<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">%----------------------------------------------------------------------------
% 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"
%----------------------------------------------------------------------------



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\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}  % &lt;&lt;&lt;

\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 $x_i&gt;x_{i+1}$. 
We say that~$i$ is a {\it decrease} of~$w$ 
if $1\le i\le n-1$ and $x_i= x_{i+1} = \cdots = x_j&gt;x_{j+1}$ 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 $x_i= x_{i+1}=
\cdots = x_j&lt;x_{j+1}$ for some~$j$ such that $i\le j\le n$
(respectively if $1\le i\le n$ and $x_i&lt;x_{i+1}$). By convention,
$x_{n+1}=+\infty$. The letter~$x_i$ is said
to be {\it an increase value} (respectively {\it a rise value})
of~$w$. Thus, the rightmost letter~$x_n$ is
always a rise value. Again, 
the set of all increases (respectively rises) is denoted
by $\INC(w)$ (respectively $\RISE(w)$).
Each rise is an increase, but not conversely.  This means that
$\RISE(w)\subset \INC(w)$.

Furthermore, a position $i$ $(1\le i \le n)$ is said to be a {\it record} 
if $x_j\le x_i$ for all~$j$
such that $1\le j\le i-1$. The letter $x_i$ is said to be a {\it record value}.
The set of all records of $w$ is denoted by $\REC(w)$.
% Let $\inrec w$ denote the {\it number} of letters of~$w$, 
% which are increase {\it and} record values. 

\smallskip

Introduce six sequences of commuting variables $(X_i)$, $(Y_i)$, 
$(Z_i)$, $(T_i)$, $(Y'_i)$, $(T'_i)$ $(i=0,1,2,\ldots\,)$, and for
each word $w=x_1x_2\ldots x_n$ from $[0,r]^*$ define the {\it
weight\/} $\psi(w)$ of
$w=x_1x_2\ldots x_n$ to be
$$
\displaylines{(1.1)\quad
\psi(w):=\prod_{i\in\hbox{\sixrm DES}}X_{x_i}
\prod_{i\in\hbox{\sixrm RISE}\setminus\hbox{\sixrm REC}}Y_{x_i}
\prod_{i\in\hbox{\sixrm DEC}\setminus\hbox{\sixrm
DES}}Z_{x_i}\hfill\cr 
\hfill{}\times
\prod_{i\in(\hbox{\sixrm
INC}\setminus\hbox{\sixrm RISE})\setminus\hbox{\sixrm
REC}}T_{x_i}
\prod_{i\in\hbox{\sixrm RISE}\cap \hbox{\sixrm REC}}Y'_{x_i}
\prod_{i\in(\hbox{\sixrm INC}\setminus\hbox{\sixrm
RISE})\cap\hbox{\sixrm REC}}T'_{x_i}, \cr
}
$$
where the argument ``$(w)$" has not been written for typographic
reasons. For example, $i\in \RISE\setminus\REC$ stands for
$i\in \RISE(w)\setminus\REC(w)$.

