\input amstex
\input amsppt.sty

\magnification=\magstep1 

\vcorrection{-0.8cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Input-Datei zum Erzeugen von Gitterpunktwegen.%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%entering a path: 
%  \Pfad(x-coordinate of starting point,y-coordinate of starting point),path
%  as 1-2-3-4-word\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 1-2-3-4-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
%
%vectors:
%  \Vektor(x-coordinate of incline, y-coordinate of incline)length(x-coordinate of starting point, y-coordinate of starting point)
%
%labelling of points:
%  \Label[location?]{[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
\newcount\@tempcntb

\newdimen\@tempdima
\newdimen\@tempdimb


\newdimen\@wholewidth
\newdimen\@halfwidth

\newcount\@xarg
\newcount\@yarg
\newcount\@yyarg
\newbox\@linechar
\newbox\@tempboxa
\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\Pfadd@cke width \@linelen depth\Pfadd@cke
\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\Vektor(#1,#2)#3(#4,#5){\unskip\leavevmode
  \xcoord#4\relax \ycoord#5\relax
      \raise\ycoord \Einheit\hbox to0pt{\hskip\xcoord \Einheit
         \Vector@(#1,#2){#3}\hss}}

\def\Vector@(#1,#2)#3{\@xarg #1\relax \@yarg #2\relax
\@tempcnta \ifnum\@xarg<0 -\@xarg\else\@xarg\fi
\ifnum\@tempcnta<5\relax
\@linelen=#3\Einheit
\ifnum\@xarg =0 \@vvector
  \else \ifnum\@yarg =0 \@hvector \else \@svector\fi
\fi
\else\@badlinearg\fi}

\def\@hvector{\@hline\hbox to 0pt{\@linefnt
\ifnum \@xarg <0 \@getlarrow(1,0)\hss\else
    \hss\@getrarrow(1,0)\fi}}

\def\@vvector{\ifnum \@yarg <0 \@downvector \else \@upvector \fi}

\def\@svector{\@sline
\@tempcnta\@yarg \ifnum\@tempcnta <0 \@tempcnta=-\@tempcnta\fi
\ifnum\@tempcnta <5
  \hskip -\wd\@linechar
  \@upordown\@clnht \hbox{\@linefnt  \if@negarg
  \@getlarrow(\@xarg,\@yyarg) \else \@getrarrow(\@xarg,\@yyarg) \fi}%
\else\@badlinearg\fi}

\def\@upline{\hbox to \z@{\hskip -.5\Pfadd@cke \vrule width \Pfadd@cke
   height \@linelen depth \z@\hss}}

\def\@downline{\hbox to \z@{\hskip -.5\Pfadd@cke \vrule width \Pfadd@cke
   height \z@ depth \@linelen \hss}}

\def\@upvector{\@upline\setbox\@tempboxa\hbox{\@linefnt\char'66}\raise
     \@linelen \hbox to\z@{\lower \ht\@tempboxa\box\@tempboxa\hss}}

\def\@downvector{\@downline\lower \@linelen
      \hbox to \z@{\@linefnt\char'77\hss}}

\def\@getlarrow(#1,#2){\ifnum #2 =\z@ \@tempcnta='33\else
\@tempcnta=#1\relax\multiply\@tempcnta \sixt@@n \advance\@tempcnta
-9 \@tempcntb=#2\relax\multiply\@tempcntb \tw@
\ifnum \@tempcntb >0 \advance\@tempcnta \@tempcntb\relax
\else\advance\@tempcnta -\@tempcntb\advance\@tempcnta 64
\fi\fi\char\@tempcnta}

