\input amstex \input amsppt.sty \nologo \magnification1200 \hsize13cm \vsize19cm %First page headline in AmS-TeX for S\'eminaire Lotharingien de Combinatoire \catcode`\@=11 \def\output@{\shipout\vbox{% \iffirstpage@ \global\firstpage@false %insert headline: \headline={\frheadline} \makeheadline \pagebody \makefootline% \else \ifrunheads@ \makeheadline \pagebody \else \pagebody \makefootline \fi \fi}% \advancepageno \ifnum\outputpenalty>-\@MM\else\dosupereject\fi} \catcode`\@=13 \font\rms=cmr8 \font\its=cmti8 \font\bfs=cmbx8 \def\frheadline{\its S\'eminaire Lotharingien de Combinatoire \bfs 45 \rms (2001), Article~B45c\hfill} \TagsOnRight \NoBlackBoxes %\def\SvanAA{9} \def\RoZeAA{8} \def\MacdAC{7} \def\LindAA{6} \def\LaWeAB{5} \def\GeViAB{4} \def\GeViAA{3} \def\FultAC{2} \def\BiLaAA{1} \catcode`\@=11 \def\dddot#1{\vbox{\ialign{##\crcr .\hskip-.5pt.\hskip-.5pt.\crcr\noalign{\kern1.5\p@\nointerlineskip} $\hfil\displaystyle{#1}\hfil$\crcr}}} \newif\iftab@\tab@false \newif\ifvtab@\vtab@false \def\tab{\bgroup\tab@true\vtab@false\vst@bfalse\Strich@false% \def\\{\global\hline@@false% \ifhline@\global\hline@false\global\hline@@true\fi\cr} \edef\l@{\the\leftskip}\ialign\bgroup\hskip\l@##\hfil&&##\hfil\cr} \def\endtab{\cr\egroup\egroup} \def\vtab{\vtop\bgroup\vst@bfalse\vtab@true\tab@true\Strich@false% \bgroup\def\\{\cr}\ialign\bgroup&##\hfil\cr} \def\endvtab{\cr\egroup\egroup\egroup} \def\stab{\D@cke0.5pt\null \bgroup\tab@true\vtab@false\vst@bfalse\Strich@true\Let@@\vspace@ \normalbaselines\offinterlineskip \openup\spreadmlines@ \edef\l@{\the\leftskip}\ialign \bgroup\hskip\l@##\hfil&&##\hfil\crcr} \def\endstab{\crcr\egroup \egroup} \newif\ifvst@b\vst@bfalse \def\vstab{\D@cke0.5pt\null \vtop\bgroup\tab@true\vtab@false\vst@btrue\Strich@true\bgroup\Let@@\vspace@ \normalbaselines\offinterlineskip \openup\spreadmlines@\bgroup} \def\endvstab{\crcr\egroup\egroup \egroup\tab@false\Strich@false} \newdimen\htstrut@ \htstrut@8.5\p@ \newdimen\htStrut@ \htStrut@12\p@ \newdimen\dpstrut@ \dpstrut@3.5\p@ \newdimen\dpStrut@ \dpStrut@3.5\p@ \def\openup{\afterassignment\@penup\dimen@=} \def\@penup{\advance\lineskip\dimen@ \advance\baselineskip\dimen@ \advance\lineskiplimit\dimen@ \divide\dimen@ by2 \advance\htstrut@\dimen@ \advance\htStrut@\dimen@ \advance\dpstrut@\dimen@ \advance\dpStrut@\dimen@} \def\Let@@{\relax\iffalse{\fi% \def\\{\global\hline@@false% \ifhline@\global\hline@false\global\hline@@true\fi\cr}% \iffalse}\fi} \def\matrix{\null\,\vcenter\bgroup \tab@false\vtab@false\vst@bfalse\Strich@false\Let@@\vspace@ \normalbaselines\openup\spreadmlines@\ialign \bgroup\hfil$\m@th##$\hfil&&\quad\hfil$\m@th##$\hfil\crcr \Mathstrut@\crcr\noalign{\kern-\baselineskip}} \def\endmatrix{\crcr\Mathstrut@\crcr\noalign{\kern-\baselineskip}\egroup \egroup\,} \def\smatrix{\D@cke0.5pt\null\, \vcenter\bgroup\tab@false\vtab@false\vst@bfalse\Strich@true\Let@@\vspace@ \normalbaselines\offinterlineskip \openup\spreadmlines@\ialign \bgroup\hfil$\m@th##$\hfil&&\quad\hfil$\m@th##$\hfil\crcr} \def\endsmatrix{\crcr\egroup \egroup\,\Strich@false} \newdimen\D@cke \def\Dicke#1{\global\D@cke#1} \newtoks\tabs@\tabs@{&} \newif\ifStrich@\Strich@false \newif\iff@rst \def\Stricherr@{\iftab@\ifvtab@\errmessage{\noexpand\s not allowed here. Use \noexpand\vstab!}% \else\errmessage{\noexpand\s not allowed here. Use \noexpand\stab!}% \fi\else\errmessage{\noexpand\s not allowed here. Use \noexpand\smatrix!