\input amstex
\input amspptbk.sty
\input sekr

\magnification1200
\input format

\input texdraw
\input path.sty

\TagsOnRight


\def\MiRRAB{8}
\def\MiRRAA{7}
\def\LaloAA{6}
\def\DaToAA{5}
\def\CiEKAA{4}
\def\CiKrAC{3}
\def\AndrAO{2}
\def\AndrAN{1}

%Christian's texdraw definitions
\def\ldreieck{\bsegment
  \rlvec(0.866025403784439 .5) \rlvec(0 -1)
  \rlvec(-0.866025403784439 .5)  
  \savepos(0.866025403784439 -.5)(*ex *ey)
        \esegment
  \move(*ex *ey)
        }
\def\rdreieck{\bsegment
  \rlvec(0.866025403784439 -.5) \rlvec(-0.866025403784439 -.5)  \rlvec(0 1)
  \savepos(0 -1)(*ex *ey)
        \esegment
  \move(*ex *ey)
        }
\def\rhombus{\bsegment
  \rlvec(0.866025403784439 .5) \rlvec(0.866025403784439 -.5) 
  \rlvec(-0.866025403784439 -.5)  \rlvec(0 1)        
  \rmove(0 -1)  \rlvec(-0.866025403784439 .5) 
  \savepos(0.866025403784439 -.5)(*ex *ey)
        \esegment
  \move(*ex *ey)
        }
\def\RhombusA{\bsegment
  \rlvec(0.866025403784439 .5) \rlvec(0.866025403784439 -.5) 
  \rlvec(-0.866025403784439 -.5) \rlvec(-0.866025403784439 .5) 
  \savepos(0.866025403784439 -.5)(*ex *ey)
        \esegment
  \move(*ex *ey)
        }
\def\RhombusB{\bsegment
  \rlvec(0.866025403784439 .5) \rlvec(0 -1)
  \rlvec(-0.866025403784439 -.5) \rlvec(0 1) 
  \savepos(0 -1)(*ex *ey)
        \esegment
  \move(*ex *ey)
        }
\def\RhombusC{\bsegment
  \rlvec(0.866025403784439 -.5) \rlvec(0 -1)
  \rlvec(-0.866025403784439 .5) \rlvec(0 1) 
  \savepos(0.866025403784439 -.5)(*ex *ey)
        \esegment
  \move(*ex *ey)
        }
\def\RhombusAsh{\bsegment
  \rlvec(0.866025403784439 .5) \rlvec(0.866025403784439 -.5) 
  \rlvec(-0.866025403784439 -.5) \rlvec(-0.866025403784439 .5) 
  \lfill f:.8
  \savepos(0.866025403784439 -.5)(*ex *ey)
        \esegment
  \move(*ex *ey)
        }
\def\RhombusASh{\bsegment
  \rlvec(0.866025403784439 .5) \rlvec(0.866025403784439 -.5) 
  \rlvec(-0.866025403784439 -.5) \rlvec(-0.866025403784439 .5) 
  \lfill f:.2
  \savepos(0.866025403784439 -.5)(*ex *ey)
        \esegment
  \move(*ex *ey)
        }
\def\RhombusBsh{\bsegment
  \rlvec(0.866025403784439 .5) \rlvec(0 -1)
  \rlvec(-0.866025403784439 -.5) \rlvec(0 1) 
  \lfill f:0.8
  \savepos(0 -1)(*ex *ey)
        \esegment
  \move(*ex *ey)
        }
\def\RhombusCsh{\bsegment
  \rlvec(0.866025403784439 -.5) \rlvec(0 -1)
  \rlvec(-0.866025403784439 .5) \rlvec(0 1) 
  \savepos(0.866025403784439 -.5)(*ex *ey)
        \esegment
  \move(*ex *ey)
        }
\def\hdSchritt{\bsegment
  \lpatt(.05 .13)
  \rlvec(0.866025403784439 -.5) 
  \savepos(0.866025403784439 -.5)(*ex *ey)
        \esegment
  \move(*ex *ey)
        }
\def\vdSchritt{\bsegment
  \lpatt(.05 .13)
  \rlvec(0 -1) 
  \savepos(0 -1)(*ex *ey)
        \esegment
  \move(*ex *ey)
        }

\def\odSchritt{\bsegment
  \lpatt(.05 .13)
  \rlvec(-0.866025403784439 -.5) 
  \savepos(-0.866025403784439 -.5)(*ex *ey)
        \esegment
  \move(*ex *ey)
        }


\def\hdaSchritt{\bsegment
  \lpatt(.05 .13)
  \rlvec(-0.866025403784439 .5) 
  \savepos(-0.866025403784439 .5)(*ex *ey)
        \esegment
  \move(*ex *ey)
        }
\def\vdaSchritt{\bsegment
  \lpatt(.05 .13)
  \rlvec(0 1) 
  \savepos(0 1)(*ex *ey)
        \esegment
  \move(*ex *ey)
        }

