\input amstex
%\documentstyle{amsppt}

%%%%Ira's definitions
\font\commentfont=cmsl10
\def\comment#1{{\commentfont#1}}
\def\GF{\operatorname{GF}}
\def\GL{\operatorname{GL}}
\def\SL{\operatorname{SL}}
\def\SU{\operatorname{SU}}
\def\Sp{\operatorname{Sp}}
\nologo
%%%%%%%%%%%%%%%
\def\K{d}

\def\StemAE{42}
\def\StanAB{41}
\def\StanAA{40}
\def\SagaAL{39}
\def\ReYaAA{38}
\def\LiMiAA{37}
\def\MilnAG{36}
\def\MilnAJ{35}
\def\MacMAA{34}
\def\MacdAB{33}
\def\LaPrAB{32}
\def\KrMoAC{31}
\def\KrMoAB{30}
\def\KratAW{29}
\def\KratAV{28}
\def\KratAP{27}
\def\KratAO{26}
\def\KratAK{25}
\def\KratAM{24}
\def\KiElAA{23}
\def\KacVAB{22}
\def\HumpAC{21}
\def\GustAA{20}
\def\GustAF{19}
\def\GeViAB{18}
\def\GeViAA{17}
\def\GaRaAA{16}
\def\FuKrAA{15}
\def\FilaAA{14}
\def\EgReAB{13}
\def\BurgAC{12}
\def\BurgAD{11}
\def\BresAK{10}
\def\BrenAG{9}
\def\BrenAE{8}
\def\BailAD{7}
\def\ABBBAA{6}
\def\AndrBI{5}
\def\AndrAF{4}
\def\AndrBH{3}
\def\AndrBG{2}
\def\AgAnAA{1}

\def\TA{{1}}
\def\TB{{2}}
\def\TC{{3}}
\def\TD{{4}}
\def\TE{{5}}
\def\TF{{6}}
\def\TG{{7}}
%\def\TH{{8}}
\def\TI{{8}}
\def\TJ{{9}}
\def\TK{{10}}
\def\TL{{11}}
\def\TM{{12}}
\def\TN{{13}}
\def\TO{{14}}
\def\TP{{15}}
\def\TQ{{16}}
\def\TR{{17}}
\def\TS{{18}}
\def\TT{{19}}
\def\TU{{20}}
\def\TV{{21}}
\def\TW{{22}}
\def\TX{{23}}
\def\TY{{24}}
\def\TZ{{25}}
\def\TTA{{26}}
\def\TTB{{27}}
\def\TTC{{28}}
\def\TTD{{29}}
\def\TTE{{30}}
\def\TTF{{31}}
\def\TTG{{32}}

\magnification1200
\hsize15.6truecm
\vsize22.8truecm
\catcode`\@=11
\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
%Ergnzung 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
%Ergnzung 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
%Ergnzung 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.%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%entering a path: 
%  \Pfad(x-coordinate of starting point,y-coordinate of starting point),path
%  as 1-2-3-4word\endPfad
%(1=step in x-direction, 2=step in y-direction, 3=upward diagonal step, 4=downward diagonal step)
%
%entering a dotted path: 
%  \SPfad(x-coordinate of starting point,y-coordinate of starting point),path
%  as 1-2-word\endSPfad
%(1=step in x-direction, 2=step in y-direction, 3=upward diagonal step, 4=downward diagonal step)
%
%coordinate-axes:
%  \Koordinatenachsen(length of positive x-axes, length of positive y-axes)(length of negative x-axes, length of negative y-axes)
%The length of negative axes are entered as negative numbers.
%
%lattice:
%  \Gitter(number of points in positive x-direction, number of points in positive y-direction)(number of points in negative x-direction, number of points in negative y-direction)
%
%diagonal lines:
%  \Diagonale(x-coordinate of SW-most point,y-coordinate of SW-most point)length of the projection on the x-axes
%
%antidiagonal lines:
%  \AntiDiagonale(x-coordinate of NW-most point,y-coordinate of NW-most point)length of the projection on the x-axes
%
%labelling of points:
%  \Label[wo?]{[label]}(x-coordinate,y-coordinate)
%where:
%  [location?]=\l,\lo,\lu,\r,\ro,\ru,\o,\u
%and l=left, r=right, u=bottom, o=top.
%In addition, if by \Einheit?cm the basic unit is changed, there exist
%\llo,\loo,\llu,\luu,\rro,\roo,\rru,\ruu.
%
%The basic unit can be changed by entering
%  \Einheit=?cm
%The default is \Einheit=0.5cm.
%
%The thickness of the paths can be changed by entering
%     \PfadDicke{?cm}
%The default is \PfadDicke=1pt.
%
%The following point sizes are available:
%\DuennPunkt, \NormalPunkt, \DickPunkt. Syntax:
%  \DickPunkt(x-coordinate,y-coordinate), etc.
%
%Besides, a circle is available by \Kreis. Syntax:
%  \Kreis(x-coordinate,y-coordinate)
%
\catcode`\@=11
\font\tenln    = line10
\font\tenlnw   = linew10

\newskip\Einheit \Einheit=0.5cm
\newcount\xcoord \newcount\ycoord
\newdimen\xdim \newdimen\ydim \newdimen\PfadD@cke \newdimen\Pfadd@cke

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%LaTeX counters, dimensions, variables for lines%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcount\@tempcnta

\newdimen\@tempdima
\newdimen\@tempdimb


\newdimen\@wholewidth
\newdimen\@halfwidth

\newcount\@xarg
\newcount\@yarg
\newcount\@yyarg
\newbox\@linechar
\newdimen\@linelen
\newdimen\@clnwd
\newdimen\@clnht

\newif\if@negarg

\def\@whilenoop#1{}
\def\@whiledim#1\do #2{\ifdim #1\relax#2\@iwhiledim{#1\relax#2}\fi}
\def\@iwhiledim#1{\ifdim #1\let\@nextwhile=\@iwhiledim
        \else\let\@nextwhile=\@whilenoop\fi\@nextwhile{#1}}

\def\@whileswnoop#1\fi{}
\def\@whilesw#1\fi#2{#1#2\@iwhilesw{#1#2}\fi\fi}
\def\@iwhilesw#1\fi{#1\let\@nextwhile=\@iwhilesw
         \else\let\@nextwhile=\@whileswnoop\fi\@nextwhile{#1}\fi}


\def\thinlines{\let\@linefnt\tenln \let\@circlefnt\tencirc
  \@wholewidth\fontdimen8\tenln \@halfwidth .5\@wholewidth}
\def\thicklines{\let\@linefnt\tenlnw \let\@circlefnt\tencircw
  \@wholewidth\fontdimen8\tenlnw \@halfwidth .5\@wholewidth}
\thinlines
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\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}}

%%%%%%%%%%%%%%%%%%%%%
%LaTeX line macros%
%%%%%%%%%%%%%%%%%%%%%
\def\Line@(#1,#2)#3{\@xarg #1\relax \@yarg #2\relax
\@linelen=#3\Einheit
\ifnum\@xarg =0 \@vline
  \else \ifnum\@yarg =0 \@hline \else \@sline\fi
\fi}

\def\@sline{\ifnum\@xarg< 0 \@negargtrue \@xarg -\@xarg \@yyarg -\@yarg
  \else \@negargfalse \@yyarg \@yarg \fi
\ifnum \@yyarg >0 \@tempcnta\@yyarg \else \@tempcnta -\@yyarg \fi
\ifnum\@tempcnta>6 \@badlinearg\@tempcnta0 \fi
\ifnum\@xarg>6 \@badlinearg\@xarg 1 \fi
\setbox\@linechar\hbox{\@linefnt\@getlinechar(\@xarg,\@yyarg)}%
\ifnum \@yarg >0 \let\@upordown\raise \@clnht\z@
   \else\let\@upordown\lower \@clnht \ht\@linechar\fi
\@clnwd=\wd\@linechar
\if@negarg \hskip -\wd\@linechar \def\@tempa{\hskip -2\wd\@linechar}\else
     \let\@tempa\relax \fi
\@whiledim \@clnwd <\@linelen \do
  {\@upordown\@clnht\copy\@linechar
   \@tempa
   \advance\@clnht \ht\@linechar
   \advance\@clnwd \wd\@linechar}%
\advance\@clnht -\ht\@linechar
\advance\@clnwd -\wd\@linechar
\@tempdima\@linelen\advance\@tempdima -\@clnwd
\@tempdimb\@tempdima\advance\@tempdimb -\wd\@linechar
\if@negarg \hskip -\@tempdimb \else \hskip \@tempdimb \fi
\multiply\@tempdima \@m
\@tempcnta \@tempdima \@tempdima \wd\@linechar \divide\@tempcnta \@tempdima
\@tempdima \ht\@linechar \multiply\@tempdima \@tempcnta
\divide\@tempdima \@m
\advance\@clnht \@tempdima
\ifdim \@linelen <\wd\@linechar
   \hskip \wd\@linechar
  \else\@upordown\@clnht\copy\@linechar\fi}

%\def\@hline{\ifnum \@xarg <0 \hskip -\@linelen \fi
%\vrule \@height \@halfwidth \@depth \@halfwidth \@width \@linelen
%\ifnum \@xarg <0 \hskip -\@linelen \fi}

\def\@getlinechar(#1,#2){\@tempcnta#1\relax\multiply\@tempcnta 8
\advance\@tempcnta -9 \ifnum #2>0 \advance\@tempcnta #2\relax\else
\advance\@tempcnta -#2\relax\advance\@tempcnta 64 \fi
\char\@tempcnta}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\def\Diagonale(#1,#2)#3{\unskip\leavevmode
  \xcoord#1\relax \ycoord#2\relax
      \raise\ycoord \Einheit\hbox to0pt{\hskip\xcoord \Einheit
         \Line@(1,1){#3}\hss}}
\def\AntiDiagonale(#1,#2)#3{\unskip\leavevmode
  \xcoord#1\relax \ycoord#2\relax \advance\xcoord by -0.05\relax
      \raise\ycoord \Einheit\hbox to0pt{\hskip\xcoord \Einheit
         \Line@(1,-1){#3}\hss}}
\def\Pfad(#1,#2),#3\endPfad{\unskip\leavevmode
  \xcoord#1 \ycoord#2 \thicklines\ZeichnePfad#3\endPfad\thinlines}
\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\ifnum#1=2
      \raise\ycoord \Einheit\hbox to0pt{\hskip\xcoord \Einheit
        \hbox{\hskip-\PfadD@cke\vrule height1 \Einheit width\PfadD@cke depth0pt}\hss}%
      \advance\ycoord by 1
    \else\ifnum#1=3
      \raise\ycoord \Einheit\hbox to0pt{\hskip\xcoord \Einheit
         \Line@(1,1){1}\hss}
      \advance\xcoord by 1
      \advance\ycoord by 1
    \else\ifnum#1=4
      \raise\ycoord \Einheit\hbox to0pt{\hskip\xcoord \Einheit
         \Line@(1,-1){1}\hss}
      \advance\xcoord by 1
      \advance\ycoord by -1
    \fi\fi\fi\fi
  \fi\next}
\def\hSSchritt{\leavevmode\raise-.4pt\hbox to0pt{\hss.\hss}\hskip.2\Einheit
  \raise-.4pt\hbox to0pt{\hss.\hss}\hskip.2\Einheit
  \raise-.4pt\hbox to0pt{\hss.\hss}\hskip.2\Einheit
  \raise-.4pt\hbox to0pt{\hss.\hss}\hskip.2\Einheit
  \raise-.4pt\hbox to0pt{\hss.\hss}\hskip.2\Einheit}
\def\vSSchritt{\vbox{\baselineskip.2\Einheit\lineskiplimit0pt
\hbox{.}\hbox{.}\hbox{.}\hbox{.}\hbox{.}}}
\def\DSSchritt{\leavevmode\raise-.4pt\hbox to0pt{%
  \hbox to0pt{\hss.\hss}\hskip.2\Einheit
  \raise.2\Einheit\hbox to0pt{\hss.\hss}\hskip.2\Einheit
  \raise.4\Einheit\hbox to0pt{\hss.\hss}\hskip.2\Einheit
  \raise.6\Einheit\hbox to0pt{\hss.\hss}\hskip.2\Einheit
  \raise.8\Einheit\hbox to0pt{\hss.\hss}\hss}}
\def\dSSchritt{\leavevmode\raise-.4pt\hbox to0pt{%
  \hbox to0pt{\hss.\hss}\hskip.2\Einheit
  \raise-.2\Einheit\hbox to0pt{\hss.\hss}\hskip.2\Einheit
  \raise-.4\Einheit\hbox to0pt{\hss.\hss}\hskip.2\Einheit
  \raise-.6\Einheit\hbox to0pt{\hss.\hss}\hskip.2\Einheit
  \raise-.8\Einheit\hbox to0pt{\hss.\hss}\hss}}
\def\SPfad(#1,#2),#3\endSPfad{\unskip\leavevmode
  \xcoord#1 \ycoord#2 \ZeichneSPfad#3\endSPfad}
\def\ZeichneSPfad#1{\ifx#1\endSPfad\let\next\relax
  \else\let\next\ZeichneSPfad
    \ifnum#1=1
      \raise\ycoord \Einheit\hbox to0pt{\hskip\xcoord \Einheit
         \hSSchritt\hss}%
      \advance\xcoord by 1
    \else\ifnum#1=2
      \raise\ycoord \Einheit\hbox to0pt{\hskip\xcoord \Einheit
        \hbox{\hskip-2pt \vSSchritt}\hss}%
      \advance\ycoord by 1
    \else\ifnum#1=3
      \raise\ycoord \Einheit\hbox to0pt{\hskip\xcoord \Einheit
         \DSSchritt\hss}
      \advance\xcoord by 1
      \advance\ycoord by 1
    \else\ifnum#1=4
      \raise\ycoord \Einheit\hbox to0pt{\hskip\xcoord \Einheit
         \dSSchritt\hss}
      \advance\xcoord by 1
      \advance\ycoord by -1
    \fi\fi\fi\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 
    \vrule height0.25pt width\xcoord \Einheit depth0.25pt\hss}}
\def\Gitter(#1,#2){\unskip \xcoord0 \ycoord0 \leavevmode
  \LOOP\ifnum\ycoord<#2
    \loop\ifnum\xcoord<#1
      \raise\ycoord \Einheit\hbox to0pt{\hskip\xcoord \Einheit\Punkt\hss}%
      \advance\xcoord by1
    \repeat
    \xcoord0
    \advance\ycoord by1
  \REPEAT}
\def\Gitter(#1,#2)(#3,#4){\unskip \xcoord#3 \ycoord#4 \leavevmode
  \LOOP\ifnum\ycoord<#2
    \loop\ifnum\xcoord<#1
      \raise\ycoord \Einheit\hbox to0pt{\hskip\xcoord \Einheit\Punkt\hss}%
      \advance\xcoord by1
    \repeat
    \xcoord#3
    \advance\ycoord by1
  \REPEAT}
\def\Label#1#2(#3,#4){\unskip \xdim#3 \Einheit \ydim#4 \Einheit
  \def\lo{\advance\xdim by-.5 \Einheit \advance\ydim by.5 \Einheit}%
  \def\llo{\advance\xdim by-.25cm \advance\ydim by.5 \Einheit}%
  \def\loo{\advance\xdim by-.5 \Einheit \advance\ydim by.25cm}%
  \def\o{\advance\ydim by.25cm}%
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%%%%%%%%%%%%%
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%%%%%%%%%%%%%
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%\NoRunningHeads
\TagsOnRight
\loadbold
\NoBlackBoxes
\topmatter 
\title Cylindric Partitions
\endtitle 
\author Ira M.~Gessel \footnote"$^1$"{\hbox{Supported in part   by
NSF grant  DMS-9306297}}\\
C.~Krattenthaler\footnote"$^2$"{Supported in part by EC's Human
Capital and Mobility Program, grant CHRX-CT93-0400 and the\linebreak 
\hbox{Austrian Science Foundation FWF, grant P10191-MAT}}
\endauthor 
\affil 
Department of Mathematics,
Brandeis University,\\
Waltham, MA 02254--9110, USA\\
e-mail: IRA\@cs.brandeis.edu\\\vskip6pt
Institut f\"ur Mathematik der Universit\"at Wien,\\
Strudlhofgasse 4, A-1090 Wien, Austria.\\
e-mail: KRATT\@Pap.Univie.Ac.At
\endaffil
\address Department of Mathematics,
Brandeis University,
Waltham, MA 02254--9110, USA.
\endaddress
\address Institut f\"ur Mathematik der Universit\"at Wien,
Strudlhofgasse 4, A-1090 Wien, Austria.
\endaddress
%\email KRATT\@Pap.Univie.Ac.At\endemail
%\dedicatory \enddedicatory
%\date \enddate
%\thanks \endthanks
\subjclass Primary 05A15;
 Secondary 05A17, 05A30, 05E05, 11P81, 33D20, 33D45
\endsubjclass
\keywords cylindric partitions, plane partitions, $\widetilde A_r$
basic hypergeometric series, symmetric functions, Schur functions\endkeywords
\abstract A new object is introduced into the theory of partitions
that generalizes plane partitions: cylindric partitions. We obtain
the generating function for cylindric partitions of a given shape
that satisfy certain row bounds as a sum of determinants of
$q$-binomial coefficients. In some special cases these determinants can
be evaluated. Extending an idea of Burge (J. Combin\. Theory Ser.~A {\bf
63} (1993), 210--222), we count cylindric partitions in two
different ways to obtain several known and new summation and transformation
formulas for 
basic hypergeometric series for the affine root system
$\widetilde A_r$. In particular, we  provide
new and elementary proofs for two $\widetilde A_r$ basic hypergeometric
summation formulas of Milne (Discrete Math\. 99 (1992),
199--246).


\endabstract
\endtopmatter
\document

\leftheadtext{I. M. Gessel and C. Krattenthaler}

\subhead 1. Introduction\endsubhead
In this paper we introduce a new
object into the theory of partitions: {\it cylindric partitions\/}. The
most basic objects in the theory of partitions are {\it linear
partitions\/}  \cite{\AndrAF}. MacMahon \cite{\MacMAA, art.~421} extended
linear partitions to {\it
plane partitions\/} (cf\. \cite{\AndrAF, Sec.~11.2; \MacMAA,
Sec.~IX, X; \GeViAA;
\GeViAB;
\KratAM; \StanAA}). Cylindric partitions are the next link in this chain.
Before proceeding to the definition of cylindric partitions, we recall the
definition of linear partitions and plane partitions. 

A {\it linear partition\/} is a linear array
$\pi=(\pi_1,\pi_2,\dots,\pi_c)$ of integers $\pi_i$ such that
$\pi_1\ge \pi_2\ge \dots\ge\pi_c$. Linear partitions are usually called
simply {\it partitions}.

Let $\La=(\la_1,\la_2,\dots,\la_r)$ and $\Mu=(\mu_1,\mu_2,\dots,\mu_r)$
be $r$-tuples of integers. A {\it plane partition of shape
$\La/\Mu$}
is a planar array $\pi$ of integers of the form
$$\matrix 
&&&\pi_{1,\mu_1+1}&\innerhdotsfor3\after\quad &\pi_{1,\la_1}\\
&&\pi_{2,\mu_2+1}\quad \dots&\pi_{2,\mu_1+1}&\innerhdotsfor2\after\quad 
&\pi_{2,\la_2}\\
&\iddots&&\vdots&&\iddots\\
\pi_{r,\mu_r+1}&\innerhdotsfor3\after\quad &\pi_{r,\la_r}
\endmatrix\tag1.1$$
such that the rows and columns are weakly decreasing, i.e.,
$$\pi_{i,j}\ge \pi_{i,j+1}\quad \text {and}\quad \pi_{i,j}\ge
\pi_{i+1,j}.\tag1.2$$
In particular, a linear partition is a one-rowed plane partition.
The left-hand array in Figure~1 shows a plane partition of shape
$(7,6,4,4)/(3,1,1,0)$.

 A cylindric partition  is a plane partition with an
additional relation between the entries of the first and the last
rows. More precisely, let $\La$ and $\Mu$ be $r$-tuples of
integers, as before, and let $\K$ be an integer. A {\it cylindric
partition of shape\/} $\La/\Mu/\K$ is a planar array of the form (1.1)
such that
$$\align \pi_{i,j}&\ge\pi_{i,j+1}\tag1.3a\\
\pi_{i,j}&\ge \pi_{i+1,j}\tag1.3b\endalign$$
and
$$\pi_{r,j}\ge \pi_{1,j+\K}.\tag1.3c
$$
Note that (1.3a) and (1.3b) are the same as (1.2), while (1.3c) is the
additional relation between the entries of the first and the last row of
$\pi$. An alternative formulation would be that a cylindric partition
is a plane partition that remains a plane partition if the last row,
shifted $\K$ units to the right, is placed above $\pi$. This is
illustrated in Figure~1.
The left-hand array $\pi_0$ in Figure~1 is a cylindric partition of shape
$(7,6,4,4)/(3,1,1,0)/4$. The
right-hand array in Figure~1 shows $\pi_0$ with the last
row, shifted by $\K=4$ units, put on top of it (displayed in italics). 
This augmented array is a plane partition.
\vskip10pt
\vbox{
$$\matrix\\ &&&9&7&6&2\\
&11&11&8&7&4\\
&10&8&8 \\
9&8&5&5\endmatrix\quad \quad \quad 
\matrix &&&&\italic9&\italic8&\italic5&\italic5\\ 
&&&9&7&6&2\\
&11&11&8&7&4\\
&10&8&8 \\
9&8&5&5
\endmatrix$$
\centerline{\eightpoint Figure 1}
}
\vskip10pt


We chose the name {\it cylindric partition\/} for these objects because
the entries of such an array~$\pi$ may be viewed as situated on the
cylinder obtained by winding up
$\pi$ and gluing together its first and last rows so that the entries
$\pi_{r,j}$ and
$\pi_{1,j+\K}$ are adjacent.

As in the theory of plane partitions, 
we cannot treat completely general shapes. Our results are subject to
the restrictions (3.1) on the shape $\La/\Mu/\K$.

We call the sum $\sum \pi_{i,j}$ of all the entries
of $\pi$ the {\it norm\/} of $\pi$ and denote it by
$n(\pi)$. The norm of the array $\pi_0$ in Figure~1 is
$n(\pi_0)=102$. 

It is well-known \cite{\GeViAA, 
\GeViAB, \StemAE, \BrenAE} that in order to
find generating functions for plane partitions it is convenient
to convert plane partitions into nonintersecting lattice paths.
In section~3 we shall show how cylindric partitions
correspond to families of nonintersecting lattice paths subject to the
additional condition that a translate of the first path does not intersect the
last path, and (in Proposition~\TA), we derive a generating function  for
these families of nonintersecting lattice paths. This is the basic
result for all that follows. Using this theorem, in section~4 we express the norm
generating function $\sum q^{n(\pi)}$ for cylindric partitions $\pi$
of a given shape, in which the entries in each row are restricted by an
upper and lower bound, as a sum of determinants whose entries are
$q$-binomial coefficients (Theorem~\TB). More generally, 
in Theorem~\TC\ we find the norm generating
function for $(\Al,\Be)$-cylindric partitions, which are 
defined in section~4 as arrays of the form (1.1) such that the entries
decrease by at least $\al_j$ along the rows from column $j$ to column
$j+1$ and by at least $\be_i$
along the columns from row $i$ to row $i+1$. In particular,
$(\0,\0)$-cylindric partitions are the (ordinary) cylindric
partitions which we have already defined. (Here $\0$ denotes
the all 0 sequence.)

In some special cases the determinants in Theorems~\TB\ and \TC\ 
can be evaluated
in closed form. These special cases are described
in section~5.

The theorems in sections~4 and 5 contain many  results in the
literature. First, many of the results about plane partitions
(without symmetries) of a given shape (see \cite{\GeViAA, 
\GeViAB, \KratAM,
\MacMAA, \StanAA}) are special cases of Theorems~\TB, \TC, \TE--\TG. Next, Burge's
generating function theorem \cite{\BurgAC} for restricted partition
pairs is the special case $r=2,\Al=\0$ of Theorem~\TC, and the generating
function theorem for partitions with prescribed hook differences of Andrews 
{\it et al\.\/} \cite{\ABBBAA} is easily derived from the special case
$r=2,\Al=(1,1,\dots,1)$ of Theorem~\TC. 
Finally, by virtue of Stanley's theory of
$(P,\om)$-partitions \cite{\StanAB}, Krattenthaler and Mohanty's
\cite{\KrMoAB} formulas for the
major counting of $n$-dimensional 
lattice paths in a particular bounded region are
corollaries of Theorem~\TC.

Burge's paper \cite{\BurgAC} is a source of
inspiration for this paper. The main emphasis in \cite{\BurgAC} lies
on deriving identities by counting ``restricted partition pairs"
(which are two-rowed $(\0,\Be)$-cylindric partitions in our language)
in two different ways. This idea carries over to $(\Al,\Be)$-cylindric
partitions with an arbitrary number of rows. Thus, in sections~6--8
we derive numerous multiple basic hypergeometric
series identities. To be more precise, we extend many of Burge's
identities in section~6. In addition, we add many more identities whose
specialized forms do not appear in Burge's paper. The identities of
section~6 form the basis for the subsequent sections~7 and 8. There we
derive identities for basic hypergeometric series for the affine root
system $\widetilde A_r$. 
Among the identities that we obtain are two identities, (7.5) and (7.11),
which were first obtained by Milne \cite{\MilnAG, Theorems~1.9 and
5.27}. In contrast to Milne's proof, which used a great deal of machinery
(especially for proving (7.5)), our proofs are purely elementary. We also 
generalize  Milne's identities  and  derive the $\widetilde A_r$ basic
hypergeometric summations (7.6) and (7.7), which are new. 
In section~8 we generalize (7.5) and (7.7) to transformation formulas
between $\widetilde A_r$ basic hypergeometric series of different
dimensions (identities (8.2) and (8.4)), a type of transformation
that does not seem to have 
appeared before. It is our belief that some of our identities
(hopefully many) will inspire future research on $\widetilde A_r$
basic hypergeometric series and eventually lead to the discovery of
more general identities for these series. Of course, we do not claim
that our methods for deriving such identities are the ``right" ones.
They have their restrictions. But they are unusual and
show connections between $\widetilde A_r$ basic hypergeometric
series and combinatorics.

There is a special case in which even more general weights for
cylindric partitions can be considered. This special case is
discussed in section~9. As a corollary we obtain the new elegant
expression (9.6) for the expansion of the monomial symmetric function
$m_{(r^c)}(\bold x)$ in terms of Schur functions 
(cf\. \cite{\MacdAB} for information about
symmetric functions), and the expansion (9.9) of the generating
function for partitions with a restricted number of repetitions in
terms of Schur functions.

Finally, in section~10 we mention some open problems and areas for
further research.

\subhead 2. Notation\endsubhead
By {\it paths\/} we always mean lattice paths in the plane integer
lattice $\Bbb Z^2$ consisting of unit horizontal and vertical steps
in the positive direction. Given points $u$ and $v$, we denote the set of all lattice
paths from $u$ to $v$ by $\Cal P(u\to v)$. If $\bold u=(u_1,\dots,u_m)$
and $\bold v=(v_1,\dots,v_m)$ are vectors of points, we denote the
set of all $m$-tuples $(P_1,\dots,P_m)$ of paths in which
$P_i$ runs from $u_i$ to $v_i$, $i=1,\dots,m$, by
$\Cal P(\bold u\to \bold v)$. 
A set of paths is said to be {\it nonintersecting\/} if no
two paths of the set have a point in common. Otherwise it is called
{\it intersecting\/}. Let $w$ be a function that to each horizontal
edge $e$ in $\Bbb Z^2$ assigns a weight $w(e)$.
Then the weight $w(P)$ of a path $P$ is defined
to be the product of the weights of all its horizontal steps. The
weight $w(\P)$ of an $m$-tuple
$\P=(P_1,\dots,P_m)$ of paths 
is defined to be the product $\prod
_{i=1} ^{m}w(P_i)$ of the weights of all the paths in the
$m$-tuple.
For any weight function $w$ defined on a set $\Cal A$, we write
$$\GF(\Cal A;w):=\sum _{x\in\Cal A} ^{}w(x)$$
for the generating function of the set $\Cal A$ with respect to the
weight $w$. 

\medpagebreak
We will use the ``$q$-notations"
$[\alpha]_q=1-q^\alpha$, $[n]_q!=[1]_q[2]_q\dotsb [n]_q$, $[0]_q!=1$,
$$\align(a;q)_k&=\prodl _{j=0} ^{k-1}(1-aq^j)\ ,\text { with
}(a;q)_0=1,\\
(a;q)_\infty &=\prodl _{j=0} ^{\infty}(1-aq^j)\ ,
\endalign$$
so that in particular $[n]_q!=(q;q)_n$, and 
$$\bmatrix n\\k\endbmatrix_q=\cases \dfrac{[n]_q
[n-1]_q\dotsb[n-k+1]_q}
{[k]_q!}&k\ge0\\
0&k<0\endcases\ .$$
Occasionally we shall use the Gasper-Rahman condensed notation
$$(a_1,a_2,\dots,a_r;q)_n:=(a_1;q)_n\, (a_2;q)_n\,\cdots\, (a_r;q)_n\ .$$
The base $q$ in $[\alpha]_q$, $[n]_q!$, $(a;q)_k$, $(a;q)_\infty$, and 
$\[\smallmatrix n\\k\endsmallmatrix\]_q$ will in
most cases be omitted. Only when the base is different from $q$ will it be
explicitly stated.

