%---------------------------------------------------------------------------- % HANPAPER_no = 54 % HANPAPER_type = article % PUBPAPER_author = D. Foata, G.-N. Han % PUBPAPER_title = Decreases and descents on words % PUBPAPER_journal= ??? % PUBPAPER_volume = ??? % PUBPAPER_year = 2007 % PUBPAPER_pages = 16 % MAKEPAPER_output= ps pdf % MAKEPAPER_4up = "-m30 -b-30" %---------------------------------------------------------------------------- % FORMAT <<< \catcode'32=9 \magnification=1200 \voffset=1cm %\hoffset=0cm \font\tenpc=cmcsc10 % 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=eufm7 at 8pt \font\eightbboard=msbm7 at 8pt \font\sevengoth=eufm7 \font\sevenbboard=msbm7 \font\sixgoth=eufm5 at 6 pt \font\fivegoth=eufm5 \font\tengoth=eufm10 \font\tenbboard=msbm10 \font\eightgoth=eufm7 at 8pt %%%%%%%%%%%%\font\eightbboard=msbm8%%%%% manque \font\eightbboard=msbm7 at 8pt \font\sevengoth=eufm7 \font\sevenbboard=msbm7 %%%%%%%%%%%%\font\sixgoth=eufm6%%%%%%%%% manque \font\sixgoth=eufm5 at 6 pt \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 \frenchspacing % format de sortie \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 %First page headline in Foata-TeX for S\'eminaire Lotharingien de Combinatoire \font\rms=cmr8 \font\its=cmti8 \font\bfss=cmbx8 \def\titleheadline{\its S\'eminaire Lotharingien de Combinatoire \bfss 58 \rms (2007), Article~B58a\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}.\quad} \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 }} %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}.\quad #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}.\quad {\sl #3}.\quad #4.\par}} %reference complementaire : \def\divers#1|#2|#3| {{\leftskip=7mm\noindent \hangindent=2mm\hangafter=1 \llap{[#1]\hskip.35em}{#2}.\quad #3.\par}} % \mathchardef\conj="0365 \def\proof{\par{\it Proof}.\quad} \def\qed{\quad\raise -2pt\hbox{\vrule\vbox to 10pt{\hrule width 4pt \vfill\hrule}\vrule}} \def\cqfd{\penalty 500 \hbox{\qed}\par\smallskip} \def\decale#1|{\par\noindent\hskip 28pt\llap{#1}\kern 5pt} %fin du format %Quelques macros \catcode`\@=11 \def\matrice#1{\null \,\vcenter {\normalbaselines \m@th \ialign {\hfil $##$\hfil &&\ \hfil $##$\hfil\crcr \mathstrut \crcr \noalign {\kern -\baselineskip } #1\crcr \mathstrut \crcr \noalign {\kern -\baselineskip }}}\,} \def\petitematrice#1{\left(\null\vcenter {\normalbaselines \m@th \ialign {\hfil $##$\hfil &&\thinspace \hfil $##$\hfil\crcr \mathstrut \crcr \noalign {\kern -\baselineskip } #1\crcr \mathstrut \crcr \noalign {\kern -\baselineskip }}}\right)} \catcode`\@=12 %%%%%%%%%%%%%%%%%%%%%%%% \def\negg{\mathop{\rm neg}\nolimits} \def\inv{\mathop{\rm inv}\nolimits} \def\imaj{\mathop{\rm imaj}\nolimits} \def\tot{\mathop{\rm tot}\nolimits} \def\L{\mathop{\rm L\kern 0pt}} \def\odd{\mathop{\rm odd}\nolimits} \def\Niw{\mathop{\rm Niw}\nolimits} \def\NIW{\mathop{{\hbox{\eightrm NIW}}}\nolimits} \def\NIWa{\mathop{{\hbox{\sixrm NIW}}}\nolimits} \def\DW{\mathop{{\hbox{\eightrm DW}}}\nolimits} \def\Der{\mathop{\rm Der}\nolimits} \def\WSP{\mathop{{\hbox{\eightrm WSP}}}\nolimits} \def\WSD{\mathop{{\hbox{\eightrm WSD}}}\nolimits} \def\fix{\mathop{\rm fix}\nolimits} \def\pix{\mathop{\rm pix}\nolimits} \def\abs{\mathop{\rm abs}\nolimits} \def\fmaj{\mathop{\rm fmaj}\nolimits} \def\fdes{\mathop{\rm fdes}\nolimits} \def\des{\mathop{\rm des}\nolimits} \def\maj{\mathop{\rm maj}\nolimits} \def\finv{\mathop{\rm finv}\nolimits} \def\Ligne{\mathop{\rm Ligne}\nolimits} \def\Iligne{\mathop{\rm Iligne}\nolimits} \def\odd{\mathop{\rm odd}\nolimits} \def\Fix{\mathop{\rm Fix}\nolimits} \def\Pix{\mathop{\rm Pix}\nolimits} \def\negg{\mathop{\rm neg}\nolimits} \def\Neg{\mathop{\rm Neg}\nolimits} \def\Cont{\mathop{\rm Cont}\nolimits} \def\fexc{\mathop{\rm fexc}\nolimits} \def\exc{\mathop{\rm exc}\nolimits} \def\Dec{\mathop{\rm Dec}\nolimits} \def\Evdec{\mathop{\rm Evdec}\nolimits} \def\dec{\mathop{\rm dec}\nolimits} \def\evdec{\mathop{\rm evdec}\nolimits} \def\odd{\mathop{\rm odd}\nolimits} \def\evweight{\mathop{\rm evweight}\nolimits} \def\EV{\mathop{{\hbox{\eightrm EV}}}\nolimits} \def\lab{\mathop{\rm lab}\nolimits} \def\bfm{{\bf m}} \def\imv{\mathop{\rm