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\topmatter
\title A Gog--Magog Conjecture
\endtitle
\author C.~Krattenthaler
\endauthor
\affil 
Fakult\"at f\"ur Mathematik der Universit\"at Wien,\\
Nordbergstra{\ss}e 15, A-1090 Wien, Austria.\\
WWW: \tt http://www.mat.univie.ac.at/\~{}kratt
\endaffil
\address Fakult\"at f\"ur Mathematik der Universit\"at Wien,
Nordbergstrasze 15, A-1090 Wien, Austria.
\endaddress
%\dedicatory \enddedicatory
%\date \enddate
%\thanks{$^\dagger$Research partially supported by NSF grant DMS-0500616.}\endthanks
%\thanks{$^\ddagger$Research partially supported by the Austrian
%Science Foundation FWF, grant S9607-N13,
%in the framework of the National Research Network
%``Analytic Combinatorics and Probabilistic Number Theory."}\endthanks
%\subjclass[2000] Primary 05A15;
% Secondary 05A16 05A17 05A19 05B45 33C20 52C20
%\endsubjclass
%\keywords rhombus tilings, lozenge tilings, plane partitions,
%nonintersecting lattice paths, determinant evaluations\endkeywords
\abstract
We present a conjecture relating magog trapezoids to gog trapezoids.
This generalizes a conjecture due to Mills, Robbins and Rumsey
[{\it J. Combin. Theory A} {\bf 42} (1986), 277--292].
In the second part of this note, we show how to simplify Act~I of
Zeilberger's celebrated proof [{\it Electron\. J. Combin.} {\bf 3} (no.~2, ``The Foata Festschrift")
(1996), Art.~\#R13] of the alternating sign matrix conjectre. 
\endabstract
\endtopmatter
\document


\definition{Definition}
An {\it $(m,n,k)$-Gog trapezoid\/} is an array of positive integers consisting
of the first $k$ columns of an array
$$
\matrix a_{11}&a_{12}&\hdotsfor2&a_{1n}\\
a_{21}&a_{22}&\dots&a_{2,n-1}\\
\hdotsfor3\\
a_{n1}
\endmatrix
$$
such that rows are strictly increasing, columns are weakly increasing,
and diagonals from lower-left to upper-right are weakly increasing,
and such that the entries in the right-most column are bounded by
$a_{1k}\le m+k$, $a_{2k}\le m+k+1$, \dots, $a_{n+1-k,k}\le m+n$.
\enddefinition

\definition{Definition}
An {\it $(m,n,k)$-Magog trapezoid\/} is an array of positive integers consisting
of the first $k$ rows of an array
$$
\matrix b_{11}&b_{12}&\innerhdotsfor2\after\quad &b_{1n}\\
b_{21}&b_{22}&\dots&b_{2,n-1}\\
\hdotsfor3\\
b_{n1}
\endmatrix
$$
such that rows are weakly increasing, columns are weakly decreasing,
and such that the entries in the first row are bounded by
$b_{11}\le m+1$, $b_{12}\le m+2$, \dots, $b_{1n}\le m+n$.
\enddefinition

\proclaim{Conjecture}The number of $(m,n,k)$-Gog trapezoids with $s$
Maxima in the right-most column and $t$ Minima in the left-most column
equals the number of $(m,n,k)$-Magog trapezoids with $t$
Maxima in the first row and $s$ Minima in the last row. Here, a
Maximum is an entry that is equal to its upper bound, whereas a
Minimum is an entry that is 1.

\endproclaim

What we would like to find is a bijection between the two sets in
question.

\smallskip
What is known about this conjecture? 

\smallskip
(1) If $m=0$ and $k=n$, and if we forget about Maxima and Minima,
then the conjecture reduces to the equality of the number of $n\times
n$ alternating sign matrices and the number of totally symmetric
self-complementary plane partitions contained in a $(2n)\times
(2n)\times (2n)$ box. As we know, this is a theorem, thanks to
Andrews \cite{\AndrAW} and Zeilberger \cite{\ZeilBD}, but no
bijection is known. The translation of alternating sign matrices into 
the relevant Gog trapezoids is explained in Section~5 of
\cite{\MiRRAB}. The translation of totally symmetric
self-complementary plane partitions into 
the relevant Magog trapezoids is explained in Section~2 of
\cite{\MiRRAC}. 

(2) If $m=0$ and if we forget about Maxima and Minima, then this is
Zeilberger's main theorem in \cite{\ZeilBD} (namely Lemma~1), but,
again, his proof is highly nonbijective.

(3) If $m=0$, then the above conjecture reduces to Conjecture~7' in
\cite{\MiRRAC}. 