\def\odaSchritt{\bsegment
  \lpatt(.05 .13)
  \rlvec(0.866025403784439 .5) 
  \savepos(0.866025403784439 .5)(*ex *ey)
        \esegment
  \move(*ex *ey)
        }



\def\hhdSchritt{\bsegment
  \lpatt(.05 .13)
  \rlvec(0.4330127 -.25) 
  \savepos(0.4330127 -.25)(*ex *ey)
        \esegment
  \move(*ex *ey)
        }

\def\hvdSchritt{\bsegment
  \lpatt(.05 .13)
  \rlvec(0 -.5) 
  \savepos(0 -.5)(*ex *ey)
        \esegment
  \move(*ex *ey)
        }

\def\hodSchritt{\bsegment
  \lpatt(.05 .13)
  \rlvec(-0.4330127 -.25) 
  \savepos(-0.4330127 -.25)(*ex *ey)
        \esegment
  \move(*ex *ey)
        }


\def\hhdaSchritt{\bsegment
  \lpatt(.05 .13)
  \rlvec(-0.4330127 .25) 
  \savepos(-0.4330127 .25)(*ex *ey)
        \esegment
  \move(*ex *ey)
        }

\def\hvdaSchritt{\bsegment
  \lpatt(.05 .13)
  \rlvec(0 .5) 
  \savepos(0 .5)(*ex *ey)
        \esegment
  \move(*ex *ey)
        }

\def\hodaSchritt{\bsegment
  \lpatt(.05 .13)
  \rlvec(0.4330127 .25) 
  \savepos(0.4330127 .25)(*ex *ey)
        \esegment
  \move(*ex *ey)
        }


\def\ringerl(#1 #2){\move(#1 #2)\fcir f:0 r:.1}


\def\hex{\bsegment
        \rlvec(1 0)  \rlvec(.5 -.866025403784439) \rlvec(-.5 -.866025403784439)
        \rlvec(-1 0) \rlvec(-.5 .866025403784439) \rlvec(.5 .866025403784439)
        \savepos(1.5 -.866025403784439)(*ex *ey)
         \esegment
        \move(*ex *ey)
}

\def\shex{\bsegment 
        \lpatt(.05 .13)
        \rlvec(1 0)  \rlvec(.5 -.866025403784439) \rlvec(-.5 -.866025403784439)
        \rlvec(-1 0) \rlvec(-.5 .866025403784439) \rlvec(.5 .866025403784439)
        \savepos(1.5 -.866025403784439)(*ex *ey)
         \esegment
        \move(*ex *ey)
}
%ende

\topmatter 
\title Descending plane partitions and rhombus tilings of a hexagon
with triangular hole
\endtitle 
\author C.~Krattenthaler$^\dagger$
\endauthor 
\affil 
Institut Girard 
Desargues, Universit\'e Claude Bernard Lyon-I,\\
21, avenue Claude Bernard, F-69622 Villeurbanne Cedex, France.\\
e-mail: kratt\@igd.univ-lyon1.fr\\
WWW: \tt http://igd.univ-lyon1.fr/\~{}kratt
\endaffil
\address Institut Girard 
Desargues, Universit\'e Claude Bernard Lyon-I, 
21, avenue Claude Bernard, F-69622 Villeurbanne Cedex, France.
\endaddress
\thanks{$^\dagger$Research partially supported by EC's IHRP Programme,
grant HPRN-CT-2001-00272, ``Algebraic Combinatorics in Europe"}\endthanks
\subjclass Primary 05A15
\endsubjclass
\keywords descending plane partitions, rhombus tilings of a hexagon
\endkeywords
\abstract 
It is shown that the descending plane partitions of Andrews 
can be geometrically realized as cyclically symmetric rhombus tilings
of a certain 
hexagon of which a centrally located equilateral triangle of side length 2 
has been removed. 
Thus, the lattice structure for descending plane
partitions, as introduced by Mills, Robbins and Rumsey, allows for an
elegant visualization.
\endabstract
\endtopmatter
\document


\subhead 1. Introduction\endsubhead
Descending plane partitions were introduced by Andrews \cite{\AndrAN}
in his attempt to prove Macdonald's conjecture on the $q$-enumeration
of cyclically symmetric plane partitions. (In \cite{\AndrAN},
Andrews only succeeded to prove the $q=1$ case of the conjecture; the
full conjecture was later proved by Mills, Robbins and Rumsey
\cite{\MiRRAA}.) It came as a big surprise, when Mills, Robbins and
Rumsey discovered in \cite{\MiRRAB} that there are, apparently, also
close connections of descending plane partitions to alternating sign
matrices. However, the corresponding conjectures of \cite{\MiRRAB} still remain
mysteries. 

In the recent paper \cite{\LaloAA}, Lalonde showed that descending
plane partitions can be nicely encoded in terms of families of
non-intersecting lattice paths. In this encoding, 
the natural lattice structure of descending plane partitions, 
that had been introduced in \cite{\MiRRAB}, has a rather
straight-forward realization.
In particular, the unique antiautomorphism of the lattice, which in
\cite{\MiRRAB} has a rather complicated definition, can be defined in
very simple terms.