imv}\nolimits} \def\oddlower{\mathop{\rm oddlower}\nolimits} \def\evenlower{\mathop{\rm evenlower}\nolimits} \def\rmin{\mathop{\rm rmin}\nolimits} \def\decval{\mathop{\rm decval}\nolimits} \def\Single{\mathop{\rm Single}\nolimits} \def\single{\mathop{\rm single}\nolimits} \def\decweight{\mathop{\rm decweight}\nolimits} \def\desweight{\mathop{\rm desweight}\nolimits} \def\pr{\mathop{\rm pr}\nolimits} \def\mult{\mathop{\rm mult}\nolimits} \def\ND{\mathop{\rm ND}\nolimits} \def\inrec{\mathop{\rm inrec}\nolimits} \def\DEC{\mathop{\hbox{\eightrm DEC}}\nolimits} \def\DES{\mathop{\hbox{\eightrm DES}}\nolimits} \def\INC{\mathop{\hbox{\eightrm INC}}\nolimits} \def\RISE{\mathop{\hbox{\eightrm RISE}}\nolimits} \def\REC{\mathop{\hbox{\eightrm REC}}\nolimits} \def\key{\mathop{\rm key}\nolimits} % >>> \titrecourant={DECREASES AND DESCENTS IN WORDS} \auteurcourant={DOMINIQUE FOATA AND GUO-NIU HAN} %\rightline{2007/09/28} \vglue 1.3cm \centerline{\bf DECREASES AND DESCENTS IN WORDS} \bigskip \centerline{ \bf Dominique Foata and Guo-Niu Han} \bigskip\bigskip {\narrower\narrower \eightpoint\noindent {\sixrm ABSTRACT}.\quad The generating function for words by a multivariable statistic involving decrease, increase, descent and rise values is explicitly calculated by using the MacMahon Master Theorem and the properties of the first fundamental transformation on words. Applications to statistical study of the symmetric group are also given. % leave a blank line } \bigskip \centerline{\bf 1. Introduction} % <<< \medskip In our recent papers [2, 3, 4, 5, 6, 7, 8, 9] that involve the calculations of factorial generating functions for the symmetric and hyperoctahedral groups, we have obtained several results on statistical distributions on {\it words}. In this paper we take up again the study of those word statistics in a more general context. They do not necessarily have counterparts on permutations, but are essential in this word calculus. The MacMahon Master Theorem [14, p.~97--98] and the first fundamental transformation on words [13, Chap.~10] will be our basic tools. The {\it number of decreases} is a crucial statistic; as such it is at the origin of our word studies. The definition of {\it decrease} slightly differs from the definition of the classical {\it descent}. Let $w=x_1x_2\cdots x_n$ be an {\it arbitrary} word, whose letters are nonnegative integers. Recall that a positive integer~$i$ is said to be a {\it descent} (or {\it descent place}) of~$w$ if $1\le i\le n-1$ and $x_i>x_{i+1}$. We say that~$i$ is a {\it decrease} of~$w$ if $1\le i\le n-1$ and $x_i= x_{i+1} = \cdots = x_j>x_{j+1}$ for some~$j$ such that $i\le j\le n-1$. The letter~$x_i$ is said to be a {\it decrease value} (respectively {\it descent value}) of~$w$. The set of all decreases (respectively descents) is denoted by $\DEC(w)$ (respectively $\DES(w)$). Each descent is a decrease, but not conversely. This means that $\DES(w)\subset \DEC(w)$. However, $\DES (w)=\DEC(w)$ when~$w$ is a word {\it without repetitions}. In the present paper our intention is to go back to the study of the number of decreases, this time associated with several other word statistics, and derive the Ur-result that should have been at the origin of several of our statistical distribution studies. This Ur-result is stated in Theorem~1.1, but has two equivalent forms, as written in Theorems~1.2 and~1.3. In parallel with the notion of decrease, we say that that a positive integer~$i$ is an {\it increase} (respectively a {\it rise\/}) of~$w$ if $1\le i\le n$ and $x_i= x_{i+1}= \cdots = x_j\sigma(j+1)\}\quad (1\le i\le n),\leqno(1.13) $$ so that $z_1$ is the {\it number of descents}, $\des\sigma$, and $\tot z=z_1+z_2+\cdots +z_n$ is the {\it major index}, $\maj\sigma$, of~$\sigma$. Also, let $\exc\sigma:=\{i:1\le i\le n-1,\sigma(i)>i\}$ be the {\it number of excedances} and $\fix\sigma$ be the {\it number of fixed points} of~$\sigma$. Finally, let $\NIW_n$ (respectively $\NIW_n(k)$) denote the set of words $c=c_1c_2\cdots c_n$, of length~$n$, whose letters are integers satisfying $c_1\ge c_1\ge\cdots \ge c_n\ge 0$ (respectively $k\ge c_1\ge c_1\ge\cdots \ge c_n\ge 0$). With each pair $(\sigma,c)\in {\goth S}_n\times\NIW_n$ we associate the monomial $$ Y_{(\sigma,c)}:=\prod_{j<\sigma(j)}Y_{c_j+z_j-1} \times \prod_{j\ge \sigma(j)}Y_{c_j+z_j}.