(4) If $k=1$ then we want to show that the number of arrays of
positive integers
$$a_1\quad a_2\quad \dots \quad a_n$$
with $a_k\le m+k$ and with $s$ Maxima and $t$ Minima is exactly the
same as the number of arrays of the same type but with $t$ Maxima and
$s$ Minima.
It is easy to show that the number in question is
$$\binom {m+2n-s-t-2}{m+n-2}-\binom {m+2n-s-t-2}{m+n-1},$$
which is indeed symmetric in $s$ and $t$.

There is also a bijective way to do this, which is as follows:
Start with the array. At each step look for the left-most entry with
minimal difference to its corresponding upper-bound (i.e., find $k$
minimal such that $(m+k)-a_k$ is minimal). Remove it (i.e., remove
$a_k$), and record this difference + 1 (i.e., record $(m+k)-a_k+1$).

Here is an example for $n=6$, $m=1$, $s=2$, $t=1$: 
The first line indicates the
upper bounds. The second line is the array with which we start.
The subsequent lines show the rest of the array after each step. In
the right-most column the difference $(m+k)-a_k$ is displayed.
$$
\smatrix\format \c&\ \c&\ \c&\ \c&\ \c&\ \c\s\quad \c\\
 2&3&4&5&6&7\\
\hlinefor8\\
1&3&3&5&5&6&&1\\
&1&3&5&5&6&&1\\
&&1&3&5&6&&2\\
&&&1&3&6&&2\\
&&&&1&3&&5\\
&&&&&1&&7\\
\endsmatrix
$$
Thus the output of the algorithm is the array
$$1\quad 1\quad 2\quad 2\quad 5\quad 7\quad ,$$
which has indeed $t=1$ Maxima and $s=2$ Minima.

It is obvious that this mapping translates Maxima into Minima. Since
it can also be shown that this map is an involution (i.e., if one
applies the same mapping to the output one recovers the input), it
follows that Minima are mapped into Maxima.

(5) If $k=2$, and if we forget about Maxima and Minima,
bijective proofs have been found by Cheballah and Biane
\cite{\ChBiAA}, and by Bettinelli \cite{\BettAA}.
The bijection of Cheballah and Biane makes use of an
involution of Sch\"utzenberger on semistandard tableaux.
It is rather involved, but it preserves certain statistics
(not the number of Maxima and Minima, though). Bettinelli's
bijection is short and elegant. It uses very simple moves of entries,
but it does not preserve any natural statistics.

\medskip
Now, let us consider $(m,n,k)$-Magog trapezoids with last entry in
each row fixed, say
$$
\matrix b_{11}&b_{12}&\innerhdotsfor3\after\quad &B_1\\
b_{21}&b_{22}&\innerhdotsfor2\after\quad &B_2\\
\hdotsfor4\\
b_{k1}&b_{k2}&\dots&B_k
\endmatrix
\tag1$$
with $B_1,B_2,\dots,B_k$ fixed.

\proclaim{Proposition}The number of $(m,n,k)$-Magog trapezoids with last
entries in each row as shown in {\rm(1)} equals
$$\CT\left(\frac {\Delta_k(m;x_1,\dots,x_k)} {\prod _{i=1}
^{k}x_i^{B_i+k-i-1}(1-x_i)^{k+n+m}}\right),\tag2$$
where
$$\align \Delta_k(m;x_1,\dots,x_k)&=\det_{1\le i,j\le
k}\left(x_j^{k-i}(1-x_j)^{m+k+i-1}-(1-x_j)^{k-i}x_j^{m+k+i-1}\right)\\
&=\sum _{g\in\Cal B_k} ^{}\sgn g\cdot g\left[\prod _{i=1}
^{k}x_i^{k-i}(1-x_i)^{m+k+i-1}\right].
\endalign$$
Here, $\Cal B_k$ denotes the hyperoctahedral group of rank~$k$,
acting on $(x_1,x_2,\dots,x_k)$ by permuting coordinates and
by replacing $x_i$ by $1-x_i$.

