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\titrecourant={EULERIAN  CALCULUS, 4}
\auteurcourant={ROBERT J. CLARKE AND DOMINIQUE FOATA}

{\eightpoint
\noindent
S\'eminaire Lotharingien de Combinatoire, B32b, 1994, 12 pp.

}

\vglue 1.5cm
\vskip -12pt 
\centerline{\bf  EULERIAN CALCULUS, IV : SPECIALIZATIONS}  
\vskip 2.8mm 
\centerline{\sevenrm BY}
\vskip 2.8mm
\centerline{{\petcap Robert} J. CLARKE
{\sevenrm AND} {\petcap Dominique} FOATA
\footnote{$(^*)$}{Supported in  part by the E.E.C
Programme on Algebraic Combinatorics, 1994-95.}} 

\vskip 1cm 
\abstract{Further excedance and descent
statistics can be defined on each rearragement
class and their generating functions explicitly calculated.
Those generating functions coincide with the classical ones
when the rearrangement class is a permutation group, but
differ as soon as the class involves repeated elements.} 

\section 1. Introduction|In our previous three papers on
Eulerian Calculus [ClFo94, ClFo95a, ClFo95b] we have
investigated further constructions of transformations on the
symmetric group and related structures, and derived explicit
formulas for classical order statistics on those structures.
Let $(a;q)_n$ denote the $q$-ascending factorial 
$$
(a;q)_n=\cases{1,&if $n=0$;\cr
(1-a)(1-aq)\ldots (1-aq^{n-1}),&if $n\ge 1$;\cr}
$$
Also let $(u_1,\ldots,u_j)$, $(v_1,\ldots, v_k)$ be two
sequences of commuting variables and
${\bf c}=(c_1,\ldots,c_j)$ and
${\bf d}=(d_1,\ldots,d_k)$ be two vectors with non-negative
integer components. Write $c=c_1+\cdots+c_j$,
$d=d_1+\cdots+d_k$, then $\bfu^\bfc$ for  $u_1^{c_1}\ldots
u_j^{c_j}$ and $\bfv^\bfd$ for $v_1^{d_1}\ldots v_k^{d_k}$,
finally $(\bfu;q)_{s+1}$ and $(-q\bfv;q)_s$ for
the two products 
$$
(u_1;q)_{s+1}\ldots (u_j;q)_{s+1}\qquad
{\rm and}\qquad(-qv_1;q)_s\ldots (-qv_k;q)_s,
$$
respectively. In our third paper [ClFo95b] we have considered
the identities of the form
$$
\leqalignno{\sum_{\bfc,\bfd}
{\bfu^{\bfc}\bfv^{\bfd}\over (t;q)_{1+c+d}}
A_{\bfc,\bfd}(t,q)
&=\sum_{s\ge 0}t^s {(-q\bfv;q)_s
\over (\bfu;q)_{s+1}} ,&(1.1)\cr
\sum_{\bfc,\bfd}
{\bfu^\bfc \bfv^\bfd\over (t;q)_{1+c+d}} 
A^I_{\bfc,\bfd}(t,q)
&=\sum_{s\ge 0}t^s{1\over
(\bfu;q)_{s+1}(q\bfv;q)_s},&(1.2)\cr
\sum_{\bfc,\bfd}
{\bfu^\bfc \bfv^\bfd\over (t;q)_{1+c+d}} 
A^{II}_{\bfc,\bfd}(t,q)
&=\sum_{s\ge 0}t^s
(-\bfu;q)_{s+1}(-q\bfv;q)_s,&(1.3)\cr
\sum_{\bfc,\bfd}
{\bfu^\bfc \bfv^\bfd\over (t;q)_{1+c+d}} 
A^{III}_{\bfc,\bfd}(t,q)
&=\sum_{s\ge 0}t^s
{(-\bfu;q)_{s+1}\over
(q\bfv;q)_s},&(1.4)
}
$$
and showed that the coefficients 
$A_{\bfc,\bfd}(t,q)$, $A^I_{\bfc,\bfd}(t,q)$,
$A^{II}_{\bfc,\bfd}(t,q)$ and $A^{III}_{\bfc,\bfd}(t,q)$ were
generating {\it polynomials} for words that are rearrangements of
the word $v=y_1y_2\ldots y_m=1^{c_1}\ldots
j^{c_j}(j+1)^{d_1}\ldots r^{d_k}$ by pairs of the
following statistics. 

Call $R(\bfc,\bfd)$ the class of those words and let
$w=x_1x_2\ldots x_m$ be a word in this class.  We say that the
word~$w$ has a $k$-{\it excedance} at~$i$ $(1\le i\le m)$, if
either $x_i>y_i$, or  $x_i=y_i$  and $x_i$ large. We also say
that $w$ has a $k$-{\it  descent} at~$i$ $(1\le i\le m)$, if
either $x>x_{i+1}$, or $x_i=x_{i+1}$ and $x_i$ large. [By
convention, $x_{m+1}=\star$, where $\star$ is an extra letter
which is bigger that the {\it small} letters $1,2,\ldots,j$, but
smaller than the {\it large} letters $(j+1),\ldots, r$.] The
numbers of $k$-excedances and $k$-descents of a word~$w$ are
denoted by $\exc_k w$ and  $\des_k w$. The 
$k$-{\it major index} of a word~$w$ is also defined to be the
{\it sum}, $\maj_k w$, of all~$i$ such that~$i$ is a 
$k$-descent in~$w$.
 