}\fi} \def\format{\ifvst@b\else\crcr\fi\egroup\iffalse{\fi\ifnum`}=0 \fi\format@} \def\format@#1\\{\def\preamble@{#1}% \def\Str@chfehlt##1{\ifx##1\s\Stricherr@\fi\ifx##1\\\let\Next\relax% \else\let\Next\Str@chfehlt\fi\Next}% \def\c{\hfil\noexpand\ifhline@@\hbox{\vrule height\htStrut@% depth\dpstrut@ width\z@}\noexpand\fi% \ifStrich@\hbox{\vrule height\htstrut@ depth\dpstrut@ width\z@}% \fi\iftab@\else$\m@th\fi\the\hashtoks@\iftab@\else$\fi\hfil}% \def\r{\hfil\noexpand\ifhline@@\hbox{\vrule height\htStrut@% depth\dpstrut@ width\z@}\noexpand\fi% \ifStrich@\hbox{\vrule height\htstrut@ depth\dpstrut@ width\z@}% \fi\iftab@\else$\m@th\fi\the\hashtoks@\iftab@\else$\fi}% \def\l{\noexpand\ifhline@@\hbox{\vrule height\htStrut@% depth\dpstrut@ width\z@}\noexpand\fi% \ifStrich@\hbox{\vrule height\htstrut@ depth\dpstrut@ width\z@}% \fi\iftab@\else$\m@th\fi\the\hashtoks@\iftab@\else$\fi\hfil}% \def\s{\ifStrich@\ \the\tabs@\vrule width\D@cke\the\hashtoks@% \fi\the\tabs@\ }% \def\sa{\ifStrich@\vrule width\D@cke\the\hashtoks@% \the\tabs@\ % \fi}% \def\se{\ifStrich@\ \the\tabs@\vrule width\D@cke\the\hashtoks@\fi}% \def\cd{\hfil\noexpand\ifhline@@\hbox{\vrule height\htStrut@% depth\dpstrut@ width\z@}\noexpand\fi% \ifStrich@\hbox{\vrule height\htstrut@ depth\dpstrut@ width\z@}% \fi$\dsize\m@th\the\hashtoks@$\hfil}% \def\rd{\hfil\noexpand\ifhline@@\hbox{\vrule height\htStrut@% depth\dpstrut@ width\z@}\noexpand\fi% \ifStrich@\hbox{\vrule height\htstrut@ depth\dpstrut@ width\z@}% \fi$\dsize\m@th\the\hashtoks@$}% \def\ld{\noexpand\ifhline@@\hbox{\vrule height\htStrut@% depth\dpstrut@ width\z@}\noexpand\fi% \ifStrich@\hbox{\vrule height\htstrut@ depth\dpstrut@ width\z@}% \fi$\dsize\m@th\the\hashtoks@$\hfil}% \ifStrich@\else\Str@chfehlt#1\\\fi% \setbox\z@\hbox{\xdef\Preamble@{\preamble@}}\ifnum`{=0 \fi\iffalse}\fi \ialign\bgroup\span\Preamble@\crcr} \newif\ifhline@\hline@false \newif\ifhline@@\hline@@false \def\hlinefor#1{\multispan@{\strip@#1 }\leaders\hrule height\D@cke\hfill% \global\hline@true\ignorespaces} \catcode`\@=13 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %Input-Datei zum Erzeugen von Gitterpunktwegen.% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %entering a path: % \Pfad(x-coordinate of starting point,y-coordinate of starting point),path % as 1-2-3-4-5-6-7-8-word\endPfad %(1=step in x-direction, 2=step in y-direction, 3=upward diagonal step, 4=downward diagonal step) %(5=step in negative x-direction, 6=step in negative y-direction, 7=downward left diagonal step, 8=upward left diagonal step) % %entering a dotted path: % \SPfad(x-coordinate of starting point,y-coordinate of starting point),path % as 1-2-3-4-5-6-7-8-word\endSPfad %(1=step in x-direction, 2=step in y-direction, 3=upward diagonal step, 4=downward diagonal step) %(5=step in negative x-direction, 6=step in negative y-direction, 7=downward left diagonal step, 8=upward left 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 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\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 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\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 \def\({\left(} \def\){\right)} \def\[{\left[} \def\]{\right]} \def\Gr{\operatorname{Gr}} \def\sgn{\operatorname{sgn}} \def\P{\Cal P} \topmatter \title On multiplicities of points on Schubert varieties in Grassmannians \endtitle \author C.~Krattenthaler\footnote"$^\dagger$"{\hbox{Partially supported by the Austrian Science Foundation FWF, grant P13190-MAT.}} \endauthor \affil Institut f\"ur Mathematik der Universit\"at Wien,\\ Strudlhofgasse 4, A-1090 Wien, Austria.