\input amstex
%\documentstyle{amsppt}
\NoRunningHeads
\catcode`\@=11
\def\item"#1"{\par\noindent\hang%
  \setbox0=\hbox{\rm#1\enspace}\ifdim\wd0>\the\parindent%
  \box0\else\hbox to \parindent{\rm#1\hfil}\enspace\fi\ignorespaces}
\let\plainitem@\item
\catcode`\@=13
\def\sgn{\operatorname{sgn}}
\magnification1200
\hsize15.6truecm
\vsize22.8truecm
\NoBlackBoxes
\topmatter 
\title Generating functions for shifted plane partitions
\endtitle 
\author C.~Krattenthaler
\endauthor 
\affil 
Institut f\"ur Mathematik der Universit\"at Wien,\\
Strudlhofgasse 4, A-1090 Wien, Austria.
\endaffil 
\address{Institut f\"ur Mathematik der Universit\"at Wien,
Strudlhofgasse 4, A-1090 Wien, Austria.}\endaddress
\subjclass{Primary 05A15, 11P57;
 Secondary 05A10, 05A15, 11B65, 11P68, 15A15.}\endsubjclass
\keywords{Generating functions, shifted plane partitions}\endkeywords
\abstract{With the help of a tableaux method, determinant formulas
for trace generating functions for various classes of shifted plane
partitions are derived. Proceeding from these determinants,
generalizations of Gansner's (J. Combin\. Theory A 30 (1981), 71--89;
Illinois J. Math\. 25 (1985), 533--554) hook formulas for shifted
plane partitions and alternative proofs of recent results of Proctor
about alternating trace generating functions are given.}\endabstract
\endtopmatter
\document
\subheading{1.~Introduction}Generating functions for certain families
of plane partitions with a given shape and given row bounds can be written
down in determinantal form. This has been shown by various methods
(\cite{Chu (1989), Gessel and Viennot (1989), Krattenthaler (1990a),
Wachs (1985)}), the most prominent being the Gessel--Viennot
\cite{(1985, 1989)}
method of nonintersecting lattice paths. In special cases these
determinants can be evaluated to result into closed forms
(\cite{Gessel and Viennot (1989), Krattenthaler (1990a, 1990b)}). Among them
 are the famous hook-content formulas of
Stanley \cite{(1971)}. 

With the help of a tableaux method, which is related to the
nonintersecting path method, and a simple determinant
evaluation, the author (\cite{Krattenthaler (1990a, 1990b)}) provided a
 uniform proof method to
establish essentially all known results concerning generating
functions for plane partitions of a given shape (without symmetries) and,
 moreover, some
new results. In this paper we continue this work. We transfer the
method of \cite{Krattenthaler (1990a)} to the case of shifted plane
 partitions. These
arise naturally when considering generating function problems for
plane partitions with symmetries (\cite{Andrews (1979), Mills et al\.
(1982, 1983, 1986), Okada (1989)}). 
Using our tableaux method we derive
determinant formulas for generating functions for shifted plane
partitions of a given shape  with prescribed entries on the main
diagonal (Theorems~1,2 below). 

Again there are special cases in which the determinants may be
evaluated. With the help of a summation formula for Schur functions
we are able to generalize Gansner's \cite{(1981a, 1981b)} hook formulas for
 trace
generating functions for two particular 
families of shifted plane partitions. 

A particular specialization of trace generating functions leads to
the alternating trace generating functions of Proctor \cite{(1990)}. 
In the case of trapezoidal shapes, there are closed forms for the
alternating trace generating function of shifted plane partitions
with prescribed entries on the main diagonal. These formulas were
first proved by Proctor [private communication]
 using group representation theory
and bideterminants. With the help of a simple determinant evaluation
(Lemma~7) we are able to provide alternative proofs which start with
the determinantal formula of Theorem~1.

We should remark that recently 
Stembridge \cite{(1990a)} discovered a powerful approach to the
computation of generating functions for shifted plane partitions,
which involves Pfaffians. For example, Gansner's hook formula 
(\cite{Gansner (1981a), Theorem~7.1}) for the
generating function of shifted reverse plane partitions 
could be easily proved that way, too. (In \cite{Stembridge (1990a)} only the
case $x\sb i=q$ is considered.) Other results about shifted plane
partitions, which we were not able to prove using our methods, occur
in \cite{Proctor (1983, 1990), and Mills et al\. (1986)}.

The next section contains all relevant definitions. In section~3 we
derive the determinant formulas for three types of generating
functions of shifted plane partitions. The generalizations of
Gansner's results are given in section~4. Finally, in section~5 the
computations concerning the 
alternating trace generating functions are discussed.


\subheading{2.~Definitions}We use the usual multidimensional notations.
For $r$-tuples of integers $\lambda
=(\lambda
\sb
1,\lambda
\sb
2, \dots,\lambda
\sb
r)$ and 
$\mu=(\mu\sb
1,\mu\sb
2, \dots,\mu\sb
r)$, we write $\lambda
+\mu$ for
$(\lambda
\sb
1+\mu\sb
1,\lambda
\sb
2+\mu\sb
2, \dots,\lambda
\sb
r+\mu\sb
r)$. If $\sigma
$ is an
element of the symmetric group $S\sb
r$ of order $r$ we define $\lambda
\sb
\sigma
$
to be the $r$-tuple $(\lambda
\sb
{\sigma
(1)},\lambda
\sb
{\sigma
(2)},
\dots,\lambda
\sb
{\sigma
(r)})$. An $r$-tuple $\lambda
$ of
integers is called a {\it partition} if
$\lambda
\sb
1\ge\lambda
\sb
2\ge\dots\ge\lambda
\sb
r\ge 0$.

Let $\alpha
,\beta
$ be arbitrary integers and $\lambda
$ a partition with
$\lambda
\sb
r\ge r$. We 
call an array $\pi$ of integers of the form
$$\matrix \pi\sb
{11}
&\pi\sb
{12}&\innerhdotsfor6\after\quad &\pi\sb
{1,\lambda
\sb
1} \\
&\pi\sb
{22}&\innerhdotsfor5\after\quad &\pi\sb
{2,\lambda
\sb
2}\\
& &\ddots& &\hphantom{mm} &
 &\mathinner{\mskip1mu\raise1pt\vbox{\hbox{.}}\mskip2mu
\raise4pt\hbox{.}\mskip2mu\raise7pt\vbox{\kern7pt\hbox{.}}\mskip1mu}\\
& & &\pi\sb
{rr}&\innerhdotsfor1\after\quad &\pi\sb
{r,\lambda
\sb
r}& & &
\endmatrix\tag2.1$$
a shifted $(\alpha
,\beta
)$--{\it plane partition of shape\/} $\lambda
$ if
$$\align\pi\sb
{ij}\ge\pi\sb
{i,j+1}+\alpha
\quad\quad &\text{for}\quad 1\le i\le r,\ 
i\le j<\lambda
\sb
i\tag2.2)(a\endalign$$
and
$$\pi\sb
{ij}\ge\pi\sb
{i+1,j}+\beta
\quad \quad \text{for}\quad 1\le i\le r-1,\ 
i\le j\le\lambda
\sb
{i+1}\ .\tag2.2)(b$$
It should be noted that in our definition of the shifted shape there
are $\lambda
\sb
i-i+1$ integers in the $i$-th row, while usually there are
$\lambda
\sb
i$ integers in the $i$-th row.