\def\@getrarrow(#1,#2){\@tempcntb=#2\relax
\ifnum\@tempcntb < 0 \@tempcntb=-\@tempcntb\relax\fi
\ifcase \@tempcntb\relax \@tempcnta='55 \or
\ifnum #1<3 \@tempcnta=#1\relax\multiply\@tempcnta
24 \advance\@tempcnta -6 \else \ifnum #1=3 \@tempcnta=49
\else\@tempcnta=58 \fi\fi\or
\ifnum #1<3 \@tempcnta=#1\relax\multiply\@tempcnta
24 \advance\@tempcnta -3 \else \@tempcnta=51\fi\or
\@tempcnta=#1\relax\multiply\@tempcnta
\sixt@@n \advance\@tempcnta -\tw@ \else
\@tempcnta=#1\relax\multiply\@tempcnta
\sixt@@n \advance\@tempcnta 7 \fi\ifnum #2<0 \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}%
  \def\ro{\advance\xdim by.5 \Einheit \advance\ydim by.5 \Einheit}%
  \def\rro{\advance\xdim by.25cm \advance\ydim by.5 \Einheit}%
  \def\roo{\advance\xdim by.5 \Einheit \advance\ydim by.25cm}%
  \def\l{\advance\xdim by-.30cm}%
  \def\r{\advance\xdim by.30cm}%
  \def\lu{\advance\xdim by-.5 \Einheit \advance\ydim by-.6 \Einheit}%
  \def\llu{\advance\xdim by-.25cm \advance\ydim by-.6 \Einheit}%
  \def\luu{\advance\xdim by-.5 \Einheit \advance\ydim by-.30cm}%
  \def\u{\advance\ydim by-.30cm}%
  \def\ru{\advance\xdim by.5 \Einheit \advance\ydim by-.6 \Einheit}%
  \def\rru{\advance\xdim by.25cm \advance\ydim by-.6 \Einheit}%
  \def\ruu{\advance\xdim by.5 \Einheit \advance\ydim by-.30cm}%
  #1\raise\ydim\hbox to0pt{\hskip\xdim
     \vbox to0pt{\vss\hbox to0pt{\hss$#2$\hss}\vss}\hss}%
}
\catcode`\@=13

\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
%Erg\"anzung des fetten Small-Capitals-Fonts:
%
 \ifsyntax@ \def\big##1{{\hbox{$\left##1\right.$}}}%
  \let\Big\big \let\bigg\big \let\Bigg\big
 \else
  \textfont\z@=\twelverm  \scriptfont\z@=\tenrm  \scriptscriptfont\z@=\sevenrm
  \textfont\@ne=\twelvei  \scriptfont\@ne=\teni  \scriptscriptfont\@ne=\seveni
  \textfont\tw@=\twelvesy \scriptfont\tw@=\tensy \scriptscriptfont\tw@=\sevensy
  \textfont\thr@@=\twelveex \scriptfont\thr@@=\tenex
        \scriptscriptfont\thr@@=\tenex
  \textfont\itfam=\twelveit \scriptfont\itfam=\tenit
        \scriptscriptfont\itfam=\tenit
  \textfont\bffam=\twelvebf \scriptfont\bffam=\tenbf
        \scriptscriptfont\bffam=\sevenbf
  \setbox\strutbox\hbox{\vrule height10.2\p@ depth4.2\p@ width\z@}%
  \setbox\strutbox@\hbox{\lower.6\normallineskiplimit\vbox{%
        \kern-\normallineskiplimit\copy\strutbox}}%
 \setbox\z@\vbox{\hbox{$($}\kern\z@}\bigsize@=1.4\ht\z@
 \fi
 \normalbaselines\rm\ex@.2326ex\jot3.6\ex@\the\twelvepoint@}

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

\font@\seventeenrm=cmr10 scaled\magstep3
\font@\seventeenit=cmti10 scaled\magstep3
\font@\seventeensl=cmsl10 scaled\magstep3
\font@\seventeensmc=cmcsc10 scaled\magstep3
\font@\seventeentt=cmtt10 scaled\magstep3
\font@\seventeenbf=cmbx10 scaled\magstep3
\font@\seventeeni=cmmi10 scaled\magstep3
\font@\seventeensy=cmsy10 scaled\magstep3
\font@\seventeenex=cmex10 scaled\magstep3
\font@\seventeenmsa=msam10 scaled\magstep3
\font@\seventeeneufm=eufm10 scaled\magstep3
\font@\seventeenmsb=msbm10 scaled\magstep3
\newtoks\seventeenpoint@
\def\seventeenpoint{\normalbaselineskip18\p@
 \abovedisplayskip21.6\p@ plus5.2\p@ minus15.4\p@
 \belowdisplayskip\abovedisplayskip
 \abovedisplayshortskip\z@ plus5.2\p@
 \belowdisplayshortskip12.1\p@ plus5.2\p@ minus7\p@
 \textonlyfont@\rm\seventeenrm \textonlyfont@\it\seventeenit
 \textonlyfont@\sl\seventeensl \textonlyfont@\bf\seventeenbf
 \textonlyfont@\smc\seventeensmc \textonlyfont@\tt\seventeentt
%Erg\"anzung 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