\medskip
The other three pairs of statistics are defined as follows :

\medskip
\noindent
$(1.5)_I$ the word $w=x_1x_2\ldots x_m$ has a
$k$-{\it descent of type}~$I$ at~$i$ 
$(1\le i\le m)$, if either $1\le i\le m-1$ and
$x_i>x_{i+1}$, or $i=m$ and $x_m$ is large. Thus in
case~$(I)$ only {\it strict} descents are counted
within the word together with a descent at the end if
the last letter is large. The number of
$k$-{\it descents of type}~$I$ in~$w$ and the sum of
all~$i$ such that~$i$ is a
$k$-{\it descent of type}~$I$ are respectively
denoted by $\des_k^I w$ and $\maj_k^I w$.

\medskip
\noindent
$(1.5)_{II}$ the word $w=x_1x_2\ldots x_m$ has a
$k$-{\it descent of type}~$II$ at~$i$ 
$(1\le i\le m)$, if either $1\le i\le m-1$ and
$x_i\ge x_{i+1}$, or $i=m$ and $x_m$ is large. Thus in
case~$(II)$ only usual descents and equalities
$x_i=x_{i+1}$ are counted within the word and one
descent at the end if the last letter is large. In the
same manner, we define $\des_k^{II} w$ and
$\maj_k^{II}$.

\medskip
\noindent
$(1.5)_{III}$ the word $w=x_1x_2\ldots x_m$ has a
$k$-{\it descent of type}~$III$ at~$i$ 
$(1\le i\le m)$, if one of the two conditions (1),
(2) holds: (1) $1\le i\le m-1$, $x_i>x_{i+1}$ and
$x_i$ large, or $x_i\ge x_{i+1}$ and $x_i$ small;
(2) $i=m$ and $x_m$ is large. Thus in
case~$(III)$  {\it strict} descents are taken into
account together with equalities $x_i=x_{i+1}$
when $x_i$ is small, with a descent at the end if the
last letter is large. In the same manner, we define
$\des_k^{III}$ and $\maj_k^{III}$.

One of the results of our third paper was to prove the following
theorem.

\th Theorem 1.1|The coefficients $A_{\bfc,\bfd}(t,q)$
occurring in identity $(1.1)$ are the following generating
polynomials
$$
A_{\bfc,\bfd}(t,q)=\sum_w t^{\des_k w}
q^{\maj_k w}\qquad (w\in R(\bfc,\bfd)),
$$
with analogous results for
$A_{\bfc,\bfd}^{I}(t,q)$, $A_{\bfc,\bfd}^{II}(t,q)$  and
$A_{\bfc,\bfd}^{III}(t,q)$ concerning identities $(1.2)$, $(1.3)$
and $(1.4)$, respectively.
\finth

We proved Theorem 1.1 in three different ways : first, by
finding {\it recurrence relations} for the generating
polynomials $A_{\bfc,\bfd}(t,q)$ that imply a system of
$q$-{\it difference equations} for their factorial generating
functions, a system that can be intregrated to yield (1.1) --
(1.4) ; second, by using a standard rearrangement technique
for biwords that goes back to MacMahon, the celebrated {\it
MacMahon Verfahren} ; third, by using the symmetric function
technique and especially the {\it Cauchy identity for Schur
functions}.

Now if we let $q=1$ in identities (1.1) -- (1.4), we obtain
$$
\leqalignno{\sum_{\bfc,\bfd}
{\bfu^{\bfc}\bfv^{\bfd}\over (1-t)^{1+c+d}}
A_{\bfc,\bfd}(t)
&=\sum_{s\ge 0}t^s {(1+\bfv)^s
\over (1-\bfu)^{s+1}} ,&(1.6)\cr
\sum_{\bfc,\bfd}
{\bfu^\bfc \bfv^\bfd\over (1-t)^{1+c+d}} 
A^I_{\bfc,\bfd}(t)
&=\sum_{s\ge 0}t^s{1\over
(1-\bfu)^{s+1}(1-\bfv)^s},&(1.7)\cr
\sum_{\bfc,\bfd}
{\bfu^\bfc \bfv^\bfd\over (1-t)^{1+c+d}} 
A^{II}_{\bfc,\bfd}(t)
&=\sum_{s\ge 0}t^s
(1+\bfu)^{s+1}(1+\bfv)^s,&(1.8)\cr
\sum_{\bfc,\bfd}
{\bfu^\bfc \bfv^\bfd\over (1-t)^{1+c+d}} 
A^{III}_{\bfc,\bfd}(t)
&=\sum_{s\ge 0}t^s
{(1+\bfu)^{s+1}\over
(1-\bfv)^s},&(1.9)
}
$$
where $A_{\bfc,\bfd}(t)$ stands for
$A_{\bfc,\bfd}(t,q)\bigm|_{q=1}$ with analogous expressions for
the other polynomials and where $(1+\bfv)^s$ stands for
$(1+v_1)^s\ldots (1+v_k)^s$ with an analagous expression for
$(1-\bfv)^{s+1}$.

The purpose of this paper is to reprove identities (1.6) --
(1.9) by using a fourth technique based on the MacMahon Master
Theorem. As there is no $q$-analogue (so far ?) of this theorem,
such a technique is not available for proving their
$q$-versions (1.1) -- (1.4).