%The basic hypergeometric series $_{p+1}\phi_p$ is defined by
%$$_{p+1}\phi_p\,\[\matrix a_1,\dots,a_{p+1}\\
%b_1,\dots,b_{p}\endmatrix;q,z\]=\sum _{i=0} ^{\infty}
%\frac {(a_1;q)_i\,(a_2;q)_i\dotsb(a_{p+1};q)_i}
%{(q;q)_i\,(b_1;q)_i\dotsb(b_p;q)_i}z^i.$$
%An excellent reference book for basic hypergeometric series is
%\cite{14}. We will frequently refer to it.





\subhead 3. Cylindric partitions and nonintersecting lattice paths\endsubhead
As pointed out in the introduction, formulas for plane partition
generating functions are most conveniently found by converting plane
partitions into nonintersecting lattice paths (cf\. \cite{\GeViAA,
\GeViAB,
\StemAE, \BrenAE}). In this section we interpret cylindric plane partitions
in terms of nonintersecting paths and count these paths.

Let $\La=(\la_1,\la_2,\dots,\la_r)$ and $\Mu=(\mu_1,\mu_2,\dots,\mu_r)$
be $r$-tuples of integers and let $\K$ be an integer such that
$$\gather
\lambda
_1\ge\lambda
_2-1\ge\lambda
_3-2\ge\dots\ge\lambda
_r-(r-1) \ge\lambda
_1-\K-r\hphantom{.}\tag3.1a\\
\mu_1\ge\mu_2-1\ge\mu_3-2\ge\dots\ge\mu_r-(r-1) \ge\mu_1-\K-r.\tag3.1b
\endgather$$
Let $a_1,a_2\dots,a_r$ and $b_1,b_2,\dots,b_r$ be integers such that
$$\gather
a_1-1\ge a_2-2\ge \dots\ge a_r-r\ge a_1-r-1\hphantom{.}\tag3.2a\\
b_1-1\ge \kern.5pt b_2-2\kern.5pt \ge \kern.5pt \dots\ge \kern.5pt b_r-r \kern.5pt \ge b_1-r-1.\tag3.2b
\endgather$$

We claim that {\sl cylindric partitions of shape $\La/\Mu/\K$ in which the
entries in row $i$ are at most $a_i$ and at least $b_i$ correspond bijectively
to families $(P_1,P_2,\dots,P_r)$ of nonintersecting
lattice paths, in which $P_i$ runs from $(-\la_i+i,b_i-i)$ to
$(-\mu_i+i,a_i-i)$, and in addition $P_r$, when shifted by $(-r-\K,r)$, does not
intersect $P_1$}. In certain cases it is necessary to add some additional
conditions to the cylindric partition: if
$\lambda_i=\lambda_{i+1}-1$ for some $i$ with $1\le i\le r-1$,
the restriction (1.3b) is also assumed to hold for 
$j=\la_i+1=\lambda_{i+1}$ with $\pi_{i,\lambda_{i}+1}:=b_i$, and if
$\la_r=\la_1-\K-1$ the restriction (1.3c) is also assumed to hold
for $j=\la_r+1=\la_1-\K$ with $\pi_{r,\la_r+1}:=b_r$. Similarly, if
$\mu_i=\mu_{i+1}-1$ for some $i$ with $1\le i\le r-1$ then  (1.3b) is
also assumed to hold for 
$j=\mu_i+1=\mu_{i+1}$ with $\pi_{i+1,\mu_{i+1}}:=a_{i+1}$, and if
$\mu_r=\mu_1-\K-1$, the restriction (1.3c) is also assumed to  hold for
$j=\mu_r+1=\mu_1-\K$ with $\pi_{1,\mu_1}:=a_1$.


The correspondence between cylindric partitions and paths is as follows. Let
$\pi$ be a cylindric partition of shape $\La/\Mu/\K$ in which the entries in row
$i$ are at most $a_i$ and  at least $b_i$. First we subtract $i$ from the entries
of the
$i$th row of $\pi$, $i=1,2,\dots,r$, obtaining the new array
$\bar\pi$. For example, consider the cylindric partition $\pi_0$ in
Figure~1. For sake of simplicity let us choose $a_i=11$ and $b_i=2$,
$i=1,2,3,4$, as row bounds. The cylindric partition $\pi_0$ with
these row bounds is displayed in the first picture in
Figure~2. (Again the italicized numbers indicate the shifted last
row.) The second picture of Figure~2 shows the result of subtracting 
$i$ from the entries of the $i$th row of $\pi_0$.

By the operation described above,  we  obtain from the cylindric partition
$\pi$ the array $\bar\pi$ that
is weakly decreasing along the rows but {\it strictly\/} decreasing
along the columns. The relation (1.3c) between the entries of the
first and the last row of $\pi$ translates into the condition
$$\bar\pi_{r,j}+r> \bar\pi_{1,j+\K}.\tag3.3$$
Let us call such an array a {\it column-strict cylindric partition with
first-last relation\/} (3.3).
Relation (3.3) is the same 
as saying that taking the last row of $\bar\pi$, adding
$r$ to each of its entries, shifting it  $\K$ positions to the right, and
putting it on top of $\bar\pi$, yields an
array that still has strictly decreasing columns. 
Of course, the new row bounds on $\bar\pi$
are $a_i-i$ and $b_i-i$.
The italicized
numbers on top of the array in the second picture of 
Figure~2 indicate the last row with $r=4$ added to each entry and
shifted by $\K=4$ units. 

\vskip10pt
\vbox{\eightpoint
$$\matrix \italic1\italic1\ge&&&&&\italic9&\italic8&\italic5&\italic5&\ge\italic2\\ 
11\ge&&&&9&7&6&2&&\ge2\\
11\ge&&11&11&8&7&4&&&\ge2\\
11\ge&&10&8&8&&&&&\ge2 \\
11\ge&9&8&5&5&&&&&\ge2
\endmatrix
\quad \quad \quad 
\matrix \italic1\italic1\ge&&&&&\italic9&\italic8&\italic5&\italic5&\ge\italic2\hphantom{-}\\ 
10\ge&&&&8&6&5&1&&\ge1\hphantom{-}\\
\hphantom{1}9\ge&&9&9&6&5&2&&&\ge0\hphantom{-}\\
\hphantom{1}8\ge&&7&5&5&&&&&\ge-1 \\
\hphantom{1}7\ge&5&4&1&1&&&&&\ge-2\endmatrix
$$
\centerline{\kern1cm\vbox{\hsize4cm\eightpoint\noindent\hangindent10pt\hangafter1
 1. cylindric partition $\pi_0$\newline
with row bounds}\quad \quad \quad \quad 
\vbox{\hsize5.5cm\eightpoint\noindent\hangindent14pt\hangafter1
 2. column-strict cylindric partition $\bar\pi_0$\newline
with row bounds}}

\vskip10pt

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\Label\r{P_1}(-5,4)
\Label\r{P_2}(-3,3)
\Label\ro{P_3}(0,4)
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\Label\ro{\italic5}(-8,5)
\Label\ro{\italic5}(-7,5)
\Label\ro{\italic8}(-6,8)
\Label\ro{\italic9}(-5,9)
$$
\centerline{\eightpoint 3. Family of nonintersecting lattice paths with
nonintersecting translate}
\vskip7pt
\centerline{\eightpoint Figure 2}
}
\vskip10pt
Next, in the usual way, the column-strict cylindric partition $\bar\pi$ is
mapped to a family of lattice paths. We recall that 
this is done by associating the $i$th row of $\bar\pi$ with a path
$P_i$ from $(-\la_i+i,b_i-i)$ to $(-\mu_i+i,a_i-i)$ where
the entries in the $i$th row are interpreted
as heights of the horizontal steps in the path
$P_i$. Thus from $\bar\pi$ we obtain the family $\P=(P_1,\dots,P_r)$
of lattice paths. The third picture of Figure~2 (without the dotted
path) displays the family of lattice paths that results in this way
from the array $\bar\pi_0$ displayed in the second
picture of Figure~2.

Clearly, the property that the columns of $\bar\pi$ are strictly
decreasing translates into the condition that $(P_1,P_2,\dots,P_r)$
is nonintersecting. The relation (3.3) translates into the condition
that $P_r$, when shifted by $(-r-\K,r)$, does not intersect
$P_1$. The dotted path in the third picture of Figure~2 indicates
this translate of the path $P_4$.

It is clear that this correspondence between cylindric partitions
of a given shape with given row bounds and families of
nonintersecting lattice paths with given starting and ending points,
in which a translate of the last path does not intersect the first path,
is a bijection. Thus, to enumerate cylindric partitions, it is sufficient to
enumerate the corresponding paths. This is done in the next proposition, which
is the basic result of this paper.

In Proposition~\TA\ we use the notation introduced in section ~2 for
lattice paths 
and for the extension of  a function on
horizontal edges of the lattice $\Bbb Z^2$ to lattice
paths and families of lattice paths.

\proclaim{Proposition \TA}Let $\S$ be the shift by $(-\al^x,\al^y)$
where $\al^x$ and $\al^y$ are positive integers. (In particular, $\S$ is
neither a horizontal nor a vertical shift.)
Let $w$ be a function on the horizontal edges of $\Bbb Z^2$
with the property that for all horizontal edges $e$ and 
some fixed indeterminate $z$
$$w(\S e)=z\cdot w(e).\tag3.4$$
Let
$\bold u=(u_1,u_2,\dots,u_r)$ and $\bold v=(v_1,v_2,\dots,v_r)$ be
$r$-tuples of points in $\Bbb Z^2$ such that $u_i$ lies to the
northwest of $u_{i+1}$ and  $v_i$ lies to the northwest of $v_{i+1}$ for
$i=1,2,\dots,r-1$; $\S u_r$ lies to the northwest of
$u_1$ and $\S v_r$ lies to the northwest of $v_1$;  and $u_i$
lies to the southwest of $v_i$ for  $i=1,2,\dots,r$. Then the generating
function  $\sum _{} ^{}w(\P)$, where the sum is over all families
$\P=(P_1,P_2,\dots,P_r)$ of lattice paths in which $P_i$ runs from $u_i$
to $v_i$ and  $(\S P_r,P_1,\dots,P_r)$ is nonintersecting,
equals
$$\sum _{k_1+\dots +k_r=0} ^{}\det_{1\le s,t\le r}\big(z^{\al^x
k_s^2/2-v_t^x k_s}\GF(\Cal P(u_s\to \S^{k_s}v_t);w)\big),
\tag3.5$$
with $v_t^x$ denoting the $x$-coordinate of $v_t$.
\endproclaim
\demo{Proof}The basic idea for the proof of this Proposition 
is the same as in the proof that the Gessel-Viennot determinant
\cite{\GeViAB, Cor.~2;
\StemAE, Theorem~1.2} is the generating function for ``ordinary"
nonintersecting lattice paths. However, the choice of the ``minimal"
intersection point must be made more carefully.

In the case of ``ordinary" nonintersecting lattice paths one considers
permutations of the end points. Since in our situation a shift
is involved, we must combine shifts with permutations of the end
points. More precisely, we consider families $(P_1,P_2,\dots,P_r)$
of lattice paths in which $P_i$ runs from $u_i$ to $\S^{k_i}v_{\si(i)}$,
where $\si$ is a permutation in $\frak S_r$, the symmetric group of order
$r$, and $k_1,k_2,\dots,k_r$ are integers with $k_1+k_2+\dots+k_r=0$.
Let us write $\bold k$ for $(k_1,k_2,\dots,k_r)$,
$\bold v_\si$ for $(v_{\si(1)},v_{\si(2)},\dots,v_{\si(r)})$
and $\S^{\bold k}\bold v$ for
$(\S^{k_1}v_1,\S^{k_2}v_2,\dots,\S^{k_r}v_r)$. Then we are
considering  families $P_1,\dots,P_r$ of lattice paths in the
disjoint union
$$\underset k_1+\dots+k_r=0\to{\bigcup _{\si\in \frak S_r} ^{}}\Cal P(\bold
u\to \S^{\bold k}\bold v_\si).$$
If $\P=(P_1,\dots,P_r)$ is an element of $\Cal P(\bold u\to \S^{\bold
k}\bold v_\si)$ we define a weight $\bar w(\P)$ by
$$\bar w(\P)=\prod _{s=1} ^{r}z^{\al^x k_s^2/2-v_{\si(s)}^x k_s}w(P_s).
\tag3.6$$

Let us extend our lattice path notation by writing $\Cal P(\bold
u\to \bold v)^+$ for the set of all families $(P_1,\dots,P_r)$ in 
$\Cal P(\bold u\to \bold v)$ for which $(\S P_r,P_1,\dots,P_r)$ is
nonintersecting and $\Cal P(\bold u\to \bold v)^-$ for the set of 
those families
in $\Cal P(\bold u\to \bold v)$ for which $(\S P_r,P_1,\dots,P_r)$ is
intersecting. We are interested in  $\GF(\Cal P(\bold u\to
\bold v)^+;w)$.
Clearly we have
$$\align \GF(\Cal P(\bold u\to \bold v)^+;w)&=
\GF(\Cal P(\bold u\to \bold v)^+;\bar w)\\
&=\GF(\Cal P(\bold u\to \bold v);\bar w)-
\GF(\Cal P(\bold u\to \bold v)^-;\bar w),\tag3.7
\endalign$$
since $w$ and $\bar w$ agree on $\Cal P(\bold u\to \S^{\bold
k}\bold v_\si)$ when $\bold k=\bold 0$.

We shall construct an involution $\ph$ on 
$$\underset k_1+\dots+k_r=0\to{\bigcup _{\si\in \frak S_r} ^{}}\Cal P(\bold
u\to \S^{\bold k}\bold v_\si)^-\tag3.8$$
that is sign-reversing and weight-preserving with respect to $\bar w$.
To be more precise, if $\P$ is in $\Cal P(\bold u\to
\S^{\bold k}\bold v_\si)^-$ and $\ph(\P)$ is in $\Cal P(\bold u\to
\S^{\bold k'}\bold v_{\si'})^-$ then we shall show that $\sgn
\si'=-\sgn \si$ and 
$$\bar w(\ph(\P))=\bar w(\P).\tag3.9$$

This will imply that
$$\underset k_1+\dots+k_r=0\to{\sum _{\si\in \frak S_r} ^{}}\sgn\si\,
\GF(\Cal P(\bold u\to \S^{\bold k}\bold v_\si)^-;\bar w)=0.\tag3.10$$
Now we observe that 
$$\Cal P(\bold u\to \S^{\bold k}\bold v_\si)^-=
\Cal P(\bold u\to \S^{\bold k}\bold v_\si)\quad \text {if
}\si\ne \id\text { or }\bold k\ne \0.\tag3.11$$
This follows from the conditions on the points
$u_1,\dots,u_r$ and $v_1,\dots,v_r$. Namely, suppose that there is a
family $(P_1,\dots,P_r)$ in $\Cal P(\bold u\to \S^{\bold k}\bold v_\si)$
such that $(\S P_r,P_1,\dots,P_r)$ is nonintersecting. By
assumption, $\S u_r, u_1,\dots,u_r$ is a sequence of points, each
lying to the northwest of its successor. Similarly, 
$$\dots,\S ^{2}
v_r,\S v_1,\dots,\S
v_r,v_1,\dots,v_r,\S^{-1}v_1,\dots,\S^{-1}v_r,\S^{-2}v_1,\dots$$
is an infinite sequence with the same property. If $(\S P_r,P_1,\dots,P_r)$
is nonintersecting,
where $P_i$ runs from $u_i$ to $\S^{k_i}v_{\si(i)}$, 
$i=1,2,\dots,r$, and therefore $\S P_r$ runs from $\S u_r$ to
$\S^{k_r+1}v_{\si(r)}$, then the sequence
$$\S^{k_r+1}v_{\si(r)},\S^{k_1}v_{\si(1)},\dots,\S^{k_r}v_{\si(r)}$$
must also be a sequence with this property. In particular,
$\S^{k_r+1}v_{\si(r)}$ lies to the northwest of
$\S^{k_i}v_{\si(i)}$, which in turn lies to the northwest of 
$\S^{k_r}v_{\si(r)}$. If $\si(i)<\si(r)$, we conclude that $k_i=k_r$, and
if $\si(i)>\si(r)$, we conclude that $k_i=k_r+1$. From
$\sum _{i=1} ^{r}k_i=0$, it follows that
$-rk_r=\vert\{i:\si(i)>\si(r)\}\vert$.
Since $k_r$ must be an
integer, $k_r$ is zero, and hence $k_i=0$ for all $i$.
Therefore $(P_1,\dots,P_r)$ is an element of 
$\Cal P(\bold u\to \bold v_\si)$. If $\si$ were not the identity, then
there would exist $i$ and $j$, with $i<j$, such that $\si(i)>\si(j)$. But
then $v_{\si(i)}$ would lie to the southeast of $v_{\si(j)}$ and
therefore $P_i$ and $P_j$ would intersect. Thus (3.11) is
established.

By consecutive use of (3.7), (3.10), (3.11), and (3.6) we obtain
$$\align \GF(\Cal P(\bold u\to \bold v)^+;w)
&=\GF(\Cal P(\bold u\to \bold v);\bar w)-
\GF(\Cal P(\bold u\to \bold v)^-;\bar w)\\
&=\GF(\Cal P(\bold u\to \bold v);\bar w)+
\underset k_1+\dots+k_r=0\to{\sum _{\si\ne\id} ^{}}\sgn\si\,
\GF(\Cal P(\bold u\to \S^{\bold k}\bold v_\si)^-;\bar w)\\
&=\underset k_1+\dots+k_r=0\to{\sum _{\si\in \frak S_r} ^{}}\sgn\si\,
\GF(\Cal P(\bold u\to \S^{\bold k}\bold v_\si);\bar w)\\
&=\underset k_1+\dots+k_r=0\to{\sum _{\si\in \frak S_r} ^{}}\sgn\si
\prod _{s=1} ^{r}z^{\al^xk_s^2/2-v_{\si(s)}^xk_s}
\GF(\Cal P(u_s\to \S^{k_s}v_{\si(s)});w)\\
&=\sum _{k_1+\dots +k_r=0} ^{}\det_{1\le s,t\le r}\big(z^{\al^x
k_s^2/2-v_t^x k_s}\GF(\Cal P(u_s\to \S^{k_s}v_t);w)\big).
\endalign$$
Thus the Proposition will be proved once we
construct the sign-reversing, $\bar
w$-weight-preserving involution $\ph$ on the set (3.8). 

To construct $\ph$, let $\P=(P_1,\dots,P_r)$ be an element of 
$\Cal P(\bold u\to \S^{\bold k}v_{\si})$ such that $(\S
P_r,P_1,\dots,P_r)$ is intersecting. Among all intersection
points of $(\S P_r,P_1,\dots,P_r)$ we choose the one that is
minimal with respect to a certain order which is induced by the
shift $\S=(-\al^x,\al^y)$. More precisely, among all intersection
points $(x,y)$ of $(\S P_r,P_1,\dots,P_r)$ we search for those
for which $\al^yx+\al^xy$ is minimal, and among all these we
choose the right-most. Denote this point by $p=(p^x,p^y)$.
Figure~3 gives a sketch of the typical situation.
\vskip10pt
\vbox{
$$
\DickPunkt(0,0)
\Label\lu{p}(0,0)
\Pfad(-5,5),44444\endPfad
\SPfad(0,0),44444\endSPfad
\DPfad(-4,4),3\endPfad
\DPfad(-3,3),33\endPfad
\DPfad(-2,2),333\endPfad
\DPfad(-1,1),3333\endPfad
\DPfad(-0,0),33333\endPfad
\DPfad(1,-1),3333\endPfad
\DPfad(2,-2),333\endPfad
\DPfad(3,-3),33\endPfad
\DPfad(4,-4),3\endPfad
\DickPunkt(-3,-2)
\Pfad(-5,0),44\endPfad
\catcode`\@=11
\hbox to0pt{\hskip-5 \Einheit
    \vrule height\Pfadd@cke width.3 \Einheit depth\Pfadd@cke\hss}
\raise-.3\Einheit\hbox to0pt{\hskip-5 \Einheit
    \hbox{\hskip-\PfadD@cke\vrule height.3 \Einheit width\PfadD@cke depth0pt}\hss}
\catcode`\@=13
\Label\u{\S=(-\al^x,\al^y)}(-5,-2)
\Label\o{\text {\eightpoint the line}}(-8,3.7)
\Label\u{\al^yx+\al^xy=\al^yp^x+\al^xp^y}(-8,4)
\Label\u{\text {other intersection points}}(3,3)
$$
\centerline{\eightpoint Figure 3}
}
\vskip10pt
Now we have two cases. Either $p$ is the intersection point of
two paths $P_i$ and $P_j$, with $i<j$, or it is the intersection
point of the translate $\S P_r$ and some path $P_i$. In the
first case we exchange the terminal portions of $P_i$ and $P_j$
beginning from
$p$, i.e., we replace $P_i$ by $P_i'$ and $P_j$ by $P_j'$ where
$$\align P_i'&=[\text {subpath of $P_i$ from $u_i$ to $p$ joined with
subpath of $P_j$ from $p$ to its end point}]\\
P_j'&=[\text {subpath of $P_j$ from $u_j$ to $p$ joined with
subpath of $P_i$ from $p$ to its end point}].\tag3.12
\endalign$$
Thus $\P=(P_1,\dots,P_i,\dots,P_j,\dots,P_r)$ is mapped to 
$$\ph(\P)=(P_1,\dots,P_i',\dots,P_j',\dots,P_r).\tag3.13$$
Clearly $\ph(\P)$ is an element of $\Cal P(\bold u\to\S^{\bold
k^{(i,j)}}\bold v_{\si\circ(ij)})$ where $\bold
k^{(i,j)}=(k_1,\dots,k_j,\dots,\mathbreak k_i,\dots,k_r)$ and $(i\ j)$
denotes the transposition that interchanges $i$ and $j$.

In the second case, we show that  $p$ is actually an intersection point
of $\S P_r$ and $P_1$. For suppose that $p$ is an intersection 
of $\S P_r$ and $P_i$, $i\ge2$. Since $\S u_r$, the starting point of $\S
P_r$, lies to the northwest of $u_1$, the starting point of
$P_1$, $\S u_r$ also lies to the northwest of $u_i$, the starting point of
$P_i$. Hence $P_1$ must intersect either $\S P_r$ or $P_i$ and this
intersection ``comes before $p$", i.e., lies to the southwest of
$p$, contradicting the fact 
that $p$ was chosen as the left-most and lowest of
all intersection points. Now, again,  we exchange terminal
portions of paths. However, the definition of $\ph$ in this case is
more delicate, since we are considering a translate of a path.
We replace $P_1$ by $P_1'$ and $P_r$ by $P_r'$ where
$$\align P_1'&=[\text {subpath of $P_1$ from $u_1$ to $p$ joined with
subpath of $\S P_r$ from $p$}\\
&\hskip7cm\text {to its end point}]\hphantom{.}\\
P_r'&=[\text {subpath of $P_r$ from $u_r$ to $\S^{-1}p$ joined with
subpath of $\S^{-1}P_1$ from $\S^{-1}p$}\\
&\hskip7cm\text {to its end point}].\tag3.14
\endalign$$
So, what happens is that we shift $P_r$ to $\S P_r$, exchange terminal
portions of $P_1$ and $\S P_r$, thus obtaining $P_1'$ and $P_r''$, and
then shift back $P_r''$ to obtain $P_r'=\S^{-1}P_r''$ (see Figure~4).
\vskip10pt
\vbox{
$$\Einheit.4cm
\SPfad(-5,5),21111122222\endSPfad
\Pfad(-3,3),222221111122\endPfad
\Pfad(0,0),21111122222\endPfad
\DickPunkt(-5,5)
\DickPunkt(-3,3)
\DickPunkt(0,0)
\DickPunkt(0,11)
\DickPunkt(2,10)
\DickPunkt(5,6)
\Kreis(-3,6)
\Label\u{\S u_r}(-5,5)
\Label\u{u_1}(-3,3)
\Label\u{u_r}(0,0)
\Label\r{\quad \S v_r}(0,11)
\Label\r{v_1}(2,10)
\Label\r{v_r}(5,6)
\Label\lo{p}(-3,6)
\Label\u{\S P_r}(-1,6)
\Label\u{P_1}(2,8)
\Label\l{P_r}(5,2)
\hskip3cm
\raise3cm\hbox{$\longrightarrow$\hskip3cm}
\SPfad(-5,5),211221111122\endSPfad
\Pfad(-3,3),22211122222\endPfad
\Pfad(0,0),211221111122\endPfad
\DickPunkt(-5,5)
\DickPunkt(-3,3)
\DickPunkt(0,0)
\DickPunkt(0,11)
\DickPunkt(2,10)
\DickPunkt(7,5)
\Kreis(-3,6)
\Label\u{\S u_r}(-5,5)
\Label\u{u_1}(-3,3)
\Label\u{u_r}(0,0)
\Label\r{\quad \S v_r}(0,11)
\Label\r{v_1}(2,10)
\Label\r{\hskip18pt\S^{-1}v_1}(7,5)
\Label\lo{p}(-3,6)
\Label\o{P_r''}(-2,8)
\Label\lu{P_1'}(0,6)
\Label\u{P_r'}(4,3)
\hskip1.7cm
$$
\centerline{\eightpoint Figure 4}
}
\vskip10pt
Thus $\P=(P_1,P_2,\dots,P_{r-1},P_r)$ is mapped to 
$$\ph(\P)=(P_1',P_2,\dots,P_{r-1},P_r').\tag3.15$$
Clearly $\ph(\P)$ is an element of $\Cal P(\bold u\to\S^{\bold
k^{(1,r)}}\bold v_{\si\circ(1r)})$ where $\bold
k^{(1,r)}=(k_r+1,k_2,\dots,\mathbreak k_{r-1},k_1-1)$.

Next we check that $\ph$ is an involution. Let $\P=(P_1,\dots,P_r)$
be an element of (3.8). Let $p=(p^x,p^y)$ be the ``minimal"
intersection point of $(\S P_r,P_1,\dots,P_r)$ and let
$\ph(\P)=(\bar P_1,\bar P_2,\dots,\bar P_r)$. We have to show that $p$ is also
the minimal intersection point for $(\S \bar P_r,\bar P_1,\dots,\bar P_r)$ since
then applying of $\ph$ to $\ph(\P)$ will give back $\P$.

It should be observed that the paths of $\P$ and $\ph(\P)$ agree with
each other up to the line $\al^yx+\al^xy=\al^yp^x+\al^xp^y$. The only changes
occur {\it strictly above\/} this line. This
follows from our assumption that the shift $\S$ is neither
horizontal nor vertical, which implies that the line
$\al^yx+\al^xy=\al^yp^x+\al^xp^y$ is neither horizontal nor vertical.
Therefore $p$ is an intersection point of $(\S \bar P_r,\bar
P_1,\dots,\bar P_r)$
that minimizes $\al^yx+\al^xy$, and among those intersection points 
it is the right-most since the intersection points of $(\S
P_r,P_1,\dots,P_r)$ and $(\S \bar P_r,\bar P_1,\dots,\bar P_r)$ are the same.

That $\ph$ changes the sign of the corresponding permutations is
clear, since the permutations are related by a transposition.
Therefore $\ph$ is sign-reversing.

Finally we have to confirm (3.9), or equivalently,
that $\ph$ is $\bar w$-weight-preserving. Again let
$\P=(P_1,\dots,P_r)$ be an element of
$\Cal P(\bold u\to \S^{\bold k}\bold v_\si)$ such that $(\S
P_r,P_1,\dots,P_r)$ is intersecting and let $p=(p^x,p^y)$ be the
minimal intersection point. 

We consider two cases. First, suppose that $p$ is an
intersection of $P_i$ and $P_j$ for some $i<j$, which implies that
$\ph(\P)$ is defined by (3.12) and (3.13). In this case $\ph(\P)$ is
an element of $\Cal P(\bold u\to \S^{\bold k^{(i,j)}}\bold
v_{\si\circ(ij)})$, where $\bold
k^{(i,j)}=(k_1,\dots,k_j,\dots,k_i,\dots,k_r)$. We have
$$w(\ph(\P))=w(\P)$$
since no shift is involved. Hence in order to prove (3.9), by the
definition (3.6) we have to verify that
$$\sum _{s=1} ^{r}(\al^x\frac
{(k^{(i,j)}_s)^2} {2}-v_{(\si\circ(ij))(s)}^xk^{(i,j)}_s)=
\sum _{s=1} ^{r}(\al^x\frac {k_s^2} {2}-v_{\si(s)}^xk_s).$$
Clearly, these sums can be transformed into each other by
interchange of the $i$th and $j$th summand; hence they agree.