\medskip
{\it
Example}. For the word
$w=3\,2\,4\,4\,5\,5\,5\,3\,1\,1\,1\,4\,1\,3\,5$ the sets $\DES$,
$\DEC$, $\INC$, $\RISE$, $\REC$ of $w$ are indicated by bullets.
$$
{
\def\c{\bullet}
\matrix{
%i    &amp; = &amp; 1 &amp; 2 &amp; 3 &amp; 4  &amp; 5 &amp; 6 &amp; 7 &amp; 8 &amp; 9 &amp; 10 &amp; 11 &amp; 12 &amp; 13 &amp; 14 &amp; 15 \cr
w   &amp; = &amp; 3 &amp; 2 &amp; 4 &amp; 4  &amp; 5 &amp; 5 &amp; 5 &amp; 3 &amp; 1 &amp; 1 &amp; 1 &amp; 4 &amp; 1 &amp; 3 &amp; 5 \cr
\DES &amp; = &amp;\c &amp;   &amp;   &amp;    &amp;   &amp;   &amp;\c &amp;\c &amp;   &amp;   &amp;   &amp;\c &amp;   &amp;   &amp;   \cr
\DEC &amp; = &amp;\c &amp;   &amp;   &amp;    &amp;\c &amp;\c &amp;\c &amp;\c &amp;   &amp;   &amp;   &amp;\c &amp;   &amp;   &amp;   \cr
\RISE&amp; = &amp;   &amp;\c &amp;   &amp; \c &amp;   &amp;   &amp;   &amp;   &amp;   &amp;   &amp; \c&amp;   &amp;\c &amp; \c&amp; \c\cr
\INC &amp; = &amp;   &amp;\c &amp;\c &amp; \c &amp;   &amp;   &amp;   &amp;   &amp;\c &amp;\c &amp; \c&amp;   &amp;\c &amp; \c&amp; \c\cr
\REC &amp; = &amp;\c &amp;   &amp;\c &amp; \c &amp; \c&amp;\c &amp;\c &amp;   &amp;   &amp;   &amp;   &amp;   &amp;   &amp;   &amp; \c\cr
}
}
$$
We have
$ \psi(w)=X_3Y_2T'_4Y'_4Z_5Z_5X_5X_3T_1T_1 Y_1X_4Y_1Y_3Y'_5$.


\medskip
Now let $C$ be the $(r+1)\times (r+1)$ matrix
$$
C=\pmatrix{0&amp;\displaystyle{X_1\over 1-Z_1}
&amp;\displaystyle{X_2\over 1-Z_2}&amp;\cdots
&amp;\displaystyle{X_{r-1}\over
1-Z_{r-1}}&amp;\displaystyle{X_r\over 1-Z_r}\cr
\displaystyle{Y_0\over 1-T_0}
&amp;0&amp;\displaystyle{X_2\over
1-Z_2}&amp;\cdots&amp;\displaystyle{X_{r-1}\over
1-Z_{r-1}}&amp;\displaystyle{X_r\over 1-Z_r}\cr
\displaystyle{Y_0\over 1-T_0}
&amp;\displaystyle{Y_1\over
1-T_1}&amp;0&amp;\cdots&amp;\displaystyle{X_{r-1}\over
1-Z_{r-1}}&amp;\displaystyle{X_r\over 1-Z_r}\cr
\vdots&amp;\vdots&amp;\vdots&amp;\ddots&amp;\vdots&amp;\vdots\cr
\noalign{\medskip}
\displaystyle{Y_0\over 1-T_0}&amp;\displaystyle{Y_1\over
1-T_1}&amp;\displaystyle{Y_2\over 1-T_2}&amp;\cdots
&amp;0&amp;\displaystyle{X_r\over 1-Z_r}\cr
\displaystyle{Y_0\over 1-T_0}&amp;\displaystyle{Y_1\over
1-T_1}&amp;\displaystyle{Y_2\over 1-T_2}&amp;\cdots 
&amp;\displaystyle{Y_{r-1}\over 1-T_{r-1}}&amp;0\cr
}.
\leqno(1.2)
$$
\medskip

\proclaim Theorem 1.1.
The generating function for the set $[0,r]^*$ by the
weight~$\psi$ is given by
\vskip-12pt
$$
\sum_{w\in [0,r]^*} \psi(w)
={\prod\limits_{0\le j\le r}
\displaystyle\Bigl(1+{Y'_j\over
1-T'_j}\Bigr)\over \det (I-C)},\leqno(1.3)
$$
where $I$ is the identity matrix of order ($r+1)$.

Of course, the expression $1/\det(I-C)$ is too redolent of
the MacMahon Master Theorem [14, p.~97--98] for not having
it play a crucial role in the proof of (1.3). It
does indeed. However we further need the properties of the first
fundamental transformation, as it is developed in Cartier-Foata
[1] and also in Lothaire [13, Chap.~10]. As
mentioned earlier, Theorem~1.1 must be regarded as our Ur-result.
Its proof is given in Section~2. Its two equivalent forms come next.