We conclude the paper by showing the the polynomials
$A_{\bfc,\bfd}(t)$, $A^I_{\bfc,\bfd}(t)$ and
$A^{II}_{\bfc,\bfd}(t)$ also have combinatorial interpretations
in terms of {\it excedances}.

\section 2. Determinantal expressions|For each $j$ and each~$k$
such that $(0\le j,k\le r)$ and $j+k=r$ denote by $B_k$ and
$B'_j$ the following two $r\times r$ matrices. The matrix $B_k$
has only 1's under the diagonal and only $t$'s above, but its
diagonal, ${\rm diag}\, B_k$, is made of $j$~1's, followed by
$k$~$t$'s, i.e., 
$$
{\rm diag}\,B_k=\vtop{\halign{\hfil#\hfil
&\hfil#\hfil\cr
$(1,\ldots,1,$&$t,\ldots,t)$\cr
\omit\upbracefill&\omit\upbracefill\cr
$j$ times&$k$ times\cr}}\leqno(2.1)
$$
In the matrix $B'_j$ the diagonal entries
consists of $j$~$t$'s followed by~$k$~1's, i.e.,
$$
{\rm diag}\,B'_j=\vtop{\halign{\hfil#\hfil
&\hfil#\hfil\cr
$(t,\ldots,t,$&$1,\ldots,1)$\cr
\omit\upbracefill&\omit\upbracefill\cr
$j$ times&$k$ times\cr}}\leqno(2.2)
$$
and the other entries consist of~$t$'s above the diagonal
and~1's under~it. Notice that $B_r=B'_r$ and $B_0=B'_0$, a
matrix that has $t$'s above the diagonal and 1's on and under
the diagonal.

Let the infinite series occurring in the right-hand sides of
(1.6), (1.7), (1.8) and (1.9) be denoted by $A(\bfu,\bfv,t)$,
 $A^I(\bfu,\bfv,t)$, $A^{II}(\bfu,\bfv,t)$ and
$A^{III}(\bfu,\bfv,t)$, respectively.
Furthermore, let $U$ be the diagonal matrix 
$U={\rm diag}(u_1,\ldots, u_j,v_1,\ldots,v_k)$.
We first prove the identities:
$$\leqalignno{
(1-t)\,A(\bfu(1-t),\bfv(1-t),t)&={1\over
\det(I_r-B_kU)};&(2.3)\cr
(1-t)\,A^I(\bfu(1-t),\bfv(1-t),t)&={(1-\bfv(1-t))\over
\det(I_r-B_0U)};&(2.4)\cr
(1-t)\,A^{II}(\bfu(1-t),\bfv(1-t),t)&={(1+\bfu(1-t))
\over \det(I_r-B_rU)};&(2.5)\cr
\qquad (1-t)\,A^{III}(\bfu(1-t),\bfv(1-t),t)
&={(1+\bfu(1-t))(1-\bfv(1-t))
\over \det (I_r-B'_j U)}.&(2.6)\cr}
$$

Identity (2.3) has been derived in
[ClFo94]. The same technique applies for the other three
identities. Let us simply prove (2.4).
First, it is routine to derive
$$\displaylines{\quad
\det(I_r-B_0U)\hfill\cr
\hfill{}=1-e_1(\bfu,\bfv)
+(1-t)e_2(\bfu,\bfv)-\cdots+(-1)^{r}(1-t)^{r-1}
e_r(\bfu,\bfv),\quad\cr}
$$
where the $e_i(\bfu,\bfv)$'s are the elementary symmetric
function in the variables $u_1,\ldots,u_j,v_1,\ldots,v_k$. Hence
$$
\displaylines{\quad
(1-t)\det(I_r-B_0U)\hfill\cr 
\hfill\eqalign{&=(1-t)-e_1(\bfu(1-t),\bfv(1-t))
+e_2(\bfu(1-t),\bfv(1-t))\cr
&\qquad\qquad\qquad{}-\cdots+(-1)^{r}
e_r(\bfu(1-t),\bfv(1-t)).\cr
&=-t+(1-\bfu(1-t))(1-\bfv(1-t)),\cr
}\qquad\cr
\noalign{\hbox{and then}}
\eqalign{{(1-\bfv(1-t))\over
\det(I_r-B_0U)}
&=(1-t)\sum_{s\ge 0}
t^s{1\over (1-\bfu(1-t))^{s+1}
(1-\bfv(1-t))^s}\cr
&=(1-t)\,A^I(\bfu(1-t),\bfv(1-t),t).\cr}\cr
}
$$
As we will make full use of the MacMahon Master Theorem 
[Mac15, p.~97] to exploit identities (2.3) - (2.6), it is
appropriate to restate this theorem now :

\medskip
{\it Let $B=(b(i,j))$ $(1\le i,j \le r)$ be a square matrix
with coefficients in a commutative ring and let $X^*$
be the free monoid generated by $X=[\,r\,]$. If 
$w=x_1x_2\ldots x_m$ is a word in~$X^*$ whose non-decreasing
rearrangement is $v=y_1y_2\ldots y_m=1^{c_1}\ldots
j^{c_j}(j+1)^{d_1}\ldots r^{d_k}$, define 
$$
\beta(w)=b(y_1,x_1)b(y_2,x_2)\ldots b(y_m,x_m)\quad
{\it and}\quad u(w)=\bfu^{\bfc}\bfv^{\bfd}.
$$
Then the following
identity holds
$$
{1\over \det (I_r-BU)}
=\sum_w \beta(w)\,u(w),\leqno(2.7)
$$
where $I_r$ is the identity matrix and where the sum is
over all words $w\in X^*$.}

