\input amstex %Sekretariats-Stildatei % %AMSPPT.STY modifiziert und erweitert von C. Krattenthaler \def\next{AMS-SEKR}\ifx\styname\next \endinput\fi \catcode`\@=11 \def\styname{AMS-SEKR} \def\styversion{2.0} {\W@{}\W@{\styname.STY - Version \styversion}\W@{}} \hyphenation{acad-e-my acad-e-mies af-ter-thought anom-aly anom-alies an-ti-deriv-a-tive an-tin-o-my an-tin-o-mies apoth-e-o-ses apoth-e-o-sis ap-pen-dix ar-che-typ-al as-sign-a-ble as-sist-ant-ship as-ymp-tot-ic asyn-chro-nous at-trib-uted at-trib-ut-able bank-rupt bank-rupt-cy bi-dif-fer-en-tial blue-print busier busiest cat-a-stroph-ic cat-a-stroph-i-cally con-gress cross-hatched data-base de-fin-i-tive de-riv-a-tive dis-trib-ute dri-ver dri-vers eco-nom-ics econ-o-mist elit-ist equi-vari-ant ex-quis-ite ex-tra-or-di-nary flow-chart for-mi-da-ble forth-right friv-o-lous ge-o-des-ic ge-o-det-ic geo-met-ric griev-ance griev-ous griev-ous-ly hexa-dec-i-mal ho-lo-no-my ho-mo-thetic ideals idio-syn-crasy in-fin-ite-ly in-fin-i-tes-i-mal ir-rev-o-ca-ble key-stroke lam-en-ta-ble light-weight mal-a-prop-ism man-u-script mar-gin-al meta-bol-ic me-tab-o-lism meta-lan-guage me-trop-o-lis met-ro-pol-i-tan mi-nut-est mol-e-cule mono-chrome mono-pole mo-nop-oly mono-spline mo-not-o-nous mul-ti-fac-eted mul-ti-plic-able non-euclid-ean non-iso-mor-phic non-smooth par-a-digm par-a-bol-ic pa-rab-o-loid pa-ram-e-trize para-mount pen-ta-gon phe-nom-e-non post-script pre-am-ble pro-ce-dur-al pro-hib-i-tive pro-hib-i-tive-ly pseu-do-dif-fer-en-tial pseu-do-fi-nite pseu-do-nym qua-drat-ics quad-ra-ture qua-si-smooth qua-si-sta-tion-ary qua-si-tri-an-gu-lar quin-tes-sence quin-tes-sen-tial re-arrange-ment rec-tan-gle ret-ri-bu-tion retro-fit retro-fit-ted right-eous right-eous-ness ro-bot ro-bot-ics sched-ul-ing se-mes-ter semi-def-i-nite semi-ho-mo-thet-ic set-up se-vere-ly side-step sov-er-eign spe-cious spher-oid spher-oid-al star-tling star-tling-ly sta-tis-tics sto-chas-tic straight-est strange-ness strat-a-gem strong-hold sum-ma-ble symp-to-matic syn-chro-nous topo-graph-i-cal tra-vers-a-ble tra-ver-sal tra-ver-sals treach-ery turn-around un-at-tached un-err-ing-ly white-space wide-spread wing-spread wretch-ed wretch-ed-ly Brown-ian Eng-lish Euler-ian Feb-ru-ary Gauss-ian Grothen-dieck Hamil-ton-ian Her-mit-ian Jan-u-ary Japan-ese Kor-te-weg Le-gendre Lip-schitz Lip-schitz-ian Mar-kov-ian Noe-ther-ian No-vem-ber Rie-mann-ian Schwarz-schild Sep-tem-ber %Zus„tze (Sind auch in besp.exc einzutragen!): form per-iods Uni-ver-si-ty cri-ti-sism for-ma-lism} \Invalid@\nofrills \Invalid@\usualspace \newif\ifnofrills@ \def\nofrills@#1#2{\relaxnext@ \DN@{\ifx\next\nofrills \nofrills@true\let#2\relax\DN@\nofrills{\nextii@}% \else \nofrills@false\def#2{#1}\let\next@\nextii@\fi \next@}} \def\usualspace@#1{\ifnofrills@\def\usualspace{#1}\fi} \def\addto#1#2{\csname \expandafter\eat@\string#1@\endcsname \expandafter{\the\csname \expandafter\eat@\string#1@\endcsname#2}} \newdimen\bigsize@ \def\big@#1#2{{\hbox{$\left#2\vcenter to#1\bigsize@{}% \right.\nulldelimiterspace\z@\m@th$}}} \def\big{\big@\@ne} \def\Big{\big@{1.5}} \def\bigg{\big@\tw@} \def\Bigg{\big@{2.5}} \def\raggedcenter@{\leftskip\z@ plus.4\hsize \rightskip\leftskip \parfillskip\z@ \parindent\z@ \spaceskip.3333em \xspaceskip.5em \pretolerance9999\tolerance9999 \exhyphenpenalty\@M \hyphenpenalty\@M \let\\\linebreak} \def\upperspecialchars{\def\ss{SS}\let\i=I\let\j=J\let\ae\AE\let\oe\OE \let\o\O\let\aa\AA\let\l\L} \def\uppercasetext@#1{% {\spaceskip1.2\fontdimen2\the\font plus1.2\fontdimen3\the\font \upperspecialchars\uctext@#1$\m@th\aftergroup\eat@$}} \def\uctext@#1$#2${\endash@#1-\endash@$#2$\uctext@} \def\endash@#1-#2\endash@{\uppercase{#1}\if\notempty{#2}--\endash@#2\endash@\fi} \def\runaway@#1{\DN@{#1}\ifx\envir@\next@ \Err@{You seem to have a missing or misspelled \string\end#1 ...}% \let\envir@\empty\fi} \newif\iftemp@ \def\notempty#1{TT\fi\def\test@{#1}\ifx\test@\empty\temp@false \else\temp@true\fi \iftemp@} \font@\tensmc=cmcsc10 \font@\sevenex=cmex7 \font@\sevenit=cmti7 \font@\eightrm=cmr8 % preloaded in plain.tex \font@\sixrm=cmr6 % preloaded in plain.tex \font@\eighti=cmmi8 \skewchar\eighti='177 % preloaded \font@\sixi=cmmi6 \skewchar\sixi='177 % preloaded \font@\eightsy=cmsy8 \skewchar\eightsy='60 % preloaded \font@\sixsy=cmsy6 \skewchar\sixsy='60 % preloaded \font@\eightex=cmex8 \font@\eightbf=cmbx8 % preloaded in plain.tex \font@\sixbf=cmbx6 % preloaded in plain.tex \font@\eightit=cmti8 % preloaded in plain.tex \font@\eightsl=cmsl8 % preloaded in plain.tex \font@\eightsmc=cmcsc8 \font@\eighttt=cmtt8 % preloaded in plain.tex %\font@\ninerm=cmr9 %\font@\ninei=cmmi9 \skewchar\ninei='177 %\font@\ninesy=cmsy9 \skewchar\ninesy='60 %\font@\nineex=cmex9 %\font@\ninebf=cmbx9 %\font@\nineit=cmti9 %\font@\ninesl=cmsl9 %\font@\ninesmc=cmcsc9 %\font@\ninemsa=msam9 %\font@\ninemsb=msbm9 %\font@\nineeufm=eufm9 %Erg„nzung des fetten Small-Capitals-Fonts: %\font@\eightbsmc=cmbcsc10 scaled 833 %\font@\tenbsmc=cmbcsc10 %\font@\twelvebsmc=cmbcsc10 scaled \magstep1 %\font@\fourteenbsmc=cmbcsc10 scaled \magstep2 %\font@\seventeenbsmc=cmbcsc10 scaled \magstep3 %\font@\twentybsmc=cmbcsc10 scaled \magstep4 \loadmsam \loadmsbm \loadeufm \UseAMSsymbols \newtoks\tenpoint@ \def\tenpoint{\normalbaselineskip12\p@ \abovedisplayskip12\p@ plus3\p@ minus9\p@ \belowdisplayskip\abovedisplayskip \abovedisplayshortskip\z@ plus3\p@ \belowdisplayshortskip7\p@ plus3\p@ minus4\p@ \textonlyfont@\rm\tenrm \textonlyfont@\it\tenit \textonlyfont@\sl\tensl \textonlyfont@\bf\tenbf \textonlyfont@\smc\tensmc \textonlyfont@\tt\tentt %Erg„nzung des fetten Small-Capitals-Fonts: \textonlyfont@\bsmc\tenbsmc % \ifsyntax@ \def\big##1{{\hbox{$\left##1\right.$}}}% \let\Big\big \let\bigg\big \let\Bigg\big \else \textfont\z@=\tenrm \scriptfont\z@=\sevenrm \scriptscriptfont\z@=\fiverm \textfont\@ne=\teni \scriptfont\@ne=\seveni \scriptscriptfont\@ne=\fivei \textfont\tw@=\tensy \scriptfont\tw@=\sevensy \scriptscriptfont\tw@=\fivesy \textfont\thr@@=\tenex \scriptfont\thr@@=\sevenex \scriptscriptfont\thr@@=\sevenex \textfont\itfam=\tenit \scriptfont\itfam=\sevenit \scriptscriptfont\itfam=\sevenit \textfont\bffam=\tenbf \scriptfont\bffam=\sevenbf \scriptscriptfont\bffam=\fivebf \setbox\strutbox\hbox{\vrule height8.5\p@ depth3.5\p@ width\z@}% \setbox\strutbox@\hbox{\lower.5\normallineskiplimit\vbox{% \kern-\normallineskiplimit\copy\strutbox}}% \setbox\z@\vbox{\hbox{$($}\kern\z@}\bigsize@=1.2\ht\z@ \fi \normalbaselines\rm\ex@.2326ex\jot3\ex@\the\tenpoint@} \newtoks\eightpoint@ \def\eightpoint{\normalbaselineskip10\p@ \abovedisplayskip10\p@ plus2.4\p@ minus7.2\p@ \belowdisplayskip\abovedisplayskip \abovedisplayshortskip\z@ plus2.4\p@ \belowdisplayshortskip5.6\p@ plus2.4\p@ minus3.2\p@ \textonlyfont@\rm\eightrm \textonlyfont@\it\eightit \textonlyfont@\sl\eightsl \textonlyfont@\bf\eightbf \textonlyfont@\smc\eightsmc \textonlyfont@\tt\eighttt %Erg„nzung des fetten Small-Capitals-Fonts: \textonlyfont@\bsmc\eightbsmc % \ifsyntax@\def\big##1{{\hbox{$\left##1\right.$}}}% \let\Big\big \let\bigg\big \let\Bigg\big \else \textfont\z@=\eightrm \scriptfont\z@=\sixrm \scriptscriptfont\z@=\fiverm \textfont\@ne=\eighti \scriptfont\@ne=\sixi \scriptscriptfont\@ne=\fivei \textfont\tw@=\eightsy \scriptfont\tw@=\sixsy \scriptscriptfont\tw@=\fivesy \textfont\thr@@=\eightex \scriptfont\thr@@=\sevenex \scriptscriptfont\thr@@=\sevenex \textfont\itfam=\eightit \scriptfont\itfam=\sevenit \scriptscriptfont\itfam=\sevenit \textfont\bffam=\eightbf \scriptfont\bffam=\sixbf \scriptscriptfont\bffam=\fivebf \setbox\strutbox\hbox{\vrule height7\p@ depth3\p@ width\z@}% \setbox\strutbox@\hbox{\raise.5\normallineskiplimit\vbox{% \kern-\normallineskiplimit\copy\strutbox}}% \setbox\z@\vbox{\hbox{$($}\kern\z@}\bigsize@=1.2\ht\z@ \fi \normalbaselines\eightrm\ex@.2326ex\jot3\ex@\the\eightpoint@} %Definition von 12-point, 14-point und 17-point Fonts \font@\twelverm=cmr10 scaled\magstep1 \font@\twelveit=cmti10 scaled\magstep1 \font@\twelvesl=cmsl10 scaled\magstep1 \font@\twelvesmc=cmcsc10 scaled\magstep1 \font@\twelvett=cmtt10 scaled\magstep1 \font@\twelvebf=cmbx10 scaled\magstep1 \font@\twelvei=cmmi10 scaled\magstep1 \font@\twelvesy=cmsy10 scaled\magstep1 \font@\twelveex=cmex10 scaled\magstep1 \font@\twelvemsa=msam10 scaled\magstep1 \font@\twelveeufm=eufm10 scaled\magstep1 \font@\twelvemsb=msbm10 scaled\magstep1 \newtoks\twelvepoint@ \def\twelvepoint{\normalbaselineskip15\p@ \abovedisplayskip15\p@ plus3.6\p@ minus10.8\p@ \belowdisplayskip\abovedisplayskip \abovedisplayshortskip\z@ plus3.6\p@ \belowdisplayshortskip8.4\p@ plus3.6\p@ minus4.8\p@ \textonlyfont@\rm\twelverm \textonlyfont@\it\twelveit \textonlyfont@\sl\twelvesl \textonlyfont@\bf\twelvebf \textonlyfont@\smc\twelvesmc \textonlyfont@\tt\twelvett %Erg„nzung des fetten Small-Capitals-Fonts: \textonlyfont@\bsmc\twelvebsmc % \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@} \def\headfonts{\twelvepoint\bf} \def\rrm{\twelvepoint\rm} \def\bbf{\twelvepoint\bf} \def\bbsmc{\twelvepoint\bsmc} \font@\fourteenrm=cmr10 scaled\magstep2 \font@\fourteenit=cmti10 scaled\magstep2 \font@\fourteensl=cmsl10 scaled\magstep2 \font@\fourteensmc=cmcsc10 scaled\magstep2 \font@\fourteentt=cmtt10 scaled\magstep2 \font@\fourteenbf=cmbx10 scaled\magstep2 \font@\fourteeni=cmmi10 scaled\magstep2 \font@\fourteensy=cmsy10 scaled\magstep2 \font@\fourteenex=cmex10 scaled\magstep2 \font@\fourteenmsa=msam10 scaled\magstep2 \font@\fourteeneufm=eufm10 scaled\magstep2 \font@\fourteenmsb=msbm10 scaled\magstep2 \newtoks\fourteenpoint@ \def\fourteenpoint{\normalbaselineskip15\p@ \abovedisplayskip18\p@ plus4.3\p@ minus12.9\p@ \belowdisplayskip\abovedisplayskip \abovedisplayshortskip\z@ plus4.3\p@ \belowdisplayshortskip10.1\p@ plus4.3\p@ minus5.8\p@ \textonlyfont@\rm\fourteenrm \textonlyfont@\it\fourteenit \textonlyfont@\sl\fourteensl \textonlyfont@\bf\fourteenbf \textonlyfont@\smc\fourteensmc \textonlyfont@\tt\fourteentt %Erg„nzung des fetten Small-Capitals-Fonts: \textonlyfont@\bsmc\fourteenbsmc % \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@} \def\chapheadfonts{\fourteenpoint\bf} \def\rrrm{\fourteenpoint\rm} \def\bbbf{\fourteenpoint\bf} \def\bbbsmc{\fourteenpoint\bsmc} \font@\seventeenrm=cmr10 scaled\magstep3 \font@\seventeenit=cmti10 scaled\magstep3 \font@\seventeensl=cmsl10 scaled\magstep3 \font@\seventeensmc=cmcsc10 scaled\magstep3 \font@\seventeentt=cmtt10 scaled\magstep3 \font@\seventeenbf=cmbx10 scaled\magstep3 \font@\seventeeni=cmmi10 scaled\magstep3 \font@\seventeensy=cmsy10 scaled\magstep3 \font@\seventeenex=cmex10 scaled\magstep3 \font@\seventeenmsa=msam10 scaled\magstep3 \font@\seventeeneufm=eufm10 scaled\magstep3 \font@\seventeenmsb=msbm10 scaled\magstep3 \newtoks\seventeenpoint@ \def\seventeenpoint{\normalbaselineskip18\p@ \abovedisplayskip21.6\p@ plus5.2\p@ minus15.4\p@ \belowdisplayskip\abovedisplayskip \abovedisplayshortskip\z@ plus5.2\p@ \belowdisplayshortskip12.1\p@ plus5.2\p@ minus7\p@ \textonlyfont@\rm\seventeenrm \textonlyfont@\it\seventeenit \textonlyfont@\sl\seventeensl \textonlyfont@\bf\seventeenbf \textonlyfont@\smc\seventeensmc \textonlyfont@\tt\seventeentt %Erg„nzung des fetten Small-Capitals-Fonts: \textonlyfont@\bsmc\seventeenbsmc % \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@} \def\rrrrm{\seventeenpoint\rm} \def\bbbbf{\seventeenpoint\bf} \def\bbbbsmc{\seventeenpoint\bsmc} \font@\rrrrrm=cmr10 scaled\magstep4 \font@\bigtitlefont=cmbx10 scaled\magstep4 \let\bbbbbf\bigtitlefont \let\bbbbbsmc=\twentybsmc \parindent1pc \normallineskiplimit\p@ \newdimen\indenti \indenti=2pc \def\pageheight#1{\vsize#1} \def\pagewidth#1{\hsize#1% \captionwidth@\hsize \advance\captionwidth@-2\indenti} \pagewidth{30pc} \pageheight{47pc} \def\topmatter{% \ifx\undefined\msafam \else\font@\eightmsa=msam8 \font@\sixmsa=msam6 \ifsyntax@\else \addto\tenpoint{\textfont\msafam=\tenmsa \scriptfont\msafam=\sevenmsa \scriptscriptfont\msafam=\fivemsa}% \addto\eightpoint{\textfont\msafam=\eightmsa \scriptfont\msafam=\sixmsa \scriptscriptfont\msafam=\fivemsa}% \fi \fi \ifx\undefined\msbfam \else\font@\eightmsb=msbm8 \font@\sixmsb=msbm6 \ifsyntax@\else \addto\tenpoint{\textfont\msbfam=\tenmsb \scriptfont\msbfam=\sevenmsb \scriptscriptfont\msbfam=\fivemsb}% \addto\eightpoint{\textfont\msbfam=\eightmsb \scriptfont\msbfam=\sixmsb \scriptscriptfont\msbfam=\fivemsb}% \fi \fi \ifx\undefined\eufmfam \else \font@\eighteufm=eufm8 \font@\sixeufm=eufm6 \ifsyntax@\else \addto\tenpoint{\textfont\eufmfam=\teneufm \scriptfont\eufmfam=\seveneufm \scriptscriptfont\eufmfam=\fiveeufm}% \addto\eightpoint{\textfont\eufmfam=\eighteufm \scriptfont\eufmfam=\sixeufm \scriptscriptfont\eufmfam=\fiveeufm}% \fi \fi \ifx\undefined\eufbfam \else \font@\eighteufb=eufb8 \font@\sixeufb=eufb6 \ifsyntax@\else \addto\tenpoint{\textfont\eufbfam=\teneufb \scriptfont\eufbfam=\seveneufb \scriptscriptfont\eufbfam=\fiveeufb}% \addto\eightpoint{\textfont\eufbfam=\eighteufb \scriptfont\eufbfam=\sixeufb \scriptscriptfont\eufbfam=\fiveeufb}% \fi \fi \ifx\undefined\eusmfam \else \font@\eighteusm=eusm8 \font@\sixeusm=eusm6 \ifsyntax@\else \addto\tenpoint{\textfont\eusmfam=\teneusm \scriptfont\eusmfam=\seveneusm \scriptscriptfont\eusmfam=\fiveeusm}% \addto\eightpoint{\textfont\eusmfam=\eighteusm \scriptfont\eusmfam=\sixeusm \scriptscriptfont\eusmfam=\fiveeusm}% \fi \fi \ifx\undefined\eusbfam \else \font@\eighteusb=eusb8 \font@\sixeusb=eusb6 \ifsyntax@\else \addto\tenpoint{\textfont\eusbfam=\teneusb \scriptfont\eusbfam=\seveneusb \scriptscriptfont\eusbfam=\fiveeusb}% \addto\eightpoint{\textfont\eusbfam=\eighteusb \scriptfont\eusbfam=\sixeusb \scriptscriptfont\eusbfam=\fiveeusb}% \fi \fi \ifx\undefined\eurmfam \else \font@\eighteurm=eurm8 \font@\sixeurm=eurm6 \ifsyntax@\else \addto\tenpoint{\textfont\eurmfam=\teneurm \scriptfont\eurmfam=\seveneurm \scriptscriptfont\eurmfam=\fiveeurm}% \addto\eightpoint{\textfont\eurmfam=\eighteurm \scriptfont\eurmfam=\sixeurm \scriptscriptfont\eurmfam=\fiveeurm}% \fi \fi \ifx\undefined\eurbfam \else \font@\eighteurb=eurb8 \font@\sixeurb=eurb6 \ifsyntax@\else \addto\tenpoint{\textfont\eurbfam=\teneurb \scriptfont\eurbfam=\seveneurb \scriptscriptfont\eurbfam=\fiveeurb}% \addto\eightpoint{\textfont\eurbfam=\eighteurb \scriptfont\eurbfam=\sixeurb \scriptscriptfont\eurbfam=\fiveeurb}% \fi \fi \ifx\undefined\cmmibfam \else \font@\eightcmmib=cmmib8 \font@\sixcmmib=cmmib6 \ifsyntax@\else \addto\tenpoint{\textfont\cmmibfam=\tencmmib \scriptfont\cmmibfam=\sevencmmib \scriptscriptfont\cmmibfam=\fivecmmib}% \addto\eightpoint{\textfont\cmmibfam=\eightcmmib \scriptfont\cmmibfam=\sixcmmib \scriptscriptfont\cmmibfam=\fivecmmib}% \fi \fi \ifx\undefined\cmbsyfam \else \font@\eightcmbsy=cmbsy8 \font@\sixcmbsy=cmbsy6 \ifsyntax@\else \addto\tenpoint{\textfont\cmbsyfam=\tencmbsy \scriptfont\cmbsyfam=\sevencmbsy \scriptscriptfont\cmbsyfam=\fivecmbsy}% \addto\eightpoint{\textfont\cmbsyfam=\eightcmbsy \scriptfont\cmbsyfam=\sixcmbsy \scriptscriptfont\cmbsyfam=\fivecmbsy}% \fi \fi \let\topmatter\relax} \def\chapterno@{\uppercase\expandafter{\romannumeral\chaptercount@}} \newcount\chaptercount@ \def\chapter{\nofrills@{\afterassignment\chapterno@ CHAPTER \global\chaptercount@=}\chapter@ \DNii@##1{\leavevmode\hskip-\leftskip \rlap{\vbox to\z@{\vss\centerline{\eightpoint \chapter@##1\unskip}\baselineskip2pc\null}}\hskip\leftskip \nofrills@false}% \FN@\next@} \newbox\titlebox@ %Uppercase ist abgestellt. \def\title{\nofrills@{\relax}\title@% \DNii@##1\endtitle{\global\setbox\titlebox@\vtop{\tenpoint\bf \raggedcenter@\ignorespaces \baselineskip1.3\baselineskip\title@{##1}\endgraf}% \ifmonograph@ \edef\next{\the\leftheadtoks}\ifx\next\empty \leftheadtext{##1}\fi \fi \edef\next{\the\rightheadtoks}\ifx\next\empty \rightheadtext{##1}\fi }\FN@\next@} \newbox\authorbox@ \def\author#1\endauthor{\global\setbox\authorbox@ \vbox{\tenpoint\smc\raggedcenter@\ignorespaces #1\endgraf}\relaxnext@ \edef\next{\the\leftheadtoks}% \ifx\next\empty\leftheadtext{#1}\fi} \newbox\affilbox@ \def\affil#1\endaffil{\global\setbox\affilbox@ \vbox{\tenpoint\raggedcenter@\ignorespaces#1\endgraf}} \newcount\addresscount@ \addresscount@\z@ \def\address#1\endaddress{\global\advance\addresscount@\@ne \expandafter\gdef\csname address\number\addresscount@\endcsname {\vskip12\p@ minus6\p@\noindent\eightpoint\smc\ignorespaces#1\par}} \def\email{\nofrills@{\eightpoint{\it E-mail\/}:\enspace}\email@ \DNii@##1\endemail{% \expandafter\gdef\csname email\number\addresscount@\endcsname {\def\usualspace{{\it\enspace}}\smallskip\noindent\eightpoint\email@ \ignorespaces##1\par}}% \FN@\next@} \def\thedate@{} \def\date#1\enddate{\gdef\thedate@{\tenpoint\ignorespaces#1\unskip}} \def\thethanks@{} \def\thanks#1\endthanks{\gdef\thethanks@{\eightpoint\ignorespaces#1.\unskip}} \def\thekeywords@{} \def\keywords{\nofrills@{{\it Key words and phrases.\enspace}}\keywords@ \DNii@##1\endkeywords{\def\thekeywords@{\def\usualspace{{\it\enspace}}% \eightpoint\keywords@\ignorespaces##1\unskip.}}% \FN@\next@} \def\thesubjclass@{} \def\subjclass{\nofrills@{{\rm2000 {\it Mathematics Subject Classification\/}.\enspace}}\subjclass@ \DNii@##1\endsubjclass{\def\thesubjclass@{\def\usualspace {{\rm\enspace}}\eightpoint\subjclass@\ignorespaces##1\unskip.}}% \FN@\next@} \newbox\abstractbox@ \def\abstract{\nofrills@{{\smc Abstract.