\input amstex
\documentstyle{amsppt}
\magnification1200
\hsize15.6truecm
\vsize22.8truecm

\font\Bf=cmbx12
\font\Rm=cmr12
\def\LL{\leavevmode\setbox0=\hbox{L}\hbox to\wd0{\hss\char'40L}}
\def\al{\alpha}
\def\be{\beta}
\def\ga{\gamma}
\def\de{\delta}
\def\ep{\varepsilon}
\def\ze{\zeta}
\def\et{\eta}
\def\th{\theta}
\def\vt{\vartheta}
\def\io{\iota}
\def\ka{\kappa}
\def\la{\lambda}
\def\rh{\rho}
\def\si{\sigma}
\def\ta{\tau}
\def\ph{\varphi}
\def\ch{\chi}
\def\ps{\psi}
\def\om{\omega}
\def\Ga{\Gamma}
\def\De{\Delta}
\def\Th{\Theta}
\def\La{\Lambda}
\def\Si{\Sigma}
\def\Ph{\Phi}
\def\Ps{\Psi}
\def\Om{\Omega}
\def\row#1#2#3{#1_{#2},\ldots,#1_{#3}}
\def\rowup#1#2#3{#1^{#2},\ldots,#1^{#3}}
\def\x{\times}
\def\crf{}            %used for crossreferencing, Tex should ignore.
\def\rf{}             %used for refencing (section-numbers)
\def\rfnew{}          %used for new-section numbers
\def\P{{\Bbb P}}
\def\R{{\Bbb R}}
\def\X{{\Cal X}}
\def\C{{\Bbb C}}
\def\Mf{{\Cal Mf}}
\def\FM{{\Cal F\Cal M}}
\def\F{{\Cal F}}
\def\G{{\Cal G}}
\def\V{{\Cal V}}
\def\T{{\Cal T}}
\def\A{{\Cal A}}
\def\N{{\Bbb N}}
\def\Z{{\Bbb Z}}
\def\Q{{\Bbb Q}}
\def\ddt{\left.\tfrac \partial{\partial t}\right\vert_0}
\def\dd#1{\tfrac \partial{\partial #1}}
\def\today{\ifcase\month\or
 January\or February\or March\or April\or May\or June\or
 July\or August\or September\or October\or November\or December\fi
 \space\number\day, \number\year}
\def\nmb#1#2{#2} %zum Nummerieren
\def\dfrac#1#2{{\displaystyle{#1\over#2}}}
\def\tfrac#1#2{{\textstyle{#1\over#2}}}
\def\iprod#1#2{\langle#1,#2\rangle}
\def\pder#1#2{\frac{\partial #1}{\partial #2}}
\def\iint{\int\!\!\int}
\def\({\left(}
\def\){\right)}
\def\[{\left[}
\def\]{\right]}
\def\supp{\operatorname{supp}}
\def\Df{\operatorname{Df}}
\def\dom{\operatorname{dom}}
\def\Ker{\operatorname{Ker}}
\def\Tr{\operatorname{Tr}}
\def\Res{\operatorname{Res}}
\def\Aut{\operatorname{Aut}}
\def\kgV{\operatorname{kgV}}
\def\ggT{\operatorname{ggT}}
\def\diam{\operatorname{diam}}
\def\Im{\operatorname{Im}}
\def\Re{\operatorname{Re}}
\def\ord{\operatorname{ord}}
\def\rang{\operatorname{rang}}
\def\rng{\operatorname{rng}}
\def\grd{\operatorname{grd}}
\def\inv{\operatorname{inv}}
\def\maj{\operatorname{maj}}
\def\des{\operatorname{des}}
\def\varmaj{\operatorname{\overline{maj}}}
\def\vardes{\operatorname{\overline{des}}}
\def\pvarmaj{\operatorname{\overline{maj}'}}
\def\pmaj{\operatorname{maj'}}
\def\ln{\operatorname{ln}}
\def\der{\operatorname{der}}
\def\Hom{\operatorname{Hom}}
\def\tr{\operatorname{tr}}
\def\Span{\operatorname{Span}}
\def\grad{\operatorname{grad}}
\def\div{\operatorname{div}}
\def\rot{\operatorname{rot}}
\def\Sp{\operatorname{Sp}}
\def\sgn{\operatorname{sgn}}
\def\liml{\lim\limits}
\def\supl{\sup\limits}
\def\bigcupl{\bigcup\limits}
\def\bigcapl{\bigcap\limits}
\def\limsupl{\limsup\limits}
\def\liminfl{\liminf\limits}
\def\intl{\int\limits}
\def\suml{\sum\limits}
\def\maxl{\max\limits}
\def\minl{\min\limits}
\def\prodl{\prod\limits}
\def\tg{\operatorname{tan}}
\def\ctg{\operatorname{cot}}
\def\arctg{\operatorname{arctan}}
\def\arccot{\operatorname{arccot}}
\def\arcctg{\operatorname{arccot}}
\def\tgh{\operatorname{tanh}}
\def\ctgh{\operatorname{coth}}
\def\arcsinh{\operatorname{arcsinh}}
\def\arccosh{\operatorname{arccosh}}
\def\arctgh{\operatorname{arctanh}}
\def\arcctgh{\operatorname{arccoth}}
\def\3{\ss}
\catcode`\@=11
\def\dddot#1{\vbox{\ialign{##\crcr
      .\hskip-.5pt.\hskip-.5pt.\crcr\noalign{\kern1.5\p@\nointerlineskip}
      $\hfil\displaystyle{#1}\hfil$\crcr}}}

\newif\iftab@\tab@false
\newif\ifvtab@\vtab@false
\def\tab{\bgroup\tab@true\vtab@false\vst@bfalse\Strich@false%
   \def\\{\global\hline@@false%
     \ifhline@\global\hline@false\global\hline@@true\fi\cr}
   \edef\l@{\the\leftskip}\ialign\bgroup\hskip\l@##\hfil&&##\hfil\cr}
\def\endtab{\cr\egroup\egroup}
\def\vtab{\vtop\bgroup\vst@bfalse\vtab@true\tab@true\Strich@false%
   \bgroup\def\\{\cr}\ialign\bgroup&##\hfil\cr}
\def\endvtab{\cr\egroup\egroup\egroup}
\def\stab{\D@cke0.5pt\null 
 \bgroup\tab@true\vtab@false\vst@bfalse\Strich@true\Let@@\vspace@
 \normalbaselines\offinterlineskip
  \openup\spreadmlines@
 \edef\l@{\the\leftskip}\ialign
 \bgroup\hskip\l@##\hfil&&##\hfil\crcr}
\def\endstab{\crcr\egroup
 \egroup}
\newif\ifvst@b\vst@bfalse
\def\vstab{\D@cke0.5pt\null
 \vtop\bgroup\tab@true\vtab@false\vst@btrue\Strich@true\bgroup\Let@@\vspace@
 \normalbaselines\offinterlineskip
  \openup\spreadmlines@\bgroup}
\def\endvstab{\crcr\egroup\egroup
 \egroup\tab@false\Strich@false}

\newdimen\htstrut@
\htstrut@8.5\p@
\newdimen\htStrut@
\htStrut@12\p@
\newdimen\dpstrut@
\dpstrut@3.5\p@
\newdimen\dpStrut@
\dpStrut@3.5\p@
\def\openup{\afterassignment\@penup\dimen@=}
\def\@penup{\advance\lineskip\dimen@
  \advance\baselineskip\dimen@
  \advance\lineskiplimit\dimen@
  \divide\dimen@ by2
  \advance\htstrut@\dimen@
  \advance\htStrut@\dimen@
  \advance\dpstrut@\dimen@
  \advance\dpStrut@\dimen@}
\def\Let@@{\relax\iffalse{\fi%
    \def\\{\global\hline@@false%
     \ifhline@\global\hline@false\global\hline@@true\fi\cr}%
    \iffalse}\fi}
\def\matrix{\null\,\vcenter\bgroup
 \tab@false\vtab@false\vst@bfalse\Strich@false\Let@@\vspace@
 \normalbaselines\openup\spreadmlines@\ialign
 \bgroup\hfil$\m@th##$\hfil&&\quad\hfil$\m@th##$\hfil\crcr
 \Mathstrut@\crcr\noalign{\kern-\baselineskip}}
\def\endmatrix{\crcr\Mathstrut@\crcr\noalign{\kern-\baselineskip}\egroup
 \egroup\,}
\def\smatrix{\D@cke0.5pt\null\,
 \vcenter\bgroup\tab@false\vtab@false\vst@bfalse\Strich@true\Let@@\vspace@
 \normalbaselines\offinterlineskip
  \openup\spreadmlines@\ialign
 \bgroup\hfil$\m@th##$\hfil&&\quad\hfil$\m@th##$\hfil\crcr}
\def\endsmatrix{\crcr\egroup
 \egroup\,\Strich@false}
\newdimen\D@cke
\def\Dicke#1{\global\D@cke#1}
\newtoks\tabs@\tabs@{&}
\newif\ifStrich@\Strich@false
\newif\iff@rst

\def\Stricherr@{\iftab@\ifvtab@\errmessage{\noexpand\s not allowed
     here. Use \noexpand\vstab!}%
  \else\errmessage{\noexpand\s not allowed here. Use \noexpand\stab!}%
  \fi\else\errmessage{\noexpand\s not allowed
     here. Use \noexpand\smatrix!}\fi}
\def\format{\ifvst@b\else\crcr\fi\egroup\iffalse{\fi\ifnum`}=0 \fi\format@}
\def\format@#1\\{\def\preamble@{#1}%
 \def\Str@chfehlt##1{\ifx##1\s\Stricherr@\fi\ifx##1\\\let\Next\relax%
   \else\let\Next\Str@chfehlt\fi\Next}%
 \def\c{\hfil\noexpand\ifhline@@\hbox{\vrule height\htStrut@%
   depth\dpstrut@ width\z@}\noexpand\fi%
   \ifStrich@\hbox{\vrule height\htstrut@ depth\dpstrut@ width\z@}%
   \fi\iftab@\else$\m@th\fi\the\hashtoks@\iftab@\else$\fi\hfil}%
 \def\r{\hfil\noexpand\ifhline@@\hbox{\vrule height\htStrut@%
   depth\dpstrut@ width\z@}\noexpand\fi%
   \ifStrich@\hbox{\vrule height\htstrut@ depth\dpstrut@ width\z@}%
   \fi\iftab@\else$\m@th\fi\the\hashtoks@\iftab@\else$\fi}%
 \def\l{\noexpand\ifhline@@\hbox{\vrule height\htStrut@%
   depth\dpstrut@ width\z@}\noexpand\fi%
   \ifStrich@\hbox{\vrule height\htstrut@ depth\dpstrut@ width\z@}%
   \fi\iftab@\else$\m@th\fi\the\hashtoks@\iftab@\else$\fi\hfil}%
 \def\s{\ifStrich@\ \the\tabs@\vrule width\D@cke\the\hashtoks@%
          \fi\the\tabs@\ }%
 \def\sa{\ifStrich@\vrule width\D@cke\the\hashtoks@%
            \the\tabs@\ %
            \fi}%
 \def\se{\ifStrich@\ \the\tabs@\vrule width\D@cke\the\hashtoks@\fi}%
 \def\cd{\hfil\noexpand\ifhline@@\hbox{\vrule height\htStrut@%
   depth\dpstrut@ width\z@}\noexpand\fi%
   \ifStrich@\hbox{\vrule height\htstrut@ depth\dpstrut@ width\z@}%
   \fi$\dsize\m@th\the\hashtoks@$\hfil}%
 \def\rd{\hfil\noexpand\ifhline@@\hbox{\vrule height\htStrut@%
   depth\dpstrut@ width\z@}\noexpand\fi%
   \ifStrich@\hbox{\vrule height\htstrut@ depth\dpstrut@ width\z@}%
   \fi$\dsize\m@th\the\hashtoks@$}%
 \def\ld{\noexpand\ifhline@@\hbox{\vrule height\htStrut@%
   depth\dpstrut@ width\z@}\noexpand\fi%
   \ifStrich@\hbox{\vrule height\htstrut@ depth\dpstrut@ width\z@}%
   \fi$\dsize\m@th\the\hashtoks@$\hfil}%
 \ifStrich@\else\Str@chfehlt#1\\\fi%
 \setbox\z@\hbox{\xdef\Preamble@{\preamble@}}\ifnum`{=0 \fi\iffalse}\fi
 \ialign\bgroup\span\Preamble@\crcr}
\newif\ifhline@\hline@false
\newif\ifhline@@\hline@@false
\def\hlinefor#1{\multispan@{\strip@#1 }\leaders\hrule height\D@cke\hfill%
    \global\hline@true\ignorespaces}
\def\Item "#1"{\par\noindent\hangindent2\parindent%
  \hangafter1\setbox0\hbox{\rm#1\enspace}\ifdim\wd0>2\parindent%
  \box0\else\hbox to 2\parindent{\rm#1\hfil}\fi\ignorespaces}
\def\ITEM #1"#2"{\par\noindent\hangafter1\hangindent#1%
  \setbox0\hbox{\rm#2\enspace}\ifdim\wd0>#1%
  \box0\else\hbox to 0pt{\rm#2\hss}\hskip#1\fi\ignorespaces}
\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