Second, suppose that $p$ is an intersection of $P_1$ and $\S
P_r$, so that $\ph(\P)$ is defined by (3.14) and (3.15).
In this case $\ph(\P)=(P_1',P_2,\dots,P_{r-1},P_r')$ is an element of 
$\Cal P(\bold u\to \S^{\bold k^{(1,r)}}\bold
v_{\si\circ(1r)})$, where $\bold
k^{(1,r)}=(k_r+1,k_2,\dots,k_{r-1},k_1-1)$. We claim that 
$$w(\ph(\P))=z^{v_{\si(r)}^x+\al^x(k_1-k_r-1)-v_{\si(1)}^x}w(\P).\tag3.16$$
In order to prove (3.16), we recall that the weight of a
horizontal edge $e$ and the shifted edge $\S e$ are related by (3.4).
By (3.14), the path $P_1'$ is identical with $P_1$ up to $p$, then it
follows $\S P_r$. We compare the  $w$-weights of the horizontal steps of
$P_1'$ with those of the corresponding steps in $P_1$ and $P_r$.
The $w$-weights of the horizontal steps of $P_1'$ up to $p$ are
the same as those of $P_1$. The $w$-weight of each of the horizontal steps
of $P_1'$ beginning from $p$ by (3.4) equals $z$ times the $w$-weight
of the corresponding step of $P_r$.
The end point of $\S P_r$ is $\S^{k_r+1}
v_{\si(r)}=v_{\si(r)}+(k_r+1)(-\al^x,\al^y)$. Hence the portion of $P_1'$
beginning from $p$ contains $v_{\si(r)}-(k_r+1)\al^x-p^x$ horizontal steps.
Similarly, the weights of the horizontal steps of $P_r'$ up to
$\S^{-1}p$
are the same as those of $P_r$, while the portion of $P_r'$ beginning
from $\S^{-1}p$ contains $v_{\si(1)}^x-k_1\al^x-p^x$ horizontal steps, the
weight of each being equal to $z^{-1}$ times the weight of the
corresponding step of $P_1$. This establishes (3.16).

Therefore, by (3.6) and (3.16), relation (3.9) in this second case
is equivalent to
$$\multline \sum _{s=1} ^{r}\(\al^x\frac {(k_s^{(1,r)})^2} {2}-
v_{(\si\circ(1r))(s)}^{x}k_s^{(1,r)}\)+(v_{\si(r)}^x+\al^x(k_1-k_r-1)-
v_{\si(1)}^x)\\
=\sum _{s=1} ^{r}\(\al^x\frac {(k_s)^2} {2}-v_{\si(s)}^{x}k_s\).
\endmultline\tag3.17$$
In (3.17) the sums on
both sides differ only in the first and last summand. So (3.17), and
hence (3.9), is equivalent to
$$\multline \al^x\frac {(k_r+1)^2} {2}-v_{\si(r)}^x(k_r+1)+ 
\al^x\frac {(k_1-1)^2} {2}-v_{\si(1)}^x(k_1-1)\\
+(v_{\si(r)}^x+\al^x(k_1-k_r-1)-v_{\si(1)}^x)=
\al^x\frac {k_1^2} {2}-v_{\si(1)}^xk_1+ 
\al^x\frac {k_r^2} {2}-v_{\si(r)}^xk_r,\endmultline$$
which is clearly true. This establishes (3.9) in the second
case. Hence $\ph$ is $\bar w$-weight-preserving.
This completes the proof of the Proposition.\quad \quad \qed
\enddemo

\remark{Remarks} 
(1) It should be noted that while in the enumeration of (ordinary)
nonintersecting lattice paths we have to consider permutations of the end
points of the lattice paths, in the previous proof we considered
permutations {\it  and shifts} of the end points. The reader who is
familiar with reflection  groups (cf\. \cite{\HumpAC, especially chapter 4}
for an introduction) will immediately notice the connection with the affine
Weyl group of type
$A$, a group that is generated by the permutations of coordinate axes and
an additional reflection with respect to a hyperplane that does not contain
the origin. In fact, the group that acts on the various configurations of
end points of lattice paths that we considered in the proof of
Proposition~\TA\ is exactly isomorphic to this Weyl group, since it is
generated by permutations of the end points and the additional operation
illustrated in Figure~4. This operation corresponds to the additional
reflection. The $n\times n$ determinants that arise in counting ordinary
nonintersecting lattice paths are alternating sums over the symmetric
group $S_n$, a Weyl group of type $A_{n-1}$; the sum of determinants in
Proposition~\TA\ is an alternating sum over the corresponding affine Weyl
group of type $\widetilde A_{n-1}$.

(2) The requirement in Proposition~\TA\ that $\S$ is
neither horizontal nor vertical is crucial. If $\S$ is a
horizontal or vertical shift, Proposition~\TA\ holds only if the shift
$\S$ carries the path $P_r$ so far that it does not restrict $P_1$ at
all. More precisely, if $\S$ is a horizontal shift, Proposition~\TA\
holds only if $\S v_r$ lies strictly to the left of $u_1$, and if $\S$
is a vertical shift, Proposition~\TA\ holds only if $\S v_r$ lies
strictly above $u_1$. However, in these two cases we are reduced to
``ordinary" nonintersecting lattice paths. Indeed, (3.5) reduces to
the usual Gessel-Viennot determinant \cite{\GeViAB, Cor.~2;
\StemAE, Theorem~1.2} because the determinants in
(3.5) in these two cases can be  non-zero only for
$k_1=k_2=\dots=k_r=0$.

The  proof of Proposition~\TA\ fails for horizontal and vertical
shifts because for them it is no longer true that the changes caused
by the mapping in this proof happen only {\it strictly above} the 
line $\al^yx+\al^xy=\al^yp^x+\al^xp^y$. So it is impossible to define
a unique  ``minimal" intersection point that remains ``minimal" after
the application of the mapping (and this is necessary to prove that
the mapping  is an involution). The problem with  horizontal and
vertical shifts is that a path can intersect its own shift. 
It is easy to check by example that Proposition~\TA\ is in fact
false for horizontal and vertical shifts (except in trivial cases).
See item (1) of the last section for open problems on this subject.

(3) The restriction that
$\al^x$ and $\al^y$ are positive is no loss of generality. If either
$\al^x$ or $\al^y$ 
is negative then $\S u_r$ could not lie to the northwest of $u_1$.

(4) Proposition~\TA\ could be generalized in several
directions.
For example, one could  consider paths with diagonal steps.
One could also consider weights that are induced not only  by horizontal
steps, but also by vertical steps or diagonal steps, and more
generally, weights that distinguish between types of steps. Results similar to
Proposition~\TA\ could be derived for these generalizations, but we omit
them since we shall not need them. See  items
(2)--(4) of the last section for some more related remarks.
\endremark


\subhead 4. The main results\endsubhead
In this section we apply the results of the previous section
to compute the norm generating function for cylindric partitions of a
given shape that satisfy given row bounds. We then generalize
this result to $(\Al,\Be)$-cylindric partitions, which are defined below.


\proclaim{Theorem \TB}Let $\La=(\la_1,\la_2,\dots,\la_r)$ and
$\Mu=(\mu_1,\mu_2,\dots,\mu_r)$ be $r$-tuples of integers and let
$\K$ be an integer with $\K>-r$ such that (3.1) is satisfied. Suppose
that $a_1,a_2,\dots,a_r$ and $b_1,b_2,\dots,b_r$ are integers
satisfying (3.2). Then the generating function 
$\sum_{}^{}q^{n(\pi)}$, where the sum is over all cylindric partitions
$\pi$ of shape $\La/\Mu/\K$, and the entries in row $i$ are at most
$a_i$ and at least $b_i$, equals
$$\multline \sum _{k_1+\dots+k_r=0} ^{}\det_{1\le s,t\le r}
\bigg(q^{b_s(\la_s-s-\mu_t+t-(r+\K)k_s)+(\mu_t-t)(s-t+rk_s)
+(r+\K)(rk_s^2/2+sk_s)}\\
\bmatrix a_t-b_s+\la_s-\mu_t-\K k_s\\
\la_s-\mu_t-s+t-(r+\K)k_s\endbmatrix\bigg).
\endmultline\tag4.1$$
If
$\lambda_i=\lambda_{i+1}-1$ for some $i$ with $1\le i\le r-1$,
the restriction (1.3b) is also assumed to hold for 
$j=\la_i+1=\lambda_{i+1}$ with $\pi_{i,\lambda_{i}+1}:=b_i$, and if
$\la_r=\la_1-\K-1$ the restriction (1.3c) is also assumed to hold for
$j=\la_r+1=\la_1-\K$ with $\pi_{r,\la_r+1}:=b_r$. Moreover, if
$\mu_i=\mu_{i+1}-1$ for some $i$ with $1\le i\le r-1$, then (1.3b) is  also
assumed to hold for 
$j=\mu_i+1=\mu_{i+1}$ with $\pi_{i+1,\mu_{i+1}}:=a_{i+1}$, and if
$\mu_r=\mu_1-\K-1$, the restriction (1.3c) is also assumed to hold for
$j=\mu_r+1=\mu_1-\K$ with $\pi_{1,\mu_1}:=a_1$.
\endproclaim
\demo{Proof}We already saw in section~3 that cylindric partitions
$\pi$  of shape $\La/\Mu/\K$ in which the entries in row $i$ are at most
$a_i$ and at least $b_i$ correspond bijectively to families
$\P=(P_1,P_2,\dots,P_r)$ of lattice paths in which $P_i$ runs from
$(-\la_i+i,b_i-i)$ to $(-\mu_i+i,a_i-i)$ and $(\S
P_r,P_1,\dots,P_r)$ is nonintersecting, where  $\S$ is the shift 
by $(-r-\K,r)$. If we define the weight $w_q$ of a horizontal edge
$e=(e_1,e_2)$ by $w_q(e):=q^{e_2}$ then this correspondence has the
property
$$q^{n(\pi)}=q^{\sum _{i=1} ^{r}i(\la_i-\mu_i)}w_q(\P).\tag4.2$$
Now we may apply Proposition~\TA\ with $u_i=(-\la_i+i,b_i-i)$,
$v_i=(-\mu_i+i,a_i-i)$, $w=w_q$, $\al^x=r+\K$, $\al^y=r$. Observe that
because of (3.1), the starting points $u_i$ and the end points $v_i$
satisfy all the properties that are required in Proposition~\TA.
Clearly, $\al^y=r$ is a positive integer, and since $\K$ is greater than
$-r$,  so is $\al^x=r+\K$. Moreover, for the weight function $w=w_q$ and
the shift $\S$ by $(-r-\K,r)$,  the relation
(3.4) holds with $z=q^r$. From Proposition~\TA\ it follows that the
generating function $\sum w(\P)$, where the sum is over all families
$\P$ of lattice paths as described above, is
$$\multline \sum _{k_1+\dots+k_r=0} ^{}\det_{1\le s,t\le r}
\big(q^{r(r+\K)k_s^2/2+r(\mu_t-t)k_s}\\
\GF(\Cal
P((-\la_s+s,b_s-s)\to(-\mu_t+t-(r+\K)k_s,a_t-t+rk_s);w_q))\big).
\endmultline\tag4.3
$$
By a standard partition generating function theorem (see
\cite{\AndrAF, Theorem~3.1}) we have
$$\GF(\Cal P((a,b)\to(c,d));w_q)=q^{(c-a)b}\bmatrix (c-a)+(d-b)\\
c-a\endbmatrix.\tag4.4$$
If we use this in (4.3) and, because of (4.2), multiply the 
expression we obtain by $q^{\sum _{i=1} ^{r}i(\la_i-\mu_i)}$, then after some
manipulation we arrive at (4.1).\quad \quad \qed
\enddemo

Now we introduce $(\Al,\Be)$-cylindric partitions. Just as (ordinary)
cylindric partitions generalize (ordinary) plane partitions,
$(\Al,\Be)$-cylindric partitions generalize $(\al,\be)$-plane
partitions \cite{\KratAM, \BrenAG}.

Let $r,\K$ be integers, $\Al=(\alpha_1,\alpha_2,\dots,\alpha_\K)$ be
a $\K$-tuple of integers, and
$\Be=(\beta_1,\be_2,\dots,\beta_r)$, 
$\La=(\lambda_1,\la_2,\dots,\lambda_r)$, $\Mu=(\mu_1,\mu_2,\dots,\mu_r)$
be $r$-tuples of integers.
An {\it $(\Al,\Be)$-cylindric partition of 
shape} $\La/\Mu/\K$ is an array $\pi$ of integers of
 the form (1.1) with
$$\pi_{i,j}\ge\pi_{i,j+1}+\al_j,\tag4.5a$$
where by convention $\al_j\equiv \al_{j\text{ mod } \K}$ for any $j$,
$$\pi_{i,j}\ge\pi_{i+1,j}+\beta_i\tag4.5b$$
and
$$\pi_{r,j}\ge\pi_{1,j+\K}+\beta_r.\tag4.5c$$
Note that the length of the sequence $\Al$ equals the length of the
``shift" $\K$ in the shape, while the length of the sequence $\Be$
equals the number of rows of the shape. Clearly, in this new terminology
``ordinary" cylindric partitions are $(\0,\0)$-cylindric
partitions (recall that $\0$ denotes the all 0 sequence). Moreover, the
various types of plane partitions are also special $(\Al,\Be)$-cylindric
partitions. For example, column-strict plane partitions (cf\.
\cite{\StanAA}) are $(\0,\1)$-cylindric partitions ($\1$ denotes the
all 1 sequence) of some shape $\La/\Mu/\K$ where $\K$ is larger than
the difference $\la_1-\mu_r$ (so that (4.5c) becomes void and so there is
no relation between the first and the last row for these $(\0,\1)$-cylindric
partitions). 

Again the {\it norm} of an $(\Al,\Be)$-cylindric partition $\pi$ is
defined to be the sum $\sum \pi_{i,j}$ of all the entries of $\pi$.

The extension of Theorem~\TB\ to $(\Al,\Be)$-cylindric partitions reads
as follows.
\proclaim{Theorem \TC}Let $\La=(\la_1,\la_2,\dots,\la_r)$ and
$\Mu=(\mu_1,\mu_2,\dots,\mu_r)$ be $r$-tuples of integers and $\K$ be
an integer with $\K>-r$ such that (3.1) is satisfied. Let
$\Al=(\al_1,\al_2,\dots,\al_\K)$ be a $\K$-tuple of integers and let
$\Be=(\be_1,\be_2,\dots,\be_r)$ be an $r$-tuple of integers with $\sum
_{k=1} ^{r}\be_k-\sum _{k=1} ^{\K}\al_k<r$.
Suppose that $a_1,a_2,\dots,a_r$ and $b_1,b_2,\dots,b_r$ are integers 
satisfying
$$\align a_i+(1-\be_i)+\sum _{k=\mu_{i+1}+1} ^{\mu_i}\al_k&\ge a_{i+1}\tag4.6a\\
b_i+(1-\be_i)+\sum _{k=\la_{i+1}} ^{\la_i-1}\al_k&\ge b_{i+1}\tag4.6b
\endalign$$
and
$$\align a_r+(1-\be_r)+\sum _{k=\mu_{1}+1} ^{\mu_r+\K}\al_k&\ge a_{1}\tag4.6c\\
b_r+(1-\be_r)+\sum _{k=\la_{1}} ^{\la_r+\K-1}\al_k&\ge b_{1}.\tag4.6d
\endalign$$
Then the generating function $\sum _{} ^{}q^{n(\pi)}$, where
the sum is over all $(\Al,\Be)$-cylindric partitions $\pi$ of shape 
$\La/\Mu/\K$ in which the first entry in row $i$ is at most $a_i$ and the
last entry in row $i$ is at least $b_i$, equals
$$\multline \sum _{k_1+\dots+k_r=0} ^{}\det_{1\le s,t\le r}
\bigg(q^{T(s,t,\bold k)}\\
\bmatrix a_t-b_s+\la_s-\mu_t-\sum _{k=\mu_t+1} ^{\la_s-1}\al_k+\sum _{k=s}
^{t-1}\be_k-k_s(\K-\sum _{k=1} ^{\K}\al_k+\sum _{k=1} ^{r}\be_k)\\
\la_s-\mu_t-s+t-(r+\K)k_s\endbmatrix\bigg),\\
\endmultline\tag4.7$$
where 
\eightpoint
$$\multline T(s,t,\bold k)=b_s(\la_s-s-\mu_t+t-(r+\K)k_s)+\sum
_{k=\mu_t} ^{\la_s-1}\al_k(k-\mu_t)-
\sum _{k=1} ^{\la_s-1}\al_k(s-t+(r+\K)k_s)\\
+(\mu_t-t)\bigg(\sum _{k=t} ^{s-1}(1-\be_k)+k_s\Big(\sum _{k=1}
^{r}(1-\be_k)+\sum _{k=1} ^{\K}\al_k\Big)\bigg)\\
+(r+\K)\bigg(\frac {k_s^2} {2}\Big(\sum _{k=1} ^{r}(1-\be_k)+\sum
_{k=1} ^{\K}\al_k\Big)+k_s\sum _{k=1}^{s-1}(1-\be_k)\bigg).
\endmultline$$
\tenpoint
The sums are interpreted by
$$\suml _{k=m} ^{n-1}\text {\rm Expr}(k)=\cases \hphantom{-}
\suml _{k=m} ^{n-1} \text {\rm Expr}(k)&n>m\\
\hphantom{-}0&n=m\\
-\suml _{k=n} ^{m-1}\text {\rm Expr}(k)&n<m.\endcases$$
If
$\lambda_i=\lambda_{i+1}-1$ for some $i$ with $1\le i\le r-1$, the
restriction (4.5b) is also assumed to hold for 
$j=\la_i+1=\lambda_{i+1}$ with $\pi_{i,\lambda_{i}+1}:=b_i-\al_{\la_i}$, and  if
$\la_r=\la_1-\K-1$ the restriction (4.5c) is also assumed to hold for
$j=\la_r+1=\la_1-\K$ with $\pi_{r,\la_r+1}:=b_r-\al_{\la_r}$. Similarly, if
$\mu_i=\mu_{i+1}-1$ for some $i$ with $1\le i\le r-1$, (4.5b) is also assumed to hold
for 
$j=\mu_i+1=\mu_{i+1}$ with
$\pi_{i+1,\mu_{i+1}}:=a_{i+1}+\al_{\mu_{i+1}}$, and if
$\mu_r=\mu_1-\K-1$ the restriction (4.5c) is also assumed to hold for $j=\mu_r+1=\mu_1-\K$ with
$\pi_{1,\mu_1}:=a_1+\al_{\mu_1}$.
\endproclaim
\demo{Proof}Following the proof idea of Theorem~\TB\ we convert
$(\Al,\Be)$-cylindric partitions to column-strict cylindric
partitions that in turn are converted into a family of
nonintersecting lattice paths with nonintersecting translate of the
last path. Then we apply Proposition~\TA.

Let $\pi$ be an $(\Al,\Be)$-cylindric partition of shape $\La/\Mu/\K$
in which the first entry in row $i$ is at most $a_i$ and the
last entry in row $i$ is at least $b_i$. Define the new array
$\bar\pi$ by
$$\bar\pi_{i,j}:=\pi_{i,j}+\sum _{k=1} ^{j-1}\al_k-\sum _{k=1}
^{i-1}(1-\be_k).\tag4.8$$
Clearly, (4.5a) translates into 
$\bar\pi_{i,j}\ge\bar\pi_{i,j+1}$, (4.5b) translates into
$\bar\pi_{i,j}>\bar\pi_{i+1,j}$, and (4.5c) translates into 
$$\bar\pi_{r,j}+\sum _{k=1} ^{\K}\al_k+\sum _{k=1} ^{r}(1-\be_k)>
\bar\pi_{1,j+\K}.\tag4.9$$
This shows that by (4.8) the $(\Al,\Be)$-cylindric partitions $\pi$
under consideration correspond bijectively to column-strict cylindric
partitions $\bar\pi$ of shape $\La/\Mu/\K$ with first-last relation
(4.9) in which the entries in the $i$th row are at most $\bar a_i$ with
$$\bar a_i=a_i+\sum _{k=1} ^{\mu_i}\al_k-\sum _{k=1} ^{i-1}(1-\be_k)$$
and at least $\bar b_i$ with
$$\bar b_i=b_i+\sum _{k=1} ^{\la_i-1}\al_k-\sum _{k=1} ^{i-1}(1-\be_k).$$
Note that by (4.8) the norms of $\pi$ and $\bar\pi$ are related by
$$\multline
n(\pi)=n(\bar\pi)-
\sum _{i=1} ^{r}\Big(\sum _{j=1} ^{\la_i}\al_j(\la_i-j+1) -\sum _{j=1}
^{\mu_i}\al_j(\mu_i-j+1)\Big)\\
 +\sum _{i=1} ^{r}\Big((\la_i-\mu_i)\sum _{k=1} ^{i-1}(1-\be_k)\Big).
\endmultline\tag4.10$$

%{} before From to prevent e-mail problems

{}From section~3 we know that these column-strict cylindric partitions
correspond bijectively to families $(P_1,P_2,\dots,P_r)$ of lattice
paths in which $P_i$ runs from $(-\la_i+i,\bar b_i)$ to $(-\mu_i+i,\bar
a_i)$ and $(\S P_r,P_1,\dots,P_r)$ is nonintersecting, where $\S$
is the shift by $(-r-\K,\sum _{k=1}^{\K}\al_k+
\sum _{k=1} ^{r}(1-\be_k))$.
Moreover, this correspondence transforms the norm of column-strict 
cylindric partitions into the weight $w_q$ (which was introduced in
the proof of Theorem~\TB) of families of lattice paths.
Therefore, in order to compute the generating function for the
$(\Al,\Be)$-cylindric partitions under consideration, we may apply
Proposition~\TA\ with $u_i=(-\la_i+i,\bar b_i)$, $v_i=(-\mu_i+i,\bar
a_i)$, $w=w_q$,\linebreak $z=q^{\sum _{k=1} ^{\K}\al_k+
\sum _{k=1} ^{r}(1-\be_k)}$, $\al^x=r+\K$, $\al^y=\sum _{k=1} ^{\K}\al_k+
\sum _{k=1} ^{r}(1-\be_k)$,
and multiply the resulting expression (3.5) by the power of $q$ that
is determined by relation (4.10). Then (4.4), with some manipulation,
leads to (4.7). Since  (3.1) and (4.6)  imply that the required
conditions for the starting points $u_i$ and  ending points $v_i$ are
satisfied, and since $\K>-r$ and $\sum _{k=1}  ^{r}\be_k-\sum _{k=1}
^{\K}\al_k<r$ imply that $\al^x>0$ and $\al^y>0$, we may now apply
Proposition~\TA.\quad \quad
\qed 
\enddemo

\remark{Remarks} (1) In each of Proposition~\TA\ and  Theorems~\TB\
and \TC, $k_s$ may be replaced by $k_t$ in the determinants
inside the multisum. This may be seen by
reordering terms and renaming the summation indices.

(2) Theorem~\TB\ is the special case $\al_i=0$, $\be_i=0$ 
of Theorem~\TC.

(3) By extending the approach to plane partitions in \cite{\KratAM} (see
also \cite{\KrMoAC}), we could also prove Theorem~\TC\ (and Theorem~\TB)  by
working directly with arrays, thus avoiding the transformation to
nonintersecting lattice paths.

(4) All  of the conditions, (3.1), (4.6), $\K>-r$,  and $\sum _{i=1}
^{r}\be_i-\sum _{k=1} ^{\K}\al_k<r$, in Theorem~\TC, are really
necessary. They ensure that after the cylindric partition is
transformed into nonintersecting lattice paths, the starting and
ending points are in the ``right" order (see the formulation of
Proposition~\TA) and that the shift is neither horizontal nor
vertical. It is easily seen that if the starting and ending points
are not in the ``right" order, Proposition~\TA\ does not hold. Also
it does not hold for a horizontal or vertical shift except for
trivial cases (see Remark~(2) after the proof of Proposition~\TA).
Hence, if any of the conditions in Theorem~\TC\ were violated,
the theorem would not hold. The same is true for Theorem~\TB.

(5) Theorem~\TC\ contains all determinantal formulas for norm generating
functions for plane partitions (without symmetries) of a given shape
\cite{\GeViAB, Theorems~15, 16; \KratAM, Theorem~6.1;
\KratAO, Theorem~1, (3.3); \StanAB, Prop.~21.2} as special cases. This is
seen by choosing $\K\ge \la_1-\mu_r$. This choice causes the
restriction (4.5c) to be void and  causes
all the determinants in the sum (4.7) to vanish  except for
$k_1=k_2=\dots=k_r=0$.

(6) In \cite{\BurgAC} Burge considers restricted partitions pairs. In
its most general form \cite{\BurgAC, p.~220} these are pairs
$$\big((p_{N_1},p_{N_1-1},\dots,p_1),(q_{N_2},q_{N_2-1},\dots,q_1)\big)$$
of partitions with
$$M_1\ge p_{N_1}\ge p_{N_1-1}\ge \dots\ge p_1\ge0,\quad 
M_2\ge q_{N_2}\ge q_{N_2-1}\ge \dots\ge q_1\ge0$$
and
$$p_i-q_{i+(1-a)}\ge 1-\al\quad \text {and}\quad q_i-p_{i+(1-b)}\ge
1-\be.$$
If we define the two-rowed array $\pi$ by $\pi_{1,i}:=p_{N_1+1-i}$
and $\pi_{2,i}:=q_{N_1+1-i+(1-a)}$ then these restricted partition
pairs  in our language are two-rowed $(\0,(1-\al,1-\be))$-cylindric
partitions of shape
$(N_1,N_1+1-a)/(0,1-a+N_1-N_2)/(a+b-2)$ in which the entries in row $i$
are at most $M_i$ and at least $0$, for $i=1,2$. Thus the $r=2$,
$\Al=\0$ special
case of Theorem~\TC\ immediately gives for the norm generating function
of these cylindric partitions
%\NoBlackBoxes %moved  to the beginning
$$\align 
&\suml _{k_1+k_2=0}
^{}q^{k_1(\alpha+\beta)-k_2(1-a+N_1-N_2-2)(\alpha+\beta)+(a+b)((\alpha
+\beta)
(k_1^2+k_2^2)/2-k_2\alpha)}\\
&\hskip1cm \bmatrix M_1-k_1(\alpha+\beta)
+N_1+(a+b)k_1\\ N_1+(a+b)k_1\endbmatrix
\bmatrix M_2-k_2(\alpha+\beta)+N_2+(a+b)k_2\\ N_2+(a+b)k_2\endbmatrix\\
&-\suml _{k_1+k_2=0}
^{}q^{-(1-a+N_1-N_2)\alpha+\alpha -k_2(1-a+N_1-N_2-2)(\alpha+\beta)
+k_2(\alpha+\beta)+(a+b)((\alpha+\beta)(k_1^2+k_2^2)/2-k_2\alpha)}\\
&\hskip1cm
\bmatrix M_2-\alpha-k_1(\alpha+\beta)+N_2+a+(a+b)k_1\\
N_2+a+(a+b)k_1\endbmatrix\\
&\hskip3cm\bmatrix M_1+\alpha-k_2(\alpha+\beta)+
N_1-a+(a+b)k_2\\ N_1-a+(a+b)k_2\endbmatrix,\tag4.11
\endalign$$
%Christian
%or in Burge's notation,
%$$\multline 
%\suml _{\mu=-\infty} ^{\infty}q^{(\alpha+\beta)(a+b)\mu^2-(a\beta-\alpha
%b-(N_1-N_2)(\alpha+\beta))\mu}g((a+b)\mu,(\alpha+\beta)\mu)\\
%-\suml _{\mu=-\infty} ^{\infty}q^{(\alpha+\beta)(a+b)\mu^2+(2a\alpha
%+a\beta+\alpha b)\mu-(N_1-N_2)((\alpha+\beta)\mu+\alpha)+a\alpha}\\
%\times g(-(a+b)\mu-a,-(\alpha+\beta)\mu-\alpha),
%\endmultline$$
%where 
%$$g(x,y)=\bmatrix M_1+x+N_1-y\\N_1+x\endbmatrix
%\bmatrix M_2-x+N_2+y\\N_2-x\endbmatrix.$$
%Christian
which agrees with
Burge's formula \cite{\BurgAC, p.~220} for his
generating function $R(N_1,M_1,\mathbreak N_2,M_2,a,b,\alpha,\beta)$. (Note that
in Burge's paper the roles of the partitions $p$ and $q$ are as in
our paper when he defines his generating function
$P(N,M,a,b,\al,\be)$;  however, they are interchanged when he defines his
generating function $R(N_1,M_1,N_2,M_2,a,b,\alpha,\beta)$.)

%Christian
Since the conditions under which formula (4.11) holds are not
addressed very clearly in \cite{\BurgAC},
we give the complete set of conditions here:
The parameters $a$ and $b$ must be nonnegative integers
(this comes from (3.1a)), with at least one of them non-zero (from
$\K>-r$). The same has to be true for $\al$ and $\be$ (from (4.6b, d) and
$\sum _{i=1} ^{r}\be_i<r$). Moreover, we must have $a\ge
N_1-N_2\ge-b$ (from (3.1b)) and $\al\ge M_2-M_1\ge -\be$ (from
(4.6a, c)).  If $a=0$ it is understood that $q_1\le \al-1$, 
if $b=0$ it is understood that $p_1\le \be-1$,
if $N_1-N_2-a=0$ it is understood that $p_{N_1}\ge M_2+1-\al$,
and if $N_2-N_1-b=0$ it is understood that $q_{N_2}\ge M_1+1-\be$
(these four conditions come from the
exceptional readings of (4.5b) and (4.5c) that are explained at the
end of the formulation of Theorem~\TC). If any of
these conditions is violated the expression (4.11) will not be the
correct generating function (see Remark~(4)).

\BlackBoxes
We note that a weaker form of Burge's partition pairs previously
was considered by Krattenthaler and Mohanty \cite{\KrMoAC} in
lattice path formulation, and is thus also covered by our Theorem~\TC.