\leqno(1.14) $$ \proclaim Theorem 1.5. We have: $$ \sum_{n\ge 0}\sum_{\scriptstyle\sigma\in {\goth S}_n \atop \scriptstyle\des\sigma\le r} R^{\fix\sigma}s^{\exc\sigma} \kern-20pt \sum_{c\in \hbox{\sixrm NIW}_n(r-\des\sigma)}\kern-20pt Y_{(\sigma,c)} ={(1-s)H_{r+1}(RY)\over H_r(sY)-sH_r(Y)}\quad (r\ge0) .\leqno(1.15) $$ When $r$ tends to infinity in (1.15), we get the identity $$ \sum_{n\ge 0}\sum_{\sigma\in {\goth S}_n} R^{\fix\sigma}s^{\exc\sigma} \sum_{c\in \hbox{\sixrm NIW}_n} Y_{(\sigma,c)} ={(1-s)H(RY)\over H(sY)-sH(Y)},\leqno(1.16) $$ derived by Shareshian and Wachs [15, Theorem~2.1] using a quasi-symmetric function approach. However, starting from (1.15), we can obtain a {\it graded form} of (1.16) as follows. Let $$ Y(\sigma;t):=\sum_{k\ge 0}t^k\sum_{c\in \hbox{\sixrm NIW}_{n}(k)} Y_{(\sigma,c)}.\leqno(1.17) $$ \proclaim Theorem 1.6. The graded form of $(1.16)$ reads: $$ \sum_{n\ge 0}\sum_{\sigma\in {\goth S}_n} R^{\fix\sigma}s^{\exc\sigma}t^{\des\sigma}Y(\sigma;t) =\sum_{r\ge 0}t^r {(1-s)H_{r+1}(RY)\over H_r(sY)-sH_r(Y)}.\leqno(1.18) $$ \goodbreak Section~5 starts with redescribing the Gessel--Reutenauer standardization [10] and showing how it is used to prove Theorem~1.5. The graded form (1.18) is deduced from Theorem~1.5 by a standard series manipulation. We end the paper by giving some specializations of our Ur-theorem and also by showing that the distribution of $(\fix,\exc,\des,\maj)$ over the symmetric groups that was found earlier in [9] using the word-analog of the Kim--Zeng transformation [11], can also be deduced from our Ur-result in two different manners. \bigskip % >>> \centerline{\bf 2. Proof of Theorem 1.1} % <<< \medskip A word $w=y_1y_2\cdots y_n\in [0,r]^*$ having no equal letters in succession is called an {\it $h$-derangement} ({\it horizontal derangement}). The set of all $h$-derangement words in $[0,r]^*$ is denoted by $[0,r]_{h}^*$. % Let $\alpha$ be the substitution of variables defined by $$ \alpha :=\{ Z_i\leftarrow 0,\quad T_i\leftarrow 0,\quad T'_i\leftarrow 0 \}. $$ Then $$ \alpha\,\psi(w)=\psi(w)=\prod_{i\in\hbox{\sixrm DES}}X_{x_i} \prod_{i\in\hbox{\sixrm RISE}\setminus \hbox{\sixrm REC}}Y_{x_i} \prod_{i\in\hbox{\sixrm RISE}\cap \hbox{\sixrm REC}}Y'_{x_i}, $$ if $w$ is an $h$-derangement and $\alpha\,\psi(w)=0$ otherwise. The following specialization of Theorem~1.1 is obtained by taking the image of identity (1.3) under~$\alpha$. \proclaim Theorem 2.1. We have $$ \sum_{w\in [0,r]_h^*} \psi(w) ={\prod\limits_{0\le j\le r} \displaystyle\Bigl(1+{Y'_j }\Bigr)\over \det (I-\alpha\,C)}.\leqno(2.1) $$ \smallskip Even though Theorem~2.1 is a special case of Theorem~1.1, it can still be used as a lemma to prove Theorem~1.1. We proceed as follows. \medskip {\it Proof of Theorem 1.1}.\quad Let $w$ be a word from $[0,r]^*$. The {\it key} of $w$ is defined to be the $h$-derangement $k$ derived from $w$ by erasing all letters~$x_i$ such that $x_i=x_{i+1}$. For instance, the key of $w=324455531114135$ is the $h$-derangement $k=3245314135$. Let $\beta$ be the substitution of variables defined by $$ \beta:=\{ X_i\leftarrow X_i/(1-Z_i),\quad Y_i\leftarrow Y_i/(1-T_i),\quad Y'_i\leftarrow Y'_i/(1-T'_i) \}. $$ Then, the generating function for the set of all $w$ whose key is $k$ by the weight $\psi$ is given by $$ \sum_{w,\, \key(w)=k} \psi(w) = \beta\,\psi(k). $$ \goodbreak Since $\beta\,\alpha\,C=C$ we have $$ \eqalignno{ \sum_{w\in [0,r]^*} \psi(w) &= \sum_{k\in [0,r]_h^*} \sum_{\key(w)=k}\psi(w) \ = \sum_{k\in [0,r]_h^*} \beta\,\psi(k) \cr & = \beta \biggl(\sum_{k\in [0,r]_h^*} \psi(k)\biggr) \ = \ \beta\, {\prod\limits_{0\le j\le r} \displaystyle\Bigl(1+{Y'_j }\Bigr)\over \det (I-\alpha\,C)}&\hbox{[by (2.1)]}\cr &={\prod\limits_{0\le j\le r} \displaystyle\Bigl(1+{\beta\,Y'_j }\Bigr)\over \det (I-\beta\,\alpha\,C)} ={\prod\limits_{0\le j\le r} \displaystyle\Bigl(1+{Y'_j\over 1-T'_j}\Bigr)\over \det (I-C)}.\qed\cr } $$ \medskip {\it Proof of Theorem 2.1}.