\input amstex
\input amsppt.sty

Well, in general, things do not work equally smoothly.
In particular, in general we do not have that $s_\ell+i_\ell$ is
equal to $d$ or $i_\ell$. In general, we can just say that
the numbers $s_\ell+i_\ell$, $\ell=1,2,\dots,d$, are
weakly increasing. Yet, we may try the same construction.
In particular, the combinatorial interpretation of the multiplicities 
in terms of nonintersecting lattice paths that I described in my
previous email is still true.

By what I wrote in my previous email,
the multiplicity $M_{\bold j}(\bold i)$ is equal to
the number of all families
$(Q_1,Q_2,\dots,Q_d)$ of nonintersecting lattice paths, where the
path $Q_\ell$ runs from $(-\ell+1,\ell-1)$,
to $(-s_{\sigma(\ell)},s_{\sigma(\ell)}+i_{\sigma(\ell)})$, 
$\ell=1,2,\dots,d$, for some permutation $\sigma\in S_d$. (As it turns
out, there is a unique permutation $\sigma$ for which such families
of nonintersecting lattice paths exist).

Let us consider an example: $d=9$, $\bold i=(4,6,7,13,14,17,19,20,21)$
and $\bold j=(1,2,4,7,10,12,13,15,16)$. Then $\bold
s=(6,6,5,2,2,0,0,0,0)$. 
Figure~1 shows a typical family for this choice of $\bold i$ and
$\bold j$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%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
    \else\ifnum#1=5
      \advance\xcoord by -1
      \raise\ycoord \Einheit\hbox to0pt{\hskip\xcoord \Einheit
         \vrule height\Pfadd@cke width1 \Einheit depth\Pfadd@cke\hss}%
    \else\ifnum#1=6
      \advance\ycoord by -1
      \raise\ycoord \Einheit\hbox to0pt{\hskip\xcoord \Einheit
        \hbox{\hskip-\PfadD@cke\vrule height1 \Einheit width\PfadD@cke depth0pt}\hss}%
    \else\ifnum#1=7
      \advance\xcoord by -1
      \advance\ycoord by -1
      \raise\ycoord \Einheit\hbox to0pt{\hskip\xcoord \Einheit
         \Line@(1,1){1}\hss}
    \else\ifnum#1=8
      \advance\xcoord by -1
      \advance\ycoord by +1
      \raise\ycoord \Einheit\hbox to0pt{\hskip\xcoord \Einheit
         \Line@(1,-1){1}\hss}
    \fi\fi\fi\fi
    \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
    \else\ifnum#1=5
      \advance\xcoord by -1
      \raise\ycoord \Einheit\hbox to0pt{\hskip\xcoord \Einheit
         \hSSchritt\hss}%
    \else\ifnum#1=6
      \advance\ycoord by -1
      \raise\ycoord \Einheit\hbox to0pt{\hskip\xcoord \Einheit
        \hbox{\hskip-2pt \vSSchritt}\hss}%
    \else\ifnum#1=7
      \advance\xcoord by -1
      \advance\ycoord by -1
      \raise\ycoord \Einheit\hbox to0pt{\hskip\xcoord \Einheit
         \DSSchritt\hss}
    \else\ifnum#1=8
      \advance\xcoord by -1
      \advance\ycoord by 1
      \raise\ycoord \Einheit\hbox to0pt{\hskip\xcoord \Einheit
         \dSSchritt\hss}
    \fi\fi\fi\fi
    \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
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\vskip10pt
\vbox{
$$
\Gitter(10,22)(0,0)
%\Koordinatenachsen(-9,10)(0,0)
\PfadDicke{.5pt}
\Pfad(9,-1),2222222222222222222222\endPfad
\Pfad(-1,0),11111111111\endPfad
\PfadDicke{1.2pt}
\Pfad(1,8),222221222112212221111\endPfad
\Pfad(2,7),222212\endPfad
\Pfad(3,6),2222\endPfad
\Pfad(4,5),2222222\endPfad
\Pfad(5,4),22222222222221221211\endPfad
\Pfad(6,3),22222222222221\endPfad
\Pfad(7,2),2222222222222\endPfad
\Pfad(8,1),2222222222222222212\endPfad
\Pfad(9,0),22222222222222222\endPfad
%\SPfad(-1,9),11111111111\endSPfad
\SPfad(1,10),112212221112211\endSPfad
\DickPunkt(1,8)
\DickPunkt(2,7)
\DickPunkt(3,6)
\DickPunkt(4,5)
\DickPunkt(5,4)
\DickPunkt(6,3)
\DickPunkt(7,2)
\DickPunkt(8,1)
\DickPunkt(9,0)
\DickPunkt(3,10)
\DickPunkt(3,12)
\DickPunkt(4,12)
\DickPunkt(7,15)
\DickPunkt(7,16)
\DickPunkt(9,17)
\DickPunkt(9,19)
\DickPunkt(9,20)
\DickPunkt(9,21)
\hskip3.5cm
$$
\centerline{\eightpoint Figure 1}
}
\vskip10pt

