%This is a plain TeX file that produces 15 output pages. {\catcode`\^^M=\active % \gdef\adrbox{\catcode`\^^M=\active % \def^^M{\egroup\hbox\bgroup}}} \def\adresse{\par\nobreak% \vskip 24pt plus 6pt minus 3pt% \hbox to \hsize\bgroup\hfill% \def\fin{\egroup\egroup\hskip2em\egroup\vfill\eject}% \adrbox\vbox\bgroup\hbox\bgroup} \def\fin{\vfill\eject} \catcode'32=9 \magnification=1200 \voffset=1cm \hoffset=0cm %\hoffset=1cm \font\tenpc=cmcsc10 %\font\eightpc=cmcsc8 % Charge des fontes de 8 et 6 points : \font\eightrm=cmr8 \font\eighti=cmmi8 \font\eightsy=cmsy8 \font\eightbf=cmbx8 \font\eighttt=cmtt8 \font\eightit=cmti8 \font\eightsl=cmsl8 \font\sixrm=cmr6 \font\sixi=cmmi6 \font\sixsy=cmsy6 \font\sixbf=cmbx6 \skewchar\eighti='177 \skewchar\sixi='177 \skewchar\eightsy='60 \skewchar\sixsy='60 % Chargement des fontes AMS \font\tengoth=eufm10 \font\tenbboard=msbm10 \font\eightgoth=eufm8 \font\eightbboard=msbm8 \font\sevengoth=eufm7 \font\sevenbboard=msbm7 \font\sixgoth=eufm6 \font\fivegoth=eufm5 \newfam\gothfam \newfam\bboardfam \catcode`\@=11 \def\raggedbottom{\topskip 10pt plus 36pt \r@ggedbottomtrue} \def\pc#1#2|{{\bigf@ntpc #1\penalty \@MM\hskip\z@skip\smallf@ntpc #2}} \def\tenpoint{% \textfont0=\tenrm \scriptfont0=\sevenrm \scriptscriptfont0=\fiverm \def\rm{\fam\z@\tenrm}% \textfont1=\teni \scriptfont1=\seveni \scriptscriptfont1=\fivei \def\oldstyle{\fam\@ne\teni}% \textfont2=\tensy \scriptfont2=\sevensy \scriptscriptfont2=\fivesy \textfont\gothfam=\tengoth \scriptfont\gothfam=\sevengoth \scriptscriptfont\gothfam=\fivegoth \def\goth{\fam\gothfam\tengoth}% \textfont\bboardfam=\tenbboard \scriptfont\bboardfam=\sevenbboard \scriptscriptfont\bboardfam=\sevenbboard \def\bboard{\fam\bboardfam}% \textfont\itfam=\tenit \def\it{\fam\itfam\tenit}% \textfont\slfam=\tensl \def\sl{\fam\slfam\tensl}% \textfont\bffam=\tenbf \scriptfont\bffam=\sevenbf \scriptscriptfont\bffam=\fivebf \def\bf{\fam\bffam\tenbf}% \textfont\ttfam=\tentt \def\tt{\fam\ttfam\tentt}% \abovedisplayskip=12pt plus 3pt minus 9pt \abovedisplayshortskip=0pt plus 3pt \belowdisplayskip=12pt plus 3pt minus 9pt \belowdisplayshortskip=7pt plus 3pt minus 4pt \smallskipamount=3pt plus 1pt minus 1pt \medskipamount=6pt plus 2pt minus 2pt \bigskipamount=12pt plus 4pt minus 4pt \normalbaselineskip=12pt \setbox\strutbox=\hbox{\vrule height8.5pt depth3.5pt width0pt}% \let\bigf@ntpc=\tenrm \let\smallf@ntpc=\sevenrm \let\petcap=\tenpc \normalbaselines\rm} \def\eightpoint{% \textfont0=\eightrm \scriptfont0=\sixrm \scriptscriptfont0=\fiverm \def\rm{\fam\z@\eightrm}% \textfont1=\eighti \scriptfont1=\sixi \scriptscriptfont1=\fivei \def\oldstyle{\fam\@ne\eighti}% \textfont2=\eightsy \scriptfont2=\sixsy \scriptscriptfont2=\fivesy \textfont\gothfam=\eightgoth \scriptfont\gothfam=\sixgoth \scriptscriptfont\gothfam=\fivegoth \def\goth{\fam\gothfam\eightgoth}% \textfont\bboardfam=\eightbboard \scriptfont\bboardfam=\sevenbboard \scriptscriptfont\bboardfam=\sevenbboard \def\bboard{\fam\bboardfam}% \textfont\itfam=\eightit \def\it{\fam\itfam\eightit}% \textfont\slfam=\eightsl \def\sl{\fam\slfam\eightsl}% \textfont\bffam=\eightbf \scriptfont\bffam=\sixbf \scriptscriptfont\bffam=\fivebf \def\bf{\fam\bffam\eightbf}% \textfont\ttfam=\eighttt \def\tt{\fam\ttfam\eighttt}% \abovedisplayskip=9pt plus 2pt minus 6pt \abovedisplayshortskip=0pt plus 2pt \belowdisplayskip=9pt plus 2pt minus 6pt \belowdisplayshortskip=5pt plus 2pt minus 3pt \smallskipamount=2pt plus 1pt minus 1pt \medskipamount=4pt plus 2pt minus 1pt \bigskipamount=9pt plus 3pt minus 3pt \normalbaselineskip=9pt \setbox\strutbox=\hbox{\vrule height7pt depth2pt width0pt}% \let\bigf@ntpc=\eightrm \let\smallf@ntpc=\sixrm \normalbaselines\rm} \let\bb=\bboard \tenpoint %\to dactylographie fran\c caise \to\to\to\to\to\to\to\to\to\to\to \catcode`\;=\active \def;{\relax\ifhmode\ifdim\lastskip>\z@ \unskip\fi\kern\fontdimen2 \font\kern -1.2 \fontdimen3 \font\fi\string;} \catcode`\:=\active \def:{\relax\ifhmode\ifdim\lastskip>\z@\unskip\fi\penalty\@M\ \fi\string:} \catcode`\!=\active \def!{\relax\ifhmode\ifdim\lastskip>\z@ \unskip\fi\kern\fontdimen2 \font \kern -1.1 \fontdimen3 \font\fi\string!} \catcode`\?=\active \def?{\relax\ifhmode\ifdim\lastskip>\z@ \unskip\fi\kern\fontdimen2 \font \kern -1.1 \fontdimen3 \font\fi\string?} \def\^#1{\if#1i{\accent"5E\i}\else{\accent"5E #1}\fi} \def\"#1{\if#1i{\accent"7F\i}\else{\accent"7F #1}\fi} \frenchspacing %\to\to\to\to\to\to format de sortie \to\to\to\to\to\to\to\to\to\to \newif\ifpagetitre \newtoks\auteurcourant \auteurcourant={\hfil} \newtoks\titrecourant \titrecourant={\hfil} \def\appeln@te{} \def\vfootnote#1{\def\@parameter{#1}\insert\footins\bgroup\eightpoint \interlinepenalty\interfootnotelinepenalty \splittopskip\ht\strutbox % top baseline for broken footnotes \splitmaxdepth\dp\strutbox \floatingpenalty\@MM \leftskip\z@skip \rightskip\z@skip \ifx\appeln@te\@parameter\indent \else{\noindent #1\ }\fi \footstrut\futurelet\next\fo@t} \pretolerance=500 \tolerance=1000 \brokenpenalty=5000 \newdimen\hmargehaute \hmargehaute=0cm \newdimen\lpage \lpage=13.3cm \newdimen\hpage \hpage=20cm \newdimen\lmargeext \lmargeext=1cm \hsize=11.25cm \vsize=18cm \parskip 0pt \parindent=12pt \def\margehaute{\vbox to \hmargehaute{\vss}}% \def\margebasse{\vss} \output{\shipout\vbox to \hpage{\margehaute\nointerlineskip \corpsdepage\margebasse} \advancepageno \global\pagetitrefalse \ifnum\outputpenalty>-20000 \else\dosupereject\fi} \def\corpsdepage{\hbox to \lpage{\hss\pagetexte\hskip\lmargeext}} \def\pagetexte{\vbox{\makeheadline\pagebody\makefootline}} \headline={\ifpagetitre\titleheadline \else \ifodd\pageno\rightheadline \else\leftheadline\fi\fi} \def\leftheadline{\eightpoint\hfil\the\auteurcourant\hfil} \def\rightheadline{\eightpoint\hfil\the\titrecourant\hfil} \def\titleheadline{\hfill} \pagetitretrue \def\footnoterule{\kern-6\p@ \hrule width 2truein \kern 5.6\p@} % the \hrule is .4pt high \let\rmpc=\sevenrm \def\pd#1#2 {\pc#1#2| } \def\pointir{\discretionary{.