\proclaim Theorem 1.2. We also have:
{\eightpoint
$$
\sum_{w\in [0,r]^*} \psi(w)
=
{\displaystyle
{\displaystyle\prod_{0\le j\le r}{1-Z_j\over 1-Z_j+X_j}
\over
\displaystyle\prod_{0\le j\le r}{1-T'_j
\over
1-T'_j+Y'_j}}
\over 
\displaystyle 1-\sum\limits_{0\le l\le r}
{\displaystyle\prod_{0\le j\le l-1}{1-Z_j\over 1-Z_j+X_j}
\over 
\displaystyle\prod_{0\le j\le l-1}{1-T_j\over
1-T_j+Y_j}}\displaystyle{X_l\over 1-Z_l+X_l}}.
\leqno\hbox{\tenrm (1.4)}
$$
}

By definition each letter equal to 0 cannot be a decrease
value. Consequently, the weight $\psi(w)$ of each
word~$w$ must not contain the variables~$X_0$, $Z_0$.
There is then another expression for the right-hand side
of (1.4) which does not involve the variables~$X_0$,
$Z_0$. To obtain it we factor out
$(1-Z_0)/(1-Z_0+X_0)$ from both numerator and denominator of
the right-hand side, as done in the next theorem.

\proclaim Theorem 1.3. We also have:
{\eightpoint
$$
\sum_{w\in [0,r]^*} \psi(w)
=
{\displaystyle
{\displaystyle\prod_{1\le j\le r}{1-Z_j\over 1-Z_j+X_j}
\over \displaystyle\prod_{0\le j\le r}{1-T'_j\over
1-T'_j+Y'_j}}
\over \displaystyle 1-\sum\limits_{1\le l\le r}
{\displaystyle\prod_{1\le j\le l-1}{1-Z_j\over 1-Z_j+X_j}
\over \displaystyle\prod_{0\le j\le l-1}{1-T_j\over
1-T_j+Y_j}}\displaystyle{X_l\over 1-Z_l+X_l}}.
\leqno\hbox{\tenrm (1.5)}
$$
}

Theorem~1.2 (and therefore Theorem~1.3) is proved in Section~3 by
using a determinantal manipulation and summing the weights
$\psi(w)$ according to their so-called {\it keys}. 
An alternate proof of Theorem~1.2, which is not reproduced
in this paper, is based on the word-analog of the Kim--Zeng transformation
[11] and follows the pattern developed in our previous paper [9].
Specializations
of those two theorems for deriving generating functions {\it on
words} only appear in Section~6. There is however a specialization
of (1.5) that deserves a special development and is now presented.

Let $\gamma$ be the homomorphism defined by the
following substitutions of variables:
$$
\gamma:=\{
X_j\leftarrow s\,Y_{j-1},\quad
Z_j\leftarrow s\,Y_{j-1},\quad
T_j\leftarrow Y_j,\quad
T'_j\leftarrow Y'_j
\}.
$$
For each word $w=x_1x_2\cdots x_n\in [0,r]^*$ we then have:
$$
\gamma\,\psi(w)=\prod_{x_i\in\hbox{\sixrm INC}\cap \hbox{\sixrm
REC}}Y'_{x_i}
\times \prod_{x_i\in \hbox{\sixrm DEC}}sY_{x_i-1}
\times \prod_{x_i\in \hbox{\sixrm INC}\setminus
\hbox{\sixrm REC}}Y_{x_i}.\leqno(1.6)
$$
Applying $\gamma$ to (1.5) we get:
$$
\sum_{w\in [0,r]^*} \gamma\,\psi(w)
={\displaystyle
{\prod\limits_{1\le j\le r}(1-sY_{j-1})
\over \prod\limits_{0\le j\le r}(1-Y'_j)}
\over\displaystyle 1-\sum\limits_{1\le l\le r}
{\prod\limits_{1\le j\le l-1}(1-sY_{j-1})
\over \prod\limits_{0\le j\le
l-1}(1-Y_j)} sY_{l-1}}.\leqno(1.7)
$$

The above right-hand side can be further simplified as stated in the following
theorem, whose proof is given in Section~4.