(7) Linear partitions with restrictions on the hook differences were
considered in a number of articles \cite{\AgAnAA, \AndrBG, \AndrBH, \AndrBI,
\ABBBAA, \BresAK, \BurgAD}. The most general result, which includes all
the others, is contained in \cite{\ABBBAA}. There Andrews
{\it et al\.\/} consider linear partitions $\La=(\la_1,\la_2,\dots,\la_M)$,
$N\ge \la_1\ge \la_2\ge \dots\ge\la_M\ge0$ where the hook differences
on diagonal $1-\be$ are at least $\-i+\be+1$ and on
diagonal $\al-1$ are at most $K-i-\al-1$. In \cite{\ABBBAA, Theorem~1}
they find the generating function $\sum q^{n(\La)}$ for these
partitions, where again $n(\La)$ is the sum $\sum _{i=1} ^{M}\la_i$ 
of all the entries of $\La$. Andrews {\it et al\.\/} give a proof using
recurrences. We shall show how their result can
be derived by using the $r=2$, $\Al=(1,1,\dots,1)$ case of our
Theorem~\TC\ and
$q$-Vandermonde summation.

Let $\La$ be a partition as described above. We write $\La$ in
Frobenius notation (cf\. \cite{\MacdAB, p.~3}) as  $\La=(\si\mid\ta)$,
$\si=(\si_1,\si_2,\dots,\si_m)$, $\ta=(\ta_1,\ta_2,\dots,\ta_m)$,
where $m$ is the (Durfee) rank of the partition $\La$. We have
$N-1\ge\si_1>\si_2>\dots>\si_m\ge 0$ and
$M-1\ge\ta_1>\ta_2>\dots>\ta_m\ge0$. The conditions on the hook
differences imply that
$$\align (\si_{j+1-\be}+j+1-\be)-(\ta_j+j)&\ge -i+\be+1\\
(\si_{j+\al-1}+j+\al-1)-(\ta_j+j)&\le K-i-\al-1.
\endalign$$
If we set $\bar\ta_j=\ta_{j+\be-1}$ then these inequalities can be
written as
$$\align \si_j&\ge \bar\ta_j-i+2\be\\
\bar\ta_j&\ge \si_{j+\al+\be-2}-K+i+2\al.
\endalign$$
Let us define the two-rowed array $\pi$ by $\pi_{1,j}:=\si_j$, $1\le
j\le m$, and $\pi_{2,j}:=\bar\ta_j$, $2-\be\le j\le m+1-\be$. Thus we
see that linear partitions $\La$,
$N\ge\la_1\ge\la_2\ge\dots\ge\la_M\ge0$ in which the hook differences
on diagonal $1-\be$ are $\ge-i+\be+1$ and on
diagonal $\al-1$ are $\le K-i-\al-1$ correspond bijectively to
$(\boldkey1,(-i+2\be,-K+i+2\al))$-cylindric partitions $\pi$ of shape
$(m,m+1-\be)/(0,1-\be)/(\al+\be-2)$, for some $m$ (where
$\boldkey1$ denotes the all 1 sequence) in which the entries
in the first row are at most $N-1$ and at least $0$ and the entries
in the second row are at most $M-1$ and at least $0$. Moreover, 
$$n(\La)=n(\pi)+m.\tag4.12$$

For fixed $m$, the  case $r=2$, $\al_i\equiv1$ of Theorem~\TC\ gives us  the
generating function $\sum q^{n(\pi)}$ for these cylindric
partitions $\pi$, namely
$$\multline \sum _{k_1=-\infty} ^{\infty}q^{m^2-m+k_1^2
(\al+\be)(K-\al-\be)-k_1i(\al+\be)+k_1K\be}\\
\bmatrix N+k_1(K-2\al-2\be)\\m-(\al+\be)k_1\endbmatrix
\bmatrix M-k_1(K-2\al-2\be)\\m+(\al+\be)k_1\endbmatrix\\
-\sum _{k_1=-\infty} ^{\infty}q^{m^2-m+k_1^2
(\al+\be)(K-\al-\be)-k_1(\al+\be)(2\be+i)-k_1K\be-\be^2+\be i}\\
\bmatrix N+i-2\be-k_1(K-2\al-2\be)\\m-\be+(\al+\be)k_1\endbmatrix
\bmatrix M-i+2\be+k_1(K-2\al-2\be)\\m+\be-(\al+\be)k_1\endbmatrix.
\endmultline$$
Finally, in order to compute the complete generating function $\sum
q^{n(\La)}$ for all linear partitions $\La$ with prescribed hook
differences as described above, by virtue of (4.12), we have to
multiply the above expression by $q^m$ and sum over all $m$. Using
the $q$-Vandermonde summation (see
e.g\. \cite{\AndrAF, (3.3.10)})
$$\sum _{L\ge0} ^{}q^{(N-L)(H-L)}\bmatrix N\\L\endbmatrix \bmatrix
M\\H-L\endbmatrix=\bmatrix N+M\\H\endbmatrix, $$
this leads to 
$$\multline \sum _{k_1=-\infty} ^{\infty}q^{k_1^2(\al+\be)K
-k_1i(\al+\be)+k_1K\be}\bmatrix N+M\\N+k_1K\endbmatrix\\
-\sum _{k_1=-\infty} ^{\infty}q^{k_1^2(\al+\be)K
-k_1i(\al+\be)-k_1K\be+\be i}\bmatrix N+M\\N+i-k_1K\endbmatrix,
\endmultline$$
which  is the expression in \cite{\ABBBAA, Theorem~1}.

In \cite{\ABBBAA} also,  some necessary conditions on the parameters are
missing. We give the complete set of conditions below and
slightly extend the range of validity of \cite{\ABBBAA, Theorem~1}.
The parameters $\al$ and $\be$ must be nonnegative integers (this comes
from (3.1)), with at least one of them non-zero (from
$\K>-r$), with $\al+\be<K$ (from $\sum _{i=1} ^{r}\be_i-\K<r$).
Moreover, we must have $-i+\be\le N-M\le K-i-\al$ (from (4.6a, c)),
$i-\be\ge 0$ (from (4.6b)), and $K-i-\al\ge0$ (from (4.6d)). (This
means that in \cite{\ABBBAA, Theorem~1} the condition $1\le i\le K/2$
has to be dropped and be replaced by the last two conditions.)
If $\al=0$ it is understood that only partitions
$\La=(\si_1,\dots,\si_m\mid \ta_1,\dots,\ta_m)$ are considered with
$\si_m\le K-i$ and $\ta_1\ge N-1-K+i$,
and if $\be=0$ it is understood that only partitions
$\La=(\si_1,\dots,\si_m\mid \ta_1,\dots,\ta_m)$ are considered with
$\si_1\ge M-1-i$ and $\ta_m\le i$
(these two conventions come from the
exceptional readings of (4.5b) and (4.5c) that are explained at the
end of the formulation of Theorem~\TC).


(8) Krattenthaler and Mohanty's lattice path enumeration results
\cite{\KrMoAB, Theorems~1, 2}, which are $q$-analogues of
\cite{\FilaAA}, are corollaries of Theorem~\TC. We
concentrate on \cite{\KrMoAB, Theorem~2} since \cite{\KrMoAB,
Theorem~1} is a special case of the first. Krattenthaler and Mohanty
consider $r$-dimensional lattice paths consisting of positive unit
steps from $\Mu=(\mu_1,\mu_2,\dots,\mu_r)$ to
$\La=(\la_1,\la_2,\dots,\la_r)$ in the region
$$x_1\ge x_2\ge\dots\ge x_r\ge x_1-\K.\tag4.13$$
In \cite{\KrMoAB, Theorem~2} they give an expression for the
generating function $\sum _{} ^{}q^{\maj_\ta Q}$ for all these
lattice paths $Q$ where $\ta$ is a fixed permutation in $\frak S_r$ and
$\maj_\ta$ is a certain statistic on lattice paths that generalizes
MacMahon's greater index (also called major index) and lesser index.
The link between our Theorem~\TC\ and Krattenthaler and Mohanty's
theorem \cite{\KrMoAB, Theorem~2} is Stanley's $(P,\om)$-partition
theorem \cite{\StanAB, Cor.~5.3+7.2}. It says that, given a poset $P$
with labelling $\om$, the generating
function $U(P,\om;q)$ for unbounded $(P,\om)$-partitions is given by 
$$U(P,\om;q)=\frac {W(P,\om;q)} {[\,\vert P\vert\,]_q!},\tag4.14$$
where
$$W(P,\om;q)=\sum _{\si} ^{}q^{\maj \si},$$
the sum being over all permutations $\si$ in the $\om$-separator
\cite{\StanAB, p.~17} of $P$. ($\vert P\vert$ denotes the cardinality
of $P$, and $\maj$ denotes the (ordinary) major index.)

To apply this theorem, we take $P$ to be the poset
$P(\Mu/\La/\K)$ with underlying set
$$\{(i,j)\in\Bbb Z^2:1\le i\le r\text { and }\mu_i<j\le \la_i\},$$
ordered by
$$(i_1,j_1)\le (i_2,j_2)\quad \text {if and only if}\quad 
\cases i_1\le i_2\text {
and }j_1\le j_2\\\text {or }i_1=r,\ i_2=1,\text { and }j_1+\K\le
j_2,\endcases$$
and labelled by $\om_\ta$, which is defined by
$$\om_\ta(\,(i,j)\,)=\sum _{t=1} ^{i-1}(\la_{\ta^{-1}(t)}-
\mu_{\ta^{-1}(t)})+j-\mu_{\ta^{-1}(i)}.$$
We may describe $\om_\ta$ by saying that $\om_\ta$ numbers the elements
of
$P$ by
$1,2,\dots,\vert P\vert$, beginning with row $\ta^{-1}(1)$,
continuing with row $\ta^{-1}(2)$, etc., with each row numbered
from left to right. Then it is not difficult to see that the
$(P(\La/\Mu/\K),\om_\ta)$-partitions are simply 
$(\0,\bold c)$-cylindric
partitions of shape $\La/\Mu/\K$, where $\bold c=(c_1,c_2,\dots,c_r)$
with $c_i=\chi(\ta(i)>\ta(i+1))$. (Here $\chi(\Cal A)$=1 if $\Cal A$ is
true and $\chi(\Cal A)$=0 otherwise.) Moreover, the
$\om_\ta$-separator of
$P(\La/\Mu/\K)$ corresponds to the lattice paths from $\Mu$ to $\La$
in the region (4.13), such that the major index of
elements of the $\om_\ta$-separator equals $\maj_\ta$ of the lattice
paths. This is seen as follows. Let $\vert \La-\Mu\vert$ denote the
cardinality of $P(\La/\Mu/\K)$ (which is $\sum _{i=1} ^{r}(\la_i-\mu_i)$)
and let $\bold l=(l_1\le l_2\le \dots\le l_{\vert \La-\Mu\vert})$ be
a linear extension of the poset $P(\La/\Mu/\K)$. 

Then $\bold l$ defines an element in the $\om_\ta$-separator of
$P(\La/\Mu/\K)$, namely
$$\om_\ta(\bold l)=\om_\ta(l_1)\om_\ta(l_2)\dots\om_\ta(l_{\vert
\La-\Mu\vert}).
$$
 Now we may define a map $\rho$ from
$P(\La/\Mu/\K)$ to
$\{1,2,\dots,r\}$ by $\rho(\,(i,j)\,):=i$ and form the sequence
$$\rho(\bold l)=\rho(l_1)\rho(l_2)\dots\rho(l_{\vert
\La-\Mu\vert}).$$
Then $\rho(\bold l)$ may be viewed as an $r$-dimensional lattice path
from
$\Mu$ to $\La$ in the region (4.13) by reading $\rho(\bold l)$ from
left to right and interpreting $i$ as a unit step in the positive
$x_i$-direction, $i=1,2,\dots,r$. By observing that 
$$\maj(\om_\ta(\bold l))=\maj_\ta(\rho(\bold l))$$
we see that $W(P(\La/\Mu/\K),\om_\ta;q)$ is the same as the
generating function $\sum q^{\maj_\ta Q}$ for these lattice paths
$Q$. Hence, by (4.14) this generating function equals\linebreak $[\,\vert
\La-\Mu\vert\,]_q!$ times the expression (4.7) with $a_i\to\infty$,
$b_i=0$, $\al_i=0$, $\be_i=c_i=\chi(\ta(i)>\ta(i+1))$, i.e.,
$$\sum _{k_1+\dots+k_r=0} ^{}[\vert \boldsymbol\lambda -
\boldsymbol\mu\vert]_q!
\det_{1 \le s , t \le r}\left(q^{T_\tau (s, t, \bold k) }/[\lambda_s - s
- \mu _t + t - (\K + r)k_ s]_q!\right)\ .$$
The exponents $T_\tau(s,t,\bold k)$ are given by
$$\multline T_\tau (s,t,\bold k)=(\mu_t-t)\sum _{k=t}
^{s-1}(1-c_k)+k_s(\mu_t-t)\sum _{k=1} ^{r}(1-c_k)\\
+(\K+r)\left(\frac {k_s^2} {2}\sum _{k=1} ^{r}(1-c_k)+k_s\sum _{k=1}
^{s-1}(1-c_k)\right)\ .
\endmultline$$
Thus we recover \cite{\KrMoAB, Theorem~2}.
\endremark


\subhead 5. Special generating functions and enumeration results\endsubhead
For some particular choices of the parameters in Theorems~\TB\ and
\TC\ the determinants can be evaluated in closed form by means of the
following three determinant evaluations (5.1)--(5.3), which are from
\cite{\KratAM, \KratAK}. All of them result from a single determinant
lemma \cite{\KratAM, Lemma~2.2}. It was shown in \cite{\KratAM, \KratAK} that
all the known product formulas for plane partitions of a given shape
(without symmetries), such as Stanley's hook-content and hook
formulas and MacMahon's generating function for plane partitions of
rectangular shape, follow from (5.1)--(5.3). In particular, the
evaluation (5.3) 
was used in \cite{\KratAK} to obtain new enumeration results for
plane partitions of staircase shape.
\proclaim{Lemma \TD}There hold
$$ \det_{1\le s,t\le r}\(\bmatrix
L_s+A+t\\L_s+t\endbmatrix\)
=q^{\sum _{i=1} ^{r}(i-1)(L_i+i)}\frac {\prod _{1\le i<j\le r}
^{}[L_i-L_j]} {\prod _{i=1} ^{r}[L_i+r]!}
\frac {\prod _{i=1} ^{r}[L_i+A+1]!} {\prod _{i=1} ^{r}[A+1-i]!},
\tag5.1$$
$$ \det_{1\le s,t\le r}\(q^{tL_s}\bmatrix
A\\L_s+t\endbmatrix\)
=q^{\sum _{i=1} ^{r}iL_i}\frac {\prod _{1\le i<j\le r}
^{}[L_i-L_j]} {\prod _{i=1} ^{r}[L_i+r]!}
\frac {\prod _{i=1} ^{r}[A+i-1]!} {\prod _{i=1} ^{r}[A-L_i-1]!},
\tag5.2$$
and
$$\multline \det_{1\le s,t\le r}\bigg(\pmatrix
BL_s+A\\L_s+t\endpmatrix\bigg)\\
=\frac {\prod _{1\le i<j\le r} ^{}(L_i-L_j)} {\prod _{i=1}
^{r}(L_i+r)!} \prod _{i=1} ^{r}\frac {(BL_i+A)!} {((B-1)L_i+A-1)!}
\prod _{i=1} ^{r}(A-Bi+1)_{i-1},
\endmultline\tag5.3$$
where $(a)_n:=a(a+1)\cdots (a+n-1)$ denotes rising factorials.
\endproclaim

\demo{Proof}For the proof of (5.1)  replace $\la_s-s$ by $L_s$ and
$a+\al-b$ by $A$ in the computation on p.~188 of \cite{\KratAM}. With
the same replacements, (5.2) follows from the computation on p.~189
of \cite{\KratAM}. Finally, (5.3) is proved by replacing $\la_s-s$ by
$L_s$, $a+\al-b$ by $A$, and $B$ by $1-B$ in the proof of Theorem~5
of \cite{\KratAK}.\quad \quad \qed
\enddemo

\remark{Remark}In fact (5.1) is equivalent to (5.2) with $A$ replaced
by $-A-1$.
\endremark

Our first theorem in this section concerns cylindric partitions of
shape $(c^r)/\Mu/\K$. The notation $(c^r)$ means the $r$-tuple
$(c,c,\dots,c)$, as usual. It generalizes Stanley's hook formula for
reverse plane partitions of a given (non-skew) shape with unrestricted
part magnitude \cite{\StanAA, Proposition~18.3}. 
\proclaim{Theorem \TE}Let 
$\Mu=(\mu_1,\mu_2,\dots,\mu_r)$ be an $r$-tuple of integers and $\K$ be
an integer with $\K>-r$ such that
$\mu_1\ge\mu_2\ge\dots\ge\mu_r\ge\mu_1-\K$. 
Then the generating function $\sum _{} ^{}q^{n(\pi)}$, where
the sum is over all cylindric partitions $\pi$ of shape 
$(c^r)/\Mu/\K$ with nonnegative entries, equals
$$\multline \sum _{k_1+\dots+k_r=0} ^{}q^{r(r+\K)\sum _{i=1}
^{r}k_i^2/2 +\K\sum _{i=1} ^{r}ik_i+r\sum _{i=1} ^{r}\mu_ik_i}\\
\frac {\prod _{1\le i<j\le r} ^{}[\mu_i+(r+\K)k_i-i-\mu_j-(r+\K)k_j+j]} {\prod
_{i=1} ^{r}[c-\mu_i-(r+\K)k_i+i-1]!}.
\endmultline\tag5.4$$

More generally, let $\Al=(\al_1,\al_2,\dots,\al_\K)$ and
$\Be=(\be_1,\be_2,\dots,\be_r)$ be integer sequences such that $\sum
_{i=1} ^{r}\be_k-\sum _{i=1} ^{\K}\al_k<r$. 
Then the generating function $\sum _{} ^{}q^{n(\pi)}$, where
the sum is over all $(\Al,\Be)$-cylindric partitions $\pi$ of shape 
$(c^r)/\Mu/\K$ with $\pi_{r,c}\ge0$, equals
$$ \sum _{k_1+\dots+k_r=0} ^{}q^{N_1(\bold k)}
\frac {\prod _{1\le i<j\le r} ^{}[\mu_i+(r+\K)k_i-i-\mu_j-(r+\K)k_j+j]} {\prod
_{i=1} ^{r}[c-\mu_i-(r+\K)k_i+i-1]!},
\tag5.5$$
where \eightpoint
$$\multline  N_1(\bold k)=(r+A-B)(r+\K)\sum _{i=1}
^{r}k_i^2/2 +(\K-A+B)\sum _{i=1} ^{r}ik_i+(r+A-B)\sum _{i=1}
^{r}\mu_ik_i\\
+\sum _{i=1} ^{r}\sum _{k=\mu_i} ^{c-1}\al_k(k-\mu_i)-\sum _{k=1}
^{r}\be_k\sum _{i=k+1} ^{r}(c-\mu_i),
\endmultline$$
\tenpoint
with $A:=\sum _{k=1} ^{\K}\al_k$ and $B:=\sum _{k=1} ^{r}\be_k$.

\endproclaim
\demo{Proof}We prove here only (5.4); (5.5) may be established, with
some more work, in the same manner. 

By Theorem~\TB\ with $\la_i=c$, $b_i=0$, $a_i\to\infty$, and using
Remark~(1) after the proof of Theorem~\TC, we see that the generating
function for cylindric partitions of shape $(c^r)/\Mu/\K$ with
nonnegative entries is given by
$$\multline \sum _{k_1+\dots+k_r=0} ^{}q^{r(r+\K)\sum _{i=1}
^{r}k_i^2/2 +\sum _{i=1} ^{r}(\mu_i-i)(rk_i-i)}\\
\det_{1\le s,t\le
r}\Big(q^{s(\mu_t-t+(r+\K)k_t)}/[c-\mu_t-s+t-(r+\K)k_t]!\Big).
\endmultline$$
Reversing the order of both rows and columns in the determinant
(i.e., replacing $s$ by $r+1-s$ and $t$ by $r+1-t$)
transforms this expression into
$$\multline \sum _{k_1+\dots+k_r=0} ^{}q^{r(r+\K)\sum _{i=1}
^{r}k_i^2/2 +\sum _{i=1} ^{r}(\mu_i-i)(rk_i-i)+(r+1)\sum _{i=1}
^{r}(\mu_i+(r+\K)k_i)-c\binom {r+1}2}\\
\det_{1\le s,t\le
r}\Big(q^{s(c-\mu_{r+1-t}-(r+\K)k_{r+1-t}-t)}/[c-\mu_{r+1-t}-
(r+\K)k_{r+1-t}-t+s]!\Big).
\endmultline$$
Now the determinant can be evaluated by (5.2) with
$L_s=c-\mu_{r+1-s}-(r+\K)k_{r+1-s}-s$, $A\to\infty$. After a few
simplifications we arrive at (5.4). 

Of course, for (5.5) we  use Theorem~\TC\ instead of
Theorem~\TB.\quad \quad \qed

\enddemo
Stanley's
formula for reverse plane partitions of a given shape with
unrestricted part magnitude \cite{\StanAA, Proposition~18.3} 
results from Theorem~\TE, 
(5.4), by setting $\K=c-\mu_r$. This
choice of $\K$ causes the condition (4.5c) to become void. Thus
we are considering ordinary plane partitions of shape $(c^r)/\Mu$,
which by a $180^\circ$ rotation become reverse plane partitions of
shape $(c-\mu_r,c-\mu_{r-1},\dots,c-\mu_1)/\0$. Moreover, for
$\K=c-\mu_r$ the sum in (5.4) consists of the single term
$k_1=k_2=\dots=k_r=0$.

\medskip
The next result concerns cylindric partitions of ``infinite
rectangular shape"\linebreak $(\infty^r)/\0/\K$. 
\proclaim{Theorem \TF}Let $\K>-r$. 
The generating function $\sum _{} ^{}q^{n(\pi)}$, where
the sum is over all cylindric partitions $\pi$ of shape
$(\infty^r)/\0/\K$
in which the entries are between $0$ and $a$, equals
$$ \sum _{k_1+\dots+k_r=0} ^{}q^{r(r+\K)\sum _{i=1}
^{r}k_i^2/2+\K\sum _{i=1} ^{r}ik_i}
\frac {\prodl _{1\le
i<j\le r} ^{}[rk_j+j-rk_i-i]} {\prodl _{i=1} ^{r}[a+rk_i+i-1]!}.
\tag5.6$$
\endproclaim

\demo{Proof}We apply Theorem~\TB\ with $\La=(c^r)$, $c\to\infty$,
$\Mu=\0$, $a_i=a$, $b_i=0$ to obtain for the desired
generating function the expression
$$ \sum _{k_1+\dots+k_r=0} ^{}\det_{1\le s,t\le r}
\bigg(q^{-t(s-t+rk_s)
+(r+\K)(rk_s^2/2+sk_s)}
\frac {1} {[a+s-t+rk_s]!}\bigg).
$$
By reversing the order of the rows and columns in the determinant and
replacing $k_i$ by $k_{r+1-i}$,
we transform this expression into
$$\multline
 \sum _{k_1+\dots+k_r=0} ^{}q^{\sum _{i=1} ^{r}(r+\K)(rk_i^2/2+ik_i)
+\sum _{i=1} ^{r}i(i-a)}\\
\det_{1\le s,t\le r}\bigg(q^{t(a-s+rk_{r+1-s})}
\frac {1} {[a+t-s+rk_{r+1-s}]!}\bigg).
\endmultline$$
Now we use (5.2) with the
replacements $A\to \infty$, $L_s\to a-s+rk_{r+1-s}$ and
obtain (5.6) after a few simplifications.
\quad \quad \qed 
\enddemo
There is an overlap
with MacMahon's result 
for the generating function for plane partitions of rectangular shape
with restricted part magnitude \cite{\MacMAA, sec.~429, proof in
sec.~494}. 
Namely, the formula for plane
partitions of arbitrary shape with restricted part magnitude (that
results by letting the width of the rectangle tend to $\infty$) is
also the $\K\to\infty$ special case of Theorem~\TF. (For $\K\to\infty$
only the $k_1=k_2=\dots=k_r=0$ term of the sum remains.)

\medskip
The last result in this section concerns staircase shapes.
\proclaim{Theorem \TG}The number of cylindric partitions $\pi$ of shape
$(\Lambda-L,\Lambda-2L,\dots,\Lambda-rL)/\0/Lr$
in which the entries are between $0$ and $a$ equals
$$\multline
\sum _{k_1+\dots+k_r=0} ^{}
(L+1)^{\binom r2}\prod _{1\le i<j\le r} ^{}(rk_j+j-rk_i-i)
\prod _{i=1} ^{r}(a+1+(\Lambda-Li)/(L+1))_{i-1} \\
\prod _{i=1} ^{r}\frac
{(a+\Lambda-L(rk_i+i))!}
{(\Lambda-(L+1)(rk_i+i)+r)!\,(a+rk_i+i-1)!},
\endmultline\tag5.7$$
where again $(a)_n:=a(a+1)\cdots (a+n-1)$ denotes the rising
factorial.
\endproclaim
\demo{Proof} We apply Theorem~\TB\ with $\la_i=\Lambda-iL$, $\mu_i=0$,
$\K=Lr$, $a_i=a$, $b_i=0$, $q\to1$ to obtain for the desired number
the expression
$$\sum _{k_1+\dots+k_r=0} ^{}\det_{1\le s,t\le r}
\bigg(\pmatrix a+\Lambda-L(rk_s+s)\\
\Lambda-(L+1)(rk_s+s)+t\endpmatrix\bigg).
$$
Here we use the determinant evaluation (5.3)
with the replacements $L_s\to
\Lambda-(L+1)(rk_s+s)$, $B\to L/(L+1)$, $A\to a+\Lambda/(L+1)$.
Thus, after some simplification we arrive at (5.7).
\quad \quad \qed
\enddemo

We know one more special choice of parameters in
which the determinants in Theorem~\TB\ can be evaluated. This is the
case of cylindric partitions of shape $(c^r)/\0/0$ with
entries between $0$ and $a$. In this case the generating
function can be written down in an even simpler form, since these
cylindric partitions are actually in bijection with certain linear
partitions. Therefore we do not state the multisum expression that
would result from Theorem~\TB\ as a separate theorem. On the
other hand, these considerations produce multisum summations that
are interesting in their own right; see (6.18) and (7.10).

Another case in which the determinants in Theorem~\TC\ could be
evaluated (by means of (5.1))
is $\Mu=\0$, $a_i=a$, $b_i=0$, $\al_i=0$, $\be_i=1$, i.e., the
case of ``column-strict" cylindric partitions of shape
$\La/\0/\K$ with entries between $0$ and $a$. Unfortunately, in this
case the restriction $\sum _{i=1} ^{r}\be_k<r$ is violated, so it is
not clear what, if anything, is actually counted by
(4.7) with this choice of parameters.



\subhead 6. Identities I: Sums of determinants\endsubhead
In this section we extend ideas of Burge \cite{\BurgAC}. He derived a
number of summation formulas by considering special cases of his
restricted partition pairs (which in our language are two-rowed
cylindric partitions; see Remark~(6) after the proof of Theorem~\TC).
He chooses  special cases in which the restricted partition pairs can
be transformed bijectively, by rearranging the parts, into linear
partitions. Since he gives the generating function for restricted
partition pairs (with certain bounds) in general and since, of
course, the generating function for linear partitions (with certain
bounds) is well-known, he obtains an identity. We do the same for
cylindric partitions with an arbitrary number of rows. We
systematically generalize all the identities that Burge derived
by this method.

We give a detailed description of the method in
the simplest case first, and then generalize. Consider cylindric
partitions of shape $(c^r)/\boldkey0/1$ with entries between $0$ and
$a$. (As before, the notation $(c^r)$ means the $r$-tuple
$(c,c,\dots,c)$. So we 
are considering cylindric partitions of rectangular shape, consisting
of $r$ rows and $c$ columns, with a shift of 1 relating the first and
last rows.) By reading such a 
cylindric partition along the columns, each column from top to bottom, and the
columns in order from left to right, it is seen that $\pi$ actually
is a linear partition with $rc$ parts that are between $0$ and $a$.
To be more precise,
by (1.3b) and (1.3c) we have for any such cylindric partition
$\pi=(\pi_{ij})$,
$$ a\ge\pi_{1,1}\ge\pi_{2,1}\ge\dots\ge \pi_{r,1}\ge\pi_{1,2}
\ge\pi_{2,2}\ge\dots\ge\pi_{r,2}
\ge\pi_{1,3}\ge\dots\ge\pi_{r,c}\ge0.
\tag6.1$$
Conversely, if $\pi$ satisfies these inequalities, then it is
a cylindric partition of shape $(c^r)/\0/1$ with entries between $0$
and $a$. Since by Theorem~\TB\ (with $\la_i=c$, $\mu_i=0$, $\K=1$,
$a_i=a$, $b_i=0$) we know the generating function $\sum _{\pi}
^{}q^{n(\pi)}$ for these cylindric partitions, and since we also know
the generating function for the linear partitions with $rc$ parts all of
which are between $0$ and $a$ (see \cite{\AndrAF, Theorem~3.1}), we
obtain the identity
$$\multline \sum _{k_1+\dots+k_r=0} ^{}q^{\binom {r+1}2 \sum _{i=1}
^{r}k_i^2+(r+1)\sum _{i=1} ^{r}ik_i}\\
\det_{1\le s,t\le r}\(q^{t(t-s-rk_s)}\bmatrix
a+c-k_s\\c-s+t-(r+1)k_s\endbmatrix\)=\bmatrix a+rc\\rc\endbmatrix.
\endmultline\tag6.2$$
Burge's doubly bounded Euler's Theorem \cite{\BurgAC, p.~216} is the
special case $r=2$, $a=M$, $c=N$  of (6.2).