\endproclaim
\remark{Remark} This generalizes Subsublemma~1.1.1 in \cite{\ZeilBD}.
At the same time, the proof below is much simpler than the one in
\cite{\ZeilBD}.
\endremark
\demo{Proof}First we convert the $(m,n,k)$-Magog trapezoids into
arrays of the same form which are {\it strictly\/} increasing along
rows, by adding $k-i$ to each entry in the $i$-th row. Thus, we obtain
arrays of the form
$$
\matrix b_{11}&b_{12}&\innerhdotsfor3\after\quad &B_1+k-1\\
b_{21}&b_{22}&\innerhdotsfor2\after\quad &B_2+k-2\\
\hdotsfor4\\
b_{k1}&b_{k2}&\dots&B_k
\endmatrix$$
with $B_1,B_2,\dots,B_k$ fixed, with weakly increasing rows, strictly
increasing columns, and such that the entries in the first row satisfy
$b_{11}\le m+k$, $b_{12}\le m+k+1$, \dots. Now, by reading row-wise, 
we convert these arrays
into families $(P_1,P_2,\dots,P_k)$ of nonintersecting lattice paths,
$P_i$ running from $A_i=(i-1,k-i+1)$ to $E_i=(n-1,B_i+k-i)$, all the
paths staying below the line $y=x+k+m$, being allowed to touch it. In
this correspondence, the $i$-th path corresponds to the $i$-th row of
the array in the obvious way (heights of horizontal steps correspond
to entries in the array, ignoring the last entry in each row,
$B_i+i-1$, which is encoded in the end point of the path anyway).

By Gessel--Viennot theory, the number of these families of
nonintersecting lattice paths equals the determinant
$$\det_{1\le i,j\le k}\big(\Cal P^+(A_i\to E_j)\big),\tag3$$
where $\Cal P^+(A\to E)$ denotes the number of lattice paths from $A$
to $E$ which stay (weakly) below the line $y=x+k+m$. By reflection
principle, these paths numbers are the difference between two path
numbers. So, writing $\Cal P(A\to E)$ for the number of paths from $A$
to $E$ (without any restriction), and $\Cal R$ for the reflection with
respect to the line $y=x+k+m+1$, expression (3) becomes
$$\align \det_{1\le i,j\le k}&\big(\Cal P^+(A_i\to E_j)\big)=
\det_{1\le i,j\le k}\big(\Cal P(A_i\to E_j)-\Cal P(\Cal RA_i\to
E_j)\big)\\
&=\det_{1\le i,j\le k}\(\binom {B_j+n-j-1}{n-i}-\binom {B_j+n-j-1}{n+m+i-1}\)\\
&=\det_{1\le i,j\le k}\(\CT\(\frac {(1-x)^{-n+i-1}} {x^{B_j+i-j-1}}-
\frac {(1-x)^{-n-m-i}} {x^{B_j-m-i-j}}\)\)\\
&=\CT\bigg(\prod _{i=1} ^{k}\frac {1} {x_i^{B_i+k-i-1}(1-x_i)^{n+m+k}}\\
&\hskip2cm\times\det_{1\le i,j\le k}\(x_j^{k-i}(1-x_j)^{m+k+i-1}-
x_j^{m+k+i-1}(1-x_j)^{k-i}\)\bigg).
\endalign$$
\line{\hbox{\hskip12cm} \qed \hss}



\enddemo
It is an easy exercise to show that in case $m=0$ the determinant
$\Delta_k$ simplifies to
$$\Delta_k(0;x_1,\dots,x_k)=\prod _{i=1} ^{k}(1-2x_i)\prod _{1\le
i<j\le k} ^{}(x_j-x_i)(x_i+x_j-1),$$
which is the expression that appears in \cite{\ZeilBD}.

\proclaim{Theorem}The number of $(m,n,k)$-Magog trapezoids 
equals
$$\CT\left(\frac {\Delta_k(m;x_1,\dots,x_k)} {\prod _{i=1}
^{k}x_i^{m+n+k-i-1}(1-x_i)^{k+n+m+1}\prod _{1\le i<j\le k}
^{}(1-x_ix_j)}\right),$$
where $\Delta_k(m;x_1,\dots,x_k)$ is defined as in the Proposition.
\endproclaim
\remark{Remark} This generalizes Sublemma~1.1 in \cite{\ZeilBD}.
\endremark

\demo{Proof}This is done in the same way as in \cite{\ZeilBD}.