Again, large parts of the initial vertical portions of the paths are
forced. To be precise, let $\{s_1,s_2,\dots,s_d\}=\{v_1,v_2,\dots,v_r\}$, 
$v_1<v_2<\dots<v_r$. (I.e., the $v_\ell$'s are the different values
that are attained by the $s_\ell$'s.) For each $\ell$,
consider the
bottom-most end point $(-s_q,s_q+i_q)$ with $s_q=v_\ell$. Then we may
cut off the vertical portions below height $s_q+i_q$ of the paths
with starting points $(-v_{\ell+1}+1,v_{\ell+1}-1)$, \dots, 
$(-v_{\ell}-1,v_{\ell}+1)$, 
$(-v_{\ell},v_{\ell})$. These cuts are indicated by the dotted line
in Figure~1. (I.e., portions below dotted lines can be
omitted.) The result, after the cuts, is shown in Figure~2.

\def\GPunkt(#1,#2){\unskip
  \raise#2 \Einheit\hbox to0pt{\hskip#1 \Einheit
          \raise-2.5pt\hbox to0pt{\hss$\ssize\bullet$\hss}\hss}}

\vskip10pt
\vbox{
$$
%\Gitter(10,12)(0,0)
%\Koordinatenachsen(-9,10)(0,0)
\PfadDicke{.5pt}
\Pfad(9,-1),2222222222222\endPfad
\Pfad(-1,0),11111111111\endPfad
\PfadDicke{1.2pt}
\Pfad(1,0),2221222112212221111\endPfad
\Pfad(2,0),212\endPfad
%\Pfad(3,6),2222\endPfad
%\Pfad(4,5),2222222\endPfad
\Pfad(5,5),221221211\endPfad
\Pfad(6,5),21\endPfad
%\Pfad(7,5),2\endPfad
\Pfad(8,7),212\endPfad
\SPfad(1,0),112212221112211\endSPfad
\DickPunkt(1,0)
\DickPunkt(2,0)
%\DickPunkt(3,6)
%\DickPunkt(4,5)
\DickPunkt(5,5)
\DickPunkt(6,5)
%\DickPunkt(7,2)
\DickPunkt(8,7)
%\DickPunkt(9,0)
\DickPunkt(3,0)
\DickPunkt(3,2)
\DickPunkt(4,2)
\DickPunkt(7,5)
\DickPunkt(7,6)
\DickPunkt(9,7)
\DickPunkt(9,9)
\DickPunkt(9,10)
\DickPunkt(9,11)
\GPunkt(1,1)
\GPunkt(2,1)
\GPunkt(3,1)
\GPunkt(1,2)
\GPunkt(2,2)
\GPunkt(1,3)
\GPunkt(2,3)
\GPunkt(3,3)
\GPunkt(4,3)
\GPunkt(1,4)
\GPunkt(2,4)
\GPunkt(3,4)
\GPunkt(4,4)
\GPunkt(1,5)
\GPunkt(2,5)
\GPunkt(3,5)
\GPunkt(4,5)
\GPunkt(1,6)
\GPunkt(2,6)
\GPunkt(3,6)
\GPunkt(4,6)
\GPunkt(5,6)
\GPunkt(6,6)
\GPunkt(1,7)
\GPunkt(2,7)
\GPunkt(3,7)
\GPunkt(4,7)
\GPunkt(5,7)
\GPunkt(6,7)
\GPunkt(7,7)
\GPunkt(8,7)
\GPunkt(1,8)
\GPunkt(2,8)
\GPunkt(3,8)
\GPunkt(4,8)
\GPunkt(5,8)
\GPunkt(6,8)
\GPunkt(7,8)
\GPunkt(8,8)
\GPunkt(9,8)
\GPunkt(1,9)
\GPunkt(2,9)
\GPunkt(3,9)
\GPunkt(4,9)
\GPunkt(5,9)
\GPunkt(6,9)
\GPunkt(7,9)
\GPunkt(8,9)
\GPunkt(9,9)
\GPunkt(1,10)
\GPunkt(2,10)
\GPunkt(3,10)
\GPunkt(4,10)
\GPunkt(5,10)
\GPunkt(6,10)
\GPunkt(7,10)
\GPunkt(8,10)
\GPunkt(9,10)
\GPunkt(1,11)
\GPunkt(2,11)
\GPunkt(3,11)
\GPunkt(4,11)
\GPunkt(5,11)
\GPunkt(6,11)
\GPunkt(7,11)
\GPunkt(8,11)
\GPunkt(9,11)
\hskip3.5cm
$$
\centerline{\eightpoint Figure 2}
}
\vskip10pt