For the next several paragraphs we fix $\K=1$. We now work with
$(\Al,\Be)$-cylindric partitions instead of just (ordinary) cylindric
partitions. The restrictions (3.1) allow
more general shapes than just rectangular ones, and the restrictions
(4.6) allow more general bounds where the same idea works to
produce a $q$-binomial summation formula. We shall consider 9 such
cases. For the ``left border" $\Mu$ of the shape 
we choose either the skew border $\Mu=(1^n,0^{r-n})$ or the ``shifted"
border $\Mu=(0^n,1^{r-n})$. For the ``right border" $\La$ of the shape 
we choose either the skew border $\La=((c+1)^m,c^{r-m})$ or the ``shifted"
border $\La=(c^m,(c+1)^{r-m})$. We shall combine the choice 
$\Mu=(1^n,0^{r-n})$ (skew border) with two different choices of upper
bounds $\bold a$, and the choice $\La=((c+1)^m,c^{r-m})$ (skew border) 
with two
different choices of lower bounds $\bold b$. Pairing the three
choices $(\Mu,\bold a)$ ``on the left" with the three choices
$(\La,\bold b)$ ``on the right", we obtain 9 cases.

Since $\K=1$, the requirement that $\al_j\equiv
\al_{j\text{ mod } \K}$ for all 
$j$ implies that all the
$\al_j$ are identical. Let us therefore write
$\al$ for $\al_j$ in what follows.

\medskip
{\it Case 1}. Take $\La=((c+1)^m,c^{r-m})$, where $0<m\le r$,
$\Mu=(1^n,0^{r-n})$, where $0\le n<r$, $\K=1$, $a_i=a-\sum _{k=1}
^{i-1}\be_k-\al\cdot \chi(i\le n)$, $b_i=-\sum _{k=1}
^{i-1}\be_k+\al\cdot \chi(i\ge m+1)$. Here and in the sequel, 
$\chi(\Cal A)$=1 if $\Cal A$ is
true and $\chi(\Cal A)$=0 otherwise. Since we want to apply Theorem~\TC,
we  require that $\sum _{k=1} ^{r}\be_k<r+\al$. Note that (3.1) is
satisfied for this $\La$ and this $\Mu$, and the choice of $a_i$
guarantees that (4.6a) is satisfied. On the other hand, (4.6c) is
satisfied if and only if $\sum _{k=1} ^{r}\be_k\le 1+\al$. Similarly,
with this choice of $b_i$ the restriction (4.6b) is satisfied, while
(4.6d) holds if and only if $\sum _{k=1} ^{r}\be_k\le 1+\al$. Thus,
if $\sum _{k=1} ^{r}\be_k\le \min\{1+\al,r-1+\al\}$ 
then the generating function $\sum
_{\pi} ^{}q^{n(\pi)}$, where the sum is over all
$(\Al,\Be)$-cylindric partitions $\pi$ of shape
$((c+1)^m,c^{r-m})/(1^n,0^{r-n})/1$, in which the first entry in row
$i$ is at most $a_i=a-\sum _{k=1} ^{i-1}\be_k-\al\cdot \chi(i\le n)$ and
the last entry in row $i$ is at least $b_i=-\sum _{k=1}
^{i-1}\be_k+\al\cdot \chi(i\ge m+1)$, equals
$$\multline \sum _{k_1+\dots+k_r=0} ^{}q^{-\sum _{k=1} ^{r}\be_k
c(r-k)-\sum _{k=1} ^{m}\be_k(m-k)+\sum _{k=1} ^{n}\be_k(n-k)+\al
r\binom {r+1}2-nc\al}\\
q^{(r+1)\big(\al+\sum _{k=1} ^{r}(1-\be_k)\big)\sum _{i=1}
^{r}k_i^2/2+(r+1)\sum _{i=1} ^{r}ik_i}\\
\hskip-4cm\det_{1\le s,t\le r}\bigg(q^{(\chi(t\le n)-t)(s-t+k_s\big(\al+\sum
_{k=1} ^{r}(1-\be_k)\big))}\\
\bmatrix a+c+\chi(s\le m)-\chi(t\le n)-\al c-k_s(1-\al+\sum _{k=1}
^{r}\be_k)\\ c+\chi(s\le m)-\chi(t\le
n)-s+t-(r+1)k_s\endbmatrix\bigg).
\endmultline\tag6.3$$

On the other hand, if, here and in the sequel, we make the convention
$\be_k\equiv\be_{k\text{ mod } r}$, we have by (4.5b) and (4.5c) and by the
row bounds for rows $n+1$ and $m$,
$$\align a-\sum _{k=1} ^{n}\be_k&\ge
\pi_{n+1,1}\ge\pi_{n+2,1}+\be_{n+1}\ge\dots\ge \pi_{r,1}+\sum _{k=n+1}
^{r-1}\be_k\ge\\
&\ge\pi_{1,2}+\sum _{k=n+1} ^{r}\be_k\ge\pi_{2,2}+\sum _{k=n+1}
^{r+1}\be_k\ge\\
&\cdots\ge\pi_{i,j}+\sum _{k=n+1}
^{rj-r+i-1}\be_k\ge\dots\ge\pi_{m,c+1}+\sum _{k=n+1} ^{rc+m-1}\be_k
\ge\sum _{k=n+1} ^{rc}\be_k.\tag6.4
\endalign$$
Conversely, let $(\pi_{i,j})$ be an array that satisfies (6.4). Let us
also assume that\linebreak $\sum _{k=1} ^{r}\be_k\ge\al$. Then it can
be seen that $\pi$
must be an $(\Al,\Be)$-cylindric partition of shape
$((c+1)^m,c^{r-m})/(1^n,0^{r-n})/1$, in which the first entry in row
$i$ is at most $a_i=a-\sum _{k=1} ^{i-1}\be_k-\al\cdot \chi(i\le n)$ and
the last entry in row $i$ is at least $b_i=-\sum _{k=1}
^{i-1}\be_k+\al\cdot \chi(i\ge m+1)$. In fact, (4.5b) and (4.5c) are
trivially satisfied. For establishing (4.5a), we observe that by (6.4)
we have
$$\pi_{i,j}+\sum _{k=n+1} ^{rj-r+i-1}\be_k\ge\pi_{i,j+1}+\sum _{k=n+1}
^{rj+i-1}\be_k\ ,$$
and since we assume $\sum _{k=1} ^{r}\be_k\ge\al$, we get
$$\pi_{i,j}\ge\pi_{i,j+1}+\sum _{k=1} ^{r}\be_k\ge \pi_{i,j+1}+\al,$$
which is (4.5a). Thus $\pi$  is indeed an $(\Al,\Be)$-cylindric
partition of shape\linebreak
$((c+1)^m,c^{r-m})/(1^n,0^{r-n})/1$. Now for
$n+1\le i\le r$, (6.4) implies that $\pi_{i,1}+\sum _{k=n+1}
^{i-1}\be_k\le a-\sum _{k=1} ^{n}\be_k$. Equivalently, for $n+1\le
i\le r$ the first entry in row $i$ is at most $a-\sum _{k=1}
^{i-1}\be_k$. Since we are assuming that $\sum _{k=1} ^{r}\be_k\ge\al$, we
may also conclude $\pi_{i,2}\le a-\sum _{k=1} ^{r+i-1}\be_k\le a-\sum
_{k=1} ^{i-1}\be_k-\al$, in particular for $1\le i\le n$. 
This establishes that indeed the first entry
in row $i$ is at most $a_i=a-\sum _{k=1} ^{i-1}\be_k-\al\cdot \chi(i\le
n)$, for all $i$. Similarly, we see that the last entry in row $i$
is at least $b_i=-\sum _{k=1} ^{i-1}\be_k+\al\cdot \chi(i\ge m+1)$.

Therefore, by setting $\bar\pi _{i,j}=\pi_{i,j}+\sum _{k=1}
^{rj+i-1}\be_k$, from the array $\pi=(\pi_{i,j})$ we obtain an array
$\bar\pi =(\bar\pi _{i,j})$ with
$$\align a+\sum _{k=1} ^{r}\be_k&\ge\bar\pi _{n+1,1}\ge\bar\pi
_{n+2,1}\ge\dots\ge\bar\pi _{r,1}\ge\\
&\ge\bar\pi _{1,2}\ge\bar\pi _{2,2}\ge\dots\ge\bar\pi _{m,c+1}\ge
(c+1)\sum _{k=1} ^{r}\be_k\ .
\endalign$$
Thus $\bar\pi =(\bar\pi _{i,j})$ is actually a linear partition with
$rc+m-n$ entries, all of which are between $(c+1)\sum _{k=1} ^{r}\be_k$
and $a+\sum _{k=1} ^{r}\be_k$. Hence, by using the partition theorem 
\cite{\AndrAF, Theorem~3.1} again, the generating function $\sum
_{\bar\pi } ^{}q^{n(\bar\pi )}$ for these arrays $\bar\pi $ is easily
computed. So, by the relation $\bar\pi =\pi_{i,j}+\sum _{k=1}
^{rj+i-1}\be_k$ we finally obtain that the generating function $\sum
_{\pi} ^{}q^{n(\pi)}$ for $(\Al,\Be)$-cylindric partitions $\pi$ of shape
$((c+1)^m,c^{r-m})/(1^n,0^{r-n})/1$, in which the first entry in row
$i$ is at most $a_i=a-\sum _{k=1} ^{i-1}\be_k-\al\cdot \chi(i\le n)$ and
the last entry in row $i$ is at least $b_i=-\sum _{k=1}
^{i-1}\be_k+\al\cdot \chi(i\ge m+1)$ also equals
$$q^{(c+1)(rc+m-n)\sum _{k=1} ^{r}\be_k-\sum _{l=r+n} ^{rc+r+m-1}\sum
_{k=1} ^{l}\be_k}
\bmatrix a-c\sum _{k=1} ^{r}\be_k+rc+m-n\\ rc+m-n\endbmatrix.
\tag6.5$$
Equating (6.3) and (6.5) we obtain a $q$-binomial identity. Replacing $a$
by $a+\al c$ and $\sum _{k=1} ^{r}\be_k$ by $\ep+\al$, and further
simplifying, we may write this identity  in the following form.
\proclaim{Theorem \TI}For $0<m\le r$, $0\le n<r$, $0\le \ep\le \min\{1,r-1\}$,
we have

\vbox{
$$\multline \sum _{k_1+\dots +k_r=0} ^{}q^{(r+1)(r-\ep)\sum _{i=1}
^{r}k_i^2/2+(r+1)\sum _{i=1} ^{r}ik_i}\\
\det_{1\le s,t\le r}\bigg(q^{(\chi(t\le n)-t)(s-t+(r-\ep)k_s)}
\bmatrix a+c+\chi(s\le m)-\chi(t\le n)-(\ep+1)k_s\\
c+\chi(s\le m)-\chi(t\le n)-s+t-(r+1)k_s\endbmatrix\bigg)\\
=q^{(r\binom {c+1}2-nc)\ep}\bmatrix
a+c(r-\ep)+m-n\\rc+m-n\endbmatrix.
\endmultline\tag6.6$$
\line{\hfil\hbox{\qed}\quad \quad }
}
\endproclaim

Identity (6.2) (with $c$ replaced by $c+1$) 
is the special case $\ep=0$, $m=r$, $n=0$  of (6.6). So, Burge's
doubly bounded Euler's Theorem \cite{\BurgAC, p.~216} for his
partition pairs generating function $P(N,M,1,2,1,1)$ is the special
case $r=2$, $\ep=0$, $m=2$, $n=0$, $a=M$, $c=N-1$  of (6.6).
Moreover, his summation at the top of p.~221 of \cite{\BurgAC} is the 
special case $r=2$, $\ep=0$, $m=1$, $n=0$, $a=M$, $c=N$  of (6.6).
%Christian
%Actually, it is just an accident that Burge found this identity. As
%already mentioned in Remark~(6) after the proof of Theorem~\TC,
%Burge's expression 
%for his $R(N_1,M_1,N_2,M_2,a,b,\al,\be)$ is incorrect, and thus the
%sum side of his identity on p.~221 of \cite{\BurgAC} 
%does not count partition pairs in 
%$R(N+1,M,N,M,1,2,1,1)$. On the other hand, as was also
%noted in Remark~(6), some conditions are missing in Burge's paper. In
%the case of $R(N+1,M,N,M,1,2,1,1)$ the fact that
%$N_1-N_2-a=(N+1)-N-1=0$ would require $p_{N_1}=p_{N+1}\ge M_2+1-\al=M$
%(cf\. Remark~(6)), and hence 
%Christian
%forces $p_{N+1}$ to be equal $M$. Thus,
%since partition pairs in $R(N+1,M,N,M,1,2,1,1)$ can be
%merged together into a linear partition with $2N+1$ parts all of
%which are between $0$ and $M$ with the largest part actually
%equal to $M$ (this was missed by Burge), the generating
%function for these partition pairs is 
%$q^M\[\smallmatrix 2N+M\\2N\endsmallmatrix\]$ 
%instead of $\[\smallmatrix 2N+M+1\\2N+1\endsmallmatrix\]$. So the
%identity on p.~221 of \cite{\BurgAC} should actually read (cf\. (4.11))
%$$\align R(N+1,&M,N,M,1,2,1,1)\\
%&=q^M\[\matrix 2N+M\\2N\endmatrix\]\\
%&=\sum _{\mu=-\infty} ^{\infty}q^{6\mu^2+3\mu}\bmatrix
%N+M+\mu+1\\N+3\mu+1\endbmatrix \bmatrix N+M-\mu\\N-3\mu\endbmatrix\\
%&\hphantom{={}}-\sum _{\mu=-\infty} ^{\infty}q^{6\mu^2+3\mu}\bmatrix
%N+M+\mu\\N+3\mu+1\endbmatrix \bmatrix N+M-\mu+1\\N-3\mu\endbmatrix.
%\tag6.7\endalign$$
%(See also the paragraph after Theorem~\TK.)
%However, the identity on p.~221 of \cite{\BurgAC} is correct.
%In fact, it comes from $R(N+1,M,N,M,2(!),1(!),1,1)$, which is
%the special case $r=2$, $\ep=0$, $m=1$, $n=0$, $a=M$, $c=N$  of (6.6)
%as was already mentioned. 

\medskip
{\it Case 2}. Take the same choices for $\La,\Mu,\bold b$ as in
Case~1 (and $\K=1$ of course), i.e., $\La=((c+1)^m,c^{r-m})$, where $0<m\le r$,
$\Mu=(1^n,0^{r-n})$, where $0\le n<r$, $\K=1$, and $b_i=-\sum _{k=1}
^{i-1}\be_k+\al\cdot \chi(i\ge m+1)$, but take $a_i=a+\sum _{k=1}
^{i-1}(1-\be_k)-\al\cdot \chi(i\le n)$. Then it is easily checked that (3.1)
and (4.6) are satisfied. If we assume $\sum _{k=1} ^{r}\be_k\ge\al+n$ it
can be seen that our $(\Al,\Be)$-cylindric partitions of shape
$((c+1)^m,c^{r-m})/(1^n,0^{r-n})/1$, in which the first entry in row
$i$ is at most $a_i=a+\sum _{k=1} ^{i-1}(1-\be_k)-\al\cdot \chi(i\le n)$ and
the last entry in row $i$ is at least $b_i=-\sum _{k=1}
^{i-1}\be_k+\al\cdot \chi(i\ge m+1)$ can be characterized by
$$\align a+\sum _{k=1} ^{n}(1-\be_k)&\ge
\pi_{n+1,1}\ge\pi_{n+2,1}+\be_{n+1}\ge\dots\ge \pi_{r,1}+\sum _{k=n+1}
^{r-1}\be_k\ge\\
&\ge\pi_{1,2}+\sum _{k=n+1} ^{r}\be_k\ge\pi_{2,2}+\sum _{k=n+1}
^{r+1}\be_k\ge\\
&\cdots\ge\pi_{i,j}+\sum _{k=n+1}
^{rj-r+i-1}\be_k\ge\dots\ge\pi_{m,c+1}+\sum _{k=n+1} ^{rc+m-1}\be_k
\ge\sum _{k=n+1} ^{rc}\be_k.\tag6.7
\endalign$$
In fact, what has to be checked is that an array $(\pi_{i,j})$
satisfying (6.7) ``obeys the upper row bounds $a_i=a+\sum _{k=1}
^{i-1}(1-\be_k)-\al\cdot \chi(i\le n)$", i.e., that $\pi_{i,1}\le a_i$ for
$n+1\le i\le r$, and that $\pi_{i,2}\le a_i$ for $1\le i\le n$.
Everything else was already checked in Case~1. Now, because of (6.7)
we have $a+\sum _{k=1} ^{n}(1-\be_k)\ge \pi_{i,1}+\sum _{k=n+1}
^{i-1}\be_k$ for $n+1\le i\le r$, or equivalently, $\pi_{i,1}\le a+\sum
_{k=1} ^{i-1}(1-\be_k)-(i-n-1)$. This implies the first assertion.
Likewise, because of (6.7) we have $a+\sum _{k=1} ^{n}(1-\be_k)\ge
\pi_{i,2}+\sum _{k=n+1} ^{r+i-1}\be_k$ for $1\le i\le n$. Using
our assumption that $\sum _{k=1} ^{r}\be_k\ge\al+n$, we conclude
$$\pi_{i,2}\le a+n -\sum _{k=1} ^{r+i-1}\be_k\le a+\sum _{k=1}
^{i-1}(1-\be_k)-\al-(i-1),$$
which implies the second assertion. 

As in Case~1, we may equate the generating function for our
$(\Al,\Be)$-cylindric partitions that results from Theorem~\TC\ with the
expression that results from the fact that by (6.7) these
$(\Al,\Be)$-cylindric partitions are essentially linear partitions. The
identity that we obtain, after again replacing $a$ by $a+\al c$
and $\sum _{k=1} ^{r}\be_k$ by $\ep+\al$, is the following.
\proclaim{Theorem \TJ}For $0<m\le r$, $0\le n\le 1$, $n\le \ep\le \min\{1,r-1\}$
we have

\NoBlackBoxes
\vbox{
$$\multline \sum _{k_1+\dots +k_r=0} ^{}q^{(r+1)(r-\ep)\sum _{i=1}
^{r}k_i^2/2+(r+1)\sum _{i=1} ^{r}ik_i}\\
\det_{1\le s,t\le r}\bigg(q^{(\chi(t\le n)-t)(s-t+(r-\ep)k_s)}
\bmatrix a+c+\chi(s\le m)-\chi(t\le n)-1+t-(\ep+1)k_s\\
c+\chi(s\le m)-\chi(t\le n)-s+t-(r+1)k_s\endbmatrix\bigg)\\
=q^{(r\binom {c+1}2-nc)\ep}\bmatrix
a+c(r-\ep)+m\\rc+m-n\endbmatrix.
\endmultline\tag6.8$$
\line{\hfil\hbox{\qed}\quad \quad }
}
\endproclaim

\BlackBoxes
\medskip
{\it Case 3}. Take the same choices for $\La,\bold b$ as in
Case~1 (and $\K=1$), i.e., $\La=((c+1)^m,c^{r-m})$, where $0<m\le r$,
$\K=1$, $b_i=-\sum _{k=1}
^{i-1}\be_k+\al\cdot \chi(i\ge m+1)$, but take $\Mu=(0^n,1^{r-n})$, where
$0\le n<r$ and $a_i=a-\sum _{k=1}
^{i-1}\be_k-\al\cdot \chi(i\ge n+1)$. 
Also, we assume here that $\al\le\sum _{k=1} ^{r}\be_k\le
\min\{1+\al,r-1+\al\}$. This guarantees that (3.1) and (4.6) hold, as is easily
checked. Again the $(\Al,\Be)$-cylindric partitions under
consideration turn out to be essentially linear partitions. However,
there is a little detail that is different from the previous two
cases. Namely, our choices force the entries in the first column to
equal the upper row bounds. More precisely, since
$\mu_n=\mu_{n+1}-1$, Theorem~\TC\ (see the last paragraph in the
statement of Theorem~\TC) gives the generating function for
$(\Al,\Be)$-cylindric partitions that also satisfy (4.5b) with
$i=n$, $j=\mu_n+1=\mu_{n+1}=1$, and $\pi_{n+1,1}=a_{n+1}+\al=a-\sum
_{k=1} ^{n}\be_k$, i.e., $\pi_{n,1}\ge a-\sum _{k=1}
^{n}\be_k+\be_n=a-\sum _{k=1} ^{n-1}\be_k$. From this and (4.5b) we
have
$$a_1=a\ge \pi_{1,1}\ge\pi_{2,1}+\be_1\ge\dots\ge\pi_{n,1}+\sum _{k=1}
^{n-1}\be_k\ge a.$$
Hence, $\pi_{i,1}=a-\sum _{k=1} ^{i-1}\be_k$ for $1\le i\le n$, as
asserted. It can be seen that our 
$(\Al,\Be)$-cylindric partitions $\pi$ of shape
$((c+1)^m,c^{r-m})/(0^n,1^{r-n})/1$, in which the first entry in row
$i$ is at most $a_i=a-\sum _{k=1} ^{i-1}\be_k-\al\cdot \chi(i\ge n+1)$ and
the last entry in row $i$ is at least $b_i=-\sum _{k=1}
^{i-1}\be_k+\al\cdot \chi(i\ge m+1)$ can be characterized by
$$\align \pi_{i,1}&=a-\sum _{k=1} ^{i-1}\be_k,\quad \quad \text {for
}1\le i\le n,\\
a-\al&\ge
\pi_{1,2}\ge\pi_{2,2}+\be_{1}\ge\dots\ge \\
&\cdots\ge\pi_{i,j}+\sum _{k=1}
^{rj-2r+i-1}\be_k\ge\dots\ge\pi_{m,c+1}+\sum _{k=1} ^{rc-r+m-1}\be_k
\ge\sum _{k=1} ^{rc-r}\be_k.\tag6.9
\endalign$$

In a manner similar to Case~1 and 2 we finally obtain the following identity.
\proclaim{Theorem \TK}For $0<m\le r$, $0\le n<r$, $0\le \ep\le \min\{1,r-1\}$
we have

\NoBlackBoxes
\vbox{
$$\multline \sum _{k_1+\dots +k_r=0} ^{}q^{(r+1)(r-\ep)\sum _{i=1}
^{r}k_i^2/2+(r+1)\sum _{i=1} ^{r}ik_i}\\
\det_{1\le s,t\le r}\bigg(q^{(\chi(t\ge n+1)-t)(s-t+(r-\ep)k_s)}
\bmatrix a+c+\chi(s\le m)-\chi(t\ge n+1)-(\ep+1)k_s\\
c+\chi(s\le m)-\chi(t\ge n+1)-s+t-(r+1)k_s\endbmatrix\bigg)\\
=q^{an+r\ep\binom {c}2}\bmatrix
a+(r-\ep)(c-1)+m\\r(c-1)+m\endbmatrix.
\endmultline\tag6.10$$
\line{\hfil\hbox{\qed}\quad \quad }
}
\endproclaim

\BlackBoxes
%Christian
%Identity (6.7) is the special case $r=2$, $\ep=0$, $m=2$, $n=1$,
%$a=M$, $c=N$  of (6.10).

\medskip
{\it Case 4}. Take $\La=((c+1)^m,c^{r-m})$, where $0<m\le r$,
$\Mu=(1^n,0^{r-n})$, where $0\le n<r$, $\K=1$, $a_i=a-\sum _{k=1}
^{i-1}\be_k-\al\cdot \chi(i\le n)$, $b_i=\sum _{k=1}
^{i-1}(1-\be_k)+\al\cdot \chi(i\ge m+1)$. We assume that $r+\al-m\le \sum
_{k=1} ^{r}\be_k\le \min\{1+\al,r-1+\al\}$. 
Then we perform analogous considerations as
before. Note that the inequality $r+\al-m\le \sum _{k=1} ^{r}\be_k$
is needed so that an array of the type (6.4), but with lower bound $m-1+\sum
_{k=n+1} ^{rc}\be_k$ instead of $\sum _{k=n+1} ^{rc}\be_k$,
automatically satisfies $\pi_{i,c}\ge \sum _{k=1}
^{i-1}(1-\be_k)+\al$
for $m+1\le i\le r$. We finally obtain the following identity.
\proclaim{Theorem \TL}For $r-1\le m\le r$, $0\le n<r$, $r-m\le \ep\le
\min\{1,r-1\}$
we have

\vbox{
$$\multline \sum _{k_1+\dots +k_r=0} ^{}q^{(r+1)(r-\ep)\sum _{i=1}
^{r}k_i^2/2}\\
\det_{1\le s,t\le r}\bigg(q^{(\chi(t\le n)-t)(r-\ep)k_s}
\bmatrix a+c+\chi(s\le m)-\chi(t\le n)-s+1-(\ep+1)k_s\\
c+\chi(s\le m)-\chi(t\le n)-s+t-(r+1)k_s\endbmatrix\bigg)\\
=q^{rc(m-1)-c\binom r2+\binom {m-n}2+(r\binom {c+1}2-nc)\ep}\bmatrix
a+c(r-\ep)+1-n\\rc+m-n\endbmatrix.
\endmultline\tag6.11$$
\line{\hfil\hbox{\qed}\quad \quad }
}
\endproclaim

\medskip
{\it Case 5}. Take the same choices for $\La,\Mu,\bold b$ as in
Case~4 (and $\K=1$ of course), i.e., $\La=((c+1)^m,c^{r-m})$, where $0<m\le r$,
$\Mu=(1^n,0^{r-n})$, where $0\le n<r$, $\K=1$, $b_i=\sum _{k=1}
^{i-1}(1-\be_k)+\al\cdot \chi(i\ge m+1)$, but take $a_i=a+\sum _{k=1}
^{i-1}(1-\be_k)-\al\cdot \chi(i\le n)$. We assume that
$\max\{\al+n,r+\al-m\}\le \sum _{k=1} ^{r}\be_k<r$. Performing
analogous considerations as before, we finally arrive at the
following identity.
\proclaim{Theorem \TM}For $0< m\le r$, $0\le n<r$, $\max\{n,r-m\}\le
\ep<r$
we have

\vbox{
$$\multline \sum _{k_1+\dots +k_r=0} ^{}q^{(r+1)(r-\ep)\sum _{i=1}
^{r}k_i^2/2}\\
\det_{1\le s,t\le r}\bigg(q^{(\chi(t\le n)-t)(r-\ep)k_s}
\bmatrix a+c+\chi(s\le m)-\chi(t\le n)-s+t-(\ep+1)k_s\\
c+\chi(s\le m)-\chi(t\le n)-s+t-(r+1)k_s\endbmatrix\bigg)\\
=q^{rc(m-1)-c\binom r2+\binom {m-n}2+(r\binom {c+1}2-nc)\ep}\bmatrix
a+c(r-\ep)+1\\rc+m-n\endbmatrix.
\endmultline\tag6.12$$
\line{\hfil\hbox{\qed}\quad \quad }
}
\endproclaim
Burge's summation for $P(N,M,1,2,0,2)$ \cite{\BurgAC, p.~216} is the 
special case $r=2$, $\ep=0$, $m=2$, $n=0$,
$a=M$, $c=N-1$  of (6.12), and his 
summation for $P(N,M,1,2,0,1)$ \cite{\BurgAC, p.~217} is the 
special case $r=2$, $\ep=1$, $m=2$, $n=0$,
$a=M$, $c=N-1$  of (6.12).


