ref{MMrefine} and \ref{BKrefine}. 

\medskip
{\sc First Step.} Bijectively we show that the generating functions
in question are the same as certain generating functions for
certain nonintersecting families of lattice paths. The ideas which
are used in these bijections are the
celebrated Knuth correspondences \cite{Knut}, one of Burge's
\cite[p.~22]{Burg} modifications of it, the geometric interpretations of Knuth's
and Burge's correspondences due to Viennot \cite{Vien} and
Desainte-Catherine and Viennot \cite{Desa} and a refinement of Choi
and Gouyou--Beauchamps \cite{Choi}. 

The first Proposition concerns Theorem~\ref{MMrefine} for even $c$.
\begin{MMrefineLem1}\label{MMrefineLem1}There is a bijection $\Delta_1$ between 
tableaux $\tau$ with $p$ odd rows, with at most $2r$ columns,
and with only odd parts which lie between $1$ and $2n-1$, and 
nonintersecting families ${\cal P}=( P_1,\ldots, P_r)$ of 
lattice paths consisting only of double steps, 
$ P_i:(2i,-2i+2)\to(2n+2i,2n+4-2i)$,
$i=1,\ldots,r-1$, $ P_r:(2r+2p,-2r-2p+2)\to(2n+2r,2n+4-2r)$,
which lie below $x=y$ (being allowed to touch $x=y$), such that 
$$n(\tau)=\maj {\Delta_1(\tau)}+p^2.\quad \quad \qed$$
\end{MMrefineLem1}
The next Proposition shows, that once we have proved the $p=0,c=2r$-case
of Theorem~\ref{MMrefine}, we have also proved the $c=2r+1$-case of
Theorem~\ref{MMrefine}.
\begin{MMrefineLem2}\label{MMrefineLem2} There is a bijection $\Delta_2$ between 
tableaux $\tau$ with $p$ odd rows, with at most $2r+1$ columns,
and with only odd parts which lie between $1$ and $2n-1$, and
pairs $(\tau_e,S)$, where $\tau_e$ is a tableaux
with even rows, with at most $2r$ columns, and with only
odd parts which lie between $1$ and $2n-1$, and where $S$ is a $p$-subset
of $\{1,3,\ldots,2n-1\}$, such that
$$n(\tau)=n(\tau_e)+\V S\quad \quad \mbox{if } 
(\tau_e,S)=\Delta_2(\tau),$$
where $\V S$ denotes the sum of all elements of $S$.\quad \quad \qed
\end{MMrefineLem2}

What concerns the case of $c$ being even in Theorem~\ref{BKrefine}, we
have the following.
\begin{BKrefineLem1}\label{BKrefineLem1}There is a bijection $\Delta_3$ between
tableaux $\tau$ with $p$
odd rows, with at most $2r$ columns, and with parts between $1$ and
$n$, and nonintersecting families ${\cal P}=
(P_1,\ldots, P_r)$ of lattice paths which lie below
$x=y$ (being allowed to touch $x=y$), 
$ P_i:(i,-i+1)\to(n+i,n+2-i)$, $i=1,2,\ldots,r-1$,
$ P_r:(r+p,-r-p+1)\to(n+r,n+2-r)$, such that 
$$n(\tau)=\ymaj_{1;0}(\Delta_3(\tau))+\binom{p+1}2.\quad \quad \qed$$
\end{BKrefineLem1}
Also here, once we have proved the $p=0,c=2r$-case
of Theorem~\ref{BKrefine}, we have also proved the $c=2r+1$-case of
Theorem~\ref{BKrefine}, as the following Proposition shows.
\begin{BKrefineLem2}\label{BKrefineLem2} There is a bijection $\Delta_4$ between 
tableaux $\tau$ with $p$
odd rows, with at most $2r+1$ columns, and with parts between $1$ and
$n$, and pairs $(\tau_e,S)$, where $\tau_e$ is a tableau with
even rows, with at most $2r$ columns, and with parts between $1$ and
$n$, and where $S$ is a $p$-subset of $\{1,2,\ldots,n\}$, such that
$$n(\tau)=n(\tau_e)+\V S, \quad \quad \mbox{if } 
(\tau_e,S)=\Delta_4(\tau).$$
where $\V S$ denotes the sum of all elements of $S$.\quad \quad \qed
\end{BKrefineLem2}