\quad A letter $x_i$ which is a record and also a rise value is called a {\it riserec} value. For each $h$-derangement $k=x_1x_2\cdots x_n$ let $w_0$ be the nondecreasing word composed of all the riserec values of $k$ and let $k_0$ be the word obtained from $k$ by erasing all the riserec values of $k$. Since the letters of $w_0$ can be uniquely inserted into the word $k_0$ for reconstructing $k$, the map $$ k\mapsto (w_0, k_0)\leqno(2.2) $$ is a bijection of the set of all $h$-derangements onto the the set of all pairs $(w_0, k_0)$ such that $w_0$ is a nondecreasing word and $k_0$ is an $h$-derangement without any riserec value. % Moreover, if a letter~$x_j$ of~$k$ is a record value and therefore becomes a letter, say,~$x_{0,i}$ of~$w_0$, then $x_{0,i}$ is a rise of~$w_0$ if and only if~$x_j$ is a rise of~$k$. Therefore $$ \psi(k)=\psi(w_0)\psi(k_0).\leqno(2.3) $$ Next apply the first fundamental transformation to $k_0$ (see [13, \S\kern 2pt 10.5]), say, $u={\bf F}_1(k_0)$. Let us recall how ${\bf F}_1$ is defined by means of an example. Start with the word $k_0=5\,3\,6\,5\,3\,2\,4\,6\,1\,2\,4\,3\,1$ and cut it before each record value to get $k_0=5\,3\mid 6\,5\,3\,2\,4\mid 6\,1\,2\,4\,3\,1=:w_1\mid w_2\mid w_3$. In each compartment move the leftmost letter to the end to obtain the cyclic shifts $\delta w_1=3\,5$, $\delta w_2=5\,3\,2\,4\,6$, $\delta w_3=1\,2\,4\,3\,1\,6$ and form the two-row matrix $\petitematrice{\delta w_1&\delta w_2&\delta w_3\cr w_1&w_2&w_3\cr}=\petitematrice{3&5&5&3&2&4&6&1&2&4&3&1&6\cr 5&3&6&5&3&2&4&6&1&2&4&3&1\cr}$. Finally, rearrange the vertical biletters of that two-row matrix in such a way that the entries on the top row are in nondecreasing order, assuming that two biletters $a\choose a'$, $b\choose b'$ can commute only when $a\not=b$. We then obtain the two-row matrix ${\bf F}_1(k_0):=\petitematrice{\overline u\cr u\cr} =\petitematrice{1&1&2&2&3&3&3&4&4&5&5&6&6\cr 6&3&3&1&5&5&4&2&2&3&6&4&1\cr}$. We can characterize the word $u=y_1y_2\cdots y_m$ when we start with an $h$-derangement~$k_0$. Let $\overline u=z_1z_2\cdots z_m$ be the nondecreasing rearrangement of $u$ (and of~$k_0$). By construction $y_i\not=z_i$ for $1\le i\le m$. Such a word~$u$ is called a {\it $v$-derangement} (vertical derangement). Denote the set of all $v$-derangements in $[0,r]^*$ by $[0,r]_{v}^*$. Then ${\bf F}_1$ provides a bijection of the set of all $h$-derangements onto the set of all pairs $(w_0, u)$ such that $w_0$ is a nondecreasing word and $u$ is a $v$-derangement: $$ k\mapsto (w_0, k_0) \mapsto (w_0, u) \quad\hbox{where}\quad {\bf F}_1(k_0) = {\overline u\choose u}. $$ $$ \leqalignno{\qquad\qquad\qquad k={\bf 2}53{\bf 5}65324612431{\bf 6} &\mapsto (w_0={\bf 256}, k_0=5365324612431)&\hbox{\it \quad Example.} \cr &\mapsto (w_0={\bf 256}, u=6331554223641), \cr {\bf F}_1(k_0)&=\petitematrice{\overline u\cr u\cr} =\petitematrice{1&1&2&2&3&3&3&4&4&5&5&6&6\cr 6&3&3&1&5&5&4&2&2&3&6&4&1&\cr}.&\hbox{since} \cr } $$ Let $\Phi$ be the homomorphism generated by $$ \Phi{i\choose j}:=\cases{{X_j},&if $j>i$;\cr \noalign{\smallskip} {Y_j},&if $j>> \centerline{\bf 3. Proof of Theorem 1.2} % <<< \medskip The proof of Theorem~1.2 is similar to the proof of Theorem~1.1 given in the previous section. First we prove the following specialization of Theorem~1.2. \proclaim Theorem 3.1. We have: $$ \sum_{w\in [0,r]_h^*} \psi(w) ={\displaystyle \prod\limits_{0\le j\le r} {1+Y'_{j}\over 1+X_j} \over \displaystyle 1-\sum\limits_{0\le l\le r} \prod\limits_{0\le j\le l-1}{1+Y_{j}\over 1+X_j} {X_l\over 1+X_l}}. \leqno(3.1) $$ \medskip {\it Proof}. Denote the left-hand side of (3.1) by {\eightrm LHS}. From Theorem~2.1 we have $$\hbox{{\eightrm LHS}} = {\prod\limits_{0\le j\le r}(1+Y'_j) \over D}, $$ where {\eightpoint $$ D = { \left|\matrix{\noalign{\medskip} 1&-{ X_1} &-{ X_2}&\cdots &-{ X_{r-1}} &- { X_r}\cr -Y_0&1&-{ X_2}&\cdots &-{ X_{r-1}}&-{ X_r}\cr -Y_0&-Y_1& 1&\cdots &-{ X_{r-1}}&-{ X_r}\cr \vdots&\vdots&\vdots&\ddots&\vdots&\vdots\cr \noalign{\medskip} -Y_0& -Y_1&-Y_2 &\cdots& 1&-{ X_r}\cr -Y_0&-Y_1 &-Y_2&\cdots &-Y_{r-1} & 1\cr }\right|}. $$ } \medskip In the above determinant subtract the $r$-th row from the $(r+1)$-st one; then the $(r-1)$-st from