\enspace}}\abstract@ \DNii@{\setbox\abstractbox@\vbox\bgroup\noindent$$\vbox\bgroup \def\envir@{abstract}\advance\hsize-2\indenti \usualspace@{{\enspace}}\eightpoint \noindent\abstract@\ignorespaces}% \FN@\next@} \def\endabstract{\par\unskip\egroup$$\egroup} \def\widestnumber#1#2{\begingroup\let\head\null\let\subhead\empty \let\subsubhead\subhead \ifx#1\head\global\setbox\tocheadbox@\hbox{#2.\enspace}% \else\ifx#1\subhead\global\setbox\tocsubheadbox@\hbox{#2.\enspace}% \else\ifx#1\key\bgroup\let\endrefitem@\egroup \key#2\endrefitem@\global\refindentwd\wd\keybox@ \else\ifx#1\no\bgroup\let\endrefitem@\egroup \no#2\endrefitem@\global\refindentwd\wd\nobox@ \else\ifx#1\page\global\setbox\pagesbox@\hbox{\quad\bf#2}% \else\ifx#1\item\setboxz@h{#2}\global\rosteritemwd\wdz@ \global\advance\rosteritemwd by.5\parindent \else\message{\string\widestnumber is not defined for this option (\string#1)}% \fi\fi\fi\fi\fi\fi\endgroup} \newif\ifmonograph@ \def\Monograph{\monograph@true \let\headmark\rightheadtext \let\varindent@\indent \def\headfont@{\bf}\def\proclaimheadfont@{\smc}% \def\demofont@{\smc}} %Bei Proclaim,...: Einrcken. \let\varindent@\indent \newbox\tocheadbox@ \newbox\tocsubheadbox@ \newbox\tocbox@ \def\toc{\toc@{Contents}} \def\newtocdefs{% \def \title##1\endtitle {\penaltyandskip@\z@\smallskipamount \hangindent\wd\tocheadbox@\noindent{\bf##1}}% \def \chapter##1{% Chapter \uppercase\expandafter{\romannumeral##1.\unskip}\enspace}% \def \specialhead##1\endspecialhead {\par\hangindent\wd\tocheadbox@ \noindent##1\par}% \def \head##1 ##2\endhead {\par\hangindent\wd\tocheadbox@ \noindent \if\notempty{##1}\hbox to\wd\tocheadbox@{\hfil##1\enspace}\fi ##2\par}% \def \subhead##1 ##2\endsubhead {\par\vskip-\parskip {\normalbaselines \advance\leftskip\wd\tocheadbox@ \hangindent\wd\tocsubheadbox@ \noindent \if\notempty{##1}\hbox to\wd\tocsubheadbox@{##1\unskip\hfil}\fi ##2\par}}% \def \subsubhead##1 ##2\endsubsubhead {\par\vskip-\parskip {\normalbaselines \advance\leftskip\wd\tocheadbox@ \hangindent\wd\tocsubheadbox@ \noindent \if\notempty{##1}\hbox to\wd\tocsubheadbox@{##1\unskip\hfil}\fi ##2\par}}} \def\toc@#1{\relaxnext@ \def\page##1% {\unskip\penalty0\null\hfil \rlap{\hbox to\wd\pagesbox@{\quad\hfil##1}}\hfilneg\penalty\@M}% \DN@{\ifx\next\nofrills\DN@\nofrills{\nextii@}% \else\DN@{\nextii@{{#1}}}\fi \next@}% \DNii@##1{% \ifmonograph@\bgroup\else\setbox\tocbox@\vbox\bgroup \centerline{\headfont@\ignorespaces##1\unskip}\nobreak \vskip\belowheadskip \fi \setbox\tocheadbox@\hbox{0.\enspace}% \setbox\tocsubheadbox@\hbox{0.0.\enspace}% \leftskip\indenti \rightskip\leftskip \setbox\pagesbox@\hbox{\bf\quad000}\advance\rightskip\wd\pagesbox@ \newtocdefs }% \FN@\next@} \def\endtoc{\par\egroup} \let\pretitle\relax \let\preauthor\relax \let\preaffil\relax \let\predate\relax \let\preabstract\relax \let\prepaper\relax \def\dedicatory #1\enddedicatory{\def\preabstract{{\medskip \eightpoint\it \raggedcenter@#1\endgraf}}} \def\thetranslator@{} \def\translator#1\endtranslator{\def\thetranslator@{\nobreak\medskip \line{\eightpoint\hfil Translated by \uppercase{#1}\qquad\qquad}\nobreak}} \outer\def\endtopmatter{\runaway@{abstract}% \edef\next{\the\leftheadtoks}\ifx\next\empty \expandafter\leftheadtext\expandafter{\the\rightheadtoks}\fi \ifmonograph@\else \ifx\thesubjclass@\empty\else \makefootnote@{}{\thesubjclass@}\fi \ifx\thekeywords@\empty\else \makefootnote@{}{\thekeywords@}\fi \ifx\thethanks@\empty\else \makefootnote@{}{\thethanks@}\fi \fi \pretitle \ifmonograph@ \topskip7pc \else \topskip4pc \fi \box\titlebox@ \topskip10pt% reset to normal value \preauthor \ifvoid\authorbox@\else \vskip2.5pc plus1pc \unvbox\authorbox@\fi \preaffil \ifvoid\affilbox@\else \vskip1pc plus.5pc \unvbox\affilbox@\fi \predate \ifx\thedate@\empty\else \vskip1pc plus.5pc \line{\hfil\thedate@\hfil}\fi \preabstract \ifvoid\abstractbox@\else \vskip1.5pc plus.5pc \unvbox\abstractbox@ \fi \ifvoid\tocbox@\else\vskip1.5pc plus.5pc \unvbox\tocbox@\fi \prepaper \vskip2pc plus1pc } \def\document{\let\fontlist@\relax\let\alloclist@\relax \tenpoint} %Modifizierte Head-Skips \newskip\aboveheadskip \aboveheadskip1.8\bigskipamount \newdimen\belowheadskip \belowheadskip1.8\medskipamount \def\headfont@{\smc} \def\penaltyandskip@#1#2{\relax\ifdim\lastskip<#2\relax\removelastskip \ifnum#1=\z@\else\penalty@#1\relax\fi\vskip#2% \else\ifnum#1=\z@\else\penalty@#1\relax\fi\fi} \def\nobreak{\penalty\@M \ifvmode\def\penalty@{\let\penalty@\penalty\count@@@}% \everypar{\let\penalty@\penalty\everypar{}}\fi} \let\penalty@\penalty \def\heading#1\endheading{\head#1\endhead} \def\subheading#1{\subhead#1\endsubhead} \def\specialheadfont@{\bf} \outer\def\specialhead{\par\penaltyandskip@{-200}\aboveheadskip \begingroup\interlinepenalty\@M\rightskip\z@ plus\hsize \let\\\linebreak \specialheadfont@\noindent\ignorespaces} \def\endspecialhead{\par\endgroup\nobreak\vskip\belowheadskip} %\outer\def\head#1\endhead{\par\penaltyandskip@{-200}\aboveheadskip % {\headfont@\raggedcenter@\interlinepenalty\@M % \ignorespaces#1\endgraf}\nobreak % \vskip\belowheadskip % \headmark{#1}} \let\headmark\eat@ \newskip\subheadskip \subheadskip\medskipamount \def\subheadfont@{\bf} \outer\def\subhead{\nofrills@{.\enspace}\subhead@ \DNii@##1\endsubhead{\par\penaltyandskip@{-100}\subheadskip \varindent@{\usualspace@{{\subheadfont@\enspace}}% \subheadfont@\ignorespaces##1\unskip\subhead@}\ignorespaces}% \FN@\next@} \outer\def\subsubhead{\nofrills@{.\enspace}\subsubhead@ \DNii@##1\endsubsubhead{\par\penaltyandskip@{-50}\medskipamount {\usualspace@{{\it\enspace}}% \it\ignorespaces##1\unskip\subsubhead@}\ignorespaces}% \FN@\next@} \def\proclaimheadfont@{\bf} \outer\def\proclaim{\runaway@{proclaim}\def\envir@{proclaim}% \nofrills@{.\enspace}\proclaim@ \DNii@##1{\penaltyandskip@{-100}\medskipamount\varindent@ \usualspace@{{\proclaimheadfont@\enspace}}\proclaimheadfont@ \ignorespaces##1\unskip\proclaim@ \sl\ignorespaces}% \FN@\next@} \outer\def\endproclaim{\let\envir@\relax\par\rm \penaltyandskip@{55}\medskipamount} \def\demoheadfont@{\it} \def\demo{\runaway@{proclaim}\nofrills@{.\enspace}\demo@ \DNii@##1{\par\penaltyandskip@\z@\medskipamount {\usualspace@{{\demoheadfont@\enspace}}% \varindent@\demoheadfont@\ignorespaces##1\unskip\demo@}\rm \ignorespaces}\FN@\next@} \def\enddemo{\par\medskip} \def\qed{\ifhmode\unskip\nobreak\fi\quad\ifmmode\square\else$\m@th\square$\fi} \let\remark\demo %\endremark=\enddemo \let\endremark\enddemo %Headfont fr Definition wie bei Beweisen. \def\definition{\runaway@{proclaim}% \nofrills@{.\demoheadfont@\enspace}\definition@ \DNii@##1{\penaltyandskip@{-100}\medskipamount {\usualspace@{{\demoheadfont@\enspace}}% \varindent@\demoheadfont@\ignorespaces##1\unskip\definition@}% \rm \ignorespaces}\FN@\next@} \def\enddefinition{\par\medskip} %\let\example\definition %\let\endexample\enddefinition %Modifikation: \let\example\demo \let\endexample\enddemo \newdimen\rosteritemwd \newcount\rostercount@ \newif\iffirstitem@ \let\plainitem@\item \newtoks\everypartoks@ \def\par@{\everypartoks@\expandafter{\the\everypar}\everypar{}} \def\roster{\edef\leftskip@{\leftskip\the\leftskip}% \relaxnext@ \rostercount@\z@ \def\item{\FN@\rosteritem@}% \DN@{\ifx\next\runinitem\let\next@\nextii@\else \let\next@\nextiii@\fi\next@}% \DNii@\runinitem {\unskip \DN@{\ifx\next[\let\next@\nextii@\else \ifx\next"\let\next@\nextiii@\else\let\next@\nextiv@\fi\fi\next@}% \DNii@[####1]{\rostercount@####1\relax \enspace{\rm(\number\rostercount@)}~\ignorespaces}% \def\nextiii@"####1"{\enspace{\rm####1}~\ignorespaces}% \def\nextiv@{\enspace{\rm(1)}\rostercount@\@ne~}% \par@\firstitem@false \FN@\next@}% \def\nextiii@{\par\par@ \penalty\@m\smallskip\vskip-\parskip \firstitem@true}% \FN@\next@} \def\rosteritem@{\iffirstitem@\firstitem@false\else\par\vskip-\parskip\fi \leftskip3\parindent\noindent \DNii@[##1]{\rostercount@##1\relax \llap{\hbox to2.5\parindent{\hss\rm(\number\rostercount@)}% \hskip.5\parindent}\ignorespaces}% \def\nextiii@"##1"{% \llap{\hbox to2.5\parindent{\hss\rm##1}\hskip.5\parindent}\ignorespaces}% \def\nextiv@{\advance\rostercount@\@ne \llap{\hbox to2.5\parindent{\hss\rm(\number\rostercount@)}% \hskip.5\parindent}}% \ifx\next[\let\next@\nextii@\else\ifx\next"\let\next@\nextiii@\else \let\next@\nextiv@\fi\fi\next@} \def\therosteritem#1{{\rm(\ignorespaces#1\unskip)}} \newif\ifnextRunin@ \def\endroster{\relaxnext@ \par\leftskip@ \penalty-50 \vskip-\parskip\smallskip \DN@{\ifx\next\Runinitem\let\next@\relax \else\nextRunin@false\let\item\plainitem@ \ifx\next\par \DN@\par{\everypar\expandafter{\the\everypartoks@}}% \else \DN@{\noindent\everypar\expandafter{\the\everypartoks@}}% \fi\fi\next@}% \FN@\next@} \newcount\rosterhangafter@ \def\Runinitem#1\roster\runinitem{\relaxnext@ \rostercount@\z@ \def\item{\FN@\rosteritem@}% \def\runinitem@{#1}% \DN@{\ifx\next[\let\next\nextii@\else\ifx\next"\let\next\nextiii@ \else\let\next\nextiv@\fi\fi\next}% \DNii@[##1]{\rostercount@##1\relax \def\item@{{\rm(\number\rostercount@)}}\nextv@}% \def\nextiii@"##1"{\def\item@{{\rm##1}}\nextv@}% \def\nextiv@{\advance\rostercount@\@ne \def\item@{{\rm(\number\rostercount@)}}\nextv@}% \def\nextv@{\setbox\z@\vbox {\ifnextRunin@\noindent\fi \runinitem@\unskip\enspace\item@~\par \global\rosterhangafter@\prevgraf}% \firstitem@false \ifnextRunin@\else\par\fi \hangafter\rosterhangafter@\hangindent3\parindent \ifnextRunin@\noindent\fi \runinitem@\unskip\enspace \item@~\ifnextRunin@\else\par@\fi \nextRunin@true\ignorespaces}% \FN@\next@} \def\footmarkform@#1{$\m@th^{#1}$} \let\thefootnotemark\footmarkform@ \def\makefootnote@#1#2{\insert\footins {\interlinepenalty\interfootnotelinepenalty \eightpoint\splittopskip\ht\strutbox\splitmaxdepth\dp\strutbox \floatingpenalty\@MM\leftskip\z@\rightskip\z@\spaceskip\z@\xspaceskip\z@ \leavevmode{#1}\footstrut\ignorespaces#2\unskip\lower\dp\strutbox \vbox to\dp\strutbox{}}} \newcount\footmarkcount@ \footmarkcount@\z@ \def\footnotemark{\let\@sf\empty\relaxnext@ \ifhmode\edef\@sf{\spacefactor\the\spacefactor}\/\fi \DN@{\ifx[\next\let\next@\nextii@\else \ifx"\next\let\next@\nextiii@\else \let\next@\nextiv@\fi\fi\next@}% \DNii@[##1]{\footmarkform@{##1}\@sf}% \def\nextiii@"##1"{{##1}\@sf}% \def\nextiv@{\iffirstchoice@\global\advance\footmarkcount@\@ne\fi \footmarkform@{\number\footmarkcount@}\@sf}% \FN@\next@} \def\footnotetext{\relaxnext@ \DN@{\ifx[\next\let\next@\nextii@\else \ifx"\next\let\next@\nextiii@\else \let\next@\nextiv@\fi\fi\next@}% \DNii@[##1]##2{\makefootnote@{\footmarkform@{##1}}{##2}}% \def\nextiii@"##1"##2{\makefootnote@{##1}{##2}}% \def\nextiv@##1{\makefootnote@{\footmarkform@{\number\footmarkcount@}}{##1}}% \FN@\next@} \def\footnote{\let\@sf\empty\relaxnext@ \ifhmode\edef\@sf{\spacefactor\the\spacefactor}\/\fi \DN@{\ifx[\next\let\next@\nextii@\else \ifx"\next\let\next@\nextiii@\else \let\next@\nextiv@\fi\fi\next@}% \DNii@[##1]##2{\footnotemark[##1]\footnotetext[##1]{##2}}% \def\nextiii@"##1"##2{\footnotemark"##1"\footnotetext"##1"{##2}}% \def\nextiv@##1{\footnotemark\footnotetext{##1}}% \FN@\next@} \def\adjustfootnotemark#1{\advance\footmarkcount@#1\relax} \def\footnoterule{\kern-3\p@ \hrule width 5pc\kern 2.6\p@} \def\captionfont@{\smc} \def\topcaption#1#2\endcaption{% {\dimen@\hsize \advance\dimen@-\captionwidth@ \rm\raggedcenter@ \advance\leftskip.5\dimen@ \rightskip\leftskip {\captionfont@#1}% \if\notempty{#2}.\enspace\ignorespaces#2\fi \endgraf}\nobreak\bigskip} \def\botcaption#1#2\endcaption{% \nobreak\bigskip \setboxz@h{\captionfont@#1\if\notempty{#2}.\enspace\rm#2\fi}% {\dimen@\hsize \advance\dimen@-\captionwidth@ \leftskip.5\dimen@ \rightskip\leftskip \noindent \ifdim\wdz@>\captionwidth@ \else\hfil\fi {\captionfont@#1}\if\notempty{#2}.\enspace\rm#2\fi\endgraf}} \def\@ins{\par\begingroup\def\vspace##1{\vskip##1\relax}% \def\captionwidth##1{\captionwidth@##1\relax}% \setbox\z@\vbox\bgroup} % start a \vbox \def\block{\RIfMIfI@\nondmatherr@\block\fi \else\ifvmode\vskip\abovedisplayskip\noindent\fi $$\def\endblock{\par\egroup$$}\fi \vbox\bgroup\advance\hsize-2\indenti\noindent} \def\endblock{\par\egroup} \def\cite#1{{\rm[{\citefont@\m@th#1}]}} \def\citefont@{\rm} \def\refsfont@{\eightpoint} \outer\def\Refs{\runaway@{proclaim}% \relaxnext@ \DN@{\ifx\next\nofrills\DN@\nofrills{\nextii@}\else \DN@{\nextii@{References}}\fi\next@}% \DNii@##1{\penaltyandskip@{-200}\aboveheadskip \line{\hfil\headfont@\ignorespaces##1\unskip\hfil}\nobreak \vskip\belowheadskip \begingroup\refsfont@\sfcode`.=\@m}% \FN@\next@} \def\endRefs{\par\endgroup} \newbox\nobox@ \newbox\keybox@ \newbox\bybox@ \newbox\paperbox@ \newbox\paperinfobox@ \newbox\jourbox@ \newbox\volbox@ \newbox\issuebox@ \newbox\yrbox@ \newbox\pagesbox@ \newbox\bookbox@ \newbox\bookinfobox@ \newbox\publbox@ \newbox\publaddrbox@ \newbox\finalinfobox@ \newbox\edsbox@ \newbox\langbox@ \newif\iffirstref@ \newif\iflastref@ \newif\ifprevjour@ \newif\ifbook@ \newif\ifprevinbook@ \newif\ifquotes@ \newif\ifbookquotes@ \newif\ifpaperquotes@ \newdimen\bysamerulewd@ \setboxz@h{\refsfont@\kern3em} \bysamerulewd@\wdz@ \newdimen\refindentwd \setboxz@h{\refsfont@ 00. } \refindentwd\wdz@ \outer\def\ref{\begingroup \noindent\hangindent\refindentwd \firstref@true \def\nofrills{\def\refkern@{\kern3sp}}% \ref@} \def\ref@{\book@false \bgroup\let\endrefitem@\egroup \ignorespaces} \def\moreref{\endrefitem@\endref@\firstref@false\ref@}% \def\transl{\endrefitem@\endref@\firstref@false \book@false \prepunct@ \setboxz@h\bgroup \aftergroup\unhbox\aftergroup\z@ \def\endrefitem@{\unskip\refkern@\egroup}\ignorespaces}% \def\emptyifempty@{\dimen@\wd\currbox@ \advance\dimen@-\wd\z@ \advance\dimen@-.1\p@ \ifdim\dimen@<\z@ \setbox\currbox@\copy\voidb@x \fi} \let\refkern@\relax \def\endrefitem@{\unskip\refkern@\egroup \setboxz@h{\refkern@}\emptyifempty@}\ignorespaces \def\refdef@#1#2#3{\edef\next@{\noexpand\endrefitem@ \let\noexpand\currbox@\csname\expandafter\eat@\string#1box@\endcsname \noexpand\setbox\noexpand\currbox@\hbox\bgroup}% \toks@\expandafter{\next@}% \if\notempty{#2#3}\toks@\expandafter{\the\toks@ \def\endrefitem@{\unskip#3\refkern@\egroup \setboxz@h{#2#3\refkern@}\emptyifempty@}#2}\fi \toks@\expandafter{\the\toks@\ignorespaces}% \edef#1{\the\toks@}} \refdef@\no{}{. } \refdef@\key{[\m@th}{] } \refdef@\by{}{} \def\bysame{\by\hbox to\bysamerulewd@{\hrulefill}\thinspace \kern0sp} \def\manyby{\message{\string\manyby is no longer necessary; \string\by can be used instead, starting with version 2.0 of \styname.STY}\by} \refdef@\paper{\ifpaperquotes@``\fi\it}{} \refdef@\paperinfo{}{} \def\jour{\endrefitem@\let\currbox@\jourbox@ \setbox\currbox@\hbox\bgroup \def\endrefitem@{\unskip\refkern@\egroup \setboxz@h{\refkern@}\emptyifempty@ \ifvoid\jourbox@\else\prevjour@true\fi}% \ignorespaces} \refdef@\vol{\ifbook@\else\bf\fi}{} \refdef@\issue{no. }{} \refdef@\yr{}{} \refdef@\pages{}{} \def\page{\endrefitem@\def\pp@{\def\pp@{pp.~}p.~}\let\currbox@\pagesbox@ \setbox\currbox@\hbox\bgroup\ignorespaces} \def\pp@{pp.~} \def\book{\endrefitem@ \let\currbox@\bookbox@ \setbox\currbox@\hbox\bgroup\def\endrefitem@{\unskip\refkern@\egroup \setboxz@h{\ifbookquotes@``\fi}\emptyifempty@ \ifvoid\bookbox@\else\book@true\fi}% \ifbookquotes@``\fi\it\ignorespaces} \def\inbook{\endrefitem@ \let\currbox@\bookbox@\setbox\currbox@\hbox\bgroup \def\endrefitem@{\unskip\refkern@\egroup \setboxz@h{\ifbookquotes@``\fi}\emptyifempty@ \ifvoid\bookbox@\else\book@true\previnbook@true\fi}% \ifbookquotes@``\fi\ignorespaces} \refdef@\eds{(}{, eds.)} \def\ed{\endrefitem@\let\currbox@\edsbox@ \setbox\currbox@\hbox\bgroup \def\endrefitem@{\unskip, ed.)\refkern@\egroup \setboxz@h{(, ed.)}\emptyifempty@}(\ignorespaces} \refdef@\bookinfo{}{} \refdef@\publ{}{} \refdef@\publaddr{}{} \refdef@\finalinfo{}{} \refdef@\lang{(}{)} \def\toappear{\nofrills\finalinfo(to appear)} \let\refdef@\relax \def\ppunbox@#1{\ifvoid#1\else\prepunct@\unhbox#1\fi} \def\nocomma@#1{\ifvoid#1\else\changepunct@3\prepunct@\unhbox#1\fi} \def\changepunct@#1{\ifnum\lastkern<3 \unkern\kern#1sp\fi} \def\prepunct@{\count@\lastkern\unkern \ifnum\lastpenalty=0 \let\penalty@\relax \else \edef\penalty@{\penalty\the\lastpenalty\relax}% \fi \unpenalty \let\refspace@\ \ifcase\count@,% usual case, do a comma \or;\or.\or % do nothing; this case is from nofrills. \or\let\refspace@\relax \else,\fi \ifquotes@''\quotes@false\fi \penalty@ \refspace@ } \def\transferpenalty@#1{\dimen@\lastkern\unkern \ifnum\lastpenalty=0\unpenalty\let\penalty@\relax \else\edef\penalty@{\penalty\the\lastpenalty\relax}\unpenalty\fi #1\penalty@\kern\dimen@} \def\endref{\endrefitem@\lastref@true\endref@ \par\endgroup \prevjour@false \previnbook@false } \def\endref@{% \iffirstref@ \ifvoid\nobox@\ifvoid\keybox@\indent\fi \else\hbox to\refindentwd{\hss\unhbox\nobox@}\fi \ifvoid\keybox@ \else\ifdim\wd\keybox@>\refindentwd \box\keybox@ \else\hbox to\refindentwd{\unhbox\keybox@\hfil}\fi\fi \kern4sp\ppunbox@\bybox@ \fi \ifvoid\paperbox@ \else\prepunct@\unhbox\paperbox@ \ifpaperquotes@\quotes@true\fi\fi \ppunbox@\paperinfobox@ \ifvoid\jourbox@ \ifprevjour@ \nocomma@\volbox@ \nocomma@\issuebox@ \ifvoid\yrbox@\else\changepunct@3\prepunct@(\unhbox\yrbox@ \transferpenalty@)\fi \ppunbox@\pagesbox@ \fi \else \prepunct@\unhbox\jourbox@ \nocomma@\volbox@ \nocomma@\issuebox@ \ifvoid\yrbox@\else\changepunct@3\prepunct@(\unhbox\yrbox@ \transferpenalty@)\fi \ppunbox@\pagesbox@ \fi \ifbook@\prepunct@\unhbox\bookbox@ \ifbookquotes@\quotes@true\fi \fi \nocomma@\edsbox@ \ppunbox@\bookinfobox@ \ifbook@\ifvoid\volbox@\else\prepunct@ vol.