\medskip
{\sc Second Step.} The preceding Propositions show that it is 
desirable to develop a theory of
counting nonintersecting lattive paths by the strange major index.
As a ``warm-up" we prove a theorem about counting {\it non-restricted}
nonintersecting lattice paths. This actually is not what we need in
order to prove Theorems~\ref{MMrefine} and \ref{BKrefine}. But also this
theorem has interesting consequences, which we will point out later.
\begin{NLPnonrestr1}\label{NLPnonrestr1}Let $\A_i=(A^{(i)}+D,-A^{(i)})$ and
$\E_i=(E_1^{(i)},E_2^{(i)})$,
$i=1,2,\ldots,r$, be
lattice points in the integer lattice $\Z^2$ such that
\alphaeqn
\begin{equation}
A^{(1)}< A^{(2)}< \cdots< A^{(r)}\ ,\label{Bed1}
\end{equation}
and 
\begin{equation}
E_1^{(1)}<
E_1^{(2)}< \cdots< E_1^{(r)} \mbox {\quad and\quad }E_2^{(1)}\ge
E_2^{(2)}\ge \cdots\ge E_2^{(r)}\ .\label{Bed2}
\end{equation}
\reseteqn
If $\gamma$ is an integer satisfying
$$D-E_1^{(1)}\le \gamma\le\min_{1\le i\le r}(E_2^{(i)}+i-1)$$
then the
generating function $\sum _{} ^{}q^{\mbox{\scriptsize ymaj}_{\beta;\gamma}({\cal P})}$, 
where the sum is over all nonintersecting
 families ${\cal P}=(P_1,\ldots,P_r)$ of lattice
paths, $P_i:\A_i\to\E_i$, $i=1,2,\ldots,r$, is equal to the expression
\begin{equation}
\det_{1\le s,t\le r}\Big(q^{s(A^{(s)}-A^{(t)})}\sum _{j\ge0}
^{}q^{j(j+\beta+\gamma+A^{(t)}-s+1)}\bmatrix -\beta\\\hphantom{-}j\endbmatrix\bmatrix
\beta+E_1^{(s)}+E_2^{(s)}-D\\E_1^{(s)}-A^{(t)}-D-j\endbmatrix
\Big).\quad \qed
\end{equation}
\end{NLPnonrestr1}

If the nonintersecting lattice paths are restricted by the line $x=y$
we have the following.
\begin{NLPrestr1}\label{NLPrestr1}Let $\A_i=(A^{(i)}+D,-A^{(i)})$ and
$\E_i=(E_1^{(i)},E_2^{(i)})$,
$i=1,2,\ldots,r$, be
lattice points in the integer lattice $\Z^2$ such that (\ref{Bed1}),
(\ref{Bed2}), and
$$2A^{(i)}+D\ge 0\mbox {\quad and\quad }E_1^{(i)}\ge
E_2^{(i)},\quad i=1,2,\ldots,r,
$$
hold. Let $\gamma$ be an integer which satisfies the inequalities
\begin{eqnarray}
D-E_1^{(1)}\le\gamma\le \min_{1\le i\le r}(E_2^{(i)}+i-1),&&
\max_{1\le i\le r}(D-E_2^{(i)}-i)\le\gamma\le E_1^{(1)}-1,\nonumber\\
\mbox { and }&&-A^{(1)}\le\gamma\le A^{(1)}+D.\nonumber
\end{eqnarray}
The generating function $\sum _{} ^{}q^{\mbox{\scriptsize ymaj}_{\beta;\gamma}({\cal P})}$, where
the sum is over all nonintersecting
 families ${\cal P}=(P_1,\ldots,P_r)$ of lattice
paths which lie below the line $x=y$ (being allowed to touch $x=y$), 
$P_i:\A_i\to\E_i$, $i=1,2,\ldots,r$, is equal to the expression
\begin{eqnarray}
\lefteqn{ \det_{1\le s,t\le r}\bigg(q^{s(A^{(s)}-A^{(t)})}\Big( \sum
_{j\ge0} ^{}q^{j(j+\beta+\gamma+A^{(t)}-s+1)}\bmatrix -\beta\\\hphantom{-}j\endbmatrix
\bmatrix
\beta+E_1^{(s)}+E_2^{(s)}-D\\E_1^{(s)}-A^{(t)}-D-j\endbmatrix}
\nonumber\\
&&-q^{s(2A^{(t)}+D+\beta+1)}\sum _{j\ge0} ^{}q^{j(j+\beta+\gamma+A^{(t)}+s+1)} 
\bmatrix -\beta\\\hphantom{-}j\endbmatrix\bmatrix
\beta+E_1^{(s)}+E_2^{(s)}-D\\E_2^{(s)}-A^{(t)}-D-j-1\endbmatrix
\Big)
\bigg).\nonumber\\
&&\hskip13cm\quad \qed
\end{eqnarray}
\end{NLPrestr1}