}{}{.\kern.35em---\kern.7em}\nobreak \hskip 0em plus .3em minus .4em } \def\resume#1{\vbox{\eightpoint \pc R\'ESUM\'E|\pointir #1}} \def\abstract#1{\vbox{\eightpoint \pc ABSTRACT|\pointir #1}} \def\titre#1|{\message{#1} \par\vskip 30pt plus 24pt minus 3pt\penalty -1000 \vskip 0pt plus -24pt minus 3pt\penalty -1000 \centerline{\bf #1} \vskip 5pt \penalty 10000 } \def\section#1|{\par\vskip .3cm {\bf #1}\pointir} \def\ssection#1|{\par\vskip .2cm {\it #1}\pointir} \long\def\th#1|#2\finth{\par\medskip {\petcap #1\pointir}{\it #2}\par\smallskip} \long\def\tha#1|#2\fintha{\par\medskip {\petcap #1.}\par\nobreak{\it #2}\par\smallskip} \def\cf{{\it cf}} \def\rem#1|{\par\medskip {{\it #1}\pointir}} \def\rema#1|{\par\medskip {{\it #1.}\par\nobreak }} % \def\ieme{\raise 1ex\hbox{\pc{}i\`eme|}} \def\omini{\raise 1ex\hbox{\pc{}o|}} \def\emini{\raise 1ex\hbox{\pc{}e|}} \def\ermini{\raise 1ex\hbox{\pc{}er|}} \def\remini{\raise 1ex\hbox{\pc{}re|}} %reference pour un article : \def\article#1|#2|#3|#4|#5|#6|#7| {{\leftskip=7mm\noindent \hangindent=2mm\hangafter=1 \llap{[#1]\hskip.35em}{#2}\pointir #3, {\sl #4}, vol.\nobreak\ {\bf #5}, {\oldstyle #6}, p.\nobreak\ #7.\par}} %reference pour un livre : \def\livre#1|#2|#3|#4| {{\leftskip=7mm\noindent \hangindent=2mm\hangafter=1 \llap{[#1]\hskip.35em}{#2}\pointir {\sl #3}\pointir #4.\par}} %reference complementaire : \def\divers#1|#2|#3| {{\leftskip=7mm\noindent \hangindent=2mm\hangafter=1 \llap{[#1]\hskip.35em}{#2}\pointir #3.\par}} % \mathchardef\conj="0365 \def\dem{\par{\it D\'emonstration}\pointir} \def\qed{\quad\raise -2pt\hbox{\vrule\vbox to 10pt{\hrule width 4pt \vfill\hrule}\vrule}} \def\virg{\raise 2pt\hbox{,}} % virgule apr\`es une fraction \def\cqfd{\penalty 500 \hbox{\qed}\par\smallskip} \def\decale#1|{\par\noindent\hskip 28pt\llap{#1}\kern 5pt} \def\decaledecale#1|{\par\noindent\hskip 34pt\llap{#1}\kern 5pt} % pour les titres en deux lignes et les sections sans point-tiret : \def\titrea#1|#2|{\message{#1 #2} \par\vskip.5cm plus .1cm minus .1cm\penalty -1000 \centerline{\bf #1} \centerline{\bf #2} \vskip 5pt \penalty 10000 } \def\sectiona#1|{\par\vskip .3cm {\bf #1.} \par\nobreak\vskip 3pt } \def\ssectiona#1|{\par\vskip .2cm {\it #1.} \par\nobreak\vskip 2pt } \catcode`\@=12 %% fin du format \pageno=31 \titrecourant={FOULKES AND LEFSCHETZ CHARACTERS} \auteurcourant={A. KERBER AND K.-J. TH\"URLINGS} \eightpoint \leftline{Publ. I.R.M.A. Strasbourg, $\oldstyle 1984$, 229/S--08} \leftline{Actes $8^e$ S\'eminaire Lotharingien, p. 31--36} \tenpoint \let\it=\sl \vskip 1.5cm \centerline{{\bf EULERIAN NUMBERS, FOULKES CHARACTERS}} \vskip 5pt \centerline{{\bf AND LEFSCHETZ CHARACTERS OF $S_n$}} \vskip 2.8mm \centerline{\sevenrm BY} \vskip 2.8mm \centerline{{\petcap Adalbert {KERBER} {\sevenrm AND} Karl-Joseph {TH\"URLINGS}}} \vskip 24pt \abstract{The aim of this talk is to point out a connection between the characters which \pd FOULKES introduced in order to give a representation theoretical generalization of Eulerian numbers and certain Lefschetz characters of $S_n$ which A.