Now we may write down a Lindstr\"om--Gessel--Viennot determinant for
these (new) starting points and (old) end points. The result would
again be a determinant of binomials. However, it seems
that it would require a considerable 
amount of notation to explicitely express
what the new starting points are. 

Moreover, this would not fully
correspond to the Lakshmibai-Weyman formula because, if you remember,
to obtain the Lakshmibai-Weyman formula, one also had to drop the
paths of length zero. If we would do that in the above picture, then
we would have to restrict the paths explicitely to the indicated
ladder-shaped region. If we would not do that then, after removal of the
starting and end points corresponding to the zero length paths (which
opens ``holes"), there are now more possibilities to connect the
(reduced set of) starting points to the (reduced set of) end points
by nonintersecting lattice paths, and the corresponding permutation
$\sigma$ would not be unique anymore. So, we could still write down a 
Lindstr\"om--Gessel--Viennot determinant (that would now correspond to
the Lakshibai-Weyman formula), however the entries would not be
binomials anymore, they would count paths with given starting and end
points that stay in this ladder-shaped region, for which no closed
formula is available.

On the other hand, there is still something that can be done with
this picture. Let us introduce what Gessel and Viennot call ``dual
paths," see Figure~3. We mark starting points $(d-\ell,i_d)$,
$\ell=1,2,\dots,d$. (They are indicated by circles in Figure~3.) 
Then, for each of these points, we move
vertically downwards, unless we hit one of the existing paths. If the
latter happens, then we continue by a diagonal step $(1,-1)$. It is
not difficult to see that in that manner we connect $(d-\ell,i_\ell)$
with $(-s_\ell,s_\ell+i_\ell)$, $\ell=1,2,\dots,d$. The
resulting paths in our running example are indicated by dotted lines 
in the left picture in Figure~3. In the right picture I have just
interchanged the roles of the two families of paths.