The preceding two theorems are proved in an (almost) purely
combinatorial way. What we use are ``strange major analogues" of the
usual ``interchanging procedure" for pairs of intersecting lattice
paths (cf.\ \cite{Gess3,Gess2,Stem2}) and (for Theorem~\ref{NLPrestr1}) a
``strange major analogue" for the reflection principle.

\medskip
What we need in order to finally prove Theorem~\ref{MMrefine} is
Theorem~\ref{NLPrestr1} with $\beta=0$, $D=1$, $A^{(i)}=i-1$,
$i=1,\ldots,r-1$, $A^{(r)}=r+p-1$, $E_1^{(i)}=n+i$,
$E_2^{(i)}=n+2-i$, and $q$ replaced by $q^2$. Likewise, in order to
prove Theorem~\ref{BKrefine} we need Theorem~\ref{NLPrestr1} with 
$\beta=1$, $\gamma=0$, $D=1$, $A^{(i)}=i-1$,
$i=1,\ldots,r-1$, $A^{(r)}=r+p-1$, $E_1^{(i)}=n+i$, and
$E_2^{(i)}=n+2-i$.


\medskip
{\sc Third step.} The expressions of
Theorems~\ref{NLPnonrestr1} and \ref{NLPrestr1} which we obtained for
the strange major generating functions for nonrestricted respectively
restricted nonintersecting lattice paths schematically are of the
form
$$\det\left(\sum _{j\ge0} \langle \mbox{COMPLICATED}\rangle\right).$$
We may use linearity of the determinant in the columns to get
$$
\det\left(\sum _{j\ge0} \langle
\mbox{COMPLICATED}\rangle\right)
=\sum _{k_1\ge 0} \sum _{k_2\ge0}\cdots\sum _{k_r\ge0}
\det( \langle \mbox{COMPLICATED}\rangle).
$$
We might hope that then we could evaluate the resulting determinants
at the right-hand side.
Indeed, if the final points of the paths are separated by
$(1,-1)$-steps, the determinants can be evaluated by means of the
following determinant lemma.
\begin{Det}\label{Det}Let $X_1,X_2,\ldots,X_r,A_2,A_3,\ldots,A_r,C$ be
indeterminates. If $p_0,p_1,\ldots, p_{r-1}$ are Laurent polynomials with
$\deg p_j\le j$ and $p_j(C/X)=p_j(X)$ for $j=0,1,\ldots,r-1$, then
\begin{eqnarray}
\lefteqn{\det_{1\le s,t\le r}\big((A_r+X_s)\cdots(A_{t+1}+X_s)
(A_r+C/X_s)\cdots(A_{t+1}+C/X_s)\cdot p_{t-1}(X_s)\big)}
\nonumber\\
&&\hskip2cm=\prod _{1\le i<j\le r} ^{}(X_i-X_j)(1-C/X_iX_j)\prod _{i=1}
^{r}A_i^{i-1}\prod _{i=1} ^{r}p_{i-1}(-A_i)\ ,\label{Deteq}
\end{eqnarray}
with the convention that empty products (like
$(A_r+X_t)\cdots(A_{s+1}+X_t)$ for $s=r$) are equal to 1. (The
indeterminate $A_1$, which occurs at the right-hand side of (\ref{Deteq}),
in fact is superflous since it occurs in the argument of a constant
polynomial.) A {\it Laurent polynomial} is a series $p(X)=\sum _{i=M}
^{N}a_ix^i$, $M,N\in \Z$, $a_i\in \R$.  Provided $a_N\ne0$ the {\it
degree} of $p$ is defined by $\deg p:=N$.\quad \quad \qed
\end{Det}
This Lemma is proved by simple row and column operations. It is a far
reaching generalization of the Vandermonde determinant, as might be
guessed from the expression $\prod _{1\le i<j\le r} ^{}(X_i-X_j)$ at
the right-hand side of (\ref{Deteq}).