\medskip
{\it Case 6}. Take the same choices for $\La,\bold b$ as in
Case~4 (and $\K=1$), i.e., $\La=((c+1)^m,c^{r-m})$, where $0<m\le r$,
$\K=1$, $b_i=\sum _{k=1}
^{i-1}(1-\be_k)+\al\cdot \chi(i\ge m+1)$, but take $\Mu=(0^n,1^{r-n})$, where
$0\le n<r$ and $a_i=a-\sum _{k=1}
^{i-1}\be_k-\al\cdot \chi(i\ge n+1)$. We assume that $r+\al-m\le \sum _{k=1}
^{r}\be_k\le \min\{1+\al,r-1+\al\}$. 
Performing analogous considerations as before we
arrive at the following identity.
\proclaim{Theorem \TN}For $r-1\le m\le r$, $0\le n<r$, $r-m\le \ep\le
\min\{1,r-1\}$
we have

\NoBlackBoxes
\vbox{
$$\multline \sum _{k_1+\dots +k_r=0} ^{}q^{(r+1)(r-\ep)\sum _{i=1}
^{r}k_i^2/2}\\
\det_{1\le s,t\le r}\bigg(q^{(\chi(t\ge n+1)-t)(r-\ep)k_s}
\bmatrix a+c+\chi(s\le m)-\chi(t\ge n+1)-s+1-(\ep+1)k_s\\
c+\chi(s\le m)-\chi(t\ge n+1)-s+t-(r+1)k_s\endbmatrix\bigg)\\
=q^{an+(c-1)\(r(m-1)-\binom r2\)+\binom m2-\binom n2+r\binom {c}2\ep}\bmatrix
a+(r-\ep)(c-1)+1\\r(c-1)+m\endbmatrix.
\endmultline\tag6.13$$
\line{\hfil\hbox{\qed}\quad \quad }
}
\endproclaim

\BlackBoxes
\medskip
{\it Case 7}. Take $\La=(c^m,(c+1)^{r-m})$, where $0<m\le r$,
$\Mu=(1^n,0^{r-n})$, where $0\le n<r$, $\K=1$, $a_i=a-\sum _{k=1}
^{i-1}\be_k-\al\cdot \chi(i\le n)$, $b_i=-\sum _{k=1}
^{i-1}\be_k+\al\cdot \chi(i\le m)$. We assume that $\al\le \sum _{k=1}
^{r}\be_k\le \min\{1+\al,r-1+\al\}$. 
Similar considerations as above give rise to the
following identity.
\proclaim{Theorem \TO}For $0<m\le r$, $0\le n<r$, $0\le \ep\le
\min\{1,r-1\}$
we have

\NoBlackBoxes
\vbox{
$$\multline \sum _{k_1+\dots +k_r=0} ^{}q^{(r+1)(r-\ep)\sum _{i=1}
^{r}k_i^2/2+(r+1)\sum _{i=1} ^{r}ik_i}\\
\det_{1\le s,t\le r}\bigg(q^{(\chi(t\le n)-t)(s-t+(r-\ep)k_s)}
\bmatrix a+c+\chi(s\ge m+1)-\chi(t\le n)-(\ep+1)k_s\\
c+\chi(s\ge m+1)-\chi(t\le n)-s+t-(r+1)k_s\endbmatrix\bigg)\\
=q^{(r\binom {c}2-n(c-1))\ep}\bmatrix
a+(r-\ep)(c-1)+r-n\\rc-n\endbmatrix.
\endmultline\tag6.14$$
\line{\hfil\hbox{\qed}\quad \quad }
}
\endproclaim

\BlackBoxes
\medskip
{\it Case 8}. Take the same choices for $\La,\Mu,\bold b$ as in
Case~7 (and $\K=1$ of course), i.e., $\La=(c^m,(c+1)^{r-m})$, where $0<m\le r$,
$\Mu=(1^n,0^{r-n})$, where $0\le n<r$, $\K=1$, $b_i=-\sum _{k=1}
^{i-1}\be_k+\al\cdot \chi(i\le m)$, but take $a_i=a+\sum _{k=1}
^{i-1}(1-\be_k)-\al\cdot \chi(i\le n)$. We assume that $\al+n\le \sum _{k=1}
^{r}\be_k\le \min\{1+\al,r-1+\al\}$. 
Similarly to above we obtain the following identity.
\proclaim{Theorem \TP}For $0<m\le r$, $0\le n\le 1$, $n\le \ep\le
\min\{1,r-1\}$
we have

\NoBlackBoxes
\vbox{
$$\multline \sum _{k_1+\dots +k_r=0} ^{}q^{(r+1)(r-\ep)\sum _{i=1}
^{r}k_i^2/2+(r+1)\sum _{i=1} ^{r}ik_i}\\
\det_{1\le s,t\le r}\bigg(q^{(\chi(t\le n)-t)(s-t+(r-\ep)k_s)}
\bmatrix a+c+\chi(s\ge m+1)-\chi(t\le n)-1+t-(\ep+1)k_s\\
c+\chi(s\ge m+1)-\chi(t\le n)-s+t-(r+1)k_s\endbmatrix\bigg)\\
=q^{(r\binom {c}2-n(c-1))\ep}\bmatrix
a+(r-\ep)(c-1)+r\\rc-n\endbmatrix.
\endmultline\tag6.15$$
\line{\hfil\hbox{\qed}\quad \quad }
}
\endproclaim

\BlackBoxes
\medskip
{\it Case 9}. Finally, take the same choices for $\La,\bold b$ as in
Case~7 (and $\K=1$), i.e., $\La=(c^m,(c+1)^{r-m})$, where $0<m\le r$,
$\K=1$, $b_i=-\sum _{k=1}
^{i-1}\be_k+\al\cdot \chi(i\le m)$, but take $\Mu=(0^n,1^{r-n})$, where
$0\le n<r$ and $a_i=a-\sum _{k=1}
^{i-1}\be_k-\al\cdot \chi(i\ge n+1)$. We assume that $\al\le \sum _{k=1}
^{r}\be_k\le \min\{1+\al,r-1+\al\}$. 
The identity that is obtained in this case is
the following.
\proclaim{Theorem \TQ}For $0<m\le r$, $0\le n< r$, $0\le \ep\le
\min\{1,r-1\}$
we have

\NoBlackBoxes
\vbox{
$$\multline \sum _{k_1+\dots +k_r=0} ^{}q^{(r+1)(r-\ep)\sum _{i=1}
^{r}k_i^2/2+(r+1)\sum _{i=1} ^{r}ik_i}\\
\det_{1\le s,t\le r}\bigg(q^{(\chi(t\ge n+1)-t)(s-t+(r-\ep)k_s)}
\bmatrix a+c+\chi(s\ge m+1)-\chi(t\ge n+1)-(\ep+1)k_s\\
c+\chi(s\ge m+1)-\chi(t\ge n+1)-s+t-(r+1)k_s\endbmatrix\bigg)\\
=q^{an+r\binom {c-1}2\ep}\bmatrix
a+(r-\ep)(c-2)+r\\r(c-1)\endbmatrix.
\endmultline\tag6.16$$
\line{\hfil\hbox{\qed}\quad \quad }
}
\endproclaim
\BlackBoxes
Burge's summation for $P(N,M,0,3,1,1)$ \cite{\BurgAC, p.~216} is the
special case $r=2$, $\ep=0$, $m=n=1$,
$a=M$, $c=N$  of (6.16).

\bigskip
Another choice of $\K$ that gives rise to identities is $d=0$. So let us
fix $d=0$ for the next several paragraphs. As before, we first explain the
method in the simplest case (which is again the rectangular shape), and
then generalize. Consider cylindric partitions of shape $(c^r)/\0/0$ with
entries between $0$ and $a$. By (1.3b) and (1.3c) we have
$$\pi_{1,j}\ge \pi_{2,j}\ge \dots\ge \pi_{r,j}\ge \pi_{1,j},\quad
\text{for any $j$,}$$
hence the entries in each column are identical. By (1.3a) we
have
$$a\ge \pi_{1,1}\ge \pi_{1,2}\ge \dots\ge\pi_{1,c}\ge 0.$$
So by adding up the entries in each column we obtain
$(r\pi_{1,1},r\pi_{1,2}, \dots,r\pi_{1,c})$, which is a linear partition with
$c$ parts, all of which are divisible by $r$ and lie between $0$ and
$ra$. This correspondence is clearly a bijection. The generating
function for these linear partitions is $\left[\smallmatrix a+c\\c
\endsmallmatrix\right]_{q^r}$. On the other hand, Theorem~\TB\ (with
$\la_i=c$, $\mu_i=0$, $d=0$, $a_i=a$, $b_i=0$) tells us the
generating function for these cylindric partitions. Since these are
in bijection with the corresponding linear partitions, we have proved 
$$\multline
\sum _{k_1+\dots+k_r=0} ^{}q^{r^2\sum _{i=1} ^{r}k_i^2/2 +r\sum
_{i=1} ^{r}ik_i} \det_{1\le s,t\le r}\(q^{t(t-s-rk_s)}\bmatrix
a+c\\c-s+t-rk_s \endbmatrix\)\\
=\bmatrix a+c\\c\endbmatrix_{q^r}.
\endmultline\tag6.17$$
Burge's summation for $P(N,M,1,1,1,1)$ \cite{\BurgAC, p.~217} is the
special case $r=2$, $a=M$, $c=N$  of (6.17).

Given the rectangular shape $(c^r)/\0/0$, the restrictions (4.6) allow
more general row bounds for which the same idea works to produce a
$q$-binomial identity. We consider the choices $a_i=a$ or $a+i$ and
$b_i=0$ or $i$. In the theorem below we list the identities that
result from these $2\times2$ possibilities. The first comes from
$a_i=a$, $b_i=0$ (and hence repeats (6.17)), the second comes from
$a_i=a+i$, $b_i=0$, the third comes from
$a_i=a$, $b_i=i$, and the fourth comes from
$a_i=a+i$, $b_i=i$.

\proclaim{Theorem \TR}There hold the identities
$$\multline
\sum _{k_1+\dots+k_r=0} ^{}q^{r^2\sum _{i=1} ^{r}k_i^2/2 +r\sum
_{i=1} ^{r}ik_i} \det_{1\le s,t\le r}\(q^{t(t-s-rk_s)}\bmatrix
a+c\\c-s+t-rk_s \endbmatrix\)\\=\bmatrix a+c\\c\endbmatrix_{q^r}.
\endmultline\tag6.18$$
$$\multline
\sum _{k_1+\dots+k_r=0} ^{}q^{r^2\sum _{i=1} ^{r}k_i^2/2 +r\sum
_{i=1} ^{r}ik_i} \det_{1\le s,t\le r}\(q^{t(t-s-rk_s)}\bmatrix
a+c+t\\c-s+t-rk_s \endbmatrix\)\\=\bmatrix a+c+1\\c\endbmatrix_{q^r}.
\endmultline\tag6.19$$
$$\multline
\sum _{k_1+\dots+k_r=0} ^{}q^{r^2\sum _{i=1} ^{r}k_i^2/2 -r\sum
_{i=1} ^{r}ik_i} \det_{1\le s,t\le r}\(\bmatrix
a+c-s\\c-s+t-rk_t \endbmatrix\)\\=q^{c\binom r2}\bmatrix a+c-r\\c\endbmatrix_{q^r}.
\endmultline\tag6.20$$
\vbox{
$$\multline
\sum _{k_1+\dots+k_r=0} ^{}q^{r^2\sum _{i=1} ^{r}k_i^2/2 -r\sum
_{i=1} ^{r}ik_i} \det_{1\le s,t\le r}\(\bmatrix
a+c-s+t\\c-s+t-rk_t \endbmatrix\)\\=q^{c\binom r2}\bmatrix
a+c+1-r\\c\endbmatrix_{q^r}. 
\endmultline\tag6.21$$
\line{\hfil\hbox{\qed}\quad \quad }
}
\endproclaim
As was already remarked, Burge's summation for $P(N,M,1,1,1,1)$
\cite{\BurgAC, p.~217} is the special case
$r=2$, $a=M$, $c=N$  of (6.18). Moreover, his 
summation for $P(N,M,1,1,0,2)$ \cite{\BurgAC, p.~217} is the
special case $r=2$, $a=M$, $c=N$  of (6.21).

One could also allow more general shapes. Let us consider cylindric
partitions of shape
$((c+1)^m,c^{r-m-m'},(c+1)^{m'})/(1^n,0^{r-n-n'},1^{n'})/0$ with
entries between $0$ and $a$. For convenience we assume  that $m+m'<r$ and
$n+n'>0$; i.e., that neither column~1 nor column~$c+1$ is completely
filled. Of course, the considerations above show that for each of
these cylindric partitions the entries in column $j$, $2\le j\le c$,
have to be identical. Moreover, Theorem~\TB\ with
$\la_i=c+1-\chi(m+1\le i\le r-m')$, $\mu_i=1-\chi(n+1\le i\le r-n)$,
$d=0$, $a_i=a$, $b_i=0$ would not give the generating function for all
these cylindric partitions, but only for those where all the $r-n-n'$
entries in column~$1$ are equal $a$ and all the $m+m'$ entries in
column $c+1$ are equal $0$. This is because of the conventions for
$\la_i=\la_{i+1} -1$ and $\mu_i=\mu_{i+1}-1$ that are explained in the
last paragraph of Theorem~\TB. Thus (4.1) with the above choices
of parameters is equal to  $q^{(r-n-n')a}$ times the generating function
for linear partitions with $c-1$ parts, all  divisible by
$r$ and between $0$ and $ra$, and we obtain the following identity.

\proclaim{Theorem \TS}Let $m,m',n,n'$ be nonnegative integers with
$0\le m+m'<r$ and $0<n+n'\le r$. Then 

\vbox{
$$\multline
\sum _{k_1+\dots+k_r=0} ^{}q^{r^2\sum _{i=1} ^{r}k_i^2/2 +r\sum
_{i=1} ^{r}ik_i} \det_{1\le s,t\le r}
\bigg(q^{(\chi(n+1\le t\le r-n')+t)(t-s-rk_s)}\\
\bmatrix a+c-\chi(m+1\le s\le r-m')+ \chi(n+1\le t\le r-n')\\
c-\chi(m+1\le s\le r-m')+\chi(n+1\le t\le r-n')-s+t-rk_s \endbmatrix\bigg)\\
=q^{(r-n-n')a}\bmatrix a+c-1\\c-1\endbmatrix_{q^r}.
\endmultline\tag6.22$$
\line{\hfil\hbox{\qed}\quad \quad }
}
\endproclaim
Note that (6.22) covers (6.18)=(6.17) (set $m=m'=0$, $n+n'=r$ and
replace $c$ by $c+1$).





\bigskip
The idea for $\K=1$ of obtaining identities by merging the entries of
appropriate cylindric partitions into linear partitions (or something
equivalent) can be extended. Again, we begin with the simplest
example, and then generalize.

Consider cylindric partitions of shape $((m\K)^r)/\0/\K$ with entries
between 0 and $a$, where $r$, $\K$, and $m$ are arbitrary positive integers. 
Figure~5.a  shows schematically the shape of such a cylindric partition,
where $m=4$. The arrows indicate the directions of (weak) decrease of
entries. The last row is  marked with a dotted line, and its shift that
dominates the top row is represented as a dotted row on top of the
shape.  We may cut such a cylindric partition into $m$ rectangular
pieces as indicated in Figure~5.a (where $m=4$). Then we take the $m$
pieces, which were originally arranged horizontally, and
rearrange them vertically, as in Figure~5.b. It is easy to see
that again there is decrease along the directions that are indicated by
the arrows. Moreover, the last column dominates the first as indicated in
Figure~5.b. Finally, we reflect the new array in some antidiagonal line
to obtain an array as in Figure~5.c. As suggested by the
picture, this array is a cylindric partition of shape $((mr)^\K)/\0/r$. 

{
\def\hSSchritt{\leavevmode\raise-.4pt\hbox to0pt{\hss.\hss}\hskip.5\Einheit
%\hskip.5\Einheit
  \raise-.4pt\hbox to0pt{\hss.\hss}\hskip.5\Einheit
%  \raise-.4pt\hbox to0pt{\hss.\hss}\hskip.2\Einheit
%  \raise-.4pt\hbox to0pt{\hss.\hss}\hskip.2\Einheit
%  \raise-.4pt\hbox to0pt{\hss.\hss}\hskip.2\Einheit
}
\def\vSSchritt{\vbox{\baselineskip.5\Einheit\lineskiplimit0pt
\hbox{.}\hbox{.}%
%\hbox{.}\hbox{.}\hbox{.}
}}
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  \hbox to0pt{\hss.\hss}\hskip.2\Einheit
  \raise.2\Einheit\hbox to0pt{\hss.\hss}\hskip.2\Einheit
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  \hbox to0pt{\hss.\hss}\hskip.2\Einheit
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  \raise-.6\Einheit\hbox to0pt{\hss.\hss}\hskip.2\Einheit
  \raise-.8\Einheit\hbox to0pt{\hss.\hss}\hss}}

\vskip10pt
\vbox{
$$
\Einheit0.2cm
\raise1cm\hbox{
\Pfad(0,0),11111111111111111111\endPfad
\SPfad(0,1),11111111111111111111\endSPfad
\Pfad(0,8),11111111111111111111\endPfad
\SPfad(5,9),11111111111111111111\endSPfad
\Pfad(0,0),22222222\endPfad
\Pfad(5,0),22222222\endPfad
\Pfad(10,0),22222222\endPfad
\Pfad(15,0),22222222\endPfad
\Pfad(20,0),22222222\endPfad
\SPfad(5,8),2\endSPfad
\SPfad(25,8),2\endSPfad
\SPfad(20,8),11111\endSPfad
\Label\ru{\rightarrow}(1,8)
\Label\ru{\downarrow}(0,7)
\Label\o{\hskip.1cm\oversetbrace\tsize\K\to{\hbox{\hskip1cm}}}(2,8)
\Label\l{\hskip.1cm\sideset r\and \to
     {\left\{\vbox{\vskip0.9cm}\right.}}(0,4)
\Label\u{\hbox{\eightpoint
    a. cylindric partition of shape $((md)^r)/\0/\K$}}(10,-5)
\hskip5cm
}
\hbox{\hskip-7cm}
\raise-3cm\hbox{
\Pfad(0,0),11111111111111111111111111111111\endPfad
\SPfad(0,1),11111111111111111111111111111111\endSPfad
\Pfad(0,5),11111111111111111111111111111111\endPfad
\SPfad(8,6),11111111111111111111111111111111\endSPfad
\Pfad(0,0),22222\endPfad
\Pfad(8,0),22222\endPfad
\Pfad(16,0),22222\endPfad
\Pfad(24,0),22222\endPfad
\Pfad(32,0),22222\endPfad
\SPfad(8,5),2\endSPfad
\SPfad(40,5),2\endSPfad
\SPfad(32,5),11111111\endSPfad
\Label\ru{\rightarrow}(1,5)
\Label\ru{\downarrow}(0,4)
\Label\o{\oversetbrace\tsize r\to{\hbox{\hskip1.6cm}}}(4,5)
\Label\l{\hskip.1cm\raise.7cm\hbox{$\sideset \K\and \to
     {\left\{\vbox{\vskip.6cm}\right.}$}}(0,2)
\Label\u{\hbox{\eightpoint
    c. after reflection, a cylindric partition of shape $((mr)^d)/\0/r$}}(16,-5)
\hskip8cm
}
\hbox{\hskip1cm}
\hbox{
\Pfad(0,-16),11111\endPfad
\Pfad(0,-8),11111\endPfad
\Pfad(0,0),11111\endPfad
\Pfad(0,8),11111\endPfad
\Pfad(0,16),11111\endPfad
\Pfad(0,-16),22222222222222222222222222222222\endPfad
\Pfad(5,-16),22222222222222222222222222222222\endPfad
\SPfad(4,-16),22222222222222222222222222222222\endSPfad
\SPfad(-1,-24),22222222222222222222222222222222\endSPfad
\SPfad(0,-24),22222222\endSPfad
\SPfad(-1,-24),1\endSPfad
\SPfad(-1,8),1\endSPfad
\Label\ru{\rightarrow}(1,16)
\Label\ru{\downarrow}(0,15)
\Label\o{\hskip.1cm\oversetbrace\tsize\K\to{\hbox{\hskip1cm}}}(2,16)
\Label\l{\hskip.1cm\sideset r\and \to
     {\left\{\vbox{\vskip0.9cm}\right.}}(0,12)
\Label\u{\hbox{\eightpoint
    b. pasted together differently}}(2,-25)
\hskip-0.6cm
}
$$
\centerline{\eightpoint Figure 5}
}}
\vskip10pt
\noindent
If we apply
the same procedure to the final array we recover our original
array. Thus, cylindric partitions of shape $((m\K)^r)/\0/\K$ with entries
between 0 and $a$ are in bijection with 
cylindric partitions of shape $((mr)^\K)/\0/r$ with entries
between 0 and $a$. (Note that $r$ and $\K$ are simply interchanged.)
Hence, their generating functions must be the same. So, from this
observation and Theorem~\TB\ we obtain the identity
$$\align &\sum _{k_1+\dots+k_r=0} ^{}q^{(r+\K)r\sum _{i=1} ^{r}k_i^2/2
+(r+\K)\sum_{i=1} ^{r}ik_i} \\
&\hskip1cm\det_{1\le s,t\le r}\(q^{t(t-s-rk_s)}\bmatrix
a+m\K-\K k_s\\m\K-s+t-(r+\K)k_s \endbmatrix\)\\
&=\sum _{l_1+\dots+l_\K=0} ^{}q^{(r+\K)\K\sum _{i=1} ^{\K}l_i^2/2
+(r+\K)\sum_{i=1} ^{\K}il_i} \\
&\hskip1.5cm\det_{1\le s,t\le \K}\(q^{t(t-s-\K l_s)}\bmatrix
a+mr-r l_s\\mr-s+t-(r+\K)l_s \endbmatrix\)
\tag6.23
\endalign$$

Next we want to apply the same technique to $(\Al,\Be)$-cylindric
partitions of ``arbitrary" shape. What we shall finally obtain is an
extension of (6.23), namely Theorem~\TT\ below. Let $\La,\Mu$ be shapes
contained in 
the rectangle $(\K^r)$ (i.e., $\la_i\le \K$ and $\mu_i\le \K$ for
$i=1,2,\dots,r$). It is not difficult to see that the above procedure
gives a correspondence between $(\Al,\Be)$-cylindric partitions $\pi$ of
shape $\La+((m\K)^r)/\Mu/\K$ and $(\Be,\Al)$-cylindric partitions $\bar\pi$ of
shape $\La'+((mr)^\K)/\Mu'/r$. (The notation $\La+((m\K)^r)$ means the
vector $(\la_1+m\K,\la_2+m\K,\dots,\la_r+m\K)$, etc. As usual, $\La'$
denotes the partition conjugate to $\La$, cf\. \cite{\MacdAB, p.~2}.)
Note that here $\Al$ and $\Be$ are interchanged, as are $r$ and $\K$,
while
$\La$ and $\Mu$ are replaced by their conjugates. In particular, the
relation between entries of $\pi$ and $\bar\pi$ is
$$\pi_{i,j}\quad \longleftrightarrow \quad \bar\pi_{j-\lfloor
(j-1)/\K\rfloor \K, i+\lfloor (j-1)/\K\rfloor r}\ .\tag6.24$$

Since we are dealing with $(\Al,\Be)$- and
$(\Be,\Al)$-cylindric partitions, we also have to consider
non-trivial bounds on the entries. This makes the situation a little
bit more delicate.

We shall see that $(\Al,\Be)$-cylindric partitions of shape
$\La+((m\K)^r)/\Mu/\K$ where the first entry in row $i$ is at most
$a-\sum _{k=1} ^{i-1}\be_k-\sum _{k=1} ^{\mu_i}\al_k$ and the last
entry in row $i$ is at least $-\sum _{k=1} ^{i-1}\be_k-\sum _{k=1}
^{\la_i-1} \al_k$ are (under suitable conditions) in bijection with
$(\Be,\Al)$-cylindric partitions of shape
$\La'+((mr)^\K)/\Mu'/r$ where the first entry in row $i$ is at most
$a-\sum _{k=1} ^{i-1}\al_k-\sum _{k=1} ^{\mu'_i}\be_k$ and the last
entry in row $i$ is at least $-\sum _{k=1} ^{i-1}\al_k-\sum _{k=1}
^{\la'_i-1} \be_k$. Observe that also with the bounds, 
$\Al$ and $\Be$, $r$ and $\K$ are interchanged, and
$\La$ and $\Mu$ are replaced by their conjugates.

Since we want to apply Theorem~\TC\ to compute the corresponding
generating functions, we have to check the relevant conditions. First,
we have to require $-d<\sum _{k=1} ^{r}\be_k-\sum _{k=1}
^{\K}\al_k<r$, and because of (4.6c) and (4.6d), $-1\le \sum _{k=1}
^{r}\be_k- \sum _{k=1} ^{\K}\al_k\le 1$. So, combining these conditions,
we assume that
$$\max\{-1,-d+1\}\le \sum _{k=1}
^{r}\be_k- \sum _{k=1} ^{\K}\al_k\le \min\{1,r-1\}.$$

Next, we investigate when an $(\Al,\Be)$-cylindric partition $\pi$
of shape $\La+((m\K)^r)/\Mu/\K$ with ``upper row bound"
$a-\sum _{k=1} ^{i-1}\be_k-\sum _{k=1} ^{\mu_i}\al_k$ does indeed yield
by the above procedure  a
$(\Be,\Al)$-cylindric partition $\bar\pi$ of shape
$\La'+((mr)^\K)/\Mu'/r$ with ``upper row bound"
$a-\sum _{k=1} ^{i-1}\al_k-\sum _{k=1} ^{\mu'_i}\be_k$, and vice
versa. So let us take an
$(\Al,\Be)$-cylindric partition $\pi$
of shape $\La+((m\K)^r)/\Mu/\K$ where the first part in row $i$ is at most
$a-\sum _{k=1} ^{i-1}\be_k-\sum _{k=1} ^{\mu_i}\al_k$. The latter
assumption means
$$\pi_{i,\mu_i+1}\le a-\sum _{k=1} ^{i-1}\be_k-\sum _{k=1}
^{\mu_i}\al_k.$$
By (4.5a) we have for $j\ge\mu_i+1$ that $\pi_{i,\mu_i+1}\ge
\pi_{i,j}+\sum _{k=\mu_i+1} ^{j-1}\al_k$, and therefore
$$\pi_{i,j}\le a-\sum _{k=1} ^{i-1}\be_k-\sum _{k=1} ^{j-1}\al_k.\tag6.25$$
In particular, for $\mu_i+1\le j\le \mu_{i-1}$ the entry $\pi_{i,j}$
lies at the (upper) border of the shape, hence after conjugation
(as illustrated in Figure~5) it lies at the (left) border of the new
shape. By (6.24) $\pi_{i,j}$ corresponds to $\bar\pi_{j,i}$, and since
this entry is at the left border, $\bar\pi_{j,i}=\bar\pi_{j,\mu'_{j}+1}$.
Thus (6.25) reads
$$\bar\pi_{j,\mu'_{j}+1}\le a-\sum _{k=1}
^{j-1}\al_k-\sum _{k=1} ^{\mu'_j}\be_k.\tag6.26$$
This holds for all $j$ with $\mu_i+1\le j\le\mu_{i-1}$, for some $i$.
Certainly there exists such an $i$ for $\mu_r+1\le j\le \K$. Hence
(6.26) would hold for all $j$ if $\mu_r=0$. However, if $\mu_r>0$
(which is the same as $\mu'_1=r$), then we have to add a further
restriction, namely $\sum _{k=1} ^{r}\be_k\le\sum _{k=1} ^{\K}\al_k$.
For if $1\le j\le \mu_r$, then by (6.25) we have the restriction for
$\pi_{1,\K+j}$ (which by (6.24) corresponds to $\bar\pi_{j,r+1}$, which
in turn is $\bar\pi_{j,\mu_j'+1}$) that 
$$\pi_{1,\K+j}\le a-\sum _{k=1} ^{\K+j-1}\al_k\le a-\sum _{k=1}
^{r}\be_k-\sum _{k=1} ^{j-1}\al_k.$$
This is equivalent to (6.26) in this case.

Since we must also be able to go in the reverse direction, by symmetry
we must assume that $\sum _{k=1} ^{r}\be_k\ge \sum _{k=1}
^{\K}\al_k$ if $\mu_1=\K$.

Similar considerations show that, given an $(\Al,\Be)$-cylindric
partition 
of shape $\La+((m\K)^r)/\Mu/\K$ with ``lower row bound"
$-\sum _{k=1} ^{i-1}\be_k-\sum _{k=1} ^{\la_i-1}\al_k$, by the conjugation
procedure we obtain a
$(\Be,\Al)$-cylindric partition of shape
$\La'+((mr)^\K)/\Mu'/r$ with ``lower row bound"
$-\sum _{k=1} ^{i-1}\al_k-\sum _{k=1} ^{\la'_i-1}\be_k$, and vice
versa, provided that $\sum _{k=1} ^{r}\be_k\le \sum _{k=1} ^{\K}\al_k$
if $\la_1<\K$ and $\sum _{k=1} ^{r}\be_k\ge \sum _{k=1} ^{\K}\al_k$
if $\la'_1<r$.

Now that we have collected all the necessary conditions, we may
substitute into Theorem ~\TC\ our two equivalent choices; first $r\to
r$, 
$\La\to\La+((m\K)^r)$, $\Mu\to\Mu$, $\K\to \K$,
$a_i\to a-\sum _{k=1} ^{i-1}\be_k-\sum _{k=1} ^{\mu_i}\al_k$, 
$b_i\to -\sum _{k=1} ^{i-1}\be_k-\sum _{k=1} ^{\la_i-1}\al_k$; and
second, $r\to\K$,
$\La\to\La'+((mr)^\K)$, $\Mu\to\Mu'$, $\K\to r$,
$a_i\to a-\sum _{k=1} ^{i-1}\al_k-\sum _{k=1} ^{\mu'_i}\be_k$, 
$b_i\to -\sum _{k=1} ^{i-1}\al_k-\sum _{k=1} ^{\la'_i-1}\be_k$. We may
then equate the two resulting expressions for the generating function
for these cylindric partitions. After cancellation, and after 
replacing $a$ by $a+m\sum _{k=1} ^{\K}\al_k$, and then $\sum _{k=1}
^{r}\be_k- \sum _{k=1} ^{\K}\al_k$ by $\ep$, we obtain the following
identity. 

\proclaim{Theorem \TT} Suppose that
$\max\{-1,-\K+1\}\le\ep\le\min\{1,r-1\}$ and let $\La$ and $\Mu$ be
shapes contained in the rectangle $(\K^r)$. If
$\la_1<\K$ or $\mu'_1=r$ then in addition we assume $\ep\le0$, and if
$\la_1'<r$ or $\mu_1=\K$ then in addition we assume $\ep\ge0$. Then

\vbox{
$$\align &\sum _{k_1+\dots+k_r=0} ^{}q^{(r+\K)(r-\ep)\sum _{i=1} ^{r}k_i^2/2
+(r+\K)\sum_{i=1} ^{r}ik_i} \\
&\hskip1cm\det_{1\le s,t\le r}\(q^{(\mu_t-t)(s-t+(r-\ep)k_s)}\bmatrix
a+m\K+\la_s-\mu_t-(\K+\ep) k_s\\m\K+\la_s-\mu_t-s+t-(r+\K)k_s \endbmatrix\)\\
&=q^{\big(r\K\binom {m+1}2-m\sum _{k=1} ^{r}\mu_k\big)\ep}\\
&\sum _{l_1+\dots+l_\K=0} ^{}q^{(r+\K)(\K+\ep)\sum _{i=1} ^{\K}l_i^2/2
+(r+\K)\sum_{i=1} ^{\K}il_i} \\
&\hskip1.5cm\det_{1\le s,t\le \K}\(q^{(\mu'_t-t)(s-t+(\K+\ep) l_s)}\bmatrix
a+mr-m\ep+\la'_s-\mu'_t-(r-\ep) l_s\\
mr+\la'_s-\mu'_t-s+t-(r+\K)l_s \endbmatrix\)\\
\tag6.27
\endalign$$
\line{\hfil\hbox{\qed}\quad \quad }
}
\endproclaim
Note that for $\K=1$ we recover Theorem~\TI. Moreover, identity (6.23)
is the special case $\La=\0$, $\Mu=\0$, $\ep=0$  of (6.27).