\nologo
\def\endemo{\qed\enddemo}
\def\ua{\underline{a}}
\def\w{\bullet}
\def\g{\frak g}
\def\h{\frak h}
\def\n{\frak n}
\def\bb{\frak b}
\def\D{\Delta}
\def\l{\lambda}
\def\Dp{\Delta^+}
\def\Dm{\Delta^-}
\def\Da{\widehat\Delta}
\def\Dap{\widehat\Delta^+}
\def\d{\delta}
\def\r{R(\h)}
\def\t{\otimes}
\def\a{\alpha}
\def\b{\beta}
\def\f{F(\g)}
\def\I{\Cal I^n}
\def\l{\lambda}
\def\d{\delta}
\def\m{\prec}
\def\th{\theta}
\def\Q{\check Q}
\def\i{{\frak i}}
\def\j{{\bold j}}
\def\N{N(w)}
\def\nd{\noindent}
\def\gg{\gamma}
\def\DD{D_\Phi}
\def\p{\Phi}
\def\nat{\Bbb N}
\def\ganz{\Bbb Z}
\def\compl{\Bbb C} 
\def\real{\Bbb R} 

\TagsOnRight


\def\formel{1.1}
\def\tabalgo{3.1}
\def\fastalgo{3.2}
\def\lM{3.3}
\def\lm{3.4}
\def\summe{4.1}
\def\decomp{4.2}
\def\Summe{4.3}
\def\formela{4.4}
\def\formelb{4.5}
\def\formelc{4.6}
\def\qtCat{6.1}
\def\eqdmin{6.2}
\def\eqdmax{6.3}
\def\eqDmin{6.4}
\def\eqDmax{6.5}
\def\ineqm{6.6}
\def\eqintm{6.7}
\def\ineqM{6.8}
\def\eqstufen{6.9}
\def\eqintM{6.10}
\def\eqintmin{6.11}
\def\ineqA{6.12}
\def\ineqa{6.13}
\def\ineqb{6.14}



\def\l{\lambda}
\def\v#1{\left\vert#1\right\vert}
\def\fl#1{\left\lfloor#1\right\rfloor}
\def\cl#1{\left\lceil#1\right\rceil}
\def\flkl#1{\lfloor#1\rfloor}
\def\clkl#1{\lceil#1\rceil}
\def\dmin#1#2{{\theta^{\text{min}}_{#1}(#2)}}
\def\dmax#1#2{{\theta^{\text{max}}_{#1}(#2)}}
\def\Dmin#1#2{{\Theta^{\text{min}}_{#1}(#2)}}
\def\Dmax#1#2{{\Theta^{\text{max}}_{#1}(#2)}}
\def\({\left(}
\def\){\right)}

My findings in case $C_n$ are based on the following observation:
Instead of making the (strict) partition corresponding to an ideal
into a self-conjugate partition and
then applying the $A_{2n-1}$ algorithm to the self-conjugate
partition, we may proceed as follows. Draw the strict partition
$(\l_1,\l_2,\dots)$. Start
as in the $A_{2n-1}$ algorithm. I.e., begin on the vertical edge on
the right of cell $(1,\l_1)$, and move 
downward until we touch the antidiagonal
$x+y=2n$. At the touching point we turn direction from 
vertical-down to horizontal-left, and move on until we touch a vertical
part of the Ferrers diagram. At the touching point we turn direction
from  horizontal-left to vertical-down. We iterate until we meet the
diagonal $x=y$. Let $k$ be the number of touching points on the
antidiagonal $x+y=2n$. If we meet the diagonal $x=y$ while travelling
horizontal-left, then the class of nilpotence is $2k$, if we meet the
diagonal while travelling vertical-downward, then the class of
nilpotence is $2k+1$. For example, if $n=9$, then for the partition
$(16,13,11,8,7,5,3)$ in Figure~1 the class of nilpotence is $2\cdot
3=6$, whereas for the partition 
$(16,13,11,8,5,3,1)$ in Figure~2 the class of nilpotence is $2\cdot
2+1=5$.