\\ e-mail: KRATT\@Ap.Univie.Ac.At\\ WWW: \tt http://www.mat.univie.ac.at/People/kratt \endaffil \address Institut f\"ur Mathematik der Universit\"at Wien, Strudlhofgasse 4, A-1090 Wien, Austria. \endaddress %\email KRATT\@Ap.Univie.Ac.At\\ %WWW: \tt http://www.mat.univie.ac.at/People/kratt\endemail %\dedicatory \enddedicatory %\date \enddate %\thanks \endthanks \subjclass Primary 14M15; Secondary 05A15 05E15 14H20 \endsubjclass \keywords Schubert varieties, singularities, multiplicities, nonintersecting lattice paths, semistandard tableaux\endkeywords \abstract We answer some questions related to multiplicity formulas by Rosenthal and Zelevinsky and by Lakshmibai and Weyman for points on Schubert varieties in Grassmannians. In particular, we give combinatorial interpretations in terms of nonintersecting lattice paths of these formulas, which makes the equality of the two formulas immediately obvious. Furthermore we provide an alternative determinantal formula for these multiplicities, and we show that they count semistandard tableaux of unusual shapes. %As an application of these combinatorial %interpretations, a short derivation of the description %of the singular locus of a Schubert variety in a Grassmannian, %originally due to Svanes, is given. \endabstract \endtopmatter \document \leftheadtext{C. Krattenthaler} \subhead 1. Introduction \endsubhead The {\it multiplicity} of a point on an algebraic variety is an important invariant that ``measures" singularity of the point. Recently, Rosenthal and Zelevinsky \cite{\RoZeAA} gave a determinantal formula for the multiplicity of a point on a Schubert variety in a Grassmannian (see Theorem~1). This formula immediately raised three questions (and, indeed, they are asked in \cite{\RoZeAA, Remark~5, paragraph after Theorem~1, Remark~7}): \roster \item Is there a direct way to see that the formula yields positive integers (that is, leaving aside the fact that the Rosenthal--Zelevinsky theorem says that it gives multiplicities of singular points)? \item The formula is in form of a binomial determinant. Such determinants are very common in combinatorics. Is there a combinatorial interpretation? \item Lakshmibai and Weyman \cite{\LaWeAB, Theorem~5.4} give a different determinantal formula in a special case (see Theorem~2). It is not immediately clear why it agrees with the formula by Rosenthal and Zelevinsky. Is there a straightforward explanation? \endroster The purpose of this note is to answer these questions. In reply to Question~1 we show that, by means of the Lindstr\"om--Gessel--Viennot theorem \cite{\LindAA, Lemma~1}, \cite{\GeViAB, Theorem~1} (see Theorem~3), the formula of Rosenthal and Zelevinsky counts certain families of nonintersecting lattice paths. Clearly, this immediately explains why the formula yields positive integers. At the same time, this also provides a first answer to Question~2. %In addition, from this nonintersecting lattice path interpretation of the %multiplicities one can immediately read off the singular locus of a %Schubert variety in a Grassmannian, thus recovering a result %originally obtained by Svanes \cite{\SvanAA}. (In contrast, it is not %possible to do this directly from the Rosenthal--Zelevinsky determinant.) % In the special case considered by Lakshmibai and Weyman, we apply an easy combinatorial transformation to the families of nonintersecting lattice paths corresponding to the Rosenthal--Zelevinsky determinant formula and thus convert them into other families of nonintersecting lattice paths. These latter families of nonintersecting lattice paths directly yield the Lakshmibai--Weyman formula, again by means of the Lindstr\"om--Gessel--Viennot theorem. This answers Question~3. In addition, we use the ``dual path" idea by Gessel and Viennot \cite{\GeViAA, Sec.