\bigskip
Using Lemma~\ref{Det}, from Theorems~\ref{NLPnonrestr1} respectively
\ref{NLPrestr1} we obtain the following two results.
\begin{NLPnonrestr2}\label{NLPnonrestr2}Let $\A_i=(A^{(i)}+D,-A^{(i)})$ and
$\E_i=(E_1+i,E_2-i)$,
$i=1,2,\ldots,r$, be
lattice points in the integer lattice $\Z^2$ such that (\ref{Bed1})
holds. 

If $\gamma$ is an integer satisfying
$$D-E_1-1\le\gamma\le E_2-1,$$
then the generating function $\sum _{} ^{}q^{\mbox{\scriptsize ymaj}_{\beta;\gamma}({\cal P})}$ 
where the sum is over all nonintersecting
 families ${\cal P}=(P_1,\ldots,P_r)$ of lattice
paths, $P_i:\A_i\to\E_i$, $i=1,2,\ldots,r$,
is equal to the expression
\begin{eqnarray}
\lefteqn{\sum _{k_1,\ldots,k_r\ge0} ^{}\bigg(\prod _{i=1}
^{r}q^{k_i(k_i+\beta+\gamma+A^{(i)}-i+1)}
\bmatrix -\beta\\\hphantom{-}k_i\endbmatrix}\nonumber\\
&&\times\frac {[\beta+E_1+E_2+i-D-1]!}
{[\beta+E_2+A^{(i)}+k_i-1]!\,[E_1+r-D-A^{(i)}-k_i]!}\prod _{1\le i<j\le r}
^{}[A^{(j)}+k_j-A^{(i)}-k_i]\bigg)\ .\nonumber\\
&&\hskip13cm \qed
\end{eqnarray}
\end{NLPnonrestr2}

\begin{NLPrestr2}\label{NLPrestr2}Let $\A_i=(A^{(i)}+D,-A^{(i)})$ and
$\E_i=(E+i,E+2-i)$,
$i=1,2,\ldots,r$, be
lattice points in the integer lattice $\Z^2$ such that (\ref{Bed1})
and
$$2A^{(i)}+D\ge 0,\quad i=1,2,\ldots,r,
$$
 hold. If $\gamma$ is an integer satisfying
$$D-E-1\le \gamma\le E\quad \mbox {and}\quad -A^{(1)}\le \gamma\le
A^{(1)}+D,$$
then the generating function $\sum _{} ^{}q^{\mbox{\scriptsize ymaj}_{\beta;\gamma}({\cal P})}$ where
the sum is over all nonintersecting
 families ${\cal P}=(P_1,\ldots,P_r)$ of lattice
paths which lie below the line $x=y$ (being allowed to touch $x=y$),
 $P_i:\A_i\to\E_i$, $i=1,2,\ldots,r$,
is equal to the expression
\begin{eqnarray}
\lefteqn{\sum _{k_1,\ldots,k_r\ge0} ^{}\prod _{i=1}
^{r}q^{k_i(k_i+\beta+\gamma+A^{(i)}-i+1)}\bmatrix
-\beta\\\hphantom{-}k_i\endbmatrix}\nonumber\\
&&\frac {[\beta+2E+2i-D]!}
{[E+r-A^{(i)}-k_i-D]!\,[\beta+E+r+1+A^{(i)}+k_i]!}\nonumber\\
&&\prod _{1\le i<j\le r} ^{}[A^{(j)}+k_j-A^{(i)}-k_i]\prod _{1\le i\le
j\le r} ^{}[A^{(i)}+k_i+A^{(j)}+k_j+D+\beta+1]\ .\,\qed
\end{eqnarray}
\end{NLPrestr2}
Setting $\beta=0$, $D=1$, $A^{(i)}=i-1$,
$i=1,\ldots,r-1$, $A^{(r)}=r+p-1$, $E=n$,
and replacing $q$ by $q^2$ in Theorem~\ref{NLPrestr2}, in view of
Proposition~\ref{MMrefineLem1}, proves the $c=2r$-case of
Theorem~\ref{MMrefine}. Because of Proposition~\ref{MMrefineLem2}
(see the remark before Proposition~\ref{MMrefineLem2}), thus also the
$c=2r+1$-case is proved.