\font@\twelverm=cmr10 scaled\magstep1
\font@\twelveit=cmti10 scaled\magstep1
\font@\twelvebf=cmbx10 scaled\magstep1
\font@\twelvei=cmmi10 scaled\magstep1
\font@\twelvesy=cmsy10 scaled\magstep1
\font@\twelveex=cmex10 scaled\magstep1

\newtoks\twelvepoint@
\def\twelvepoint{\normalbaselineskip15\p@
 \abovedisplayskip15\p@ plus3.6\p@ minus10.8\p@
 \belowdisplayskip\abovedisplayskip
 \abovedisplayshortskip\z@ plus3.6\p@
 \belowdisplayshortskip8.4\p@ plus3.6\p@ minus4.8\p@
 \textonlyfont@\rm\twelverm \textonlyfont@\it\twelveit
 \textonlyfont@\sl\twelvesl \textonlyfont@\bf\twelvebf
 \textonlyfont@\smc\twelvesmc \textonlyfont@\tt\twelvett
%Erg„nzung des fetten Small-Capitals-Fonts:
%
 \ifsyntax@ \def\big##1{{\hbox{$\left##1\right.$}}}%
  \let\Big\big \let\bigg\big \let\Bigg\big
 \else
  \textfont\z@=\twelverm  \scriptfont\z@=\tenrm  \scriptscriptfont\z@=\sevenrm
  \textfont\@ne=\twelvei  \scriptfont\@ne=\teni  \scriptscriptfont\@ne=\seveni
  \textfont\tw@=\twelvesy \scriptfont\tw@=\tensy \scriptscriptfont\tw@=\sevensy
  \textfont\thr@@=\twelveex \scriptfont\thr@@=\tenex
        \scriptscriptfont\thr@@=\tenex
  \textfont\itfam=\twelveit \scriptfont\itfam=\tenit
        \scriptscriptfont\itfam=\tenit
  \textfont\bffam=\twelvebf \scriptfont\bffam=\tenbf
        \scriptscriptfont\bffam=\sevenbf
  \setbox\strutbox\hbox{\vrule height10.2\p@ depth4.2\p@ width\z@}%
  \setbox\strutbox@\hbox{\lower.6\normallineskiplimit\vbox{%
        \kern-\normallineskiplimit\copy\strutbox}}%
 \setbox\z@\vbox{\hbox{$($}\kern\z@}\bigsize@=1.4\ht\z@
 \fi
 \normalbaselines\rm\ex@.2326ex\jot3.6\ex@\the\twelvepoint@}

\font@\fourteenrm=cmr10 scaled\magstep2
\font@\fourteenit=cmti10 scaled\magstep2
\font@\fourteensl=cmsl10 scaled\magstep2
\font@\fourteensmc=cmcsc10 scaled\magstep2
\font@\fourteentt=cmtt10 scaled\magstep2
\font@\fourteenbf=cmbx10 scaled\magstep2
\font@\fourteeni=cmmi10 scaled\magstep2
\font@\fourteensy=cmsy10 scaled\magstep2
\font@\fourteenex=cmex10 scaled\magstep2
\font@\fourteenmsa=msam10 scaled\magstep2
\font@\fourteeneufm=eufm10 scaled\magstep2
\font@\fourteenmsb=msbm10 scaled\magstep2
\newtoks\fourteenpoint@
\def\fourteenpoint{\normalbaselineskip15\p@
 \abovedisplayskip18\p@ plus4.3\p@ minus12.9\p@
 \belowdisplayskip\abovedisplayskip
 \abovedisplayshortskip\z@ plus4.3\p@
 \belowdisplayshortskip10.1\p@ plus4.3\p@ minus5.8\p@
 \textonlyfont@\rm\fourteenrm \textonlyfont@\it\fourteenit
 \textonlyfont@\sl\fourteensl \textonlyfont@\bf\fourteenbf
 \textonlyfont@\smc\fourteensmc \textonlyfont@\tt\fourteentt
%Erg„nzung des fetten Small-Capitals-Fonts:
%
 \ifsyntax@ \def\big##1{{\hbox{$\left##1\right.$}}}%
  \let\Big\big \let\bigg\big \let\Bigg\big
 \else
  \textfont\z@=\fourteenrm  \scriptfont\z@=\twelverm  \scriptscriptfont\z@=\tenrm
  \textfont\@ne=\fourteeni  \scriptfont\@ne=\twelvei  \scriptscriptfont\@ne=\teni
  \textfont\tw@=\fourteensy \scriptfont\tw@=\twelvesy \scriptscriptfont\tw@=\tensy
  \textfont\thr@@=\fourteenex \scriptfont\thr@@=\twelveex
        \scriptscriptfont\thr@@=\twelveex
  \textfont\itfam=\fourteenit \scriptfont\itfam=\twelveit
        \scriptscriptfont\itfam=\twelveit
  \textfont\bffam=\fourteenbf \scriptfont\bffam=\twelvebf
        \scriptscriptfont\bffam=\tenbf
  \setbox\strutbox\hbox{\vrule height12.2\p@ depth5\p@ width\z@}%
  \setbox\strutbox@\hbox{\lower.72\normallineskiplimit\vbox{%
        \kern-\normallineskiplimit\copy\strutbox}}%
 \setbox\z@\vbox{\hbox{$($}\kern\z@}\bigsize@=1.7\ht\z@
 \fi
 \normalbaselines\rm\ex@.2326ex\jot4.3\ex@\the\fourteenpoint@}

\font@\seventeenrm=cmr10 scaled\magstep3
\font@\seventeenit=cmti10 scaled\magstep3
\font@\seventeensl=cmsl10 scaled\magstep3
\font@\seventeensmc=cmcsc10 scaled\magstep3
\font@\seventeentt=cmtt10 scaled\magstep3
\font@\seventeenbf=cmbx10 scaled\magstep3
\font@\seventeeni=cmmi10 scaled\magstep3
\font@\seventeensy=cmsy10 scaled\magstep3
\font@\seventeenex=cmex10 scaled\magstep3
\font@\seventeenmsa=msam10 scaled\magstep3
\font@\seventeeneufm=eufm10 scaled\magstep3
\font@\seventeenmsb=msbm10 scaled\magstep3
\newtoks\seventeenpoint@
\def\seventeenpoint{\normalbaselineskip18\p@
 \abovedisplayskip21.6\p@ plus5.2\p@ minus15.4\p@
 \belowdisplayskip\abovedisplayskip
 \abovedisplayshortskip\z@ plus5.2\p@
 \belowdisplayshortskip12.1\p@ plus5.2\p@ minus7\p@
 \textonlyfont@\rm\seventeenrm \textonlyfont@\it\seventeenit
 \textonlyfont@\sl\seventeensl \textonlyfont@\bf\seventeenbf
 \textonlyfont@\smc\seventeensmc \textonlyfont@\tt\seventeentt
%Erg„nzung des fetten Small-Capitals-Fonts:
%
 \ifsyntax@ \def\big##1{{\hbox{$\left##1\right.$}}}%
  \let\Big\big \let\bigg\big \let\Bigg\big
 \else
  \textfont\z@=\seventeenrm  \scriptfont\z@=\fourteenrm  \scriptscriptfont\z@=\twelverm
  \textfont\@ne=\seventeeni  \scriptfont\@ne=\fourteeni  \scriptscriptfont\@ne=\twelvei
  \textfont\tw@=\seventeensy \scriptfont\tw@=\fourteensy \scriptscriptfont\tw@=\twelvesy
  \textfont\thr@@=\seventeenex \scriptfont\thr@@=\fourteenex
        \scriptscriptfont\thr@@=\fourteenex
  \textfont\itfam=\seventeenit \scriptfont\itfam=\fourteenit
        \scriptscriptfont\itfam=\fourteenit
  \textfont\bffam=\seventeenbf \scriptfont\bffam=\fourteenbf
        \scriptscriptfont\bffam=\twelvebf
  \setbox\strutbox\hbox{\vrule height14.6\p@ depth6\p@ width\z@}%
  \setbox\strutbox@\hbox{\lower.86\normallineskiplimit\vbox{%
        \kern-\normallineskiplimit\copy\strutbox}}%
 \setbox\z@\vbox{\hbox{$($}\kern\z@}\bigsize@=2\ht\z@
 \fi
 \normalbaselines\rm\ex@.2326ex\jot5.2\ex@\the\seventeenpoint@}

\catcode`\@=13

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Input-Datei zum Erzeugen von Gitterpunktwegen.%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Pfad-Eingabe: 
%  \Pfad(x-Koordinate des Anf.Punkts,y-Koordinate des Anf.Punkts),Pfad
%  als 1-2-Wort\endPfad
%(1=Schritt in x-Richtung, 2=Schritt in y-Richtung)
%
%Koordinatenachsen:
%  \Koordinatenachsen(L„nge der x-Achse, L„nge der y-Achse)
%
%Gitter:
%  \Gitter(Anzahl der Punkte in x-Richtung, Anzahl der Punkte in x-Richtung)
%
%Bezeichnung von Punkten:
%  \Label[wo?]{[Bezeichnung]}(x-Koordinate,y-Koordinate)
%wobei:
%  [wo?]=\l,\lo,\lu,\r,\ro,\ru,\o,\u
%und l=links, r=rechts, u=unten, o=oben.
%
%Die Einheit kann durch Eingabe von
%  \Einheit=?cm
%ver„ndert werden.
%
%Die Dicke der Pfade kann durch Eingabe von
%     \PfadDicke{?cm}
%ver„ndert werden.
%
%Es stehen die folgenden Punktgr”áen zur Verf�gung:
%\DuennPunkt, \NormalPunkt, \DickPunkt. Eingabe:
%  \DickPunkt(x-Koordinate,y-Koordinate), etc.
%
%Weiters steht mit \Kreis ein Kreis zur Verf�gung. Eingabe:
%  \Kreis(x-Koordinate,y-Koordinate)
%
\catcode`\@=11
\newskip\Einheit \Einheit=0.5cm
\newcount\xcoord \newcount\ycoord
\newdimen\xdim \newdimen\ydim \newdimen\PfadD@cke \newdimen\Pfadd@cke
\PfadD@cke1pt \Pfadd@cke0.5pt
\def\PfadDicke#1{\PfadD@cke#1 \divide\PfadD@cke by2 \Pfadd@cke\PfadD@cke
 \multiply\PfadD@cke by2}
\long\def\LOOP#1\REPEAT{\def\BODY{#1}\ITERATE}
\def\ITERATE{\BODY \let\next\ITERATE \else\let\next\relax\fi \next}
\let\REPEAT=\fi
\def\Punkt{\hbox{\raise-2pt\hbox to0pt{\hss$\ssize\bullet$\hss}}}
\def\DuennPunkt(#1,#2){\unskip
  \raise#2 \Einheit\hbox to0pt{\hskip#1 \Einheit
          \raise-2.5pt\hbox to0pt{\hss$\bullet$\hss}\hss}}
\def\NormalPunkt(#1,#2){\unskip
  \raise#2 \Einheit\hbox to0pt{\hskip#1 \Einheit
          \raise-3pt\hbox to0pt{\hss\twelvepoint$\bullet$\hss}\hss}}
\def\DickPunkt(#1,#2){\unskip
  \raise#2 \Einheit\hbox to0pt{\hskip#1 \Einheit
          \raise-4pt\hbox to0pt{\hss\fourteenpoint$\bullet$\hss}\hss}}
\def\Kreis(#1,#2){\unskip
  \raise#2 \Einheit\hbox to0pt{\hskip#1 \Einheit
          \raise-4pt\hbox to0pt{\hss\fourteenpoint$\circ$\hss}\hss}}
\def\Pfad(#1,#2),#3\endPfad{\unskip\leavevmode
  \xcoord#1 \ycoord#2 \ZeichnePfad#3\endPfad}
\def\ZeichnePfad#1{\ifx#1\endPfad\let\next\relax
  \else\let\next\ZeichnePfad
    \ifnum#1=1
      \raise\ycoord \Einheit\hbox to0pt{\hskip\xcoord \Einheit
         \vrule height\Pfadd@cke width1 \Einheit depth\Pfadd@cke\hss}%
      \advance\xcoord by 1
    \else
      \raise\ycoord \Einheit\hbox to0pt{\hskip\xcoord \Einheit
        \hbox{\hskip-1pt\vrule height1 \Einheit width\PfadD@cke
 depth0pt}\hss}%
      \advance\ycoord by 1
    \fi
  \fi\next}
\def\Koordinatenachsen(#1,#2){\unskip
 \hbox to0pt{\hskip-.5pt\vrule height#2 \Einheit width.5pt depth1 \Einheit}%
 \hbox to0pt{\hskip-1 \Einheit \xcoord#1 \advance\xcoord by1
    \vrule height0.25pt width\xcoord \Einheit depth0.25pt\hss}}
\def\Koordinatenachsen(#1,#2)(#3,#4){\unskip
 \hbox to0pt{\hskip-.5pt \ycoord-#4 \advance\ycoord by1
    \vrule height#2 \Einheit width.5pt depth\ycoord \Einheit}%
 \hbox to0pt{\hskip-1 \Einheit \hskip#3\Einheit 
    \xcoord#1 \advance\xcoord by1 \advance\xcoord by-#3 
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%\showthe\abovedisplayskip
\def\WachAA{32}
\def\VienAA{31}
\def\SunaAA{30}
\def\SulaAE{29}
\def\StemAE{28}
\def\SaStAA{27}
\def\RobyAA{26}
\def\ProcAD{25}
\def\ProcAB{24}
\def\MacdAA{23}
\def\LiMiAB{22}
\def\KulkAB{21}
\def\KulkAC{20}
\def\KrSuAA{19}
\def\KratAP{18}
\def\KnutAA{17}
\def\GustPR{16}
\def\GustAA{15}
\def\GoulAD{14}
\def\GordAA{13}
\def\GeViAB{12}
\def\GessAL{11}
\def\GessAH{10}
\def\GaRaAA{9}
\def\FomiAB{8}
\def\CoHeAA{7}
\def\ChGoAA{6}
\def\BurgAA{5}
\def\AndrAK{4}
\def\AndrAJ{3}
\def\AndrAF{2}
\def\AbhyAB{1}
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\def\labu#1{\raise-6pt\hbox to20pt{\hskip-2pt\eightpoint$#1$\hss}}
\def\P{{\Cal P}}
\def\x{{\bold x}}
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\def\oddrows{\operatorname{oddrows}}
\def\oddcolumns{\operatorname{oddcolumns}}