~\pd BJ\"ORNER mentioned at his talk in Feuerstein and which were described in detail by R. \pd STANLEY in [1]. The missing link is a theorem on the irreducible constituents of Foulkes' characters.} \section 1. Eulerian numbers|Let $\pi =\bigl(\pi (1)\ldots \pi (n)\bigr)$ be an element of the symmetric group $S_n$, e.g. $(13248765)\in S_8$ (list notation!) with the {\it up-and-down-sequence} $A(\pi)$ (rises indicated by $+,$ falls denoted by $-$), for example $$A((13248765))=\ +\,-\,+\,+\,-\,-\,-\ .$$ The number of permutations with a given number of rises, i.e. of entries $+$ in its up-down sequence, defines an {\it Eulerian number:} $$A(n,k):=\left|\{\pi \in S_n\mid A(\pi )\ {\rm has}\ k\ {\rm rises} \}\right|.$$ According to \pd FOULKES, $A(\pi )$ yields a skew diagram via the rule $$\matrix{ +\leftarrow&\times\cr &\downarrow-\cr} $$ This means, that to an entry $+$ of $A(\pi)$ there corresponds a node $\times$ that has to be added at the left of the last node added, and in the same row. Correspondingly to an entry $-$ there corresponds a node that has to be added just below the last node. For example the sequence $(+-++---)$ mentioned above gives $$\matrix{ &&\times&\times\cr \times&\times&\times\cr \times\cr \times\cr \times\cr} \ \ {according\ to}\ \ \ \matrix{ &&+&\times\cr +&+&-\cr -\cr -\cr -\cr} .$$ This resulting skew diagram is the rim part of $\lambda (A)$ : $R^{\lambda (A)}_{11}=(431^3)/(2)$. Recall the definition of {\it skew representation} by the Littlewood-Richardson rule: $$\displaystyle [\lambda /\mu ] :=\sum _\nu ([\mu ][\nu ],[\lambda ])[\nu ].$$ \th Theorem {(Foulkes)}|The number of permutations with given up-down sequence $A$ can be identified with a dimension of a skew representation: $$\left|\{\pi \mid A(\pi )=A\}\right| =dim[R^{\lambda (A)}_{11}].$$ \finth For example, if we put $A:={+\,-\,+\,+\,-\,-\,-}$ then the number of permutations with this sequence isthe dimension of $[(431^3)/(2)]$ which has the decomposition $[421^2]+[41^4]+[3^21^2]+[321^3]$ and therefore the dimension $245.$ More generally we have the following generalization of Foulkes' result: \th Theorem {(K/Th)}| The decomposition of $[R^{\lambda (A)}_{11}]$ is obtained from $A$ by successive applications of the rule $$\matrix{ \times&\nearrow\!+\cr \swarrow\!-\cr} $$ \finth By this pictorial description we mean that to an entry $+$ of $A$ there corresponds a node $\times$ which has to be added to the right of the last node, maybe in a higher row, while to an entry $-$ there corresponds a node added to the left of the last node or in a lower row. Consider once more the example $A=(+-++---)$. We start with a node $\times$, and the first entry of $A$ is a $+$, so the corresponding node has to be added, according to the rule, to the right of the starting node, i.e. we obtain the diagram $\times\otimes$, where the last node added is encircled. Now the second entry of $A$ is a minus sign, hence the corresponding addition of a node is again uniquely determined, and we get the diagram $$\matrix{ \times&\times\cr \otimes\cr} .