the $r$-th row; \dots, the first row from the second. We obtain: \medskip { \eightpoint $$D={ \left|\matrix{\noalign{\medskip} 1&-{ X_1} &-{ X_2}&\kern-.3cm\cdots &\kern-.3cm-{ X_{r-1}} &- { X_r}\cr -1-Y_0&{1+X_1 }&{ 0}&\cdots &{0}&{ 0}\cr 0& -1-Y_1& {1+X_2 }&\cdots & 0& 0\cr \vdots&\vdots&\vdots&\ddots&\vdots&\vdots\cr \noalign{\medskip} 0& 0&0 &\cdots& 1+X_{r-1}& 0\cr 0&0 &0&\cdots &-1-Y_{r-1} &{1+X_r }\cr }\right|}. $$ } \bigskip Now expand the determinant by the cofactors of the {\it first row}. We get: $$ D=\prod_{1\le j\le r}{(1+X_j) } -\sum_{1\le l\le r}\Bigl(\prod_{0\le j\le l-1}(1+Y_j) \prod_{l+1\le j\le r}{(1+X_j) }\Bigr)X_l. $$ We further have: $$\leqalignno{ D&=\prod_{1\le j\le r}{(1+X_j) } +X_0\prod_{1\le j\le r}{(1+X_j)}\cr &\kern2cm{}- \sum_{0\le l\le r}\Bigl(\prod_{0\le j\le l-1}(1+Y_j) \prod_{l\le j\le r}{(1+X_j) }\Bigr){X_l\over 1+X_l}.\cr } $$ Hence, $$\hbox{\eightrm LHS} ={\displaystyle\prod_{0\le j\le r}{(1+Y'_j)\over (1+X_j)}\over\displaystyle 1-\sum_{0\le l\le r}\prod_{0\le j\le l-1} {(1+Y_j)\over (1+X_j)}{X_l\over 1+X_l}}.\qed\leqno(3.2) $$ {\it Proof of Theorem 1.2.}\quad As in the proof of Theorem~1.1 we may write: $$ \sum_{w\in [0,r]^*} \psi(w) = \sum_{k\in [0,r]_h^*} \sum_{\key(w)=k}\psi(w) = \sum_{k\in [0,r]_h^*} \beta\,\psi(k) = \beta(\hbox{\eightrm LHS}). $$ Using (3.2) it is immediate to verify that $\beta(\hbox{\eightrm LHS}) $ is equal to the right-hand side of (1.4).\qed \bigskip % >>> % \vfill\eject \centerline{\bf 4. Proof of Theorem 1.4} % <<< \medskip First, we may check that {\eightpoint $$ \leqalignno{ &\prod\limits_{0\leq j\leq r}(1-sY_j) - \prod\limits_{0\leq j\leq r}(1-Y_j)\cr =&\sum_{0\leq l \leq r} \prod\limits_{0\leq j\leq l}(1-sY_j) \prod\limits_{l+1\leq j\leq r} (1-Y_j) - \sum_{0\leq l \leq r} \prod\limits_{0\leq j\leq l-1}(1-sY_j) \prod\limits_{l\leq j\leq r} (1-Y_j)\cr =& (1-s)\sum_{0\leq l \leq r} Y_l \prod\limits_{0\leq j\leq l-1}(1-sY_j) \prod\limits_{l+1\leq j\leq r} (1-Y_j).\cr } $$ } \goodbreak {\it Proof of Theorem 1.4}.\quad Using (1.7) we have: {%\eightpoint $$ \leqalignno{ & { 1 \over 1-s\sum\limits_{0\leq l\leq r-1} Y_l {\prod\limits_{0\leq j\leq l-1}(1-sY_j)/\prod\limits_{0\leq j\leq l}(1-Y_j)} }\cr &= { \prod\limits_{0\leq j \leq r-1} (1-Y_j) \over \prod\limits_{0\leq j\leq r-1} (1-Y_j) -s\sum\limits_{0\leq l\leq r-1} Y_l {\prod\limits_{0\leq j\leq l-1}(1-sY_j)\prod\limits_{l+1\leq j\leq r-1}(1-Y_j)} }\cr &= { \prod\limits_{0\leq j \leq r-1} (1-Y_j) \over \prod\limits_{0\leq j\leq r-1} (1-Y_j) -s { \bigl(\prod\limits_{0\leq j\leq r-1}(1-sY_j) - \prod\limits_{0\leq j\leq r-1}(1-Y_j)\bigr)/ ( 1-s ) } } \cr &= { (1-s) \prod\limits_{0\leq j \leq r-1} (1-Y_j) \over \prod\limits_{0\leq j\leq r-1} (1-Y_j) -s \prod\limits_{0\leq j\leq r-1}(1-sY_j) }.\qed \cr } $$ } \bigskip % >>> \centerline{\bf 5. Proofs of Theorems 1.5 and 1.6} % <<< \medskip An updated version of the Gessel--Reutenauer standardization [10] is fully described in our previous paper [9], Section~5. The standardization consists of mapping each word~$w$ from $[0,r]^*$ of length~$n$ onto a pair $(\sigma,c)$, where $\sigma\in {\goth S}_n$ and $c=c_1c_2\cdots c_n$ is a word of length~$n$, whose letters are nonnegative integers having the property: $r-\des\sigma\ge c_1\ge c_2\ge \cdots \ge c_n\ge 0$, the symbol $\des\sigma$ being the {\it number of descents} of~$\sigma$. We recall the construction of the inverse $(\sigma,c)\mapsto w$ by means of an example. {\eightpoint $$\matrice{ {\rm Id}&=&1&2&3&4&5&6&7&8&9&10&11&12&13 &14&15&16&17&18&19&20&21&22\cr \rightarrow\sigma&=&\bf1&\underline 5&\underline6&\underline8&\underline{13}&\underline{14}& \bf7&\underline{17}&4&\bf10&\underline{15}&\underline{18}& \underline{19}&2&9&\bf16&\underline{20}&\underline{22} &3&11&12&21\cr \quad z&=&4&4&4&4&4&4&3&3&2&2&2&2&2&1&1&1&1&1&0&0&0&0\cr \rightarrow c&=&2&2&2&2&2&2&2&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1\cr \quad\overline c&=&6&6&6&6&6&6&5&4&3&3&3&3&3&2&2&2&2&2&1&1&1&1\cr \quad \sigma&=& (\bf16)&(12&18&22&21)&(\bf 10)&(\bf 7)& (4&8&17&20&11&15&9)&(2&5&13&19&3&6&14)&(\bf1)\cr \quad \check\sigma&=& \bf16&12&18&22&21&\bf 10&\bf 7& 4&8&17&20&11&15&9&2&5&13&19&3&6&14&\bf1\cr \mapsto w&=& 2\ \mid&\underline 3&\underline 2&1&1\mid& 3\ \mid& 5\mid& \underline 6&\underline 4&\underline 2&1&\underline 3&2&3\mid &\underline 6&\underline 6&\underline 3&1&\underline 6&\underline 6&2\mid &6\cr} $$ } \goodbreak \medskip In the example $n=22$. The second row contains the values $\sigma(i)$ $(i=1,2,\ldots,n)$ of the {\it starting} permutation~$\sigma$. The {\it excedances} $\sigma(i)>i$ are underlined, while the {\it fixed points} $\sigma(i)=i$ are written in boldface. The third row is the vector $z=z_1z_2\cdots z_n$ defined by (1.13) so that $z_1=\des\sigma$ and $\tot z=z_1+z_2+\cdots+z_n$ is the {\it major index} of~$\sigma$ denoted by $\maj\sigma$, as already mentioned in the Introduction. The fourth row is the {\it starting} nonincreasing word $c=c_1c_2\cdots c_n$. The fifth row $\overline c=\overline c_1\overline c_2\cdots \overline c_n$ is the word defined by $$ \overline c_i:=z_i+c_i\quad (1\le i\le n). $$ In the sixth row the permutation~$\sigma$ is represented as the product of its disjoint cycles. Each cycle starts with its {\it minimum} element and those minimum elements are in {\it decreasing} order when reading the whole word from left to right. When removing the parentheses in the sixth row we obtain the seventh row denoted by $\check\sigma=\check\sigma(1)\check\sigma(2)\cdots \check\sigma(n)$. The bottom row is the word~$w=x_1x_2\cdots x_n$ corresponding to the pair $(\sigma,c)$ defined by $$ x_i:=\overline c_{\check\sigma(i)} \quad(1\le i\le n). $$ For instance, $\check\sigma(9)=8$ and $\overline c(8)=4$. Hence $x_9=4$. The decrease values of~$w$ have been underlined. It can be verified that all the above steps are reversible and that $w\mapsto (\sigma,c)$ is a bijection of the set of all words from $[0,r]^*$ of length~$n$ onto the set of pairs $(\sigma,c)$ such that $\sigma\in{\goth S}_n$, $\des\sigma\le r$ and $c=c_1c_2\cdots c_n$ is a word satisfying $r-\des\sigma\ge c_1\ge c_2\ge\cdots \ge c_n\ge 0$. Furthermore, $x_i$ is a decrease value of~$w$ if and only if $\check\sigma(i)<\check\sigma(i+1)$, if and only if $\check\sigma(i)<\sigma(\check\sigma(i))$. Also $x_i$ is an increase and record value of~$w$ if and only if $\check\sigma(i)$ is a fixed point of~$\sigma$. Hence, $$\leqalignno{ \psi_R(w)&= R^{\inrec w}s^{\dec w}\kern-6pt \prod\limits_{x_i\in \hbox{\sixrm DEC}}Y_{x_i-1} \prod\limits_{x_i\in \hbox{\sixrm INC}} Y_{x_i}\cr &=R^{\fix \sigma}s^{\exc\sigma}\kern-6pt \prod_{\check\sigma(i)<\sigma(\check\sigma(i))} Y_{\overline c_{\check\sigma(i)} -1} \prod_{\check\sigma(i)\ge\sigma(\check\sigma(i))} Y_{\overline c_{\check\sigma(i)}}\cr &=R^{\fix \sigma}s^{\exc\sigma} \prod_{j<\sigma(j)} Y_{\overline c_{j} -1} \prod_{j\ge\sigma(j)} Y_{\overline c_{j}}\cr &=R^{\fix \sigma}s^{\exc\sigma} \prod_{j<\sigma(j)}Y_{c_j+z_j-1} \prod_{j\ge \sigma(j)}Y_{c_j+z_j}\cr &=R^{\fix\sigma}s^{\exc\sigma}Y_{(\sigma,c)}.\cr} $$ Consequently, $$ \sum_{w\in [0,r]^*}\psi_R(w) =\sum_{n\ge 0}\sum_{\scriptstyle\sigma\in {\goth S}_n \atop\scriptstyle \des\sigma\le r} R^{\fix\sigma}s^{\exc\sigma} \kern-15pt \sum_{c\in \hbox{\sixrm NIW}_n(r-\des\sigma)}\kern-15pt Y_{(\sigma,c)}.\leqno(5.1) $$ This achieves the proof of Theorem~1.5 by taking identity (1.12) into account.\cqfd \goodbreak \medskip For the proof of Theorem~1.6 we multiply both sides of (1.15) by~$t^r$ and sum over $r\ge0$. We obtain: $$ \displaylines{\quad \sum_{r\ge 0}t^r {(1-s)H_{r+1}(RY)\over H_r(sY)-sH_r(Y)}\hfill\cr \qquad\quad{}=\sum_{r\ge 0}t^r \sum_{n\ge 0}\sum_{\scriptstyle\sigma\in {\goth S}_n \atop \scriptstyle\des\sigma\le r} R^{\fix\sigma}s^{\exc\sigma} \kern-20pt \sum_{c\in \hbox{\sixrm NIW}_n(r-\des\sigma)}\kern-20pt Y_{(\sigma,c)}\hfill\cr \qquad\quad{}=\sum_{n\ge 0}\sum_{r\ge 0}t^r\sum_{0\le j\le r} \sum_{\scriptstyle\sigma\in {\goth S}_n \atop \scriptstyle\des\sigma=r-j} R^{\fix\sigma}s^{\exc\sigma} \kern-10pt \sum_{c\in \hbox{\sixrm NIW}_n(j)}\kern-10pt Y_{(\sigma,c)}\hfill\cr \qquad\quad{}=\sum_{n\ge 0}\sum_{j\ge 0}t^j\sum_{r\ge j}t^{r-j} \sum_{\scriptstyle\sigma\in {\goth S}_n \atop \scriptstyle\des\sigma=r-j} R^{\fix\sigma}s^{\exc\sigma} \kern-10pt \sum_{c\in \hbox{\sixrm NIW}_n(j)}\kern-10pt Y_{(\sigma,c)}\hfill\cr \qquad\quad{}=\sum_{n\ge 0}\sum_{j\ge 0}t^j\sum_{k\ge 0}t^{k} \sum_{\scriptstyle\sigma\in {\goth S}_n \atop \scriptstyle\des\sigma=k} R^{\fix\sigma}s^{\exc\sigma} \kern-10pt \sum_{c\in \hbox{\sixrm NIW}_n(j)}\kern-10pt Y_{(\sigma,c)}\hfill\cr \qquad\quad{}=\sum_{n\ge 0} \sum_{\sigma\in {\goth S}_n} R^{\fix\sigma}s^{\exc\sigma}t^{\des\sigma} \sum_{j\ge 0}t^j\kern-5pt \sum_{c\in \hbox{\sixrm NIW}_n(j)}\kern-10pt Y_{(\sigma,c)}\hfill\cr \qquad\quad{}=\sum_{n\ge 0} \sum_{\sigma\in {\goth S}_n} R^{\fix\sigma}s^{\exc\sigma}t^{\des\sigma} Y(\sigma;t).