~\unhbox\volbox@ \fi\fi \ppunbox@\publbox@ \ppunbox@\publaddrbox@ \ifbook@ \ppunbox@\yrbox@ \ifvoid\pagesbox@ \else\prepunct@\pp@\unhbox\pagesbox@\fi \else \ifprevinbook@ \ppunbox@\yrbox@ \ifvoid\pagesbox@\else\prepunct@\pp@\unhbox\pagesbox@\fi \fi \fi \ppunbox@\finalinfobox@ \iflastref@ \ifvoid\langbox@.\ifquotes@''\fi \else\changepunct@2\prepunct@\unhbox\langbox@\fi \else \ifvoid\langbox@\changepunct@1% \else\changepunct@3\prepunct@\unhbox\langbox@ \changepunct@1\fi \fi } \outer\def\enddocument{% \runaway@{proclaim}% \ifmonograph@ % do nothing \else \nobreak \thetranslator@ \count@\z@ \loop\ifnum\count@<\addresscount@\advance\count@\@ne \csname address\number\count@\endcsname \csname email\number\count@\endcsname \repeat \fi \vfill\supereject\end} %Modifizierte Fonts fr Proclaim,... \def\headfont@{\headfonts} \def\proclaimheadfont@{\bf} \def\specialheadfont@{\bf} \def\subheadfont@{\bf} \def\demoheadfont@{\smc} %Kontrollsequenzen fr Inhaltsverzeichnis und Index: \newif\ifThisToToc \ThisToTocfalse \newif\iftocloaded \tocloadedfalse \def\C@L{\noexpand\Cal}\def\B@B{\noexpand\Bbb}\def\fR@K{\noexpand\frak} \def\S@{\noexpand\S}\def\P@P{\noexpand\"} \def\xpar{\\} \def\writetoc#1{\iftocloaded\ifThisToToc\begingroup\def\totoc{} \def\Cal{\noexpand\C@L}\def\Bbb{\noexpand\B@B} \def\frak{\noexpand\fR@K}\def\goth{\frak}\def\S{\noexpand\S@} \def\"{\noexpand\P@P} \def\xpar{\par\penalty100000 }\def\idx##1{##1}\def\\{\xpar} \edef\next@{\write\toc{\noindent#1\leaderfill\noexpand\folio\par}}% \next@\endgroup\global\ThisToTocfalse\fi\fi} \def\leaderfill{\leaders\hbox to 1em{\hss.\hss}\hfill} \newif\ifindexloaded \indexloadedfalse \def\idx#1{\ifindexloaded\begingroup\def\ign{}\def\it{}\def\/{}% \def\smc{}\def\bf{}\def\tt{}% \def\Cal{\noexpand\C@L}\def\Bbb{\noexpand\B@B}% \def\frak{\noexpand\fR@K}\def\goth{\frak}\def\S{\noexpand\S@}% \def\"{\noexpand\P@P}% {\edef\next@{\write\index{#1, \noexpand\folio}}\next@}% \endgroup\fi{#1}} \def\ign#1{} \def\loadindex{\input amspptb.idx\relax} \def\loadtoc{\input amspptb.toc\relax} \def\totoc{\global\ThisToToctrue} \outer\def\head#1\endhead{\par\penaltyandskip@{-200}\aboveheadskip {\headfont@\raggedcenter@\interlinepenalty\@M \ignorespaces#1\endgraf}\nobreak \vskip\belowheadskip \headmark{#1}\writetoc{#1}} \outer\def\chaphead#1\endchaphead{\par\penaltyandskip@{-200}\aboveheadskip {\chapheadfonts\raggedcenter@\interlinepenalty\@M \ignorespaces#1\endgraf}\nobreak \vskip3\belowheadskip \headmark{#1}\writetoc{#1}} \def\folio{{\foliofont@\ifnum\pageno<\z@ \romannumeral-\pageno \else\number\pageno \fi}} %\def\foliofont@{\eightrm} %\def\headlinefont@{\eightpoint} %\def\leftheadline{\rlap{\folio}\hfill \iftrue\topmark\fi \hfill} %\def\rightheadline{\hfill \expandafter\iffalse\botmark\fi % \hfill \llap{\folio}} \newtoks\leftheadtoks \newtoks\rightheadtoks %Uppercase ist abgestellt: \def\leftheadtext{\nofrills@{\relax}\lht@ \DNii@##1{\leftheadtoks\expandafter{\lht@{##1}}% \mark{\the\leftheadtoks\noexpand\else\the\rightheadtoks} \ifsyntax@\setboxz@h{\def\\{\unskip\space\ignorespaces}% \headlinefont@##1}\fi}% \FN@\next@} %Uppercase ist abgestellt: \def\rightheadtext{\nofrills@{\relax}\rht@ \DNii@##1{\rightheadtoks\expandafter{\rht@{##1}}% \mark{\the\leftheadtoks\noexpand\else\the\rightheadtoks}% \ifsyntax@\setboxz@h{\def\\{\unskip\space\ignorespaces}% \headlinefont@##1}\fi}% \FN@\next@} %\headline={\def\chapter#1{\chapterno@. }% % \def\\{\unskip\space\ignorespaces}\headlinefont@ % \ifodd\pageno \rightheadline \else \leftheadline\fi} \def\NoRunningHeads{\global\runheads@false\global\let\headmark\eat@} \def\NoPageNumbers{\gdef\folio{}} %\def\logo@{\baselineskip2pc \hbox to\hsize{\hfil\eightpoint Typeset by % \AmSTeX}} \newif\iffirstpage@ \firstpage@true \newif\ifrunheads@ \runheads@true %Erg„nzungen zu Runningheads und Pagenumbers: \newdimen\fullhsize \fullhsize=\hsize \newdimen\fullvsize \fullvsize=\vsize \def\fullline{\hbox to\fullhsize} \let\nopagenumbers\NoPageNumbers \def\pagenumbers{\gdef\folio{\folio@}} \let\norunningheads\NoRunningHeads \def\userunningheads{\global\runheads@true} %Default: Seitennumerierung unten. \norunningheads \headline={\def\chapter#1{\chapterno@. }% \def\\{\unskip\space\ignorespaces}\ifrunheads@\headlinefont@ \ifodd\pageno\rightheadline \else\leftheadline\fi \else\hfil\fi\ifNoRunHeadline\global\NoRunHeadlinefalse\fi} \let\folio@\folio \def\foliofont@{\foliofont} \def\foliofont{\eightrm} \def\headlinefont@{\headlinefont} \def\headlinefont{\eightpoint\smc} \def\leftheadline{\rlap{\folio}\hfill \ifNoRunHeadline\else\iftrue\topmark\fi\fi \hfill} \def\rightheadline{\hfill\ifNoRunHeadline \else \expandafter\iffalse\botmark\fi\fi \hfill \llap{\folio}} \footline={{\eightpoint\bottremark}% \ifrunheads@\else\hfil{\let\foliofont\tenrm\folio}\fi\hfil} \def\bottremark{} %Definition von \norunninghead: \newif\ifNoRunHeadline \def\norunninghead{\global\NoRunHeadlinetrue} %\norunninghead \output={\output@} %\def\output@{\shipout\vbox{% % \iffirstpage@ \global\firstpage@false % \pagebody \logo@ \makefootline% % \else \ifrunheads@ \makeheadline \pagebody % \else \pagebody \makefootline \fi % \fi}% % \advancepageno \ifnum\outputpenalty>-\@MM\else\dosupereject\fi} % %Modifizierter Output + Index-Output: \newif\ifoffset\offsetfalse \output={\output@} \def\output@{% \ifoffset \ifodd\count0\advance\hoffset by0.5truecm \else\advance\hoffset by-0.5truecm\fi\fi \shipout\vbox{% \makeheadline \pagebody \makefootline }% \advancepageno \ifnum\outputpenalty>-\@MM\else\dosupereject\fi} \def\indexoutput#1{% \ifoffset \ifodd\count0\advance\hoffset by0.5truecm \else\advance\hoffset by-0.5truecm\fi\fi \shipout\vbox{\makeheadline \vbox to\fullvsize{\boxmaxdepth\maxdepth% \ifvoid\topins\else\unvbox\topins\fi% #1 % \ifvoid\footins\else % footnote info is present \vskip\skip\footins \footnoterule \unvbox\footins\fi \ifr@ggedbottom \kern-\dimen@ \vfil \fi}% \baselineskip2pc \makefootline}% \global\advance\pageno\@ne \ifnum\outputpenalty>-\@MM\else\dosupereject\fi} \newbox\partialpage \newdimen\halfsize \halfsize=0.5\fullhsize \advance\halfsize by-0.5em \def\begindoublecolumns{\output={\indexoutput{\unvbox255}}% \begingroup \def\line{\fullline} \output={\global\setbox\partialpage=\vbox{\unvbox255\bigskip}}\eject \output={\doublecolumnout}\hsize=\halfsize \vsize=2\fullvsize} \def\enddoublecolumns{\output={\balancecolumns}\eject \endgroup \pagegoal=\fullvsize% \output={\output@}} \def\doublecolumnout{\splittopskip=\topskip \splitmaxdepth=\maxdepth \dimen@=\fullvsize \advance\dimen@ by-\ht\partialpage \setbox0=\vsplit255 to \dimen@ \setbox2=\vsplit255 to \dimen@ \indexoutput{\pagesofar} \unvbox255 \penalty\outputpenalty} \def\pagesofar{\unvbox\partialpage \wd0=\hsize \wd2=\hsize \hbox to\fullhsize{\box0\hfil\box2}} \def\balancecolumns{\setbox0=\vbox{\unvbox255} \dimen@=\ht0 \advance\dimen@ by\topskip \advance\dimen@ by-\baselineskip \divide\dimen@ by2 \splittopskip=\topskip {\vbadness=10000 \loop \global\setbox3=\copy0 \global\setbox1=\vsplit3 to\dimen@ \ifdim\ht3>\dimen@ \global\advance\dimen@ by1pt \repeat} \setbox0=\vbox to\dimen@{\unvbox1} \setbox2=\vbox to\dimen@{\unvbox3} \pagesofar} \def\indexitems{\catcode`\^^M=\active \def\^^M={\par}% \everypar{\parindent=0pt\hangindent=1em\hangafter=1}% \noindent} \def\beginindex{\eightpoint\begindoublecolumns \indexitems\begingroup \raggedright\pretolerance10000} \def\endindex{\endgroup\enddoublecolumns\hsize=\fullhsize \vsize=\fullvsize\tenpoint} \tenpoint \catcode`\@=\active \def\nmb#1#2{#2} \def\smallheadings{\let\chapheadfonts\tenpoint\let\headfonts\tenpoint} %\def\qed{\ifhmode\unskip\nobreak\fi\ifmmode\ifinner\else\hskip5\p@\fi\fi % \hbox{\hskip5\p@\vrule width4\p@ height6\p@ depth1.5\p@\hskip\p@}} \tenpoint \catcode`\@=\active \font\Bf=cmbx12 \font\Rm=cmr12 \def\LL{\leavevmode\setbox0=\hbox{L}\hbox to\wd0{\hss\char'40L}} \def\al{\alpha} \def\be{\beta} \def\ga{\gamma} \def\de{\delta} \def\ep{\varepsilon} \def\ze{\zeta} \def\et{\eta} \def\th{\theta} \def\vt{\vartheta} \def\io{\iota} \def\ka{\kappa} \def\la{\lambda} \def\rh{\rho} \def\si{\sigma} \def\ta{\tau} \def\ph{\varphi} \def\ch{\chi} \def\ps{\psi} \def\om{\omega} \def\Ga{\Gamma} \def\De{\Delta} \def\Th{\Theta} \def\La{\Lambda} \def\Si{\Sigma} \def\Ph{\Phi} \def\Ps{\Psi} \def\Om{\Omega} \def\row#1#2#3{#1_{#2},\ldots,#1_{#3}} \def\rowup#1#2#3{#1^{#2},\ldots,#1^{#3}} \def\x{\times} \def\crf{} %used for crossreferencing, Tex should ignore. \def\rf{} %used for refencing (section-numbers) \def\rfnew{} %used for new-section numbers \def\P{{\Bbb P}} \def\R{{\Bbb R}} \def\X{{\Cal X}} \def\C{{\Bbb C}} \def\Mf{{\Cal Mf}} \def\FM{{\Cal F\Cal M}} \def\F{{\Cal F}} \def\G{{\Cal G}} \def\V{{\Cal V}} \def\T{{\Cal T}} \def\A{{\Cal A}} \def\N{{\Bbb N}} \def\Z{{\Bbb Z}} \def\Q{{\Bbb Q}} \def\ddt{\left.\tfrac \partial{\partial t}\right\vert_0} \def\dd#1{\tfrac \partial{\partial #1}} \def\today{\ifcase\month\or January\or February\or March\or April\or May\or June\or July\or August\or September\or October\or November\or December\fi \space\number\day, \number\year} \def\nmb#1#2{#2} %zum Nummerieren \def\iprod#1#2{\langle#1,#2\rangle} \def\pder#1#2{\frac{\partial #1}{\partial #2}} \def\iint{\int\!\!\int} \def\({\left(} \def\){\right)} \def\[{\left[} \def\]{\right]} \def\supp{\operatorname{supp}} \def\Df{\operatorname{Df}} \def\dom{\operatorname{dom}} \def\Ker{\operatorname{Ker}} \def\Tr{\operatorname{Tr}} \def\Res{\operatorname{Res}} \def\Aut{\operatorname{Aut}} \def\kgV{\operatorname{kgV}} \def\ggT{\operatorname{ggT}} \def\diam{\operatorname{diam}} \def\Im{\operatorname{Im}} \def\Re{\operatorname{Re}} \def\ord{\operatorname{ord}} \def\rang{\operatorname{rang}} \def\rng{\operatorname{rng}} \def\grd{\operatorname{grd}} \def\inv{\operatorname{inv}} \def\maj{\operatorname{maj}} \def\des{\operatorname{des}} \def\varmaj{\operatorname{\overline{maj}}} \def\vardes{\operatorname{\overline{des}}} \def\pvarmaj{\operatorname{\overline{maj}'}} \def\pmaj{\operatorname{maj'}} \def\ln{\operatorname{ln}} \def\der{\operatorname{der}} \def\Hom{\operatorname{Hom}} \def\tr{\operatorname{tr}} \def\Span{\operatorname{Span}} \def\grad{\operatorname{grad}} \def\div{\operatorname{div}} \def\rot{\operatorname{rot}} \def\Sp{\operatorname{Sp}} \def\sgn{\operatorname{sgn}} \def\liml{\lim\limits} \def\supl{\sup\limits} \def\bigcupl{\bigcup\limits} \def\bigcapl{\bigcap\limits} \def\limsupl{\limsup\limits} \def\liminfl{\liminf\limits} \def\intl{\int\limits} \def\suml{\sum\limits} \def\maxl{\max\limits} \def\minl{\min\limits} \def\prodl{\prod\limits} \def\tg{\operatorname{tan}} \def\ctg{\operatorname{cot}} \def\arctg{\operatorname{arctan}} \def\arccot{\operatorname{arccot}} \def\arcctg{\operatorname{arccot}} \def\tgh{\operatorname{tanh}} \def\ctgh{\operatorname{coth}} \def\arcsinh{\operatorname{arcsinh}} \def\arccosh{\operatorname{arccosh}} \def\arctgh{\operatorname{arctanh}} \def\arcctgh{\operatorname{arccoth}} \def\3{\ss} \catcode`\@=11 \def\dddot#1{\vbox{\ialign{##\crcr .\hskip-.5pt.\hskip-.5pt.\crcr\noalign{\kern1.5\p@\nointerlineskip} $\hfil\displaystyle{#1}\hfil$\crcr}}} \newif\iftab@\tab@false \newif\ifvtab@\vtab@false \def\tab{\bgroup\tab@true\vtab@false\vst@bfalse\Strich@false% \def\\{\global\hline@@false% \ifhline@\global\hline@false\global\hline@@true\fi\cr} \edef\l@{\the\leftskip}\ialign\bgroup\hskip\l@##\hfil&&##\hfil\cr} \def\endtab{\cr\egroup\egroup} \def\vtab{\vtop\bgroup\vst@bfalse\vtab@true\tab@true\Strich@false% \bgroup\def\\{\cr}\ialign\bgroup&##\hfil\cr} \def\endvtab{\cr\egroup\egroup\egroup} \def\stab{\D@cke0.5pt\null \bgroup\tab@true\vtab@false\vst@bfalse\Strich@true\Let@@\vspace@ \normalbaselines\offinterlineskip \openup\spreadmlines@ \edef\l@{\the\leftskip}\ialign \bgroup\hskip\l@##\hfil&&##\hfil\crcr} \def\endstab{\crcr\egroup \egroup} \newif\ifvst@b\vst@bfalse \def\vstab{\D@cke0.5pt\null \vtop\bgroup\tab@true\vtab@false\vst@btrue\Strich@true\bgroup\Let@@\vspace@ \normalbaselines\offinterlineskip \openup\spreadmlines@\bgroup} \def\endvstab{\crcr\egroup\egroup \egroup\tab@false\Strich@false} \newdimen\htstrut@ \htstrut@8.5\p@ \newdimen\htStrut@ \htStrut@12\p@ \newdimen\dpstrut@ \dpstrut@3.5\p@ \newdimen\dpStrut@ \dpStrut@3.5\p@ \def\openup{\afterassignment\@penup\dimen@=} \def\@penup{\advance\lineskip\dimen@ \advance\baselineskip\dimen@ \advance\lineskiplimit\dimen@ \divide\dimen@ by2 \advance\htstrut@\dimen@ \advance\htStrut@\dimen@ \advance\dpstrut@\dimen@ \advance\dpStrut@\dimen@} \def\Let@@{\relax\iffalse{\fi% \def\\{\global\hline@@false% \ifhline@\global\hline@false\global\hline@@true\fi\cr}% \iffalse}\fi} \def\matrix{\null\,\vcenter\bgroup \tab@false\vtab@false\vst@bfalse\Strich@false\Let@@\vspace@ \normalbaselines\openup\spreadmlines@\ialign \bgroup\hfil$\m@th##$\hfil&&\quad\hfil$\m@th##$\hfil\crcr \Mathstrut@\crcr\noalign{\kern-\baselineskip}} \def\endmatrix{\crcr\Mathstrut@\crcr\noalign{\kern-\baselineskip}\egroup \egroup\,} \def\smatrix{\D@cke0.5pt\null\, \vcenter\bgroup\tab@false\vtab@false\vst@bfalse\Strich@true\Let@@\vspace@ \normalbaselines\offinterlineskip \openup\spreadmlines@\ialign \bgroup\hfil$\m@th##$\hfil&&\quad\hfil$\m@th##$\hfil\crcr} \def\endsmatrix{\crcr\egroup \egroup\,\Strich@false} \newdimen\D@cke \def\Dicke#1{\global\D@cke#1} \newtoks\tabs@\tabs@{&} \newif\ifStrich@\Strich@false \newif\iff@rst \def\Stricherr@{\iftab@\ifvtab@\errmessage{\noexpand\s not allowed here. Use \noexpand\vstab!}% \else\errmessage{\noexpand\s not allowed here. Use \noexpand\stab!}% \fi\else\errmessage{\noexpand\s not allowed here. Use \noexpand\smatrix!}\fi} \def\format{\ifvst@b\else\crcr\fi\egroup\iffalse{\fi\ifnum`}=0 \fi\format@} \def\format@#1\\{\def\preamble@{#1}% \def\Str@chfehlt##1{\ifx##1\s\Stricherr@\fi\ifx##1\\\let\Next\relax% \else\let\Next\Str@chfehlt\fi\Next}% \def\c{\hfil\noexpand\ifhline@@\hbox{\vrule height\htStrut@% depth\dpstrut@ width\z@}\noexpand\fi% \ifStrich@\hbox{\vrule height\htstrut@ depth\dpstrut@ width\z@}% \fi\iftab@\else$\m@th\fi\the\hashtoks@\iftab@\else$\fi\hfil}% \def\r{\hfil\noexpand\ifhline@@\hbox{\vrule height\htStrut@% depth\dpstrut@ width\z@}\noexpand\fi% \ifStrich@\hbox{\vrule height\htstrut@ depth\dpstrut@ width\z@}% \fi\iftab@\else$\m@th\fi\the\hashtoks@\iftab@\else$\fi}% \def\l{\noexpand\ifhline@@\hbox{\vrule height\htStrut@% depth\dpstrut@ width\z@}\noexpand\fi% \ifStrich@\hbox{\vrule height\htstrut@ depth\dpstrut@ width\z@}% \fi\iftab@\else$\m@th\fi\the\hashtoks@\iftab@\else$\fi\hfil}% \def\s{\ifStrich@\ \the\tabs@\vrule width\D@cke\the\hashtoks@% \fi\the\tabs@\ }% \def\sa{\ifStrich@\vrule width\D@cke\the\hashtoks@% \the\tabs@\ % \fi}% \def\se{\ifStrich@\ \the\tabs@\vrule width\D@cke\the\hashtoks@\fi}% \def\cd{\hfil\noexpand\ifhline@@\hbox{\vrule height\htStrut@% depth\dpstrut@ width\z@}\noexpand\fi% \ifStrich@\hbox{\vrule height\htstrut@ depth\dpstrut@ width\z@}% \fi$\dsize\m@th\the\hashtoks@$\hfil}% \def\rd{\hfil\noexpand\ifhline@@\hbox{\vrule height\htStrut@% depth\dpstrut@ width\z@}\noexpand\fi% \ifStrich@\hbox{\vrule height\htstrut@ depth\dpstrut@ width\z@}% \fi$\dsize\m@th\the\hashtoks@$}% \def\ld{\noexpand\ifhline@@\hbox{\vrule height\htStrut@% depth\dpstrut@ width\z@}\noexpand\fi% \ifStrich@\hbox{\vrule height\htstrut@ depth\dpstrut@ width\z@}% \fi$\dsize\m@th\the\hashtoks@$\hfil}% \ifStrich@\else\Str@chfehlt#1\\\fi% \setbox\z@\hbox{\xdef\Preamble@{\preamble@}}\ifnum`{=0 \fi\iffalse}\fi \ialign\bgroup\span\Preamble@\crcr} \newif\ifhline@\hline@false \newif\ifhline@@\hline@@false \def\hlinefor#1{\multispan@{\strip@#1 }\leaders\hrule height\D@cke\hfill% \global\hline@true\ignorespaces} \def\Item "#1"{\par\noindent\hangindent2\parindent% \hangafter1\setbox0\hbox{\rm#1\enspace}\ifdim\wd0>2\parindent% \box0\else\hbox to 2\parindent{\rm#1\hfil}\fi\ignorespaces} \def\ITEM #1"#2"{\par\noindent\hangafter1\hangindent#1% \setbox0\hbox{\rm#2\enspace}\ifdim\wd0>#1% \box0\else\hbox to 0pt{\rm#2\hss}\hskip#1\fi\ignorespaces} \def\item"#1"{\par\noindent\hang% \setbox0=\hbox{\rm#1\enspace}\ifdim\wd0>\the\parindent% \box0\else\hbox to \parindent{\rm#1\hfil}\enspace\fi\ignorespaces} \let\plainitem@\item \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 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 \hsize13cm \vsize19cm \newdimen\fullhsize \newdimen\fullvsize \newdimen\halfsize \fullhsize13cm \fullvsize19cm \halfsize=0.5\fullhsize \advance\halfsize by-0.5em \magnification1200 %First page headline in AmS-TeX for S\'eminaire Lotharingien de Combinatoire %\norunninghead in amspptbk.sty musz ausgeschaltet werden!!! \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 54 \rms (2006), Article~B54l\hfill} \userunningheads \TagsOnRight \def\AnBBAA{1} \def\ArmDAA{2} \def\AthaAI{3} \def\AtReAA{4} \def\AtTzAA{5} \def\BesDAA{6} \def\BjBrAB{7} \def\BRWaAA{8} \def\BRWaAB{9} \def\CartAA{10} \def\ChaFAA{11} \def\EdelAA{12} \def\FoReAA{13} \def\FoReAB{14} \def\GoJaAS{15} \def\HumpAC{16} \def\KratCB{17} \def\KrewAC{18} \def\PropAI{19} \def\ReivAG{20} \def\SimiAD{21} \def\StanBZ{22} \def\StanAP{23} \def\StemAZ{24} %equation numbers \def\AA{2.1} \def\AB{2.2} \def\AC{3.1} \def\AD{4.1} \def\AE{4.2} \def\AF{4.3} \def\AFa{4.4} \def\AFb{4.5} \def\AFc{4.6} \def\AG{4.7} \def\AGa{4.8} \def\AGb{4.9} \def\AGc{4.10} \def\AGd{4.11} \def\AR{4.12} \def\SIa{4.13} \def\SIb{4.14} \def\SIc{4.15} \def\SId{4.16} \def\AHa{4.17} \def\AH{4.18} \def\AHb{4.19} \def\AHc{4.20a} \def\AHd{4.20b} \def\AI{5.1} \def\AJ{5.2} \def\AKa{6.1} \def\AKb{6.2} \def\AKc{6.3} %\def\AKd{6.4} \def\AKe{6.4} \def\AKf{6.5} \def\AKg{6.6} %\def\AKh{6.7} \def\AKi{6.7} \def\AKj{6.8} %\def\AKk{6.11} \def\AK{6.9} \def\ALa{7.1} \def\ALb{7.2} \def\ALc{7.3} \def\ALd{7.4} \def\ALe{7.5} \def\ALf{7.6} \def\ALg{7.7} \def\ALh{7.8} \def\ALgg{7.9} \def\ALhh{7.10} \def\ALi{7.11} \def\ALj{7.12} \def\ALk{7.13} \def\ALl{7.14} \def\ALm{7.15} \def\ALmm{7.16} \def\ALmmm{7.17} \def\ALn{7.18} \def\ALo{7.19} \def\ALp{7.20} \def\ALq{7.21} \def\ALpp{7.22} \def\ALr{7.23} \def\AL{7.24} \def\AM{8.1} \def\AMa{8.2} \def\AN{8.3} \def\AO{9.1} \def\AOa{9.2} \def\AOb{9.3} \def\AOc{9.4} \def\AOd{9.6} \def\AOe{9.5} \def\AT{A.1} \def\AU{A.2} \def\AV{A.3} \def\AW{A.4} \def\AX{A.5} \def\AP{A.6} \def\AS{A.7} %theorem numbers \def\PA{1} \def\PB{2} \def\PE{3} \def\PF{4} %\def\LB{5} \def\LBa{5} \def\PC{6} \def\PD{7} \def\TA{8} \def\CB{9} \def\al{\alpha} \def\be{\beta} \def\coef#1{\left\langle#1\right\rangle} \def\rk{\operatorname{rk}} \def\Nar{\operatorname{Nar}} \topmatter \title The $M$-triangle of generalised non-crossing partitions for the types $E_7$ and $E_8$ \endtitle \author C.