\topmatter 
\title Non-crossing two-rowed arrays and summations for Schur
functions\\
\tenpoint (Summary)
\endtitle 
\author C.~Krattenthaler
\endauthor 
\affil 
Institut f\"ur Mathematik der Universit\"at Wien,\\
Strudlhofgasse 4, A-1090 Wien, Austria.\\
e-mail: KRATT\@PAP.UNIVIE.AC.AT
\endaffil 
%\address Institut f\"ur Mathematik der Universit\"at Wien,
%Strudlhofgasse 4, A-1090 Wien, Austria.
%\endaddress
\email KRATT\@Pap.Univie.Ac.At (In), KRATT\@AWIRAP (Bitnet)\endemail
%\dedicatory \enddedicatory
%\date \enddate
%\thanks \endthanks
\subjclass Primary 05E05;
 Secondary 05A10, 05A15, 05A17, 05A30, 05E10, 33D20.
\endsubjclass
\keywords Tableaux, plane partitions, symmetric functions, Schur
functions, generating functions,
skew Knuth-correspondence, basic hypergeometric series in $U(n)$ and
$Sp(n)$, basic hypergeometric series for $A_l$ and $C_l$\endkeywords
\abstract
In the first part of this paper (sections~1,2) we give combinatorial
proofs for determinantal formulas for sums of Schur functions ``in a strip" that
were originally obtained by Gessel, respectively Goulden, using
algebraic methods. The combinatorial analysis involves
certain families of two-rowed arrays, asymmetric
variations of Sagan and Stanley's skew
Knuth-correspondence, and variations of
one of Burge's correspondences.
In the third section we specialize the
parameters in these determinants to compute norm generating functions
for tableaux in a strip. In case we can get rid of the determinant we
obtain multifold summations that are basic hypergeometric series for
$A_r$ and $C_r$ respectively. In some cases these sums can be evaluated. Thus in
particular, an alternative proof for refinements of the Bender-Knuth
and MacMahon \hbox{(ex-)}Conjectures, which were first obtained in another
paper by the author, is provided. Although there are some parallels
with the original proof, perhaps this proof is easier accessible.
Finally, in section~4, we record further applications of our methods
to the enumeration of paths with respect to weighted turns.
\endabstract
\endtopmatter
\rightheadtext{Non-crossing two-rowed arrays}

\document
\abovedisplayskip=6pt plus2pt minus 4pt
\belowdisplayskip=6pt plus2pt minus 4pt

\subheading{1. Generating functions for non-crossing two-rowed arrays}
We consider two-rowed arrays $P=(p\mid q)$ of the form
$$\matrix p_{-a}&p_{-a+1}&\hdots&p_{-1}&p_1&\hdots &p_k\\
&&&&q_1&\hdots &q_k&q_{-1}&\hdots&q_{-b+1}&q_{-b}\endmatrix\ ,\tag1.1$$
where $a,k,b$ are some nonnegative integers and where 
the entries $p_i,q_i$ are positive integers such that both rows
of the array are weakly increasing. (To be precise, 
if $k=0$, i.e\. the ``middle
part" of the array is empty, for a $c\le \min\{a,b\}$ 
we also allow the entries
$p_{-1},\dots,p_{-c}$ and $q_{-1},\dots,q_{-c}$ to be ``empty".) 
We say that $P$ is {\it of the 
type} $(a,b)$ and {\it of the shape} $(a,k,b)$. If both rows of $P$
are strictly increasing then we call $P$ a {\it strict} two-rowed
array. Given an array $P_1=(p^{(1)}\mid q^{(1)})$ of the shape
$(a_1,k_1,b_1)$ and an array $P_2=(p^{(2)}\mid q^{(2)})$ of the shape
$(a_2,k_2,b_2)$, we say that $P_1$ {\it dominates} (resp\. {\it
strictly dominates}) $P_2$ if the following three conditions hold:
\roster
\item "(D1)" $a_1\le a_2$ and $p_l^{(1)}\le p_l^{(2)} $ (resp\. $p_l^{(1)}<
 p_l^{(2)} $) for all $l=-1,-2,\dots,-\min\{a_1,a_2\}$. 
(By convention, these inequalities are
also violated if $p_l^{(1)}$ should be an ``empty" entry.)
\item "(D2)" $b_1\le b_2$ and $q_l^{(1)}\ge q_l^{(2)} $ (resp\. $q_l^{(1)}>
 q_l^{(2)} $) for all $l=-1,-2,\dots,-\min\{b_1,b_2\}$. 
(By convention, these inequalities are
also violated if $q_l^{(1)}$ should be an ``empty" entry.)
\item "(D3)" For each $m$, $1\le m\le k_2$, there is an $l$, $1\le
l\le k_1$, such that $p_l^{(1)}\le p_m^{(2)}$ and $q_l^{(1)}\ge
q_m^{(2)}$ (resp\. $p_l^{(1)}< p_m^{(2)}$ and $q_l^{(1)}>
q_m^{(2)}$).
\endroster
Let $\al=(\al_r,\al_{r-1},\dots,\al_1)$ and $\be=(\be_r,\be_{r-1},\dots,\be_1)$
be $r$-tupels of nonnegative integers. A family $\P=(P_1,\dots,P_r)$ of 
two-rowed arrays is called {\it of the type}
$(\al,\be)$ if $P_i$ is of the type $(\al_i,\be_i)$, $i=1,2,\dots,r$.
$\P$ is called {\it strict} if all $P_i$, $i=1,\dots,r$, are
strict. The family $\P$ is called {\it non-crossing}
(resp\. {\it strictly non-crossing}) if $P_i$
dominates (resp\. strictly dominates) $P_{i+1}$, $i=1,\dots,r-1$.

Our objects of interest are non-crossing strict families of two-rowed
arrays (see the example in section~2) and strictly non-crossing (ordinary) 
two-rowed arrays. If 
$\al$ and $\be$ are partitions, the most convenient way to look at a
non-crossing strict family $\P=(P_1,\dots,P_r)=((p^{(1)}\mid
q^{(1)}),\dots,(p^{(r)}\mid q^{(r)}))$ of the type $(\al,\be)$ is by
rephrasing the three conditions (D1)--(D3) of dominance in terms of
the following three properties:
\roster
\item"(NS1)" The {\it front part}, the array $(p_{-i}^{(r-j+1)})_{i,j\ge 1}$ is a 
column-strict plane partition of shape $\al'/\nu$. 
\item"(NS2)" The {\it tail}, the array $(q_{-i}^{(r-j+1)})_{i,j\ge 1}$ is a
tableau of shape $\be'/\nu$. $\nu$ is the same partition as in (NS1).
\item"(NS3)" The {\it middle part} of $\P$, interpreted as multiset
$\{(p_j^{(i)},q_j^{(i)}):1\le i\le r,\ 1\le j\le k_i\}$, can be
viewed as a family of $r$ pairwise non-crossing lattice paths in
$\Z^2$ consisting of unit horizontal and vertical steps in the
positive direction, all of which starting at $(0,0)$ and ending with
an infinite horizontal part. (Thus the final points of the paths might
be considered to be of the form $(\infty,y_i)$, for some nonnegative
integers $y_i$.) This is seen as follows: For fixed $i$ consider the
points $(p_j^{(i)},q_j^{(i)})$, $j=1,\dots,k_i$. Now to these points
apply Viennot's \cite{\VienAA} light and shadow procedure (with the sun
being located in the North-West). For each $i$ this yields a lattice
path of the described type. The condition (D3) (ordinary dominance)
simply says that these $r$ paths are pairwise non-crossing (from here
we are lead to call the arrays under consideration ``non-crossing"), 
the first
path lying ``avove" the second, the second ``above" the third, etc.
\endroster

\NoBlackBoxes
A similar interpretation holds for strictly non-crossing (ordinary)
families $\P=(P_1,\dots,P_r)$. This time the conditions (D1)--(D3)
can be rephrased in the following way.
\roster
\item"(SN1)" The {\it front part}, the array $(p_{-j}^{(r-i+1)})_{i,j\ge 1}$ is a 
column-strict plane partition of shape $\al/\nu$. 
($\nu$ is allowed to differ from $0$
only if the middle part (see (SN3)) of $\Cal P$ is empty.)
\item"(SN2)" The {\it tail}, the array $(q_{-j}^{(r-i+1)})_{i,j\ge 1}$ is a
tableau of shape $\be/\nu$. $\nu$ is the same partition as in (SN1).
($\nu$ is allowed to differ from $0$
only if the middle part (see (SN3)) of $\Cal P$ is empty.)
\item"(SN3)" 
The {\it middle part} of $\P$, interpreted as multiset
$\{(p_j^{(i)},q_j^{(i)}):1\le i\le r,\ 1\le j\le k_i\}$, can be
viewed as a family of $r$ pairwise non-touching path-like objects,
which were called {\it pathoids} by Kulkarni \cite{\KulkAB}. This is done
in the same way as before. We omit the details.
\endroster
Finally let $\x=(x_1,x_2,\dots)$ and $\y=(y_1,y_2,\dots)$ be two
infinite sequences of indeterminates. We define the {\it weight}
$w_{\x,\y}(P)$ of an array $P$ of the form (1.1) to be the product
$\prod x_\ep\prod y_\et$ where $\ep$ runs over all elements of the
first row of $P$ and $\et$ runs over all elements of the second row
of $P$. The weight $w_{\x,\y}(\P)$ of a family $\P=(P_1,\dots,P_r)$
is defined to be the product over the weights of all arrays of $\P$,
$\prod _{i=1} ^{r}w_{\x,\y}(P_i)$.

\BlackBoxes
Now we are in the position to formulate the main theorems of this
section.
\proclaim{Theorem 1}Let $\al,\be$ be partitions. The generating
function $\sum w_{\x,\y}(\P)$ for non-crossing strict families $\P$
of the type $(\al,\be)$ is
$$\det_{1\le s,t\le r}(f_{\al_s+s-\be_t-t}(\x,\y)),\tag1.2$$
where $f_m(\x,\y)=\sum _{k} ^{}e_{k+m}(\x)e_k(\y)$ with 
$e_n(\z)$ being the elementary symmetric function of order $n$ in
the variables $z_1,z_2,\dots$. 

