%----------------------------------------------------------------------------
% HANPAPER_no     = 42
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% PUBPAPER_author = D. Foata, G.-N. Han
% PUBPAPER_title  = Signed words and permutations,~III;\\
%                            the MacMahon Verfahren
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\auteurcourant={DOMINIQUE FOATA AND GUO-NIU HAN}
\titrecourant={SIGNED WORDS AND PERMUTATIONS, III}

%\rightline{Jan. 3, 2006}
\vglue 1.6cm
\centerline{\bf SIGNED WORDS AND PERMUTATIONS, III;}
\smallskip
\centerline{\bf THE MACMAHON VERFAHREN}
\bigskip
\centerline{\sl Dominique Foata and Guo-Niu Han}

\bigskip
\line{\hfil\hbox to 9.5cm{\hskip3cm\vtop{\eightpoint
\eightsl\hsize=3cm\parindent=0pt
Glory to Viennot,

Wizard of Bordeaux,

A prince in Physics,

In Mathematics,

Combinatorics,

Even in Graphics,

Sure,   in Viennotics.

He builds bijections,

Top calculations.

}\hfil
\vtop{\eightpoint
\eightsl\hsize=3.5cm\parindent=0pt
No one can beat him.

Only verbatim

Can we follow him.

He has admirers,

Who have got down here.

They all celebrate

Such a happy fate.

Sixty years have gone,

He still is our don.


\medskip
\noindent
Dedicated to Xavier.

\noindent
Lucelle, April 2005.
}}}

\bigskip
{\eightpoint
\narrower\narrower

\noindent
{\bf Abstract}.\quad
The MacMahon Verfahren is fully exploited to derive 
a five-variable generating polynomial over signed permutations, 
where the parameters are: ``neg" (number of negative values), ``fdes" (flag 
descent number), ``fmaj" (flag major index), ``ifdes" and ``ifmaj" 
(fdes and fmaj for the inverse permutation); then a three-variable 
generating polynomial over signed words with given letter multiplicities, 
where the parameters are ``neg", ``fmaj", and ``fdes". 

}

\bigskip

\centerline{\bf 1. Introduction}

\medskip
To paraphrase Leo Carlitz [Ca56], the present paper could have been
entitled ``Expansions of certain products," as we want to expand the product
$$\displaylines{
\rlap{(1.1)}\hfill
K_\infty(u):=\prod\limits_{i\ge 0,j\ge 0}
{1\over 
(1-uZ_{ij}q_1^iq_2^j)},\hfill\cr
\noalign{\hbox{in its infinite version, and}}
\rlap{(1.2)}\hfill
K_{r,s}(u):= \prod\limits_{0\le i\le r,\, 0\le j\le s}
{1\over 
(1-uZ_{ij}q_1^{i}q_2^{j})},\hfill\cr
\noalign{\hbox{in its graded version, where}}
Z_{ij}:=\cases{Z,&if $i$ and $j$ are both odd;\cr
1,&if $i$ and $j$ are both even;\cr
0,&if $i$ and $j$ have different parity.\cr}\cr}
$$

The second pair under study, which depends on~$r$ variables
$u_1$, \dots~, $u_r$, reads
$$\displaylines{\noalign{\vskip -6pt}
\rlap{(1.3)}\hfill
L_\infty(u_1,\ldots, u_r):=\prod_{1\le i\le r}
{1\over (u_i;q^2)_{\infty}}
{1\over (u_iqZ;q^2)_{\infty}},\hfill\cr
\rlap{(1.4)}\hfill
L_s(u_1,\ldots,u_r):=
 \prod_{1\le i\le r}
{1\over (u_i;q^2)_{\lfloor s/2\rfloor +1}}
{1\over (u_iqZ;q^2)_{\lfloor (s+1)/2\rfloor}}.\hfill\cr}
$$

In those expressions we have used the usual notations
for the $q$-{\it ascending factorial}
$$\leqalignno{\noalign{\vskip-8pt}
(a;q)_n&:=\cases{1,&if $n=0$;\cr
(1-a)(1-a q)\,\ldots\,(1-a q^{n-1}),&if
$n\geq 1$;\cr} &(1.5)\cr
\noalign{\hbox{in its {\it finite} form and}}
(a;q)_{\infty }&:=\lim\nolimits_n(a;q)_n=
{\textstyle \prod\limits _{n\geq 0}(1-a
q^n)};&(1.6)\cr}
$$
in its {\it infinite} form.