\vskip10pt
\vbox{
$$
\PfadDicke{.5pt}
\Pfad(9,-1),2222222222222\endPfad
\Pfad(-1,0),11111111111\endPfad
\PfadDicke{1.2pt}
\Pfad(1,0),2221222112212221111\endPfad
\Pfad(2,0),212\endPfad
\Pfad(5,5),221221211\endPfad
\Pfad(6,5),21\endPfad
\Pfad(8,7),212\endPfad
%\SPfad(1,0),112212221112211\endSPfad
\SPfad(1,11),66666666464\endSPfad
\SPfad(2,11),666664666\endSPfad
\SPfad(3,11),666664666\endSPfad
\SPfad(4,11),666444\endSPfad
\SPfad(5,11),46466\endSPfad
\SPfad(6,11),4464\endSPfad
\SPfad(7,11),44\endSPfad
\SPfad(8,11),4\endSPfad
\Kreis(1,11)
\Kreis(2,11)
\Kreis(3,11)
\Kreis(4,11)
\Kreis(5,11)
\Kreis(6,11)
\Kreis(7,11)
\Kreis(8,11)
\Kreis(9,11)
\DickPunkt(1,0)
\DickPunkt(2,0)
\DickPunkt(5,5)
\DickPunkt(6,5)
\DickPunkt(8,7)
\DickPunkt(3,0)
\DickPunkt(3,2)
\DickPunkt(4,2)
\DickPunkt(7,5)
\DickPunkt(7,6)
\DickPunkt(9,7)
\DickPunkt(9,9)
\DickPunkt(9,10)
%\DickPunkt(9,11)
\GPunkt(1,1)
\GPunkt(2,1)
\GPunkt(3,1)
\GPunkt(1,2)
\GPunkt(2,2)
\GPunkt(1,3)
\GPunkt(2,3)
\GPunkt(3,3)
\GPunkt(4,3)
\GPunkt(1,4)
\GPunkt(2,4)
\GPunkt(3,4)
\GPunkt(4,4)
\GPunkt(1,5)
\GPunkt(2,5)
\GPunkt(3,5)
\GPunkt(4,5)
\GPunkt(1,6)
\GPunkt(2,6)
\GPunkt(3,6)
\GPunkt(4,6)
\GPunkt(5,6)
\GPunkt(6,6)
\GPunkt(1,7)
\GPunkt(2,7)
\GPunkt(3,7)
\GPunkt(4,7)
\GPunkt(5,7)
\GPunkt(6,7)
\GPunkt(7,7)
\GPunkt(8,7)
\GPunkt(1,8)
\GPunkt(2,8)
\GPunkt(3,8)
\GPunkt(4,8)
\GPunkt(5,8)
\GPunkt(6,8)
\GPunkt(7,8)
\GPunkt(8,8)
\GPunkt(9,8)
\GPunkt(1,9)
\GPunkt(2,9)
\GPunkt(3,9)
\GPunkt(4,9)
\GPunkt(5,9)
\GPunkt(6,9)
\GPunkt(7,9)
\GPunkt(8,9)
\GPunkt(9,9)
\GPunkt(1,10)
\GPunkt(2,10)
\GPunkt(3,10)
\GPunkt(4,10)
\GPunkt(5,10)
\GPunkt(6,10)
\GPunkt(7,10)
\GPunkt(8,10)
\GPunkt(9,10)
\GPunkt(1,11)
\GPunkt(2,11)
\GPunkt(3,11)
\GPunkt(4,11)
\GPunkt(5,11)
\GPunkt(6,11)
\GPunkt(7,11)
\GPunkt(8,11)
\GPunkt(9,11)
\hbox{\hskip7cm}
\PfadDicke{.5pt}
\Pfad(9,-1),2222222222222\endPfad
\Pfad(-1,0),11111111111\endPfad
\PfadDicke{1.2pt}
\SPfad(1,0),2221222112212221111\endSPfad
\SPfad(2,0),212\endSPfad
\SPfad(5,5),221221211\endSPfad
\SPfad(6,5),21\endSPfad
\SPfad(8,7),212\endSPfad
%\Pfad(1,0),112212221112211\endPfad
\Pfad(1,11),66666666464\endPfad
\Pfad(2,11),666664666\endPfad
\Pfad(3,11),666664666\endPfad
\Pfad(4,11),666444\endPfad
\Pfad(5,11),46466\endPfad
\Pfad(6,11),4464\endPfad
\Pfad(7,11),44\endPfad
\Pfad(8,11),4\endPfad
\Kreis(1,11)
\Kreis(2,11)
\Kreis(3,11)
\Kreis(4,11)
\Kreis(5,11)
\Kreis(6,11)
\Kreis(7,11)
\Kreis(8,11)
\Kreis(9,11)
\DickPunkt(1,0)
\DickPunkt(2,0)
\DickPunkt(5,5)
\DickPunkt(6,5)
\DickPunkt(8,7)
\DickPunkt(3,0)
\DickPunkt(3,2)
\DickPunkt(4,2)
\DickPunkt(7,5)
\DickPunkt(7,6)
\DickPunkt(9,7)
\DickPunkt(9,9)
\DickPunkt(9,10)
%\DickPunkt(9,11)
\GPunkt(1,1)
\GPunkt(2,1)
\GPunkt(3,1)
\GPunkt(1,2)
\GPunkt(2,2)
\GPunkt(1,3)
\GPunkt(2,3)
\GPunkt(3,3)
\GPunkt(4,3)
\GPunkt(1,4)
\GPunkt(2,4)
\GPunkt(3,4)
\GPunkt(4,4)
\GPunkt(1,5)
\GPunkt(2,5)
\GPunkt(3,5)
\GPunkt(4,5)
\GPunkt(1,6)
\GPunkt(2,6)
\GPunkt(3,6)
\GPunkt(4,6)
\GPunkt(5,6)
\GPunkt(6,6)
\GPunkt(1,7)
\GPunkt(2,7)
\GPunkt(3,7)
\GPunkt(4,7)
\GPunkt(5,7)
\GPunkt(6,7)
\GPunkt(7,7)
\GPunkt(8,7)
\GPunkt(1,8)
\GPunkt(2,8)
\GPunkt(3,8)
\GPunkt(4,8)
\GPunkt(5,8)
\GPunkt(6,8)
\GPunkt(7,8)
\GPunkt(8,8)
\GPunkt(9,8)
\GPunkt(1,9)
\GPunkt(2,9)
\GPunkt(3,9)
\GPunkt(4,9)
\GPunkt(5,9)
\GPunkt(6,9)
\GPunkt(7,9)
\GPunkt(8,9)
\GPunkt(9,9)
\GPunkt(1,10)
\GPunkt(2,10)
\GPunkt(3,10)
\GPunkt(4,10)
\GPunkt(5,10)
\GPunkt(6,10)
\GPunkt(7,10)
\GPunkt(8,10)
\GPunkt(9,10)
\GPunkt(1,11)
\GPunkt(2,11)
\GPunkt(3,11)
\GPunkt(4,11)
\GPunkt(5,11)
\GPunkt(6,11)
\GPunkt(7,11)
\GPunkt(8,11)
\GPunkt(9,11)
\hskip3.5cm
$$
\centerline{\eightpoint Figure 3}
}
\vskip10pt