We have to sum the expressions (2) over $m+n-1\ge B_1\ge B_2\ge
\dots\ge B_k\ge -\infty$. That is,
\NoBlackBoxes
$$\align &\sum _{m+n-1\ge B_1\ge
\dots\ge B_k\ge -\infty} ^{}\CT\left(\frac {\Delta_k(m;x_1,\dots,x_k)} {\prod _{i=1}
^{k}x_i^{B_i+k-i-1}(1-x_i)^{k+n+m}}\right)\\
&\quad =\sum _{0\le \lambda_1\le\dots\le \lambda_k} ^{}
\CT\left(\frac {\Delta_k(m;x_1,\dots,x_k)} {\prod _{i=1}
^{k}x_i^{m+n+k-i-1-\lambda_i}(1-x_i)^{k+n+m}}\right)\\
&\quad =\CT\(\frac {\Delta_k(m;x_1,\dots,x_k)} {\prod _{i=1}
^{k}x_i^{m+n+k-i-1}(1-x_i)^{k+n+m}}\frac {1}
{(1-x_k)(1-x_{k-1}x_k)\cdots(1-x_1\cdots x_k)}\)\\
&\quad =\frac {1} {k!}\CT\Bigg(\frac {\Delta_k(m;x_1,\dots,x_k)} {\prod _{i=1}
^{k}x_i^{m+n+k-1}(1-x_i)^{k+n+m}}\\
&\hskip3cm \times\sum _{\pi\in S_k} ^{}\sgn\pi\cdot
\pi\left[\frac {x_1x_2^2\cdots x_k^k} {(1-x_k)\cdots (1-x_1x_2\cdots
x_k)}\right]\Bigg)\\
&\quad =\frac {1} {k!}\CT\(\frac {\Delta_k(m;x_1,\dots,x_k)} {\prod _{i=1}
^{k}x_i^{m+n+k-1}(1-x_i)^{k+n+m}}
\frac {x_1\cdots x_k\prod _{1\le i<j\le k} ^{}(x_i-x_j)} {\prod _{i=1}
^{k}(1-x_i)\prod _{1\le i<j\le k} ^{}(1-x_ix_j)}\)\\
&\quad =\frac {1} {k!}\CT\Bigg(\frac {\Delta_k(m;x_1,\dots,x_k)} {\prod _{i=1}
^{k}x_i^{m+n+k-1}(1-x_i)^{k+n+m}}
\frac {x_1\cdots x_k} {\prod _{i=1}
^{k}(1-x_i)\prod _{1\le i<j\le k} ^{}(1-x_ix_j)}\\
&\hskip3cm \times\sum _{\pi\in S_k}
^{}\sgn\pi\cdot \pi\left[x_1^0x_2^1\cdots x_k^{k-1}\right]\Bigg)\\
&\quad =\frac {1} {k!}\sum _{\pi\in S_k} ^{}
\CT\(\pi\left[\frac {\Delta_k(m;x_1,\dots,x_k)} {\prod _{i=1}
^{k}x_i^{m+n+k-1}(1-x_i)^{k+n+m+1}}
\frac {x_1x_2^2\cdots x_k^k} {\prod _{1\le i<j\le k} ^{}(1-x_ix_j)}
\right]\)\\
&\quad =\frac {1} {k!}\sum _{\pi\in S_k} ^{}
\CT\(\frac {\Delta_k(m;x_1,\dots,x_k)} {\prod _{i=1}
^{k}x_i^{m+n+k-1}(1-x_i)^{k+n+m+1}}
\frac {x_1x_2^2\cdots x_k^k} {\prod _{1\le i<j\le k} ^{}(1-x_ix_j)}
\)\\
&\quad =\CT\(\frac {\Delta_k(m;x_1,\dots,x_k)} {\prod _{i=1}
^{k}x_i^{m+n+k-1}(1-x_i)^{k+n+m+1}}
\frac {x_1x_2^2\cdots x_k^k} {\prod _{1\le i<j\le k} ^{}(1-x_ix_j)}
\).
\endalign$$
\line{\hbox{\hskip7cm}\qed \hss}
\enddemo



\Refs

\ref\no \AndrAW\by G. E. Andrews \yr 1994 
\paper Plane partitions V: The t.s.s.c.p.p\. conjecture
\jour J.~Combin\. Theory Ser\. A \vol 66 
\pages 28--39\endref

\ref\no \BettAA\by J. Bettinelli \yr 2016
\paper A simple explicit bijection between $(n,2)$-Gog and Magog 
trapezoids
\jour S\'em\. Lothar\. Combin\. \vol B75
\pages Article~B75e, 9~pp\endref

\ref\no \ChBiAA\by H. Cheballah and P. Biane \yr 2012
\paper Gog and Magog triangles, and the Sch\"utzenberger involution
\jour S\'em\. Lothar\. Combin\. \vol B66
\pages Article~B66d, 20~pp\endref

\ref\no \MiRRAB\by W. H. Mills, D. H. Robbins and H. Rumsey \yr 1983 
\paper Alternating sign matrices and descending plane partitions
\jour J. Combin\. Theory Ser.~A\vol 34
\pages 340--359\endref

\ref\no \MiRRAC\by W. H. Mills, D. H. Robbins and H. Rumsey \yr 1986 
\paper Self--complementary totally symmetric plane partitions
\jour J. Combin. Theory A\vol 42
\pages 277--292 \endref

\ref\no \ZeilBD\by D.    Zeilberger \yr 1996 
\paper Proof of the alternating sign matrix conjecture
\jour Electron\. J. Combin\.\vol 3 \rm(no.~2, ``The Foata Festschrift")
\pages Article~\#R13, 84~pp\endref

\endRefs
\end{document}