~4} to derive an alternative determinantal formula for the multiplicities in the general case. As a bonus, this enables us to also find a combinatorial description of the multiplicities as the numbers of semistandard tableaux of unusual shapes, thus providing another answer to Question~2. In the next section we review the basic definitions and the formulas by Rosenthal and Zelevinsky and by Lakshmibai and Weyman. Then, in Section~3, we first recall the Lindstr\"om--Gessel--Viennot theorem on nonintersecting paths, and then explain how to interpret the formulas by Rosenthal and Zelevinsky and by Lakshmibai and Weyman in terms of nonintersecting lattice paths, and why this immediately explains that they are equivalent in the relevant special case. We conclude this section by showing how the description of the singular locus of a Schubert variety in a Grassmannian follows directly from the combinatorics of nonintersecting lattice paths. Finally, in Section~4, we derive the alternative determinantal formula for the multiplicities (see Theorem~5) and its interpretation in terms of semistandard tableaux (see Corollary~6). \subhead 2. The multiplicity formulas by Rosenthal and Zelevinsky and by Lakshmibai and Weyman \endsubhead Let $d$ and $n$ be positive integers with $0\le d\le n$. The {\it Grassmannian} $\Gr_d(V)$ is the variety of all $d$-dimensional subspaces in an $n$-dimensional vector space $V$ (over some algebraically closed field of arbitrary characteristic). Given an integer vector $\bold i=(i_1,i_2,\dots,i_d)$, $1\le i_1d$, the path $Q_1$ starts with $d+1$ vertical steps. This forces all the other paths (if they want to be nonintersecting) to also be vertical until they reach height $d+1$. If they do not reach height $d+1$, then they terminate in a point $(i_\ell-d,d)$. All this is clearly visible in Figure~2. Hence, we may without loss of any information cut off the path portions until height $d+1$, respectively remove the paths that even do not reach that height. (The dotted line in Figure~2 indicates the line of height $d+1=8$ along which the cut is performed.) What remains is a family of paths that is familiar from the formula of Lakshmibai and Weyman, see Figure~1. This makes it obvious why formula (1) with $\bold j=(1,2,\dots,d)$ and formula (2) agree. \remark{Remark} The above nonintersecting lattice path interpretation of the multiplicity $M_{\bold j}(\bold i)$ can be used to determine all Schubert cells $X_{\bold j}^\circ$ in the Schubert variety $X_{\bold i}$ which consist of singular points. For, it is well-known that this amounts to determine, for fixed $\bold i$, those $\bold j$ which satisfy $M_{\bold j}(\bold i)>1$. In terms of the lattice path picture, this means to find those $\bold j$ which guarantee that the end points $(-s_{\ell},s_{\ell}+i_{\ell})$, $\ell=1,2,\dots,d$, of the paths are located in a manner such that there is more than one family of nonintersecting lattice paths connecting the starting points $(-\ell+1,\ell-1)$, $\ell=1,2,\dots,d$, to these end points. This is easily done, as it is easy to characterize the location of end points which yield just one such family of nonintersecting lattice paths. (In contrast, it is not possible to do this directly from the Rosenthal--Zelevinsky determinant.) Thus one would recover the well-known description of the singular locus of the Schubert variety $X_{\bold i}$ (cf\. e.g\. \cite{\BiLaAA, Theorem~9.3.1}). We leave the details to the reader for the sake of brevity. \endremark %\smallskip %A point in a variety is nonsingular if and only if its multiplicity %is 1. Equivalently, a point in a variety is singular if and only if %its multiplicity is greater than 1. %Hence, formulas for the multiplicity should in principle enable one %to determine the singular locus of a variety. Here the disadvantage %of the determinantal formulas by Rosenthal and Zelevinsky (given in %Theorem~1) and by Lakshmibai and Weyman (given in Theoremm~2) becomes %apparent, because it is hard to tell whether a determinant would %evaluate to 1 or not. (As was said earlier, it is even not directly %obvious why these formulas