~Krattenthaler$^{\dagger}$ \endauthor \affil Fakult\"at f\"ur Mathematik, Universit\"at Wien,\\ Nordbergstra{\ss}e~15, A-1090 Vienna, Austria.\\ WWW: \tt http://www.mat.univie.ac.at/\~{}kratt \endaffil \address Fakult\"at f\"ur Mathematik, Universit\"at Wien, Nordbergstra{\ss}e~15, A-1090 Vienna, Austria. WWW: \tt http://www.mat.univie.ac.at/\~{}kratt \endaddress \dedicatory D\'edi\'e \`a Xavier Viennot \enddedicatory \thanks $^\dagger$Research partially supported by the Austrian Science Foundation FWF, grant S9607-N13, in the framework of the National Research Network ``Analytic Combinatorics and Probabilistic Number Theory," and by the ``Algebraic Combinatorics" Programme during Spring 2005 of the Institut Mittag--Leffler of the Royal Swedish Academy of Sciences% \endthanks \subjclass Primary 05E15; Secondary 05A05 05A15 05A19 06A07\linebreak 20F55 \endsubjclass \keywords root systems, generalised non-crossing partitions, M\"obius function, $M$-triangle, generalised cluster complex, face numbers, $F$-triangle \endkeywords \abstract The $M$-triangle of a ranked locally finite poset $P$ is the generating function $\sum _{u,w\in P} ^{}\mu(u,w)\,x^{\rk u}y^{\rk w}$, where $\mu(.,.)$ is the M\"obius function of $P$. We compute the $M$-triangle of Armstrong's poset of $m$-divisible non-crossing partitions for the root systems of type $E_7$ and $E_8$. For the other types except $D_n$ this had been accomplished in the earlier paper ``The $F$-triangle of the generalised cluster complex" [in: ``Topics in Discrete Mathematics,'' M.~Klazar, J.~Kratochvil, M.~Loebl, J.~Matou\v sek, R.~Thomas and P.~Valtr, eds., Springer--Verlag, Berlin, New York, 2006, pp.~93--126]. Altogether, this almost settles Armstrong's $F=M$ Conjecture, predicting a surprising relation between the $M$-triangle of the $m$-divisible partitions poset and the $F$-triangle (a certain refined face count) of the generalised cluster complex of Fomin and Reading, the only gap remaining in type $D_n$. Moreover, we prove a reciprocity result for this $M$-triangle, again with the possible exception of type $D_n$. Our results are based on the calculation of certain decomposition numbers for the reflection groups of types $E_7$ and $E_8$, which carry in fact finer information than the $M$-triangle does. The decomposition numbers for the other exceptional reflection groups had been computed in the earlier paper. As an aside, we show that there is a closed form product formula for the type $A_n$ decomposition numbers, leaving the problem of computing the type $B_n$ and type $D_n$ decomposition numbers open. \endabstract \endtopmatter \vskip.5cm \document \leftheadtext{C. Krattenthaler} %\topmark{C. Krattenthaler} \subhead 1. Introduction\endsubhead The lattice of non-crossing partitions of Kreweras \cite{\KrewAC} is a now classical object of study in combinatorics with many fascinating properties (see \cite{\SimiAD}). In \cite{\EdelAA}, Edelman generalised non-crossing partitions to $m$-divisible non-crossing partitions and showed that they, too, have many beautiful properties. Recently, Bessis \cite{\BesDAA} and Brady and Watt \cite{\BRWaAA} gave a uniform definition of non-crossing partitions associated to root systems, which includes Kreweras non-crossing partitions as ``type $A_n$" non-crossing partitions, as well as Reiner's \cite{\ReivAG} ``type $B_n$" non-crossing partitions. (As it turned out, Reiner did not have the ``right" definition of ``type $D_n$" non-crossing partitions). The question whether Edelman's \cite{\EdelAA} $m$-divisible non-crossing partitions also allow for an extension to root systems was answered by Armstrong \cite{\ArmDAA}, who introduced $m$-divisible non-crossing partitions for all root systems uniformly. (He calls them also ``generalised" non-crossing partitions for root systems.) Extending an earlier conjecture of Chapoton \cite{\ChaFAA} (which, in the meantime, has become a theorem due to Athanasiadis \cite{\AthaAI}), Armstrong predicted a relation between the ``$M$-triangle" of his $m$-divisible non-crossing partitions poset and the ``$F$-triangle" of the generalised cluster complex of Fomin and Reading \cite{\FoReAA}. (Roughly speaking, the $M$-triangle is a bi-variate generating function involving the M\"obius function, while the $F$-triangle is a refined face count. The reader is referred to Section~2 for the definition of the $M$-triangle, and to Section~8 for the definition of the $F$-triangle.) In the sequel, we shall refer to Armstrong's conjecture as the ``$F=M$ Conjecture." In \cite{\KratCB}, the $F$-triangle of the generalised cluster complex was computed for all types, and the $M$-triangle of the $m$-divisible non-crossing partitions poset was computed for all types except for the types $D_n$, $E_7$, and $E_8$. Thus, the $F=M$ Conjecture could be verified for all types except the afore-mentioned three. The purpose of this paper is to compute the $M$-triangle of the $m$-divisible non-crossing partitions poset for the exceptional types $E_7$ and $E_8$. Thus, the $F=M$ Conjecture remains open only for $D_n$. It should be noted, however, that a conjectural expression for the $M$-triangle of the $m$-divisible non-crossing partitions poset appears in \cite{\KratCB, Sec.~11, Prop.~D}. Moreover, since, in the case of type $D_n$, Section~11 of \cite{\KratCB} contains a proof for $m=1$ (independent from the one in \cite{\AthaAI}), the results from \cite{\KratCB} together with the results of the present paper provide a case-by-case proof of the $m=1$ case of the $F=M$ Conjecture, that is, of Chapoton's (ex-)conjecture. In order to compute the $M$-triangle of the $m$-divisible partitions poset for $E_7$ and $E_8$, we compute certain decomposition numbers in the corresponding reflection groups (see Section~3 for the definition). These decomposition numbers carry, in fact, finer information than the $M$-triangle. They are therefore of intrinsic interest. For the other reflection groups of exceptional type, they had been already computed in \cite{\KratCB}, but not for the types $A_n$, $B_n$, and $D_n$. In Section~10, we point out that, in fact, a result of Goulden and Jackson \cite{\GoJaAS, Theorem~3.2} on the minimal factorization of a long cycle implies a closed form product formula for the type $A_n$ decomposition numbers. We plan to address the problem of computing the type $B_n$ and type $D_n$ decomposition numbers in a future publication. Our paper is organised as follows. The next section contains the definition and basics of non-crossing partitions for root systems and the $m$-divisible non-crossing partitions. There, we define also the main object of this paper, the $M$-triangle of the $m$-divisible non-crossing partitions poset. Section~3 recalls the formula from \cite{\KratCB} which expresses the $M$-triangle in terms of the afore-mentioned decomposition numbers and the characteristic polynomials of non-crossing partitions posets of lower rank. Section~4 explains our strategy of how to compute these decomposition numbers for the types $E_7$ and $E_8$. (This strategy is actually applicable for any specific reflection group, except that the computational difficulties increase significantly with the rank of the reflection group.) The intermediate Section~5 recalls from \cite{\KratCB} how these decomposition numbers can be used to compute the characteristic polynomial of the non-crossing partitions poset corresponding to the reflection group. The programme from Section~4 is then implemented in Section~6 to compute the $M$-triangle of the $m$-divisible non-crossing partitions poset of type $E_7$ and in Section~7 to compute the $M$-triangle of the $m$-divisible non-crossing partitions poset of type $E_8$. The purpose of Section~8 is to briefly explain the $F=M$ Conjecture, and why the results from Sections~6 and 7 together with results from \cite{\KratCB} prove it for the types $E_7$ and $E_8$. Section~9 presents a curious observation which results from our explicit expressions in this paper and in \cite{\KratCB} for the $M$-triangle of the $m$-divisible non-crossing partitions poset: a reciprocity relation which sets the $M$-triangle with parameter $m$ in relation with the $M$-triangle with parameter $-m$. It would be interesting to find an intrinsic explanation of this phenomenon. We conclude the paper by addressing the type $A_n$ decomposition numbers. Theorem~9 in Section~10 states the afore-mentioned result by Goulden and Jackson in the language of the present paper, and Theorem~10 gives the implied result on arbitrary type $A_n$ decomposition numbers. An Appendix lists the decomposition numbers for the types $A_1,A_2,A_3,A_4,A_5,A_6,A_7,D_4,D_5,D_6,D_7,E_6$, which are needed in our calculations of Sections~6 and 7. \subhead 2. Generalised non-crossing partitions\endsubhead In this section we recall the definition of Armstrong's \cite{\ArmDAA} $m$-divisible non-crossing partitions poset, and we define its $M$-triangle. Let $\Phi$ be a finite root system of rank $n$. (We refer the reader to \cite{\HumpAC} for all root system terminology.) For an element $\al\in\Phi$, let $t_\al$ denote the reflection in the central hyperplane perpendicular to $\al$. Let $W=W(\Phi)$ be the group generated by these reflections. By definition, any element $w$ of $W$ can be represented as a product $w=t_1t_2\cdots t_\ell$, where the $t_i$'s are reflections. We call the minimal number of reflections which is needed for such a product representation the {\it absolute length\/} of $w$, and we denote it by $\ell_T(w)$. We then define the {\it absolute order} on $W$, denoted by $\le_T$, by $$u\le_T w\quad \text{if and only if}\quad \ell_T(w)=\ell_T(u)+\ell_T(u^{-1}w).$$ It can be shown that this is equivalent to the statement that any shortest product representation of $u$ by reflections occurs as an initial segment in some shortest product representation of $w$ by reflections. We can now define the {\it non-crossing partition lattice $NC(\Phi)$}. Let $c$ be a {\it Coxeter element\/} in $W$, that is, the product of all reflections corresponding to the simple roots. Then $NC(\Phi)$ is defined to be the restriction of the partial order $\le_T$ to the set of all elements which are less than or equal to $c$ in absolute order. This definition makes sense because, regardless of the chosen Coxeter element $c$, the resulting poset is always the same up to isomorphism. It can be shown that $NC(\Phi)$ is in fact a lattice (see \cite{\BRWaAB} for a uniform proof), and moreover self-dual (this is obvious from the definition). Clearly, the minimal element in $NC(\Phi)$ is the identity element in $W$, which we denote by $\ep$, and the maximal element in $NC(\Phi)$ is the chosen Coxeter element $c$. The term ``non-crossing partition lattice" is used because $NC(A_n)$ is isomorphic to the lattice of non-crossing partitions originally introduced by Kreweras \cite{\KrewAC} (see also \cite{\FoReAB}), and because also $NC(B_n)$ and $NC(D_n)$ can be realised as lattices of non-crossing partitions (see \cite{\AtReAA, \ReivAG}). The poset of {\it $m$-divisible non-crossing partitions} has as a ground-set the following subset of $(NC(\Phi))^{m+1}$, $$\multline NC^m(\Phi)=\{(w_0;w_1,\dots,w_m):w_0w_1\cdots w_m=c\text{ and }\\ \ell_T(w_0)+\ell_T(w_1)+\dots+\ell_T(w_m)=\ell_T(c)\}. \endmultline\tag\AA$$ The order relation is defined by $$(u_0;u_1,\dots,u_m)\le(w_0;w_1,\dots,w_m)\quad \text{if and only if}\quad u_i\ge_T w_i,\ 1\le i\le m.$$ We emphasize that, according to this definition, $u_0$ and $w_0$ need not be related in any way. The poset $NC^m(\Phi)$ is graded by the rank function $$\rk\big((w_0;w_1,\dots,w_m)\big)=\ell_T(w_0).$$ Thus, there is a unique maximal element, namely $(c;\ep,\dots,\ep)$, where $\ep$ stands for the identity element in $W$, but, if $m>1$, there are several different minimal elements. In particular, there is no global minimum in $NC^m(\Phi)$ if $m>1$ and, hence, $NC^m(\Phi)$ is not a lattice for $m>1$. (It is, however, a graded join-semilattice, see \cite{\ArmDAA, Theorem~3.4.4}.) The central object in the present paper is the ``$M$-triangle" of $NC^m(\Phi)$, which is the polynomial defined by $$ M^m_\Phi(x,y)=\sum _{u,w\in NC^m(\Phi)} ^{}\mu(u,w)\,x^{\rk u}y^{\rk w},$$ where $\mu(u,w)$ is the M\"obius function in $NC^m(\Phi)$. It is called ``triangle" because the M\"obius function $\mu(u,w)$ vanishes unless $u\le w$, and, thus, the only coefficients in the polynomial which may be non-zero are the coefficients of $x^ky^l$ with $0\le k\le l\le n$. An equivalent object is the {\it dual $M$-triangle}, which is defined by $$ (M^m_\Phi)^*(x,y)= \sum _{u,w\in(NC^m(\Phi))^*} ^{}\mu^*(u,w)\,x^{\rk^*w}y^{\rk^*u}, $$ where $(NC^m(\Phi))^*$ denotes the poset {\rm dual} to $NC^m(\Phi)$ {\rm(}i.e., the poset which arises from $NC^m(\Phi)$ by reversing all order relations{\rm)}, where $\mu^*$ denotes the M\"obius function in\linebreak $(NC^m(\Phi))^*$, and where $\rk^*$ denotes the rank function in $(NC^m(\Phi))^*$. It is equivalent since, obviously, we have $$ (M^m_\Phi)^*(x,y)=(xy)^nM^m_\Phi(1/x,1/y). \tag\AB$$ The dual $M$-triangle of $NC^m(\Phi)$ (and, thus, its $M$-triangle as well) was computed explicitly in \cite{\KratCB} for all types, except for $D_n$ (a conjectural expression appears, however, in \cite{\KratCB, Sec.~11, Prop.~D}), for $E_7$, and for $E_8$. In Sections~6 and 7 below, we fill this gap for $E_7$ and $E_8$. Thus, it is only the case of $D_n$ which remains open. \subhead 3. How to compute the $M$-triangle of the generalised non-crossing partitions for a specific root system\endsubhead We follow the strategy outlined in \cite{\KratCB, Sec.~12}, which, however, has to be complemented by additional ideas. These additional ideas will be described in the next section. Let us recall from \cite{\KratCB} that the dual $M$-triangle (and, thus, the $M$-triangle as well) can be expressed in terms of certain {\it decomposition numbers} and {\it characteristic polynomials} of non-crossing partitions posets, which we define now. The decomposition number $N_\Phi(T_1,T_2,\dots,T_d)$ is the number of ``minimal" products $c_1c_2\cdots c_d$ less than or equal to the Coxeter element $c$ in absolute order, ``minimal" meaning that all the $c_i$'s are different from $\ep$ and that $\ell_T(c_1)+\ell_T(c_2)+\dots+\ell_T(c_d)=\ell_T(c_1c_2\cdots c_d)$, such that the type of $c_i$ as a parabolic Coxeter element is $T_i$, $i=1,2,\dots,d$. (Here, the term ``parabolic Coxeter element" means a Coxeter element in some parabolic subgroup. The reader must recall that it follows from \cite{\BesDAA, Lemma~1.4.3} that any element $c_i$ is indeed a Coxeter element in a parabolic subgroup of $W=W(\Phi)$. By definition, the type of $c_i$ is the type of this parabolic subgroup.) On the other hand, we denote by $\chi^*_{NC(\Psi)}(y)$ the reciprocal polynomial of the characteristic polynomial of $NC(\Psi)$, that is, using self-duality of $NC(\Psi)$, $$\chi^*_{NC(\Psi)}(y)=\sum _{u\in NC(\Psi)} \mu(\hat 0_{NC(\Psi)},u)\,y^{\rk \Psi-\rk u}=\sum _{u\in NC(\Psi)} \mu(u,\hat 1_{NC(\Psi)})\,y^{\rk u},$$ where $\hat 0_{NC(\Psi)}$ stands for the minimal element and $\hat 1_{NC(\Psi)}$ stands for the maximal element in $NC(\Psi)$. (The reader should recall from Section~2 that $\hat 0_{NC(\Psi)}$ is the identity element in $W(\Psi)$ and that $\hat 1_{NC(\Psi)}$ is the chosen Coxeter element in $W(\Psi)$.) Using this notation, a combination of Eqs.