\qed\hfill\cr } $$ \bigskip \centerline{\bf 6. Specializations and $q$-Calculus} \medskip In Theorem~1.4, replace $Y_j, Y_j'$ $(j\geq 0)$ by $u$. We get the identity $$ \sum_{w\in [0,r]^*}s^{\dec w}u^{\lambda w}= {1-s\over (1-u)^{r+1}(1-us)^{-r}-s(1-u)},\leqno(6.1) $$ where $\lambda w$ denotes the length of~$w$. Now replace $X_j$ $(j\ge 0)$ by $us$ and the other variables by~$u$ in Theorem~1.2. We then recover the {\it classical} generating function for words by number of descents: $$ \sum_{w\in [0,r]^*}s^{\des w}u^{\lambda w}= {1-s\over (1-u+us)^{r+1}-s}.\leqno(6.2) $$ As was proved in our paper [9] (formula (1.15)), when multiplying (6.1) (and not (6.2)) by $t^r$ and summing over $r\ge 0$ we get the generating function for the pair $(\exc,\des)$ over the symmetric groups: $$ \sum_{r\ge 0}t^r\sum_{w\in [0,r]^*}s^{\dec w}u^{\lambda w}= \sum_{n\ge 0}{u^n\over (1-t)^{n+1}}\sum_{\sigma\in {\goth S}_n} s^{\exc\sigma}t^{\des\sigma}.\leqno(6.3) $$ Now, recall the traditional notation of the $q$-ascending factorial $$ (a;q)_n=\cases{1,&if $n=0$;\cr (1-a)(1-aq)\cdots (1-aq^{n-1}),&if $n\ge1$.\cr} $$ \proclaim Theorem 6.1. The factorial generating function for the distributions of the vector $(\fix,\exc,\des,\maj)$ over the symmetric groups~${\goth S}_n$ is given by $$ \displaylines{(6.4)\quad \sum_{n\ge 0}{u^n\over (t;q)_{n+1}} \sum_{\sigma\in {\goth S}_n} R^{\fix\sigma}s^{\exc\sigma}t^{\des\sigma}q^{\maj\sigma}\ \hfill\cr \hfill{}=\sum_{r\ge 0}t^r {1\over (uR;q)_{r+1}} {(1-sq)\,(u;q)_r\,(usq;q)_r \over ((u;q)_r-sq(usq;q)_r)}.\quad\cr} $$ The theorem was proved in our previous paper [9] by means of the word-analog of the Kim--Zeng transformation~[11] and the Gessel--Reutenauer standardization. Here, it is a simple consequence of Theorem~1.6 by applying the homomorphism $\phi$ generated by $\phi(Y_j):=uq^j$ $(j\ge 0)$ and $\phi(s):=sq$ to both sides of (1.18). We proceed as follows. First, $\smash{\phi H_r(Y)=\prod\limits_{0\le j\le r-1}(1-uq^j)^{-1} =1/(u;q)_r}$ and $$\displaylines{\eqalign{ \phi {(1-s)H_{r+1}(RY)\over H_r(sY)-sH_r(Y)}&={(1-sq)/(uR;q)_{r+1}\over 1/(usq;q)_r-sq/(u;q)_r}\cr &={(1-sq)(u;q)_r\,(usq;q)_r \over (uR;q)_{r+1}\,(\bigl((u;q)_r-sq(usq;q)_r\bigr)}.\cr}\cr \noalign{\hbox{Then, for $(\sigma,c)\in {\goth S}_n\times \NIW_n$}} \eqalign{ \phi Y_{(\sigma,c)}&=u^nq^{\tot c+\tot z-\exc\sigma} =q^{\maj\sigma-\exc\sigma}u^nq^{\tot c};\cr \phi Y(\sigma;t)&=q^{\maj\sigma-\exc\sigma} u^n\sum_{j\ge 0} t^j\kern-5pt \sum_{c\in \hbox{\sixrm NIW}_n(j)}\kern-10pt q^{\tot c}=q^{\maj\sigma-\exc\sigma}{u^n\over (t;q)_{n+1}};\cr }\cr \noalign{\hbox{so that}} \phi\bigl( R\fix^{\sigma}s^{\exc\sigma}t^{\des\sigma}Y(\sigma;t) \bigr)=R^{\fix\sigma}(sq)^{\exc\sigma}t^{\des\sigma} q^{\maj\sigma-\exc\sigma}{u^n\over (t;q)_{n+1}}.\cr} $$ Hence, the image of identity (1.18) under $\phi$ gives back (6.4).\cqfd \medskip There is still another proof, which we now describe. In identity (1.4) make the substitutions $X_j\leftarrow usq^j$, $Y_j\leftarrow uq^j$, $Z_j\leftarrow usq^j$, $T_j\leftarrow uq^j$, $Y'_j\leftarrow uRq^j$, $T'_j\leftarrow uRq^j$. The weight $\psi(w)$ becomes $s^{\dec w}R^{\inrec w}u^{\lambda w}q^{\tot w}$, and (1.4) yields the identity $$ \sum_{w\in [0,r]^*}\kern-8pt s^{\dec w}R^{\inrec w} u^{\lambda w}q^{\tot w} ={\displaystyle{(us;q)_{r+1}\over (uR;q)_{r+1}} \over \displaystyle 1-\kern-6pt \sum\limits_{0\le l\le r} {(us;q)_{l}\over (u;q)_{l}}usq^l}.\leqno(6.5) $$ Now, use the $q$-telescoping argument provided by Krattenthaler~[12]: $$\displaylines{ {(us;q)_{l}\over (u;q)_{l}}usq^l ={sq\over 1-sq}\Bigl({(us;q)_{l+1}\over (u;q)_l} -{(us;q)_l\over (u;q)_{l-1}}\Bigr) \quad (1\le l\le r).