\medskip
Make the substitution $B\leftarrow B_k$ in identity (2.7).
Then the monomial $\beta(w)$ is simply equal to~$t^{\exc_k w}$,
so that the MacMahon Master Theorem yields the identity
$$
{1\over \det (I_r-B_k U)}=\sum_w t^{\exc_k w} 
\bfu^{\bfc}\bfv^{\bfd}. 
\leqno(2.8)
$$
With the substitution $B\leftarrow B'_j$ we get the identity
$$
{1\over \det (I_r-B_j'U)}=\sum_w t^{\exc^j w} 
\bfu^{\bfc}\bfv^{\bfd},
\leqno(2.9)
$$
where

\medskip
\noindent
(2.10) $\exc^j w$ is the number of $i$ such that $1\le i\le m$
and  either $x_i> y_i$, or $x_i=y_i$ and $x_i\le j$.

\medskip
Thus the inverses of the denominators occurring on the
right-hand sides of (2.3) - (2.6) are interpreted in terms of
excedences. To get an interpretation in terms of descents we
need to recall the construction of the transformation that maps
excedences onto descents.

\section 3. Descents|In our first paper [ClFo94] we have
constructed a bijection $\Phi_k$ of each rearrangement class
$R(\bfc,\bfd)$ onto itself such that
$$
\des_k w=\exc_k\Phi_k(w) \leqno(3.1)
$$
for each word $w$ in $R(\bfc,\bfd)$. Hence (2.8) and (3.1) imply
$$
{1\over \det (I_r-B_k U)}=\sum_w t^{\des_k w} 
\bfu^{\bfc}\bfv^{\bfd}
=\sum_{\bfc,\bfd} \bfu^{\bfc}\bfv^{\bfd} A_{\bfc,\bfd}(t)
\leqno(3.2)
$$
To do the counterpart for identity (2.9) and write
$$
{1\over \det (I_r-B'_j U)}=\sum_w t^{\des^j w} 
\bfu^{\bfc}\bfv^{\bfd}, \leqno(3.3)
$$
we need an appropriate definition for ``$\des^j$" and a new
transformation $\Psi^j$ of $R(\bfc,\bfd)$ onto itself having the
property
$$
\des^jw=\exc^j \Psi^j(w).\leqno(3.4) 
$$
For ``$\des^j$ we take the definition
\medskip
\noindent
(3.5) Let $w=x_1x_2\ldots x_m$ be a word ; then $\des^j w$ is
the number of $i$ such that $0\le i\le m-1$ and either
$x_i>x_{i+1}$, or $x_i=x_{i+1}$ and $x_i\le j$. [By
convention, $x_0=x_1$.]

\medskip\noindent
As for $\Psi^j$ we take the {\it conjugate} of $\Phi_k$ in a
sense that will be made more precise. First, recall the
construction of $\Phi_k$ using a running example (see
[ClFo94, \S\kern2pt 5]).