~(12.3) and (12.4) from \cite{\KratCB} reads as follows. \proclaim{Proposition \PA} For any finite root system $\Phi$ of rank $n$, we have $$\multline (M^m_\Phi)^*(x,y)= \sum _{d=0} ^{n} \sum _{(T_1,\dots,T_d)} ^{}x^{\rk T_1+\dots+\rk T_d}\cdot N_\Phi(T_1,T_2,\dots,T_d)\\ \cdot\chi^*_{NC(T_1)}(y) \chi^*_{NC(T_2)}(y)\cdots \chi^*_{NC(T_d)}(y)\binom md, \endmultline\tag\AC$$ where the inner sum is over all possible $d$-tuples $(T_1,T_2,\dots,T_d)$ of types {\rm(}not necessarily irreducible types{\rm)}. In {\rm(\AC)}, and in the sequel, the notation $NC(T)$ means $NC(\Psi)$, where $\Psi$ is a root system of type $T$, and $\rk T$ denotes the rank of $\Psi$. \endproclaim So, what we have to do to apply Formula~(\AC) to compute the (dual) $M$-triangle is, first, to determine all the decomposition numbers $N_\Phi(T_1,T_2,\dots,T_d)$, and, taking advantage of the multiplicativity $$\chi^*_{NC(\Psi_1*\Psi_2)}(y) =\chi^*_{NC(\Psi_1)\times NC(\Psi_2)}(y) =\chi^*_{NC(\Psi_1)}(y)\cdot\chi^*_{NC(\Psi_2)}(y)$$ of the characteristic polynomial (here, $\Psi_1*\Psi_2$ denotes the direct sum of the root systems $\Psi_1$ and $\Psi_2$), second, one needs a list of the characteristic polynomials $\chi^*_{NC(\Psi)}(y)$ for all {\it irreducible} root systems $\Psi$. We describe in the following section how we compute the decomposition numbers, and in the subsequent section how to use this knowledge to compute as well the characteristic polynomials that we need. \subhead 4. How to compute the decomposition numbers\endsubhead Our strategy to compute the decomposition numbers $N_\Phi(T_1,T_2,\dots,T_d)$ for a fixed root system $\Phi$ is to find as many linear relations between them as possible, and eventually solve this system of linear equations. Indeed, the decomposition numbers have many relations between themselves, some of which have already been stated and used in \cite{\KratCB}. We recall these here in Proposition~\PB\ below. Equation~(\AD) says that the order of the types $T_1$, $T_2$, \dots, $T_d$ is not relevant. Equation~(\AE) reduces the computation to the computation of the decomposition numbers $N_\Phi(T_1,T_2,\dots,T_d)$ with ``full" rank, that is, with $\rk T_1+\rk T_2+\dots+\rk T_d=\rk\Phi$. In Equation~(\AF) we add another family of relations. They involve the decomposition numbers $N_T(T'_1,T'_2,\dots,T'_e)$ of ``smaller" root systems than $\Phi$, that is, decomposition numbers with $\rk T<\rk\Phi$. (Similarly to Proposition~\PA, by abuse of notation, $N_T(T'_1,T'_2,\dots,T'_e)$ stands for the decomposition number $N_\Psi(T'_1,T'_2,\dots,T'_e)$, where $\Psi$ is a root system of type $T$.) If we had already computed these beforehand, then the relations (\AF) are linear relations between full rank decomposition numbers $N_\Phi(T_1,T_2,\dots,T_d)$. Proposition~\PD\ allows one to reduce these beforehand computations to irreducible root systems of smaller rank. A further set of linear relations comes from an identity featuring the zeta polynomials of (ordinary and generalized) non-crossing partitions posets given in Proposition~\PE. Indeed, as we explain in Remark~(2) after the proof of Proposition~\PE, all the zeta polynomials appearing in (\AG) are explicitly known, so that, by comparing coefficients of $m^iz^j$ on both sides of (\AG), we obtain a set of linear relations between the decomposition numbers $N_\Phi(T_1,T_2,\dots,T_d)$. While the number of equations which result from Propositions~\PB\ and \PE\ exceeds the number of variables (that is, the number of decomposition numbers of full rank) by far, it turns out that they are not sufficient to determine them uniquely. To remedy this somewhat, we add Proposition~\PF\ which provides the values for three special decomposition numbers, and we add Proposition~\PC\ which, as we illustrate in the Remark after the statement of the proposition, allows one to compute all the (full rank) decomposition numbers $N_\Phi(T,A_1)$ with $\rk T=\rk \Phi-1$. As we shall see in Sections~6 and 7, the system of linear equations resulting from Propositions~\PB, \PE, \PF\ and \PC\ still does not yield a unique solution for the decomposition numbers for $\Phi=E_7$ and $\Phi=E_8$. However, they allow one to come very close, so that, upon adding some arithmetic considerations and, in type~$E_8$, a {\sl Maple} calculation using Stembridge's {\tt coxeter} package \cite{\StemAZ}, one eventually succeeds in finding all the decomposition numbers. \proclaim{Proposition \PB} Let $\Phi$ be a finite root system. Then, for any permutation $\si$ of $\{1,2,\dots,d\}$, we have $$ N_\Phi(T_{\si(1)},T_{\si(2)},\dots,T_{\si(d)})=N_\Phi(T_1,T_2,\dots,T_d). \tag\AD$$ Furthermore, $$ N_\Phi(T_1,T_2,\dots,T_d)= \sum _{T} ^{}N_\Phi(T_1,T_2,\dots,T_d,T), \tag\AE$$ where the sum is over all types $T$ of rank $\rk\Phi-\rk T_1-\rk T_2- \dots -\rk T_d$. If $\rk T_1+\rk T_2+\dots +\rk T_d+ \rk T'_1+\rk T'_2+\dots +\rk T'_e=\rk\Phi$, then $$ N_\Phi(T_1,T_2,\dots,T_d,T'_1,T'_2,\dots,T'_e)=\sum _{T} ^{} N_T(T'_1,T'_2,\dots,T'_e)N_\Phi(T_1,T_2,\dots,T_d,T), \tag\AF$$ where the sum is over all types $T$ of rank $\rk T'_1+\rk T'_2+ \dots +\rk T'_e$. Finally, if one of $T_1,T_2,\dots,T_d$ is not the type of a sub-diagram of the Dynkin diagram of $\Phi$, then $N_\Phi(T_1,T_2,\dots,T_d)=0$. \endproclaim \example{Examples} An example of a relation of the form (\AE) is (\ALi). To be precise, Equation~(\ALi) is the special case of (\AE) where $\Phi=E_8$, $d=1$, and $T_1=D_4$. An example of a relation of the form (\AF) is (we make also use of (\AD)) $$\align N_{E_7}(A_1*A_3,A_1^2,A_1) &=N_{A_1^5}(A_1*A_3,A_1)N_{E_7}(A_1^5,A_1^2)\\ &\kern1cm+ N_{A_1^3*A_2}(A_1*A_3,A_1)N_{E_7}(A_1^3*A_2,A_1^2)\\ &\kern1cm+ N_{A_1*A_2^2}(A_1*A_3,A_1)N_{E_7}(A_1*A_2^2,A_1^2)\\ &\kern1cm+ N_{A_1^2*A_3}(A_1*A_3,A_1)N_{E_7}(A_1^2*A_3,A_1^2)\\ &\kern1cm+ N_{A_2*A_3}(A_1*A_3,A_1)N_{E_7}(A_2*A_3,A_1^2)\\ &\kern1cm+ N_{A_1*A_4}(A_1*A_3,A_1)N_{E_7}(A_1*A_4,A_1^2)\\ &\kern1cm+ N_{A_1*D_4}(A_1*A_3,A_1)N_{E_7}(A_1*D_4,A_1^2)\\ &\kern1cm+ N_{A_5}(A_1*A_3,A_1)N_{E_7}(A_5,A_1^2)\\ &\kern1cm+ N_{D_5}(A_1*A_3,A_1)N_{E_7}(D_5,A_1^2)\\ &=2N_{E_7}(A_1^2*A_3,A_1^2)+ 3N_{E_7}(A_2*A_3,A_1^2)+ 5N_{E_7}(A_1*A_4,A_1^2)\\ &\kern1cm+ 9N_{E_7}(A_1*D_4,A_1^2)+ 6N_{E_7}(A_5,A_1^2)+ 4N_{E_7}(D_5,A_1^2). \endalign$$ To be precise, the above equation is the special case of (\AF) where $\Phi=E_7$, $d=1$, $e=2$, $T_1=A_1^2$, $T_1'=A_1*A_3$ and $T_2'=A_1$. The decomposition numbers $N_\Psi(\dots)$ with $\Psi=A_1^5,A_1^3*A_2,A_1*A_2^2, A_1^2*A_3,A_2*A_3,A_1*A_4,A_1*D_4,A_5,D_5$ which are needed here are computed by using Proposition~\PD\ and the decomposition numbers for the irreducible root systems $A_1,A_2,A_3,A_4,A_5,D_4,D_5$ tabulated in the Appendix. \endexample \demo{Proof of Proposition \PB} Equation (\AD) is \cite{\KratCB, Eq.~(45)}, while Equation~(\AE) is \cite{\KratCB, Eq.~(46)}. In order to see (\AF), we recall that, by definition and by taking into account the rank assumption, the number $N_\Phi(T_1,T_2,\dots,T_d,T'_1,T'_2,\dots,T'_e)$ is the number of decompositions $$c=c_1c_2\cdots c_dc'_1c'_2\cdots c'_e,\tag\AFa$$ with $c_i$ of type $T_i$ and $c'_i$ of type $T'_i$ for all $i$. The decompositions (\AFa) can also be determined by first decomposing $$c=c_1c_2\cdots c_dc',\tag\AFb$$ with $c_i$ of type $T_i$ for all $i$, and subsequently $$c'=c'_1c'_2\cdots c'_e,\tag\AFc$$ with $c'_i$ of type $T'_i$ for all $i$. If we fix the type of $c'$, $T$ say, then the number of decompositions (\AFb) is $N_\Phi(T_1,T_2,\dots,T_d,T)$. For determining the number of decompositions (\AFc), we recall from \cite{\BesDAA, Cor.~1.6.2 and Def.~1.6.3} that $c'$ is a Coxeter element in a parabolic subgroup, denoted by $W_{c'}$ in \cite{\BesDAA}, and that the absolute order $\le_T$ on $W=W(\Phi)$ restricted to $W_{c'}$ is identical with absolute order on $W_{c'}$. In other words, the decompositions (\AFc) with $c'_i\in W$ are identical with the decompositions (\AFc) with $c'_i\in W_{c'}$. Hence, the number of decompositions (\AFc) is the number $N_T(T'_1,T'_2,\dots,T'_e)$. Finally, in order to get the {\it total\/} number $N_\Phi(T_1,T_2,\dots,T_d)$, we must sum the products $$N_T(T'_1,T'_2,\dots,T'_e)N_\Phi(T_1,T_2,\dots,T_d,T)$$ over all possible types $T$, thus arriving at Equation~(\AF). The last assertion follows again from the fact \cite{\BesDAA, Lemma~1.4.3} that any element which is less than or equal to a Coxeter element $c$ of $W$ is the Coxeter element in some parabolic subgroup of $W$. Hence, its type must by a sub-type of $\Phi$, the type of $W$.\quad \quad \qed \enddemo More equations come from the following proposition, featuring the zeta polynomial of the non-crossing partitions posets (ordinary and generalized). Recall that, given a poset $P$, its {\it zeta polynomial\/} $Z_P(z)$ is the number of multichains $x_1\le x_2\le \dots\le x_{z-1}$ in $P$. (It can be shown that this is indeed a polynomial in $z$. The reader should consult \cite{\StanAP, Sec.~3.11} for more information on this topic.) \proclaim{Proposition \PE} For any finite root system $\Phi$ of rank $n$, we have $$\multline Z_{NC^m(\Phi)}(z) =\sum _{d=0} ^{n} \sum _{(T_1,\dots,T_d)} ^{} N_\Phi(T_1,T_2,\dots,T_d)\\ \cdot Z_{NC(T_1)}(z-1) Z_{NC(T_2)}(z-1)\cdots Z_{NC(T_d)}(z-1)\binom md, \endmultline\tag\AG$$ where the inner sum is over all possible $d$-tuples $(T_1,T_2,\dots,T_d)$ of types, and where $N_\Phi(T_1,T_2,\dots,T_d)$ is as before. \endproclaim \demo{Proof} Let $\hat 1_{NC^m(\Phi)}$ denote the maximal element $(c;\ep,\dots,\ep)$ in $NC^m(\Phi)$. If, given a multichain $x_1\le x_2\le \dots\le x_{z-1}$ in $NC^m(\Phi)$, we remove the minimal element $w=x_1$, then a multichain remains which is by $1$ shorter. Hence, by summing over all possible $w$'s, we obtain $$Z_{NC^m(\Phi)}(z)=\sum _{w\in NC^m(\Phi)} ^{} Z_{[w,\hat 1_{NC^m(\Phi)}]}(z-1).\tag\AGa$$ Now, by definition of $NC^m(\Phi)$, $w$ is of the form $(w_0;w_1,\dots,w_m)$ with $w_0w_1\cdots w_m=c$ and $\ell_T(w_0)+\ell_T(w_1)+\dots+\ell_T(w_m)=\ell_T(c)=n$. Moreover, as we already recalled in Section~3, it follows from \cite{\BesDAA, Lemma~1.4.3} that any element $w_j$ is a Coxeter element in a parabolic subgroup of $W=W(\Phi)$ (or, in the language used earlier, a ``parabolic Coxeter element"). On the other hand, we have $$[w,\hat 1_{NC^m(\Phi)}]\cong[\ep,w_{1}]\times[\ep,w_{2}]\times \dots\times[\ep,w_{m}],\tag\AGb$$ where each interval $[\ep,w_{j}]$ is an interval in $NC(\Phi)$. In fact, some of the $w_j$'s may be equal to the identity $\ep$. If we denote by $w_{i_1}, w_{i_2},\dots,w_{i_d}$ those among the $w_j$'s which are {\it not\/} equal to $\ep$, then (\AGb) reduces to $$[w,\hat 1_{NC^m(\Phi)}]\cong[\ep,w_{i_1}]\times[\ep,w_{i_2}]\times \dots\times[\ep,w_{i_d}].$$ More precisely, since each $w_{i_j}$ is a parabolic Coxeter element, each interval $[\ep,w_{i_j}]$ is isomorphic to some non-crossing partition lattice $NC(\Psi)$, where $\Psi$ is the root system of this parabolic subgroup. The zeta polynomial being multiplicative, this implies $$Z_{[w,\hat 1_{NC^m(\Phi)}]}(z-1)= Z_{[\ep,w_{i_1}]}(z-1)Z_{[\ep,w_{i_2}]}(z-1) \cdots Z_{[\ep,w_{i_d}]}(z-1).$$ If we take into account that the number of possibilities to choose the indices $\{i_10$. If we substitute this in (\AKe) and (\AKf), then we obtain $$\align N_{E_7}(A_1^3*A_2, A_1^2) &=79-70(x+y),\\ N_{E_7}(A_1^3*A_2, A_2) &=\frac {4} {3}(35(x+y)-26). \endalign$$ Since the decomposition numbers must be non-negative integers, we conclude that $x+y=1$. Thus, we have $X=29-10y$ and $Y=30y-6$, with $y=1$ or $y=2$. To decide which of the two values is the true value, we appeal to Lemma~\LBa. Given a decomposition $c=c_1c_2$, where $c_1$ is of type $A_1^4$ and $c_2$ is of type $A_1^3$, the entire orbit $\Cal O=\{(c^kc_1c^{-k},c^kc_2c^{-k}):k\in\Bbb Z\}$ consists of pairs $(x,y)$ with $c=xy$, where $x$ is of type $A_1^4$ and $y$ is of type $A_1^3$. Since Lemma~\LBa\ in the case of $E_7$ says that the orbits of reflections have all size $9$, the size of $\Cal O$ must be a divisor of $9$. We claim that it cannot be $1$. For, in that case $c_1$ and $c_2$ commute with the Coxeter element $c$. By a result of Carter \cite{\CartAA, Prop.~30}, the centralizer of $c$ is known to be the cyclic group (of order $h=18$) generated by $c$. Hence, $c_1$ and $c_2$ would have to be powers of $c$. Since $c_1$ is of type $A_1^4$ and $c_2$ is of type $A_1^3$, both $c_1$ and $c_2$ are of order $2$. However, the only element of order $2$ in the cyclic group generated by $c$ is $c^9$, whence $c^9=c_1=c_2$, which is absurd. The conclusion is that $3$ divides the orbit $\Cal O$. Thus, $3$ must divide $X=N_{E_7}(A_1^4,A_1^3)$. Out of the two possible values for $y$, we must therefore choose $y=2$, and so $X=9$ and $Y=N_{E_7}(A_1^2*A_2,A_1^3)=54$. Now we substitute these values in the expressions in terms of $X$ and $Y$ found by {\sl Mathematica} for the other full rank decomposition numbers. The result is that $N_{E_7}(E_7)=1$, $N_{E_7}(E_6, A_1) = 9$, $N_{E_7}(D_6, A_1) = 9$, $N_{E_7}(A_6, A_1) = 9$, $N_{E_7}(A_1*D_5, A_1) = 9$, $N_{E_7}(A_1*A_5, A_1) = 9$, $N_{E_7}(A_2*D_4, A_1) = 0$, $N_{E_7}(A_2*A_4, A_1) = 9$, $N_{E_7}(A_1^2*D_4, A_1) = 0$, $N_{E_7}(A_1^2*A_4, A_1) = 0$, $N_{E_7}(A_3^2, A_1) = 0$, $N_{E_7}(A_1*A_2*A_3, A_1) = 9$, $N_{E_7}(A_1^3*A_3, A_1) = 0$, $N_{E_7}(A_2^3, A_1) = 0$, $N_{E_7}(A_1^2*A_2^2, A_1) = 0$, $N_{E_7}(A_1^4*A_2, A_1) = 0$, $N_{E_7}(A_1^6, A_1) = 0$, $N_{E_7}(D_5, A_2) = 18$, $N_{E_7}(A_5, A_2) = 30$, $N_{E_7}(A_1*A_4, A_2) = 54$, $N_{E_7}(A_1*D_4, A_2) = 9$, $N_{E_7}(A_2*A_3, A_2) = 36$, $N_{E_7}(A_1^2*A_3, A_2) = 36$, $N_{E_7}(A_1*A_2^2, A_2) = 36$, $N_{E_7}(A_1^3*A_2, A_2) = 12$, $N_{E_7}(A_1^5, A_2) = 0$, $N_{E_7}(D_5, A_1^2) = 54$, $N_{E_7}(A_5, A_1^2) = 63$, $N_{E_7}(A_1*D_4, A_1^2) = 27$, $N_{E_7}(A_1*A_4, A_1^2) = 81$, $N_{E_7}(A_2*A_3, A_1^2) = 27$, $N_{E_7}(A_1^2*A_3, A_1^2) = 27$, $N_{E_7}(A_1*A_2^2, A_1^2) = 27$, $N_{E_7}(A_1^3*A_2, A_1^2) = 9$, $N_{E_7}(A_1^5, A_1^2) = 0$, $N_{E_7}(D_5, A_1, A_1) = 162$, $N_{E_7}(A_5, A_1, A_1) = 216$, $N_{E_7}(A_1*D_4, A_1, A_1) = 81$, $N_{E_7}(A_1*A_4, A_1, A_1) = 324$, $N_{E_7}(A_2*A_3, A_1, A_1) = 162$, $N_{E_7}(A_1^2*A_3, A_1, A_1) = 162$, $N_{E_7}(A_1*A_2^2, A_1, A_1) = 162$, $N_{E_7}(A_1^3*A_2, A_1, A_1) = 54$, $N_{E_7}(A_1^5, A_1, A_1) = 0$, $N_{E_7}(D_4,A_3)=9$, $N_{E_7}(A_4,A_3)=54$, $N_{E_7}(A_1*A_3, A_3) = 135$, $N_{E_7}(A_2^2, A_3) = 54$, $N_{E_7}(A_1^2*A_2, A_3) = 162$, $N_{E_7}(A_1^4, A_3) = 27$, $N_{E_7}(D_4, A_1*A_2) = 45$, $N_{E_7}(A_4, A_1*A_2) = 162$, $N_{E_7}(A_1*A_3, A_1*A_2) = 243$, $N_{E_7}(A_2^2, A_1*A_2) = 54$, $N_{E_7}(A_1^2*A_2, A_1*A_2) = 162$, $N_{E_7}(A_1^4, A_1*A_2) = 27$, $N_{E_7}(D_4, A_1^3) = 30$, $N_{E_7}(A_4, A_1^3) = 99$, $N_{E_7}(A_1*A_3, A_1^3) = 126$, $N_{E_7}(A_2^2, A_1^3) = 18$, $N_{E_7}(A_1^2*A_2, A_1^3) = 54$, $N_{E_7}(A_1^4, A_1^3) = 9$, $N_{E_7}(D_4, A_2, A_1) = 81$, $N_{E_7}(A_4, A_2, A_1) = 378$, $N_{E_7}(A_1*A_3, A_2, A_1) = 783$, $N_{E_7}(A_2^2, A_2, \mathbreak A_1) = 270$, $N_{E_7}(A_1^2*A_2, A_2, A_1) = 810$, $N_{E_7}(A_1^4, A_2, A_1) = 135$, $N_{E_7}(D_4, A_1^2, A_1) = 243$, $N_{E_7}(A_4, A_1^2, A_1) = 891$, $N_{E_7}(A_1*A_3, A_1^2, A_1) = 1377$, $N_{E_7}(A_2^2, A_1^2, A_1) = 324$,\linebreak $N_{E_7}(A_1^2*A_2, A_1^2, A_1) = 972$, $N_{E_7}(A_1^4, A_1^2, A_1) = 162$, $N_{E_7}(D_4, A_1, A_1, A_1) = 729$, $N_{E_7}(A_4, \mathbreak A_1, A_1, A_1) = 2916$, $N_{E_7}(A_1*A_3, A_1, A_1, A_1) = 5103$, $N_{E_7}(A_2^2, A_1, A_1, A_1) = 1458$,\linebreak $N_{E_7}(A_1^2*A_2, A_1, A_1, A_1) = 4374$, $N_{E_7}(A_1^4, A_1, A_1, A_1) = 729$, $N_{E_7}(A_3, A_3, A_1) = 486$, $N_{E_7}(A_3, A_1*A_2, A_1) = 1458$, $N_{E_7}(A_3, A_1^3, A_1) = 891$, $N_{E_7}(A_1*A_2, A_1*A_2, A_1) = 2430$, $N_{E_7}(A_1*A_2, A_1^3, A_1) = 1215$, $N_{E_7}(A_1^3, A_1^3, A_1) = 540$, $N_{E_7}(A_3, A_2, A_2) = 432$,\linebreak $N_{E_7}(A_1*A_2, A_2, A_2) = 1188$, $N_{E_7}(A_1^3, A_2, A_2) = 711$, $N_{E_7}(A_3, A_2, A_1^2) = 1053$,\linebreak $N_{E_7}(A_1*A_2, A_2, A_1^2) = 2349$, $N_{E_7}(A_1^3, A_2, A_1^2) = 1323$, $N_{E_7}(A_3, A_1^2, A_1^2) = 2430$,\linebreak $N_{E_7}(A_1*A_2, A_1^2, A_1^2) = 3402$, $N_{E_7}(A_1^3, A_1^2, A_1^2) = 1539$, $N_{E_7}(A_3, A_2, A_1, A_1) = 3402$, $N_{E_7}(A_1*A_2, A_2, A_1, A_1) = 8262$, $N_{E_7}(A_1^3, A_2, A_1, A_1) = 4779$, $N_{E_7}(A_3, A_1^2, A_1, A_1) = 8019$, $N_{E_7}(A_1*A_2, A_1^2, A_1, A_1) = 13851$, $N_{E_7}(A_1^3, A_1^2, A_1, A_1) = 7047$, $N_{E_7}(A_3, A_1, A_1, A_1, \mathbreak A_1) = 26244$, $N_{E_7}(A_1*A_2, A_1, A_1, A_1, A_1) = 52488$, $N_{E_7}(A_1^3, A_1, A_1, A_1, A_1) = 28431$, $N_{E_7}(A_2, A_2, A_2, A_1) = 2916$, $N_{E_7}(A_2, A_2, A_1^2, A_1) = 6561$, $N_{E_7}(A_2, A_1^2, A_1^2, A_1) = 13122$, $N_{E_7}(A_1^2, A_1^2, A_1^2, A_1) = 19683$, $N_{E_7}(A_2, A_2, A_1, A_1, A_1) = 21870$, $N_{E_7}(A_2, A_1^2, A_1, A_1, \mathbreak A_1) = 45927$, $N_{E_7}(A_1^2, A_1^2, A_1, A_1, A_1) = 78732$, $N_{E_7}(A_2, A_1, A_1, A_1, A_1, A_1) = 157464$, $N_{E_7}(A_1^2, A_1, A_1, A_1, A_1, A_1) = 295245$, $N_{E_7}(A_1,A_1,A_1,A_1,A_1,A_1,A_1)=1062882$, plus the assignments implied by (\AD) and (\AE), all other numbers $N_{E_7}(T_1,\dots,T_d)$ being zero. Finally, we substitute the values found for the decomposition numbers and the formulae for the characteristic polynomials from (\AJ) in (\AC), and we obtain {\eightpoint $$\allowdisplaybreaks\multline (M^m_{E_7})^*\(x,y\)= \frac {1} {280}{{m( 9 m -2) ( 9 m -4) ( 9 m -5) ( 9 m -8) ( 3 m -1) ( 3 m -2) {x^7}{y^7}} }\\ - \frac {9} {40}{{ {m^2}( 9 m -2) ( 9 m -5) ( 3 m -1) ( 3 m -2) ( 9 m -4) {x^7}{y^6}}} \\+ \frac {3} {40}{{ {m^2}( 9 m -2) ( 9 m -4) ( 3 m -1) ( 243 {m^2} -81m-14) {x^7}{y^5}}} \\- \frac {3} {8}{{ {m^2}( 3 m -1) ( 9 m -2) ( 9 m +2) ( 81 {m^2} -9m-4) {x^7}{y^4}}} \\+ \frac {3} {8}{{ {m^2}( 9 m -2) ( 3 m +1) ( 9 m +2) ( 81 {m^2} +9m-4) {x^7}{y^3}}} \\- \frac {3} {40}{{ {m^2}( 3 m +1) ( 9 m +2) ( 9 m +4) ( 243 {m^2} +81m-14) {x^7}{y^2}}} \\+ \frac {9} {40}{{ {m^2} ( 3 m +1) ( 3 m +2) ( 9 m +2) ( 9 m +4) ( 9 m +5) {x^7}y}} \\- \frac {1} {280}{{m ( 3 m +1) ( 3 m +2) ( 9 m +2) ( 9 m +4) ( 9 m +5) ( 9 m +8) {x^7}}} \\+ \frac {9} {40}{{ m( 9 m -2) ( 9 m -5) ( 3 m -1) ( 3 m -2) ( 9 m -4) {x^6}{y^6}}} \\- \frac {27} {20}{{ {m^2}( 27 m -13) ( 9 m -2) ( 9 m -4) ( 3 m -1) {x^6}{y^5}} } \\+ \frac {27} {8}{{ {m^2}( 3 m -1) ( 9 m -2) ( 243 {m^2} -45m-10) {x^6}{y^4}}} - \frac {27} {2}{{ {m^2}( 9 m -2) ( 9 m +2) ( 27 {m^2} -1) {x^6}{y^3}}} \\+ \frac {27} {8}{{ {m^2}( 3 m +1) ( 9 m +2) ( 243 {m^2} +45m-10) {x^6}{y^2}}} - \frac {27} {20}{{ {m^2} ( 3 m +1) ( 9 m +2) ( 9 m +4) ( 27 m +13) {x^6}y}} \\+ \frac {9} {40}{{ m ( 3 m +1) ( 3 m +2) ( 9 m +2) ( 9 m +4) ( 9 m +5) {x^6}}} + \frac {3} {40}{{ m( 207 m -103) ( 9 m -2) ( 9 m -4) ( 3 m -1) {x^5}{y^5}} } \\- \frac {27} {8}{{ {m^2}( 207 m -71) ( 3 m -1) ( 9 m -2) {x^5}{y^4}}} + \frac {9} {4}{{ {m^2}( 9 m -2) ( 1863 {m^2} -144m-55) {x^5}{y^3}}} \\- \frac {9} {4}{{ {m^2}( 9 m +2) ( 1863 {m^2} +144m-55) {x^5}{y^2}}} + \frac {27} {8}{{ {m^2} ( 3 m +1) ( 9 m +2) ( 207 m +71) {x^5}y}} \\- \frac {3} {40}{{ m ( 3 m +1) ( 9 m +2) ( 9 m +4) ( 207 m +103) {x^5}}} + \frac {21} {8}{{ m( 63 m -23) ( 3 m -1) ( 9 m -2) {x^4}{y^4}}} \\- \frac {189} {2}{{ {m^2}( 21 m -5) ( 9 m -2) {x^4}{y^3}}} + \frac {63} {4}{{ {m^2} ( 1701 {m^2} -37) {x^4}{y^2}}} - \frac {189} {2}{{ {m^2} ( 9 m +2) ( 21 m +5) {x^4}y}} \\+ \frac {21} {8}{{ m ( 3 m +1) ( 9 m+2 ) ( 63 m +23) {x^4}}} + \frac {21} {2}{{ m( 27 m-7 ) ( 9 m -2) {x^3}{y^3}}} - \frac {189} {2}{{ {m^2} ( 81 m -13) {x^3}{y^2}}} \\+ \frac {189} {2}{{ {m^2} ( 81 m +13) {x^3}y}} - \frac {21} {2}{{ m ( 9 m +2) ( 27 m +7) {x^3}}} + \frac {21} {2}{{m ( 63 m -11) {x^2}{y^2}}} \\- 1323 {m^2}{x^2}y + \frac {21} {2}{{ m ( 63 m +11) {x^2}}} + 63 m xy - 63 m x + 1 . \endmultline\tag{\tenpoint\AK}$$}% \subhead 7. The $M$-triangle of generalised non-crossing partitions of type $E_8$\endsubhead In this section we implement the programme outlined in Sections~4 and 5 to compute the decomposition numbers for $E_8$ and, thus, via (\AC) and (\AJ), the $M$-triangle of the $m$-divisible non-crossing partitions of type $E_8$. Again, to begin with, we have to compute the decomposition numbers for types of smaller ranks, that is, of ranks $\le 7$. The decomposition numbers for the irreducible types of rank $\le 7$ that we need, for $A_1$, $A_2$, $A_3$, $A_4$, $A_5$, $A_6$, $A_7$, $D_4$, $D_5$, $D_6$, $D_7$, $E_6$, and $E_7$, are given in the Appendix, respectively in Section~6. Those for the reducible types of rank $\le 7$ then can be computed by using Proposition~\PD.