\proclaim Theorem 1.4. We have:
$$\displaylines{(1.8)\quad
\sum_{w\in [0,r]^*} \gamma\,\psi(w)\hfill\cr
\noalign{\vskip-10pt}
\hfill{}={(1-s)\prod\limits_{0\le j\le r-1}(1-Y_j)
\prod\limits_{0\le j\le r-1}(1-s\,Y_j)\over
\prod\limits_{0\le j\le r}(1-Y'_j)\,
\Bigl(\,\prod\limits_{0\le j\le r-1}(1-Y_j)
-s  \prod\limits_{0\le j\le r-1}(1-s\,Y_j)\Bigr)}.\quad\cr}
$$

For each word $w=x_1x_2\cdots x_n$ let $\inrec w$ denote the
{\it number} of letters of~$w$, which are increase {\it and}
record values. Also let $\dec w$ be the number of decreases in
$w$  and
$\tot w=x_1+x_2+\cdots +x_n$ be the {\it sum} of the
letters of~$w$. Also make the substitution $Y_j'\leftarrow RY_j$,
where~$R$ is a new variable and let
$\psi_R:=\gamma\,\psi\mid_{Y'_j\leftarrow R\,Y_j}$, so that (1.6)
becomes:
$$
\psi_R(w)=R^{\inrec w}s^{\dec w}
\prod_{x_i\in \hbox{\sixrm DEC}}Y_{x_i-1}
\times \prod_{x_i\in \hbox{\sixrm INC}}Y_{x_i}.\leqno(1.9)
$$
On the other hand, let 
$$
\leqalignno{H(Y)&amp;:=\prod_{i\ge 0}(1-Y_i)^{-1};&amp;(1.10)\cr
H_r(Y)&amp;:=\prod_{0\le i\le r-1}(1-Y_i)^{-1}
\quad (r\ge 0).&amp;(1.11)\cr}
$$
Using the homomorphism $\psi_R$, identity (1.8) may be rewritten
as:
$$
\sum_{w\in [0,r]^*}\psi_R(w)={(1-s)H_{r+1}(RY)\over
H_r(sY)-sH_r(Y)}.\leqno(1.12)
$$

The left-hand side of (1.12) can be further expressed as a series
over the symmetric groups ${\goth S}_n$ $(n=0,1,\ldots\,)$. If
$\sigma=\sigma(1)\sigma(2)\cdots \sigma(n)$ is a permutation of
$12\cdots n$, let $z=z_1z_2\cdots z_n$ be the word defined by:
$$
z_i:=\#\{j:i\le j\le n-1,\, \sigma(j)&gt;\sigma(j+1)\}\quad
(1\le i\le n),\leqno(1.13)
$$
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 $\exc\sigma:=\{i:1\le i\le
n-1,\sigma(i)&gt;i\}$ 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
$$
Y_{(\sigma,c)}:=\prod_{j&lt;\sigma(j)}Y_{c_j+z_j-1}
\times \prod_{j\ge \sigma(j)}Y_{c_j+z_j}.\leqno(1.14)
$$

\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
% &gt;&gt;&gt;
\centerline{\bf 2. Proof of Theorem 1.1} % &lt;&lt;&lt;

\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
$$
\eqalignno{
\sum_{w\in [0,r]^*} \psi(w)
&amp;= \sum_{k\in [0,r]_h^*} \sum_{\key(w)=k}\psi(w) 
\ = \sum_{k\in [0,r]_h^*} \beta\,\psi(k) \cr
&amp; = \beta \biggl(\sum_{k\in [0,r]_h^*} \psi(k)\biggr)
\ = \ \beta\,
{\prod\limits_{0\le j\le r}
\displaystyle\Bigl(1+{Y'_j
}\Bigr)\over \det (I-\alpha\,C)}&amp;\hbox{[by (2.1)]}\cr
&amp;={\prod\limits_{0\le j\le r}
\displaystyle\Bigl(1+{\beta\,Y'_j
}\Bigr)\over \det (I-\beta\,\alpha\,C)}
={\prod\limits_{0\le j\le r}
\displaystyle\Bigl(1+{Y'_j\over
1-T'_j}\Bigr)\over \det (I-C)}.\qed\cr
}
$$