Now I deform the lattice slightly, so that the newly introduced paths
become orthogonal paths, see Figure~4.

\vskip10pt
\vbox{
$$
\PfadDicke{.5pt}
%\Pfad(9,-1),2222222222222\endPfad
%\Pfad(-1,0),11111111111\endPfad
\PfadDicke{1.2pt}
\SPfad(1,0),2223222332232223333\endSPfad
\SPfad(2,1),232\endSPfad
\SPfad(5,9),223223233\endSPfad
\SPfad(6,10),23\endSPfad
\SPfad(8,14),232\endSPfad
%\SPfad(1,0),112212221112211\endSPfad
\Pfad(1,11),66666666161\endPfad
\Pfad(2,12),666661666\endPfad
\Pfad(3,13),666661666\endPfad
\Pfad(4,14),666111\endPfad
\Pfad(5,15),16166\endPfad
\Pfad(6,16),1161\endPfad
\Pfad(7,17),11\endPfad
\Pfad(8,18),1\endPfad
\Kreis(1,11)
\Kreis(2,12)
\Kreis(3,13)
\Kreis(4,14)
\Kreis(5,15)
\Kreis(6,16)
\Kreis(7,17)
\Kreis(8,18)
\Kreis(9,19)
\DickPunkt(1,0)
\DickPunkt(2,1)
\DickPunkt(5,9)
\DickPunkt(6,10)
\DickPunkt(8,14)
\DickPunkt(3,2)
\DickPunkt(3,4)
\DickPunkt(4,5)
\DickPunkt(7,11)
\DickPunkt(7,12)
\DickPunkt(9,15)
\DickPunkt(9,17)
\DickPunkt(9,18)
%\DickPunkt(9,11)
\GPunkt(1,0)
\GPunkt(1,1)
\GPunkt(1,2)
\GPunkt(1,3)
\GPunkt(1,4)
\GPunkt(1,5)
\GPunkt(1,6)
\GPunkt(1,7)
\GPunkt(1,8)
\GPunkt(1,9)
\GPunkt(1,10)
\GPunkt(1,11)
\GPunkt(2,1)
\GPunkt(2,2)
\GPunkt(2,3)
\GPunkt(2,4)
\GPunkt(2,5)
\GPunkt(2,6)
\GPunkt(2,7)
\GPunkt(2,8)
\GPunkt(2,9)
\GPunkt(2,10)
\GPunkt(2,11)
\GPunkt(2,12)
\GPunkt(3,2)
\GPunkt(3,3)
\GPunkt(3,4)
\GPunkt(3,5)
\GPunkt(3,6)
\GPunkt(3,7)
\GPunkt(3,8)
\GPunkt(3,9)
\GPunkt(3,10)
\GPunkt(3,11)
\GPunkt(3,12)
\GPunkt(3,13)
\GPunkt(4,5)
\GPunkt(4,6)
\GPunkt(4,7)
\GPunkt(4,8)
\GPunkt(4,9)
\GPunkt(4,10)
\GPunkt(4,11)
\GPunkt(4,12)
\GPunkt(4,13)
\GPunkt(4,14)
\GPunkt(5,9)
\GPunkt(5,10)
\GPunkt(5,11)
\GPunkt(5,12)
\GPunkt(5,13)
\GPunkt(5,14)
\GPunkt(5,15)
\GPunkt(6,10)
\GPunkt(6,11)
\GPunkt(6,12)
\GPunkt(6,13)
\GPunkt(6,14)
\GPunkt(6,15)
\GPunkt(6,16)
\GPunkt(7,11)
\GPunkt(7,12)
\GPunkt(7,13)
\GPunkt(7,14)
\GPunkt(7,15)
\GPunkt(7,16)