An array $\pi$ of the form (2.1) is called a {\it shifted
$(\alpha
,\beta
)$--reverse plane partition of shape $\lambda
$}, if it satisfies
$$\pi\sb
{ij}\le\pi\sb
{i,j+1}+\alpha
\quad\quad \text{for}\quad 1\le i\le r,\ 
i\le j<\lambda
\sb
i\tag2.3)(a$$
and
$$\pi\sb
{ij}\le\pi\sb
{i+1,j}+\beta
\quad \quad \text{for}\quad 1\le i\le r-1,\ 
i\le j\le\lambda
\sb
{i+1}\ .\tag2.3)(b$$

To give an example, the array in Figure~1 below is a shifted $(0,2)$--plane 
partition of shape (8,6,6,4).
$$\matrix \format &\quad \r\\11&9&7&7&6&6&3&1\\
&6&5&5&4&3\\ & &3&2&1&1\\ & & &\hskip-8pt -1
\endmatrix$$\centerline{\eightpoint Figure 1}

\medpagebreak
The entries of $\pi$ are called {\it parts\/} of $\pi$. The sum of
all parts of $\pi$, $\sum\pi\sb
{ij}$, is called the {\it norm} of $\pi$,
for which we write $n(\pi)$. The {\it $l$-diagonal} of an array $\pi$ of
the form (2.1) is the sequence of parts $\pi\sb
{ij}$ with $j-i=l$. The
{\it $l$-trace} of $\pi$, denoted by $t\sb
l(\pi)$, is the sum of the
parts of the $l$-diagonal of $\pi$. 

We shall consider three types of
generating functions for shifted $(\alpha
,\beta
)$--plane partitions subject
to certain restrictions. The {\it part generating function} for a
certain class $C$ of shifted $(\alpha
,\beta
)$--plane partitions is 
$\sum \sb
{\pi}\prod x\sb
{\pi\sb
{ij}}\,,$
where the sum is over all $\pi$ belonging to $C$ and the product is over
all parts of $\pi$. The {\it norm generating function} for $C$ is
$\sum \sb
{\pi}x\sp
{n(\pi)}\,,$
while the {\it trace generating function} for $C$ is 
$\sum \sb
{\pi}\prod \sb
{l\in{\Bbb
Z}}x\sb
l\sp
{t\sb
l(\pi)}\,,$
where in both cases the sum is over all $\pi$ being an element of
that class $C$. (${\Bbb
Z}$ is the set of integers.) Of course, 
the norm generating function comes out of both the part and the trace
generating function by setting $x\sb
i=x\sp
i$, respectively $x\sb
i=x$.