\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
$\petitematrice{\delta w_1&amp;\delta w_2&amp;\delta w_3\cr
w_1&amp;w_2&amp;w_3\cr}=\petitematrice{3&amp;5&amp;5&amp;3&amp;2&amp;4&amp;6&amp;1&amp;2&amp;4&amp;3&amp;1&amp;6\cr
5&amp;3&amp;6&amp;5&amp;3&amp;2&amp;4&amp;6&amp;1&amp;2&amp;4&amp;3&amp;1\cr}$. 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
${\bf F}_1(k_0):=\petitematrice{\overline u\cr u\cr}
=\petitematrice{1&amp;1&amp;2&amp;2&amp;3&amp;3&amp;3&amp;4&amp;4&amp;5&amp;5&amp;6&amp;6\cr
6&amp;3&amp;3&amp;1&amp;5&amp;5&amp;4&amp;2&amp;2&amp;3&amp;6&amp;4&amp;1\cr}$.

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: 
$$
k\mapsto (w_0, k_0) \mapsto (w_0, u)
\quad\hbox{where}\quad {\bf F}_1(k_0) = {\overline u\choose u}.
$$
$$
\leqalignno{\qquad\qquad\qquad
k={\bf 2}53{\bf 5}65324612431{\bf 6} 
&amp;\mapsto (w_0={\bf 256}, k_0=5365324612431)&amp;\hbox{\it
\quad Example.}
\cr &amp;\mapsto (w_0={\bf 256}, u=6331554223641), \cr
{\bf F}_1(k_0)&amp;=\petitematrice{\overline u\cr u\cr}
=\petitematrice{1&amp;1&amp;2&amp;2&amp;3&amp;3&amp;3&amp;4&amp;4&amp;5&amp;5&amp;6&amp;6\cr
6&amp;3&amp;3&amp;1&amp;5&amp;5&amp;4&amp;2&amp;2&amp;3&amp;6&amp;4&amp;1&amp;\cr}.&amp;\hbox{since}
\cr
}
$$

Let $\Phi$ be the homomorphism generated by 
$$
\Phi{i\choose j}:=\cases{{X_j},&amp;if
$j&gt;i$;\cr
\noalign{\smallskip}
{Y_j},&amp;if
$j&lt;i$.\cr
}
$$
By the property [13, Chap.~10] of the first fundamental transformation, we have
$$
\psi(k)=\psi(w_0) \psi(k_0)
=\psi(w_0) \Phi {\overline u\choose u}.\leqno(2.4)
$$


Let $\ND(r)$ be the
set of all {\it non-decreasing} words from $[0,r]^*$. It follows
from the properties of the above bijections that
$$\leqalignno{
\sum_{w\in [0,r]_h^*} \psi(w)  
&amp;=
\sum_{w_0\in \ND(r)} \psi(w_0) 
\sum_{u\in [0,r]_v^*} \Phi {\overline u\choose u};\cr
\sum_{w_0\in \ND(r)} \psi(w_0) &amp;= \prod_{0\le j\le r} (1+Y'_j).\cr
}
$$

There remains to prove the identity:
$$
\sum_{u\in [0,r]_v^*} \Phi{\overline u\choose u}
=
{1 \over \det (I-\alpha\,C)}.\leqno(2.5)
$$

The proof is based on the celebrated MacMahon
Master Theorem, using the noncommutative version developed
in [1, Chap.~4]. Also see [13, Chap.~10]. Consider
the matrix
$$
C''=\pmatrix{0&amp;{0\choose 1}
&amp;{0\choose 2}&amp;\cdots
&amp;{0\choose{r-1}}&amp;{0\choose r}\cr
\noalign{\smallskip}
{1\choose 0}&amp;{0}
&amp;{1\choose 2}&amp;\cdots
&amp;{1\choose{r-1}}&amp;{1\choose r}\cr
\noalign{\smallskip}
{2\choose 0}&amp;{2\choose 1}
&amp;{0}&amp;\cdots
&amp;{2\choose{r-1}}&amp;{2\choose r}\cr
\noalign{\smallskip}
\vdots&amp;\vdots&amp;\vdots&amp;\ddots&amp;\vdots&amp;\vdots\cr
\noalign{\medskip}
{r-1\choose 0}&amp;{r-1\choose 1}&amp;{r-1\choose 2}&amp;\cdots
&amp;0&amp;{r-1\choose r}\cr
\noalign{\smallskip}
{r\choose 0}&amp;{r\choose 1}&amp;{r\choose 2}&amp;\cdots 
&amp;{r\choose {r-1}}&amp;0\cr
}.
$$