$$ The next entry of $A$ is a plus sign, so that there are two places open for an additional node which are to the right of the node which was added last time: $$\matrix{ \times&\times&\otimes\cr \times\cr} \ \ {and}\ \ \matrix{ \times&\times\cr \times&\otimes\cr} .$$ The next steps yield the following cascade of diagrams: $$\matrix{ \times&\times&\times&\otimes\cr \times\cr &\swarrow&\searrow\cr} \ \ \ \ \ \ \ \ \ \ \matrix{ \times&\times&\otimes\cr \times&\times\cr &\swarrow&\searrow\cr} $$ $$\matrix{ \times&\times&\times&\times\cr \times\cr \otimes\cr &&\downarrow\cr} \ \ \matrix{ \times&\times&\times&\times\cr \times&\otimes\cr \cr &\downarrow\cr} \ \ \matrix{ \times&\times&\times\cr \times&\times\cr \otimes\cr &\downarrow\cr} \ \ \matrix{ \times&\times&\times\cr \times&\times&\otimes\cr \cr &\downarrow\cr} $$ $$\matrix{ \times&\times&\times&\times\cr \times\cr \times\cr \times\cr \times\cr} \ \ \matrix{ \times&\times&\times&\times\cr \times&\times\cr \times\cr \times\cr} \ \ \matrix{ \times&\times&\times\cr \times&\times\cr \times\cr \times\cr \times\cr} \ \ \matrix{ \times&\times&\times\cr \times&\times&\times\cr \times\cr \times\cr} $$ Hence from $A=(+-++---)$ we obtain the diagrams $$[4,1^4],[4,2,1^2],[3,2,1^3],[3^2,1^2],$$ and each one of them exactly once. {\it Proof}\pointir ``By example'' $$\displaylines{\quad R^{\lambda (({+}{-}{-}{+}{+}{-}{-}{+}{+}{+}))}_{11} =\det\pmatrix{[2]&[3]&[6]&[7]&[11]\cr 1&[1]&[4]&[5]&[9]\cr 0&1&[3]&[4]&[8]\cr 0&0&1&[1]&[5]\cr 0&0&0&1&[4]\cr}\hfill\cr \hfill\eqalign{&=[4]\det\pmatrix{[2]&[3]&[6]&[7]\cr 1&[1]&[4]&[5]\cr 0&1&[3]&[4]\cr 0&0&1&[1]\cr} -\det\pmatrix{[2]&[3]&[6]&[11]\cr 1&[1]&[4]&[9]\cr 0&1&[3]&[8]\cr 0&0&1&[5]\cr}\cr &=[4][(7,6,6,4)/(5,5,3,3)]-[(8,7,7,5)/(6,6,4)]\cr &=[4][R^{\lambda (({+}{-}{-}{+}{+}{-}))}_{11}] -[R^{\lambda (({+}{-}{-}{+}{+}{-}{+}{+}{+}{+}))}_{11}].\cr} \quad\cr} $$ Hence, the following lemma completes the proof. \th Lemma|Let $A$ denote an up-and-down sequence. Then, for each $k\in {\bf N}$ we have $$[k+1][R^{\lambda (A)}_{11}]= [R^{\lambda ((A{+}{+}\cdots{+}))}_{11}] +[R^{\lambda ((A{-}{+}\cdots{+}))}_{11}].$$ \finth \section 2. Foulkes characters| Foulkes' result gives the following interpretation of Eulerian numbers as sums of dimensions of skew representations: $$A(n,k)=\sum _{A,k\ ups} {\rm dim}[R^{\lambda (A)}_{11}],$$ while the above generalization gives a more general result in terms of characters: $$\chi ^{n,k}:=\sum _{A,k\ ups} \chi ^{R^{\lambda (A)}_{11}}.$$ We suggest to call these characters {\it Foulkes\ characters.