\GPunkt(7,17)
\GPunkt(8,14)
\GPunkt(8,15)
\GPunkt(8,16)
\GPunkt(8,17)
\GPunkt(8,18)
\GPunkt(9,15)
\GPunkt(9,16)
\GPunkt(9,17)
\GPunkt(9,18)
\GPunkt(9,19)
\hbox{\hskip7cm}
\PfadDicke{.5pt}
%\Pfad(9,-1),2222222222222\endPfad
%\Pfad(-1,0),11111111111\endPfad
\PfadDicke{1.2pt}
\Pfad(1,11),66666666161\endPfad
\Pfad(2,12),666661666\endPfad
\Pfad(3,13),666661666\endPfad
\Pfad(4,14),666111\endPfad
\Pfad(5,15),16166\endPfad
\Pfad(6,16),1161\endPfad
\Pfad(7,17),11\endPfad
\Pfad(8,18),1\endPfad
\Kreis(1,11)
\Kreis(2,12)
\Kreis(3,13)
\Kreis(4,14)
\Kreis(5,15)
\Kreis(6,16)
\Kreis(7,17)
\Kreis(8,18)
\Kreis(9,19)
\DickPunkt(3,2)
\DickPunkt(3,4)
\DickPunkt(4,5)
\DickPunkt(7,11)
\DickPunkt(7,12)
\DickPunkt(9,15)
\DickPunkt(9,17)
\DickPunkt(9,18)
%\DickPunkt(9,11)
\GPunkt(1,0)
\GPunkt(1,1)
\GPunkt(1,2)
\GPunkt(1,3)
\GPunkt(1,4)
\GPunkt(1,5)
\GPunkt(1,6)
\GPunkt(1,7)
\GPunkt(1,8)
\GPunkt(1,9)
\GPunkt(1,10)
\GPunkt(1,11)
\GPunkt(2,1)
\GPunkt(2,2)
\GPunkt(2,3)
\GPunkt(2,4)
\GPunkt(2,5)
\GPunkt(2,6)
\GPunkt(2,7)
\GPunkt(2,8)
\GPunkt(2,9)
\GPunkt(2,10)
\GPunkt(2,11)
\GPunkt(2,12)
\GPunkt(3,2)
\GPunkt(3,3)
\GPunkt(3,4)
\GPunkt(3,5)
\GPunkt(3,6)
\GPunkt(3,7)
\GPunkt(3,8)
\GPunkt(3,9)
\GPunkt(3,10)
\GPunkt(3,11)
\GPunkt(3,12)
\GPunkt(3,13)
\GPunkt(4,5)
\GPunkt(4,6)
\GPunkt(4,7)
\GPunkt(4,8)
\GPunkt(4,9)
\GPunkt(4,10)
\GPunkt(4,11)
\GPunkt(4,12)
\GPunkt(4,13)
\GPunkt(4,14)
\GPunkt(5,9)
\GPunkt(5,10)
\GPunkt(5,11)
\GPunkt(5,12)
\GPunkt(5,13)
\GPunkt(5,14)
\GPunkt(5,15)
\GPunkt(6,10)
\GPunkt(6,11)
\GPunkt(6,12)
\GPunkt(6,13)
\GPunkt(6,14)
\GPunkt(6,15)
\GPunkt(6,16)
\GPunkt(7,11)
\GPunkt(7,12)
\GPunkt(7,13)
\GPunkt(7,14)
\GPunkt(7,15)
\GPunkt(7,16)
\GPunkt(7,17)
\GPunkt(8,14)
\GPunkt(8,15)
\GPunkt(8,16)
\GPunkt(8,17)
\GPunkt(8,18)
\GPunkt(9,15)
\GPunkt(9,16)
\GPunkt(9,17)
\GPunkt(9,18)
\GPunkt(9,19)
\hskip3.5cm
$$
\centerline{\eightpoint Figure 4}
}
\vskip10pt