The generating function $\sum w_{\x,\x}(\P)$ for non-crossing strict
families $\P$ of the type $(\al,\be)$ with the additional property
$p_j^{(i)}\ge q_j^{(i)}$, $i=1,\dots,r$ and $j=1,\dots,k_i$, is
$$\det_{1\le s,t\le r}(f_{\al_s+s-\be_t-t}(\x,\x)-
f_{\al_s+s+\be_t+t}(\x,\x)).\tag1.3$$
\endproclaim
\proclaim{Theorem 2}Let $\al,\be$ be partitions. The generating
function $\sum w_{\x,\y}(\P)$ for strictly non-crossing families $\P$
of the type $(\al,\be)$ is
$$\det_{1\le s,t\le r}(g_{\al_s+s-\be_t-t}(\x,\y)),\tag1.4$$
where $g_m(\x,\y)=\sum _{k} ^{}h_{k+m}(\x)h_k(\y)$ with 
 $h_n(\z)$ being the complete homogenous symmetric function of order $n$ in
the variables $z_1,z_2,\dots$. 

The generating function $\sum w_{\x,\x}(\P)$ for strictly non-crossing 
families $\P$ of the type $(\al,\be)$ with the additional property
$p_j^{(i)}> q_j^{(i)}$, $i=1,\dots,r$ and $j=1,\dots,k_i$, is
$$\det_{1\le s,t\le r}(g_{\al_s+s-\be_t-t}(\x,\x)-
g_{\al_s+s+\be_t+t}(\x,\x)).\tag1.5$$
\endproclaim
\demo{Sketch of Proof} We imitate the usual procedure with
nonintersecting lattice paths (\cite{\GeViAB}, see also
\cite{\StemAE, section~1}. For the proof of (1.2) we
consider strict families $\P=(P_1,\dots,P_r)$ of two-rowed arrays
where $P_i$ is of the type $(a_{\si(i)}+\si(i)-i,\be_i)$,
$i=1,2,\dots,r$, for some
permutation $\si\in\frak S_r$. As with nonintersecting lattice paths
we set up a weight-preserving involution on the crossing families of
two-rowed arrays (where ``crossing" means a violation against one of
the conditions (D1)--(D3)), such that the corresponding permutations
differ only by a transposition. After having observed that for
$\si\ne \text {id}$ there are no non-crossing strict families of the
above type, the usual arguments lead to (1.2).

In order to prove (1.3) we consider strict families
$\P=(P_1,\dots,P_r)$ of two-rowed arrays where $P_i$ is of the type
$(\et_i(\al_{\si(i)}+\si(i))-i,\be_i)$, $i=1,2,\dots,r$,
for some permutation $\si\in
\frak S_r$ and $\et_i\in\{-1,1\}$, $i=1,\dots,r$. Also here we give a
weight-preserving involution on the crossing families (where
``crossings of the main diagonal", i.e\. violations against
$p_j^{(i)}\ge q_j^{(i)}$, have to be considered, too) such that
either the corresponding permutations differ by a transposition and
the corresponding $\et_i$'s are identical, or the corresponding
permutations are identical and exactly one of the corresponding 
$\et_i$'s changes its sign.

The proofs for (1.4) and (1.5) are similar. Only at all places the
roles of weak and strict order have to be exchanged.

The bijections of this section are inspired by ideas from
\cite{\KratAP}.
\quad \quad \qed
\enddemo

\subheading{2. Summations for Schur functions ``in a strip"}
In this section we relate Theorems~1 and 2 to summations for Schur
functions. In the following given a partition $\mu$ the symbol $\mu'$
denotes the conjugate of $\mu$, as usual. The number of odd parts of
$\mu$ is denoted by $\oddrows(\mu)$, the number of odd parts of $\mu'$
is denoted by $\oddcolumns(\mu)$. (This terminology stems from
visualizing partitions as Ferrer's boards.)
\proclaim{Theorem 3} Let $\al,\be$ be partitions.
$$\gather
\sum _{\la,\la_1\le r} ^{}s_{\la/\be'}(\x) s_{\la/\al'}(\y)=
\det_{1\le s,t\le r}(f_{\al_s+s-\be_t-t}(\x,\y))\tag2.1\\
\sum _{\la,\la_1\le c,\oddrows(\la)=p} ^{}s_\la(\x)= 
\cases \det\limits_{1\le s,t\le r}(f_{p\cdot\ch(s=r)+s-t}(\x,\x)\\
\hskip1.8cm -f_{p\cdot\ch(s=r)+s+t}(\x,\x))&c=2r\\
e_p(\x)\det\limits_{1\le s,t\le r}(f_{s-t}(\x,\x)
 -f_{s+t}(\x,\x))&c=2r+1\endcases\\\tag2.2\\
\sum _{\la,\ell(\la)\le r} ^{}s_{\la/\be}(\x) s_{\la/\al}(\y)=
\det_{1\le s,t\le r}(g_{\al_s+s-\be_t-t}(\x,\y))\tag2.3\\
\sum _{\la,\ell(\la)\le c,\oddcolumns(\la)=p} ^{}s_\la(\x)= 
\cases \det\limits_{1\le s,t\le r}(g_{p\cdot\ch(s=r)+s-t}(\x,\x)\\
\hskip1.8cm -g_{p\cdot\ch(s=r)+s+t}(\x,\x))&c=2r\\
h_p(\x)\det\limits_{1\le s,t\le r}(g_{s-t}(\x,\x)
-g_{s+t}(\x,\x))&c=2r+1\endcases\\\tag2.4
\endgather$$
where $\ch$ is the usual truth function, $\ch(\Cal A)=1$ if $\Cal A$
is true and $\ch(\Cal A)=0$ otherwise.
\endproclaim
\remark{Remark} Identity (2.3) is due to Gessel \cite{\GessAH, Theorem~16,
cf\. the paragraph just before Theorem~16}, while (2.4)
is due to Goulden \cite{\GoulAD, Theorems~2.4 and 2.6}. Clearly (2.1) and
(2.2) follow from (2.3) and (2.4), respectively, by application of
the homomorphism on symmetric functions that interchanges the roles
of elementary and complete homogenous symmetric functions (cf\.
\cite{\MacdAA, pp.~14/15}). However, it is our goal to give {\it combinatorial} proofs for
{\it all} of these identities.
\endremark
\demo{Sketch of Proof} For a proof of (2.1) 
we set up a ``part"-preserving bijection
between non-crossing strict families $\P=(P_1,\dots,P_r)$ of the type
$(\al,\be)$ and pairs $(\pi,\ta)$ of a column-strict plane partition
$\pi$ of shape $\la/\be'$ and a tableaux $\ta$ of shape $\la/\al'$,
with $\la$ being a partition with $\la_1\le r$. (Of course ``part"-preserving
means that the multiset of the elements in $P_1,\dots,P_r$ is
identically with the multiset of parts in $\pi$ and $\ta$.) This
bijection is a variation of Sagan and Stanley's \cite{\SaStAA} skew Knuth
correspondence. In the explanation of the bijection we refer to the
Fomin--Roby \cite{\RobyAA, section~4.1; \FomiAB} description of the skew Knuth correpondence.

The bijection is best explained with a running example.
Consider the family $\P^{(0)}=(P_1^{(0)},P_2^{(0)},P_3^{(0)})$ where
$$P_1^{(0)}=\matrix 1\ 3\ \hphantom{1}\\3\ 5\ 6\endmatrix\ ,\quad 
P_2^{(0)}=\matrix 2\ 4\ 5\ \hphantom{2\ 2}\\\hphantom{1}\ 1\ 2\ 3\
5\endmatrix\ ,\quad 
P_3^{(0)}=\matrix 1\ 2\ 4\ 5\ \hphantom{1\ 1}\\
\hphantom{1\ 1\ 1}\ 2\ 3\ 4\endmatrix\ .$$
It is a non-crossing strict family of the type
$(\al^{(0)},\be^{(0)})=((3,1,0),(2,2,1))$. Now, given a non-crossing strict family
interpret the front part as column-strict plane partition $\tilde \pi$
of shape $\al'/\nu$ and the tail as tableau $\tilde \ta$ of shape $\be'/\nu$ 
as explained in (NS1) and (NS2), respectively, where $\nu$ is some
partition. In our
example the front part $\tilde\pi^{(0)}$ and the tail $\tilde\ta^{(0)}$ are
as follows,
$$\tilde\pi^{(0)}=\smatrix\format\sa\c\s\c\se\\\hlinefor5\\
& 4&&2&\\\hlinefor5\\&2&\\\hlinefor3\\&1&\\\hlinefor3\endsmatrix
\quad \text {and}\quad 
\tilde\ta^{(0)}=\smatrix\format\sa\c\s\c\s\c\se\\\hlinefor7\\
& 3&&3&&6&\\\hlinefor7\\&4&&5&\\\hlinefor5\endsmatrix.$$
Next write $\tilde\pi$ and $\tilde\ta$ as multichains in Young's lattice,
i.e\. as sequences of partitions,
the length of the sequences being determined by $m+1$, where $m$ is
the largest element in $\P$. For the tableau $\tilde\ta$ this is
standard (cf\. \cite{\MacdAA, pp.~4/5; \RobyAA, pp.~12/13}), for the column-strict plane partition
$\tilde\pi$ the $i$-th partition in the sequence, $i$ running from $0$
to $m$,
corresponds to the shape of the column-strict plane partition
that results from $\tilde\pi$ by deleting the numbers
$\{1,2,\dots,m-i\}$ in $\tilde\pi$. So in our example the sequence for
$\tilde\pi^{(0)}$ is $0,0,0,1,1,21,211$, while
the sequence for $\tilde\ta^{(0)}$ is
$0,0,0,2,21,22,32$. (Here we use a short
notation for partitions. For instance, $211$ is short for $(2,1,1)$,
$0$ is short for the empty partition.)
\vskip10pt
\vbox{\noindent
$$\smatrix \format\r\s\l\s\l\s\l\s\l\s\l\s\l\s\l\\
\labu{3^32^21}\hskip-4pt&\omit&\hskip20pt&\omit&\labu{3^22^3}&\omit& \labu{3^22^2}&\omit& 
\labu{32^3}&\omit&
\labu{322}&\omit& \labu{32}&\omit&\\
\omit&\hlinefor{13}\\
\vphantom{\vtop to20pt{e\vss}}
&&\vphantom{f}&&\vphantom{f}&&\vphantom{f}&&\vphantom{f}&&\vphantom{f}&&
\vphantom{f}&& \lab{32}\\
\omit&\hlinefor{13}\\
\labr{3^22^31}&&\vphantom{\vtop to20pt{e\vss}}&&&&&&
\hskip-19pt\vtop to5pt{\vss\noindent\hsize14pt$1$}&&&&&& \lab{22}\\
\omit&\hlinefor{13}\\
\labr{32^31^2}&&\vphantom{\vtop to20pt{e\vss}}&&&&&&&&&&&& \lab{21}\\
\omit&\hlinefor{13}\\
\labr{32^31}&&\vphantom{\vtop to20pt{e\vss}}&&
\hskip-19pt\vtop to5pt{\vss\noindent\hsize14pt$1$}&&&&&&&&&& \lab{2}\\
\omit&\hlinefor{13}\\
\labr{2^31}&&\vphantom{\vtop to20pt{e\vss}}&&&&&&&&&&
\hskip-19pt\vtop to5pt{\vss\noindent\hsize14pt$2$}&& \lab{0}\\
\omit&\hlinefor{13}\\
\labr{221}&&\vphantom{\vtop to20pt{e\vss}}&&&&&&&&
\hskip-19pt\vtop to5pt{\vss\noindent\hsize14pt$1$}&&&& \lab{0}\\
\omit&\hlinefor{13}\\
\labr{211}&\omit&&\omit& \lab{21}&\omit& \lab{1}&\omit& \lab{1}&\omit& 
\lab{0}&\omit& \lab{0}&\omit& \lab{0}
\endsmatrix
\quad \quad 
\pi^{(0)}=\smatrix \format\sa\c\s\c\s\c\se\\
\hlinefor7\\
&\vphantom{fg}&&&&&\\
\hlinefor7\\
&&&&&3&\\
\hlinefor7\\
&5&&5&&1&\\
\hlinefor7\\
&4&&4&\\
\hlinefor5\\
&2&&2&\\
\hlinefor5\\
&1&\\
\hlinefor3
\endsmatrix,
\quad 
\ta^{(0)}=\smatrix \format\sa\c\s\c\s\c\se\\
\hlinefor7\\
&&&&&3&\\
\hlinefor7\\
&&&1&&5&\\
\hlinefor7\\
&&&2&&6&\\
\hlinefor7\\
&2&&3&\\
\hlinefor5\\
&3&&5&\\
\hlinefor5\\
&4&\\
\hlinefor3
\endsmatrix$$
\centerline{\eightpoint Figure 1}
}
\vskip10pt
Now we are able to fill the appropriate Fomin--Roby picture.
The lower border corresponds to $\tilde\pi$, the right border to
$\tilde\ta$, the $(s,t)$-entry in the matrix corresponds to the number
of pairs $(p_j^{(i)},q_j^{(i)})$ in the middle part of $\P$ that
equal $(s,t)$. The left-hand diagram in Figure~1 shows the
Fomin--Roby picture corresponding to our running example. The
subdivision of rows and columns that contain ``multiple entries" is
done in direction South-East, (and not in direction North-East
as in \cite{\RobyAA, section~4.1}).
Figure~1a contains the ``subdivided"
Fomin--Roby picture that corresponds to the diagram in Figure~1.
Also, in the diagram we are working upwards and to the left (and
not upwards and to the right as in \cite{\RobyAA, section~4.1}). The rules for the
algorithm are the same as in \cite{\RobyAA, Example~2.6.3}.
From the upper border, in the same way as explained above,
we read a column-strict plane partition $\pi$ of
shape $\la/\be'$, from the left border a tableau $\ta$ of shape
$\la/\al'$, where $\la$ is some partition. Examining the properties
of this mapping it is not too difficult to see that thus we indeed
obtain a part-preserving bijection between 
non-crossing strict families $\P=(P_1,\dots,P_r)$ of the type
$(\al,\be)$ and pairs $(\pi,\ta)$ where $\pi$ is a column-strict plane partition
of shape $\la/\be'$ and $\ta$ is a tableaux of shape $\la/\al'$,
with $\la$ being a partition with $\la_1\le r$.
The pair $(\pi^{(0)},\ta^{(0)})$ resulting from our example
is exhibited in Figure~1.