In order to finally prove Theorem~\ref{BKrefine}, we need
Theorem~\ref{NLPrestr2} with $\beta=1$, $\gamma=0$, $D=1$, $A^{(i)}=i-1$,
$i=1,\ldots,r-1$, $A^{(r)}=r+p-1$, and $E=n$.

\medskip
{\sc Fourth Step.} If $\beta=1$ and $\gamma=(D-1)/2$,
the following multifold basic hypergeometric summation enables us to
evaluate the sum in Theorem~\ref{NLPrestr2}. 
\begin{Gust}For $r\ge 1$ there holds the summation formula
\begin{eqnarray}
\lefteqn{\sum _{k_1,\ldots,k_r\ge0} ^{}\prod _{i=1} ^{r}\left(\frac
{\sqrt q} {q^iA}\right)^{k_i}\prod _{i=1} ^{r}\frac {(m_iA)_{k_i}}
{(qm_i/A)_{k_i}}}\hskip4cm\nonumber\\
&&\times\prod _{1\le i<j\le r} ^{}\frac {1-\frac {m_j}
{m_i}q^{k_j-k_i}} {1-\frac {m_j} {m_i}}\prod  _{1\le i\le j\le r}
\frac {1-m_im_jq^{k_i+k_j}}
{1-m_im_j}\nonumber\\
&&\hskip-2cm=\prod  _{1\le i<j\le r} \frac {1-m_im_j/q} {1-m_im_j}\prod _{i=1}
^{r}\frac {(1-m_i/\sqrt q)(1-m_i/A)} {(1-m_i^2)(1-\sqrt q/q^iA)}\
\label{Gusteq}
\end{eqnarray}
provided that there exist nonnegative integers $n_i$ with
$n_1>n_2>\cdots >n_r$ such that $m_i A=q^{-n_i}$ for all
$i=1,2,\ldots,r$.\quad \quad \qed
\end{Gust}
This summation is a special case of Gustafson's
$C_r$ $_6\psi_6$ summation (or in the old notation: 
$Sp(r)$ $_6\psi_6$ summation) \cite[Theorem~5.1]{Gust1}. The derivation of
(\ref{Gusteq}) from Gustafson's $C_r$ $_6\psi_6$ sum can be found in
\cite{Lill1}. 

\medskip
Using this summation we finally obtain.
\begin{NLPrestr3}\label{NLPrestr3}Provided the assumptions of Theorem~\ref{NLPrestr2}
hold, the generating function $\sum _{} ^{}q^{\mbox{\scriptsize ymaj}_{1;(D-1)/2}({\cal P})}$ where
the sum is over all nonintersecting
 families ${\cal P}=(P_1,\ldots,P_r)$ of lattice
paths which lie below the line $x=y$ (being allowed to touch $x=y$),
 $P_i:\A_i\to\E_i$, $i=1,2,\ldots,r$,
is equal to the expression
\begin{eqnarray}
\lefteqn{\prod _{i=1} ^{r}\frac {[2E+2i-D+1]!}
{[E+r-A^{(i)}-D]!\,[E+r+1+A^{(i)}]!}
\prod _{1\le i<j\le r} ^{}[A^{(j)}-A^{(i)}]}\nonumber\\
&&\times \prod _{1\le i< j\le r} ^{}[A^{(i)}+A^{(j)}+D+1]
\prod _{i=1} ^{r}\frac {[A^{(i)}+(D+1)/2]} {[E+i-(D-1)/2]}\ .\quad
\quad \qed
\end{eqnarray}
\end{NLPrestr3}
The proof of Theorem~\ref{BKrefine} can now be completed. For the
$c=2r$-case, in Theorem~\ref{NLPrestr3} set $D=1$, $A^{(i)}=i-1$,
$i=1,\ldots,r-1$, $A^{(r)}=r+p-1$, $E=n$, and combine with
Proposition~\ref{BKrefineLem1}. Again, because of Proposition~\ref{BKrefineLem2}
(see the remark before Proposition~\ref{BKrefineLem2}), thus also the
$c=2r+1$-case is proved.