\footnote{See Footnote~1.} Then we use (\AD) and (\AE) to express all the decomposition numbers in terms of full rank decomposition numbers $N_\Phi(T_1,T_2,\dots,T_d)$ (that is, those with $\rk\Phi=\rk T_1+\rk T_1+\dots+\rk T_d$), in which the types $T_i$ are ordered (for example, according to lexicographic order). Subsequently we produce the equations which one obtains from (\AF) (using the earlier computed decomposition numbers of smaller rank) and from comparing coefficients of $m^iz^j$, $i,j\in\{0,1,\dots,8\}$, on both sides of (\AG). Finally, we use Proposition~\PF\ to determine $N_{E_8}(E_8)$, $N_{E_8}(A_1)$ and $N_{E_8}(A_1,A_1,A_1,A_1,A_1,A_1,A_1,A_1)$, Proposition~\PC\ to compute the numbers $N_{E_8}(T,A_1)$ with $\rk T=7$, and we use the last assertion in Proposition~\PB\ for the types $A_1^6$, $A_1^5$, $A_1^3*A_3$, $A_2^3$ and $A_1^2*D_4$ (which do not occur as the type of a sub-diagram of the Dynkin diagram of $E_8$). To wit, the latter leads to the following special values of decomposition numbers: $$\allowdisplaybreaks\gather N_{E_8}(E_8)=1,\\ N_{E_8}(A_1)=120,\\ N_{E_8}(A_1,A_1,A_1,A_1,A_1,A_1,A_1,A_1)=37968750,\\ \matrix N_{E_8}(E_7,A_1)= N_{E_8}(D_7,A_1)= N_{E_8}(A_7,A_1)= N_{E_8}(A_1*E_6,A_1)\hfill\\ \kern2cm= N_{E_8}(A_1*A_6,A_1)= N_{E_8}(A_2*D_5,A_1)\hfill\\ \kern2cm= N_{E_8}(A_3*A_4,A_1)= N_{E_8}(A_1*A_2*A_4,A_1)=15,\hfill \endmatrix\\ \matrix N_{E_8}(A_1*D_6,A_1)= N_{E_8}(A_2*A_5,A_1)= N_{E_8}(A_1^2*D_5,A_1)= N_{E_8}(A_1^2*A_5,A_1)\hfill\\ \kern2cm= N_{E_8}(A_3*D_4,A_1)= N_{E_8}(A_1*A_2*D_4,A_1)= N_{E_8}(A_1^3*D_4,A_1)\hfill\\ \kern2cm= N_{E_8}(A_1^3*A_4,A_1)= N_{E_8}(A_1*A_3^2,A_1)= N_{E_8}(A_2^2*A_3,A_1)\hfill\\ \kern2cm= N_{E_8}(A_1^2*A_2*A_3,A_1)= N_{E_8}(A_1^4*A_3,A_1)= N_{E_8}(A_1*A_2^3,A_1)\hfill\\ \kern2cm= N_{E_8}(A_1^3*A_2^2,A_1)= N_{E_8}(A_1^5*A_2,A_1)= N_{E_8}(A_1^7,A_1)\hfill\\ \kern2cm= N_{E_8}(A_1^6,A_2)= N_{E_8}(A_1^6,A_1^2)= N_{E_8}(A_1^6,A_1,A_1)= N_{E_8}(A_1^5,A_3)\hfill\\ \kern2cm= N_{E_8}(A_1^5,A_1*A_2)= N_{E_8}(A_1^5,A_1^3)= N_{E_8}(A_1^5,A_2,A_1)\hfill\\ \kern2cm= N_{E_8}(A_1^5,A_1^2,A_1)= N_{E_8}(A_1^5,A_1,A_1,A_1)= N_{E_8}(A_1^3*A_3,A_2)\hfill\\ \kern2cm= N_{E_8}(A_1^3*A_3,A_1^2)= N_{E_8}(A_1^3*A_3,A_1,A_1)= N_{E_8}(A_2^3,A_2)\hfill\\ \kern2cm= N_{E_8}(A_2^3,A_1^2)= N_{E_8}(A_1^4*A_2,A_2)= N_{E_8}(A_1^4*A_2,A_1^2)\hfill\\ \kern2cm= N_{E_8}(A_1^2*D_4,A_2)= N_{E_8}(A_1^2*D_4,A_1^2)=0.\hfill \endmatrix \endgather$$ Again, we let {\sl Mathematica~5.2} solve the described system of linear equations for the full rank decomposition numbers.\footnote{\NoBlackBoxes The {\sl Mathematica} input is available at {\tt http://www.mat.univie.ac.at/\~{}kratt/artikel/cluster2.html}.} Although this is now a system of more than 600 equations with only about 250 variables, the situation here is even worse as the solution space is four-dimensional. {\sl Mathematica} expresses all the variables in terms of $X=N_{E_8}(A_5,\mathbreak A_1*A_2)$, $Y=N_{E_8}(D_5,A_1*A_2)$, $N_{E_8}(A_4,A_1*A_3)$, and $N_{E_8}(D_4,A_4)$. For the subsequent considerations, we work with the following selection of relations output by {\sl Mathematica}: $$\align N_{E_8}(A_5, A_3) &= \frac {1} {4}(750-X),\tag\ALa\\ N_{E_8}(D_5, A_3) &= \frac {1} {4}(375-Y),\tag\ALb\\ N_{E_8}(A_5, A_1^3) &= \frac {5} {6}(750-X),\tag\ALc\\ N_{E_8}(D_5, A_1^3) &= \frac {5} {6}(375-Y),\tag\ALd\\ N_{E_8}(A_2*A_3, A_1*A_2) &= \frac {3} {25}(4125-8X+16Y),\tag\ALe\\ N_{E_8}(A_2*A_3, A_1^3) &= \frac {1} {5}(1125+4X-8Y),\tag\ALf\\ N_{E_8}(A_1*D_4, A_3) &= Y-150,\tag\ALg\\ N_{E_8}(A_1^2*A_3, A_1^3) &= \frac {1} {5}(49875 - 56X - 138Y),\tag\ALh\\ N_{E_8}(A_1*A_4, A_1*A_2) &= 2 (2295 - 3 X - 4 Y ),\tag\ALgg\\ N_{E_8}(A_1^3*A_2, A_1^3) &= \frac {1} {15}(112 X + 226 Y - 86625).\tag\ALhh \endalign $$ There is one further relation which we shall make use of, $$\align N_{E_8}(D_4)&= N_{E_8}(D_4,D_4)+N_{E_8}(D_4,A_4)+N_{E_8}(D_4,A_3*A_1)\\ &\kern2cm+ N_{E_8}(D_4,A_2^2)+N_{E_8}(D_4,A_2*A_1^2)+N_{E_8}(D_4,A_1^4)\tag\ALi\\ &=\frac {27263} {168} - \frac {1} {40}N_{E_8}(A_4, A_1*A_3) + \frac {1} {40} N_{E_8}(D_4, A_4) + \frac {283} {1500}X + \frac {7957} {15750} Y,\tag\ALj \endalign$$ where the first line is an instance of (\AE), and where the second line follows from the first upon substituting the relations for the full rank decomposition numbers output by {\sl Mathematica}. \BlackBoxes Relations (\ALa) and (\ALc) imply $$X\equiv 6\pmod{12},\tag\ALk$$ and Relations (\ALb) and (\ALd) imply $$Y\equiv 3\pmod{12}.\tag\ALl$$ Next, Relation (\ALe) implies $$X\equiv 2Y\pmod{25}.\tag\ALm$$ Solving (\ALk), (\ALl) and (\ALm) for $X$ and $Y$ yields $$\align X&=300x+24y+6,\tag\ALmm\\ Y&=12y+3,\tag\ALmmm \endalign$$ for some integers $x$ and $y$ with $y\ge0$. If we substitute this in (\ALa), (\ALf)--(\ALhh), then we obtain $$\align N_{E_8}(A_5, A_3) &=3(62-25x-2y),\tag\ALn\\ N_{E_8}(A_2*A_3, A_1^3) &=225+240x,\tag\ALo\\ N_{E_8}(A_1*D_4, A_3) &=12y-147,\tag\ALp\\ N_{E_8}(A_1^2*A_3, A_1^3) &=15(655-224x-40y),\tag\ALq\\ N_{E_8}(A_1*A_4, A_1*A_2) &= 30(151 - 60x - 8y),\tag\ALpp\\ N_{E_8}(A_1^3*A_2, A_1^3) &= 5(448x + 72y-1137).\tag\ALr \endalign$$ From (\ALn) we infer $x\le 2$, while (\ALo) implies $x\ge0$. Furthermore, from (\ALp) we infer $y\ge 13$, while (\ALq) implies $y\le16$. If $x\ge1$, then (\ALpp) implies $y\le 91/8<13$, a contradiction. Hence, $x=0$. In this case, Equation~(\ALr) implies $y\ge 1137/72> 15$, and so $y=16$. Substituting this in (\ALmm) and (\ALmmm), we obtain $X=N_{E_8}(A_5,A_1*A_2)=390$ and $Y=N_{E_8}(D_5,A_1*A_2)=195$. Unfortunately, similar arithmetic and positivity considerations do not suffice to determine the remaining two ``free variables," the decomposition numbers $N_{E_8}(A_4,A_1*A_3)$ and $N_{E_8}(D_4,A_4)$. Instead, by a 12~days {\sl Maple} computation using Stembridge's {\tt coxeter} package, we found that $N_{E_8}(D_4)=325$ and $N_{E_8}(D_4,A_4)=15$. If we use these values, together with the already determined values for $X$ and $Y$, in (\ALj), we eventually find that $N_{E_8}(A_4,A_1*A_3)=390$. We now substitute the above values for $N_{E_8}(A_5,A_1*A_2)$, $N_{E_8}(D_5,A_1*A_2)$, $N_{E_8}(A_4,\mathbreak A_1*A_3)$, and $N_{E_8}(D_4,A_4)$ in the expressions found by {\sl Mathematica} for the other full rank decomposition numbers. The result is that $N_{E_8}(E_8)= 1$, $N_{E_8}(E_7, A_1)= 15$, $N_{E_8}(D_7, A_1)= 15$, $N_{E_8}(A_7, A_1)= 15$, $N_{E_8}(A_1*E_6, A_1)= 15$, $N_{E_8}(A_1*D_6, A_1)= 0$, $N_{E_8}(A_1*A_6, A_1)= 15$, $N_{E_8}(A_2*D_5, A_1)= 15$, $N_{E_8}(A_2*A_5, A_1)= 0$,%\linebreak $N_{E_8}(A_1^2*D_5, A_1)= 0$, $N_{E_8}(A_1^2*A_5, A_1)= 0$, $N_{E_8}(A_3*D_4, A_1)= 0$, $N_{E_8}(A_3*A_4, A_1)= 15$, $N_{E_8}(A_1*A_2*D_4, A_1)= 0$, $N_{E_8}(A_1*A_2*A_4, A_1)= 15$, $N_{E_8}(A_1^3*D_4, A_1)= 0$, $N_{E_8}(A_1^3*A_4, A_1)= 0$, $N_{E_8}(A_1*A_3^2, A_1)= 0$, $N_{E_8}(A_2^2*A_3, A_1)= 0$, $N_{E_8}(A_1^2*A_2*A_3, A_1)= 0$, $N_{E_8}(A_1^4*A_3, A_1)= 0$, $N_{E_8}(A_1*A_2^3, A_1)= 0$, $N_{E_8}(A_1^3*A_2^2, A_1)= 0$, $N_{E_8}(A_1^5*A_2, A_1)= 0$, $N_{E_8}(A_1^7, A_1)= 0$, $N_{E_8}(E_6, A_2)= 20$, $N_{E_8}(D_6, A_2)= 15$, $N_{E_8}(A_6, A_2)= 60$, $N_{E_8}(A_1*D_5, A_2)= 60$, $N_{E_8}(A_1*A_5, A_2)= 60$, $N_{E_8}(A_2*D_4, A_2)= 20$, $N_{E_8}(A_2*A_4, A_2)= 90$, $N_{E_8}(A_3^2, A_2)= 45$, $N_{E_8}(A_1^2*D_4, A_2)= 0$, $N_{E_8}(A_1^2*A_4, A_2)= 90$, $N_{E_8}(A_1*A_2*A_3, A_2)= 90$, $N_{E_8}(A_1^3*A_3, A_2)= 0$, $N_{E_8}(A_2^3, A_2)= 0$, $N_{E_8}(A_1^2*A_2^2, A_2)= 45$, $N_{E_8}(A_1^4*A_2, A_2)= 0$, $N_{E_8}(A_1^6, A_2)= 0$, $N_{E_8}(E_6, A_1^2)= 45$, $N_{E_8}(D_6, A_1^2)= 90$, $N_{E_8}(A_6, A_1^2)= 135$, $N_{E_8}(A_1*D_5, A_1^2)= 135$, $N_{E_8}(A_1*A_5, A_1^2)= 135$, $N_{E_8}(A_2*D_4, A_1^2)= 45$, $N_{E_8}(A_2*A_4, A_1^2)= 90$, $N_{E_8}(A_3^2, A_1^2)= 45$, $N_{E_8}(A_1^2*D_4, A_1^2)= 0$, $N_{E_8}(A_1^2*A_4, A_1^2)= 90$, $N_{E_8}(A_1*A_2*A_3, A_1^2)= 90$, $N_{E_8}(A_1^3*A_3, A_1^2)= 0$, $N_{E_8}(A_2^3, A_1^2)= 0$, $N_{E_8}(A_1^2*A_2^2, A_1^2)= 45$, $N_{E_8}(A_1^4*A_2, A_1^2)= 0$, $N_{E_8}(A_1^6, A_1^2)= 0$, $N_{E_8}(E_6, A_1, A_1)= 150$, $N_{E_8}(D_6, A_1, A_1)= 225$, $N_{E_8}(A_6, A_1, A_1)= 450$, $N_{E_8}(A_1*D_5, A_1, A_1)= 450$, $N_{E_8}(A_1*A_5, A_1, A_1)= 450$, $N_{E_8}(A_2*D_4, A_1, A_1)= 150$, $N_{E_8}(A_2*A_4, A_1, A_1)= 450$, $N_{E_8}(A_3^2, A_1, A_1)= 225$, $N_{E_8}(A_1^2*D_4, A_1, A_1)= 0$, $N_{E_8}(A_1^2*A_4, A_1, A_1)= 450$, $N_{E_8}(A_1*A_2*A_3, A_1, A_1)= 450$, $N_{E_8}(A_1^3*A_3, A_1, A_1)= 0$, $N_{E_8}(A_2^3, A_1, A_1)= 0$, $N_{E_8}(A_1^2*A_2^2, A_1, A_1)= 225$, $N_{E_8}(A_1^4*A_2, A_1, A_1)= 0$, $N_{E_8}(A_1^6, A_1, A_1)= 0$, $N_{E_8}(D_5, A_3)= 45$, $N_{E_8}(A_5, A_3)= 90$, $N_{E_8}(A_1*A_4, A_3)= 315$, $N_{E_8}(A_1*D_4, A_3)= 45$, $N_{E_8}(A_2*A_3, A_3)= 270$, $N_{E_8}(A_1^2*A_3, A_3)= 270$, $N_{E_8}(A_1*A_2^2, A_3)= 225$, $N_{E_8}(A_1^3*A_2, A_3)= 225$, $N_{E_8}(A_1^5, A_3)= 0$, $N_{E_8}(D_5, A_1*A_2)= 195$, $N_{E_8}(A_5, A_1*A_2)= 390$, $N_{E_8}(A_1*A_4, A_1*A_2)= 690$, $N_{E_8}(A_1*D_4, A_1*A_2)= 195$, $N_{E_8}(A_2*A_3, A_1*A_2)= 495$, $N_{E_8}(A_1^2*A_3, A_1*A_2)= 495$, $N_{E_8}(A_1*A_2^2, A_1*A_2)= 300$, $N_{E_8}(A_1^3*A_2, A_1*A_2)= 300$, $N_{E_8}(A_1^5, A_1*A_2)= 0$, $N_{E_8}(D_5, A_1^3)= 150$, $N_{E_8}(A_5, A_1^3)= 300$, $N_{E_8}(A_1*A_4, A_1^3)= 375$, $N_{E_8}(A_1*D_4, A_1^3)= 150$, $N_{E_8}(A_2*A_3, A_1^3)= 225$, $N_{E_8}(A_1^2*A_3, A_1^3)= 225$, $N_{E_8}(A_1*A_2^2, A_1^3)= 75$, $N_{E_8}(A_1^3*A_2, A_1^3)= 75$, $N_{E_8}(A_1^5, A_1^3)= 0$, $N_{E_8}(D_5, A_2, A_1)= 375$, $N_{E_8}(A_5, A_2, A_1)= 750$, $N_{E_8}(A_1*A_4, A_2, A_1)= 1950$, $N_{E_8}(A_1*D_4, A_2, A_1)= 375$, $N_{E_8}(A_2*A_3, A_2, A_1)= 1575$, $N_{E_8}(A_1^2*A_3, A_2, A_1)= 1575$, $N_{E_8}(A_1*A_2^2, A_2, A_1)= 1200$, $N_{E_8}(A_1^3*A_2, A_2, A_1)= 1200$, $N_{E_8}(A_1^5, A_2, A_1)= 0$, $N_{E_8}(D_5, A_1^2, A_1)= 1125$, $N_{E_8}(A_5, A_1^2, A_1)= 2250$, $N_{E_8}(A_1*A_4, A_1^2, A_1)= 3825$, $N_{E_8}(A_1*D_4, A_1^2, A_1)= 1125$, $N_{E_8}(A_2*A_3, A_1^2, A_1)= 2700$, $N_{E_8}(A_1^2*A_3, A_1^2, A_1)= 2700$, $N_{E_8}(A_1*A_2^2, A_1^2, A_1)= 1575$, $N_{E_8}(A_1^3*A_2, A_1^2, A_1)= 1575$, $N_{E_8}(A_1^5, A_1^2, A_1)= 0$, $N_{E_8}(D_5, A_1, A_1, A_1)= 3375$, $N_{E_8}(A_5, A_1, A_1, A_1)= 6750$, $N_{E_8}(A_1*A_4, A_1, A_1, A_1)= 13500$, $N_{E_8}(A_1*D_4, A_1, A_1, A_1)= 3375$, $N_{E_8}(A_2*A_3, A_1, A_1, A_1)= 10125$, $N_{E_8}(A_1^2*A_3, A_1, A_1, A_1)= 10125$, $N_{E_8}(A_1*A_2^2, A_1, A_1, A_1)= 6750$, $N_{E_8}(A_1^3*A_2, A_1, A_1, A_1)= 6750$, $N_{E_8}(A_1^5, A_1, A_1, A_1)= 0$,\linebreak $N_{E_8}(D_4, D_4)= 5$, $N_{E_8}(D_4, A_4)= 15$, $N_{E_8}(A_4, A_4)= 138$, $N_{E_8}(D_4, A_1*A_3)= 105$, $N_{E_8}(A_4, A_1*A_3)= 390$, $N_{E_8}(A_1*A_3, A_1*A_3)= 1155$, $N_{E_8}(D_4, A_2^2)= 35$, $N_{E_8}(A_4, A_2^2)= 180$, $N_{E_8}(A_1*A_3, A_2^2)= 360$, $N_{E_8}(A_2^2, A_2^2)= 95$, $N_{E_8}(D_4, A_1^2*A_2)= 135$, $N_{E_8}(A_4, A_1^2*A_2)= 630$, $N_{E_8}(A_1*A_3, A_1^2*A_2)= 1035$, $N_{E_8}(A_2^2, A_1^2*A_2)= 270$, $N_{E_8}(A_1^2*A_2, A_1^2*A_2)= 495$, $N_{E_8}(D_4, A_1^4)= 30$, $N_{E_8}(A_4, A_1^4)= 165$, $N_{E_8}(A_1*A_3, A_1^4)= 255$, $N_{E_8}(A_2^2, A_1^4)= 60$, $N_{E_8}(A_1^2*A_2, A_1^4)= 135$, $N_{E_8}(A_1^4, A_1^4)= 30$, $N_{E_8}(D_4, A_3, A_1)= 225$, $N_{E_8}(A_4, A_3, A_1)= 1215$, $N_{E_8}(A_1*A_3, A_3, A_1)= 4050$, $N_{E_8}(A_2^2, A_3, A_1)= 1575$, $N_{E_8}(A_1^2*A_2, A_3, A_1)= 5400$, $N_{E_8}(A_1^4, A_3, A_1)= 1350$, $N_{E_8}(D_4, A_1*A_2, A_1)= 975$, $N_{E_8}(A_4, A_1*A_2, A_1)= 4590$, $N_{E_8}(A_1*A_3, A_1*A_2, A_1)= 10800$, $N_{E_8}(A_2^2, A_1*A_2, A_1)= 3450$, $N_{E_8}(A_1^2*A_2, A_1*A_2, A_1)= 9900$, $N_{E_8}(A_1^4, A_1*A_2, A_1)= 2475$, $N_{E_8}(D_4, A_1^3, A_1)= 750$, $N_{E_8}(A_4, A_1^3, A_1)= 3375$, $N_{E_8}(A_1*A_3, A_1^3, A_1)= 6750$, $N_{E_8}(A_2^2, A_1^3, A_1)= 1875$, $N_{E_8}(A_1^2*A_2, A_1^3, A_1)= 4500$, $N_{E_8}(A_1^4, A_1^3, A_1)= 1125$, $N_{E_8}(D_4, A_2, A_2)= 175$, $N_{E_8}(A_4, A_2, A_2)= 1140$, $N_{E_8}(A_1*A_3, \mathbreak A_2, A_2)= 3300$, $N_{E_8}(A_2^2, A_2, A_2)= 1300$, $N_{E_8}(A_1^2*A_2, A_2, A_2)= 4500$, $N_{E_8}(A_1^4, A_2, A_2)= 1125$, $N_{E_8}(D_4, A_2, A_1^2)= 675$, $N_{E_8}(A_4, A_2, A_1^2)= 3015$, $N_{E_8}(A_1*A_3, A_2, A_1^2)= 8550$, $N_{E_8}(A_2^2, A_2, A_1^2)= 2925$, $N_{E_8}(A_1^2*A_2, A_2, A_1^2)= 9000$, $N_{E_8}(A_1^4, A_2, A_1^2)= 2250$, $N_{E_8}(D_4, \mathbreak A_1^2, A_1^2)= 1800$, $N_{E_8}(A_4, A_1^2, A_1^2)= 8640$, $N_{E_8}(A_1*A_3, A_1^2, A_1^2)= 17550$, $N_{E_8}(A_2^2, A_1^2, \mathbreak A_1^2)= 5175$, $N_{E_8}(A_1^2*A_2, A_1^2, A_1^2)= 13500$, $N_{E_8}(A_1^4, A_1^2, A_1^2)= 3375$, $N_{E_8}(D_4, A_2, A_1, \mathbreak A_1)= 1875$, $N_{E_8}(A_4, A_2, A_1, A_1)= 9450$, $N_{E_8}(A_1*A_3, A_2, A_1, A_1)= 27000$, $N_{E_8}(A_2^2, A_2, \mathbreak A_1, A_1)= 9750$, $N_{E_8}(A_1^2*A_2, A_2, A_1, A_1)= 31500$, $N_{E_8}(A_1^4, A_2, A_1, A_1)= 7875$, $N_{E_8}(D_4, \mathbreak A_1^2, A_1, A_1)= 5625$, $N_{E_8}(A_4, A_1^2, A_1, A_1)= 26325$, $N_{E_8}(A_1*A_3, A_1^2, A_1, A_1)= 60750$, $N_{E_8}(A_2^2, A_1^2, A_1, A_1)= 19125$, $N_{E_8}(A_1^2*A_2, A_1^2, A_1, A_1)= 54000$, $N_{E_8}(A_1^4, A_1^2, A_1, A_1)= 13500$, $N_{E_8}(D_4, A_1, A_1, A_1, A_1)= 16875$, $N_{E_8}(A_4, A_1, A_1, A_1, A_1)= 81000$, $N_{E_8}(A_1*A_3, \mathbreak A_1, A_1, A_1, A_1)= 202500$, $N_{E_8}(A_2^2, A_1, A_1, A_1, A_1)= 67500$, $N_{E_8}(A_1^2*A_2, A_1, A_1, A_1, \mathbreak A_1)= 202500$, $N_{E_8}(A_1^4, A_1, A_1, A_1, A_1)= 50625$, $N_{E_8}(A_3, A_3, A_2)= 1350$, $N_{E_8}(A_3, \mathbreak A_1*A_2, A_2)= 5175$, $N_{E_8}(A_3, A_1^3, A_2)= 3825$, $N_{E_8}(A_1*A_2, A_1*A_2, A_2)= 15000$,\linebreak $N_{E_8}(A_1*A_2, A_1^3, A_2)= 9825$, $N_{E_8}(A_1^3, A_1^3, A_2)= 6000$, $N_{E_8}(A_3, A_3, A_1^2)= 4050$, $N_{E_8}(A_3, \mathbreak A_1*A_2, A_1^2)= 13500$, $N_{E_8}(A_3, A_1^3, A_1^2)= 9450$, $N_{E_8}(A_1*A_2, A_1*A_2, A_1^2)= 30825$,\linebreak $N_{E_8}(A_1*A_2, A_1^3, A_1^2)= 17325$, $N_{E_8}(A_1^3, A_1^3, A_1^2)= 7875$, $N_{E_8}(A_3, A_3, A_1, A_1)= 12150$, $N_{E_8}(A_3, A_1*A_2, A_1, A_1)= 42525$, $N_{E_8}(A_3, A_1^3, A_1, A_1)= 30375$, $N_{E_8}(A_1*A_2, A_1*A_2, A_1, \mathbreak A_1)= 106650$, $N_{E_8}(A_1*A_2, A_1^3, A_1, A_1)= 64125$, $N_{E_8}(A_1^3, A_1^3, A_1, A_1)= 33750$, $N_{E_8}(A_3, \mathbreak A_2, A_2, A_1)= 10575$, $N_{E_8}(A_3, A_2, A_1^2, A_1)= 29700$, $N_{E_8}(A_3, A_1^2, A_1^2, A_1)= 76950$,\linebreak $N_{E_8}(A_1*A_2, A_2, A_2, A_1)= 35700$, $N_{E_8}(A_1*A_2, A_2, A_1^2, A_1)= 84825$, $N_{E_8}(A_1*A_2, A_1^2, A_1^2, \mathbreak A_1)= 171450$, $N_{E_8}(A_1^3, A_2, A_2, A_1)= 25125$, $N_{E_8}(A_1^3, A_2, A_1^2, A_1)= 55125$, $N_{E_8}(A_1^3, A_1^2, \mathbreak A_1^2, A_1)= 94500$, $N_{E_8}(A_3, A_2, A_1, A_1, A_1)= 91125$, $N_{E_8}(A_3, A_1^2, A_1, A_1, A_1)= 243000$, $N_{E_8}(A_1*A_2, A_2, A_1, A_1, A_1)= 276750$, $N_{E_8}(A_1*A_2, A_1^2, A_1, A_1, A_1)= 597375$, $N_{E_8}(A_1^3, \mathbreak A_2, A_1, A_1, A_1)= 185625$, $N_{E_8}(A_1^3, A_1^2, A_1, A_1, A_1)= 354375$, $N_{E_8}(A_3, A_1, A_1, A_1, A_1, \mathbreak A_1)= 759375$, $N_{E_8}(A_1*A_2, A_1, A_1, A_1, A_1, A_1)= 2025000$, $N_{E_8}(A_1^3, A_1, A_1, A_1, A_1, A_1)= 1265625$, $N_{E_8}(A_2, A_2, A_2, A_2)= 9350$, $N_{E_8}(A_2, A_2, A_2, A_1^2)= 24975$, $N_{E_8}(A_2, A_2, A_1^2, \mathbreak A_1^2)= 64350$, $N_{E_8}(A_2, A_1^2, A_1^2, A_1^2)= 