The purpose of this note is to point out that there is an even more striking
realization of descending plane partitions, in terms of rhombus
tilings of a hexagon of which a centrally located 
equilateral triangle of side length 2 
has been removed. In this picture, the lattice
structure, and particularly the antiautomorphism, become even more
transparent. In addition, this interpretation extends effortlessly to
the more general $d$-descending plane partitions of Andrews (the case
$d=0$ corresponding to descending plane partitions), thus implying
analogous results for these more general objects.

\subhead 2. Descending plane partitions\endsubhead
A {\it descending plane partition}
is an array $\pi$ of positive integers of the form
$$\matrix \pi\sb
{1,1}
&\pi\sb
{1,2}&\innerhdotsfor6\after\quad &\pi\sb
{1,\lambda
\sb
1} \\
&\pi\sb
{2,2}&\innerhdotsfor5\after\quad &\pi\sb
{2,\lambda
\sb
2}\\
& &\ddots& &\hphantom{mm} &
 &\mathinner{\mskip1mu\raise1pt\vbox{\hbox{.}}\mskip2mu
\raise4pt\hbox{.}\mskip2mu\raise7pt\vbox{\kern7pt\hbox{.}}\mskip1mu}\\
& & &\pi\sb
{k,k}&\innerhdotsfor1\after\quad &\pi\sb
{k,\lambda
\sb
k}& & &
\endmatrix\tag2.1$$
such that

\roster 
\item $\la_1\ge\la_2\ge \dots\ge\la_k\ge k$,
\item the entries along rows are weakly decreasing,
\item the entries along columns are strictly decreasing,
\item the first entry in each row does not exceed the number of
entries in the preceding row but is greater than the number of entries
in its own row.
\endroster
\noindent
A typical example is the array
$$\matrix 6&6&6&4&2\\
&5&3&2&1\\
&&2\endmatrix\quad .
\tag2.2$$
(This is, in fact, the running example $D_0$ in \cite{\LaloAA}.)
Andrews \cite{\AndrAN, Theorem~10} 
has shown that the number of descending plane partitions with
entries $\le n$ is equal to
$$
\prod _{i=1} ^{n}
\prod _{j=1} ^{n}\frac {n+i+j-1} {2i+j-1}.$$

Mills, Robbins and Rumsey also defined certain statistics on
descending plane partitions. (The enumeration with respect to these
statistics is conjecturally identical with the enumeration of
alternating sign matrices with respect to certain other statistics, see
\cite{\MiRRAB, Conj.~3}.) Let us call an entry $\pi_{i,j}$ of a
descending plane partition {\it special}, if $\pi_{i,j}\le j-i$. 
Thus, the special entries in the descending plane partition in (2.2) are
the 2 in the first row and the 2 and 1 in the second row.
For a given descending plane partition $\pi$ with
entries $\le n$, the statistics introduced in \cite{\MiRRAB} are

\roster 
\item "$\bullet$"the number of entries of $\pi$ which equal
to $n$, denoted by $r(\pi)$;
\item "$\bullet$"the number of special entries of $\pi$, denoted by $s(\pi)$;
\item "$\bullet$"the total number of entries of $\pi$, denoted by $i(\pi)$.
\endroster

For the example in (2.2), we have $r(D_0)=3$, $s(D_0)=3$, and 
$i(D_0)=10$.

The lattice structure on descending plane partitions with entries $\le
n$ is given by the
following partial order:
given two descending plane partitions $\pi$ and $\si$, we define
$\pi\le \si$ if, for all $i$ and $j$, whenever $\pi_{i,j}$ exists
then $\si_{i,j}$ also exists and, in addition, $\pi_{i,j}\le
\si_{i,j}$. We shall not recall the definition of the
antiautomorphism $\ta$ given in \cite{\MiRRAB} here. For our
purposes, it will suffice to note that it is shown in \cite{\MiRRAB}
that it is unique, and that, for any descending plane partition $\pi$
with entries $\le n$ it satisfies the following properties:
$$\align r(\pi)+r(\ta(\pi))&=n-1,\tag2.3\\
s(\ta(\pi))&=s(\pi),\tag2.4\\
i(\pi)+i(\ta(\pi))&=\binom n2+s(\pi).\tag2.5
\endalign$$


\subhead 3. Rhombus tilings and descending plane partitions\endsubhead
In this section we show that descending plane partitions with entries
$\le n$ are in bijection with cyclically symmetric 
rhombus tilings of a hexagon with side
lengths $n-1,n+1,n-1,n+1,n-1,n+1$ of which a centrally located
equilateral triangle
with side length 2 has been removed. (In the hexagon,
we assume that the angles between sides are $120^\circ$.
See Figure~1 for the case that $n=6$.) By a
rhombus we always mean a rhombus with side lengths 1 and angles of
$60^\circ$ and $120^\circ$. Finally, a rhombus tiling is called {\it
cyclically symmetric}, if it is invariant under rotation by
$120^\circ$. A cyclically symmetric rhombus tiling of the hexagon with
triangular hole in the case $n=6$
is shown in Figure~2. (At this point, all shadings, as well as the
thick and dotted lines should be ignored.)