It is well-known that the skew Schur function $s_{\la/\be'}$ can be
either defined as generating function for tableaux of shape $\la/\be'$
or as generating function for column-strict plane partitions of shape
$\la/\be'$. (There is also a bijection basing on jeu de taquin which
settles this equivalence of definitions for the Schur function.) Thus
the above bijection by (1.2) proves (2.1).
\vskip10pt
\vbox{\noindent
$$\smatrix \format\sa\l\s\l\s\l\s\l\s\l\s\l\s\l\s\l\s\l\s\l\s\l\\
\hlinefor{21}\\
&\vphantom{\vtop to20pt{e\vss}}\lab{3^32^21}&&\lab{3^22^31}&&\lab{3^22^3}&&\lab{3^22^21}&&\lab{3^22^2}
&&\lab{32^3}&&\lab{32^21}&&\lab{32^2}&&\lab{321}&&\lab{32}&&
\lab{32}\\
\hlinefor{21}\\
&\vphantom{\vtop to20pt{e\vss}}\lab{3^22^31}&& \lab{32^41}&&
\lab{32^4}&& \lab{32^31}&&
\lab{32^3}\hskip-16pt\vtop to8pt{\vss\noindent\hsize14pt\seventeenpoint$\times$}&& 
\lab{2^4}&& \lab{2^31}&&
\lab{2^3}&& \lab{2^21}&& \lab{22}&& \lab{22}\\
\hlinefor{21}\\
&\vphantom{\vtop to20pt{e\vss}}\lab{32^41}&& \lab{2^51}&& \lab{2^5}&&
\lab{2^41}&& \lab{2^4}&& \lab{2^4}&& \lab{2^31}&& \lab{2^3}&&
\lab{2^21}&& \lab{22}&& \lab{22}\\
\hlinefor{21}\\
&\vphantom{\vtop to20pt{e\vss}}\lab{32^31^2}&& \lab{2^41^2}&&
\lab{2^41}&& \lab{2^31^2}&& \lab{2^31}&& \lab{2^31}&& 
\lab{2^21^2}&& \lab{2^21}&&
\lab{21^2}&& \lab{21}&& \lab{21}\\
\hlinefor{21}\\
&\vphantom{\vtop to20pt{e\vss}}\lab{32^31}
\hskip-16pt\vtop to8pt{\vss\noindent\hsize14pt\seventeenpoint$\times$}
&& \lab{2^41}&& \lab{2^4}&&
\lab{2^31}&& \lab{2^3}&& \lab{2^3}&& \lab{2^21}&& \lab{22}&&
\lab{21}&& \lab{2}&& \lab{2}\\
\hlinefor{21}\\
&\vphantom{\vtop to20pt{e\vss}}\lab{2^41}&& \lab{2^41}&& \lab{2^4}&&
\lab{2^31}&& \lab{2^3}&& \lab{2^3}&&
 \lab{2^21}&& \lab{22}&& \lab{21}&& \lab{2}&&
\lab{2}\\
\hlinefor{21}\\
&\vphantom{\vtop to20pt{e\vss}}\lab{2^31^2}&& \lab{2^31^2}&&
\lab{2^31}&& \lab{2^21^2}&& \lab{2^21}&& \lab{2^21}&&
\lab{21^2}&& \lab{21}&& \lab{11}&& \lab{1}&& \lab{1}\\
\hlinefor{21}\\
&\vphantom{\vtop to20pt{e\vss}}\lab{2^31}&& \lab{2^31}&& \lab{2^3}&&
\lab{2^21}&& \lab{22}&& \lab{22}&& \lab{21}&& \lab{2}
\hskip-16pt\vtop to8pt{\vss\noindent\hsize14pt\seventeenpoint$\times$}
&& \lab{1}&&
\lab{0}&& \lab{0}\\
\hlinefor{21}\\
&\vphantom{\vtop to20pt{e\vss}}\lab{2^21^2}&& \lab{2^21^2}&&
\lab{2^21}&& \lab{21^2}&& \lab{21}&& \lab{21}&& \lab{11}&& \lab{1}&&
\lab{1}\hskip-16pt\vtop to8pt{\vss\noindent\hsize14pt\seventeenpoint$\times$}
&& \lab{0}&& \lab{0}\\
\hlinefor{21}\\
&\vphantom{\vtop to20pt{e\vss}}\lab{2^21}&& \lab{2^21}&& \lab{22}&&
\lab{21}&& \lab{2}&& \lab{2}
\hskip-16pt\vtop to8pt{\vss\noindent\hsize14pt\seventeenpoint$\times$}
&& \lab{1}&& \lab{0}&& \lab{0}&&
\lab{0}&& \lab{0}\\
\hlinefor{21}\\
\omit&\vphantom{\vtop to20pt{e\vss}}\lab{211}&\omit& \lab{211}&\omit& 
\lab{21}&\omit&
\lab{11}&\omit& \lab{1}&\omit& \lab{1}&\omit& \lab{1}&\omit& \lab{0}&\omit& 
\lab{0}&\omit&
\lab{0}&\omit& \lab{0}
\endsmatrix$$
\centerline{\eightpoint Figure 1a}
}
\vskip10pt

\medskip
For the $c=2r$ case of (2.2) we use a bijection due to Choi and
Gouyou--Beauchamps \cite{\ChGoAA, proof of Th\'eor\`eme~3} (cf\. \cite{\KratAP,
Proposition~32} for a detailed description)
between tableaux with $p$ odd rows and with at most $2r$ columns, and
pairs $(\P,S)$, where $\P=(P_1,\dots,P_r)$ is a non-crossing strict
family of $r$ two-rowed arrays of the shape $(0,0)$,
and $S$ is a $p$-subset of $\{1,2,\dots,h-1\}$ where $h$ is the
smallest element of the first row of $P_r$. (Also here $0$ denotes
the empty partition. But in this context of course it means that it
is coded by the $r$-tuple $(0,0,\dots,0)$.) Clearly $S$ can be put at
the beginning of the first row of $P_r$ thus forming an array of the
type $(p,0)$. Thus one obtains a bijection between tableaux with $p$
odd rows and at most $2r$ columns and non-crossing strict families of
the shape $((p,0,\dots,0),0)$. Use of (1.3) with this shape
establishes the $c=2r$ case of (2.2).

For the $c=2r+1$ case of (2.2) we use a bijection (cf\.
\cite{\KratAP, proof of Theorem~21, last paragraph})
between tableaux with $p$ odd rows and with at most $2r+1$ columns
and pairs $(\P,S)$ where $\P$ is a non-crossing strict family of $r$
two-rowed arrays of the shape $(0,0)$ and $S$ is a
$p$-subset of the positive integers. By (1.3) with $\al=\be=0$
this yields the $c=2r+1$ case of (2.2), where $S$ produces
$h_p(\x)$ and $\P$ produces the determinant.

The bijections for establishing (2.2) base on one of Burge's
\cite{\BurgAA, p.~22}
variations of the Knuth correspondences \cite{\KnutAA}.

The arguments for proving (2.3) and (2.4) are similar.\quad \quad
\qed
\enddemo
Also (1.3) and (1.5), for generic $\al,\be$, have interesting
interpretations in terms of tableaux generating functions. 
For example, we can prove the following theorem.
\proclaim{Theorem 4}The generating function $\sum _{\ta} ^{}w(\ta)$
over all oscillating semistandard tableaux $\ta=(\ta^{(i)})$,
$\ta:\be\to\al$ (cf\. \cite{\SunaAA,
\GessAL, \RobyAA}), with at most $r$ rows equals (1.5).
The weight $w(\ta)$ is defined by $\prod _{} ^{}x_i^{
\vert\ta^{(2i-1)}-\ta^{(2i)}\vert+\vert\ta^{(2i-1)}-\ta^{(2i-2)}\vert}$.
\endproclaim

\subheading{3. Norm generating functions for tableaux}
Given a tableau $\ta$ we define the {\it norm}, $n(\ta)$, of $\ta$ to
be the sum of all the entries of $\ta$. In this section we list
several results for norm generating functions that are obtained from
(2.1)--(2.4) by specializing the indeterminates $\x$ and $\y$.
Clearly, whenever we set $x_i=q^{M_i}$ and $y_i=q^{N_i}$ in
(2.1)--(2.4) we obtain a determinant for the norm generating function
for some family of tableaux (resp\. pairs of tableaux). However, the
cases of interest are only those where the determinant simplifies.
For the simplification of the determinants we use special cases of
the following lemma from \cite{\KratAP, Lemma~34}.
\proclaim{Lemma 5}Let $X_1,X_2,\dots,X_r,A_2,A_3,\dots,A_r,C$ be
indeterminates. If $p_0,p_1,\dots, p_{r-1}$ are Laurent polynomials with
$\deg p_j\le j$ and $p_j(C/X)=p_j(X)$ for $j=0,1,\dots,r-1$, then
$$\multline \det_{1\le s,t\le r}\big((A_r+X_s)\dotsb(A_{t+1}+X_s)
(A_r+C/X_s)\dotsb(A_{t+1}+C/X_s)\cdot p_{t-1}(X_s)\big)\\
=\prodl _{1\le i<j\le r} ^{}(X_i-X_j)(1-C/X_iX_j)\prodl _{i=1}
^{r}A_i^{i-1}\prodl _{i=1} ^{r}p_{i-1}(-A_i)\ ,
\endmultline\tag 3.1
$$
with the convention that empty products (like
$(A_r+X_t)\dotsb(A_{s+1}+X_t)$ for $s=r$) are equal to 1. (The
indeterminate $A_1$, which occurs at the right-hand side of (5.5.1),
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

\endproclaim
By this lemma we arrive at multifold sums. They are basically basic
hypergeometric series for $A_r$ respectively $C_r$. Sometimes even
these series can be evaluated. These computations are very similar to
those in \cite{\KratAP}.