As in the proofs of Theorems~\TI--\TQ, one could also try other row
bounds here. However, one is much more restricted in this more general
case. Still, a choice of row bounds  different from 
that in the proof of Theorem~\TT\ will be used in the proof of
Theorem~\TTC.

\bigskip
To conclude this section, we apply a different trick to obtain a
different type of identity. Consider $(\0,(0,0,\dots,-1))$-cylindric
partitions of shape $(c^r)/\0/0$ with entries between $0$ and $a$.
Because of our choice of $\Be$ and because of (4.5b) and (4.5c), each
column of such a 
$(\0,(0,0,\dots,-1))$-cylindric partition contains either only one integer
all over or two consecutive integers. In the latter case, the larger
integer occupies the top entries in this column, the lesser integer
the bottom entries. Let $k_j$ denote the number of $j$'s in the first
row, $j=a,a-1,\dots,1,0$. Then it is not difficult to see that
a $(\0,(0,0,\dots,-1))$-cylindric
partition under consideration can be visualized as in Figure~6.

\vskip10pt
\vbox{
$$
\Einheit.3cm
\Pfad(0,0),111111111111111111111111111111\endPfad
\Pfad(0,8),111111111111111111111111111111\endPfad
\Pfad(0,0),22222222\endPfad
\SPfad(5,0),22222222\endSPfad
\SPfad(10,0),22222222\endSPfad
\SPfad(20,0),22222222\endSPfad
\SPfad(25,0),22222222\endSPfad
\Pfad(30,0),22222222\endPfad
\SPfad(0,0),2212112212212\endSPfad
\SPfad(5,0),2122112212212\endSPfad
\SPfad(20,0),2212122112212\endSPfad
\Label\o{a}(2,6)
\Label\u{a-1}(3,2)
\Label\o{a-1}(7,6)
\Label\u{a-2}(8,2)
\Label\o{1}(22,6)
\Label\u{0}(23,2)
\Label\r{0}(26,4)
\Label\o{\hskip.25cm\oversetbrace k_{a}\to{\hbox{\hskip1.5cm}}}(2,8)
\Label\o{\hskip.25cm\oversetbrace k_{a-1}\to{\hbox{\hskip1.5cm}}}(7,8)
\Label\o{\hskip.25cm\oversetbrace k_{1}\to{\hbox{\hskip1.5cm}}}(22,8)
\Label\o{\hskip.25cm\oversetbrace k_{0}\to{\hbox{\hskip1.5cm}}}(27,8)
\Label\o{\hskip.2cm\hbox to3cm{\leaderfill}}(15,8)
\hskip9cm$$
\centerline{\eightpoint Figure 6}
}
\vskip10pt
\noindent
In this figure each of the
displayed regions contains only entries equal to the label assigned to
that region. Each region is a Ferrers diagram (cf\.
\cite{\SagaAL, p.~54}), which is
equivalent to a linear partition. Hence it
is easy to write down an expression for these cylindric partitions by
again appealing to the standard partition theorem \cite{\AndrAF,
Theorem~3.1}, 
$$\sum _{k_0+k_1+\dots+k_a=c} ^{}q^{\sum _{i=1}
^{a}(rk_i(i-1)+k_i)}\prod _{i=1} ^{r}\bmatrix
k_i+r-1\\r-1\endbmatrix.$$
Equating this expression with the one obtained from Theorem~\TC\ (with
$\al_i=0$, $\be_i=0$ for $1\le i\le r-1$, $\be_r=-1$, $\la_i=c$,
$\mu_i=0$, $\K=0$, $a_i=a$, $b_i=0$) we obtain another identity.

\proclaim{Theorem \TU}There holds

\vbox{
$$\multline \sum _{k_1+\dots+k_r=0} ^{}q^{\binom {r+1}2 \sum _{i=1}
^{r}k_i^2+r\sum _{i=1} ^{r}ik_i}
\det_{1\le s,t\le r}\(q^{t(t-s-(r+1)k_s)}\bmatrix
a+c+k_s\\c-s+t-rk_s\endbmatrix\)\\
=\sum _{l_1+\dots+l_a\le c} ^{}q^{\sum _{i=1}
^{a}(rl_i(i-1)+l_i)}\prod _{i=1} ^{r}\bmatrix
l_i+r-1\\r-1\endbmatrix.
\endmultline\tag6.28$$
\line{\hfil\hbox{\qed}\quad \quad }
}
\endproclaim

\subhead 7. Identities II: Summations for $\widetilde A_r$ basic
hypergeometric series\endsubhead
In this section we consider special cases of Theorems~\TI--\TS\ of
section~6 in which the determinants can be evaluated by means of
(5.1), (5.2), or (5.3). We obtain summations for multiple basic
hypergeometric series of a particular type which we now
explain. For an introduction to (one-dimensional) basic
hypergeometric series the reader should consult the book of Gasper and
Rahman \cite{\GaRaAA}.

A {\it basic hypergeometric series for the root system $A_{r-1}$}
(also called a {\it basic hypergeometric series for $U(r)$}) is a
series of the form
$$\sum _{k_1,\dots,k_r=-\infty} ^{\infty}\(\prod _{1\le i<j\le r} ^{}\frac
{1-q^{k_i-k_j}x_i/x_j} {1-x_i/x_j}\cdot \big(\text{basic
hypergeometric stuff}\big)\).\tag7.1$$
The reader who is familiar with the root system of type $A_{r-1}$
(cf\. \cite{\HumpAC} for an introduction to root systems) will
have no problem in associating the positive roots of $A_{r-1}$ with the
terms $x_i/x_j$, $1\le i<j\le r$.

By a {\it basic hypergeometric series for the \rm(\it affine\/\rm) \it
root system $\widetilde A_{r-1}$} we mean a series of the form
$$\sum _{k_1+\dots+k_r=0} ^{}\(\prod _{1\le i<j\le r} ^{}\frac
{1-q^{k_i-k_j}x_i/x_j} {1-x_i/x_j}\cdot \big(\text{basic
hypergeometric stuff}\big)\).\tag7.2$$
Note that the difference lies in the range of summation. Such a
series is usually called a {\it basic hypergeometric
series for $\SU(r)$\/}. However, in our context (in particular cf\.
Remark~(1) after the proof of Proposition~\TA) it is more natural to
call it a basic hypergeometric series 
for $\widetilde A_{r-1}$, or simply an {\it $\widetilde A_{r-1}$ basic
hypergeometric series\/}. For work on this subject (mainly undertaken
by Milne and Gustafson) see \cite{\MilnAG, \LiMiAA, \GustAA} 
and the references cited therein.

\medskip
For our first summation we let $c\to\infty$ in (6.2). The resulting
determinants on the left-hand side were already evaluated in the proof
of Theorem~\TF\ (set $\K=1$). Thus the left-hand side equals (5.6) with
$\K=1$. This gives the following theorem.

\proclaim{Theorem \TV} Let $a\ge0$. Then 

\vbox{
$$\sum _{k_1+\dots+k_r=0} ^{}q^{\binom {r+1}2\sum _{i=1} ^{r}k_i^2+
\sum _{i=1} ^{r}ik_i} \frac {\prod _{1\le i<j\le r}
^{}[rk_j+j-rk_i-i]} {\prod _{i=1} ^{r}[a+rk_i+i-1]!} 
=\frac {1} {[a]!}.
\tag7.3$$
\line{\hfil\hbox{\qed}\quad \quad }
}
\endproclaim
Before continuing, we want to point out that this is indeed a series
of type (7.2). Namely, in (7.2) replace $q$ by $q^{-r}$ and $x_i$ by
$q^{-i}$ to obtain
$$\sum _{k_1+\dots+k_r=0} ^{}
\frac {\prod _{1\le i<j\le r}
^{}[rk_j+j-rk_i-i]} {\prod _{1\le i<j\le r}
^{}[j-i]} \cdot \big(\text{other stuff}\big).
$$
The numerator of the above product is clearly in (7.3). The
denominator, which equals $\prod _{j=1} ^{r}[j-1]!$ is absorbed in the
denominator of the summand in (7.3). This remark also shows that the
expressions (5.4)--(5.6) that we gave for the generating functions of
special sets of cylindric partitions are all $\widetilde A_{r-1}$
basic hypergeometric series. Moreover, (5.7) is an $\widetilde A_{r-1}$
(ordinary) hypergeometric series.

If one wants to know if an identity like (7.3) is already known one
has to consult the papers of Milne and Gustafson. In \cite{\MilnAG}
one finds Theorem~1.9, which in our notation can be rewritten as
$$\multline
\sum _{k_1+\dots+k_r=S} ^{}(-1)^{(r-1)S}
q^{\binom {r+1}2\sum _{i=1} ^{r}k_i^2+
\sum _{i=1} ^{r}ik_i} \frac {\prod _{1\le i<j\le r}
^{}[rk_j+j-rk_i-i]} {\prod _{i=1} ^{r}[rk_i+i-1]!} \\
=\frac {q^{(r+1)\binom {S+1}2}} {[S]!}, 
\endmultline\tag7.4$$
provided $S\ge0$. This identity is strikingly similar to
(7.3). In fact, it turns out that (7.3) and (7.4) are equivalent. This
is seen by what we call the ``rotation trick". Let $S$ be some fixed
integer. Division of $S$ by $r$ gives a unique
representation $S=Qr+R$ where $Q,R$ are integers with $0\le R\le
r$. Then in (7.3) replace $k_1$ by $k_{1+R}-Q$, \dots, $k_{r-R}$ by
$k_r-Q$, $k_{r-R+1}$ by $k_1-Q-1$, \dots, $k_r$ by $k_R-Q-1$. So the
effect is a rotation of the summation indices, combined with a certain
shift. If we rewrite (7.3) after these replacements and finally replace
$a$ by $a+S$, we obtain a summation that covers both (7.3) and (7.4).

\proclaim{Theorem \TW} Suppose that $a+S\ge0$. Then 

\vbox{
$$\multline
\sum _{k_1+\dots+k_r=S} ^{}(-1)^{(r-1)S}
q^{\binom {r+1}2\sum _{i=1} ^{r}k_i^2+
\sum _{i=1} ^{r}ik_i} \frac {\prod _{1\le i<j\le r}
^{}[rk_j+j-rk_i-i]} {\prod _{i=1} ^{r}[a+rk_i+i-1]!} \\
=\frac {q^{(r+1)\binom {S+1}2}} {[a+S]!}, 
\endmultline\tag7.5$$
\line{\hfil\hbox{\qed}\quad \quad }
}
\endproclaim
Setting $S=0$ in (7.5) yields (7.3)  and setting $a=0$  yields (7.4).
Of course, the rotation trick could also be used to derive (7.5)
from (7.4). So (7.3) and (7.4) are indeed
equivalent. This is remarkable, since Milne used a great deal of machinery
to prove (7.4) (in fact a large part of his paper \cite{\MilnAG} is devoted
to the proof of this identity), while our proof of (7.3) is rather
elementary. The ingredients are nonintersecting lattice paths, the simple
bijection between cylindric partitions and linear partitions, and the
determinant lemma (5.2).

In addition, our method also produces new identities. Recall that (6.2) is
just a special case of (6.6). Now, in (6.6) set $m=r$, $n=0$, $\ep=1$,
and again let $c\to\infty$. Then a similar computation gives a
summation related to (7.3) that appears to be new.

\proclaim{Theorem \TX}Let $a\ge0$. Then 

\NoBlackBoxes
\vbox{
$$\sum _{k_1+\dots+k_r=0} ^{}q^{(r+1)(r-1)\sum _{i=1} ^{r}k_i^2/2+
2\sum _{i=1} ^{r}ik_i} \frac {\prod _{1\le i<j\le r}
^{}[(r-1)k_j+j-(r-1)k_i-i]} {\prod _{i=1} ^{r}[a+(r-1)k_i+i-1]!} 
=0.
\tag7.6$$
\line{\hfil\hbox{\qed}\quad \quad }
}
\endproclaim
\BlackBoxes
See Theorem~\TTB\ for a generalization of this identity.

The next theorem results from Theorem~\TM\ by letting $a\to\infty$.

\proclaim{Theorem \TY} Let $c\ge0$, $0\le n<r$, and $0\le
\ep<r$. Then 
$$\multline
\sum _{k_1+\dots+k_r=0} ^{}q^{(r+1)(r-\ep)\sum _{i=1} ^{r}k_i^2/2+
(\ep+1)\sum _{i=1} ^{r}ik_i+(r-\ep)\sum _{i=1} ^{n}k_i} \\
\frac {\prod _{1\le i<j\le r} ^{}
[\chi(i\le n)+(r+1)k_i-i-\chi(j\le n)-(r+1)k_j+j]} {\prod _{i=1}
^{r}[c-\chi(i\le n)-(r+1)k_i+i-1]!} \\
=\frac {q^{\ep(r\binom c2-n(c-1))}} {[rc-n]!}.
\endmultline\tag7.7$$
\endproclaim

\demo{Proof}To prove (7.7) for $n\le\ep<r$ we set $m=r$ and replace
$c$ by $c-1$ in
Theorem~\TM, and then let $a\to\infty$. 
With these replacements, and by changing $k_s$ into $k_t$ (see
Remark~(1) after the proof of Theorem~\TC), (6.12) reads
$$\multline \sum _{k_1+\dots +k_r=0} ^{}q^{(r+1)(r-\ep)\sum _{i=1}
^{r}k_i^2/2-(r-\ep)\sum _{i=1} ^{r}ik_i +(r-\ep)\sum _{i=1} ^{n}k_i}\\
\det_{1\le s,t\le r}\big(1/
[c-\chi(t\le n)-s+t-(r+1)k_t]!\big)
=q^{\ep(r\binom c2-n(c-1))+c\binom r2+\binom {n+1}2-nr }
\frac {1} {[rc-n]!}.
\endmultline$$
Now in the determinant reverse the order of both rows and columns
(i.e., replace $s$ by $r+1-s$ and $t$ by $r+1-t$) and then evaluate it
by means of (5.1) with $L_i=c-\chi(r+1-i\le n)-i-(r+1)k_{r+1-i}$ and
$A\to\infty$. A few simplifications then lead to (7.7) in the case
$n\le \ep<r$.

To establish (7.7) for $0\le\ep<n$ also, we have to examine where the
condition $n\le\ep$ is needed in the proof of (6.12). Recall that
$\ep$ actually stands for $\sum _{k=1} ^{r}\be_k-\al$. So $n\le\ep$ is
equivalent to $\sum _{k=1} ^{r}\be_k\ge\al+n$. It is not elaborated in
Case~5, though it is in Case~2 (see the passage around (6.7)), that $\sum
_{k=1} ^{r}\be_k\ge\al+n$  is needed so that the linear partitions
that are counted by the right-hand side of (6.12) will automatically
``obey the upper row bounds $a_i=a+\sum _{k=1}
^{i-1}(1-\be_k)-\al\cdot \chi(i\le n)$" when interpreted as
$(\Al,\Be)$-cylindric partitions. Therefore, if we drop the
condition $\sum _{k=1} ^{r}\be_k\ge \al+n$ there might be a difference
between the two sides of (6.12). But this difference would be due to
linear partitions that (as $(\Al,\Be)$-cylindric partitions) do {\it
not\/} obey these row bounds. Therefore, these linear partitions must
contain at least one part that violates such an upper row bound. But as
$a$ increases to infinity, so does each upper row bound, so in the limit
as $a\to\infty$ the difference between the two sides of (6.12) vanishes, for
all $\ep$ with
$0\le\ep<r$. 
\quad \quad \qed
\enddemo
Let us have a closer look at (7.7). For convenience, write $y_i$ for
$\chi(i\le n)-i$. Then, for $r=2$, the series in (7.7) reduces to a
single sum that can be written in basic hypergeometric notation (cf\.
section~2) as
$$\multline \frac {(1-q^{y_1-y_2})}
{(q;q)_{c-y_1-1}\,(q;q)_{c-y_2-1}}\\
\sum _{k=-\infty} ^{\infty}
\frac {(q^{3+(y_1-y_2)/2},-q^{3+(y_1-y_2)/2}, q^{1+y_1-c},
q^{2+y_1-c}, q^{3+y_1-c} ;q^3)_k} {(q^{(y_1-y_2)/2},-q^{(y_1-y_2)/2},
q^{2-y_2+c}, q^{1-y_2+c}, q^{y_2+c} ;q^3)_k}\\
(-1)^kq^{(3-6\ep)\binom k2}\(q^{-3-3\ep-(1+\ep)y_1+(\ep-2)y_2+3c}\)^k.
\endmultline$$
So we see that (7.7) for $r=2$ is a special case of
Bailey's very well-poised $_6\psi_6$-summation \cite{\BailAD, (4.7);
\GaRaAA, (5.3.1)} (cf\. \cite{\GaRaAA, (5.1.5)} for notation)
$$\multline _6\psi_6\!\left[\matrix q\sqrt a,-q\sqrt a,b,c,d,e\\
\sqrt a,-\sqrt a,aq/b, aq/c, aq/d, aq/e\endmatrix ;
q,qa^2/bcde\right]\\
=\frac {(qa,q,q/a, qa/bc, qa/bd, qa/be, qa/cd, qa/ce, qa/de;q)_\infty} {(qa/b,
qa/c, qa/d, qa/e, q/b, q/c, q/d, q/e, qa^2/bcde;q)_\infty}
\endmultline\tag7.8$$
(take $a=q^{y_1-y_2}$, $b=q^{1+y_1-c}$, $c=q^{2+y_1-c}$,
$d=q^{3+y_1-c}$, and if $\ep=0$ let $e\to\infty$, while if $\ep=1$ let
$e\to0$.) Therefore, for general $r$, (7.7) should come from
Gustafson's 
$\SU(r)$-extension of Bailey's $_6\psi_6$-sum 
\cite{\GustAF, Theorem~1.15; \GustAA, Theorem~4.1},
$$\multline \sum _{k_1+\dots+k_r=0} ^{}q^{\sum _{i=1} ^{r}ik_i}\prod
_{1\le i<j\le r} ^{}\frac {1-q^{k_i-k_j}x_i/x_j} {1-x_i/x_j} \prod
_{i,j=1} ^{r}\frac {(a_ix_j;q)_{k_j}} {(b_ix_j;q)_{k_j}}\\
=\frac {(q;q)_\infty^{r-1}\,\prod _{i,j=1} ^{r}(b_i/a_j;q)_\infty
\prod _{1\le i<j\le r}
^{}(qx_i/x_j;q)_\infty\,(qx_j/x_i;q)_\infty} {(q^{1-r}\prod
_{i=1} ^{r} (b_i/a_i);q)_\infty\prod _{i,j=1}
^{r}(q/a_ix_j;q)_\infty\,(b_ix_j;q)_\infty }\\
\tsize\cdot \big(q/\prod _{i=1} ^{r}a_ix_i;q\big)_\infty\,\big(q^{1-r}
\prod _{i=1} ^{r} b_ix_i;q\big)_\infty.
\endmultline\tag7.9$$
However, it apparently does not. In basic hypergeometric notation the
series in (7.7) would read
$$\multline \frac {\prod _{1\le i<j\le r} ^{}(1-q^{\chi(i\le
n)-i-\chi(j\le n)+j})} {\prod _{i=1} ^{r}(q;q)_{c-\chi(i\le n)+i-1}}\\
\hskip-2cm\sum _{k_1+\dots+k_r=0} ^{}q^{-(r+1)(\ep+1)\sum _{i=1} ^{r}k_i^2/2
+(r+\ep+2)\sum _{i=1} ^{r} ik_i-(\ep+1)\sum _{i=1} ^{n}k_i}\\
\prod _{1\le i<j\le r} ^{}\frac {1-q^{(r+1)k_i-(r+1)k_j+\chi(i\le
n)-i-\chi(j\le n)+j}} {1-q^{\chi(i\le
n)-i-\chi(j\le n)+j}}
\prod _{i=1} ^{r}(q^{1-c+\chi(i\le n)-i};q)_{(r+1)k_i}.
\endmultline$$
\NoBlackBoxes
In order to generate the summand of
the above series from (7.9), one would have to replace $q$ by $q^{r+1}$,
$x_i$ by $q^{\chi(i\le n)-i}$, and $a_i$ by $q^{-c+i}$. This is
fine with the product $\prod _{1\le i<j\le r} ^{}$ in the summand. But it
doesn't quite yield the product\linebreak
$\prod _{i=1} ^{r}(q^{1-c+\chi(i\le n)-i};q)_{(r+1)k_i}$ 
in the numerator that is needed, not to mention the powers of $q$, and at this
point one is stuck. So there is some mystery as to the relation between
(7.7) and (7.9). 
It seems that one has to look for an identity involving a sum with
``bigger" expressions, very likely for an $\SU(r)$-extension of an
$_8\psi_8$-identity, such as \cite{\GaRaAA, (5.6.2) or Ex.~5.12}, for
example.
%
(See also item~(8) of the last section.)

\BlackBoxes
\medskip
Finally we consider (6.17)=(6.18). We may directly apply (5.2) (with
$A=a+c$, $L_i=c-i-rk_i$) to evaluate the determinants on the left-hand
side. Thus we obtain the identity 
$$\multline \sum _{k_1+\dots+k_r=0} ^{}q^{r^2\sum _{i=1}
^{r}k_i^2/2}\frac {\prod _{1\le i<j\le r} ^{}[rk_j+j-rk_i-i]} {\prod
_{i=1} ^{r} [c-rk_i+r-i]!}\\
\frac {\prod _{i=1} ^{r}[a+c+i-1]!} {\prod _{i=1} ^{r}[a+rk_i+i-1]!}
=\bmatrix a+c\\a\endbmatrix_{q^r}\ .
\endmultline\tag7.10$$
The rotation trick works here also: replace 
$k_1$ by $k_{1+R}-Q$, \dots, $k_{r-R}$ by
$k_r-Q$, $k_{r-R+1}$ by $k_1-Q-1$, \dots, $k_r$ by $k_R-Q-1$, and then
replace
$a$ by $a+S$, and $c$ by $c-S$, and rewrite. This leads to
$$\multline \sum _{k_1+\dots+k_r=S} ^{}(-1)^{(r-1)S}q^{r^2\sum _{i=1}
^{r}k_i^2/2}\frac {\prod _{1\le i<j\le r} ^{}[rk_j+j-rk_i-i]} {\prod
_{i=1} ^{r} [c-rk_i+r-i]!}\\
\frac {\prod _{i=1} ^{r}[a+c+i-1]!} {\prod _{i=1} ^{r}[a+rk_i+i-1]!}
=q^{rS^2/2}\bmatrix a+c\\a+S\endbmatrix_{q^r}\ .
\endmultline\tag7.11$$
This identity is less exciting than (7.5) or (7.7) since it is indeed just
a special case of Gustafson's $_6\psi_6$-summation (7.9) for $\SU(r)$
(replace $q$ by $q^r$, $x_i$ by $q^{-i-rS\cdot\chi(i=r)}$, $a_i$ by
$q^{-a+i}$, $b_i$ by $q^{c+r+i}$).
%

\subhead 8. Identities III: Transformation formulas\endsubhead
In this section we proceed in the spirit of the previous section, but
apply determinant evaluations to special cases of Theorems~\TT\ and
\TU. Here we obtain transformation formulas between $\widetilde A_r$
basic hypergeometric series of different dimensions, one more summation
formula for $\widetilde A_r$
basic hypergeometric series, and a transformation formula between
an $\widetilde A_r$
basic hypergeometric series and an ordinary multiple basic
hypergeometric series. All the identities in this section are new. In
particular, transformation formulas between series of different
dimensions do not seem to have appeared before.

We begin with setting $\ep=0$ and $\Mu=\0$ in Theorem~\TT, and then
let $m\to\infty$. The resulting determinants (on both sides) were
already evaluated in the proof of Theorem~\TF. We thus obtain the
following transformation formula.

\proclaim{Theorem \TZ} For $a\ge0$ we have

\vbox{
$$\multline \sum _{k_1+\dots+k_r=0} ^{}q^{r(r+\K)\sum _{i=1} ^{r}k_i^2/2+
\K \sum _{i=1} ^{r}ik_i} \frac {\prod _{1\le i<j\le r}
^{}[rk_j+j-rk_i-i]} {\prod _{i=1} ^{r}[a+rk_i+i-1]!} \\
=\sum _{l_1+\dots+l_\K=0} ^{}q^{\K(r+\K)\sum _{i=1} ^{\K}l_i^2/2+
r \sum _{i=1} ^{\K}il_i} \frac {\prod _{1\le i<j\le \K}
^{}[\K l_j+j-\K l_i-i]} {\prod _{i=1} ^{\K}[a+\K l_i+i-1]!} .
\endmultline\tag8.1$$
\line{\hfil\hbox{\qed}\quad \quad }
}
\endproclaim
Once more, the rotation trick can be applied (see the paragraph before
Theorem~\TW), this time for both sums simultaneously. This gives rise
to the following generalization of Theorem~\TZ.
\proclaim{Theorem \TTA} For $a\ge0$ we have

\vbox{
$$\multline \sum _{k_1+\dots+k_r=S} ^{}
(-1)^{(r-1)S}q^{r(r+\K)\sum _{i=1} ^{r}k_i^2/2+
\K \sum _{i=1} ^{r}ik_i-\K\binom {S+1}2-rS(S+\K)/2} \\
\times\frac {\prod _{1\le i<j\le r}
^{}[rk_j+j-rk_i-i]} {\prod _{i=1} ^{r}[a-S+rk_i+i-1]!} \\
=\sum _{l_1+\dots+l_\K=T} ^{}
(-1)^{(\K-1)T}q^{\K(r+\K)\sum _{i=1} ^{\K}l_i^2/2+
r \sum _{i=1} ^{\K}il_i-r\binom {T+1}2-\K T(T+r)/2}\\
\times \frac {\prod _{1\le i<j\le \K}
^{}[\K l_j+j-\K l_i-i]} {\prod _{i=1} ^{\K}[a-T+\K l_i+i-1]!} .
\endmultline\tag8.2$$
\line{\hfil\hbox{\qed}\quad \quad }
}
\endproclaim
Note that by setting $\K=1$ and $T=0$ in Theorem~\TTA\ we recover
Theorem~\TW\ (and similarly, by setting $\K=1$ in Theorem~\TZ\ we
recover Theorem~\TV). 

We might also start with setting $\ep=1$ and $\Mu=\0$ in Theorem~\TT,
and then let $m\to\infty$. Note that because of the assumptions of
Theorem~\TT, the choice $\ep=1$ forces $r$ to be at least 2. In this
case the right-hand side vanishes, while the determinants on the
left-hand side can be evaluated like those in the proof of
Theorem~\TF. This leads to the following summation.
\proclaim{Theorem \TTB} For $r\ge2$ and $a\ge0$ we have

\vbox{
$$\multline 
\sum _{k_1+\dots+k_r=0} ^{}q^{(r+\K)(r-1)\sum _{i=1} ^{r}k_i^2/2+
(\K+1) \sum _{i=1} ^{r}ik_i} \\
\times\frac {\prod _{1\le i<j\le r}
^{}[(r-1)k_j+j-(r-1)k_i-i]} {\prod _{i=1} ^{r}[a+(r-1)k_i+i-1]!} 
=0.
\endmultline
\tag8.3$$
\line{\hfil\hbox{\qed}\quad \quad }
}
\endproclaim
This identity generalizes (7.6). The fact that $\K$ appears only in
the exponent of $q$ implies that in the sum we must actually have
term-wise cancellation.