\midinsert
\vskip10pt
\vbox{
$$
\Pfad(-9,9),1111111111111111\endPfad
\Pfad(-3,2),11121212211212112\endPfad
\Pfad(-9,8),2\endPfad
\Pfad(-9,8),1\endPfad
\Pfad(-8,7),2\endPfad
\Pfad(-8,7),1\endPfad
\Pfad(-7,6),2\endPfad
\Pfad(-7,6),1\endPfad
\Pfad(-6,5),2\endPfad
\Pfad(-6,5),1\endPfad
\Pfad(-5,4),2\endPfad
\Pfad(-5,4),1\endPfad
\Pfad(-4,3),2\endPfad
\Pfad(-4,3),1\endPfad
\Pfad(-3,2),2\endPfad
\SPfad(4,7),1112\endSPfad
\SPfad(2,4),1122\endSPfad
\SPfad(-1,1),1122\endSPfad
\thinlines
\Diagonale(-1,-1){11}
\Label\r{x+y=2n}(10,8)
\AntiDiagonale(-10,10){11}
\Label\o{x=y}(-10,10)
\DuennPunkt(1,1)
\DuennPunkt(4,4)
\DuennPunkt(7,7)
\hskip0cm
$$
\centerline{\eightpoint Figure 1}
}
\vskip10pt
\endinsert

\midinsert
\vskip10pt
\vbox{
$$
\Pfad(-9,9),1111111111111111\endPfad
\Pfad(-3,2),12121211211212112\endPfad
\Pfad(-9,8),2\endPfad
\Pfad(-9,8),1\endPfad
\Pfad(-8,7),2\endPfad
\Pfad(-8,7),1\endPfad
\Pfad(-7,6),2\endPfad
\Pfad(-7,6),1\endPfad
\Pfad(-6,5),2\endPfad
\Pfad(-6,5),1\endPfad
\Pfad(-5,4),2\endPfad
\Pfad(-5,4),1\endPfad
\Pfad(-4,3),2\endPfad
\Pfad(-4,3),1\endPfad
\Pfad(-3,2),2\endPfad
\SPfad(4,7),1112\endSPfad
\SPfad(-1,4),1111122\endSPfad
\SPfad(-1,1),22\endSPfad
\thinlines
\Diagonale(-1,-1){11}
\Label\r{x+y=2n}(10,8)
\AntiDiagonale(-10,10){11}
\Label\o{x=y}(-10,10)
\DuennPunkt(4,4)
\DuennPunkt(7,7)
\hskip0cm
$$
\centerline{\eightpoint Figure 2}
}
\vskip10pt
\endinsert