From (2.1) we derive the following results.
\proclaim{Theorem 6}Let $\al$ be a fixed partition.
The generating function $\sum q^{n(\ta_1)+\ta_2}$
for pairs $(\ta_1,\ta_2)$ of tableaux where $\ta_1$ is of shape $\la$
and $\ta_2$ is of shape $\la/\al'$ with $\la$ being some partition
with $\la_1\le r$, and where the parts of $\ta_1$ are $\equiv m_1\pmod
a$ and between $m_1$ and $m_1+(M_1-1)a$, and where 
the parts of $\ta_2$ are $\equiv m_2\pmod
a$ and between $m_2$ and $m_2+(M_2-1)a$, is
$$%\multline 
\sum _{\la,\la_1\le r}
^{}s_\la(q^{m_1},q^{m_1+a},\dots,q^{m_1+(M_1-1)a})
s_{\la/\al'}(q^{m_2},q^{m_2+a},\dots,q^{m_2+(M_2-1)a})=\hskip2cm$$
$$\multline
=q^{a\sum _{i=1} ^{r}\(\binom {\al_i}2+m_1\al_i/a\)}\sum
_{k_1,\dots,k_r\ge0} ^{}\prod _{i=1} ^{r}q^{ak_i(k_i+\al_i+\mu-1)}
\bmatrix -\mu+1\\k_i\endbmatrix_{q^a}%\hskip1.5cm
\\
\frac {(q^{a(M_2+\mu+\al_i+k_i+i-1)};q^a)_{M_1-\al_i-k_i}}
{(q^a;q^a)_{M_1+r-\al_i-k_i-i}}\prod _{1\le i<j\le r}
^{}[\al_j+k_j+j-\al_i-k_i-i]_{q^a},
\endmultline\tag3.2$$
where $\mu:=(m_1+m_2)/a$, $(A;q)_n:=(A;q)_\infty/(Aq^n;q)_\infty$
with $(A;q)_\infty=\prod _{i=0} ^{\infty}(1-Aq^i)$,
and $[n]_{q}:=(1-q^n)$, by definition.
An alternative expression for the right-hand side of (3.2) is
$$\multline
q^{a\sum _{i=1} ^{r}\(\binom {\al_i}2+m_1\al_i/a\)}
\sum _{k_1,\dots,k_r\ge0} ^{}\prod _{i=1}
^{r}q^{ak_i(k_i+\al_i+\mu-1)}\bmatrix M_2\\k_i\endbmatrix_{q^a}\\
\frac {[M_1+i-1]_{q^a}!}
{[M_1+r-\al_i-k_i-i]_{q^a}!\,[\al_i+k_i+i-1]_{q^a}!}
\prod _{1\le i<j\le r}
^{}[\al_j+k_j+j-\al_i-k_i-i]_{q^a}\ .
\endmultline\tag3.3$$
If $m_1+m_2=a$ the identities (3.2) (respectively (3.3)) reduce to
$$\multline \sum _{\la,\la_1\le r}
^{}s_\la(q^{m_1},q^{m_1+a},\dots,q^{m_1+(M_1-1)a})
s_{\la/\al'}(q^{a-m_1},q^{2a-m_1},\dots,q^{M_2a-m_1})\\
=q^{a\sum _{i=1} ^{r}\(\binom{\al_i}2 +m_1\al_i/a\)}
\prod _{i=1} ^{r}\frac {[M_1+M_2+i-1]_{q^a}!}
{[M_2+\al_i+i-1]_{q^a}!\,[M_1+r-\al_i-i]_{q^a}!}
\prod _{1\le i<j\le r} ^{}[\al_j+j-\al_i-i]_{q^a}.
\endmultline\tag3.4$$
In particular, for $\al=0$ this can be written as
$$\multline \sum _{\la,\la_1\le r}
^{}s_\la(q^{m_1},q^{m_1+a},\dots,q^{m_1+(M_1-1)a})
s_{\la}(q^{a-m_1},q^{2a-m_1},\dots,q^{M_2a-m_1})\\
=\prod _{1\le i\le M_1,1\le j\le M_2} ^{}\frac {[r+i+j-1]_{q^a}}
{[i+j-1]_{q^a}}.
\endmultline\tag3.5$$
\endproclaim
\remark{Remark} The special cases $a=1$, $m_1=m_2=1$, $M_1=M_2$ of
(3.2) and $a=2$, $m_1=1$, $M_1=M_2$ of (3.5) first appeared in
\cite{\KratAP, Theorems~20, 9}.
\endremark
\demo{Sketch of Proof} In (2.1) set $x_i=q^{m_1+(i-1)a}$ for
$i=1,\dots,M_1$, $x_i=0$ for $i>M_1$, and $y_i=q^{m_2+(i-1)a}$ for
$i=1,\dots,M_2$, $y_i=0$ for $i>M_2$. With these specializations the
elementary symmetric functions
reduce to $q$-binomial coefficients times some power of $q$ (cf.
\cite{\MacdAA, p.~19, Ex.~3}). Next to each of the entries (which are $q$-binomial
summations) of the determinant one of Heine's
$_2\phi_1$-transformations \cite{\GaRaAA, Appendix (III.2)} is applied.
In the resulting determinant we use the linearity in the
rows to take out the summations, thus arriving at a multifold sum of
determinants. The determinants are evaluated by taking some factors
out of the determinants and then applying Lemma~5 with $C\to0$,
$X_s=-q^{-a(k_s+\al_s+s)}$, $A_t=q^{-a(M_1+t)}$, $p_{i-1}(X)=\prod
_{j=2} ^{i}(q^{a(M_2+\mu-j)}+X)$. This
gives (3.2) after some simplification. 
The expression in (3.3) results from the
following $A_r$ analogue of one of Heine's
$_2\phi_1$-transformations \cite{\GaRaAA, Appendix (III.2)}, 
newly discovered by Gustafson
\cite{\GustPR},\NoBlackBoxes
$$%\multline
\sum _{k_1,\dots,k_r\ge0} ^{}\left(\prod _{i=1} ^{r}q^{k_i(1-i)}
Z^{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}}=\hskip2cm$$
$$\multline
=\prod _{i=1} ^{r}\frac {(Cq^{i-r}/B)_\infty\,(BZX_i)_\infty}
{(Zq^{i-r})_\infty\,(CX_i)_\infty}\hskip7cm\\
\sum _{k_1,\dots,k_r\ge0} ^{}\left(\prod _{i=1} ^{r}q^{k_i(1-i)}
\(\frac {C} {B}\)^{k_i}\frac {(ABZ/C)_{k_i}\,(BX_i)_{k_i}}
{(q)_{k_i}\,(BZX_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}}.\endmultline$$

If $m_1+m_2=a$, i.e\. $\mu=1$, due the $q$-binomial coefficient the
series in (3.2) only consists of the term for $k_1=\dots=k_r=0$. This
immediately gives (3.4). For $\al=0$ the product can be written in
the form (3.5).\quad \quad \qed
\enddemo
Now we turn to implications of (2.2).
\proclaim{Theorem 7}The generating function $\sum q^{n(\ta)}$
for tableaux $\ta$ with $p$ odd rows, with at most $c$ columns,
and with parts $\equiv m\pmod a$ and between $m$ and $m+(M-1)a$, 
for $c=2r$ is
$$\multline \sum _{\la,\la_1\le 2r,\oddrows(\la)=p}
^{}s_\la(q^m,q^{m+a},\dots,q^{m+(M-1)a})\\
=q^{a\binom p2+mp}\sum _{k_1,\dots,k_r\ge0} ^{}\prod _{i=1}
^{r}q^{ak_i(k_i-1+p\cdot\ch(i=r)+\mu)}\bmatrix
-\mu+1\\k_i\endbmatrix_{q^a}\hskip5cm\\
\frac {(q^{a(M+r+p\cdot\ch(i=r)+k_i+i+\mu)};q^a)
_{M-r-p\cdot\ch(i=r)-k_i+i-1}} {(q^a;q^a)_{M+r-p\cdot\ch(i=r)-k_i-i}}\\
\prod _{1\le i<j\le r}
^{}[p\cdot\ch(j=r)+k_j+j-p\cdot\ch(i=r)-k_i-i]_{q^a}\\
\prod _{1\le i\le j\le r} ^{}
[p\cdot\ch(j=r)+k_j+j+p\cdot\ch(i=r)+k_i+i+\mu-1]_{q^a},
\endmultline\tag3.6a$$
where $\mu=2m/a$, and for $c=2r+1$
$$\multline \sum _{\la,\la_1\le 2r+1,\oddrows(\la)=p}
^{}s_\la(q^m,q^{m+a},\dots,q^{m+(M-1)a})\\
=q^{a\binom p2+mp}\bmatrix M\\p\endbmatrix_{q^a}
\sum _{k_1,\dots,k_r\ge0} ^{}\prod _{i=1}
^{r}q^{ak_i(k_i-1+\mu)}\bmatrix
-\mu+1\\k_i\endbmatrix_{q^a}\frac {(q^{a(M+r+k_i+i+\mu)};q^a)
_{M-r-k_i+i-1}} {(q^a;q^a)_{M+r-k_i-i}}\\
\prod _{1\le i<j\le r}
^{}[k_j+j-k_i-i]_{q^a}
\prod _{1\le i\le j\le r} ^{}
[k_j+j+k_i+i+\mu-1]_{q^a},
\endmultline\tag3.6b$$

 In case $2m=a$ the identities (3.6a,b) reduce to
$$\multline \sum _{\la,\la_1\le 2r,\oddrows(\la)=p}
^{}s_\la(q,q^3,\dots,q^{2M-1})=
q^{p^2} \frac {[2r+2p]_{q^2}\,[r]_{q^2}}
{[2r+p]_{q^2}\,[r+p]_{q^2}}\bmatrix M\\p\endbmatrix_{q^2}
\frac {\bmatrix M+2r\\M\endbmatrix_{q^2}\,} {\bmatrix
M+2r+p\\M\endbmatrix_{q^2}}\\
\times\prod _{i=1} ^{M}\frac {[r+i]_{q^2}} {[i]_{q^2}}
\prod _{1\le i<j\le M} ^{}\frac {[2r+i+j]_{q^2}} {[i+j]_{q^2}}
\endmultline\tag3.7a$$
and
$$\sum _{\la,\la_1\le 2r+1,\oddrows(\la)=p}
^{}s_\la(q,q^3,\dots,q^{2M-1})=
q^{p^2}\bmatrix M\\p\endbmatrix_{q^2}
\prod _{i=1} ^{M}\frac {[r+i]_{q^2}} {[i]_{q^2}}
\prod _{1\le i<j\le M} ^{}\frac {[2r+i+j]_{q^2}} {[i+j]_{q^2}}
\tag3.7b$$
where without loss of generality we let $a=2$.

In case $m=a$ the identities (3.6a,b) simplify to
$$\multline \sum _{\la,\la_1\le 2r,\oddrows(\la)=p}
^{}s_\la(q,q^2,\dots,q^M)\\=
q^{\binom {p+1}2}\frac {[2r]_q} {[2r+p]_q}\bmatrix M\\p\endbmatrix_q
\frac {\dsize \bmatrix M+2r\\M\endbmatrix_q} {\dsize\bmatrix 
M+2r+p\\M\endbmatrix_q}\prod _{1\le i\le j\le M} ^{}\frac {[2r+i+j]_q}
{[i+j]_q}\endmultline\tag3.8a$$
and
$$\sum _{\la,\la_1\le 2r+1,\oddrows(\la)=p}
^{}s_\la(q,q^2,\dots,q^M)=
q^{\binom{p+1}2}\bmatrix M\\p\endbmatrix_q\prod _{1\le i\le j\le M} ^{}
\frac {[2r+i+j]_q} {[i+j]_q},
\tag3.8b$$
where without loss of generality we let $a=1$.
\endproclaim
\demo{Sketch of Proof} To prove (3.6a) we do the same as before in
the proof of (3.2). The only difference is that now Lemma~5 with
$C=1$, $X_s=q^{a(p\cdot\ch(s=r)+k_s+s+\mu/2-1/2)}$,
$A_t=-q^{a(-M-\mu/2+1/2-t)}$, 
$
p_{i-1}(X):=\Big(\prod _{j=1} ^{i}(1/A_j+1/X)
\prod _{j=-i+2} ^{0}(A_j+X)
-\prod _{j=1} ^{i}(1/A_j+X)
\prod _{j=-i+2} ^{0}(A_j+1/X)\Big)\bigg/\(X-\frac {1} {X}\)
$
has to be applied. Obviously, the $p=0$ case of (3.6a) immediately
yields (3.6b). For $m=2a$, i.e\. $\mu=2$, due to the $q$-binomial
coefficient the sums in (3.6a,b) reduce to the
respective terms for $k_1=\dots=k_r=0$. This leads to (3.7a,b). For
$m=a$, i.e\. $\mu=1$, the sums in (3.6a,b) can be evaluated by a
special case of Gustafson's \cite{\GustAA, Theorem~5.1} 
$C_r$ $_6\ps_6$ summation (see also \cite{\LiMiAB; \KratAP,
subsection~5.6}). After some simplification
one arrives at (3.8a,b).\quad \quad \qed
\enddemo