For $x$ being an indeterminate, we introduce the ``$x$-notations"
$[\beta
]=1-x\sp
\beta
$, $[n]!=[1][2]\dotsb [n]$, $[0]!=1$,
$$\align\bmatrix n\\k\endbmatrix&=\frac {[n]!} {[k]!\,[n-k]!}\ ,\\
(a;x)\sb
\infty &=\prod \sb
{j=0} \sp
{\infty}(1-ax\sp
j)\ ,\endalign$$
and
$$(a;x)\sb
\beta
=(a;x)\sb
\infty/(ax\sp
\beta
;x)\sb
\infty\ .
$$
Finally, for the integers $m,n$, $m\le n$, we set 
$$x(m,n)=\prod \sb
{j=m+1} \sp
{n}x\sb
j\ ,$$
with the empty product being equal to 1 by convention.
\subheading{3.~The determinant formulas}In this section we are going
to prove the following theorems.
\proclaim{Theorem~1}Let $\alpha
,\beta
$ be arbitrary integers, $\lambda
$ be a
partition with $\lambda
\sb
r\ge r$, and let $a,b$  be $r$-tuples of integers
satisfying
$$a\sb
i-\alpha
-\beta
\ge a\sb
{i+1}\quad \text {and}\quad
b\sb
i+\alpha
(\lambda
\sb
i-\lambda
\sb
{i+1})+(1-\beta
)\ge b\sb
{i+1}\tag3.1$$
for $i=1,2,\dots,r-1$. Then the trace generating function for shifted
$(\alpha
,\beta
)$-plane partitions of shape $\lambda
$ in which the first part in row $i$ is
equal to $a\sb
i$ and the last part in row $i$ is at least $b\sb
i$ is 
$$\multline x\sb
0\sp
{\sum \sb
{i=1} \sp
{r}a\sb
i}\prod \sb
{i=1}
\sp
{r}\bigg(x(0,\lambda
\sb
i-i)\sp
{b\sb
i}\prod \sb
{l=0}
\sp
{\lambda
\sb
i-i-1}x(0,l)\sp
\alpha
\bigg)\\
\times \det \sb
{1\le s,t\le r} \Big(t(\lambda
\sb
s-s;a\sb
t+(1-\alpha
-\beta
)(s-t)-\alpha
(\lambda
\sb
s-s)-b\sb
s)\Big)\ , \endmultline\tag3.2$$
where $t(p;m)$ is the coefficient of $z\sp
m$ in $1\Big/\prod \sb
{l=0}
\sp
{p}(1-x(0,l)z)$.
The norm generating function for the same class of shifted
$(\alpha
,\beta
)$-plane partitions is
$$x\sp
{N\sb
1}\det \sb
{1\le s,t\le r}\left(\bmatrix (\lambda
\sb
s-s)(1-\alpha
)+(1-\alpha
-\beta
)
(s-t)+a\sb
t-b\sb
s\\
\lambda
\sb
s-s\endbmatrix\right)\ ,\tag3.3$$
where $N\sb
1=\sum \sb
{i=1} \sp
{r}\left(b\sb
i(\lambda
\sb
i-i)+a\sb
i+\alpha
\binom{\lambda
\sb
i-i}{2}\right)$. If $\beta
=1-\alpha
$, the part generating function
for this class is given by
$$ \prod \sb
{i=1} \sp
{r}x\sb
{a\sb
i}\det \sb
{1\le s,t\le r}\Big(h \sb
{\lambda
\sb
s-s}
\sp
{(\alpha
)}(\bold x;a\sb
t,b\sb
s)\Big)\ ,\tag3.4$$
where $h\sb
n\sp
{(\alpha
)}(\bold x;A,B)=\sum x\sb
{i\sb
1}\dotsb x\sb
{i\sb
n}$, 
the sum being over all $i\sb
1,\dots,i\sb
n$ satisfying $i\sb
1\le A$, $i\sb
n\ge
B$, and $i\sb
j\ge i\sb
{j+1}+\alpha
$ for $j=1,2,\dots,n-1$.
\endproclaim
\proclaim{Theorem~2}Let $\alpha
,\beta
$ be arbitrary integers, $\lambda
$ be a
partition with $\lambda
\sb
r\ge r$, and let $a,b$  be $r$-tuples of integers
satisfying
$$a\sb
i+\alpha
(\lambda
\sb
{i-1}-\lambda
\sb
{i})+(1-\beta
)\ge a\sb
{i-1}\quad \text {and}\quad 
b\sb
i-\alpha
-\beta
\ge b\sb
{i-1}
\tag3.5$$
for $i=1,2,\dots,r-1$. Then the trace generating function for shifted
$(\alpha
,\beta
)$-reverse plane partitions of shape $\lambda
$ in which the first part in row $i$ is
equal to $b\sb
i$ and the last part in row $i$ is at most $a\sb
i$ is 
$$\multline \prod \sb
{i=1} \sp
{r}\prod \sb
{l=0} \sp
{\lambda
\sb
i-i-1}
x(l,\lambda
\sb
i-i)\sp
\alpha
\, \det\sb
{1\le s,t\le r}
 \Big(x(-1,\lambda
\sb
s-s)\sp
{b\sb
t-(s-t)(1-\alpha
-\beta
)}\\
\cdot t'(\lambda
\sb
s-s;a\sb
s+(1-\alpha
-\beta
)(s-t)-\alpha
(\lambda
\sb
s-s)-b\sb
t)\Big)\ , \endmultline\tag3.6$$
where $t'(p;m)$ is the coefficient of $z\sp
m$ in $1\Big/\prod \sb
{l=0}
\sp
{p}(1-x(l,p)z)$.
The norm generating function for the same class of shifted
$(\alpha
,\beta
)$-plane partitions is
$$x\sp
{N\sb
2}\det \sb
{1\le s,t\le r}\left(x\sp
{(\lambda
\sb
s-s+1)(b\sb
t+t(1-\alpha
-\beta
))}
\bmatrix (\lambda
\sb
s-s)(1-\alpha
)+(1-\alpha
-\beta
)
(s-t)+a\sb
s-b\sb
t\\
\lambda
\sb
s-s\endbmatrix\right),\tag3.7$$
where $N\sb
2=\sum \sb
{i=1} \sp
{r}\left(\alpha
\binom{\lambda
\sb
i-i+1}{2} -i(\lambda
\sb
i-i+1)(1-\alpha
-\beta
)\right)$. 
If $\beta
=1-\alpha
$, the part generating function
for this class is given by
$$ \prod \sb
{i=1} \sp
{r}x\sb
{b\sb
i}\det \sb
{1\le r,s\le t}\Big(h \sb
{\lambda
\sb
s-s}
\sp
{(\alpha
)}(\bold x;a\sb
s,b\sb
t)\Big)\ .\tag3.8$$
\endproclaim
In order to prove these theorems, we shall apply Lemma~2.1 of
\cite{Krattenthaler (1990a)},
which we restate here without proof.
\proclaim{Lemma 3}Let $\{M\sb
\sigma
:\sigma
\in S\sb
r\}$ be a collection of
pairwise disjoint sets with a weight function $w$ acting on $\bigcup
\sb
{\sigma
\in S\sb
r}M\sb
\sigma
$. Let $M\sb
{\text {id}}$ be partitioned into two
disjoint subsets, $M\sb
{\text {id}}\sp
+$ and $M\sb
{\text {id}}\sp
-$, $M\sb
{\text
{id}}=M\sb
{\text {id}}\sp
+\cup M\sb
{\text {id}}\sp
-$. Assume a
weight-preserving involution $\varphi
$ acting on $M\sb
{\text {id}}\sp
-\cup\bigcup
\sb
{\sigma
\ne\text {id}}M\sb
\sigma
$ satisfying the following property, holding
for all $\sigma
\in S\sb
r$: For every $\pi$ on which $\varphi
$ is applicable,
$\pi\in M\sb
\sigma
$, there exists a transposition $(i,j)$, $1\le i,j\le
r$, for which $\varphi
(\pi)\in M\sb
{\sigma
(i,j)}$. Then 
$$F(M\sb
{\text {id}}\sp
+;w)=\sum \sb
{\sigma
\in S\sb
r}\sgn\sigma
\,F(M\sb
\sigma
;w)\
,\tag3.9$$
where $F(M;w)$ denotes the generating function $\sum \sb
{\pi\in
M}w(\pi)$.\qed
\endproclaim

\demo{Proof of Theorems}In analogy with \cite{Krattenthaler (1990a)} we first
 define a shifted
$\alpha
$-tableaux of shape $\lambda
$ to be an array of the form (2.1) such that (2.2)(a)
holds. Thus, for shifted $\alpha
$-tableaux there are only restrictions
on the rows but none on the columns. All the notions of section~2, parts,
 norm, trace, etc\., are
extended to shifted $\alpha
$-tableaux in the obvious way. 

For the set of all shifted $\alpha
$-tableaux of shape $\lambda
$ in which the
first part in row $i$ is equal to $a\sb
i$ and the last part in row $i$
is at least $b\sb
i$ we write $ST\sb
\alpha
(\lambda
;a,b)$. The subset of $ST\sb
\alpha
(\lambda
;a,b)$
containing all shifted $(\alpha
,\beta
)$-plane partitions of $ST\sb
\alpha
(\lambda
;a,b)$
is denoted by $ST\sp
+\sb
\alpha
(\lambda
;a,b)$, its complement by
$ST\sp
-\sb
\alpha
(\lambda
;a,b)$. 