Let $\gamma_n(h,k)$ be the number of ideals in type $C_n$ with
dimension $h$ and class of nilpotence $k$.
If we follow the line of argument of the proof of Theorem~6.1, then we
obtain 
$$\multline
\sum _{h\ge0} ^{}\gamma_n(h,2k)t^h=
\sum_{n<i_1<\dots<i_k<i_{k+1}=2n} 
\prod_{j=1}^{k-1}t^{i_{j+1}(i_{j+3}-i_{j+2})}
\bmatrix {i_{j+2}-i_j-1}\\{i_{j+1}-i_j}\endbmatrix_t\\
\cdot
\sum _{\ell=0} ^{i_2-i_1-1}\bmatrix i_1+i_2-2n-1\\\ell\endbmatrix
t^{i_1(i_3-i_2)-\binom{2n-i_2+1}2+\binom {l+1}2}
\endmultline$$
(where $i_{k+2}=2n+1$) for the generating function for ideals with
nilpotence $2k$, and
$$\multline
\sum _{h\ge0} ^{}\gamma_n(h,2k-1)t^h=
\sum_{2n-i_2<i_1\le n<i_2<\dots<i_k<i_{k+1}=2n} 
t^{i_1(i_3-i_2)-\binom{2n-i_2+1}2}\\
\cdot
(1+t)(1+t^2)\cdots(1+t^{i_1+i_2-2n-1})
\prod_{j=1}^{k-1}t^{i_{j+1}(i_{j+3}-i_{j+2})}
\bmatrix {i_{j+2}-i_j-1}\\{i_{j+1}-i_j}\endbmatrix_t
\endmultline$$
(where again $i_{k+2}=2n+1$) for the generating function for ideals with
nilpotence $2k-1$. After having done the substitution $i_j\to i_j+n$
and combining both formulas, this can be ``compactly'' written as follows:
$$\multline
\sum _{h,K\ge0} ^{}\gamma_n(h,K)t^hq^{K}=
\sum_{-i_2<i_1<i_2<\dots<i_k<i_{k+1}=n} q^{2k-\chi(i_1\le0)}\\
\cdot \bigg(
\prod_{j=1}^{k-1}t^{(i_{j+1}+n)(i_{j+3}-i_{j+2})}
\bmatrix {i_{j+2}-i_j-1}\\{i_{j+1}-i_j}\endbmatrix_t\\
\cdot
\sum _{\ell=0} ^{i_2-i_1-1}\bmatrix i_1+i_2-1\\\ell\endbmatrix
t^{(i_1+n)(i_3-i_2)-\binom{n-i_2+1}2+\binom {l+1}2}\bigg),
\endmultline$$
where $i_{k+2}=n+1$.

\medskip
In the following let us forget about the dimension, that is, we
specialize $t$ to $1$.

What I want to compute next is the number of ideals with class of
nilpotence at most $2k$. In principle, I could sum up the above
formulas with $t=1$, but it is more convenient to use an algorithm
which determines whether the class of nilpotence is at most $2k$. This
algorithm works as follows. Again we are given a strict partition
$\l=(\l_1,\l_2,\dots)$. This time we start at the diagonal $x=y$ (!),
at the lowest intersection point with the Ferrers diagram of $\l$. We
move right until we touch the antidiagonal $x+y=2n$. At the
touching point we turn direction from horizontal-right to vertical-up,
and move on until we touch a horizontal part of the Ferrers
diagram. At the touching point we turn direction from vertical-up to
horizontal-right. Etc. The class of nilpotence is at most twice the
number of touching points on $x+y=2n$. See Figure~3, where I apply
this procedure to the partition of Figure~1.

\midinsert
\vskip10pt
\vbox{
$$
\Pfad(-9,9),1111111111111111\endPfad
\Pfad(-3,2),11121212211212112\endPfad
\Pfad(-9,8),2\endPfad
\Pfad(-9,8),1\endPfad
\Pfad(-8,7),2\endPfad
\Pfad(-8,7),1\endPfad
\Pfad(-7,6),2\endPfad
\Pfad(-7,6),1\endPfad
\Pfad(-6,5),2\endPfad
\Pfad(-6,5),1\endPfad
\Pfad(-5,4),2\endPfad
\Pfad(-5,4),1\endPfad
\Pfad(-4,3),2\endPfad
\Pfad(-4,3),1\endPfad
\Pfad(-3,2),2\endPfad
\SPfad(7,8),12\endSPfad
\SPfad(3,6),11122\endSPfad
\SPfad(-2,2),1111222\endSPfad
\thinlines
\Diagonale(-1,-1){11}
\Label\r{x+y=2n}(10,8)
\AntiDiagonale(-10,10){11}
\Label\o{x=y}(-10,10)
\DuennPunkt(2,2)
\DuennPunkt(6,6)
\DuennPunkt(8,8)
\hskip0cm
$$
\centerline{\eightpoint Figure 3}
}
\vskip10pt
\endinsert