\remark{Remark} Identity (3.7) is a refinement of the MacMahon
(ex-)Conjecture which was first proved in
another paper of the author \cite{\KratAP, Theorem~11}. (The MacMahon Conjecture was
proved by Andrews \cite{\AndrAK},
Macdonald \cite{\MacdAA, pp.~51/52, Ex.~16,17}, and Proctor \cite{\ProcAD,
Proposition~7.3}.) Likewise, identity (3.8) is a
refinement of the Bender--Knuth (ex-)Conjecture 
which the author also first proved in the same
paper \cite{\KratAP, Theorem~21}. (The Bender--Knuth Conjecture was proved by 
Andrews \cite{\AndrAJ}, Gordon \cite{\GordAA}, Macdonald
\cite{\MacdAA, pp.~51--53, Ex.~16,18},
and Proctor \cite{\ProcAD, Proposition~7.2}.) As was remarked in
\cite{\KratAP} by summing the
expressions in (3.7) respectively (3.8) with respect to $p$ we get
new proofs of the MacMahon respectively the Bender--Knuth Conjecture. We
formulate the MacMahon Conjecture in an equivalent form. The original
formulation is in terms of symmetric plane partitions.
\proclaim{Corollary 8}The generating function for
tableaux with at most $c$ columns,
and with only odd parts which lie between $1$ and $2M-1$, is given by
$$\sum _{\la,\la_1\le c}
^{}s_\la(q,q^3,\dots,q^{2M-1})=
\prod _{i=1} ^{M}\frac {[c+2i-1]_q} {[2i-1]_q}
\prod _{1\le i<j\le M} ^{}\frac {[c+i+j-1]_{q^2}} {[i+j-1]_{q^2}}\ .
\tag {MacMahon Conjecture}$$
The generating function for
tableaux with at most $c$ columns,
and with parts which lie between $1$ and $M$, is given by
$$\prod _{1\le i\le j\le M} ^{}\frac {[c+i+j-1]_q} {[i+j-1]_q}\ .\tag
{Bender--Knuth Conjecture}$$

\endproclaim
\demo{Sketch of Proof} In order to obtain the MacMahon Conjecture the
sum over all $p$ of the expressions in (3.7a) is evaluated by a
special case of the very well-poised $_6\phi_5$-summation (cf\.
\cite{\GaRaAA, Appendix (II.21)}) while the sum over all $p$ of the expressions in (3.7b) is
evaluated by the $q$-binomial summation (cf\. \cite{\AndrAF, (3.3.7)}). Similarly,
in order to obtain the Bender--Knuth Conjecture the sum over all $p$
of the expressions in (3.8a) is evaluated by the $q$-Kummer summation
(cf\. \cite{\GaRaAA, Appendix (II.9)}), while the sum over all $p$ of the expressions in (3.8b)
is again evaluated by the $q$-binomial summation.\quad \quad \qed
\enddemo
\endremark

\bigskip
Finally we turn to the implications of (2.3) and (2.4). In the same
special cases as before we can get rid of the determinants thus
arriving at $A_r$- respectively $C_r$-type sums. But unfortunately
here it does not seem to be possible to find any nontrivial cases in
which these sums can be evaluated to result into closed forms. But it
should be noted that these sums are {\it finite}.
\proclaim{Theorem 9}Let $\al$ be a fixed partition.
The generating function $\sum q^{n(\ta_1)+\ta_2}$
for pairs $(\ta_1,\ta_2)$ of tableaux where $\ta_1$ is of shape $\la$
and $\ta_2$ is of shape $\la/\al$ with $\la$ being some partition
with $\ell(\la)\le r$, and where the parts of $\ta_1$ are $\equiv m_1\pmod
a$ and between $m_1$ and $m_1+(M_1-1)a$, and where 
the parts of $\ta_2$ are $\equiv m_2\pmod
a$ and between $m_2$ and $m_2+(M_2-1)a$, for $M_1\ge r-\al_1$ is
$$\multline \sum _{\la,\ell(\la)\le r}
^{}s_\la(q^{m_1},q^{m_1+a},\dots,q^{m_1+(M_1-1)a})
s_{\la/\al}(q^{m_2},q^{m_2+a},\dots,q^{m_2+(M_2-1)a})\\
=\frac {q^{m_1\sum _{i=1} ^{r}\al_i}}
{(q^a;q^a)_{M_1-1}^r\,(q^{m_1+m_2};q^a)_{M_2}^r}\hskip8cm\\
\sum _{k_1,\dots,k_r\ge0} ^{}\prod _{i=1} ^{r}q^{ak_i(M_1+\al_i)}\frac
{(q^{a(1-M_1)};q^a)_{k_i}\,(q^{m_1+m_2};q^a)_{k_i}}
{(q^a;q^a)_{k_i}\,(q^{aM_2+m_1+m_2};q^a)_{k_i}}\prod _{1\le i<j\le
r} ^{}(1-q^{a(k_j-k_i)}).
\endmultline\tag3.9$$

\endproclaim
\BlackBoxes

\remark{Remark} The restriction $M_1\ge r-\al_1$ actually is not a
serious restriction. Obviously the tableaux $\ta_1$ have at most $M_1$
rows. Therefore, if $M_1<r$ we may replace $r$ by $M_1$ and then
apply (3.9). 
\endremark
\demo{Sketch of Proof} We proceed in analogy with the proof of
Theorem~6. In (2.3) set $x_i=q^{m_1+(i-1)a}$ for
$i=1,\dots,M_1$, $x_i=0$ for $i>M_1$, and $y_i=q^{m_2+(i-1)a}$ for
$i=1,\dots,M_2$, $y_i=0$ for $i>M_2$. With these specializations the
complete homogenous symmetric functions
reduce to $q$-binomial coefficients times some power of $q$ (cf.
\cite{\MacdAA, p.~19, Ex.~3}). This time a limiting case of another one of Heine's
$_2\phi_1$-transformations \cite{\GaRaAA, Appendix (III.1)} is applied. 
In the resulting determinant we use the linearity in the
rows to take out the summations thus arriving at a multifold sum of
determinants. Upon taking some factors out of these determinants,
they reduce to Vandermonde determinants and are therefore easily
evaluated.\quad \quad \qed
\enddemo
\proclaim{Theorem 10}The generating function $\sum q^{n(\ta)}$
for tableaux $\ta$ with $p$ odd columns, with at most $c$ rows,
and with parts $\equiv m\pmod a$ and between $m$ and $m+(M-1)a$, 
for $c=2r$ and $M\ge r$ is
$$\multline \sum _{\la,\ell(\la)\le 2r,\oddcolumns(\la)=p}
^{}s_\la(q^m,q^{m+a},\dots,q^{m+(M-1)a})\\
=\frac {q^{mp}} {(q^a;q^a)_{M-1}^r\,(q^{2m};q^a)_M^r}\sum
_{k_1,\dots,k_r\ge0} ^{}\prod _{i=1} ^{r}q^{ak_i(M+p\cdot\ch(i=r))}
\frac {(q^{a(1-M)};q^a){k_i}\,(q^{2m};q^a)_{k_i}}
{(q^a;q^a)_{k_i}\,(q^{aM+2m})_{k_i}}\\
\times\prod _{1\le i<j\le r} ^{}(1-q^{a(k_j-k_i)})\prod _{1\le i\le j\le
r} ^{}(1-q^{a(k_i+k_j)+2m}),
\endmultline\tag3.10a$$
and for $c=2r+1$ and $M\ge r$
$$\multline \sum _{\la,\ell(\la)\le 2r+1,\oddcolumns(\la)=p}
^{}s_\la(q^m,q^{m+a},\dots,q^{m+(M-1)a})\\
=\frac {q^{mp}} {(q^a;q^a)_{M-1}^r\,(q^{2m};q^a)_M^r}
\bmatrix M+p-1\\p\endbmatrix_{q^a}
\sum_{k_1,\dots,k_r\ge0} ^{}\prod _{i=1} ^{r}q^{ak_iM}
\frac {(q^{a(1-M)};q^a){k_i}\,(q^{2m};q^a)_{k_i}}
{(q^a;q^a)_{k_i}\,(q^{aM+2m})_{k_i}}\\
\times\prod _{1\le i<j\le r} ^{}(1-q^{a(k_j-k_i)})\prod _{1\le i\le j\le
r} ^{}(1-q^{a(k_i+k_j)+2m}).
\endmultline\tag3.11a$$

\endproclaim

\remark{Remark} Again the restriction $M\ge r$ is not serious since
if $M< r$ we may replace $r$ by $\lceil\frac {M} {2}\rceil$.
\endremark\demo{Sketch of Proof} We do the same as before in the proof of
Theorem~9. Here the determinants which have to be computed are
$$\det_{1\le s,t\le r}(q^{-at(m+k_s)}-q^{at(m+k_s)}).$$
This is done by using Lemma~5 with $C=1$, $X_s=q^{-a(m+k_s)}$, $A_t\to0$,
$p_{i-1}(X)=(X^{i}-1/X^{i})/(X-1/X)$ (see also \cite{\ProcAB,
p.~10, (I)}).\quad \quad \qed
\enddemo

\subhead 4. ``Flagged" non-crossing two-rowed arrays and enumeration
of nonintersecting lattice paths with given number of turns\endsubhead
Theorems~1 and 2 can be generalized to ``flagged" families of non-crossing
two-rowed arrays by using the same proof. By ``flagged" we mean that the 
entries in the $i$-th two-rowed array $P_i$ of a family $\Cal P$ are
bounded by different lower and upper bounds. (This resembles the
definition for flagged Schur functions (cf\. e.g\. \cite{\WachAA}) in
terms of tableaux with different lower and upper bounds on the
entries in each row of the tableaux.)
We do not have the space to write down the results. Instead, we
record three interesting applications of these results in the enumeration
of nonintersecting lattice paths with a given number of turns. These
results are related to the computation of Hilbert polynomials of
determinantal ideals (cf\. \cite{\KulkAC, \CoHeAA}). Similar results
can be derived for the path-like objects that were called pathoids
in \cite{\KulkAB}. For the interested reader we remark that
enumeration results for pairs of nonintersecting lattice paths with a
given number of EN-corners (see the definition below) of the ``upper"
path and a given number of NE-corners (see the definition below) of
the ``lower" path are given in \cite{\SulaAE, \KrSuAA}.