\bigskip
Finally we come to the promised consequences of
Theorems~\ref{NLPnonrestr1} and \ref{NLPnonrestr2}. First we can prove
the following two identities about the summation of squares of Schur
functions.
\begin{Schursquare}There hold
\begin{eqnarray}
\lefteqn{\sum _{\lambda,\lambda_1\le r}
^{}s_\lambda^2(q^n,q^{n-1},\dots,q)}\nonumber\\
&&=\sum _{k_1,\dots,k_r\ge0} ^{}\prod _{i=1} ^{r}(-1)^{k_i}
q^{\binom {k_i+1}2}\frac {[2n+i]!} {[n+i+k_i]!\,[n+r-i-k_i]!}
\prod _{1\le i<j\le r} ^{}[j+k_j-i-k_i]\quad 
\end{eqnarray}
and
\begin{equation}
\sum _{\lambda,\lambda_1\le r} ^{}s^2_\lambda(q^{2n-1},q^{2n-3},\dots,q)
=\prod _{1\le i,j\le n} ^{}\frac {[r+i+j-1]_{q^2}} {[i+j-1]_{q^2}}\ .
\quad \quad \qed
\end{equation}
\end{Schursquare}
Secondly, we implicitely proved a new $A_r$ $q$-Gau{\ss} summation.
\begin{qGauss}There holds
\begin{eqnarray}
\lefteqn{\sum _{k_1,\dots,k_r\ge0} ^{}\left(\prod _{i=1} ^{r}q^{k_i(1-i)}
\left(\frac {C} {AB}\right)^{k_i}\frac {(A)_{k_i}\,(BX_i)_{k_i}}
{(q)_{k_i}\,(CX_i)_{k_i}}\right)\prod _{1\le i<j\le r} ^{}\frac {1-\frac
{X_j} {X_i}q^{k_j-k_i}} {1-\frac
{X_j} {X_i}}}\hskip4cm\nonumber\\
&&\hskip3cm=\prod _{i=1} ^{r}\frac {(\frac {C} {B}q^{i-r})_{\infty}\,(\frac {C}
{A}X_i)_{\infty}} {(CX_i)_{\infty}\,(\frac {C} {AB}q^{i-r})_\infty},
\end{eqnarray}
provided that none of the denominators vanish, $\vert q\vert<1$, and $\vert
C/AB\vert<\vert q^{r-1}\vert$.\quad \quad \qed
\end{qGauss}

\setcounter{equation}{0}
\renewcommand{\theequation}{\mbox{A.\arabic{equation}}}
\section{Appendix: Definitions}
An $r$-tuple $\lambda=(\lambda_1,\dots,\lambda_r)$ of nonnegative integers
satisfying $\lambda_1\ge\lambda_2\ge\dots\ge\lambda_r$ is called a
{\it partition}. The components $\lambda_i$ are called {\it parts} of the
partition. Let $\lambda$ be a partition.
A {\it tableau $\tau$ of shape $\lambda$} is an array 
\begin{equation}
\begin{array}{cccccc}
\tau_{11}&\tau_{12}&\multicolumn{3}{c}\dotfill &\tau_{1\lambda_1}\\
\tau_{21}&\tau_{22}&\multicolumn{2}{c}\dotfill  &\tau_{2\lambda_2}\\
\vdots&\multicolumn{2}{c}\dotfill  &\iddots\\
\tau_{r1}&\ldots&\tau_{r\lambda_r}
\end{array}\label{tableau}
\end{equation}
of positive integers $\tau_{ij}$, $1\le i\le r$, $1\le j\le \lambda_i$,
such that the rows are weakly  and the columns are strictly
increasing.
The entries of $\tau$ are called
{\it parts} of the tableau. The sum of all the parts of a tableau $\tau$
is called the {\it norm}, in symbols $n(\tau)$, of the tableau. Given
a set $T$ of tableaux, the {\it norm generating  function for $T$} is
defined to be $\sum _{\tau\in T} ^{}q^{n(\tau)}$. If we
speak of the generating function for some set of tableaux we always
mean the norm generating function.