As shown in the references just mentioned,
there holds the identity
$$
{1\over \det(I-C'')}
=\sum_{u\in [0,r]_v^*} {\overline u\choose u}.\leqno(2.6)
$$
Applying $\Phi$ to both sides of (2.6) yields (2.5).\qed

\bigskip

% &gt;&gt;&gt;
\centerline{\bf 3. Proof of Theorem 1.2} % &lt;&lt;&lt;

\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
$$ D =
{
\left|\matrix{\noalign{\medskip}
 1&amp;-{ X_1} &amp;-{ X_2}&amp;\cdots &amp;-{ X_{r-1}} &amp;- { X_r}\cr
-Y_0&amp;1&amp;-{ X_2}&amp;\cdots &amp;-{ X_{r-1}}&amp;-{ X_r}\cr 
-Y_0&amp;-Y_1&amp; 1&amp;\cdots &amp;-{ X_{r-1}}&amp;-{ X_r}\cr
\vdots&amp;\vdots&amp;\vdots&amp;\ddots&amp;\vdots&amp;\vdots\cr
\noalign{\medskip}
-Y_0&amp; -Y_1&amp;-Y_2 &amp;\cdots&amp; 1&amp;-{ X_r}\cr 
-Y_0&amp;-Y_1 &amp;-Y_2&amp;\cdots &amp;-Y_{r-1} &amp; 1\cr
}\right|}.
$$
}



\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
$$D={
\left|\matrix{\noalign{\medskip}
 1&amp;-{ X_1}
&amp;-{ X_2}&amp;\kern-.3cm\cdots
&amp;\kern-.3cm-{ X_{r-1}} &amp;- { X_r}\cr
-1-Y_0&amp;{1+X_1
}&amp;{ 0}&amp;\cdots
&amp;{0}&amp;{ 0}\cr
 0&amp; -1-Y_1&amp;
{1+X_2
}&amp;\cdots
&amp; 0&amp; 0\cr
\vdots&amp;\vdots&amp;\vdots&amp;\ddots&amp;\vdots&amp;\vdots\cr
\noalign{\medskip}
0&amp; 0&amp;0
&amp;\cdots&amp; 1+X_{r-1}&amp;
0\cr 
0&amp;0
&amp;0&amp;\cdots
&amp;-1-Y_{r-1}
&amp;{1+X_r }\cr
}\right|}.
$$
}

\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:
$$\leqalignno{
D&amp;=\prod_{1\le j\le r}{(1+X_j)
}
+X_0\prod_{1\le j\le
r}{(1+X_j)}\cr
&amp;\kern2cm{}-
\sum_{0\le l\le r}\Bigl(\prod_{0\le j\le
l-1}(1+Y_j)
\prod_{l\le j\le r}{(1+X_j)
}\Bigr){X_l\over 1+X_l}.\cr
}
$$
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
% &gt;&gt;&gt;
% \vfill\eject
\centerline{\bf 4. Proof of Theorem 1.4} % &lt;&lt;&lt;

\medskip
First, we may check that
{\eightpoint
$$
\leqalignno{
&amp;\prod\limits_{0\leq j\leq r}(1-sY_j) -
\prod\limits_{0\leq j\leq r}(1-Y_j)\cr
=&amp;\sum_{0\leq l \leq r} \prod\limits_{0\leq j\leq l}(1-sY_j)
\prod\limits_{l+1\leq j\leq r} (1-Y_j)
-
\sum_{0\leq l \leq r} \prod\limits_{0\leq j\leq l-1}(1-sY_j)
\prod\limits_{l\leq j\leq r} (1-Y_j)\cr
=&amp; (1-s)\sum_{0\leq l \leq r} Y_l \prod\limits_{0\leq j\leq l-1}(1-sY_j)
\prod\limits_{l+1\leq j\leq r} (1-Y_j).\cr
}
$$
}