\midinsert
\centertexdraw{
  \drawdim truecm \setunitscale.8 \linewd.02
  \rhombus \rhombus \rhombus \rhombus \rhombus \ldreieck
  \move (-0.866025403784439 -.5)
  \rhombus \rhombus \rhombus \rhombus \rhombus \rhombus \ldreieck
  \move (-1.732050807568877 -1)
  \rhombus \rhombus \rhombus \rhombus \rhombus \rhombus \rhombus \ldreieck
  \move (-2.598076211353316 -1.5)
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  \move (-3.4641 -2)
  \rhombus \rhombus \rhombus \rhombus \rhombus \rhombus 
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  \move (-4.3301 -2.5)
  \rhombus \rhombus \rhombus \rhombus \rhombus 
  \rmove (.866025 .5)
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  \move (-5.19615 -3)
  \rhombus \rhombus \rhombus \rhombus \rhombus 
  \rmove (1.73205 0)
  \rdreieck \rhombus \rhombus \rhombus \rhombus \ldreieck
  \move (-5.19615 -3)
  \rdreieck
  \rhombus \rhombus \rhombus  \rhombus \rhombus \rhombus \rhombus 
  \rhombus \rhombus \rhombus  \rhombus 
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  \rdreieck
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  \rhombus \rhombus \rhombus  \rhombus 
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  \rdreieck
  \rhombus \rhombus \rhombus \rhombus \rhombus \rhombus 
  \rhombus \rhombus \rhombus 
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  \rdreieck
  \rhombus \rhombus \rhombus \rhombus \rhombus \rhombus 
  \rhombus \rhombus 
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  \rdreieck
  \rhombus \rhombus \rhombus \rhombus \rhombus 
  \rhombus \rhombus 
}
\vskip6pt
\centerline{\eightpoint Figure 1}
\bigskip
\endinsert


\midinsert
\bigskip
\centertexdraw{
\drawdim truecm \setunitscale.8
\linewd.05
\move(0 0)
\RhombusAsh \RhombusA \RhombusA 
\RhombusB \RhombusB \RhombusA \RhombusB \RhombusB 
\RhombusA \RhombusB \RhombusB \RhombusBsh 
\move(-.866025 -.5)
\RhombusA \RhombusAsh \RhombusB \RhombusB
\RhombusA \RhombusB \RhombusA \RhombusB \RhombusA \RhombusB
\RhombusBsh \RhombusB 
\move(-1.73205 -1)
\RhombusA \RhombusA \RhombusB \RhombusB
\RhombusAsh \RhombusB \RhombusB \RhombusBsh \RhombusB
\RhombusA \RhombusA \RhombusB
\move(-2.598 -1.5)
\RhombusB \RhombusA \RhombusA \RhombusB \RhombusA 
\move(-3.4641 -2)
\RhombusB \RhombusA \RhombusB \RhombusA \RhombusA 
\move(-4.3301 -2.5)
\RhombusB \RhombusB \RhombusA \RhombusB
\RhombusA \RhombusA \RhombusA \RhombusB
\RhombusA \RhombusB \RhombusA \RhombusA 
\move(-5.19615 -3)
\RhombusB \RhombusB \RhombusA \RhombusB
\RhombusA \RhombusA \RhombusB \RhombusB
\RhombusA \RhombusA \RhombusA \RhombusA
\move(-1.73205 -1)
\RhombusC
\move(1.73205 -1)
\RhombusC
\move(1.73205 -5)
\RhombusC
\move(1.73205 -6)
\RhombusC \RhombusC
\move(1.73205 -7)
\RhombusC \RhombusC
\move(3.4641 -1)
\RhombusC \RhombusC
\move(3.4641 -2)
\RhombusC \RhombusC
\move(4.3301 -3.5)
\RhombusC 
\move(4.3301 -4.5)
\RhombusC 
\move(.866025 -8.5)
\RhombusC 
\move(-1.73205 -8)
\RhombusC 
\move(-2.598 -4.5)
\RhombusC \RhombusC \RhombusC \RhombusC 
\move(-5.19615 -5)
\RhombusC 
\move(-5.19615 -6)
\RhombusC \RhombusC \RhombusC 
\move(-5.19615 -7)
\RhombusC \RhombusC \RhombusC 
%
\move(1.29904 -.25)
\hdSchritt \hdSchritt \vdSchritt \vdSchritt 
\hdSchritt \vdSchritt \vdSchritt \hdSchritt 
\vdSchritt \vdSchritt 
\move(1.29904 -1.25)
\vdSchritt \vdSchritt \hdSchritt \vdSchritt 
\hdSchritt \vdSchritt \hdSchritt \vdSchritt 
\move(1.29904 -4.25)
\vdSchritt \vdSchritt
%
\linewd.1
\move (0.8660254 0.5)
\rlvec(0 -7)
\rlvec(4.33013 -2.5)
}
\centerline{\eightpoint Figure 2}
\bigskip
\endinsert