} They have the following remarkable properties: \th Theorem {(Foulkes)}| \vskip 3pt \indent\indent (i) {\it no. of cycles of $\pi ={}$no. of cycles of $\rho \Rightarrow \chi ^{n,k}(\pi )=\chi ^{n,k}(\rho )$} ; \vskip 3pt \indent\indent (ii) $\chi ^{n,0}=\zeta ^{(1^n)}$,\quad $\chi ^{n,n-1}=\zeta ^{(n)}$,\quad $\chi ^{n,k}=\zeta ^{(1^n)}\otimes \chi ^{n,n-1-k}$ ; \vskip 3pt \indent\indent (iii) {\it The Foulkes characters satisfy the following recursion:} $$\chi^{n,k}_\mu=\chi^{n-1,k-1}_{\mu^*}-\chi^{ n-1,k}_{\mu^*}, \mu^*:=(\mu_1,\ldots,\mu_{i-1},\mu_i-1,\mu_{i+1},\ldots).$$\finth \vskip 5pt {\petcap Further Properties}. \vskip 3pt \indent \indent (i) $(\chi ^{n,k},\zeta ^\lambda )>0 \Rightarrow \lambda _1\le k+1$,\quad $\lambda '_1\le n-k$ ; \vskip 3pt \indent \indent (ii) $(\chi ^{n,k},\zeta ^{(j+1,1^{n-j-1})})>0\Leftrightarrow j=k$ ; \vskip 3pt \indent \indent (iii) {\it The $\chi ^{n,k}$ are linearly independent;} \vskip 3pt \indent \indent (iv) {\it If $\chi :S_n\rightarrow {\bf C}$ denotes a character, depending only on the \indent\indent number of cyclic factors, then we have} $$\chi =\sum _i {(\chi ,\zeta ^{(i+1,1^{n-i-1})}) \over f^{(i+1,1^{n-i-1})}}\chi^{n,i}.$$ %\finth \noindent %For example, $\chi ^{5,2}=6\zeta ^{(3,1^2)}+3\zeta ^{(2^2,1)} %+3\zeta ^{(3,2)}$, %take $(1/3)\chi ^{5,2}$. Using 5.8.30 in KERBER-TH\"URLINGS we obtain \th Theorem| The ``P\'olya-char\-ac\-ter'' $\chi,$ defined by $$\chi (\pi ):=m^{\hbox{\sevenrm no. of cycles of }\pi }$$ has the following decomposition into irreducibles: $$\chi =\sum _k {m+k\choose n} \chi ^{n,k}.$$ \finth \section 3. Foulkes tables| This section contains the Foulkes tables $\displaystyle F_i:=(\chi _j^{n,k})$ of the symmetric groups $S_n$, for $n\leq7$. We recall that the $j$-th column of the Foulkes table contains in its $i$-th row the value of the Foulkes characters $\chi^{n,i}$ on the classes of elements which consist of $j$ cyclic factors. The 0-th row indicates the column numbers $j,$ while the $0$-th column shows the row numbers $i.$ $$F_1=\matrix{ i\backslash j&1\cr 0&1\cr} ,\ F_2= \matrix{ i\backslash j&2&1\cr 0&1&-1\cr 1&1&1}, \ F_3= \matrix{ i\backslash j&3&2&1\cr 0&1&-1&1\cr 1&4&0&-2\cr 2&1&1&1}, $$ $$ F_4=\matrix{ i\backslash j&4&3&2&1\cr 0&1&-1&1&-1\cr 1&11&-3&-1&3\cr 2&11&3&-1&-3\cr 3&1&1&1&1}, \ F_5= \matrix{ i\backslash j&5&4&3&2&1\cr 0&1&-1&1&-1&1\cr 1&26&-10&2&2&-4\cr 2&66&0&-6&0&6\cr 3&26&10&2&-2&-4\cr 4&1&1&1&1&1}, $$ $$F_6=\matrix{ i\backslash j&6&5&4&3&2&1\cr 0&1&-1&1&-1&1&-1\cr 1&57&-25&9&-1&-3&5\cr 2&302&-40&-10&8&2&-10\cr 3&302&40&-10&-8&2&10\cr 4&57&25&9&1&-3&-5\cr 5&1&1&1&1&1&1}, $$ $$ F_7= \matrix{ i\backslash j&7&6&5&4&3&2&1\cr 0&1&-1&1&-1&1&-1&1\cr 1&120&-56&24&-8&0&4&-6\cr 2&1191&-245&15&19&-9&-5&15\cr 3&2416&0&-80&0&16&0&-20\cr 4&1191&245&15&-19&-9&5&15\cr 5&120&56&24&8&0&-4&-6\cr 6&1&1&1&1&1&1&1}. $$ \section 4. The connection with a result of Stanley|We consider groups acting on posets $M$ such that $x\le y\Leftrightarrow gx\le gy$. An important example is the action of $S_n$ on $2^n,$ the power set of $n,$ with the inclusion as partial order. Denote by $R$ a subset of the set of ranks, and by $K_R(M)$ a set of rank selected chains. Put $K_R(M,{\bf C}):={\bf C}^{K_R(M)}$ and denote by $H_i(M_R,{\bf C})$ the {\it homology group.