\goodbreak

{\it Proof of Theorem 1.4}.\quad Using (1.7) we have:
{%\eightpoint
$$
\leqalignno{
&amp;
{
1
\over
1-s\sum\limits_{0\leq l\leq r-1} Y_l 
{\prod\limits_{0\leq j\leq l-1}(1-sY_j)/\prod\limits_{0\leq j\leq l}(1-Y_j)}
}\cr
&amp;=
{
\prod\limits_{0\leq j \leq r-1} (1-Y_j)
\over
\prod\limits_{0\leq j\leq r-1} (1-Y_j)
-s\sum\limits_{0\leq l\leq r-1} Y_l 
{\prod\limits_{0\leq j\leq l-1}(1-sY_j)\prod\limits_{l+1\leq j\leq r-1}(1-Y_j)}
}\cr
&amp;=
{
\prod\limits_{0\leq j \leq r-1} (1-Y_j)
\over
\prod\limits_{0\leq j\leq r-1} (1-Y_j)
-s
{
\bigl(\prod\limits_{0\leq j\leq r-1}(1-sY_j) -
\prod\limits_{0\leq j\leq r-1}(1-Y_j)\bigr)/
( 1-s )
}
} \cr
&amp;=
{
(1-s) \prod\limits_{0\leq j \leq r-1} (1-Y_j)
\over
\prod\limits_{0\leq j\leq r-1} (1-Y_j)
-s \prod\limits_{0\leq j\leq r-1}(1-sY_j) 
}.\qed \cr
}
$$
}
\bigskip

% &gt;&gt;&gt;
\centerline{\bf 5. Proofs of Theorems 1.5 and 1.6} % &lt;&lt;&lt;

\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
$$\matrice{
{\rm Id}&amp;=&amp;1&amp;2&amp;3&amp;4&amp;5&amp;6&amp;7&amp;8&amp;9&amp;10&amp;11&amp;12&amp;13
&amp;14&amp;15&amp;16&amp;17&amp;18&amp;19&amp;20&amp;21&amp;22\cr
\rightarrow\sigma&amp;=&amp;\bf1&amp;\underline
5&amp;\underline6&amp;\underline8&amp;\underline{13}&amp;\underline{14}&amp;
\bf7&amp;\underline{17}&amp;4&amp;\bf10&amp;\underline{15}&amp;\underline{18}&amp;
\underline{19}&amp;2&amp;9&amp;\bf16&amp;\underline{20}&amp;\underline{22}
&amp;3&amp;11&amp;12&amp;21\cr
\quad z&amp;=&amp;4&amp;4&amp;4&amp;4&amp;4&amp;4&amp;3&amp;3&amp;2&amp;2&amp;2&amp;2&amp;2&amp;1&amp;1&amp;1&amp;1&amp;1&amp;0&amp;0&amp;0&amp;0\cr
\rightarrow
c&amp;=&amp;2&amp;2&amp;2&amp;2&amp;2&amp;2&amp;2&amp;1&amp;1&amp;1&amp;1&amp;1&amp;1&amp;1&amp;1&amp;1&amp;1&amp;1&amp;1&amp;1&amp;1&amp;1\cr
\quad\overline
c&amp;=&amp;6&amp;6&amp;6&amp;6&amp;6&amp;6&amp;5&amp;4&amp;3&amp;3&amp;3&amp;3&amp;3&amp;2&amp;2&amp;2&amp;2&amp;2&amp;1&amp;1&amp;1&amp;1\cr
\quad \sigma&amp;=&amp;
(\bf16)&amp;(12&amp;18&amp;22&amp;21)&amp;(\bf 10)&amp;(\bf 7)&amp;
(4&amp;8&amp;17&amp;20&amp;11&amp;15&amp;9)&amp;(2&amp;5&amp;13&amp;19&amp;3&amp;6&amp;14)&amp;(\bf1)\cr
\quad \check\sigma&amp;=&amp;
\bf16&amp;12&amp;18&amp;22&amp;21&amp;\bf 10&amp;\bf 7&amp;
4&amp;8&amp;17&amp;20&amp;11&amp;15&amp;9&amp;2&amp;5&amp;13&amp;19&amp;3&amp;6&amp;14&amp;\bf1\cr
\mapsto w&amp;=&amp;
2\ \mid&amp;\underline 3&amp;\underline 2&amp;1&amp;1\mid&amp; 3\ \mid&amp; 5\mid&amp;
\underline 6&amp;\underline 4&amp;\underline 2&amp;1&amp;\underline 3&amp;2&amp;3\mid
&amp;\underline 6&amp;\underline 6&amp;\underline 3&amp;1&amp;\underline
6&amp;\underline 6&amp;2\mid &amp;6\cr}
$$