Suppose $j=2$, $k=4$, $r=6$ and let $\star$ be an extra
letter between 2 (the largest small letter) and 3 (the
smallest large letter). Next consider the word
$$
w= 2, 1, 1, 3, 1, 3, 3, 5, 5, 2, 3, 3, 2, 1, 4, 5, 4, 6, 6, 1, 3.
$$
(a) Add $\star$ at the end of $w$ :
$$
w{\star }
= 2, 1, 1, 3, 1, 3, 3, 5, 5, 2, 3, 3, 2, 1, 4, 5, 4, 6, 6, 1, 3,{\star };
$$
(b) cut $w$ before each left to right upper record :
$$w{\star }
=\sep 2,1,1\sep 3,1\sep 3\sep 3\sep 5\sep
5,2,3,3,2,1,4\sep 5,4\sep 6\sep 6,1,3,{\star }\sep; 
$$
(c) change the mutual orders of all factors beginning with the
{\it same large} letter (here 3, 5, 6) :
$$
w'=\sep 2,1,1\sep 3\sep 3\sep 3,1\sep 5,4\sep 5,2,3,3,2,1,4
\sep 5\sep 6,1,3,{\star }\sep 6\sep;
$$
(d) form the following biword where the top word in each factor
is simply equal to bottom factor shifted to the left, the first
letter being moved to the end :
$$
\left(\displaystyle{\Delta w'\atop w'}\right)
=\left(\matrix{1\ 1\ 2\cr
2\ 1\ 1\cr}\right|
\matrix{3\cr 3\cr}
\left|\matrix{3\cr 3\cr}\right|
\matrix{1\ 3\cr 3\ 1\cr}
\left|\matrix{4\ 5\cr 5\ 4\cr}\right.\kern-3pt
\left|\matrix{2\ 3\ 3\ 2\ 1\ 4\ 5\cr
5\ 2\ 3\ 3\ 2\ 1\ 4\cr}\right|
\matrix{5\cr 5\cr}
\left|\matrix{1\ 3\ {\star }\ 6\cr
6\ 1\ 3\ {\star }\cr}\right|\kern-3pt
\left.\matrix{6\cr 6\cr}\right);
$$
(e) remove the vertical bars and
rearrange the columns of the latter biword in such a way
that the top row is {\it non-decreasing} and the mutual order of
all the biletters having the {\it same top letter} is preserved :
$$
\pmatrix{\dots\cr\Phi(w')\cr}=
\left(\matrix{1\ 1\ 1\ 1\ 1\ 2\ 2\ 2\ {\star}\ 3\ 3\ 3\ 3\ 3\ 3\ 
4\ 4\ 5\ 5\ 5\ 6\ 6\cr
2\ 1\ 3\ 2\ 6\ 1\ 5\ 3\ 3\ 3\ 3\ 1\ 2\ 3\ 1\ 5\ 1\ 4\ 4\ 5\
{\star }\ 6
\cr}\right);
$$
$({\rm e}')$ cut the biword before each change of top letter :
$$
\pmatrix{\dots\cr\Phi(w')\cr}=
\left(\matrix{1\ 1\ 1\ 1\ 1\cr
2\ 1\ 3\ 2\ 6\cr}\right|
\left.\matrix{2\ 2\ 2\cr 1\ 5\ 3\cr}\right|
\matrix{\star\cr
3\cr}
\left|\matrix{3\ 3\ 3\ 3\ 3\ 3\cr
3\ 3\ 1\ 2\ 3\ 1\cr}\right|\kern-3pt
\left.\matrix{4\ 4\cr 5\ 1\cr}\right|
\matrix{5\ 5\ 5\cr 4\ 4\ 5\cr}
\left|\matrix{6\ 6\cr
{\star }\ 6\cr}\right);
$$
(f) within each factor of the biword whose {\it top word} has
only (equal) {\it large} letters reverse (i.e., take the
mirror-image of) the {\it bottom word} :
$$
\pmatrix{\dots\cr  w''\cr}=
\left(\matrix{1\ 1\ 1\ 1\ 1\cr
2\ 1\ 3\ 2\ 6\cr}\right|
\left.\matrix{2\ 2\ 2\cr 1\ 5\ 3\cr}\right|
\matrix{\star\cr
3\cr}
\left|\matrix{3\ 3\ 3\ 3\ 3\ 3\cr
1\ 3\ 2\ 1\ 3\ 3\cr}\right|\kern-3pt
\left.\matrix{4\ 4\cr 1\ 5\cr}\right|
\matrix{5\ 5\ 5\cr 5\ 4\ 4\cr}
\left|\matrix{6\ 6\cr
6\ {\star }\cr}\right);
$$
(g) remove the vertical bars, delete $\star$ that necessarily
occurs at the end of the bottom word ; the remaining
{\it bottom} word is, by definition, $\Phi_k(w)$, i.e.,
$\Phi_k(w){\star}=w''$ :
$$
\pmatrix{\dots\cr  \Phi_k(w)\cr}=
\left(\matrix{1\ 1\ 1\ 1\ 1\ 2\ 2\ 2\ 
3\ 3\ 3\ 3\ 3\ 3\ 4\ 4\ 5\ 5\ 5\ 6\ 6\cr
2\ 1\ 3\ 2\ 6\ 1\ 5\ 3\ 
3\ 1\ 3\ 2\ 1\ 3\ 3\ 1\ 5\ 5\ 4\ 4\ 6\cr}\right).
$$
In particular, $\des_k w=\exc_k \Phi_k(w)=12$.

Furthermore, let
$c=c_1+\cdots+c_j$, $d=d_1+\cdots + d_k$ and let $x$ be the
{\it last letter} of $w$, so that $w=w_1x$. Then,
$\Phi_k(w_1x)$ admits the factorization $(w_2,x,w_3)$, where
$w_2$ and $w_3$ are words of length $l(w_2)=c$ and
$l(w_3)=d-1$. (If $d=0$, $l(w_2)=c-1$.)  Thus 

\medskip
\noindent (3.5) {\it the last letter of $w$ is equal to the
$(c+1)$-st letter of $\Phi_k(w)$.}

\medskip
In particular, 
$$
\Phi_0(w_2x)=w_3x
\qquad{\rm and}\qquad
\Phi_r(w_2x)=xw_4.\leqno(3.6)
$$

The construction of $\Psi^j$ will also be given by means of an
example with $j=4$, $k=2$, $r=6$.

\medskip
\noindent
(a) start with a word $w_5=z_1z_2\ldots z_m$ in $R(\bfc,\bfd)$
(remember that all the letters are taken from the alphabet
$[\,r\,]$) and for each $i=1,2,\ldots, m$ define 
$$
x_i=r+1-z_{m+1-i}\qquad
{\rm and\ form\ the\ word}\qquad w_6=x_1x_2\ldots x_m. 
$$
Clearly $\des^k w_5=\des_k w_6$. Under this transformation the
image of 
$$v=4,6,1,1,3,2,3,6,5,4,4,5,2,2,4,4,6,4,6,6,5$$
is precisely the word $w$ in the preceding example.