If we now use the idea of the proof of Theorem~6.1 to this situation
(in a slightly modified form),
then we get 
$$\underset j_1,j_2,\dots,j_k\ge1,\,j_{k+1}\ge0\to
{\sum _{j_1+j_2+\dots+j_{k+1}=n}} ^{}\binom {j_1+j_2-1}{j_1}
\binom {j_2+j_3-1}{j_2}\cdots \binom {j_{k-1}+j_k-1}{j_{k-1}}
\binom {j_{k}+2j_{k+1}}{j_{k}}$$
for the number of ideals with class of nilpotence $2k$ or $2k-1$, and
$$\underset j_1,j_2,\dots,j_k,j_{k+1}\ge0\to
{\sum _{j_1+j_2+\dots+j_{k+1}=n}} ^{}\binom {j_1+j_2-1}{j_1}
\binom {j_2+j_3-1}{j_2}\cdots \binom {j_{k-1}+j_k-1}{j_{k-1}}
\binom {j_{k}+2j_{k+1}}{j_{k}}$$
for the number of ideals with class of nilpotence at most $2k$.
Let us denote the latter number by $\bar\gamma_n(2k)$. Now we can
easily compute the generating function $\sum _{n\ge0}
^{}\bar\gamma_n(2k)x^n$ for the ideals of nilpotence at most $2k$. We
substitute the previous expression in the generating function,
$$\multline
{\sum _{j_1,j_2,\dots,j_k,j_{k+1}\ge0}} ^{}
x^{j_1+\cdots+j_k+j_{k+1}}
\binom {j_1+j_2-1}{j_1}
\binom {j_2+j_3-1}{j_2}\\
\cdots \binom {j_{k-1}+j_k-1}{j_{k-1}}
\binom {j_{k}+2j_{k+1}}{j_{k}}.
\endmultline$$
Now we perform the summation over $j_1$ and obtain $(1/(1-x))^{j_2}$
in the sum. Next the summation over $j_2$ is performed, etc. If we use
the fact that 
$$
\cfrac 1\\1-
\cfrac x\\1-
\cfrac x\\
\cfrac \ddots\\1-x\endcfrac\endcfrac\endcfrac\endcfrac =
\frac{U_{h}\(1/2\sqrt x\)}{\sqrt x\,U_{h+1}\(1/2\sqrt x\)},
$$
where $x$ occurs $h$ times in the continued fraction, then we finally
end up with
$$\sum _{j_{k+1}\ge0} ^{}\frac {1} {\sqrt x}\(\frac {U_k\(1/2\sqrt
x\)} {U_{k+1}\(1/2\sqrt x\)}\)^{2j_{k+1}+1}.$$
The sum is easily evaluated as it is just a geometric series. This
gives
$$\frac {U_k\(1/2\sqrt x\)U_{k+1}\(1/2\sqrt x\)} 
{\sqrt x\,\(U_{k+1}^2\(1/2\sqrt x\)-U_{k}^2\(1/2\sqrt x\)\)}.$$
Now, it is easy to verify that
$$U_k(x)U_{k+1}(x)=U_{2k+1}(x)+U_{2k-1}(x)+\dots+U_1(x)$$
and
$$U_{k+1}^2(x)-U_k^2(x)=U_{2k+2}(x).$$
Hence, our generating function equals
$$\frac {U_{2k+1}\(1/2\sqrt x\)+U_{2k-1}\(1/2\sqrt
x\)+\dots+U_1\(1/2\sqrt x\)} 
{\sqrt x\,U_{2k+2}\(1/2\sqrt x\)}.$$

As it is also known that the generating function $\sum _{P}
^{}x^{\ell(P)/2}$ for Dyck paths starting at the origin, never
exceeding height $2k+1$, and ending
at height $h$ ($\ell(P)$ denoting the length of $P$) is given by
$$\frac {U_{2k+1-h}\(1/2\sqrt x\)} {\sqrt x\,U_{2k+2}\(1/2\sqrt x\)},$$
(see for example Theorem~A2 in my Dyck path/restricted permutations paper)
it follows that the number of ideals in type $C_n$ with class of
nilpotence at most $2k$ is equal to the number of Dyck paths of length
$2n$ which start at the origin and never exceed height $2k+1$.