To construct the announced bijection, given a rhombus tiling, 
we restrict the tiling to
a fundamental region (see the region cut out by the thick lines in
Figure~2), and we read off a family $(P_1,P_2,\dots,P_k)$ (for some $k$) of
non-intersecting lattice paths (the notion ``non-intersecting" will be
explained in a moment) consisting of horizontal unit steps in the
positive direction and vertical unit steps in the
negative direction, where $P_i$ runs from $(0,x_i+2)$ to
$(x_i,0)$, $0\le x_1<x_2<\dots<x_k$, in a way which is now standard
(see e.g\. \cite{\CiKrAC, \CiEKAA}). That is, we mark the rhombi of
the tiling that are cut in two by the borders of the fundamental
region (these are the shaded rhombi in Figure~2), and subsequently we
connect the latter rhombi which are located along the left border 
with those rhombi which are located along the bottom border by paths.
The paths result by connecting, for each horizontally oriented rhombus
and for each rhombus oriented south-west/north-east, 
the midpoints of the two edges of the rhombus
pointing from south-west to north-east. (In Figure~2, the
paths are indicated by dotted lines.) These paths are finally deformed
so that they become proper orthogonal lattice paths, see
Figure~3. (The labels should be ignored at this point.) 
Since they come from a tiling, they are {\it
non-intersecting}, meaning that any two paths have no point in common.

\midinsert
$$
\PfadDicke{2pt}
\Einheit.7cm
\Gitter(6,7)(0,0)
\Koordinatenachsen(6,7)(0,0)
\Pfad(0,6),1166166166\endPfad
\Pfad(0,5),66161616\endPfad
\Pfad(0,2),66\endPfad
\DickPunkt(0,2)
\DickPunkt(0,5)
\DickPunkt(0,6)
\DickPunkt(0,0)
\DickPunkt(3,0)
\DickPunkt(4,0)
\Label\ro{6}(0,6)
\Label\ro{6}(1,6)
\Label\ro{4}(2,4)
\Label\ro{2}(3,2)
\Label\ro{3}(0,3)
\Label\ro{2}(1,2)
\Label\ro{1}(2,1)
\Label\lo{6}(0,6)
\Label\lo{5}(0,5)
\Label\lo{2}(0,2)
\hskip4.2cm
$$
\centerline{\eightpoint Figure 3}
\endinsert

To extract the descending plane partition corresponding to the rhombus
tiling, we read the heights of the horizontal steps along each path,
including the height of the starting point of the path,
but ignoring heights of $0$ (in Figure~3,
the heights to be read are the numbers in the figure), each path
contributing one row of the corresponding descending plane partition. 
It is easy to verify that, in that manner, we do indeed obtain a
descending plane partition, as well as that this correspondence is a
bijection. 
In particular, as can be seen from a comparison of (2.2) and Figure~3, the
descending plane partition corresponding to the rhombus tiling of
Figure~2 is the array (2.2). 
(It can now be realized that the path picture is identical with the
path figure in Figure~1 in \cite{\LaloAA}, except that the first vertical
strip of the grid there 
has been removed here, the latter being not necessary anyway, as
it is forced). 

It is easy to see that the number $r(\pi)$ of entries of size
$n$ is the number of horizontally oriented rhombi in the upper-most
strip of the fundamental region
(should it be contained in the tiling,
the horizontally oriented rhombus situated in the top-most
position of the hexagon also counts;
in the tiling in Figure~2 there are 3 horizontally oriented rhombi 
in the upper-most strip of the fundamental region), 
that the number $s(\pi)$ of special
entries is the number of horizontally oriented rhombi in the wedge
cut off by the rays which emanate from the midpoint of one side of
the triangle (see Figure 4; the rhombi corresponding to special
entries are the three dark ones), and that the total number of entries $i(\pi)$
is the number of horizontally oriented rhombi in the fundamental
region except for rhombi which touch the lower border of the
fundamental region
(but including horizontally oriented 
rhombi cut in two by the border of the fundamental region; in the tiling
in Figure~2 there are 10 such horizontally oriented rhombi in the fundamental
region). We have to exclude the rhombi touching the lower border of
the fundamental region because, under the translation between rhombus tilings
and descending plane partitions via non-intersecting lattice paths,
they would correspond to horizontal steps of height $0$, which
are ignored in the construction.