So, what we obtain finally 
is a family $(P_1,P_2,\dots,P_d)$ of nonintersecting lattice paths, where the
path $P_\ell$ is a path consisting of positive horizontal unit steps
and negative vertical unit steps and runs from $(-d+\ell,i_d+\ell)$,
to $(-s_\ell,i_\ell+d)$, $\ell=1,2,\dots,d$. Hence, again by the
Lindstr\"om--Gessel--Viennot theorem, we obtain that the multiplicity
$M_{\bold j}(\bold i)$ is equal to the determinant
$$\det_{1\le p,q\le d}\left(\binom {i_d-i_p-s_q}{d-p-s_q}\right).$$
(In fact, we could restrict this determinant to $1\le p,q\le d-1$,
because $\binom {i_d-i_d-s_q}{d-d-s_q}=\binom
{-s_q}{-s_q}=\delta_{q,d}$. This corresponds to the combinatorial
fact that path $P_d$ is a zero length path. See the top-right of Figure~4.)

\catcode`\@=11
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\def\endvstab{\crcr\egroup\egroup
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\newdimen\htstrut@
\htstrut@8.5\p@
\newdimen\htStrut@
\htStrut@12\p@
\newdimen\dpstrut@
\dpstrut@3.5\p@
\newdimen\dpStrut@
\dpStrut@3.5\p@
\def\openup{\afterassignment\@penup\dimen@=}
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\def\Let@@{\relax\iffalse{\fi%
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\newdimen\D@cke
\def\Dicke#1{\global\D@cke#1}
\newtoks\tabs@\tabs@{&}
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\newif\iff@rst

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\def\format{\ifvst@b\else\crcr\fi\egroup\iffalse{\fi\ifnum`}=0 \fi\format@}
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 \def\r{\hfil\noexpand\ifhline@@\hbox{\vrule height\htStrut@%
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\vskip10pt
\vbox{
$$
\PfadDicke{.5pt}
%\Pfad(9,-1),2222222222222\endPfad
%\Pfad(-1,0),11111111111\endPfad
\PfadDicke{1.2pt}
\Pfad(1,11),66666666161\endPfad
\Pfad(2,12),666661666\endPfad
\Pfad(3,13),666661666\endPfad
\Pfad(4,14),666111\endPfad
\Pfad(5,15),16166\endPfad
\Pfad(6,16),1161\endPfad
\Pfad(7,17),11\endPfad
\Pfad(8,18),1\endPfad
\Kreis(1,11)
\Kreis(2,12)
\Kreis(3,13)
\Kreis(4,14)
\Kreis(5,15)
\Kreis(6,16)
\Kreis(7,17)
\Kreis(8,18)
\Kreis(9,19)
\DickPunkt(3,2)
\DickPunkt(3,4)
\DickPunkt(4,5)
\DickPunkt(7,11)
\DickPunkt(7,12)
\DickPunkt(9,15)
\DickPunkt(9,17)
\DickPunkt(9,18)
%\DickPunkt(9,11)
\GPunkt(1,0)
\GPunkt(1,1)
\GPunkt(1,2)
\GPunkt(1,3)
\GPunkt(1,4)
\GPunkt(1,5)
\GPunkt(1,6)
\GPunkt(1,7)
\GPunkt(1,8)
\GPunkt(1,9)
\GPunkt(1,10)
\GPunkt(1,11)
\GPunkt(2,1)
\GPunkt(2,2)
\GPunkt(2,3)
\GPunkt(2,4)
\GPunkt(2,5)
\GPunkt(2,6)
\GPunkt(2,7)
\GPunkt(2,8)
\GPunkt(2,9)
\GPunkt(2,10)
\GPunkt(2,11)
\GPunkt(2,12)
\GPunkt(3,2)
\GPunkt(3,3)
\GPunkt(3,4)
\GPunkt(3,5)
\GPunkt(3,6)
\GPunkt(3,7)
\GPunkt(3,8)
\GPunkt(3,9)
\GPunkt(3,10)
\GPunkt(3,11)
\GPunkt(3,12)
\GPunkt(3,13)
\GPunkt(4,5)
\GPunkt(4,6)
\GPunkt(4,7)
\GPunkt(4,8)
\GPunkt(4,9)
\GPunkt(4,10)
\GPunkt(4,11)
\GPunkt(4,12)
\GPunkt(4,13)
\GPunkt(4,14)
\GPunkt(5,9)
\GPunkt(5,10)
\GPunkt(5,11)
\GPunkt(5,12)
\GPunkt(5,13)
\GPunkt(5,14)
\GPunkt(5,15)
\GPunkt(6,10)
\GPunkt(6,11)
\GPunkt(6,12)
\GPunkt(6,13)
\GPunkt(6,14)
\GPunkt(6,15)
\GPunkt(6,16)
\GPunkt(7,11)
\GPunkt(7,12)
\GPunkt(7,13)
\GPunkt(7,14)
\GPunkt(7,15)
\GPunkt(7,16)
\GPunkt(7,17)
\GPunkt(8,14)
\GPunkt(8,15)
\GPunkt(8,16)
\GPunkt(8,17)
\GPunkt(8,18)
\GPunkt(9,15)
\GPunkt(9,16)
\GPunkt(9,17)
\GPunkt(9,18)
\GPunkt(9,19)
\Label\ro{1}(5,15)
\Label\ro{1}(6,16)
\Label\ro{1}(7,17)
\Label\ro{1}(8,18)
\Label\ro{2}(8,17)
\Label\ro{2}(7,16)
\Label\ro{3}(6,14)
\Label\ro{4}(8,15)
\Label\ro{4}(4,11)
\Label\ro{5}(5,11)
\Label\ro{6}(6,11)
\Label\ro{6}(2,7)
\Label\ro{6}(3,8)
\Label\ro{9}(1,3)
\Label\ro{11}(2,2)
\hskip7cm
\raise5cm\hbox{$
\smatrix \format\sa\c\s\c\s\c\s\c\s\c\s\c\s\c\s\c\se\\
\hlinefor{17}\\
&1&&1&&1&&1&&4&&6&&6&&9&\\
\hlinefor{17}\\
\omit&&&2&&2&&3&&5&& &\omit& &&11&\\
\omit& \omit&\hlinefor9& \omit&\omit& \omit&\hlinefor3\\
\omit& &\omit& &&4&& &&6&& \\
\omit& \omit&\omit& \omit&\hlinefor3& \omit &\hlinefor3
\endsmatrix$}
$$
\centerline{\eightpoint Figure 5}
}
\vskip10pt