A {\it plane partition of shape $\lambda$} is an array $\tau$ of positive
integers $\tau_{ij}$ of the form
(\ref{tableau}) such that the rows and the columns are weakly decreasing. 
The notions part, norm, generating function are used for plane
partitions in the same sense as with tableaux. A plane partition is
called {\it symmetric}, if it is symmetric with respect to the main
diagonal.

The {\it Schur function} $s_\lambda(x_1,x_2,\dots)=s_\lambda({\bf x})$ is a
symmetric function (cf.\ \cite{Macd,Stan}) in the variables $x_1,x_2,\dots$ 
and is combinatorially defined by
$$s_\lambda({\bf x})=\sum _{\tau} ^{}\prod _{} ^{}x_{\tau_{ij}}\ ,$$
where the sum is over all tableaux $\tau$ of shape $\lambda$ and the
product is over all parts $\tau_{ij}$ of $\tau$. 
The vector $\bf x$ of variables can be finite or infinite.

\medskip
\vskip10pt 
\vbox{\noindent
$$\Koordinatenachsen(8,8)(0,0)
\Gitter(8,8)(0,0)
\Pfad(1,-1),221221112122\endPfad
\DickPunkt(1,-1)
\DickPunkt(6,6)
\Label\ro{P_0}(3,3)
\hskip4cm
$$
\nobreak
\centerline{\small Figure 1}
}
\vskip10pt

In this paper we always consider lattice
paths in the plane 
consisting of unit horizontal and vertical steps
in the positive direction. We frequently call them shortly
{\it paths}.
A family of lattice paths is called {\it intersecting} if 
there are  two
paths in the family which have a point in common, if not the family
is called {\it nonintersecting}.

Any path in a natural way corresponds to a multiset permutation
consisting of $1$'s and $2$'s. Let $P$ be a path from
$\A =( A_1, A_2)$ to $\E =(E_1,E_2)
$. We frequently abbreviate the fact that a path $P$ goes from
$\A$ to $\E$ by $P:\A\to\E$.
$P$ may be represented by a pair $(\A ,\pi)$, where 
$\A $ is the
starting point of $P$ and
$\pi=\pi\sb 1\pi\sb 2\dots\pi\sb { E
\sb 1+ E
\sb 2- A\sb 1- A\sb 2}$, 
where $\pi\sb i=1$
if the $i$'th step in the path $P$ is a horizontal one and $\pi\sb
i=2$
if the $i$'th step in the path $P$ is a vertical one. $\pi$ is a
multiset permutation consisting of $ E
\sb 1- A\sb 1$ entries of $1$ and
$ E
\sb 2- A\sb 2$ entries of $2$. For example, the path $P_0$ in Figure 1 is
represented by $((1,-1),\, 221221112122)$. Of course, this representation of
paths is unique. Hence, we may identify each path with its
representation.

The {\it major index} (or ``greater index") of a multiset permutation
$\pi=\pi\sb 1\pi\sb 2\dots 
\pi\sb n$, $\pi\sb i\in {\bf
N}$ (set of positive
integers), is defined by
$$\maj \pi =\sum \sb {i=1} \sp {n-1}i\cdot \chi
 (\pi\sb i>\pi\sb {i+1})\ ,$$
where $\chi$ is the usual truth function,
$\chi
 (A)=1$ if $A$ is true, and $\chi
 (A)=0$ otherwise.


Given a path $P=(\A ,\pi)$, we extend 
the major index
 to $P$ by defining $\maj P:=\maj \pi $. For our path in Figure~1 we
have $\maj P_0=2+5+9=16$. 