\midinsert
\bigskip
\centertexdraw{
\drawdim truecm \setunitscale.8
\linewd.05
\move(0 0)
\RhombusAsh \RhombusA \RhombusA 
\RhombusB \RhombusB \RhombusA \RhombusB \RhombusB 
\RhombusASh \RhombusB \RhombusB \RhombusBsh 
\move(-.866025 -.5)
\RhombusA \RhombusAsh \RhombusB \RhombusB
\RhombusA \RhombusB \RhombusASh \RhombusB \RhombusASh \RhombusB
\RhombusBsh \RhombusB 
\move(-1.73205 -1)
\RhombusA \RhombusA \RhombusB \RhombusB
\RhombusAsh \RhombusB \RhombusB \RhombusBsh \RhombusB
\RhombusA \RhombusA \RhombusB
\move(-2.598 -1.5)
\RhombusB \RhombusA \RhombusA \RhombusB \RhombusA 
\move(-3.4641 -2)
\RhombusB \RhombusA \RhombusB \RhombusA \RhombusA 
\move(-4.3301 -2.5)
\RhombusB \RhombusB \RhombusA \RhombusB
\RhombusA \RhombusA \RhombusA \RhombusB
\RhombusA \RhombusB \RhombusA \RhombusA 
\move(-5.19615 -3)
\RhombusB \RhombusB \RhombusA \RhombusB
\RhombusA \RhombusA \RhombusB \RhombusB
\RhombusA \RhombusA \RhombusA \RhombusA
\move(-1.73205 -1)
\RhombusC
\move(1.73205 -1)
\RhombusC
\move(1.73205 -5)
\RhombusC
\move(1.73205 -6)
\RhombusC \RhombusC
\move(1.73205 -7)
\RhombusC \RhombusC
\move(3.4641 -1)
\RhombusC \RhombusC
\move(3.4641 -2)
\RhombusC \RhombusC
\move(4.3301 -3.5)
\RhombusC 
\move(4.3301 -4.5)
\RhombusC 
\move(.866025 -8.5)
\RhombusC 
\move(-1.73205 -8)
\RhombusC 
\move(-2.598 -4.5)
\RhombusC \RhombusC \RhombusC \RhombusC 
\move(-5.19615 -5)
\RhombusC 
\move(-5.19615 -6)
\RhombusC \RhombusC \RhombusC 
\move(-5.19615 -7)
\RhombusC \RhombusC \RhombusC 
%
\linewd.1
\move (0.8660254 -5.5)
\rlvec(4.33013 -2.5)
\move (0.8660254 -5.5)
\rlvec(4.33013 2.5)
}
\centerline{\eightpoint Figure 4}
\bigskip
\endinsert

The partial order on descending plane partitions is also 
easily described on the corresponding rhombus tilings: a
tiling $T_2$ covers another tiling $T_1$ if it results
from $T_1$ by a local replacement

\bigskip

\centertexdraw{
\drawdim truecm \setunitscale.7
\linewd.05
\move(0 0)
\RhombusB \RhombusA
\move(.866025 .5)
\RhombusC
\htext(3 -.5){$\longrightarrow$}
\move(5 0)
\bsegment
\RhombusA \RhombusB
\move(0 0)
\RhombusC
\esegment
}
\bigskip

\noindent
in the fundamental region, or from the ``special" replacement

\bigskip

\centertexdraw{
\drawdim truecm \setunitscale.7
\linewd.05
\move(.866025 1.5)
\RhombusC
\move(.866025 2.5)
\RhombusC
\move(.866025 3.5)
\RhombusC
\move(0 0)
\RhombusA
\move(.866025 0.5)  \rlvec(-1.73205 1) \rlvec(1.73205 1)
\htext(3 1.5){$\longrightarrow$}
\move(6 0)
\bsegment
\move(.866025 0.5)  \rlvec(-1.73205 1) \rlvec(1.73205 1)
\RhombusB
\RhombusB
\move(0 0)
\move(.866025 3.5)  \rlvec(-.866025 -.5) \rlvec(.866025 -.5)
\lfill f:.8
\rlvec(.866025 .5) \rlvec(-.866025 .5)
\lfill f:.8
\move(.866025 .5)  \rlvec(.866025 .5) \rlvec(0 -1)
\lfill f:.8
\rlvec(-.866025 -.5) \rlvec(0 1)
\lfill f:.8
\esegment
}

\bigskip

\noindent
in the neighbourhood of the removed triangle. 
The former does also include the replacement 

\bigskip

\centertexdraw{
\drawdim truecm \setunitscale.7
\linewd.05
\move(0 0)
\RhombusB
\RhombusA
\move(.866025 .5)
\RhombusC
\htext(3 -.5){$\longrightarrow$}
\move(.866025 -.5)  \rlvec(-.866025 -.5) \rlvec(.866025 -.5)
\lfill f:.8
\rlvec(.866025 .5) \rlvec(-.866025 .5)
\lfill f:.8
\move(5 0)
\bsegment
\RhombusA \RhombusB
\move(0 0)
\RhombusC
\move(.866025 .5)  \rlvec(-.866025 -.5) \rlvec(.866025 -.5)
\lfill f:.8
\rlvec(.866025 .5) \rlvec(-.866025 .5)
\lfill f:.8
\esegment
}

\bigskip

\noindent
along the left border of the fundamental region.

There is a canonical antiautomorphism on the rhombus tilings:
the reflection of the tiling in the horizontal symmetry axis
of the hexagon with triangular hole. (In fact,
because of the cyclic symmetry of the tilings, the reflection in
{\it any} of the 3 symmetry axes of the tiled region would yield the
same result.) Since Mills, Robbins and Rumsey have shown in
\cite{\MiRRAB, Sec.~3} that there is a {\it unique} antiautomorphism
on the lattice of descending plane partitions with entries $\le n$,
denoted $\ta$ in \cite{\MiRRAB},
and since we have shown that this lattice is isomorphic to the lattice
on rhombus tilings with the covering relations defined as described
above, this reflection {\it must\/} correspond under this isomorphism
to the antiautomorphism $\ta$ of \cite{\MiRRAB}.