} Using these notions we can introduce $$\kappa _R(g):=\hbox{trace of $g$ on }K_R(M,{\bf C}), \gamma _{R,i}(g):=\hbox{trace of $g$ on }H_i(M_R,{\bf C}),$$ and $$\nu _R(g):=\sum _{i=0}^r (-1)^{\left|R\right|-i} \gamma _{R,i}(g),$$ the {\it Lefschetz\ character}. Then, to begin with, we have the following well known facts: $$ \kappa _R=\sum _{T\subseteq R}\nu _T,\ \ or,\ equivalently,\ \ \nu _R =\sum _{T\subseteq R}(-1)^{\left|R\setminus T\right|}T.$$ \th Theorem {(Stanley)}|If $R:=(n_1,\ldots,n_k)_<$, $\rho :=(n_1,n_2-n_1,\ldots,n_k-n_{k-1},n-n_k)$ ; $\rho ^*:=\hbox{partition obtained by reordering}$, then {\rm (i)} $\kappa _R=\xi ^{\rho ^*}$, the Young character, $\displaystyle {}=\sum _{\lambda \vdash n}\left|ST^{\lambda '} (\rho ^*)\right|\zeta ^\lambda $ (standard tableaux, shape $\lambda '$, content $\rho ^*$). {\rm (ii)} $\displaystyle \nu _R =\sum _{\lambda \vdash n}\left|ST_R^{\lambda '}(1^n)\right| \zeta ^\lambda $ (standard Young tableaux with $R$ as set of ascents).\finth %\vskip 2cm %\rightline{then $\mu _1$ is an ascent.\qquad} %\vskip 2cm Hence we obtain from the above discussion of Foulkes characters : \vskip 5pt {\petcap Theorem.} $$\chi ^{n,n-k-1}=\sum _R \nu _R\qquad(\left|R\right|=k).$$ This shows the connection between Foulkes characters and the Lefschetz characters of $S_n$ on $2^n$. \goodbreak \vskip 1cm \eightpoint \centerline{REFERENCES} \nobreak \vskip.5cm \article 1|\pd FOULKES (H.O.)|Eulerian numbers, Newcomb's problem and representations of symmetric groups|Discrete Math.|30|1980|3--49| \divers 2|\pd KERBER (Adalbert) and \pd TH\"URLINGS (Karl-Josef)|Symmetrieklassen von Funktionen und ihre Abz\"ahlungstheorie~II (in print), III (in preparation), {\sl Bayreuther Math. Schriften}| \article 3|\pd STANLEY (Richard)|Some aspects of groups acting on finite posets|J. Combinatorial Theory, Ser.~A|32|1982|132--161| \nobreak \vskip 1cm \line{\quad\vtop{\hbox{Adalbert {\petcap Kerber} and Karl-Josef {\petcap Th\"urlings},} \hbox{Lehrstuhl II f\"ur Mathematik,} \hbox{Universit\"at Bayreuth,} \hbox{D-95440 Bayreuth, Deutschland} \hbox{email : {\eightrm kerber@btm2x6.mat.uni-bayreuth.de}}}\ %hfill %\vtop{\hbox{Dominique {\petcap Foata,}} %\hbox{D\'epartement de math\'ematique,} %\hbox{Universit\'e Louis-Pasteur,} %\hbox{7, rue Ren\'e-Descartes,} %\hbox{F-67084 Strasbourg, France.} %\hbox{email : {\eightrm %foata@math.u-strasbg.fr}}}\quad} %\tenpoint %\adresse %{{\petcap {Adalbert} Kerber} and Karl-Josef {\petcap Th\"urlings}, %Lehrstuhl II f\"ur Mathematik, %Universit\"at Bayreuth, %Postfach 3008, %D-8580 Bayreuth. % %\fin } \bye **************************endofpaper********************************