By definition each couple $21$ that occurs in a multiset
permutation $\pi$, and only these, makes a contribution to the
major index of $\pi$. Given a path $P=(\A ,\pi)$,
the occurence of $21$ in $\pi$ means that a vertical step is followed
by a horizontal one. The point which is the end point of this vertical
step (and at the same time
 the starting point of this horizontal step) will be called
a {\it North-East corner} of the path $P$. 
The North-East corners of
our path in Figure~1 are $(1,1)$, $(2,3)$, and $(5,4)$.
By the above
consideration we see that only North-East corners of a path make a
contribution to the major index. Besides, the contribution of the
North-East corner $(a,b)$ is the number of steps from the starting
point of the path to $(a,b)$, or in symbols $a+b-A_1-A_2$ provided
that the starting point is $(A_1,A_2)$. 

Finally we introduce the {\it strange major index}
$\ymaj_{\beta;\gamma}$ of a path and of
families of paths. 
Let $\beta$ be some real number, $\gamma$ be an integer, 
and $P$ be a path from $\A=(A_1,A_2)$ to
$\E=(E_1,E_2)$. We define
$\ymaj_{\beta;\gamma}$
by
$$ \ymaj _{\beta;\gamma}(P)=\left\{\begin{array}{ll}
\maj P+\beta\cdot\v{\{(p_1,p_2):(p_1,p_2)\in NE(P)\mbox {
and }p_2>\gamma\}}
&\mbox {if }\gamma\ge A_2\\ 
\maj P+\beta\cdot\vert\{(p_1,p_2):(p_1,p_2)\in NE(P)&\\
\hskip2.5cm\mbox {
and }p_1\ge A_1+A_2-\gamma\}\vert+\beta(A_2-\gamma)
&\mbox {if }\gamma<A_2\end{array}\right.
$$
where $NE(P)$ denotes the set of North-East corners of $P$.
The idea is that every North-East corner which 
lies strictly above respectively to the right-hand 
side of a fixed horizontal respectively vertical line contributes an
extra weight to the ordinary major index. Clearly,
$\ymaj_{0;\gamma}$ is identically with the major index itself.
For example for the path $P_0$ in our example in Figure~1 we have
$\ymaj_{\beta;2}(P_0)=16+2\beta$, $\ymaj_{\beta;3}(P_0)=16+\beta$, or
$\ymaj_{\beta;-2}(P_0)=16+2\beta+\beta=16+3\beta$. 

The strange major index is extended to 
families ${\cal P}=(P_1,\dots,P_r)$, $P_i:\A_i\to\E_i$,
$i=1,2,\dots,r$, by
$$\ymaj_{\beta;\gamma}({\cal P})=\sum _{i=1}
^{r}\ymaj_{\beta;\gamma-i+1}(P_i).$$
Similarly, the major index of the family 
$\cal P$ is defined by $\maj {\cal P}:=\sum _{i=1} ^{r}\maj P_i$.

\medskip
The $q$-notations which are used are
$[\alpha]_q=1-q^\alpha$, $[n]_q!=[1]_q[2]_q\cdots [n]_q$, $[0]_q!=1$,
\begin{eqnarray*}(a;q)_k&=&\prod _{j=0} ^{k-1}(1-aq^j)\ ,\mbox { and
}(a;q)_0=1,\\
(a;q)_\infty &=&\prod _{j=0} ^{\infty}(1-aq^j)\ ,
\end{eqnarray*}
so that in particular $[n]_q!=(q;q)_n$, and 
$$\bmatrix n\\k\endbmatrix_q=\left\{\begin{array}{ll}\displaystyle
 \frac{[n]_q\cdot
[n-1]_q\cdots[n-k+1]_q}
{[k]_q!}&k\ge0\\
0&k<0\end{array}\right. .$$
The base $q$ in $[\alpha]_q$, $[n]_q!$, $(a;q)_k$, $(a;q)_\infty$, and 
$\left[ {n\atop k}\right]_q$ in
most cases is omitted. Only if the base is different from $q$ it
is explicitely stated.


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