The properties (2.3)--(2.5) can now rather easily been read off
the geometric picture in terms of rhombus tilings. While
going through the arguments below, the reader
should always have an eye on Figures~2 and 4.

We begin with (2.3). We have seen that the statistics
$r(\pi)$ corresponds to the number of horizontally oriented rhombi 
in the upper-most strip of the fundamental region. In this upper-most
strip, there are always exactly $n-1$ rhombi which touch the upper-right
border of the hexagon along an edge, of which the first few are horizontally
oriented, and the others are oriented downwards (to the right).
Say, that there are $k$ which are horizontally oriented.
We now apply the reflection with respect to the horizontal symmetry axis.
Thereby, these $k$ rhombi are sent to the bottom-right border. 
By the cyclic symmetry, they correspond to $k$ rhombi in the
upper-most strip of the fundamental region which touch the upper-right
border of the hexagon along an edge and which are oriented downwards.
Consequently, there must be $n-1-k$ horizontally oriented rhombi
in the reflected tiling. The relation (2.3) follows.

Next we address (2.4). We have seen that the statistics
$s(\pi)$ corresponds to the number of horizontally oriented rhombi in the wedge
cut off by the rays which emanate from the midpoint of one side of
the triangle. If we apply the reflection with respect to the
horizontal symmetry axis, then these rhombi stay in the wedge, and
they are still horizontally oriented. This establishes the
relation (2.4).

Finally, we come to the most subtle statistics (in terms of rhombus
tilings), $i(\pi)$. We have seen that $i(\pi)$ corresponds to the
number of horizontally oriented rhombi in the fundamental
region except for rhombi which touch the lower border of the
fundamental region. We now imagine that we superimpose the fundamental region,
with the bottom-most strip excluded, and its mirror image 
with respect to the horizontal symmetry axis. Clearly, the union of 
the two is the region to the right of the vertical line which connects
the top-most and the bottom-most point of the hexagon. 
For convenience, let us denote this region by $R$.
We claim that, in $R$,
the number of horizontally oriented rhombi (in this count we also 
include horizontally rhombi which are cut in two by the border of
the fundamental region) is always the same, namely $\binom n2$.
This is best seen by observing that this is the case for the special tiling
in which all the horizontally oriented rhombi sit in the equilateral triangle
with side length $n-1$ on the top of $R$, and that all other tilings
can be obtained from this special tiling by applying the local replacements
explained just before, which do not change the number of horizontally oriented
rhombi in $R$. Thus, $i(\pi)+i(\tau(\pi))$ is equal to $\binom n2$ plus
the number of horizontally oriented rhombi in the intersection of
the fundamental region
(with bottom strip excluded) with its mirror image, because the latter rhombi
are counted twice in  $i(\pi)+i(\tau(\pi))$. However, this intersection
is exactly the previously described wedge, and the number of
horizontally oriented rhombi in the wedge is $s(\pi)$. This explains
the relation (2.5).

\remark{Remark}
The correspondence which is described in this section is, in some sense,
``implicit in the literature."
It could be extracted (with some effort) 
by a very attentive reading and comparison
of Andrews' original papers \cite{\AndrAN, \AndrAO}, 
David and Tomei's observation \cite{\DaToAA} of the equivalence of
plane partitions and rhombus tilings, and 
observations from \cite{\CiKrAC, Figure~3.2 and Proof of Lemma~3.1} 
and \cite{\CiEKAA, Figures~6 and 7 and Proof of Theorem~7}.
\endremark

\subhead 4. $d$-Descending plane partitions\endsubhead
In the original papers \cite{\AndrAN, \AndrAO}, Andrews did in fact
also consider more general objects, which we shall call here {\it
$d$-descending plane partitions}. (Andrews did not give them a name.
The objects of which we are talking, are the objects in the sets
$De(d;m,n)$ in \cite{\AndrAN, \AndrAO}.) For $d\le 1$, we define a 
$d$-descending plane partition to be an array $\pi$ of positive
integers of the form (2.1), such that

\roster 
\item $\la_1\ge\la_2\ge \dots\ge\la_k\ge k$,
\item the entries along rows are weakly decreasing,
\item the entries along columns are strictly decreasing,
\item the first entry in each row does not exceed the number of
entries in the preceding row less $d$ 
but is greater than the number of entries
in its own row less $d$.
\endroster