Given a permutation $\sigma
\in S\sb
r$, let the $r$-tuple $e\sb
\sigma
$ be
defined by $(1-\sigma
(1),2-\sigma
(2),\dots,r-\sigma
(r))$.
Next we construct an involution $\varphi
\sb
{\alpha
,\beta
}$ acting on $\bigcup
\sb
{\sigma
\in S\sb
r}ST\sb
\alpha
\sp
-(\lambda
;a\sb
\sigma
+(1-\alpha
-\beta
)e\sb
\sigma
,b)$ which satisfies
the properties required in Lemma~3. For a shifted
$\alpha
$-tableaux $\tau
$ of shape $\lambda
 $ this involution is introduced
in two steps:
\roster
\item "(A)" Consider all pairs $(\tau
\sb
{ij},\tau
\sb
{i+1,j})$ of parts of
$\tau
$ with
$$\tau
\sb
{ij}<\tau
\sb
{i+1,j}+\beta
\ .\tag3.10$$
Let $J$ be the largest integer with $\tau
\sb
{iJ}<\tau
\sb
{i+1,J}+\beta
$ for
some $i$. Let $I$ be the smallest integer with
$\tau
\sb
{IJ}<\tau
\sb
{I+1,J}+\beta
$. (The idea is to search for the right-most
pairs and, among them, for the highest pair satisfying (3.10) in the tableaux
 $\tau
$.)
\item "(B)" Let $K$ be the smallest integer with
$\tau
\sb
{IK}<\tau
\sb
{I+1,K}+\beta
$. The image of $\tau
$ under application of
$\varphi
\sb
{\alpha
,\beta
}$ is the tableaux $\tau
'$, which is obtained by leaving
all rows of $\tau
$ unchanged, with the exception of the $I$'th and
$(I+1)$'th row. Here we set
$$\tau
\sb
{Ij}'=\cases \tau
\sb
{Ij}&j\ge K\\\tau
\sb
{I+1,j+1}-(1-\alpha
-\beta
)&j<K\endcases$$
and
$$\tau
'\sb
{I+1,j}=\cases \tau
\sb
{I+1,j}&j>K\\\tau
\sb
{I,j-1}+(1-\alpha
-\beta
)&j\le
K\endcases\ .$$
(For the geometrical interpretation of the action of $\varphi
\sb
{\alpha
,\beta
}$
confer \cite{Krattenthaler (1990a), Def.~4.1}.)
\endroster 
Let $\tau
\in ST\sb
\alpha
\sp
-(\lambda
;a\sb
\sigma
+(1-\alpha
-\beta
)e\sb
\sigma
,b) $ and let $I,K$ be
the integers being determined in (A). It is clear from the definition of
$\varphi
\sb
{\alpha
,\beta
}$ that $\varphi
\sb
{\alpha
,\beta
}(\tau
)$ again is a shifted
$(\alpha
,\beta
)$-plane partition of shape $\lambda
$ with diagonal entries
being given by the vector $a\sb
{\sigma
(I,I+1)}+(1-\alpha
-\beta
)e\sb
{\sigma
(I,I+1)}$.
Besides, the last part in row $i$ of $\varphi
\sb
{\alpha
,\beta
}(\tau
)$ is again at
least $b\sb
i$: For $i\ne I+1$ this is obvious, since the last parts in
each row of $\tau
$ remain unchanged under application of
$\varphi
\sb
{\alpha
,\beta
}$, with the exception of the $(I+1)$-th row. Here the
last part might be changed, namely in case $K=\lambda
\sb
{I+1}$, where the
last part of row $(I+1)$ of $\varphi
\sb
{\alpha
,\beta
}(\tau
)$ is given by
$\tau
\sb
{I,\lambda
\sb
{I+1}-1}+(1-\alpha
-\beta
)$. Use of (2.2)(a) yields
$$\align\tau
\sb
{I,\lambda
\sb
{I+1}-1}&\ge \tau
\sb
{I,\lambda
\sb
I}+\alpha
(\lambda
\sb
I-\lambda
\sb
{I+1}+1)\\
&\ge b\sb
i+\alpha
(\lambda
\sb
I-\lambda
\sb
{I+1}+1)\ .
\endalign$$
Hence, by the second part of (3.1), we obtain
$$\tau
\sb
{I,\lambda
\sb
{I+1}-1}+(1-\alpha
-\beta
)\ge b\sb
{i+1}\ ,$$
just as being claimed. 

Another fact is that, if $\sigma
\ne \text {id}$, we have
$$ST\sb
\alpha
(\lambda
;a\sb
\sigma
+(1-\alpha
-\beta
)e\sb
\sigma
,b)
=ST\sb
\alpha
\sp
-(\lambda
;a\sb
\sigma
+(1-\alpha
-\beta
)e\sb
\sigma
,b)\ .$$
Since $\sigma
\ne \text {id}$, 
there must exist an
integer $i$, $1\le i\le r-1$, such that $\sigma
(i)>\sigma
(i+1)$. Using the
inequality
$$a\sb
R< a\sb
S+(R-S)(1-\alpha
-\beta
), \quad \text {for }r\ge R>S\ge 1,\tag 3.11$$
which is equivalent with (3.1), it can be shown that
$$a\sb
{\sigma
(i)}+(1-\alpha
-\beta
)(i-\sigma
(i))<a\sb
{\sigma
(i+1)}+(1-\alpha
-\beta
)(i+1-\sigma
(i+1))
+\alpha
+\beta
\ .$$
This implies that for any $\tau
\in ST\sb
\alpha
(\lambda
;a\sb
\sigma
+(1-\alpha
-\beta
)e\sb
\sigma
,b)$
the inequality $\tau
\sb
{ii}<\tau
\sb
{i+1,i+1}+\alpha
+\beta
$ is true, hence
$\tau
\sb
{i,i+1}<\tau
\sb
{i+1,i+1}+\beta
$, and thus $\tau
\in 
ST\sb
\alpha
\sp
-(\lambda
;a\sb
\sigma
+(1-\alpha
-\beta
)e\sb
\sigma
,b)$.

Summarizing we have the following properties of $\varphi
\sb
{\alpha
,\beta
}$:
\roster
\item $\varphi
\sb
{\alpha
,\beta
}$ is an involution on
$$ST \sb
{\alpha
} \sp
{-}(\lambda
;a,b)\cup\underset\sigma
\ne\text {id}
\to{\bigcup \sb
{\sigma
\in S\sb
r}}ST\sb
\alpha
(\lambda
;a\sb
\sigma
+(1-\alpha
-\beta
)e\sb
\sigma
,b)\ .$$
If $\tau
\in ST\sb
\alpha
(\lambda
;a\sb
\sigma
+(1-\alpha
-\beta
)e\sb
\sigma
,b)$ and 
$\tau
$ is an element of the domain of $\varphi
\sb
{\alpha
,\beta
}$, then there
exists a transposition $(I,I+1)$ with
$$\varphi
\sb
{\alpha
,\beta
}(\tau
)\in
ST\sb
\alpha
(\lambda
,a\sb
{\sigma
(I,I+1)}+(1-\alpha
-\beta
)e\sb
{\sigma
(I,I+1)},b)\ .$$
\item The $l$-trace and norm are invariant under application of
$\varphi
\sb
{\alpha
,\beta
}$. If $\alpha
+\beta
=1$ the set of parts remains unchanged
under application of $\varphi
\sb
{\alpha
,\beta
}$, too.