143100$, $N_{E_8}(A_1^2, A_1^2, A_1^2, A_1^2)= 261225$, $N_{E_8}(A_2, A_2, \mathbreak A_2, A_1, A_1)= 78000$, $N_{E_8}(A_2, A_2, A_1^2, A_1, A_1)= 203625$, $N_{E_8}(A_2, A_1^2, A_1^2, A_1, A_1)= \mathbreak 479250$, $N_{E_8}(A_1^2, A_1^2, A_1^2, A_1, A_1)= 951750$, $N_{E_8}(A_2, A_2, A_1, A_1, A_1, A_1)= 641250$,\linebreak $N_{E_8}(A_2, A_1^2, A_1, A_1, A_1, A_1)= 1569375$, $N_{E_8}(A_1^2, A_1^2, A_1, A_1, A_1, A_1)= 3341250$, $N_{E_8}(A_2, \mathbreak A_1, A_1, A_1, A_1, A_1, A_1)= 5062500$, $N_{E_8}(A_1^2, A_1, A_1, A_1, A_1, A_1, A_1)= 11390625$, $N_{E_8}(A_1, \mathbreak A_1, A_1, A_1, A_1, A_1, A_1, A_1)= 37968750$, plus the assignments implied by (\AD) and (\AE), all other numbers $N_{E_8}(T_1,\dots,T_d)$ being zero. Finally, we substitute the values found for the decomposition numbers and the formulae for the characteristic polynomials from (\AJ) in (\AC), and we obtain {\eightpoint $$\allowdisplaybreaks\multline (M^m_{E_8})^*\(x,y\)= \frac {1} {1344}{{m( 15 m -8) ( 15 m -11) ( 15 m -14) ( 5 m -1) ( 5 m -2) ( 3 m -1) ( 5 m -3) {x^8}{y^8}}} \\- \frac {5} {56}{{ {m^2}( 15 m -8) ( 15 m -11) ( 5 m -1) ( 5 m -2) ( 5 m -3) ( 3 m -1) {x^8}{y^7}} } \\+ \frac {5} {48}{{ {m^2}( 15 m -8) ( 5 m -1) ( 3 m -1) ( 5 m -2) ( 225 {m^2} -90m-13) {x^8}{y^6}}} \\- \frac {15} {8}{{ {m^2}( 5 m -1) ( 5 m -2) ( 3 m -1) ( 5 m +1) ( 75 {m^2} -15m-4) {x^8}{y^5}}} \\+ \frac {1} {32}{{{m^2} ( 5 m +1) ( 5 m -1) ( 84375 {m^4} - 12375 {m^2} +436) {x^8}{y^4}}} \\- \frac {15} {8}{{ {m^2}( 5 m -1) ( 3 m +1) ( 5 m +1) ( 5 m +2) ( 75 {m^2}+15m-4 ) {x^8}{y^3}}} \\+ \frac {5} {48}{{ {m^2}( 3 m +1) ( 5 m +1) ( 5 m +2) ( 15 m +8) ( 225 {m^2} +90m-13) {x^8}{y^2}}} \\- \frac {5} {56}{{ {m^2} ( 3 m +1) ( 5 m +1) ( 5 m +2) ( 5 m +3) ( 15 m +8) ( 15 m +11) {x^8}y}} \\+ \frac {1} {1344}{{m ( 3 m +1) ( 5 m +1) ( 5 m +2) ( 5 m +3) ( 15 m +8) ( 15 m +11) ( 15 m +14) {x^8}}} \\+ \frac {5} {56}{{ m( 15 m -8) ( 15 m -11) ( 5 m -1) ( 5 m -2) ( 5 m -3) ( 3 m -1) {x^7}{y^7}}} \\- \frac {25} {8}{{ {m^2}( 5 m -1) ( 3 m -1) ( 5 m -2) {{( 15 m -8) }^2} {x^7}{y^6}}} \\+ \frac {375} {8}{{ {m^2}( 5 m -1) ( 5 m -2) ( 3 m -1) ( 45 {m^2} -12m-2) {x^7}{y^5}}} \\- \frac {15} {8}{{ {m^2}( 5 m +1) ( 5 m -1) ( 5625 {m^3} - 2250 {m^2} -75m+74) {x^7}{y^4}}} \\+ \frac {15} {8}{{ {m^2}( 5 m -1) ( 5 m +1) ( 5625 {m^3} +2250m^2-75m-74) {x^7}{y^3}} } \\- \frac {375} {8}{{ {m^2}( 3 m +1) ( 5 m +1) ( 5 m +2) ( 45 {m^2} +12m-2) {x^7}{y^2}}} \\+ \frac {25} {8}{{ {m^2} ( 3 m +1) ( 5 m +1) ( 5 m +2) {{( 15 m +8) }^2}{x^7}y} } \\- \frac {5} {56}{{ m ( 3 m +1) ( 5 m +1) ( 5 m +2) ( 5 m +3) ( 15 m +8) ( 15 m +11) {x^7}}} \\+ \frac {5} {48}{{ m( 15 m -8) ( 5 m -1) ( 5 m -2) ( 3 m -1) ( 195 m -107) {x^6}{y^6}}} \\- \frac {375} {8}{{{m^2} ( 39 m -16) ( 5 m -1) ( 5 m -2) ( 3 m -1) {x^6}{y^5}}} \\+ \frac {5} {16}{{ {m^2} ( 5 m -1) ( 219375 {m^3} - 103500 {m^2} + 3675 m +2342 ) {x^6}{y^4}}} \\- \frac {75} {4}{{ {m^2}(5 m -1) ( 5 m +1) ( 975 {m^2} -32) {x^6}{y^3}} } \\+ \frac {5} {16}{{ {m^2}( 5 m +1) ( 219375 {m^3} + 103500 {m^2} +3675 m -2342) {x^6}{y^2}}} \\- \frac {375} {8}{{ {m^2} ( 3 m +1) ( 5 m +1) ( 5 m +2) ( 39 m +16) {x^6}y}} \\+ \frac {5} {48}{{ m ( 3 m +1) ( 5 m +1) ( 5 m +2) ( 15 m +8) ( 195 m +107) {x^6}}} + 15 m( 30 m -13) ( 5 m -1) ( 5 m -2) ( 3 m -1) {x^5}{y^5} \\- 15 {m^2} ( 5 m -1) ( 2250 {m^2} - 1395 m+218 ) {x^5}{y^4} + 75 {m^2}( 5 m -1) ( 900 {m^2} -66m-23) {x^5}{y^3} \\- 75 {m^2}( 5 m +1) ( 900 {m^2} +66m-23) {x^5}{y^2} + 15 {m^2} ( 5 m +1) ( 2250 {m^2} + 1395 m+218) {x^5}y \\- 15 m ( 3 m +1) ( 5 m +1) ( 5 m +2) ( 30 m +13) {x^5} + \frac {1} {2}{{m ( 5 m -1) ( 10350 {m^2} - 6675 m+1084) {x^4}{y^4}}} \\- 15{m^2} ( 1380 m -307) ( 5 m -1) {x^4}{y^3} + 5 {m^2} ( 31050 {m^2} -577) {x^4}{y^2} - 15 {m^2} ( 5 m +1) ( 1380 m +307) {x^4}y \\+ \frac {1} {2}{{m ( 5 m +1) ( 10350 {m^2} + 6675 m+1084 ) {x^4}}} + 45 m( 45 m -11) ( 5 m -1) {x^3}{y^3} - 1125 {m^2} ( 27 m -4) {x^3}{y^2} \\+ 1125 {m^2} ( 27 m +4) {x^3}y - 45 m ( 5 m +1) ( 45 m +11) {x^3} + \frac {35} {2}{{ m ( 105 m -17) {x^2}{y^2}}} - 3675 {m^2}{x^2}y \\+ \frac {35} {2}{{ m ( 105 m +17) {x^2}}} + 120 m xy - 120 m x + 1 . \endmultline\tag{\tenpoint\AL}$$}% \subhead 8. The $F=M$ Conjecture\endsubhead In this section we describe briefly the $F=M$ Conjecture of Armstrong \cite{\ArmDAA} predicting a surprising relation between the $M$-triangle of the $m$-divisible partitions poset and the $F$-triangle of the generalised cluster complex of Fomin and Reading \cite{\FoReAA}, we review the progress from \cite{\KratCB}, and we explain that our results from Sections~6 and 7 provide proofs of the $F=M$ Conjecture for $E_7$ and $E_8$. If we put this together with the results from \cite{\KratCB}, then it remains only the $D_n$ case of the conjecture which is not completely proven. For a non-negative integer $m$, the generalised cluster complex $\De^m(\Phi)$ is a certain simplicial complex on a certain set of ``coloured" roots, the roots being from $\Phi$. The precise definition will not be important here, we refer the reader to \cite{\FoReAA, Sec.~2}. The only fact which is important here is that some of the coloured roots can be positive, others negative. Let $f_{k,l}(\Phi,m)$ denote the number of faces of $\De^m(\Phi)$ which contain exactly $k$ positive and $l$ negative coloured roots. Define the $F$-triangle of $\De^m(\Phi)$, denoted by $F^m_\Phi(x,y)$, as the two-variable polynomial $$F^m_\Phi(x,y)= \sum _{k,l\ge0} ^{}f_{k,l}(\Phi,m)\,x^ky^l. \tag\AM$$ It is called ``triangle" because all faces have cardinality at most $n$ and, thus, in the summation in (\AM) we can restrict the summation indices to the triangle $k+l\le n$, $k,l\ge0$. The generalised version of Chapoton's (ex-)conjecture \cite{\ChaFAA, Conjecture~1}, due to Armstrong \cite{\ArmDAA, Conjecture~5.3.2}, is the following. \proclaim{Conjecture FM} For any finite root system $\Phi$ of rank $n$, we have $$F^m_\Phi(x,y)=y^n\,M^m_\Phi\(\frac {1+y} {y-x},\frac {y-x} {y}\). \tag\AMa$$ Equivalently, $$ (1-xy)^n F^m_\Phi\left(\frac {x(1+y)} {1-xy},\frac {xy} {1-xy}\right)=(M^m_\Phi)^*(-x,-y).\tag\AN $$ \endproclaim It is easy to see (cf\. \cite{\KratCB, Sec.~8}) that it is enough to prove the conjecture for the irreducible root systems. In \cite{\KratCB}, this has been done for the root systems of type $A_n$, $B_n$, $I_2(a)$, $H_3$, $H_4$, $F_4$, and $E_6$. The paper contains also a partial proof for the root system of type $D_n$. Given that we computed the $M$-triangle of the $m$-divisible non-crossing partitions poset for $E_7$ and $E_8$ and that the $F$-triangle of the generalised cluster complex has been computed in \cite{\KratCB} for {\it all\/} irreducible root systems (thus, in particular, for $E_7$ and $E_8$), the verification of (\AN) for $E_7$ and $E_8$ is pure routine on a computer algebra system. If we combine these observations with the proof of the $m=1$ case of the $F=M$ Conjecture by Athanasiadis \cite{\AthaAI} (an alternative case-by-case proof is provided by the results in \cite{\KratCB} and in the present paper), then we obtain the following theorem. \proclaim{Theorem FM} Conjecture FM is true with the possible exception of root systems containing a copy of the root system $D_k$ for some $k$. If $m=1$, Conjecture FM is true unconditionally. \endproclaim \subhead 9. A reciprocity phenomenon\endsubhead By staring at the explicit expressions for the $M$-triangles of the $m$-divisible non-crossing partitions posets for the irreducible root systems which we computed in \cite{\KratCB} and in the present paper (in the $D_n$ case, this expression is only conjectural), one observes a curious reciprocity relation between the original $M$-triangle and the $M$-triangle with $m$ replaced by $-m$. From the outset, the $M$-triangle with $-m$ in place of $m$ has no combinatorial meaning since there is no meaning for ``$(-m)$-divisible non-crossing partitions." Nevertheless, it would be interesting to find an intrinsic explanation of the phenomenon given in the theorem below. In any case, examples of situations where combinatorial meaning was given to non-combinatorial parameters are not so uncommon; for example, there exist reciprocity theorems for {\it $P$-partitions} and the {\it Ehrhart quasi-polynomial} of polytopes (cf.\ \cite{\StanAP, Theorems~4.5.7, 4.6.26, Cor.~4.5.15}), which have their explanation in Stanley's reciprocity theorem for linear homogeneous diophantine equations \cite{\StanAP, Theorem~4.6.14}, and there exist reciprocity theorems for monomer-dimer coverings of rectangles (cf.\ \cite{\AnBBAA, \PropAI, \StanBZ}). \proclaim{Theorem \TA} For any finite root system of rank $n$, except possibly for those which contain a copy of $D_k$ for some $k$, we have $$ y^nM^{-m}_\Phi(xy,1/y)=M^m_\Phi(x,y). \tag\AO$$ If the (conjectural) expression for $M^m_{D_n}(x,y)$ in \cite{\KratCB, Sec.~11, Prop.~D} is correct, then {\rm(\AO)} is true without any exceptions. Phrased differently, if the $F=M$ Conjecture is true, then {\rm(\AO)} is true unconditionally. \endproclaim Formula (\AO) contains a reciprocity between numbers of maximal simplices in the generalised cluster complex $\De^m(\Phi)$, observed by Fomin and Reading \cite{\FoReAA, Eq.~(11.2) and Prop.~11.4}, and revealed as an instance of Ehrhart reciprocity by Athanasiadis and Tza\-na\-ki \cite{\AtTzAA, Sec.~7}, as a special case. (Again, for root systems containing a copy of $D_k$ for some $k$, this is a conditional statement.) To see this, we first translate (\AO) into a reciprocity relation for the $F$-triangle $F^m_\Phi(x,y)$ via (\AMa): $$F^m_\Phi(x,y)=(1+x)^n\, F^{-m}_\Phi\(-\frac {x} {1+x},\frac {y-x} {1+x}\). \tag\AOa$$ If we compare coefficients of $x^ny^0$ on both sides of this equation, then we obtain the relation $$f_{n,0}(\Phi,m)= \sum _{k=0} ^{n}(-1)^kf_k(\Phi,-m),\tag\AOb$$ where $f_k(\Phi,m)= \sum _{l=0} ^{k}f_{l,k-l}(\Phi,m)$ is the {\it total\/} number of faces in $\De^m(\Phi)$ containing exactly $k$ roots (positive or negative). In the notation of \cite{\AtTzAA, \FoReAA}, we have $f_{n,0}(\Phi,m)=N^+(\Phi,m)$, and by \cite{\FoReAA, Eq.~(10.2)} the sum on the right-hand side of (\AOb) is equal to $(-1)^nf_n(\Phi,-m-1)$ ($(-1)^nN(\Phi,-m-1)$ in the notation of \cite{\FoReAA}). Thus, we arrive at the reciprocity relation $$N^+(\Phi,m)=(-1)^{n}N(\Phi,-m-1),$$ which is exactly \cite{\FoReAA, Eq.~(11.2)}. The combinatorial significance of (\AOa) in general is less clear. Comparison of coefficients of $x^ky^l$ on both sides leads to the identity $$f_{k,l}(\Phi,m)= \sum _{r,s\ge0} ^{}(-1)^{r+s+l}\binom {n-r-s}{k+l-r-s}\binom sl f_{r,s}(\Phi,-m).\tag\AOc$$ \subhead 10. A formula for the $A_n$ decomposition numbers\endsubhead Since the decomposition numbers $N_\Phi(T_1,T_2,\dots,T_d)$ carry so much information on the enumerative structure of (ordinary and generalised) non-crossing partitions (see, for example, Propositions~\PA\ and \PE), it would be of intrinsic interest to compute these numbers also for the classical root systems. As a matter of fact, the type $A_n$ decomposition numbers of full rank are known due to a result of Goulden and Jackson \cite{\GoJaAS, Theorem~3.2} on the minimal factorization of a long cycle. (The condition on the sum $l(\al_1)+l(\al_2)+\dots+l(\al_m)$ is misstated throughout the latter paper. It should be replaced by $l(\al_1)+l(\al_2)+\dots+l(\al_m)=(m-1)n+1$.) In the following theorem, we state this result in our language. \proclaim{Theorem 9} Let $\Phi=A_n$, and let $T_1,T_2,\dots,T_d$ be types with $\rk T_1+\rk T_2+\dots+\rk T_d=n$, where $$T_i=A_1^{m_1^{(i)}}*A_2^{m_2^{(i)}}*\dots*A_n^{m_n^{(i)}},\quad i=1,2,\dots,d.$$ Then $$ N_{A_n}(T_1,T_2,\dots,T_d)=(n+1)^{d-1}\prod _{i=1} ^{d} \frac {1} {n-\rk T_i+1}\binom {n-\rk T_i+1}{m_1^{(i)},m_2^{(i)},\dots, m_n^{(i)}}, \tag\AOe$$ where the multinomial coefficient is defined by $$\binom {M}{m_1,m_2,\dots,m_n}=\frac {M!} {m_1!\,m_2!\cdots m_n!\,(M-m_1-m_2-\dots-m_n)!}.$$ \endproclaim This result allows one to derive a compact formula for {\it all\/} type $A_n$ decomposition numbers. \proclaim{Theorem 10} Let $\Phi=A_n$, and let the types $T_1,T_2,\dots,T_d$ be given, where $$T_i=A_1^{m_1^{(i)}}*A_2^{m_2^{(i)}}*\dots*A_n^{m_n^{(i)}},\quad i=1,2,\dots,d.$$ Then $$\multline N_{A_n}(T_1,T_2,\dots,T_d)=(n+1)^{d-1} \binom {n+1}{\rk T_1+\rk T_2+\dots+\rk T_d+1}\\ \times \prod _{i=1} ^{d} \frac {1} {n-\rk T_i+1}\binom {n-\rk T_i+1}{m_1^{(i)},m_2^{(i)},\dots, m_n^{(i)}}. \endmultline\tag\AOd$$ \endproclaim \demo{Proof} If we write $r$ for $n-\rk T_1-\rk T_2-\dots-\rk T_d$, then the relation (\AE) becomes $$N_{A_n}(T_1,T_2,\dots,T_d)= {\sum _{T:\rk T=r}} ^{}N_{A_n}(T_1,T_2,\dots,T_d,T). $$ Upon letting $T=A_1^{m_1}*A_2^{m_2}*\dots*A_n^{m_n}$, substitution of (\AOe) in the above equation yields $$\align N_{A_n}(T_1,T_2,\dots,T_d)&=\sum _{m_1+2m_2+\dots+nm_n=r} ^{} (n+1)^{d}\frac {1} {n-r+1}\binom {n-r+1}{m_1,m_2,\dots,m_n}\\ &\kern2cm\cdot \prod _{i=1} ^{d} \frac {1} {n-\rk T_i+1}\binom {n-\rk T_i+1}{m_1^{(i)},m_2^{(i)},\dots, m_n^{(i)}}. \endalign$$ Now, by comparison of coefficients of $z^r$ on both sides of $$(1+z+z^2+z^3+\cdots)^M=(1-z)^{-M},$$ we infer $$\sum _{m_1+2m_2+\dots+rm_r=r} ^{}\binom M{m_1,m_2,\dots,m_r}= \binom {M+r-1}r. $$ If we use this identity with $M=n-r+1$, we obtain our claim after little simplification.\quad \quad \qed \enddemo The reader should note that formula (\AOd) generalises Kreweras' formula \cite{\KrewAC, Theorem~4} for the number of non-crossing partitions of $n+1$ with given block sizes, to which it reduces for $d=1$. \subhead Appendix: The decomposition numbers for types $A_n$ and $D_n$ for small $n$, and for $E_6$\endsubhead In this appendix we list the full rank decomposition numbers, that is, the decomposition numbers $N_{\Phi}(T_1,T_2,\dots,T_d)$, where $\rk T_1+\rk T_2+\dots+\rk T_d=\rk \Phi$, for $\Phi=A_1,A_2,A_3,A_4,A_5,A_6,A_7,D_4,D_5,D_6,D_7,E_6$. These numbers are required for setting up the system of equations (\AF) in Sections~6 and 7. We do not list decomposition numbers which are zero. It was explained in Section~4 how to compute these numbers by setting up a system of linear equations for them. Whenever, aside from the listing of the numbers, there is no further comment, then the computation of the decomposition numbers is either trivial (for $A_1$ and $A_2$), or the equations in Propositions~\PB\ and \PE, together with the assignments from Propositions~\PF\ and \PC\ determine them already uniquely. If not, then we mention the additional assignments, respectively considerations, which are required for the computation.\footnote{The {\sl Mathematica} inputs for the computations are available at\newline {\tt http://www.mat.univie.ac.at/\~{}kratt/artikel/cluster2.html}.} Clearly, the decomposition numbers for $A_n$, $n=1,2,\dots,$ can be computed directly from Theorem~9. \NoBlackBoxes \subsubhead The decomposition numbers for $A_1$\endsubsubhead $N_{A_1}(A_1)=1$. \subsubhead The decomposition numbers for $A_2$\endsubsubhead $N_{A_2}(A_2)=1$, $N_{A_2}(A_1,A_1)=3$. \subsubhead The decomposition numbers for $A_3$\endsubsubhead $N_{A_3}(A_3)=1$, $N_{A_3}(A_2,A_1)=4$, $N_{A_3}(A_1^2,A_1)=2$, $N_{A_3}(A_1,A_1,A_1)=16$. \subsubhead The decomposition numbers for $A_4$\endsubsubhead $N_{A_4}(A_4)=1$, $N_{A_4}(A_3,A_1)=5$, $N_{A_4}(A_1*A_2,A_1)=5$, %$N_{A_4}(A_1^3,A_1)=0$, $N_{A_4}(A_2,A_2)=5$, $N_{A_4}(A_2,A_1^2)=5$, $N_{A_4}(A_1^2,A_1^2)=5$, $N_{A_4}(A_2,A_1,A_1)=25$, $N_{A_4}(A_1^2,\mathbreak A_1,A_1)=25$, $N_{A_4}(A_1,A_1,A_1,A_1)=125$. \subsubhead The decomposition numbers for $A_5$\endsubsubhead $N_{A_5}(A_5)=1$, $N_{A_5}(A_4,A_1)=6$, $N_{A_5}(A_1*A_3,A_1)=6$, $N_{A_5}(A_2^2,A_1)=3$, %$N_{A_5}(A_1^2*A_2,A_1)=0$, %$N_{A_5}(A_1^4,A_1)=0$, $N_{A_5}(A_3,A_2)=6$, $N_{A_5}(A_1*A_2,A_2)=12$, $N_{A_5}(A_1^3,A_2)=2$, $N_{A_5}(A_3,\mathbreak A_1^2)=9$, $N_{A_5}(A_1*A_2,A_1^2)=18$, $N_{A_5}(A_1^3,A_1^2)=3$, $N_{A_5}(A_3,A_1,A_1)=36$, $N_{A_5}(A_1*A_2,\mathbreak A_1,A_1)=72$, $N_{A_5}(A_1^3,A_1,A_1)=12$, $N_{A_5}(A_2,A_2,A_1)=36$, $N_{A_5}(A_2,A_1^2,A_1)=54$, $N_{A_5}(A_1^2,A_1^2,A_1)=81$, $N_{A_5}(A_2,A_1,A_1,A_1)=216$, $N_{A_5}(A_1^2,A_1,A_1,A_1)=324$, $N_{A_5}(A_1,\mathbreak A_1,A_1,A_1,A_1)=1296$. \subsubhead The decomposition numbers for $A_6$\endsubsubhead %In addition to the described equations and assignments, we use also %the last assertion in Proposition~\PB\ for the type $A_1^4$, that is, %$N_{A_6}(A_1^4,\mathbreak A_2)=N_{A_6}(A_1^4,A_1^2)=0$. %This yields $N_{A_6}(A_6)=1$, $N_{A_6}(A_5, A_1) = 7$, $N_{A_6}(A_1*A_4, A_1) = 7$, $N_{A_6}(A_2*A_3, A_1) = 7$, $N_{A_6}(A_4, A_2) = 7$, $N_{A_6}(A_1*A_3, A_2) = 14$, $N_{A_6}(A_2^2, A_2) = 7$, $N_{A_6}(A_1^2*A_2, A_2) = 7$, $N_{A_6}(A_4, A_1^2) = 14$, $N_{A_6}(A_1*A_3, A_1^2) = 28$, $N_{A_6}(A_2^2, A_1^2) = 14$, $N_{A_6}(A_1^2*A_2, A_1^2) = 14$, $N_{A_6}(A_3, A_3) = 7$, $N_{A_6}(A_3, A_1*A_2) = 21$, $N_{A_6}(A_3, A_1^3) = 7$, $N_{A_6}(A_1*A_2, A_1*A_2) = 63$, $N_{A_6}(A_1*A_2, A_1^3) = 21$, $N_{A_6}(A_1^3, A_1^3) = 7$, $N_{A_6}(A_4, A_1, A_1) = 49$, $N_{A_6}(A_1*A_3, A_1, A_1) = 98$, $N_{A_6}(A_2^2, A_1, A_1) = 49$, $N_{A_6}(A_1^2*A_2, A_1, A_1) = 49$, $N_{A_6}(A_3, A_2, A_1) = 49$, $N_{A_6}(A_3, A_1^2, A_1) = 98$, $N_{A_6}(A_1*A_2, A_2, A_1) = 147$, $N_{A_6}(A_1*A_2, A_1^2, A_1) = 294$, $N_{A_6}(A_1^3, A_2, A_1) = 49$, $N_{A_6}(A_1^3, A_1^2, A_1) = 98$, $N_{A_6}(A_3, A_1, A_1, A_1) = 343$, $N_{A_6}(A_1*A_2, A_1, A_1, A_1) = 