Clearly, descending plane partitions are the special case
$d=0$.\footnote{In \cite{\AndrAN, \AndrAO}, the value $d=2$ is also
allowed. The corresponding objects should be in bijection with the
cyclically symmetric plane partitions, which are at the beginning of
the whole story. However, there is a slight inaccuracy there. In order
to make the bijection with cyclically symmetric plane partitions work, and in
order to make the claimed enumeration formulae in \cite{\AndrAN,
\AndrAO} true, the definition of $d$-descending plane partitions has
to be modified in case $d=2$: first of all, one must also allow
0 as a possible entry, and, second, one must require that, in each
row, the first
entry {\it is equal\/} to the number of entries in its row less $d-1$.
We could also adopt this definition for arbitrary $d$, because the new
objects would be in bijection with the objects as defined originally
by Andrews by appending an appropriate number of 0s to each row
of the latter objects. In any case,
under this modification, the assertions of this section apply also for
$d=2$.} As in the case of descending plane partitions, there is a
closed form product formula for the number of all $d$-descending plane
partitions with entries $\le n$, see \cite{\AndrAN}, \cite{\AndrAO,
Theorem~8}. 

In the same way as in the preceding section, it can be seen that 
$d$-descending plane partitions with entries $\le n$ are in bijection
with cyclically symmetric 
rhombus tilings of a hexagon with side
lengths $n+d-1,n+1,n+d-1,n+1,n+d-1,n+1$ of which a centrally
located equilateral triangle
with side length $2-d$ has been removed, see Figure~5
for an example with $n=4$ and $d=-1$.

\midinsert
\centertexdraw{
\drawdim truecm \setunitscale.8
\linewd.05
\move(0 0)
\RhombusA \RhombusB \RhombusB \RhombusB \RhombusA \RhombusB 
\RhombusB 
\move(-.866025 -.5)
\RhombusA \RhombusB
\move(-1.73205 -1)
\RhombusB \RhombusA
\move(-2.598 -1.5)
\RhombusB \RhombusA
\move(-3.4641 -2)
\RhombusB \RhombusA \RhombusA \RhombusB \RhombusA \RhombusA \RhombusA 
\move(1.73205 0)
\RhombusC
\move(1.73205 -1)
\RhombusC
\move(1.73205 -2)
\RhombusC
\move (0.8660254 -3.5)
\RhombusC
\move (-0.8660254 -3.5)
\RhombusC \RhombusC \RhombusC 
\move (-3.4641 -3)
\RhombusC \RhombusC 
}
\centerline{\eightpoint Figure 5}
\bigskip
\endinsert

A lattice structure can be introduced on $d$-descending plane
partitions in the same way as in the preceding section. It is obvious
that, again, the reflection in the horizontal symmetry axis of the
hexagon with triangular hole is an antiautomorphism of this
lattice. Since it can be shown in the same manner as in \cite{\MiRRAB,
Sec.~3} that for the lattice of $d$-descending plane partitions with
entries $\le n$ there is a unique antiautomorphism, the
antiautomorphism defined by the above reflection
generates this unique antiautomorphism of the lattice
of $d$-descending plane partitions.



\remark{Acknowledgement} I am indebted to the anonymous referee for
pointing out an oversight in the original interpretation of the
statistics $i(\pi)$.
\endremark



\Refs

\ref\no \AndrAN\by G. E. Andrews \yr 1979 \paper Plane partitions
(III): The weak Macdonald conjecture\jour Inventiones Math.\vol
53\pages 193--225 \endref

\ref\no \AndrAO\by G. E. Andrews \yr 1980 
\paper Macdonald's conjecture and descending plane partitions
\inbook Combinatorics, representation theory and statistical methods in groups, Young Day Proceedings\eds T.~V.~Narayana, R.~M.~Mathsen, J.~G.~Williams
\publ Lecture Notes in Pure Math\., vol.~57, Marcel Dekker
\publaddr New York, Basel
\pages 91--106\endref

\ref\no \CiKrAC\by M.    Ciucu and C. Krattenthaler \yr 2000 
\paper Plane partitions II: $5\frac {1}{2}$ symmetry classes
\inbook Combinatorial Methods in Representation Theory\eds M.~Kashiwara, K.~Koike, S.~Okada, I.~Terada, H.~Yamada
\publ Advanced Studies in Pure Mathematics, vol.~28
\publaddr RIMS, Kyoto
\pages 83--103\endref

\ref\no \CiEKAA\by M.    Ciucu, T. Eisenk\"olbl, C. Krattenthaler and D. Zare \yr 2001 
\paper Enumeration of lozenge tilings of hexagons with a central triangular hole
\jour J. Combin\. Theory Ser.~A \vol 95
\pages 251--334\endref

\ref\no \DaToAA\by G.    David and C. Tomei \yr 1989 
\paper The problem of the calissons
\jour Amer\. Math\. Monthly\vol 96
\pages 429--431\endref

\ref\no \LaloAA\by P.    Lalonde \yr 2003 \paper Lattice paths and the
antiautomorphism of the poset of descending plane partitions\jour
Discrete Math\. \vol 271 \pages 311-319\endref 

\ref\no \MiRRAA\by W. H. Mills, D. H. Robbins and H. Rumsey \yr 1982
\paper Proof of the Macdonald conjecture\jour Inventiones Math.\vol
66\pages 73--87 \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 

\endRefs
\enddocument