\endroster

\medskip
Since, because of (3.11), the map $\sigma
\to a\sb
\sigma
+(1-\alpha
-\beta
)e\sb
\sigma
$ is
injective,
the sets
$ST\sb
\alpha
(\lambda
;a\sb
\sigma
+(1-\alpha
-\beta
)e\sb
\sigma
,b)$ are pairwise disjoint. Hence, all
assumptions of Lemma~3 are satisfied with
$M\sb
\sigma
=ST\sb
\alpha
(\lambda
;a\sb
\sigma
+(1-\alpha
-\beta
)e\sb
\sigma
,b)$, etc\., 
and the trace weight $w(\tau
)=\prod
\sb
{l\in{\Bbb
Z}}x\sb
l\sp
{t\sb
l(\tau
)}$.  The generating functions
$F\big(ST\sb
\alpha
(\lambda
;a\sb
\sigma
+(1-\alpha
-\beta
)e\sb
\sigma
,b)\big)$ are equal to the
product of the corresponding generating functions of the rows.
Therefore, for this choice of $M\sb
\sigma
$,
 the sum on the right-hand side of (3.9) can be written in
the form of a determinant, which finally, after taking some factors
out of this determinant, results into the right-hand side of 
(3.2).

Formula (3.3) is obtained
by setting $x\sb
i=x$ for all $i$ in (3.2) and using the well-known
expansion for $1/(z;x)\sb
{p+1}$ (\cite{Andrews (1976), (3.3.7)}). Formula
 (3.4) is proved in
analogy with (3.2) by observing that if $\alpha
+\beta
=1$, all assumptions
of Lemma~3 are satisfied with $M\sb
\sigma
=ST\sb
\alpha
(\lambda
;a\sb
\sigma
,b)$, etc\., and
$w(\tau
)=\prod x\sb
{\tau
\sb
{ij}}$.

Theorem~2 can be proved by constructing a similar involution, which
acts on shifted $(\alpha
,\beta
)$-reverse tableaux, or by using the
transformation $\pi\sb
{ij}\to -\pi\sb
{ij}$, which maps shifted
$(\alpha
,\beta
)$-plane partitions onto shifted
$(\alpha
,\beta
)$-reverse plane partitions, and transforming the results of
Theorem~1 into the ``reverse" case. \qed
\enddemo
The special case $\alpha
=0$, $\beta
=1$, $b\sb
i=0$ of (3.3) was used to obtain
generating functions for symmetry classes of plane partitions
(\cite{Andrews (1979), Mills et al\. (1982, 1983, 1986), Okada
(1989)}). It is also interesting to note that a slight
modification of $\varphi
\sb
{01}$ serves to provide an alternative proof of
Okada's \cite{(1990)} Theorem about partially strict shifted plane
partitions.

\subheading{4.~Trace generating functions with no upper part
bounds}The main tool which is used in this section to obtain closed
forms for generating functions for shifted $(\alpha
,\beta
)$-(reverse) plane
partitions with no upper bound on the parts in each row, is the
following summation theorem, which is proved by inserting the
bideterminantal form of Schur functions (\cite{Macdonald (1979), p.~24,
(3.1)}) into the 
summation formula (\cite{Mcdonald (1979), p.~45, Ex.~4}) for Schur functions.

\proclaim{Lemma~4}Let $X\sb
1,X\sb
2,\dots,X\sb
r$ be indeterminates. Then
$$\sum \sb
{a\sb
1>a\sb
2>\dotsb>a\sb
r\ge 0}\det\sb
{1\le s,t\le r}\left(X\sb
s\sp
{a\sb
t}\right)=
\frac {\prod
\sb
{1\le i<j\le r}(X\sb
i-X\sb
j)} {\prod \sb
{i=1} \sp
{r}(1-X\sb
i)\prod
\sb
{1\le i<j\le r}(1-X\sb
iX\sb
j)}\ .\quad \lower10pt\hbox{\qed}$$
\endproclaim
Since it is more straight-forward to deal with, we first consider the
case of shifted $(\alpha
,\beta
)$-reverse plane partitions.

\proclaim{Theorem~5}Let $K,B$ be integers, $K\ge 1$. 
The trace generating function for shifted
$(\alpha
,\beta
)$-reverse plane partitions $\pi$ of shape $\lambda
$ in which the
first part in row $i$ is $\equiv -i(1-\alpha
-\beta
)\pmod K$ and
$\pi\sb
{11}\ge (KB-(1-\alpha
-\beta
))$ is
$$\multline\prod \sb
{i=1} \sp
{r}x(-1,\lambda
\sb
i-i)\sp
{-i(1-\alpha
-\beta
)+K(B+i-1)}\prod
\sb
{i=1} \sp
{r}\prod
\sb
{l=0} \sp
{\lambda
\sb
i-i-1}\frac{x(l,\lambda
\sb
i-i)\sp
\alpha
} {(1-x(l,\lambda
\sb
i-i))}\\
\times\frac {\prod \sb
{1\le j<i\le r} (1-x(\lambda
\sb
i-i,\lambda
\sb
j-j)\sp
K)} {\prod
\sb
{i=1} \sp
{r}(1-x(-1,\lambda
\sb
i-i)\sp
K)
\prod \sb
{1\le i<j\le r} (1-x(-1,\lambda
\sb
i-i)\sp
Kx(-1,\lambda
\sb
j-j)\sp
K)}.\endmultline$$
\endproclaim

\demo{Proof}In (3.6) set $b\sb
t=KB\sb
t-t(1-\alpha
-\beta
)$ and let
$a\sb
s\to\infty$. Using the fact that, if $\lim\sb
{i\to\infty}c\sb
i=c$ then
also $\lim\sb
{z\to 1\sp
-}(1-z)\sum\sb
{j}c\sb
jz\sp
j=c$, we obtain, after taking some
factors out of the determinant, for the trace
generating function for shifted $(\alpha
,\beta
)$-reverse plane partitions
of shape $\lambda
$ in which the first part in row $i$ is equal to
$KB\sb
i-i(1-\alpha
-\beta
)$ the expression
$$\frac{\prod \sb
{i=1} \sp
{r}\prod
\sb
{l=0} \sp
{\lambda
\sb
i-i-1}x(l,\lambda
\sb
i-i)\sp
\alpha
\prod \sb
{i=1}
\sp
{r}x(-1,\lambda
\sb
i-i)\sp
{-i(1-\alpha
-\beta
)}}{\prod \sb
{i=1} \sp
{r}\prod
\sb
{l=0} \sp
{\lambda
\sb
i-i-1}(1-x(l,\lambda
\sb
i-i))}
\det\left(x(-1,\lambda
\sb
s-s)\sp
{KB\sb
t}\right).$$
When taking into account (3.5), one observes that, to obtain the
desired generating function, these
 expressions have to be summed up over all $B\sb
1,B\sb
2,\dots,B\sb
r$
with\linebreak $B\le B\sb
1<\dotsb<B\sb
r$. Application of Lemma~4 after some
manipulation yields the claimed expression. \qed
\enddemo
 Gansner's \cite{(1981a)} Theorem~7.1 is the special case
$\alpha
=\beta
=0$, $K=B=1$ of Theorem~5.
\medpagebreak