The above has also a combinatorial interpretation in terms of
certain semistandard tableaux. If we label horizontal steps along diagonals
by 1, 2, \dots, respectively, as indicated in Figure~5 (i.e., along the
first diagonal of horizontal steps immediately to the right of the
starting points, the latter indicated by circles, horizontal steps
are labelled by 1, along the next
diagonal horizontal steps are labelled by 2, etc.), and then read the labels along each
path and form columns out of it, then we obtain an array, as shown in
Figure~5 in our example, which has the following properties:
\roster
\item The length of column $\ell$ is $\ell-s_{d-\ell}$,
$\ell=1,2,\dots,d-1$.
\item The entries along rows are weakly increasing.
\item The entries along columns are strictly increasing.
\item If at the bottom of column $d-\ell$ we write
$i_d-i_{\ell}-s_\ell+1$, $\ell=1,2,\dots,d-1$ 
(see Figure~6 for the corresponding extended
array in our running example), then rows are still weakly
increasing and columns are still strictly increasing.
\endroster



\vskip10pt
\vbox{
$$
\smatrix \format\sa\c\s\c\s\c\s\c\s\c\s\c\s\c\s\c\se\\
\hlinefor{17}\\
&1&&1&&1&&1&&4&&6&&6&&9&\\
\hlinefor{17}\\
&\boldkey 2&&2&&2&&3&&5&& \boldkey1\boldkey0&&\boldkey1\boldkey0 &&11&\\
\hlinefor{17}\\
\omit& && \boldkey3&&4&& \boldkey6&&6&& &\omit& &&\boldkey1\boldkey2&\\
\omit& \omit&\hlinefor9& \omit&\omit& \omit&\hlinefor3\\
\omit& &\omit& &&\boldkey5&& &&\boldkey7&& \\
\omit& \omit&\omit& \omit&\hlinefor3& \omit &\hlinefor3
\endsmatrix
$$
\centerline{\eightpoint Figure 6}
}
\vskip10pt

  With best wishes,

    Christian Krattenthaler



\end