\medskip
A different type of transformation formula is obtained by replacing $m$
by $m-1$ and $\la_i$ by $\K$ in Theorem~\TT, and then letting $a\to
\infty$. The resulting determinants (on both sides) have already been
evaluated in the proof of Theorem~\TE. This proves the next theorem
when $\max\{-1,-\K+1\}\le \ep\le \min\{1,r-1\}$. It requires a
different argument to show the theorem for $-\K<\ep<r$.
\proclaim{Theorem \TTC} Let $-\K<\ep<r$ and let
$\Mu$ be a shape contained in the rectangle $(\K^r)$. Then 

\vbox{
$$\multline \sum _{k_1+\dots+k_r=0} ^{}q^{(r+\K)(r-\ep)\sum _{i=1} ^{r}k_i^2/2+
(\K+\ep) \sum _{i=1} ^{r}ik_i+(r-\ep)\sum _{i=1} ^{r}\mu_ik_i}\\
\times \frac {\prod _{1\le i<j\le r}
^{}[\mu_i+(r+\K)k_i-i-\mu_j-(r+\K)k_j+j]} {\prod _{i=1}
^{r}[m\K-\mu_i-(r+\K)k_i+i-1]!} \\ 
=q^{\big(rd\binom m2-(m-1)\sum _{i=1} ^{r}\mu_i\big)\ep}
\sum _{l_1+\dots+l_\K=0} ^{}q^{(r +\K)(\K +\ep)\sum _{i=1} ^{\K }l_i^2/2+
(r-\ep) \sum _{i=1} ^{\K }il_i+(\K +\ep)\sum _{i=1} ^{\K }\mu'_il_i}\\
\times \frac {\prod _{1\le i<j\le \K }
^{}[\mu'_i+(r+\K)l_i-i-\mu'_j-(r+\K)l_j+j]} {\prod _{i=1}
^{\K }[mr-\mu'_i-(r+\K)l_i+i-1]!} .
\endmultline\tag8.4$$
\line{\hfil\hbox{\qed}\quad \quad }
}
\endproclaim

\demo{Proof}Consider $(\Al,\Be)$-cylindric partitions of shape
$((m\K)^r)/\Mu/\K$ where the last entry in row $i$ is at least $-\sum
_{k=i} ^{r-1}(1-\be_k)$ and where there is no upper bound on the
entries. By the cut-and-paste procedure that led to a proof of
Theorem~\TT\ and is illustrated in Figure~5, we see that these
$(\Al,\Be)$-cylindric partitions are in bijection with
$(\Be,\Al)$-cylindric partitions of shape $((mr)^\K)/\Mu'/r$ where the
last entry in row $i$ is at least $-\sum _{k=i} ^{\K-1}(1-\al_k)$ and
where there is again no upper bound on the entries. Hence, the
generating functions for these two sets of cylindric partitions must
be the same. By Theorem~\TC\ with the replacements $r\to r$, $\la_i\to
m\K$, $\mu_i\to\mu_i$, $\K\to\K$, $a_i\to\infty$, $b_i\to-\sum _{k=i}
^{r-1}(1-\be_k)$, the generating function for the first set equals
$$\sum _{k_1+\dots+k_r=0} ^{}q^{N_2(\bold
k)}\det\big(1/[m\K-\mu_t-s+t-(r+\K)k_t]!\big),$$
where \eightpoint
$$\multline N_2(\bold k)=(r+\K)(r-B+A)\sum _{i=1}
^{r}k_i^2/2+(r-B+A)\sum _{i=1} ^{r} (\mu_i-i)k_i-(m\K r-\sum _{i=1}
^{r}\mu_i)(\be_r+\al_\K) \\
-m\K\binom r2+ A\Big(r\K\binom m2-(m-1)\sum _{i=1} ^{r}\mu_i\Big)
+\sum _{i=1} ^{r}(r-i)\mu_i \\
+m\K \sum _{k=1} ^{r}k\be_k +mr\sum _{k=1} ^{\K}k\al_k-\sum _{k=1} ^{r}
\be_k\sum _{i=1} ^{k}\mu_i -\sum _{k=1} ^{\K}\al_k\sum _{i=1} ^{k}\mu'_i,
\endmultline$$
\tenpoint
with $A:=\sum _{k=1} ^{\K}\al_k$ and $B:=\sum _{k=1} ^{r}\be_k$. In
the determinant we reverse the order of both row and columns (i.e., we
replace $s$ by $r+1-s$ and $t$ by $r+1-t$). Then the determinant can
be evaluated using (5.1) with $L_i=m\K-\mu_{r+1-i}-i-(r+\K)k_{r+1-i}$
and $A\to\infty$. After a few manipulations we obtain for the
generating function under consideration the expression
$$\sum _{k_1+\dots+k_r=0} ^{}q^{N_3(\bold k)}\frac {\prod _{1\le
i<j\le r} ^{} [\mu_i+(r+\K)k_i-i-\mu_j-(r+\K)k_j+j]} {\prod _{i=1}
^{r}[m\K-\mu_i-(r+\K)k_i+i-1]!},$$
where \eightpoint
$$\multline N_3(\bold k)=(r+\K)(r-B+A)\sum _{i=1}
^{r}k_i^2/2+(\K+B-A)\sum _{i=1} ^{r}ik_i+(r-B+A)\sum _{i=1} ^{r}
\mu_ik_i\\
-(m\K r-\sum _{i=1} 
^{r}\mu_i)(\be_r+\al_\K) 
+ A\Big(r\K\binom m2-(m-1)\sum _{i=1} ^{r}\mu_i\Big)\\
+m\K \sum _{k=1} ^{r}k\be_k +mr\sum _{k=1} ^{\K}k\al_k-\sum _{k=1} ^{r}
\be_k\sum _{i=1} ^{k}\mu_i -\sum _{k=1} ^{\K}\al_k\sum _{i=1} ^{k}\mu'_i.
\endmultline$$
\tenpoint
Obviously, the generating function for the second set is the same
expression but with $r$ and $\K$, $\Al$ and $\Be$ interchanged, and
with $\Mu$ replaced by $\Mu'$. If we equate these two expressions and
finally replace $B-A$ by $\ep$ we obtain (8.4) after some
cancellation.
\quad \quad \qed
\enddemo

Note that this transformation formula generalizes the summation in
Theorem~\TY\ 
(set $\K=1$ in (8.4)). It is safe to speculate
that (8.4) must be a special case of a much more general
transformation formula between $\widetilde A_r$ basic
hypergeometric series of different dimensions. See the last section,
item~(9) for more precise comments on this subject.

\medskip
Our last computation in this section starts with Theorem~\TU. If we let
$c\to\infty$, the determinants on the left-hand side factor by the
determinant evaluation in the proof of Theorem~\TF\ (with $r$ replaced
by $r+1$). This gives a transformation between an $\widetilde A_{r-1}$
basic hypergeometric series and an ordinary multiple hypergeometric
series. 

\proclaim{Theorem~\TTD}For $a\ge0$ we have

\vbox{
$$\multline
\sum _{k_1+\dots+k_r=0} ^{}q^{\binom {r+1}2\sum _{i=1} ^{r}k_i^2-
\sum _{i=1} ^{r}ik_i} \frac {\prod _{1\le i<j\le r}
^{}[(r+1)k_j+j-(r+1)k_i-i]} {\prod _{i=1} ^{r}[a+(r+1)k_i+i-1]!} \\
=\sum _{l_1,l_2,\dots,l_a\ge0} ^{}q^{\sum _{i=1}
^{a}(rl_i(i-1)+l_i)}\prod _{i=1} ^{r}\bmatrix
l_i+r-1\\r-1\endbmatrix.
\endmultline
\tag8.5$$
\line{\hfil\hbox{\qed}\quad \quad }
}
\endproclaim

\subhead 9. Some Schur function expansions\endsubhead
A standard application of nonintersecting lattice paths is to prove
the Jacobi-Trudi identities for Schur functions (cf\. \cite{%
\GeViAB;
\SagaAL, 154--159; \FuKrAA, sec.~3}). In this brief section we apply 
nonintersecting lattice paths {\it with a nonintersecting translate of the last
path\/} in a similar way to obtain  expansions in  Schur
functions (see (9.6) and (9.9) below).

Recall that in the proof of the Jacobi-Trudi identity for Schur functions
\cite{\MacdAB, I, (3.4)} by means of nonintersecting lattice paths,
the weight assigned to a horizontal
edge $e$ from $(i-1,j)$ to $(i,j)$ is $w(e):=x_j$. Given this
weight, the only way to satisfy (3.4) is to choose $\S$
to be a horizontal shift. But this is forbidden if we want to apply
Proposition~\TA. Hence, this weight assignment does not make sense in
our context.

On the other hand, in the proof of the dual Jacobi-Trudi identity
\cite{\MacdAB, I, (3.5)} (the
N\"agelsbach-Kostka identity, see (9.2) below) the weight 
assigned to a horizontal edge $e$ from 
$(i-1,j)$ to $(i,j)$ is $w(e):=x_{i+j}$, so that the weight
is invariant along the antidiagonal direction. Therefore we may take
this weight and choose $\S$ to be a shift in the antidiagonal direction, say
$\S=(-r,r)$. Then (3.4) is satisfied with $z=1$, and, given an
appropriate choice of starting points $\bold u=(u_1,u_2,\dots,u_r)$
and end points $\bold v=(v_1,v_2,\dots,v_r)$, we may apply
Proposition~\TA. We choose $u_i=(-i+1,i-1)$ and $v_i=(c-i+1,n-c+i-1)$.
Then by (3.5) the generating function $\sum _{} ^{}w(\P)$ for all
families $\P=(P_1,P_2,\dots,P_r)$ of lattice paths, where $P_i$ runs
from $u_i=(-i+1,i-1)$ to $v_i=(c-i+1,n-c+i-1)$, and where $(\S
P_r,P_1,\dots,P_r)$ is nonintersecting, equals
$$\sum _{k_1+\dots+k_r=0} ^{}\det_{1\le s,t\le
r}\big(e_{c+s-t-rk_s}(x_1,\dots,x_n)\big),
\tag9.1$$ 
where $e_m(x_1,\dots,x_n):=\sum _{1\le i_1<\dots< i_m\le n}
^{}x_{i_1}\cdots x_{i_m}$ denotes the {\it elementary symmetric
function\/} of order $m$ in $x_1,\dots,x_n$ (cf\. \cite{\MacdAB,
p.~12}). By changing again $k_s$ into $k_t$ (see Remark~(1) after the
proof of Theorem~\TC), and by using the dual Jacobi-Trudi identity
for Schur functions 
$s_{\La}(x_1,\dots,x_n)$ \cite{\MacdAB, I, (3.5)},
$$s_{\La'}(x_1,\dots,x_n)=\det_{s,t}\big(e_{\la_t-t+s}(x_1,\dots,x_n)\big),
\tag9.2$$
we may write (9.1) as
$$\sum _{k_1+\dots+k_r=0}
^{}s_{(c-rk_1,c-rk_2,\dots,c-rk_r)'}(x_1,\dots,x_n) .
\tag9.3$$
The reader should be warned when reading this expression that the
subscripts for the Schur functions in (9.3) will not always be
partitions (i.e., the components of $(c-rk_1,\dots,c-rk_r)$ may be
out of order) so one must apply the modification rule
$$s_{(\dots,\la_i,\la_{i+1},\dots)'}(x_1,\dots,x_n)=
-s_{(\dots,\la_{i+1}-1,\la_i+1,\dots)'}(x_1,\dots,x_n),
\tag9.4$$
perhaps repeatedly, to obtain a sum of Schur functions that are
really indexed by partitions. (See the example below.)

On the other hand, because of our choice of starting points $\bold u$,
end points $\bold v$, and shift $\S$, all $r$ lattice paths are forced to
run in parallel. Hence, the generating function under consideration
also equals
$$e_c(x_1^r,x_2^r,\dots,x_n^r)=m_{(r^c)}(x_1,x_2,\dots,x_n),\tag9.5$$
where 
$$m_{(\la_1,\dots,\la_m)}(x_1,\dots,x_n):=\sum _{1\le
i_1,\dots,i_m\le n,\ i_j\text{ pairwise distinct}} ^{}
x_{i_1}^{\la_1}\cdots x_{i_m}^{\la_m}$$
denotes the {\it monomial symmetric function\/} in $x_1,\dots,x_n$
(cf\. \cite{\MacdAB, I, (2.1)}). Hence again, by equating (9.3) and (9.5),
we have an identity. The most convenient way to write it is after
taking the limit $n\to\infty$ and changing the
summation indices from $k_i$ to $-k_i$.

\proclaim{Theorem \TTE} Let $\bold x=(x_1,x_2,\dots)$. Then
$$m_{(r^c)}(\bold x)=\sum _{k_1+\dots+k_r=0}
^{}s_{(c+rk_1,c+rk_2,\dots,c+rk_r)'} (\bold x),\tag9.6$$
where the right-hand side has to be understood according to the
modification rule (9.4).\quad \quad \qed
\endproclaim
Let us consider an example. Take $c=3$ and $r=2$. Then (9.6) and (9.4) give
$$\align m_{(2^3)}(\bold x)&=s_{(3,3)'}(\bold x)+s_{(5,1)'}(\bold x)
+s_{(1,5)'}(\bold x)+s_{(-1,7)'}(\bold x)\\
&=s_{(3,3)'}(\bold x)+s_{(5,1)'}(\bold x)-s_{(4,2)'}(\bold
x)-s_{(6)'}(\bold x)\\
&=s_{(2,2,2)}(\bold x)+s_{(2,1,1,1,1)}(\bold x)-s_{(2,2,1,1)}(\bold
x)- s_{(1,1,1,1,1,1)}(\bold x).
\endalign$$
We remark that it is not too hard to recover (9.6) from the
combinatorial description of the inverse Kostka matrix
given by E\u{g}ecio\u{g}lu and Remmel \cite{\EgReAB}. Thus 
in particular, it is also seen that when straightened out by (9.4), the 
expansion (9.6) is multiplicity-free; i.e., the coefficients on the
right-hand side are $1,-1$ or $0$.

By applying the isomorphism $\om$ (cf\. \cite{\MacdAB, pp.~14--17})
to both sides of (9.6), or by making use of a duality
principle in \cite{\LaPrAB, (4.2), (4.3); \ReYaAA,
Theorem~2.6(a)(b)}, we obtain the related identity
$$f_{(r^c)}(\bold x)=
(-1)^{(r-1)c}h_{c}(x_1^r,x_2^r,\dots)=\sum _{k_1+\dots+k_r=0}
^{}s_{(c+rk_1,c+rk_2,\dots,c+rk_r)} (\bold x).\tag9.7$$
(Here, $f_m(\bold x)$ denotes the ``forgotten symmetric function",
cf\. \cite{\MacdAB, p.~15}.) What is interesting about this identity
is that it suggests what the expression (3.5) means for
horizontal or vertical shifts. More precisely, in Proposition~1 take
$u_i=(i,1)$, $v_i=(c+i,\infty)$, $i=1,2,\dots,r$, $\S=(-r,0)$ (which
is a horizontal shift), and the weight $w(e)$ of a horizontal edge $e$
from $(i-1,j)$ to $(i,j)$ to be $w(e):=x_j$
(which implies $z=1$). Then, by the
Jacobi-Trudi identity (see \cite{\MacdAB, I, (3.4)})
$$s_{\La}(\bold x)=\det_{s,t}\big(h_{\la_t-t+s}(\bold x)\big),
\tag9.8$$
(3.5) reduces exactly to the right-hand side of (9.7) (up to sign). Thus the
left-hand side of (9.7) tells us that in this special case the
expression (3.5) has a very simple meaning, despite the fact that
the shift $\S$ is horizontal, so that Proposition~1 does not
apply. It would be interesting to figure out what is going on
here. (See also item (1) of the last section.)
%

\medskip
Another way of looking at what we did 
in deriving (9.6) 
is to say that we counted
{\it row-strict\/} cylindric partitions (i.e., $(\1,\0)$-cylindric
partitions, where $\1$ denotes the all 1 sequence) of rectangular
shape $(c^r)/\0/0$ with entries between $1$ and $n$. The weight of
such a row-strict cylindric partition $\pi$ is defined by $w(\pi):= 
\prod _{} ^{} x_{\pi_{i,j}}$ with $\pi_{i,j}$ running over all entries
of $\pi$. This is seen by the translation of $(\Al,\Be)$-cylindric
partitions into families of lattice paths that we gave in the proof of
Theorem~\TC.

So, let us next consider row-strict cylindric partitions of shape
$(c^r)/\0/1$ with entries between $1$ and $n$. Using again the
translation into families of lattice paths, Proposition~\TA, in
combination with (9.2), tells us
that the generating function for these cylindric partitions equals
$$\sum _{k_1+\dots+k_r=0}
^{}s_{(c-(r+1)k_1,c-(r+1)k_2,\dots,c-(r+1)k_r)'}(x_1,\dots,x_n) .
$$
On the other hand, given any such row-strict cylindric partition
$\pi$, we may line up the entries of $\pi$ by reading along the
columns as in (6.1). Thus we obtain a linear partition
$\boldsymbol\rho=(\rho_1,\rho_2,\dots,\rho_{rc})$ with the additional
property that $\rho_i$ is strictly larger than $\rho_{i+r}$. This
property is the same as saying that no part of $\rho$ may be repeated
more than $r$ times. Clearly, the generating function for these linear
partitions (with weight defined by $w(\rho):=\prod _{i=1}
^{rc}x_{\rho_i}$) is the homogeneous part of degree $rc$ in 
$\prod _{i=1} ^{n} (1+x_i+x_i^2+\dots+x_i^r)$. Of course, we can also
play the same game with row-strict cylindric partitions of shape
$\La/\0/1$, provided $\La=(\la_1,\dots,\la_r)$ is some partition with
$\la_1-\la_r\le1$. Taking $n\to\infty$, we obtain the following
generating function theorem.

\proclaim{Theorem \TTF}For the generating function of all linear
partitions having the property that no part is repeated more than $r$
times we have the expansion in terms of Schur functions
$$\prod _{i=1} ^{\infty}(1+x_i+x_i^2+\dots+x_i^r)=\sum
_{k_1+\dots+k_r=0} ^{} \ \sum _{\La\text{ with }\la_1-\la_r\le 1}
^{}s_{
(\La+(r+1)\bold k)'}(\bold x) \ ,\tag9.9$$
where $\bold k=(k_1,\dots,k_r)$, $\La=(\la_1,\la_2,\dots,\la_r)$, 
and $\La+(r+1)\bold k$ is short for
component-wise addition and scalar multiplication. Again, the
right-hand side of (9.9) must be understood according to the
modification rule (9.4).\quad \quad \qed
\endproclaim

As an example, the case $r=2$ of Theorem \TTF\ gives
$$\prod_{i=1}^\infty (1+x_i+x_i^2)
=\sum_{j,k\ge 0}\xi(k)s_{(2^j,1^k)},$$
where $$\xi(k)=\cases \hphantom{-} 1,&\text{if $k\equiv 0$ or $1 \pmod 6$}\cr 
-1,&\text{if $k\equiv 3$ or $4 \pmod 6$}\cr
\hphantom{-}0,&\text{if $k\equiv 2$ or $5 \pmod 6$}\cr
\endcases$$

By the duality principle in \cite{\LaPrAB, (4.2), (4.3); \ReYaAA,
Theorem~2.6(a)(b)}, (9.9) is equivalent to 
$$\prod _{i=1} ^{\infty}(1+x_i+x_i^2+\dots+x_i^r)^{-1}=\sum
_{k_1+\dots+k_r=0} ^{} \ (-1)^{\sum _{i=1} ^{r}\la_i}
\sum _{\La\text{ with }\la_1-\la_r\le 1}
^{}s_{\La+(r+1)\bold k}(\bold x) \ .\tag9.10$$
This corrects the expansion (4.8) of Lascoux and Pragacz
\cite{\LaPrAB}. However, their argument (specialize the Cauchy
identity for Schur functions), if carefully completed, would also
give a proof of (9.10), after some work. It should also be remarked, that
by a general theorem of Remmel and Yang \cite{\ReYaAA, Theorem~3.1},
the expansion (9.9) (and hence also (9.10)) is multiplicity-free.

It should be noted that (9.10) is related to a special case of
(3.5) with a horizontal shift $\S$. More precisely, in Proposition~1
take $u_i=(i,1)$, $v_i=(\la_i+i,\infty)$, $i=1,2,\dots,r$, where $\La$
is a partition with $\la_1-\la_r\le 1$, $\S=(-r-1,0)$ (which is a
horizontal shift), and the weight $w(e)$ of a horizontal edge $e$
from $(i-1,j)$ to $(i,j)$ to be $w(e):=x_j$. Then, again by the
Jacobi-Trudi identity (9.8), the expression
(3.5) reduces exactly to the homogeneous component of degree
$\la_1+\dots+\la_r$ in the right-hand side of (9.10) (up to sign).
Again, this could be a hint of what the expression (3.5) means for
horizontal or vertical shifts $\S$. (See also item (1) of the last section.)
%

\medskip
What we did so far in this section was to transfer the ideas that lead
to Theorems~\TI--\TS\ to counting row-strict cylindric partitions.
What about the method that lead to (6.23) and finally to Theorem~\TT?
In particular, what happens if we consider row-strict cylindric
partitions of shape $((m\K)^r)/\0/\K$, with arbitrary $\K$? The
cut-and-paste method illustrated in Figure~5 should again give us an
identity. When applying cut-and-paste to such a row-strict cylindric
partition we obtain a column-strict  cylindric partition, i.e., a
$(\0,\1)$-cylindric partition, of shape $((mr)^\K)/\0/r$.
Unfortunately, after translating the latter into lattice paths, 
we would have to consider a horizontal shift $\S$, and Proposition~\TA\
does not apply in this case. So the problem remains of extending
Proposition~\TA\ to horizontal (and also vertical) shifts.

\subhead 10. Future directions and open problems\endsubhead
(1) Extend Proposition~\TA\ so that it also holds for horizontal and
vertical shifts $\S$. Alternatively, is there anything reasonable
counted by (3.5) when $\S$ 
is a horizontal or vertical shift? 
The two cases discussed in
section~9 (see the remarks concerning (9.7) and (9.10)) should give
hints in this direction.
%

(2) In 
%Christian
\cite{\BrenAE, \BrenAG} Brenti considers what he calls {\it dotted plane
partitions}. These can be translated into nonintersecting lattice
paths consisting of 3 types of steps (horizontal, vertical, and diagonal). He
gives a number of very interesting applications, in particular in
connection with Hall-Littlewood functions and $q$-Stirling numbers.
So it should be a nice project to explore the implications of
considering nonintersecting lattice paths {\it with a nonintersecting
translate of the last path\/}, where the lattice paths consist of $3$
types of steps, and also of considering weights different from ours
(this is for the $q$-Stirling part).

(3) In \cite{\KratAP} it was shown by one of the authors how to
enumerate nonintersecting lattice paths with respect to the major
index, and, more generally, in \cite{\KratAW} with respect to weighted
turns. Currently, M.~Prohaska is working out the analogous theory for
enumerating nonintersecting lattice paths {\it with nonintersecting
translate of the last path\/} by major index and weighted turns. In
particular, this will give rise to more identities.

(4) In \cite{\KratAP, \FuKrAA, \KratAW, \KratAV}  the enumeration
of nonintersecting lattice paths that are bounded by a line (diagonal
or vertical) was studied. This was shown to have interesting
applications to tableau and plane partition counting. Therefore it
is natural to ask for enumerating nonintersecting lattice paths {\it
with nonintersecting translate of the last path\/} which in addition
are bounded by a line. In particular, since the enumeration done in
\cite{\KratAP, \KratAW} related to $C_r$ basic hypergeometric series,
this should produce $\widetilde C_r$ basic hypergeometric series
identities. This project is also currently being undertaken by M.~Prohaska. 

(5) Is there any representation theoretic meaning of cylindric
partitions? A few remarks are in order to 
motivate this question. As
is well-known, column-strict plane partitions of a given shape index
the weights of $\GL$- (or $\SL$-) characters. When translated into
lattice path language, column-strict plane partitions correspond to
nonintersecting lattice paths (see \cite{\GeViAA, 
\GeViAB}). The Weyl group
associated with $\GL$-characters is the symmetric group, the Weyl group
of type $A$. When counting nonintersecting lattice paths it is exactly
this group that acts on the end points (or equivalently, starting points)
of the lattice paths. Next, symplectic tableaux (introduced by King and
El-Sharkaway \cite{\KiElAA}) of a given shape index the weights of
$\Sp$-characters. It was shown in
\cite{\FuKrAA} that, when translated into lattice path language,
symplectic tableaux correspond to nonintersecting lattice paths that
are bounded by a diagonal line. The Weyl group associated with
$\Sp$-characters is the hyperoctahedral group, the Weyl group of type
$C$ (which is the same as the Weyl group of type $B$). It is formulated
slightly differently in \cite{\FuKrAA}, but what  happens in essence in the
enumeration of nonintersecting lattice paths that are bounded by a
diagonal line (cf\. \cite{\KratAP, sec.~2.2}) is that exactly this group
acts on the end points (or on the starting points, as formulated in
\cite{\KratAP}).

Now, in this paper we considered cylindric partitions that correspond
to nonintersecting lattice paths {\it with nonintersecting translate
of the last path\/}. As already mentioned in Remark~(1) after the
proof of
Proposition~\TA, the {\it affine\/} Weyl group of type $A$ acts on the
end points of the lattice paths. This Weyl group appears in the
representation theory of Kac-Moody algebras of type $A_r^{(1)}$ (see
\cite{\KacVAB}). Hence, is there a connection between Kac-Moody
algebras of type $A_r^{(1)}$ and cylindric partitions? Even more
speculatively, is there a connection between Kac-Moody algebras of type
$C_r^{(1)}$ and nonintersecting lattice paths {\it with nonintersecting
translate of the last path\/} that in addition are bounded by a line?
This question is motivated by the fact that exactly the Weyl group
corresponding to this Kac-Moody algebra (the affine Weyl group of
type $C$) would act on the end points (or starting points) of the
lattice paths.

(6) In \cite{\BurgAC, pp.~217--218} 
%\cite{\BurgAC, p.~217/218}
Burge develops sort of a Bailey
chain for his partition pair generating functions. The most
interesting results of his paper, two doubly bounded summations
\cite{\BurgAC, pp.~219--220} 
%\cite{\BurgAC, p.~219/220} 
that contain both Rogers' and Schur's previously unrelated
proofs of the celebrated Rogers-Ramanujan identities, is
derived by his Bailey chain argument. It is very unfortunate
that we failed to find a Bailey chain for $r$-rowed cylindric
partitions, for general $r$. Once found, it would give rise to
many more interesting identities.

(7) Since cylindric partitions proved a special case of 
Gustafson's $_6\psi_6$-summation (7.9) for $\SU(r)$): Are there
more general weights for lattice paths which would lead to a proof of
the full formulas (7.9) or at least of some terminating cases such as
Milne's $_6\phi_5$-summation for $\SU(r)$ \cite{\MilnAJ, Theorem~1.49}?

(8) What is the relation between our summation (7.7) and Gustafson's
$_6\psi_6$-sum (7.9) for $\SU(r)$? Is there a generalization of (7.9)
that would also cover (7.7)? 
As mentioned in section~7, the answer to these questions would very
likely be an $\SU(r)$-extension of some $_8\psi_8$-identity, such as 
\cite{\GaRaAA, (5.6.2) or Ex.~5.12}, for
example.

(9) The transformation formula (8.4) must be a special case of a
more general formula. What is it? In particular, by setting
$y_{ij}:=\chi($the cell $(i,j)$ is contained in the shape
$\Mu$), we could rewrite (8.4) in basic hypergeometric notation as
$$\multline 
\frac {\prod _{1\le i<j\le r} ^{}(1-q^{\sum _{l=1} ^{\K}y_{il}-i-\sum
_{l=1} ^{\K}y_{jl}+j})} {\prod _{i=1} ^{r}(q;q)_{m\K-\sum _{l=1}
^{\K}y_{il}+i-1}}\\
\hskip-1cm
\sum _{k_1+\dots+k_r=0} ^{}q^{-(r+\K)(\K+\ep)\sum _{i=1} ^{r}k_i^2/2+
(r+2\K+\ep) \sum _{i=1} ^{r}ik_i-(\K+\ep)\sum _{i=1} ^{r}\sum _{j=1}
^{\K}y_{ij}k_i}\\ 
\prod _{1\le i<j\le r} ^{}\frac {1-q^{(r+\K)k_i-(r+\K)k_j
+\sum _{l=1} ^{\K}y_{il}-i-\sum
_{l=1} ^{\K}y_{jl}+j}} {1-q^{\sum _{l=1} ^{\K}y_{il}-i-\sum
_{l=1} ^{\K}y_{jl}+j}}
 {\prod _{i=1} ^{r}(q^{1-m\K+\sum _{l=1}
^{\K}y_{il}-i};q)_{(r+\K)k_i}}\\
\hskip-4cm =q^{\big(rd\binom m2-(m-1)\sum _{i=1} ^{r}\mu_i\big)\ep}
\frac {\prod _{1\le i<j\le \K } ^{}(1-q^{\sum _{l=1}
^{r}y_{li}-i-\sum 
_{l=1} ^{r}y_{lj}+j})} {\prod _{i=1} ^{\K }(q;q)_{mr-\sum _{l=1}
^{r}y_{li}+i-1}}\\
\hskip-1cm
\sum _{l_1+\dots+l_\K =0} ^{}q^{-(r+\K)(r-\ep)\sum _{i=1} ^{\K }l_i^2/2+
(\K+2r-\ep) \sum _{i=1} ^{\K }il_i-(r-\ep)\sum _{i=1} ^{\K }\sum _{j=1}
^{r}y_{ji}l_i}\\ 
\prod _{1\le i<j\le \K } ^{}\frac {1-q^{(r+\K)l_i-(r+\K)l_j
+\sum _{l=1} ^{r}y_{li}-i-\sum
_{l=1} ^{r}y_{lj}+j}} {1-q^{\sum _{l=1} ^{r}y_{li}-i-\sum
_{l=1} ^{r}y_{lj}+j}}
{\prod _{i=1} ^{\K }(q^{1-mr+\sum _{l=1}
^{r}y_{li}-i};q)_{(r+\K)l_i}}
\endmultline\tag10.1$$
Is there a transformation connecting the two series appearing in
(10.1) for arbitrary $y_{ij}$?

(10) Are there more transformation formulas between $\widetilde A_r$
basic hypergeometric series and ordinary multiple basic hypergeometric
series? In particular, can (8.5) be generalized?

(11) Many identities in sections 6--8 are subject to rather strange
restrictions on the parameters (in particular on $\ep$). Special cases
show that they are actually wrong outside the range of these
restrictions. There must be some more general identities that
would explain the ``difference" occurring outside this range.


\remark{Acknowledgment} We thank Jeff Remmel for helpful
conversations regarding the symmetric function expansions in
section~9.
\endremark




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\endRefs

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