yield positive numbers.) However, our %interpretation in terms of nonintersecting lattice paths makes it %possible. %Let $\bold i$ and $\bold j$ be given. The question that we are asking %is whether $M_{\bold j}(\bold i)$ is 1 or larger. In combinatorial %terms, we are asking whether there is just one family %$(Q_1,Q_2,\dots,Q_d)$ of nonintersecting lattice paths, where paths %connect the starting points $(-\ell+1,\ell-1)$, $\ell=1,2,\dots,d$, %with the end points $(-s_{\ell},s_{\ell}+i_{\ell})$, %$\ell=1,2,\dots,d$, in some order (cf\. Figure~2). %For convenience, let us call a group of points %$$\multline %(a,b), (a,b-1), \dots, (a,b-d_1), \\ %(a-1,b-d_1), (a-2,b-d_1), \dots, %(a-c_1,b-d_1), \\ %(a-c_1,b-d_1-1), \dots, (a-c_1,b-d_1-d_2),\\ %(a-c_1-1,b-d_1-d_2), \dots, (a-c_1-c_2,b-d_1-d_2), \\ %\dots, %(a-c_1-\dots-c_h,b-d_1-\dots-d_h), %\endmultline$$ %for some integers $a$ and $b$ and nonnegative integers $c_k$ and $d_k$, %$k=1,2,\dots,h$, a %{\it zig-zag chain}, and denote it by %$Z(a,b,\bold c,\bold d)$, where $\bold c$ stands for $(c_1,\dots,c_h)$, %and similarly for $\bold d$. Taking into %account that $s_{\ell}+i_\ell$ is a nondecreasing function in $\ell$, %it costs no amount of time to see that the only way that this can %happen is when the end points come in groups that form %zig-zag chains $Z_1$, %$Z_2$, \dots, %$Z_N$, for some $N$, %with the additional property that the number of %points in $Z_k$ is exactly equal to the difference %of $x$-ccordinates of the ``right-most" %points in the zig-zag chains $Z_{k-1}$ and $Z_{k}$, i.e., if %$Z_{k-1}=Z(a_{k-1},b_{k-1},\bold c^{(k-1)},d^{(k-1)})$ and %$Z_k=Z(a_k,b_k,\bold c^{(k)},\bold d^{(k)})$, then the number of %points in $Z_k$ is $a_k-a_{k-1}$, $k=1,2,\dots,N$, with the convention %$a_0=-d$. This situation is exemplified in Figure~3, which %shows the only family of nonintersecting lattice paths for the choice %of $\bold i=(4,5,6,7,8,9,10,11,15,16,17,18,19,20)$ and $\bold %j=(1,2,3,4,5,6,8,9,12,13,14,15,16,17)$, so that $\bold %s=(10,9,8,8,7,6,6,6,2,1,0,0,0,0)$, and the occurring zig-zag chains are %$Z(-6,17,(2,1),(2,2))$ and $Z(0,20,(2),(3))$. %\midinsert %\vskip8pt %\vbox{ %$$ %\Einheit.35cm %\Gitter(1,21)(-14,0) %\PfadDicke{.5pt} %\Pfad(0,0),222222222222222222222\endPfad %\Pfad(-15,0),1111111111111111\endPfad %\PfadDicke{1.2pt} %\Pfad(-13,13),22221111111\endPfad %\Pfad(-12,12),2222111111\endPfad %\Pfad(-11,11),2222111\endPfad %\Pfad(-10,10),2222\endPfad %\Pfad(-9,9),22222\endPfad %\Pfad(-8,8),222222\endPfad %\Pfad(-7,7),22222222\endPfad %\Pfad(-6,6),222222222\endPfad %\Pfad(-5,5),22222222222222211111\endPfad %\Pfad(-4,4),2222222222222221111\endPfad %\Pfad(-3,3),222222222222222111\endPfad %\Pfad(-2,2),222222222222222\endPfad %\Pfad(-1,1),2222222222222222\endPfad %\Pfad(0,0),22222222222222222\endPfad %\DickPunkt(-13,13) %\DickPunkt(-12,12) %\DickPunkt(-11,11) %\DickPunkt(-10,10) %\DickPunkt(-9,9) %\DickPunkt(-8,8) %\DickPunkt(-7,7) %\DickPunkt(-6,6) %\DickPunkt(-5,5) %\DickPunkt(-4,4) %\DickPunkt(-3,3) %\DickPunkt(-2,2) %\DickPunkt(-1,1) %\DickPunkt(0,0) %\DickPunkt(-10,14) %\DickPunkt(-9,14) %\DickPunkt(-8,14) %\DickPunkt(-8,15) %\DickPunkt(-7,15) %\DickPunkt(-6,15) %\DickPunkt(-6,16) %\DickPunkt(-6,17) %\DickPunkt(-2,17) %\DickPunkt(-1,17) %\DickPunkt(0,17) %\DickPunkt(0,18) %\DickPunkt(0,19) %\DickPunkt(0,20) %\hskip-4.5cm %$$ %\centerline{\eightpoint Figure 3} %} %\vskip8pt %\endinsert %Such a situation arises if and only if the following is true: %Divide the sequence $\bold i$ into maximal strings of consecutive %integers, say %$$\bold i=(i_1,\dots,i_{k_1},i_{k_1+1},\dots,i_{k_2},\dots, %i_{k_{t-1}+1},\dots,i_{d}),\tag4$$ %where $i_{\mu+1}-i_{\mu}=1$ as long as $\mu$ is not equal to one of %the $k_u$'s. Then $i_{k_u}