1029$, $N_{A_6}(A_1^3, A_1, A_1, A_1) = 343$, $N_{A_6}(A_2, A_2, A_2) = 49$, $N_{A_6}(A_2, A_2, A_1^2) = 98$, $N_{A_6}(A_2, A_1^2, A_1^2) = 196$, $N_{A_6}(A_1^2, A_1^2, A_1^2) = 392$, $N_{A_6}(A_2, A_2, A_1,\mathbreak A_1) = 343$, $N_{A_6}(A_2, A_1^2, A_1, A_1) = 686$, $N_{A_6}(A_1^2, A_1^2, A_1, A_1) = 1372$, $N_{A_6}(A_2, A_1, A_1, A_1,\mathbreak A_1) = 2401$, $N_{A_6}(A_1^2, A_1, A_1, A_1, A_1) = 4802$, $N_{A_6}(A_1, A_1, A_1, A_1, A_1, A_1) = 16807$. \subsubhead The decomposition numbers for $A_7$\endsubsubhead %In addition to the described equations and assignments, we use also %the last assertion in Proposition~\PB\ for the type $A_1^5$, that is, %$N_{A_7}(A_1^5,\mathbreak A_2)=N_{A_7}(A_1^5,A_1^2)=0$. %This yields $N_{A_7}(A_7)=1$, $N_{A_7}(A_6, A_1) = 8$, $N_{A_7}(A_1*A_5, A_1) = 8$, $N_{A_7}(A_2*A_4, A_1) = 8$, $N_{A_7}(A_3^2, A_1) = 4$, $N_{A_7}(A_5, A_2) = 8$, $N_{A_7}(A_1*A_4, A_2) = 16$, $N_{A_7}(A_2*A_3, A_2) = 16$, $N_{A_7}(A_1^2*A_3, A_2) = 8$, $N_{A_7}(A_1*A_2^2, A_2) = 8$, $N_{A_7}(A_5, A_1^2) = 20$, $N_{A_7}(A_1*A_4, A_1^2) = 40$, $N_{A_7}(A_2*A_3, A_1^2) = 40$, $N_{A_7}(A_1^2*A_3, A_1^2) = 20$, $N_{A_7}(A_1*A_2^2, A_1^2) = 20$, $N_{A_7}(A_5, A_1, A_1) = 64$, $N_{A_7}(A_1*A_4, A_1, A_1) = 128$, $N_{A_7}(A_2*A_3, A_1, A_1) = 128$, $N_{A_7}(A_1^2*A_3, A_1, A_1) = 64$, $N_{A_7}(A_1*A_2^2, A_1, A_1) = 64$, $N_{A_7}(A_4, A_3) = 8$, $N_{A_7}(A_1*A_3, A_3) = 24$, $N_{A_7}(A_2^2, A_3) = 12$, $N_{A_7}(A_1^2*A_2, A_3) = 24$, $N_{A_7}(A_1^4, A_3) = 2$, $N_{A_7}(A_4, A_1*A_2) = 32$, $N_{A_7}(A_1*A_3, A_1*A_2) = 96$, $N_{A_7}(A_2^2, A_1*A_2) = 48$, $N_{A_7}(A_1^2*A_2, A_1*A_2) = 96$, $N_{A_7}(A_1^4, A_1*A_2) = 8$, $N_{A_7}(A_4, A_1^3) = 16$, $N_{A_7}(A_1*A_3, A_1^3) = 48$, $N_{A_7}(A_2^2, A_1^3) = 24$, $N_{A_7}(A_1^2*A_2, A_1^3) = 48$, $N_{A_7}(A_1^4, A_1^3) = 4$, $N_{A_7}(A_4, A_2, A_1) = 64$, $N_{A_7}(A_1*A_3, A_2, A_1) = 192$, $N_{A_7}(A_2^2, A_2, A_1) = 96$, $N_{A_7}(A_1^2*A_2, A_2, A_1) = 192$, $N_{A_7}(A_1^4, A_2, A_1) = 16$, $N_{A_7}(A_4,\mathbreak A_1^2, A_1) = 160$, $N_{A_7}(A_1*A_3, A_1^2, A_1) = 480$, $N_{A_7}(A_2^2, A_1^2, A_1) = 240$, $N_{A_7}(A_1^2*A_2, A_1^2, A_1) = 480$, $N_{A_7}(A_1^4, A_1^2, A_1) = 40$, $N_{A_7}(A_4, A_1, A_1, A_1) = 512$, $N_{A_7}(A_1*A_3, A_1, A_1, A_1) = 1536$, $N_{A_7}(A_2^2, A_1, A_1, A_1) = 768$, $N_{A_7}(A_1^2*A_2, A_1, A_1, A_1) = 1536$, $N_{A_7}(A_1^4, A_1, A_1, A_1) = 128$, $N_{A_7}(A_3, A_3, A_1) = 64$, $N_{A_7}(A_3, A_1*A_2, A_1) = 256$, $N_{A_7}(A_3, A_1^3, A_1) = 128$, $N_{A_7}(A_1*A_2, A_1*A_2, A_1) = 1024$, $N_{A_7}(A_1*A_2, A_1^3, A_1) = 512$, $N_{A_7}(A_1^3, A_1^3, A_1) = 256$, $N_{A_7}(A_3, A_2,\mathbreak A_2) = 64$, $N_{A_7}(A_3, A_2, A_1^2) = 160$, $N_{A_7}(A_3, A_1^2, A_1^2) = 400$, $N_{A_7}(A_1*A_2, A_2, A_2) = 256$, $N_{A_7}(A_1*A_2, A_2, A_1^2) = 640$, $N_{A_7}(A_1*A_2, A_1^2, A_1^2) = 1600$, $N_{A_7}(A_1^3, A_2, A_2) = 128$, $N_{A_7}(A_1^3, A_2, A_1^2) = 320$, $N_{A_7}(A_1^3, A_1^2, A_1^2) = 800$, $N_{A_7}(A_3, A_2, A_1, A_1) = 512$, $N_{A_7}(A_3, A_1^2,\mathbreak A_1, A_1) = 1280$, $N_{A_7}(A_1*A_2, A_2, A_1, A_1) = 2048$, $N_{A_7}(A_1*A_2, A_1^2, A_1, A_1) = 5120$, $N_{A_7}(A_1^3, A_2, A_1, A_1) = 1024$, $N_{A_7}(A_1^3, A_1^2, A_1, A_1) = 2560$, $N_{A_7}(A_3, A_1, A_1, A_1, A_1) =\mathbreak 4096$, $N_{A_7}(A_1*A_2, A_1, A_1, A_1, A_1) = 16384$, $N_{A_7}(A_1^3, A_1, A_1, A_1, A_1) = 8192$, $N_{A_7}(A_2, A_2,\mathbreak A_2, A_1) = 512$, $N_{A_7}(A_2, A_2, A_1^2, A_1) = 1280$, $N_{A_7}(A_2, A_1^2, A_1^2, A_1) = 3200$, $N_{A_7}(A_1^2, A_1^2, A_1^2,\mathbreak A_1) = 8000$, $N_{A_7}(A_2, A_2, A_1, A_1, A_1) = 4096$, $N_{A_7}(A_2, A_1^2, A_1, A_1, A_1) = 10240$, $N_{A_7}(A_1^2,\mathbreak A_1^2, A_1, A_1, A_1) = 25600$, $N_{A_7}(A_2, A_1, A_1, A_1, A_1, A_1) = 32768$, $N_{A_7}(A_1^2, A_1, A_1, A_1, A_1,\mathbreak A_1) = 81920$, $N_{A_7}(A_1, A_1, A_1, A_1, A_1, A_1, A_1) = 262144$. \subsubhead The decomposition numbers for $D_4$\endsubsubhead $N_{D_4}(D_4)=1$, $N_{D_4}(A_3,A_1)=9$, %$N_{D_4}(A_1*A_2,A_1)=0$, $N_{D_4}(A_1^3,A_1)=3$, $N_{D_4}(A_2,A_2)=6$, $N_{D_4}(A_2,A_1^2)=9$, %$N_{D_4}(A_1^2,A_1^2)=0$, $N_{D_4}(A_2,A_1,A_1)=36$, $N_{D_4}(A_1^2,A_1,A_1)=27$, $N_{D_4}(A_1,A_1,A_1,A_1)=162$. \subsubhead The decomposition numbers for $D_5$\endsubsubhead $N_{D_5}(D_5)=1$, $N_{D_5}(D_4,A_1)=4$, $N_{D_5}(A_4,A_1)=8$, $N_{D_5}(A_1*A_3,A_1)=4$, $N_{D_5}(A_1^2*A_2,A_1)=4$, $N_{D_5}(A_3,A_2)=12$, $N_{D_5}(A_1*A_2,A_2)=16$, $N_{D_5}(A_1^3,A_2)=8$, $N_{D_5}(A_3,A_1^2)=22$, $N_{D_5}(A_1*A_2,A_1^2)=8$, $N_{D_5}(A_1^3,A_1^2)=4$, $N_{D_5}(A_3,A_1,A_1)=80$, $N_{D_5}(A_1*A_2,A_1,A_1)=64$, $N_{D_5}(A_1^3,A_1,A_1)=32$, $N_{D_5}(A_2,A_2,\mathbreak A_1)=64$, $N_{D_5}(A_2,A_1^2,A_1)=96$, $N_{D_5}(A_1^2,A_1^2,A_1)=80$, $N_{D_5}(A_2,A_1,A_1,A_1)=384$, $N_{D_5}(A_1^2,A_1,A_1,A_1)=448$, $N_{D_5}(A_1,A_1,A_1,A_1,A_1)=2048$. \subsubhead The decomposition numbers for $D_6$\endsubsubhead Unfortunately, the solution space of the system of linear equations does not consist of a unique solution but is two-dimensional.\linebreak {\sl Mathematica~5.2} expresses all the decomposition numbers in terms of $X=N_{D_6}(A_1^3,A_1^3)$ and $Y=N_{D_6}(A_1^4,A_1^2)$. In the sequel we make use of the two relations $$\align N_{D_6}(A_1^2*A_2, A_1^2) &= 25 - \hphantom{1}\frac {27}{40}X - 3Y, \\ N_{D_6}(A_3, A_3) &= 20 + \frac9{100}X. \endalign$$ They imply $X\equiv0$ (mod $200$), $X\le 1000/27<200$, $Y\equiv2$ (mod $3$), and $Y\le 25/3<9$. Hence, we have $X=N_{D_6}(A_1^3,A_1^3)=0$, and the three possibilities $Y=2,5,8$. To decide which of the three values is the true value, we argue as in Section~6 when we had to decide which of two possible values for $N_{E_7}(A_1^2*A_2,A_1^3)$ was the correct one (see the paragraph after (\AKj)). Since Lemma~\LBa\ in the case of $D_6$ says that the orbits of reflections have all size $5$, here the conclusion is that $5$ must divide $Y=N_{D_6}(A_1^4,A_1^2)$. Thus, $N_{D_6}(A_1^4,A_1^2)=5$. If we substitute this in the relations for the decomposition numbers found by {\sl Mathematica}, then we obtain $N_{D_6}(D_6)=1$, $N_{D_6}(D_5, A_1) = 5$, $N_{D_6}(A_5, A_1) = 10$, $N_{D_6}(A_1*D_4, A_1) = 5$, $N_{D_6}(A_2*A_3, A_1) = 5$, $N_{D_6}(A_1^2*A_3, A_1) = 5$, $N_{D_6}(D_4, A_2) = 5$, $N_{D_6}(A_4, A_2) = 10$, $N_{D_6}(A_1*A_3, A_2) = 30$, $N_{D_6}(A_2^2, A_2) = 10$, $N_{D_6}(A_1^2*A_2, A_2) = 10$, $N_{D_6}(A_1^4, A_2) = 5$, $N_{D_6}(D_4, A_1^2) = 5$, $N_{D_6}(A_4, A_1^2) = 35$, $N_{D_6}(A_1*A_3, A_1^2) = 30$, $N_{D_6}(A_2^2, A_1^2) = 10$, $N_{D_6}(A_1^2*A_2, A_1^2) = 10$, $N_{D_6}(A_1^4, A_1^2) = 5$, $N_{D_6}(A_3, A_3) = 20$, $N_{D_6}(A_3,\mathbreak A_1*A_2) = 45$, $N_{D_6}(A_3, A_1^3) = 25$, $N_{D_6}(A_1*A_2, A_1*A_2) = 70$, $N_{D_6}(A_1*A_2, A_1^3) = 25$, $N_{D_6}(D_4, A_1, A_1) = 25$, $N_{D_6}(A_4, A_1, A_1) = 100$, $N_{D_6}(A_1*A_3, A_1, A_1) = 150$, $N_{D_6}(A_2^2, A_1,\mathbreak A_1) = 50$, $N_{D_6}(A_1^2*A_2, A_1, A_1) = 50$, $N_{D_6}(A_1^4, A_1, A_1) = 25$, $N_{D_6}(A_3, A_2, A_1) = 125$, $N_{D_6}(A_3, A_1^2, A_1) = 250$, $N_{D_6}(A_1*A_2, A_2, A_1) = 250$, $N_{D_6}(A_1*A_2, A_1^2, A_1) = 375$, $N_{D_6}(A_1^3,\mathbreak A_2, A_1) = 125$, $N_{D_6}(A_1^3, A_1^2, A_1) = 125$, $N_{D_6}(A_3, A_1, A_1, A_1) = 875$, $N_{D_6}(A_1*A_2, A_1, A_1,\mathbreak A_1) = 1500$, $N_{D_6}(A_1^3, A_1, A_1, A_1) = 625$, $N_{D_6}(A_2, A_2, A_2) = 100$, $N_{D_6}(A_2, A_2, A_1^2) = 225$, $N_{D_6}(A_2, A_1^2, A_1^2) = 350$, $N_{D_6}(A_1^2, A_1^2, A_1^2) = 475$, $N_{D_6}(A_2, A_2, A_1, A_1) = 750$, $N_{D_6}(A_2, A_1^2,\mathbreak A_1, A_1) = 1375$, $N_{D_6}(A_1^2, A_1^2, A_1, A_1) = 2000$, $N_{D_6}(A_2, A_1, A_1, A_1, A_1) = 5000$, $N_{D_6}(A_1^2,\mathbreak A_1, A_1, A_1, A_1) = 8125$, $N_{D_6}(A_1, A_1, A_1, A_1, A_1, A_1) = 31250$. \subsubhead The decomposition numbers for $D_7$\endsubsubhead In addition to the described equations and assignments, we use also the last assertion in Proposition~\PB\ for the type $A_1^5$, that is, $N_{D_7}(A_1^5,\mathbreak A_2)=N_{D_7}(A_1^5,A_1^2)=0$. Unfortunately, the solution space of the system of linear equations does not consist of a unique solution but is two-dimensional. {\sl Mathematica~5.2} expresses all the decomposition numbers in terms of $X=N_{D_7}(A_1^4,A_1^3)$ and $Y=N_{D_7}(A_1^2*A_2,A_1^3)$. In the sequel we make use of the two relations $$\align N_{D_7}(A_1^4, A_1*A_2) &= \hphantom{5}\frac65(36 - X),\tag\AT \\ N_{D_7}(A_1^2*A_2, A_3) &= \frac3{10}(Y+162),\tag\AU\\ N_{D_7}(A_1*A_3, A_1^3) &= 84 - \frac {1} {3}Y, \tag\AV\\ N_{D_7}(A_1*A_2^2, A_2) &= \frac{14}{15}(36 - 3X - Y),\tag\AW\\ N_{D_7}(A_1*A_2^2, A_1^2) &= \hphantom{7}\frac75(3X + Y-36).\tag\AX \endalign$$ Relation (\AT) implies $X\equiv1$ (mod $5$), while Relations~(\AU) and (\AV) imply $Y\equiv18$ (mod~$30$). Since decomposition numbers must be non-negative, Relations~(\AW) and (\AX) force $3X+Y$ to be equal to $36$. Hence, we must have $X=N_{D_7}(A_1^4,A_1^3)=6$ and $Y=N_{D_7}(A_1^2*A_2,A_1^3)=18$. If we substitute this in the relations for the decomposition numbers found by {\sl Mathematica}, then we obtain $N_{D_7}(D_7)=1$, $N_{D_7}(D_6, A_1) = 6$, $N_{D_7}(A_6, A_1) = 12$, $N_{D_7}(A_1*D_5, A_1) = 6$, $N_{D_7}(A_2*D_4, A_1) = 6$, $N_{D_7}(A_1^2*A_4, A_1) = 6$, $N_{D_7}(A_3^2, A_1) = 6$, $N_{D_7}(D_5, A_2) = 6$, $N_{D_7}(A_5, A_2) = 12$, $N_{D_7}(A_1*D_4, A_2) = 12$, $N_{D_7}(A_1*A_4, A_2) = 24$, $N_{D_7}(A_2*A_3, A_2) = 36$, $N_{D_7}(A_1^2*A_3, A_2) = 18$, $N_{D_7}(A_1^3*A_2, A_2) = 12$, $N_{D_7}(D_5, A_1^2) = 9$, $N_{D_7}(A_5, A_1^2) = 54$, $N_{D_7}(A_1*D_4, A_1^2) = 18$, $N_{D_7}(A_1*A_4, A_1^2) = 36$, $N_{D_7}(A_2*A_3, A_1^2) = 54$, $N_{D_7}(A_1^2*A_3, A_1^2) = 27$, $N_{D_7}(A_1^3*A_2, A_1^2) = 18$, $N_{D_7}(D_5, A_1, A_1) = 36$, $N_{D_7}(A_5, A_1, A_1) = 144$, $N_{D_7}(A_1*D_4, A_1, A_1) = 72$, $N_{D_7}(A_1*A_4, A_1, A_1) = 144$, $N_{D_7}(A_2*A_3, A_1, A_1) = 216$, $N_{D_7}(A_1^2*A_3, A_1, A_1) = 108$, $N_{D_7}(A_1^3*A_2, A_1, A_1) = 72$, $N_{D_7}(D_4, A_3) = 6$, $N_{D_7}(A_4, A_3) = 18$, $N_{D_7}(A_1*A_3, A_3) = 72$, $N_{D_7}(A_2^2, A_3) = 27$, $N_{D_7}(A_1^2*A_2, A_3) = 54$, $N_{D_7}(A_1^4, A_3) = 18$, $N_{D_7}(D_4, A_1*A_2) = 12$, $N_{D_7}(A_4, A_1*A_2) = 72$, $N_{D_7}(A_1*A_3, A_1*A_2) = 180$, $N_{D_7}(A_2^2, A_1*A_2) = 72$, $N_{D_7}(A_1^2*A_2, A_1*A_2) = 108$, $N_{D_7}(A_1^4, A_1*A_2) = 36$, $N_{D_7}(D_4, A_1^3) = 2$, $N_{D_7}(A_4, A_1^3) = 60$, $N_{D_7}(A_1*A_3, A_1^3) = 78$, $N_{D_7}(A_2^2, A_1^3) = 36$, $N_{D_7}(A_1^2*A_2, A_1^3) = 18$, $N_{D_7}(A_1^4, A_1^3) = 6$, $N_{D_7}(D_4, A_2, A_1) = 36$, $N_{D_7}(D_4, A_1^2, A_1) = 54$, $N_{D_7}(A_4, A_2, A_1) = 144$, $N_{D_7}(A_4, A_1^2, A_1) = 432$, $N_{D_7}(A_1*A_3, A_2, A_1) = 468$, $N_{D_7}(A_1*A_3, A_1^2, A_1) = 918$, $N_{D_7}(A_2^2, A_2, A_1) = 180$, $N_{D_7}(A_2^2, A_1^2, A_1) = 378$, $N_{D_7}(A_1^2*A_2, A_2, A_1) = 324$, $N_{D_7}(A_1^2*A_2, A_1^2, A_1) = 486$, $N_{D_7}(A_1^4, A_2, A_1) = 108$, $N_{D_7}(A_1^4, A_1^2, A_1) = 162$, $N_{D_7}(D_4, A_1, A_1, A_1) = 216$, $N_{D_7}(A_4, A_1, A_1, A_1) = 1296$, $N_{D_7}(A_1*A_3, A_1, A_1, A_1) = 3240$, $N_{D_7}(A_2^2, A_1, A_1, A_1) = 1296$, $N_{D_7}(A_1^2*A_2, A_1, A_1, A_1) = 1944$, $N_{D_7}(A_1^4, A_1, A_1, A_1) = 648$, $N_{D_7}(A_3, A_3, A_1) = 216$, $N_{D_7}(A_3, A_1*A_2, A_1) = 648$, $N_{D_7}(A_3, A_1^3, A_1) = 396$, $N_{D_7}(A_1*A_2, A_1*A_2, A_1) = 1728$, $N_{D_7}(A_1*A_2, A_1^3, A_1) = 864$, $N_{D_7}(A_1^3, A_1^3, A_1) = 240$, $N_{D_7}(A_3, A_2, A_2) = 180$, $N_{D_7}(A_1*A_2, A_2, A_2) = 576$, $N_{D_7}(A_1^3, A_2, A_2) = 384$, $N_{D_7}(A_3, A_2, A_1^2) = 486$, $N_{D_7}(A_1*A_2, A_2, A_1^2) = 1296$, $N_{D_7}(A_1^3, A_2, A_1^2) = 648$, $N_{D_7}(A_3, A_1^2, A_1^2) = 1053$, $N_{D_7}(A_1*A_2, A_1^2,\mathbreak A_1^2) = 2592$, $N_{D_7}(A_1^3, A_1^2, A_1^2) = 1080$, $N_{D_7}(A_3, A_2, A_1, A_1) = 1512$, $N_{D_7}(A_3, A_1^2, A_1, A_1) = 3564$, $N_{D_7}(A_1*A_2, A_2, A_1, A_1) = 4320$, $N_{D_7}(A_1*A_2, A_1^2, A_1, A_1) = 9072$, $N_{D_7}(A_1^3, A_2, A_1,\mathbreak A_1) = 2448$, $N_{D_7}(A_1^3, A_1^2, A_1, A_1) = 4104$, $N_{D_7}(A_3, A_1, A_1, A_1, A_1) = 11664$, $N_{D_7}(A_1*A_2, A_1, A_1, A_1, A_1) = 31104$, $N_{D_7}(A_1^3, A_1, A_1, A_1, A_1) = 15552$, $N_{D_7}(A_2, A_2, A_2, A_1) = 1296$, $N_{D_7}(A_2, A_2, A_1^2, A_1) = 3240$, $N_{D_7}(A_2, A_1^2, A_1^2, A_1) = 6804$, $N_{D_7}(A_1^2, A_1^2, A_1^2, A_1) = 13122$, $N_{D_7}(A_2, A_2, A_1, A_1, A_1) = 10368$, $N_{D_7}(A_2, A_1^2, A_1, A_1, A_1) = 23328$, $N_{D_7}(A_1^2, A_1^2,\mathbreak A_1, A_1, A_1) = 46656$, $N_{D_7}(A_2, A_1, A_1, A_1, A_1, A_1) = 77760$, $N_{D_7}(A_1^2, A_1, A_1, A_1, A_1, A_1) = 163296$, $N_{D_7}(A_1, A_1, A_1, A_1, A_1, A_1, A_1) = 559872$. \subsubhead The decomposition numbers for $E_6$\endsubsubhead These numbers have already been computed in \cite{\KratCB, Sec.~17} using Stembridge's {\tt coxeter} package \cite{\StemAZ}. An alternative, independent way to do this is by following the method described in Section~4. Here, in addition to the equations and assignments described at the beginning of the Appendix, we use also the last assertion in Proposition~\PB\ for the type $A_1^4$, that is, $N_{E_6}(A_1^4,A_2)=N_{E_6}(A_1^4,A_1^2)=0$. Unfortunately, the solution space of the system of linear equations does not consist of a unique solution but is one-dimensional. {\sl Mathematica~5.2} expresses all the decomposition numbers in terms of $X=N_{E_6}(A_1^3,A_1^3)$. In the sequel we make use of the two relations $$\align N_{E_6}(A_3,A_3)&=\frac {1} {100}(2592+9X),\tag\AP\\ N_{E_6}(A_1^3,A_1^3)&=\hphantom{11}\frac {1} {5}(192-6X).\tag\AS \endalign$$ From (\AP) we infer that $X\equiv12\ (\text {mod 100})$, while (\AS) implies that $X\le 32$. Hence, we have $X=N_{E_6}(A_1^3,A_1^3)=12$. If we substitute this in the relations for the decomposition numbers found by {\sl Mathematica}, then we obtain $N_{E_6}(E_6)=1$, $N_{E_6}(D_5, A_1) = 12$, $N_{E_6}(A_5, A_1) = 6$, $N_{E_6}(A_1*A_4, A_1) = 12$, $N_{E_6}(A_1*A_2^2, A_1) = 6$, $N_{E_6}(A_1^2*A_2, A_2) = 36$, $N_{E_6}(D_4, A_2) = 4$, $N_{E_6}(A_4, A_2) = 24$, $N_{E_6}(A_1*A_3, A_2) = 24$, $N_{E_6}(A_2^2, A_2) = 8$, $N_{E_6}(D_4, A_1^2) = 18$, $N_{E_6}(A_4, A_1^2) = 36$, $N_{E_6}(A_1*A_3, A_1^2) = 36$, $N_{E_6}(A_1^2*A_2, A_1^2) = 18$, $N_{E_6}(A_3, A_3) = 27$, $N_{E_6}(A_3, A_1*A_2) = 72$, $N_{E_6}(A_3, A_1^3) = 36$, $N_{E_6}(A_1*A_2, A_1*A_2) = 48$, $N_{E_6}(A_1*A_2, A_1^3) = 24$, $N_{E_6}(A_1^3, A_1^3) = 12$, $N_{E_6}(D_4, A_1, A_1) = 48$, $N_{E_6}(A_4, A_1, A_1) = 144$, $N_{E_6}(A_1*A_3, A_1, A_1) = 144$, $N_{E_6}(A_2^2, A_1, A_1) = 24$, $N_{E_6}(A_1^2*A_2, A_1, A_1) = 144$, $N_{E_6}(A_3, A_2, A_1) = 180$, $N_{E_6}(A_3, A_1^2, A_1) = 378$, $N_{E_6}(A_1*A_2, A_2, A_1) = 336$, $N_{E_6}(A_1*A_2, A_1^2, A_1) = 360$, $N_{E_6}(A_1^3, A_2, A_1) = 168$, $N_{E_6}(A_1^3, A_1^2, A_1) = 180$, $N_{E_6}(A_2, A_2, A_2) = 160$, $N_{E_6}(A_2, A_2, A_1^2) = 288$, $N_{E_6}(A_2, A_1^2, A_1^2) = 504$, $N_{E_6}(A_1^2, A_1^2, A_1^2) = 432$, $N_{E_6}(A_2, A_2,\mathbreak A_1, A_1) = 1056$, $N_{E_6}(A_2, A_1^2, A_1, A_1) = 1872$, $N_{E_6}(A_1^2, A_1^2, A_1, A_1) = 2376$, $N_{E_6}(A_3, A_1,\mathbreak A_1, A_1) = 1296$, $N_{E_6}(A_1*A_2, A_1, A_1, A_1) = 1728$, $N_{E_6}(A_1^3, A_1, A_1, A_1) = 864$, $N_{E_6}(A_2, A_1,\mathbreak A_1, A_1, A_1) = 6912$, $N_{E_6}(A_1^2, A_1, A_1, A_1, A_1) = 10368$, $N_{E_6}(A_1, A_1, A_1, A_1, A_1, A_1) = \mathbreak 41472$. \bigskip \remark{Acknowledgment} I am indebted to Sergey Fomin for suggesting to me to embark on this project and, thus, to the Institut Mittag--Leffler, Anders Bj\"orner and Richard Stanley for hosting both of us at the same time at the Institut during the ``Algebraic Combinatorics" programme in Spring 2005. Moreover, I wish to thank Christos Athanasiadis and Vic Reiner for some helpful exchanges that I had with him on the subject matter. Last, but not least, I thank the referee for an extremely careful reading of the manuscript, and for pointing out that Lemma~1.3.4 of \cite{\BesDAA} simplifies the proof of Proposition~\PC. \endremark { \eightpoint \subhead Notes\endsubhead After the first version of this paper was distributed, Eleni Tzanaki found a uniform proof of Armstrong's $F=M$ Conjecture (presented here in Conjecture~FM) in ``Faces of generalized cluster complexes and noncrossing partitions," {\tt ar$\chi$iv:math.CO/0605785}. Thus, our Theorem~\TA\ becomes an unconditional theorem, that is to say, the reciprocity relation (\AO) holds for any finite root system $\Phi$. Furthermore, the explicit form of the $M$-triangle in type $D_n$ is given by \cite{\KratCB, Prop.~D} up to a simple substitution of variables. The problem of computing the decomposition numbers for the type $B_n$ is solved implicitly in ``Enumeration of $m$-ary cacti" (Adv. Appl. Math. {\bf 24} (2000), 22--56) by Mikl\'os B\'ona, Michel Bousquet, Gilbert Labelle and Pierre Leroux. This is explained in detail in the article ``Decomposition numbers for finite Coxeter groups and generalised non-crossing partitions" by Thomas M\"uller and the author, which also solves the problem for the remaining type $D_n$. In particular, the results in that paper, together with the results in the present paper and in \cite{\KratCB}, constitute an independent --- case-by-case --- proof of the $F=M$ Conjecture. } \Refs \ref\no \AnBBAA\by N. Anzalone, J. Baldwin, I. Bronshtein and T. Kyle Petersen \yr 2003 \paper A reciprocity theorem for monomer-dimer coverings\inbook Discrete models for complex systems, DMCS '03 (Lyon)\publ Discrete Math. Theor. Comput. Sci. Proc., AB \pages 179--193\endref \ref\no \ArmDAA\by D. D. 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