\cr \noalign{\hbox{We obtain:}} (6.6)\quad \kern-10pt\sum_{w\in [0,r]^*}\kern-8pt s^{\dec w}R^{\inrec w} u^{\lambda w}q^{\tot w} ={1\over (uR;q)_{r+1}} {(1-sq)\,(u;q)_r\,(usq;q)_r \over ((u;q)_r-sq(usq;q)_r)}.\hfill\cr} $$ The summation can also be made over the symmetric groups by using the Gessel--Reutenauer standardization $w\mapsto (\sigma,c)$. This time only the following properties are needed: $$ \dec w=\exc\sigma;\quad \tot w=\maj\sigma+\tot c;\quad \inrec w=\fix\sigma. $$ Multiply (6.6) by $t^r$ and sum over $r\ge 0$. We get: $$\displaylines{\quad \sum_{r\ge 0}t^r\sum_{n\ge 0}\sum_{\scriptstyle\sigma\in {\goth S}_n\atop \scriptstyle \des\sigma\le r} \sum_{c\in \hbox{\sixrm NIW}_n(r-\des\sigma)} s^{\exc\sigma}R^{\fix\sigma}u^nq^{\maj\sigma+\tot c}\hfill\cr \hfill{}=\sum_{r\ge 0}t^r {1\over (uR;q)_{r+1}} {(1-sq)\,(u;q)_r\,(usq;q)_r \over ((u;q)_r-sq(usq;q)_r)}.\quad\cr} $$ Following the same pattern as in the proof of Theorem~1.6 we again derive identity (6.4).\cqfd {\bf Acknowledgements}. The authors should like to thank Christian Krattenthaler and Jiang Zeng for their careful readings and knowledgeable remarks. \vfill\eject \vglue 1cm \centerline{\bf References} \bigskip {\eightpoint \livre 1|Pierre Cartier, Dominique Foata|Probl\`emes combinatoires de commutation et r\'earrangements| Berlin, Springer-Verlag, 1969 (Lecture Notes in Math., 85), 88 pages. Also freely available on the {\sl S\'emin. Lothar. Combin.} website| \article 2|Dominique Foata, Guo-Niu Han|Signed words and permutations, I: A fundamental transformation|Proc. Amer. Math. Soc.|135|2007|31--40| \divers 3|------, ------|Signed words and permutations, II; The Euler--Mahonian polynomials, {\sl Electronic J. Combin.}, {\bf11(2)}, \#R22, {\oldstyle 2005}, 18 pages| \divers 4|------, ------|Signed words and permutations, III; The MacMahon Verfahren, {\sl S\'emin. Lothar. Combin.}, {\bf 54}, [B25a], {\oldstyle 2006}, 20 pages| \divers 5|------, ------|Signed words and permutations, IV; Fixed and pixed points, preprint 21 pages, {\sl Israel J. Math.}, {\oldstyle 2006}, (to appear)| \divers 6|------, ------|Signed words and permutations, V; A sextuple distribution, preprint 24 pages, {\sl Ramanujan J.}, {\oldstyle 2007}, (to appear)| \divers 7|------, ------|Fix-Mahonian Calculus, I: Two transformations, preprint 16 pages, {\oldstyle 2006}, {\sl Europ. J. Combin.} (to appear)| \divers 8|------, ------|Fix-Mahonian Calculus, II: Further statistics, preprint 13 pages, {\oldstyle 2006}, {\sl J. Combinatorial Theory, Ser.~A} (to appear)| \divers 9|------, ------|Fix-Mahonian Calculus, III: A quadruple distribution, preprint 26~p., {\oldstyle 2007}, {\sl Monatshefte f\"ur Math.} (to appear)| \article 10|Ira Gessel, Christophe Reutenauer|Counting permutations with given cycle structure and descent set| J. Combin. Theory Ser. A|64|1993|189--215| \article 11|Dongsu Kim, Jiang Zeng|A new decomposition of derangements|J. Combin. Theory Ser. A|96|2001|192--198| \divers 12|Christian Krattenthaler|Personal communication, {\oldstyle 2007}| \livre 13|M. Lothaire|Combinatorics on Words|Addison-Wesley, London {\oldstyle 1983} (Encyclopedia of Math. and its Appl., {\bf 17})| \livre 14|Percy Alexander MacMahon|Combinatory Analysis, {\rm vol.~1 and~2}|Cambridge, Cambridge Univ. Press, {\oldstyle 1915}. (Reprinted by Chelsea, New York, {\oldstyle 1955})| \divers 15|John Shareshian, Michelle Wachs|$q$-Eulerian polynomials, excedance number and major index, {\sl Electronic Research Announcements of the Amer. Math. Soc.}, vol.~{\bf13}, {\oldstyle 2007}, p.~33-45. See also the proceedings of the FPSAC 2007, Tianjin| } \bigskip\bigskip \hbox{\vtop{\halign{#\hfil\cr Dominique Foata \cr Institut Lothaire\cr 1, rue Murner\cr F-67000 Strasbourg, France\cr \noalign{\smallskip} {\tt foata@math.u-strasbg.fr}\cr}} \qquad \vtop{\halign{#\hfil\cr Guo-Niu Han\cr I.R.M.A. UMR 7501\cr Universit\'e Louis Pasteur et CNRS\cr 7, rue Ren\'e-Descartes\cr F-67084 Strasbourg, France\cr \noalign{\smallskip} {\tt guoniu@math.u-strasbg.fr}\cr}} } \bye % vim: fmr=<<<,>>> foldmethod=marker % vim: set sw=2: set ts=2: set cindent