\medskip
\noindent
(b) apply $\Phi_k$ to $w_6$ to get
$$
\pmatrix{\dots\cr  \Phi_k(w)\cr}
=\pmatrix{\overline w_7\cr w_7}
=\left(\matrix{1\ 1\ 1\ 1\ 1\ 2\ 2\ 2\ 
3\ 3\ 3\ 3\ 3\ 3\ 4\ 4\ 5\ 5\ 5\ 6\ 6\cr
2\ 1\ 3\ 2\ 6\ 1\ 5\ 3\ 
3\ 1\ 3\ 2\ 1\ 3\ 3\ 1\ 5\ 5\ 4\ 4\ 6\cr}\right);
$$
(c) replace each entry $z$ in the above biword by
$(r+1-z)$ to obtain
$$
\alpha=\left(\matrix{6\ 6\ 6\ 6\ 6\ 5\ 5\ 5\ 
4\ 4\ 4\ 4\ 4\ 4\ 3\ 3\ 2\ 2\ 2\ 1\ 1\cr
5\ 6\ 4\ 5\ 1\ 6\ 2\ 4\ 
4\ 6\ 4\ 5\ 6\ 4\ 4\ 6\ 2\ 2\ 3\ 3\ 1\cr}\right)
$$
(d) consider the above biword as a {\it circuit} in the
terminology developed in [CaFo69]. Such a circuit can be
expressed as a product of true cycles, and the factorization is
unique except for the order of the factors (see [CaFo69, chap.~4,
Proposition~4.1]). The true cycles are sorted out to the left
one by one. With the running example we get :
$$
\left[\matrix{6\ 5\cr 5\ 6\cr}\right]\!
\left[\matrix{6\cr 6\cr}\right]\!
\left[\matrix{4\cr 4\cr}\right]\!
\left[\matrix{6\ 4\cr 4\ 6\cr}\right]\!
\left[\matrix{2\cr 2\cr}\right]\!
\left[\matrix{2\cr 2\cr}\right]\!
\left[\matrix{4\cr 4\cr}\right]\!
\left[\matrix{5\ 2\ 3\ 4\cr 2\ 3\ 4\ 5\cr}\right]\!
\left[\matrix{6\ 5\ 4\cr 5\ 4\ 6\cr}\right]\!
\left[\matrix{4\cr 4\cr}\right]\!
\left[\matrix{6\ 1\ 3\cr 1\ 3\ 6\cr}\right]\!
\left[\matrix{1\cr 1\cr}\right]\!;
$$
(e) take the inverse of each true cycle, i.e., exchange top and
bottom words within each true cycle :
$$
\left[\matrix{5\ 6\cr 6\ 5\cr}\right]\!
\left[\matrix{6\cr 6\cr}\right]\!
\left[\matrix{4\cr 4\cr}\right]\!
\left[\matrix{4\ 6\cr 6\ 4\cr}\right]\!
\left[\matrix{2\cr 2\cr}\right]\!
\left[\matrix{2\cr 2\cr}\right]\!
\left[\matrix{4\cr 4\cr}\right]\!
\left[\matrix{2\ 3\ 4\ 5\cr 5\ 2\ 3\ 4\cr}\right]\!
\left[\matrix{5\ 4\ 6\cr 6\ 5\ 4\cr}\right]\!
\left[\matrix{4\cr 4\cr}\right]\!
\left[\matrix{1\ 3\ 6\cr 6\ 1\ 3\cr}\right]\!
\left[\matrix{1\cr 1\cr}\right]\!;
$$
(f) suppress the brackets in the above product and reorder the
columns to get a circuit whose top row is non-decreasing, i.e.,
$$
\pmatrix{\overline w_8\cr  w_8\cr}=
\left(\matrix{1\ 1\ 2\ 2\ 2\ 
3\ 3\ 4\ 4\ 4\ 4\ 4\ 4\ 5\ 5\ 5\ 6\ 6\ 6\ 6\ 6\cr
6\ 1\ 2\ 2\ 5\ 2\ 1\ 4\ 
6\ 4\ 3\ 5\ 4\ 6\ 4\ 6\ 5\ 6\ 4\ 4\ 3\cr}\right).
$$ 
The bottom word is by definition $\Psi^k(v)$.

It is readily verified that $\exc_k w_7=\exc^k w_6$, and so
$\des^k v=\des_k w_6=\exc_k \Psi_k(w_6)\ (=\exc_k w_7)
=\exc^k w_8=\exc^k \Psi^k(v)$.
In the running example all those quantities are equal to~12.

Thus (3.4) is verified and identity (3.3) holds.

\section 4. The proofs of (1.6) - (1.9)|Identity (1.6) is a
consequence of (2.3) and (3.2).

For the proof of (1.7) we proceed as follows. Let $\des=\des_0$
and $D=\det(I_r-B_0U)$. Then
$(1/D)v_k=\sum_w t^{\des w}u(w)v_k$ by (3.2) for $k=0$. As~$r$
is the maximum letter of the alphabet, it does not create a
descent, if it is placed at the end of a word: $\des wr=\des w$.
But $w\mapsto wr$ is a bijection of the set $X^*$ of all words
onto the set $X^*r$ of words ending with~$r$. Therefore,
$$\eqalign{{1\over D}-{1\over  D}v_k
&=\sum_w t^{\des w}u(w)-\sum_w t^{\des w}u(w)v_k\cr
&=\sum_w t^{\des w} u(w)-\sum_w t^{\des wr}u(wr)
=\sum_{w\in X^*\setminus X^*r} t^{\des w}u(w),\cr }$$
where the last summation is over all words {\it not} ending
with~$r$.