In the case of ordinary shifted plane partitions there is only a
closed form result analogous to the previous one in case of a
triangular shape $(r\sp
r)$, the shape $\lambda
$ with $\lambda
\sb
i=r$ for all $i$.
\proclaim{Theorem~6}Let $K,B$ be integers, $K\ge 1$.
The trace generating function for shifted $(\alpha
,\beta
)$-plane partitions
$\pi$ of shape $(r\sp
r)$ in which  the first part in row $i$ is $\equiv
i(1-\alpha
-\beta
)\pmod K$ and $\pi\sb
{rr}\ge (KB+r(1-\alpha
-\beta
))$ is 
$$\prod \sb
{i=0} \sp
{r-1}x\sb
i\sp
{N\sb
3(i)}\prod \sb
{0\le i<j\le r-1} \frac
{(1-x(i,j)\sp
K)} {(1-x(i,j))(1-x(-1,i)\sp
Kx(-1,j)\sp
K)}\prod \sb
{i=0} \sp
{r-1}\frac
{1} {(1-x(-1,i)\sp
K)},$$
where $N\sb
3(i)=(r-i)(r+i+1)/2+(KB-r\alpha
)(r-i)+(\alpha
+K)\binom{r-i}2-\beta
\binom{r-i+1}2$.
\endproclaim

\demo{Proof} Let $\hat B=KB-r\alpha
$.
First we consider the generating function of shifted $(\alpha
,\beta
)$-plane
partitions of shape $(r\sp
r)$ where the first part in row $i$ is equal
to $a\sb
i$ and the last part in row $i$ is at least $\hat B+i(1-\beta
)$. If
these choices are inserted into (3.2), we get
$$x\sb
0\sp
{\sum a\sb
i}\prod \sb
{i=1} \sp
{r}x(0,r-i)\sp
{\hat B+i(1-\beta
)}\prod \sb
{l=0}
\sp
{r-1}x(0,l)\sp
{\alpha
(r-l-1)}\det\Big(t(r-s;a\sb
t-t(1-\alpha
-\beta
)-\alpha
r-\hat B)\Big)\ ,\tag4.1$$
where $t(p;m)$ is the coefficient of $z\sp
m$ in the power series
expansion given in Theorem~1. Using the partial fraction expansion of
this power series, it turns out that the determinant in (4.1) by
simple row manipulations may be simplified to
$$\det\left(x(0,r-s)\sp
{a\sb
t-t(1-\alpha
-\beta
)-\alpha
 r-\hat B}\prod \sb
{l=0}
\sp
{r-s-1}\frac {(-1)\sp
{r-s}x(l,r-s)} {(1-x(l,r-s))}\right)\ .\tag4.2$$
Because of (2.2)(b), any shifted $(\alpha
,\beta
)$-plane partition 
of shape $(r\sp
r)$
with $\pi\sb
{rr}\ge KB+r(1-\alpha
-\beta
)$ automatically 
possesses the property that the last
part in row $i$ is greater than $\hat B+i(1-\beta
)$. Therefore, by setting
$a\sb
i=KA\sb
i+i(1-\alpha
-\beta
)$ in (4.1) and summing up over all
$A\sb
1,\dots,A\sb
r$ with $A\sb
1>\dotsb>A\sb
r\ge KB+r(1-\alpha
-\beta
)$, the desired
generating function will be obtained. Writing the determinant in (4.1)
in the form (4.2), it is seen that again Lemma~4 may be applied. Some
tedious manipulations finally result into the claimed expression.
\qed
\enddemo
The case $K=2$ of Theorem~6 is somewhat related to a result of Proctor
\cite{(1990) (``CYI" case of Theorem~1)} 
(cf\. also \cite{Stembridge (1990b)}), although there is no overlap.
Gansner's \cite{(1981b) Theorem~4.2} is obtained from Theorem~6 by
setting $\alpha
=\beta
=0$, $K=1$, and $B=-r$.
\subheading{5.~Alternating trace generating functions}Recently, 
\cite{Proctor (1990)}
considered unusual weightings for shifted 
plane partitions $\pi$, namely
$w\sb
1(\pi)=x\sp
{\sum \sb
{i\ge0}(-1)\sp
{i-1}it\sb
i(\pi)}$ and 
$w\sb
2(\pi)=x\sp
{\sum \sb
{i\ge0}(-1)\sp
{i}(i+1)t\sb
{i}(\pi)}$, which he called
``alternating trace" weightings. He also observed [private
communication]
that with these weightings there are closed expressions for
trapezoidal shapes, the shapes $\lambda
$, where $\lambda
\sb
i=\Lambda
-i$ for some
$\Lambda
$.  His proofs base on the
representation theory of the 
symplectic group. We give more elementary proofs below, which use the
determinants in Theorems~1,2 and the following computational lemma.
Special cases of these results earlier appeared in
\cite{Proctor (1988), Prop.~4.1, and Proctor (1990), Theorem~1}.
\proclaim{Lemma~7}Let $A\sb
1,A\sb
2,\dots,A\sb
r,B\sb
2,\dots,B\sb
r$ be
indeterminates. Then we have
$$\multline\det \left(( {C}/ {A\sb
t}+B\sb
r)\dotsb( {C}/
{A\sb
t}+B\sb
{s+1})(A\sb
t+B\sb
r)\dotsb(A\sb
t+B\sb
{s+1})\right)\\
=\prod \sb
{i=2} \sp
{r}B\sb
i\sp
{i-1}\prod \sb
{1\le i<j\le r}
(A\sb
i-A\sb
j)(1-C/A\sb
iA\sb
j)\ .\endmultline\tag5.1$$
\endproclaim