As in \cite{\KratAP} a point $X$ of a lattice path is called {\it
NE-corner}, if $X$ is the end point of a step in North direction and
at the same time starting point of a step in East direction. Analogously, 
$X$ is called {\it EN-corner}, if $X$ is the end point of a step in East 
direction and at the same time starting point of a step in North direction.
\proclaim{Theorem 11} Let $\Cal A\sb
i=(A\sb
1\sp
{(i)},A\sb
2\sp
{(i)})$ and $\Cal E\sb
i=(E\sb
1\sp
{(i)},E\sb
2\sp
{(i)})$ be lattice points satisfying
$A\sb
1\sp
{(1)}\le A\sb
1\sp
{(2)}\le\dotsb\le A\sb
1\sp
{(r)},\quad
A\sb
2\sp
{(1)}>A\sb
2\sp
{(2)}>\dotsb>A\sb
2\sp
{(r)},$
and
$E\sb
1\sp
{(1)}< E\sb
1\sp
{(2)}<\dotsb< E\sb
1\sp
{(r)},\quad
E\sb
2\sp
{(1)}\ge E\sb
2\sp
{(2)}\ge \dotsb\ge E\sb
2\sp
{(r)}.$
The number of all families $\Cal P=(P\sb
1,\dots,P\sb
r)$ of nonintersecting lattice paths $P\sb
i:\Cal A\sb
i\to\Cal E\sb
i$,
such that the paths of $\Cal P$ altogether contain exactly $K$ NE-corners, is
$$\sum \sb
{k\sb
1+\dotsb+k\sb
r=K} \sp
{}\det\sb
{1\le s,t\le r}
\bigg(\binom{E\sb
1\sp
{(t)}-A\sb
1\sp
{(s)}+s-t}{k\sb
s+s-t}
\binom{E\sb
2\sp
{(t)}-A\sb
2\sp
{(s)}-s+t}{k\sb
s}\bigg).\quad \quad \qed\tag4.1$$
\endproclaim
\remark{Remark} A special case of this result
is of relevance in the computation of Hilbert
polynomials of determinantal ideals. In fact, Kulkarni \cite{\KulkAC,
Main Theorem~5}
derived this special case ($r=p$, $K=E$, $\Cal
A\sb
i=(0,a\sb
{p-i+1})$, $\Cal E\sb
i=(m(2)-b\sb
{p-i+1},m(1))$) from Abhyankar's formula \cite{\AbhyAB,
(20.14.4), p.~484},
while Conca and Herzog \cite{\CoHeAA} used it to give an alternative
proof of Abhyankar's formula. A special case of the ``pathoid" analogue
of Theorem~11 is also related to Abhyankar's formula
(cf\. \cite{\KulkAB, Main Theorem~7}).
\endremark
\proclaim{Theorem 12}Let $\Cal A\sb
i=(A\sb
1\sp
{(i)},A\sb
2\sp
{(i)})$ and
$\Cal E\sb
i=(E\sb
1\sp
{(i)},E\sb
2\sp
{(i)})$ be lattice points satisfying
$A\sb
1\sp
{(1)}\le A\sb
1\sp
{(2)}\le\dotsb\le A\sb
1\sp
{(r)},\quad
A\sb
2\sp
{(1)}>A\sb
2\sp
{(2)}>\dotsb>A\sb
2\sp
{(r)},$
$
E\sb
1\sp
{(1)}< E\sb
1\sp
{(2)}<\dotsb< E\sb
1\sp
{(r)},\quad
E\sb
2\sp
{(1)}\ge E\sb
2\sp
{(2)}\ge \dotsb\ge E\sb
2\sp
{(r)},$
and
$A\sb
1\sp
{(i)}\ge A\sb
2\sp
{(i)},\quad E\sb
1\sp
{(i)}\ge E\sb
2\sp
{(i)},\quad
i=1,\dots,r.$
The number of all families $\Cal P=(P\sb
1,\dots,P\sb
r)$ of nonintersecting lattice paths $P\sb
i:\Cal A\sb
i\to\Cal E\sb
i$, which do not cross the line $x=y$, and where the paths of
$\Cal P$ altogether
contain exactly $K$ NE-corners, is
$$\multline
\sum \sb
{k\sb
1+\dotsb+k\sb
r=K} \sp
{}\det\sb
{1\le s,t\le r}
\bigg(\binom{E\sb
1\sp
{(t)}-A\sb
1\sp
{(s)}+s-t}{k\sb
s+s-t}
\binom{E\sb
2\sp
{(t)}-A\sb
2\sp
{(s)}-s+t}{k\sb
s}\\
-\binom{E\sb
1\sp
{(t)}-A\sb
2\sp
{(s)}-s-t+1}{k\sb
s-t}
\binom{E\sb
2\sp
{(t)}-A\sb
1\sp
{(s)}+s+t-1}{k\sb
s+s}\bigg).\quad \quad \qed
\endmultline\tag4.2$$
\endproclaim
\proclaim{Theorem 13}Let $\Cal A\sb
i=(A\sb
1\sp
{(i)},A\sb
2\sp
{(i)})$ and 
$\Cal E\sb
i=(E\sb
1\sp
{(i)},E\sb
2\sp
{(i)})$ be lattice points satisfying 
$A\sb
1\sp
{(1)}< A\sb
1\sp
{(2)}<\dotsb< A\sb
1\sp
{(r)},\quad
A\sb
2\sp
{(1)}\ge A\sb
2\sp
{(2)}\ge \dotsb\ge A\sb
2\sp
{(r)},$
$
E\sb
1\sp
{(1)}\le  E\sb
1\sp
{(2)}\le \dotsb\le  E\sb
1\sp
{(r)},\quad
E\sb
2\sp
{(1)}> E\sb
2\sp
{(2)}> \dotsb> E\sb
2\sp
{(r)},$
and
$A\sb
1\sp
{(i)}\ge  A\sb
2\sp
{(i)},\quad E\sb
1\sp
{(i)}\ge  E\sb
2\sp
{(i)},\quad
i=1,\dots,r.$
The number of all families $\Cal P=(P\sb
1,\dots,P\sb
r)$ of nonintersecting lattice paths $P\sb
i:\Cal A\sb
i\to\Cal E\sb
i$, $P\sb
i:\Cal A\sb
i\to\Cal E\sb
i$, which do not cross the line $x=y$, and where the paths of
$\Cal P$ altogether
contain exactly $K$ EN-corners, is
$$\multline
\sum \sb
{k\sb
1+\dotsb+k\sb
r=K} \sp
{}\det\sb
{1\le  s,t\le  r}
\bigg(\binom{E\sb
1\sp
{(t)}-A\sb
1\sp
{(s)}+s-t}{k\sb
s+s-t}
\binom{E\sb
2\sp
{(t)}-A\sb
2\sp
{(s)}-s+t}{k\sb
s}\\
-\binom{E\sb
1\sp
{(t)}-A\sb
2\sp
{(s)}-s-t+3}{k\sb
s-t+1}
\binom{E\sb
2\sp
{(t)}-A\sb
1\sp
{(s)}+s+t-3}{k\sb
s+s-1}\bigg).\quad \quad \qed
\endmultline\tag4.3$$
\endproclaim


\NoBlackBoxes
\Refs

\ref\no \AbhyAB\by S. S. Abhyankar \yr 1988 \book Enumerative combinatorics 
of Young tableaux\publ Marcel Dekker\publaddr New York, Basel\endref

\ref\no \AndrAF\by G. E. Andrews \yr 1976 
\book The Theory of Partitions
\publ Addison--Wesley
\publaddr Reading 
\bookinfo Encyclopedia of Mathematics and its Applications, Vol.~2\endref

\ref\no \AndrAJ\by G. E. Andrews \yr 1977 
\paper Plane partitions (II): The equivalence of the Bender--Knuth and the MacMahon conjectures
\jour Pacific J. Math.\vol 72
\pages 283--291 \endref

\ref\no \AndrAK\by G. E. Andrews \yr 1978 
\paper Plane partitions (I): The MacMahon conjecture
\inbook Studies in Foundations and Combinatorics\ed G.-C. Rota
\publ Adv. in Math. Suppl. Studies, Vol.1
\pages 131--150 \endref

\ref\no \BurgAA\by W. H. Burge \yr 1974 
\paper Four correspondences between graphs and generalized Young tableaux
\jour J. Combin\. Theory~A\vol 17
\pages 12---30\endref

\ref\no \ChGoAA\by S. H. Choi and D. Gouyou--Beauchamps \yr 1991 
\paper Enum\'eration de tableaux de Young semi-standard
\inbook Actes de 3eme Colloque de Series Formelles et Combinatoire Algebrique, Bordeaux 1991
\publ M.~Delest, G.~Jacob, P.~Leroux, Eds.
\publaddr LabRI, Universit\'e Bordeaux~I
\pages 229---243\endref

\ref\no \CoHeAA\by A.    Conca and J. Herzog \paper On the Hilbert 
function of determinantal rings and their canonical module
\jour Proc\. Amer\. Math\. Soc\. \vol 122 \yr 1994\pages 677--681\endref

\ref\no \FomiAB\by S.    Fomin 
\paper Schensted algorithms for graded graphs
\jour J. Alg\. Combin\.\vol 4\yr 1995\pages 5--45\endref

\ref\no \GaRaAA\by G.    Gasper and M. Rahman \yr 1990 
\book Basic hypergeometric series
\publ Encyclopedia of Mathematics And Its Applications~35, Cambridge University Press
\publaddr Cambridge\endref

\ref\no \GessAH\by I. M. Gessel \yr 1990 
\paper Symmetric functions and $P$-recursiveness
\jour J. Combin\. Theory A\vol 53
\pages 257---285\endref

\ref\no \GessAL\by I. M. Gessel \yr 1993 \paper Counting paths in Young's 
lattice\jour J. Statist\. Plann\. Inference\vol 34 \endref

\ref\no \GeViAB\by I. M. Gessel and G. Viennot \yr 1989 
\paper Determinants, paths, and plane partitions 
\paperinfo preprint\endref

\ref\no \GordAA\by B.    Gordon \yr 1971 
\paper Notes on plane partitions IV. Multirowed partitions with strict decrease along columns
\inbook Combinatorics, Proc\. Sympos\. Pure Math\. vol.~XIX, Univ\. California, Los Angeles, California, 1968
\publ Amer\. Math\. Soc\.
\publaddr Providence, R.~I.
\pages 91--100\endref

\ref\no \GoulAD\by I. P. Goulden \yr 1992 
\paper A linear operator for symmetric functions and tableaux in a strip with given trace
\jour Discrete Math\.\vol 99
\pages 69---77\endref

\ref\no \GustAA\by R. A. Gustafson \yr 1989 
\paper The Macdonald identities for affine root systems of classical type and hypergeometric series very-well-poised on semisimple Lie algebras
\inbook Ramanujan International Symposium on Analysis (Dec.~26th to 28th, 1987, Pune, India)\ed N.~K.~Thakare 
\pages 187---224\endref

\ref\no \GustPR\by R. A. Gustafson 
\paper private communication\endref

\ref\no \KnutAA\by D. E. Knuth \yr 1970 
\paper Permutations, matrices, and generalized Young tableaux
\jour Pacific J. Math\. \vol 34
\pages 709---727\endref

\ref\no \KratAP\by C.    Krattenthaler 
\paper The major counting of nonintersecting lattice paths and generating functions for tableaux 
\yr 1995
\publ Mem\. Amer\. Math\. Soc\. 115, no.~552\publaddr Providence, R.~I. \endref

\ref\no \KrSuAA\by C.    Krattenthaler and R. A. Sulanke \paper 
Counting pairs of nonintersecting lattice paths with respect to weighted 
turns\paperinfo presented at the poster session of the 5th Conference
for Formal Power Series and Algebraic Combinatorics, Florence 1993 \endref

\ref\no \KulkAC\by D. M. Kulkarni 
\paper Counting of paths and coefficients of Hilbert polynomial of a
 determinantal ideal\yr 1996
\jour Discrete Math\.\vol 154\pages 141--151\endref

\ref\no \KulkAB\by D. M. Kulkarni 
\paper Counting of path-like objects and coefficients of Hilbert polynomial of a determinantal ideal
\paperinfo preprint \endref

\ref\no \LiMiAB\by G. M. Lilly and S. C. Milne 
\paper The $C_l$ Bailey transform and Bailey Lemma \yr 1993
\jour Constr\. Approx\. \vol 9\pages 473--500\endref

\ref\no \MacdAA\by I. G. Macdonald \yr 1979 
\book Symmetric Functions and Hall Polynomials 
\publ Oxford University Press
\publaddr New York/Lon\-don\endref

\ref\no \ProcAD\by R. A. Proctor \yr 1984
\paper Bruhat lattices, plane partitions generating functions, 
and minuscule representations
\jour Europ\. J. Combin\.\vol 5
\pages 331--350\endref

\ref\no \ProcAB\by R. A. Proctor \yr 1990 
\paper New symmetric plane partition identities from invariant theory work of DeConcini and Procesi
\jour Europ\. J. Combin\.\vol 11
\pages 289---300\endref

\ref\no \RobyAA\by T. W. Roby \yr 1991 
\book Applications and extensions of Fomin's generalization of the Robinson--Schensted correspondence to differential posets
\publ Ph.D. thesis, M.I.T.
\publaddr Cambridge, Massachusetts\endref

\ref\no \SaStAA\by B. E. Sagan and R. P. Stanley \yr 1990 
\paper Robinson--Schensted algorithms for skew tableaux
\jour J. Combin\. Theory Ser.~A\vol 55
\pages 161--193\endref

\ref\no \StemAE\by J. R. Stembridge \yr 1990 
\paper Nonintersecting paths, pfaffians and plane partitions
\jour Adv\. in Math\.\vol 83
\pages 96---131\endref

\ref\no \SulaAE\by R. A. Sulanke \yr 1993 \paper Refinements of the 
Narayana numbers\jour Bull\. Inst\. Combin\. Appl\.\vol 7\pages 60--66\endref

\ref\no \SunaAA\by S.    Sundaram \yr 1986 \book On the combinatorics of 
representations of $Sp(2n,\Bbb C)$\publ Ph\. D. Thesis\publaddr 
Massachusetts Institute of Technology\endref

\ref\no \VienAA\by G.    Viennot \yr 1977 
\paper Une forme g\'eom\'etrique de la correspondance de Robinson--Schensted
\inbook Combinatoire et Repr\'esentation du Groupe Sym\'etrique\ed D. Foata 
\publ Lecture Notes in Math\., Vol.~579, Springer--Ver\-lag
\publaddr New York
\pages 29---58\endref

\ref\no \WachAA  \by M. L. Wachs \yr 1985 \paper Flagged Schur functions, 
Schubert polynomials, and symmetrizing operators\jour J. Combin\. 
Theory Ser.~A\vol 40\pages 276--289\endref

\endRefs
\enddocument