}

\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}
$\sigma(i)&gt;i$ 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 
$\check\sigma(i)&lt;\check\sigma(i+1)$, if and only if
$\check\sigma(i)&lt;\sigma(\check\sigma(i))$. 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,
$$\leqalignno{
\psi_R(w)&amp;=
R^{\inrec w}s^{\dec w}\kern-6pt
\prod\limits_{x_i\in \hbox{\sixrm DEC}}Y_{x_i-1}
\prod\limits_{x_i\in \hbox{\sixrm INC}} Y_{x_i}\cr
&amp;=R^{\fix \sigma}s^{\exc\sigma}\kern-6pt
\prod_{\check\sigma(i)&lt;\sigma(\check\sigma(i))}
Y_{\overline c_{\check\sigma(i)} -1}
\prod_{\check\sigma(i)\ge\sigma(\check\sigma(i))}
Y_{\overline c_{\check\sigma(i)}}\cr
&amp;=R^{\fix \sigma}s^{\exc\sigma}
\prod_{j&lt;\sigma(j)}
Y_{\overline c_{j} -1}
\prod_{j\ge\sigma(j)}
Y_{\overline c_{j}}\cr
&amp;=R^{\fix \sigma}s^{\exc\sigma}
\prod_{j&lt;\sigma(j)}Y_{c_j+z_j-1}
\prod_{j\ge \sigma(j)}Y_{c_j+z_j}\cr
&amp;=R^{\fix\sigma}s^{\exc\sigma}Y_{(\sigma,c)}.\cr}
$$

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
$$
(a;q)_n=\cases{1,&amp;if $n=0$;\cr
(1-a)(1-aq)\cdots (1-aq^{n-1}),&amp;if $n\ge1$.\cr}
$$

\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
$$\displaylines{\eqalign{
\phi {(1-s)H_{r+1}(RY)\over
H_r(sY)-sH_r(Y)}&amp;={(1-sq)/(uR;q)_{r+1}\over
1/(usq;q)_r-sq/(u;q)_r}\cr
&amp;={(1-sq)(u;q)_r\,(usq;q)_r
\over (uR;q)_{r+1}\,(\bigl((u;q)_r-sq(usq;q)_r\bigr)}.\cr}\cr
\noalign{\hbox{Then, for $(\sigma,c)\in {\goth S}_n\times
\NIW_n$}}
\eqalign{
\phi Y_{(\sigma,c)}&amp;=u^nq^{\tot c+\tot z-\exc\sigma}
=q^{\maj\sigma-\exc\sigma}u^nq^{\tot c};\cr
\phi Y(\sigma;t)&amp;=q^{\maj\sigma-\exc\sigma}
u^n\sum_{j\ge 0}
t^j\kern-5pt
\sum_{c\in \hbox{\sixrm
NIW}_n(j)}\kern-10pt q^{\tot
c}=q^{\maj\sigma-\exc\sigma}{u^n\over (t;q)_{n+1}};\cr }\cr
\noalign{\hbox{so that}}
\phi\bigl( R\fix^{\sigma}s^{\exc\sigma}t^{\des\sigma}Y(\sigma;t)
\bigr)=R^{\fix\sigma}(sq)^{\exc\sigma}t^{\des\sigma}
q^{\maj\sigma-\exc\sigma}{u^n\over (t;q)_{n+1}}.\cr}
$$
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




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