\demo{Proof} The determinant on the left-hand side of (5.1) is a
polynomial in  $A\sb
1,\dots,A\sb
r$ of degree $3\binom r2$ divided by $\prod
\sb
{i} A\sb
i\sp
{r-1}$. Since, for $1\le{t\sb
1}< {t\sb
2}\le r$, the determinant is
zero for $A\sb
{t\sb
1}=A\sb
{t\sb
2}$ or $A\sb
{t\sb
1}=C/A\sb
{t\sb
2}$, respectively, 
the left-hand
side is equal to
$$p(A\sb
1,\dots,A\sb
r)\prod \sb
{i<j}(A\sb
i-A\sb
j)(A\sb
iA\sb
j-C)\Big/\prod
\sb
{i}A\sb
i\sp
{r-1}\ ,$$
where $p(A\sb
1,\dots,A\sb
r)$ is some polynomial in the $A\sb
i$'s. Comparing
the degrees yields that $p$ is a constant, which is computed by
comparing coefficients of $\prod \sb
{i}A\sb
i\sp
{r-i}$ on both sides of
(5.1). \qed
\enddemo

\proclaim{Theorem~8}Let $\lambda
$ be a trapezoidal shape, 
$\lambda
\sb
i=\Lambda
-i$, with $\Lambda
\ge 2r$. Given an integer $B$ and an
$r$-tuple $a$ satisfying $(3.1)$, the generating function with respect to the
alternating trace 
weight $w\sb
1$ for shifted
$(\alpha
,\beta
)$-plane partitions of shape $\lambda
$ in which the first part
in row $i$ is equal to $a\sb
i$ and the last part in row $i$ is at least
$B-\beta
 i+\alpha
 i$ is
$$\multline x\sp
{N\sb
4}\prod \sb
{i=1} \sp
{r}\frac {[\Lambda
 -r+a\sb
i-i(1-\alpha
-\beta
)-\Lambda
\alpha
-B]!}
 {[\Lambda
-2i]!\,[r+a\sb
i-i(1-\alpha
-\beta
)-\Lambda
\alpha
-B]!}\times\\
\prod \sb
{1\le i<j\le r}
[a\sb
i-a\sb
j-(i-j)(1-\alpha
-\beta
)][a\sb
i+a\sb
j-(i+j)(1-\alpha
-\beta
)
-\Lambda
(2\alpha
-1)-2B+1]\ ,
\endmultline$$
where 
$$\multline\tsize N\sb
4=\sum \sb
{i=1} \sp
{r}a\sb
i(i-\lfloor \frac {\Lambda
} {2}\rfloor)+\alpha
\left(r\Lambda
\lfloor 
\frac {\Lambda
} {2}\rfloor-\frac {\Lambda
} {2}r(2r+1)+\frac {r(4r\sp
2+3r-1)}
{6}\right)+\chi(\Lambda
 \text { \rm odd})\\
\tsize\cdot\left(\alpha
(\Lambda
\frac {r\sp
2}
{2}-4\binom{r+1}3)+\beta
(\frac {r(4r\sp
2+3r-1)} {6}-r(r+1)\lfloor \frac {\Lambda
}
{2}\rfloor)
+B(\Lambda
 r-r(r+1))\right)\ .
\endmultline$$
($\chi $ is the usual truth function.)
\endproclaim

\demo{Proof} The alternating trace weight $w\sb
1$ comes out of the
general trace weighting $\prod \sb
{l\in{\Bbb
Z}}x\sb
l\sp
{t\sb
l(\pi)}$ 
by setting
$x\sb
i=x\sp
{(-1)\sp
{i-1}i}$. Since for this choice of the variables $x\sb
i$,
we have 
$$t(p;m)=x\sp
{-m\lfloor \frac {p} {2}\rfloor}\bmatrix p+m\\p\endbmatrix\
,$$
which is again shown using \cite{Andrews (1976), (3.3.7)}, by (3.2) the
desired generating function allows the form
$$x\sp
{N\sb
5}\det\left(x\sp
{(-\lfloor \frac {\Lambda
} {2}\rfloor+s)(a\sb
t-t(1-\alpha
-\beta
))}
\bmatrix \Lambda
(1-\alpha
)-s+a\sb
t-t(1-\alpha
-\beta
)-B\\\Lambda
-2s\endbmatrix\right),\tag5.2$$
where we suppress the exact expression for $N\sb
5$ for brevity. Taking
some factors out of the determinant, this expression may be written
$$\multline x\sp
{N\sb
6}\prod \sb
{i=1} \sp
{r}\frac {[\Lambda
-r+\hat A\sb
i]!}
{[\Lambda
-2i]!\,[r+\hat A\sb
i]}
\det\Big((x\sp
{-\Lambda
-1-\hat A\sb
t}-x\sp
{-r})\dotsb
(x\sp
{-\Lambda
-1-\hat A\sb
t}-x\sp
{-s-1})\\
\cdot(x\sp
{\hat A\sb
t}-x\sp
{-r})\dotsb(x\sp
{\hat A\sb
t}-x\sp
{-s-1})\Big)\
 ,
\endmultline$$
where $\hat A\sb
t=a\sb
t-t(1-\alpha
-\beta
)-\Lambda
\alpha
-B$. The determinant can be
evaluated using Lemma~7 with $A\sb
i=x\sp
{\hat A\sb
i}$, $B\sb
i=-x\sp
{-i}$, and
$C=x\sp
{-\Lambda
-1}$.
Simplifications finally lead to the claimed expression. \qed
\enddemo
For the weighting $w\sb
2$ we only obtained a closed form expression for
the first row length of the shifted shape being even. In case of odd
first row length, it does not seem to be possible to evaluate the
determinant.
\proclaim{Theorem~9}Let $\lambda
$ be a trapezoidal shape, 
$\lambda
\sb
i=2\Lambda
-1-i$, with $\Lambda
> r$. Given an integer $B$ and an
$r$-tuple $a$ satisfying $(3.1)$, the generating function with respect to the
alternating trace 
weight $w\sb
2$ for shifted
$(\alpha
,\beta
)$-plane partitions of shape $\lambda
$ in which the first part
in row $i$ is equal to $a\sb
i$ and the last part in row $i$ is at least
$B-\beta
 i+\alpha
 i$ is
$$\multline x\sp
{N\sb
7}\prod \sb
{i=1} \sp
{r}\frac {(1+x\sp
{a\sb
i-i(1-\alpha
-\beta
)
-\alpha
(2\Lambda
-1)-B+\Lambda
})} {(1+x\sp
{\Lambda
-i})}\\
\times\frac {[2\Lambda
-1 -r+a\sb
i-i(1-\alpha
-\beta
)-(2\Lambda
-1)\alpha
-B]!}
 {[2\Lambda
-1
-2i]!\,[r+a\sb
i-i(1-\alpha
-\beta
)-(2\Lambda
-1)\alpha
-B]!}\\
\times\prod \sb
{1\le i<j\le r}
[a\sb
i-a\sb
j-(i-j)(1-\alpha
-\beta
)][a\sb
i+a\sb
j-(i+j)(1-\alpha
-\beta
)
-(2\Lambda
-1)(2\alpha
-1)-2B+1]\ ,
\endmultline$$
where 
$N\sb
7=\sum \sb
{i=1} \sp
{r}a\sb
i(i- {\Lambda
})+\alpha
\left(2\Lambda
\sp
2r-\Lambda
r(3r+2)+\frac {r(2r\sp
2+3r-2)} {3}\right)+\Lambda
 r\sp
2+r
$.
\endproclaim