In the same manner, $(1/D)tv_k$ is the generating function
for  words ending with~$r$ by ``$1+\des$." Define
$$
\des' w=\cases{1+\des w,&if $w$ ends with $r$;\cr
\des w,&otherwise.\cr}
$$
Then
$$
{1-v_k(1-t)\over D}
={1\over D}-{1\over D}v_k+{1\over D}tv_k
=\sum_w t^{\des ' w} u(w).\leqno(4.1)
$$
Now  continue with other factors:
$$
{(1-v_{k-1}(1-t))(1-v_k(1-t))\over D}
=\Bigl(\sum_w t^{\des ' w} u(w)\Bigr)(1-v_{k-1}+tv_{k-1}).
$$
In the same manner, $w\mapsto w(r-1)$ maps $X^*$ onto the
set $X^*(r-1)$ of all words ending with $(r-1)$. Furthermore,
$\des' w=\des' w(r-1)$, for, either $w$ ends with~$r$ and
its last letter is counted  as a descent, or $w$ ends   with a
letter $\le (r-1)$ and no other  descent is created.
Again, 
$$
\sum_w t^{\des' w}u(w)-\sum_wt^{\des' w} u(w)v_{k-1}
=\sum_{w\in X^*\setminus X^*(r-1)} t^{\des' w}u(w)
$$
and the factor $\bigl(\sum _w t^{\des' w}u(w)\bigr)tv_{k-1}$
adds back all the words ending with $(r-1)$ and an extra
descent is to be counted for  all those words. Define
$$\displaylines{
\des'' w=\cases{1+\des' w,&if $w$ ends with
$(r-1)$;\cr
\des' w,&otherwise;\cr}\cr
\noalign{\hbox{or, in an equivalent manner,}}
\des '' w=\cases{1+\des w,&if $w$ ends with
$(r-1)$, $r$;\cr
\des w,&otherwise.\cr}\cr
\noalign{\hbox{Then}}
{1-v_{k-1}(1-t)(1-v_k(1-t))\over D}
=\sum_w t^{\des'' w}u(w).\cr}
$$
More generally, let
$$\displaylines{(4.2)\qquad
\des^{I} w=\cases{1+\des w,&if $w$ ends with a {\it large}
letter;\cr 
\des w,&if $w$ ends with a {\it small} letter;\cr}\hfill\cr
\noalign{\hbox{and let}}
(7.11)\qquad
A^I_{\bfc,\bfd}(t)=\sum_w t^{\des^I w}\qquad 
(w\in
R(\bfc,\bfd)).\hfill\cr
}
$$
Then, by induction, we can prove the
identity 
$$
\sum_{\bfc,\bfd} \bfu^{\bfc}\bfv^{\bfd}A^I_{\bfc,\bfd}(t)
=(1-t)\,A^I(\bfu(1-t),\bfv(1-t),t)={(1-\bfv(1-t)) \over
\det(I_r-B_0U)},\leqno(4.3) 
$$
i.e., we have proved (1.7).


Next, $1/(\det(I_r-B_rU))$ is the generating function for 
$X^*$ by ``$\des_r$" and the statistic ``$\des_r$" always
includes one descent at the end of each non-empty word~$w$.
In  the same manner, the
product $(1-tu_1+u_1)\ldots(1-tu_j+u_j)$ may be regarded  as
an operator that kills the ultimate descent in each non-empty
word, if the last letter is small. 

Hence if we let
$$
A^{II}_{\bfc,\bfd}(t)=\sum_w t^{\des^{II}} w\qquad 
(w\in R(\bfc,\bfd)),
$$
the following identity holds
$$
\sum_{\bfc,\bfd} \bfu^{\bfc}\bfv^{\bfd}A^{II}_{\bfc,\bfd}(t)
=(1-t)\,A^{II}(\bfu(1-t),\bfv(1-t),t)={(1+\bfu(1-t)) \over
\det(I_r-B_rU)},
$$
i.e., identity (1.8) is proved.

\medskip
Now $1/\det(I_r-B_j'U)=\sum_w t^{\des^j w}u(w)$. The
numerator of the fraction on the right-hand side of (2.6) is an
operator that kills a descent at the beginning of a non-empty
word~$w$, if its first letter is small,  but adds a descent at
the end if its last letter is large. 

Let  
$$
A^{III}_{\bfc,\bfd}(t)=\sum_w t^{\des^{III}} w\qquad 
(w\in R(\bfc,\bfd))
$$
Then the following identity holds:
$$
\sum_{\bfc,\bfd} \bfu^{\bfc}\bfv^{\bfd}A^{III}_{\bfc,\bfd}(t)
=(1-t)\,A^{III}(\bfu(1-t),\bfv(1-t),t)=\sum_{s\ge 0} t^s
{(1+\bfu)^{s+1}\over
(1-\bfv)^s},
$$
and this proves (1.9).

\goodbreak
\section 5. Excedance statistics|The final problem is to
associate {\it excedance statistics}  to those descent
statistics. We will only do it for $\des^I$ and $\des^{II}$.
Let $\Phi=\Phi_0$.