In order to have the same conclusion for class of nilpotence at most
$2k-1$, I compute the generating function for ideals with class of 
nilpotence {\it exactly} equal $2k-1$. That is, if
$\tilde\gamma_n(2k-1)$ denotes the number of ideals in $C_n$ with
class of nilpotence $2k-1$, then I compute $\sum _{n\ge0}
^{}\tilde\gamma(2k-1)x^n$. The case $k=1$ has to be done
separately, but this is trivial. (This is already included in
Peterson's result.) So let us assume $k\ge2$. If we substitute the
formula (with $t=1$) from the first page in the generating function, we obtain
$$\multline
\sum _{n\ge0} ^{}
\sum_{2n-i_2<i_1\le n<i_2<\dots<i_k<2n} x^n
\pmatrix {2n-i_{k-1}-1}\\{i_{k}-i_{k-1}}\endpmatrix
\pmatrix {i_{k}-i_{k-2}-1}\\{i_{k-1}-i_{k-2}}\endpmatrix\\
\cdots
\pmatrix {i_{3}-i_1-1}\\{i_{2}-i_1}\endpmatrix
2^{i_1+i_2-2n-1}
\endmultline$$
for our generating function. As before, after having replaced $i_\ell$ by
$i_\ell+n$ for all $\ell$, the sums over
$n,i_k,i_{k-1},\dots,i_3$ are easily computed. On using again that the
resulting continued fractions can be expressed in terms of
Chebyshev polynomials, we obtain 
$$\sum _{-i_2<i_1\le 0<i_2} ^{}x^{i_2+k-1}\(\frac {U_1\(1/2\sqrt x\)}
{x^{k/2-1}\,U_{k-1}\(1/2\sqrt x\)}\)^2 \(\frac {U_{k-1}\(1/2\sqrt x\)} 
{\sqrt x\,U_k\(1/2\sqrt x\)}\)^{i_2-i_1+1}2^{i_1+i_2-1}.$$
Next we sum over $i_1$, and subsequently over $i_2$. In both cases, it
is just geometric series that have to be summed, 
a terminating series when summing over $i_1$
and two nonterminating series when summing over $i_2$. The result,
after cancellation of factors, is
$$\frac {U_k\(1/2\sqrt x\)} {\(U_k\(1/2\sqrt x\)-2\sqrt
x\,U_{k-1}\(1/2\sqrt x\)\)\(U_k^2\(1/2\sqrt x\)-U_{k-1}^2\(1/2\sqrt
x\)\)}.$$
It is now routine to verify that this is equal to
$$\multline
\frac {U_{2k}\(1/2\sqrt x\)+U_{2k-2}\(1/2\sqrt
x\)+\dots+U_0\(1/2\sqrt x\)} 
{\sqrt x\,U_{2k+1}\(1/2\sqrt x\)}\\
-\frac {U_{2k-1}\(1/2\sqrt x\)+U_{2k-3}\(1/2\sqrt
x\)+\dots+U_1\(1/2\sqrt x\)} 
{\sqrt x\,U_{2k}\(1/2\sqrt x\)}.
\endmultline$$
On the other hand, by the result on Dyck paths that I quoted before,
this is the generating function for Dyck paths, starting at the
origin, ending arbitrarily, with height {\it exactly} $2k$. (The first
part counts the paths with height at most $2k$, whereas the part which
is subtracted counts the paths with height at most $2k-1$.)
Therefore, if we combine, we obtain that the number of ideals in type
$C_n$ with class of nilpotence exactly $K$ is equal to te number
of Dyck paths of length $2n$, which start at the origin and end
arbitrarily, with height exactly $K+1$.
By using the iterated reflection formula out of Mohanty's book, 
one obtains that this number
is equal to
$$\sum _{s=0} ^{\fl{h/2}}\sum _{k\in\Bbb Z} ^{}\frac {1+2s+2k(h+2)}
{2n+1} \binom {2n+1}{n-s-k(h+2)}.$$

\proclaim{Corollary}
The number of ideals in $C_n$ with class of
nilpotence at most $2$ is $F_{2n}$, and for $n\ge1$
the number of ideals in $C_n$ with class of
nilpotence at most $3$ is $2\cdot 3^{n-1}$.
\endproclaim


\end