\demo{Proof} The alternating trace weight $w\sb
2$ comes out of the
general trace weighting by setting $x\sb
i=x\sp
{(-1)\sp
i(i+1)}$. For this
choice of the variables $x\sb
i$, we get, another time using
\cite{Andrews (1976), (3.3.7)},
$$\align t'(p;m)&=x\sp
{-m\lfloor \frac {p+3} {2}\rfloor}\bmatrix
p+m+1\\p+1\endbmatrix -x\sp
{-1-(m-1)\lfloor \frac {p+3} {2}\rfloor }
\bmatrix p+m\\p+1\endbmatrix\\
&=x\sp
{-m\lfloor \frac {p+3} {2}\rfloor}\bmatrix
p+m\\p\endbmatrix\frac {[\lfloor \frac {p+1} {2}\rfloor]+x\sp
{m+\lfloor \frac
{p+1} {2}\rfloor}[\lfloor \frac {p+2} {2}\rfloor]} {[p+1]}\ .
\endalign$$
Using this identity, by (3.2) we obtain, after taking some factors
out of the determinant, for the desired generating function 
$$\multline x\sp
{N\sb
7}
\prod \sb
{i=1} \sp
{r}\frac {(1+x\sp
{a\sb
i-i(1-\alpha
-\beta
)
-\alpha
(2\Lambda
-1)-B+\Lambda
})} {(1+x\sp
{\Lambda
-i})}\det\bigg(x\sp
{(-\Lambda
+1+s)
(a\sb
t-t(1-\alpha
-\beta
))}\\
\cdot \bmatrix (2\Lambda
-1)(1-\alpha
)-s+a\sb
t-t(1-\alpha
-\beta
)-B\\
2\Lambda
-1-2s\endbmatrix\bigg)\ .
\endmultline$$
The determinant in this expression is a special case of the
determinant in (5.2) (replace $\Lambda
$ by $2\Lambda
-1$ in (5.2)). \qed
\enddemo


\Refs
\ref\by Andrews, G.~E.\book The Theory of Partitions\publ
Addison--Wesley\publaddr Reading\yr 1976\bookinfo Encyclopedia of
Mathematics and its Applications, Vol.~2\endref

\ref \by Andrews, G.~E.\paper Plane partitions (III): The weak Macdonald 
conjecture\jour \vol 53\jour Inventiones Math.\yr 1979\pages 193--225
 \endref
\ref\by Chu, W. C.\paper Dominance method for plane partitions.
III. Flagged skew plane partitions\jour 
Acta Math\. Sinica\vol 5\yr 1989\pages 321--328\endref
\ref \by Gansner, E.\paper The Hillman--Grassl correspondence 
and the enumeration of reverse plane partitions\jour J. Combin. Theory 
A\vol 30
\yr 1981a
\pages 71--89 \endref
\ref \by Gansner, E.\paper The enumeration of plane partitions via the 
Burge correspondence\jour Illinois J. Math.\vol 25\yr 1981b\pages 
533--554 \endref

\ref\by Gessel, I., G. Viennot \paper Binomial
determinants, paths, and hook length formulae\jour Adv\. in Math\. 
\vol 58\yr 1985\pages 300--321\endref
\ref \by Gessel, I., G. Viennot\paper Determinants, paths, and 
plane partitions\paperinfo preprint\yr 1989 \endref
\ref \by Krattenthaler, C.\paper Generating functions for plane 
partitions of a given shape\jour Manuscripta Math\. \vol 69\yr1990a\pages
 173--202
 \endref
\ref\by Krattenthaler, C.\paper A determinant evaluation and some enumeration
results for plane partitions\inbook Number-Theoretic Analysis"\publ 
 E.~Hlawka,
R.~F.~Tichy eds., 
     Lect\. Notes in Math. 1452, Springer-Verlag\publaddr Berlin, New
York\pages 121--131\yr 1990b \endref
\ref \by Macdonald, I. G. \book Symmetric Functions and Hall 
Polynomials \publ Oxford University Press\publaddr New 
York/London\yr1979\endref
\ref\by Proctor, R. A.\paper Shifted plane partitions of
trapezoidal shape\jour 
Proc\. Amer\. Math\. Soc\.\vol 89\yr 1983\pages 553--559\endref
\ref\by Proctor, R. A.\paper Odd symplectic group\jour 
Inventiones Math\.\vol 92\yr 1988\pages 307--332\endref
\ref\by Proctor, R. A.\paper New symmetric plane partition identities from
invariant theory work of De Concini and Procesi\jour 
Europ\. J. Combinatorics\vol 11\yr 1990\pages 289--300 \endref
\ref \by Mills, W. H., D. H. Robbins,  H. Rumsey\paper Proof 
of the Macdonald conjecture\jour Inventiones Math.\vol 66\yr 
1982\pages 73--87 \endref
\ref\by Mills, W. H., D. H. Robbins,  H. Rumsey\paper Alternating sign
 matrices and descending plane
partitions\jour 
J. Combin\. Theory A\vol 34\yr 1983\pages 340--359\endref
\ref \by Mills, W. H., D. H. Robbins,  H. Rumsey\paper Self--complementary
 totally symmetric plane 
partitions\jour J. Combin. Theory A\vol 42\yr 1986\pages 277--292 
 \endref
\ref \by Okada, S.\paper On the generating functions for certain 
classes of plane partitions\jour J. Combin. Theory A\vol 51\yr 
1989\pages 1--23 \endref
\ref\by Okada, S.\paper Partially strict shifted plane partitions\jour 
J. Combin\. Theory A\vol 53\yr 1990\pages 143--156\endref
\ref \by Stanley, R. P.\paper Theory and applications of plane 
partitions: Part 2\jour Stud. Appl. Math\vol 50\yr 1971\pages 
259--279 \endref
\ref\by Stembridge, J.\paper Nonintersecting paths, Pfaffians, and
plane partitions\jour Adv\. in Math\. \vol 83\yr1990a\pages 96--131\endref
\ref\by Stembridge, J.\paper Hall-Littlewood functions, plane
partitions, and the Rogers-Ramanujan identities\jour 
Trans\. Amer\. Math\. Soc\.\vol 319\yr 1990b\pages 469--498\endref
\ref\by Wachs, M. L.\paper Flagged Schur functions, Schubert
polynomials, and symmetrizing operators\jour 
J. Combin\. Theory A\vol 40\yr 1985\pages 276--289\endref


\enddocument