It follows from (3.1) and $(1.5)_I$ that
$$
\des^I w'x=\cases{
1+\des w'x=1+\exc\Phi(w'x)=1+\exc w_1x,&if
$x$ is large;\cr
\des w'x=\exc \Phi(w'x)
=\exc w_1x,&if $x$ is small.\cr}
$$
We are then led to define 
$$
\exc^I w=\cases{
1+\exc w,&if
the last letter of $w$ is large;\cr
\exc w,&if the last letter of $w$ is small.\cr
}\leqno(5.1)
$$
We have then proved the following proposition

\th Proposition 5.1|The transformation
$\Phi$ also satisfies
$\exc^I  \Phi(w)=\des^I w$, and consequently
$$
A^I_{\bfc,\bfd}(t)=\sum_w t^{\des^I w}=
\sum_w t^{\exc^I w}\qquad (w\in R(\bfc,\bfd)).
$$
\finth

Now using (3.6),
$\des^{II} w'x$ is equal to
$$
\cases{\des_r w'x-1=\exc_r
\Phi_r(w'x)-1 =\exc_r xw_2-1,&if $x$ is small;\cr
\des_r w'x=\exc_r\Phi_r(w'x)=\exc_r xw_2,&if
$x$ is large.\cr}
$$
Again, for each word $w=x_1x_2\ldots x_m$ whose
non-decreasing rearrangement is
$\overline w=y_1y_2\ldots y_m$, we are led to define 

\medskip
\noindent
(5.2) $\exc^{II}w$ to be the number of $i$ such that
$1\le i\le m$ and $x_i\ge y_i$, {\it minus one} if $x_1$
is small.

\medskip
We have then the following result.

\th Proposition 5.2|The transformation
$\Phi_r$ also satisfies
$$\displaylines{
\exc^{II}  \Phi_r(w)=\des^{II} w,\cr
\noalign{\hbox{and consequently}}
A^{II}_{\bfc,\bfd}(t)=\sum_w t^{\des^{II} w}=
\sum_w t^{\exc^{II} w}\qquad (w\in R(\bfc,\bfd)).\cr
}$$
\finth

\rem Remark|Keep the same notations, in particular,
consider the letters 1, \dots~,~$j$ as being small,
and the other letters large. Furthermore, keep the
natural ordering on $[\,r\,]$. When restricted to
permutations, the statistics  ``$\des^I$,"
``$\des^{II}$" ``$\des^{III}$" coincide with the 
definition of ``$\des_k$." However ``$\exc_k$," ``$\exc^I$," and
``$\exc^{II}$" differ from one another. Consequently,
$\Phi_k$, $\Phi$ and $\Phi_r$ provide three different
bijections that map the same ``$\des_k$" onto three 
different excedance statistics.

In the following table we have displayed the actions of
those three transformations on the elements of 
${\cal S}_3$ with one  small letter ($j=1$) and two
large letters 2 and~3 ($k=2$) The letters giving rise to
excedances in the various cases have been printed in
bold-face.

$$
\vbox{\def\un{{\bf 1}} 
\def\deu{{\bf 2}}\def\tro{{\bf 3}} 
\halign{\vrule\ \hfil$#$\hfil\ \vrule
&\strut\ \hfil$#$\hfil\ \vrule 
&\ \hfil$#$\hfil\ \vrule 
&\ \hfil$#$\hfil\ \vrule  
&\ \hfil$#$\hfil\ \vrule \cr
\noalign{\hrule}
w&\des_2 w&\Phi_2&\Phi&\Phi_3\cr
\noalign{\hrule}
1,2,3&1&1,\tro,2&1,2,\tro&\tro,1,2\cr
1,3,2&2&1,\deu,\tro&1,\tro,\deu&\deu,1,\tro\cr
2,1,3&2&\deu,\tro,1&\deu,1,\tro&\tro,\deu,1\cr
2,3,1&1&\tro,1,2&\tro,2,1&1,\tro,2\cr
3,1,2&2&\tro,\deu,1&\tro,1,\deu&\deu,\tro,1\cr
3,2,1&2&\deu,1,\tro&\deu,\tro,1&1,\deu,\tro\cr
\noalign{\hrule}
}}
$$

\bigskip
{\eightpoint
\centerline{REFERENCES}
\bigskip

 \livre
CaFo69|Pierre Cartier and Dominique Foata|Probl\`emes
combinatoires de commutation et r\'earrangements|Berlin,
Springer-Verlag, $1969$ ({\sl Lecture Notes in Math.}, {\bf
85})|

\article ClFo94|Robert J. Clarke and Dominique
Foata|Eulerian Calculus, I: univariable 
statistics|Europ. J. Combinatorics|15|1994|345--362|

\article ClFo95a|Robert J. Clarke and Dominique
Foata|Eulerian Calculus, II: an  extension of Han's
fundamental transformation|Europ. J.
Combinatorics|16|1995|221--252|


\divers ClFo95b|Robert J. Clarke and Dominique
Foata|Eulerian
Calculus, III : The ubiquitous Cauchy formula, to appear in {\sl
Europ. J. Combinatorics}, $\oldstyle 1995$|


\livre Mac15|(Major) P.A. MacMahon|Combinatory
Analysis, {\rm vol.~1}|Cambridge, Cambridge Univ. Press,
$\oldstyle 1915$ (Reprinted by Chelsea, New York,
$\oldstyle 1955$)|

\nobreak
\vskip 1cm
\line{\quad\vtop{\hbox{Robert J. {\petcap Clarke},}
             \hbox{Pure Mathematics Department,}
              \hbox{University of Adelaide,}
             \hbox{Adelaide, South Australia 5005, Australia}
            \hbox{email : {\eightrm
rclarke@maths.adelaide.edu.au}}}\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}


}

\bye