All those products will be the basic ingredients for
deriving the distributions of various statistics attached to signed
permutations and words. By {\it signed word} we understand a
word $w=x_1x_2\ldots x_m$, whose letters are integers, positive or
negative. If $\bfm=(m_1,m_2,\ldots, m_r)$ is a sequence of
nonnegative integers such that $m_1+m_2+\cdots+m_r=m$, let
$B_{\bfm}$ be the set of rearrangements $w=x_1x_2\ldots x_m$
of the sequence $1^{m_1}2^{m_2}\ldots r^{m_r}$, with the
convention that some letters $i$ $(1\le i\le r)$ may be replaced by
their opposite values $-i$. For typographical reasons we shall use
the notation $\overline i:=-i$ in the sequel. When
$m_1=m_2=\cdots=m_r=1$, the class $B_{\bfm}$ is simply the
hyperoctahedral group $B_m$ (see [Bo68], p.~252-253) of the
signed permutations of order~$m$ $(m=r$).

Using the $\chi$-notation that maps each
statement~$A$ onto the value $\chi(A)=1$ or~0 depending on
whether~$A$ is true or not, we recall that the usual
{\it inversion number}, $\inv w$,  of each signed word
$w=x_1x_2\ldots x_n$ is defined by
$$\leqalignno{\noalign{\vskip-7pt}
\inv w&:=\sum_{1\le j\le n}\sum_{i<j}\chi(x_i>x_j).\cr
\noalign{\hbox{It also makes sense to introduce}}
\imv w&:=\sum_{1\le j\le n}\;\sum_{i<j}\chi(\overline
x_i>x_j),\cr}
$$
and define the {\it flag-inversion number} of ~$w$ by
$$
\finv w:=\inv w+\imv w+\negg w,
$$
where
$\negg w:=\sum\limits_{1\le j\le n}\chi(x_j<0)$. As noted in our
previous paper [FoHa05a], the flag-inversion number coincides with the
traditional {\it length function}~$\ell$, when applied to each
signed permutation.

\goodbreak
The {\it flag-major index} ``fmaj" and the {\it flag descent
number} ``fdes", which were introduced by Adin and Roichman~[AR01]
for signed permutations, are also valid for signed words. They read
$$\leqalignno{\noalign{\vskip-2pt}
\fmaj w&:=2\maj w+\negg w;\cr
\fdes w&:=2\des w+\chi(x_1<0);\cr}
$$
where $\maj w:=\sum_jj\,\chi(x_j>x_{j+1})$ denotes the
usual {\it major index} of~$w$ and $\des w$ the
{\it number of descents}
$\des w:=\sum_j\chi(x_j>x_{j+1})$.
Finally, for each signed {\it
permutation}~$w$ let
$w^{-1}$ denote the inverse of~$w$ and define
$\ifdes w:=\fdes w^{-1}$ and $\ifmaj:=\fmaj w^{-1}$.

\rem Notations|In the sequel $B_n$ (resp. $B_{\bfm}$) designates
the hyperoctahedral group of order~$n$ (resp. the set of 
signed words of multiplicity $\bfm=(m_1,m_2,\ldots, m_r)$), as
defined above. Each generating polynomial for $B_n$
(resp. for~$B_\bfm$) by some $k$-variable statistic will be
denoted by
$B_n(t_1,\ldots,t_k)$ (resp. $B_\bfm(t_1,\ldots,t_k)$). When the
variable~$t_i$ is missing in the latter expression, it means that
the variable $t_i$ is given the value~1.

\medskip
The main two results of this paper corresponding to the
two pairs of products earlier introduced can be stated as
follows.

\proclaim Theorem 1.1. Let
$$
B_n(t_1,t_2,q_1,q_2,Z):=\sum_{w\in B_n}
t_1^{\fdes w}t_2^{\ifdes w}
q_1^{\fmaj w}q_2^{\ifmaj w}Z^{{\rm neg}\,w}\leqno(1.7)
$$
be the generating polynomial for the group~$B_n$ by the
five-variable statistic
$(\fdes,\ifdes,\fmaj,\ifmaj,\negg)$. Then,
$$\displaylines{
(1.8)\quad
\sum_{n\ge 0}{u^n\over (t_1^2;q_1^2)_{n+1}(t_2^2;q_2^2)_{n+1}}
(1+t_1)(1+t_2)B_n(t_1,t_2,q_1,q_2,Z)\hfill\cr
\noalign{\vskip-8pt}
\hfill{}=\sum_{r\ge 0, s\ge 0}t_1^rt_2^s\, K_{r,s}(u),\quad\cr
\noalign{\vskip-15pt}}
$$
where $K_{r,s}(u)$ is defined in $(1.2)$.

\proclaim Theorem 1.2.
For each sequence $\bfm=(m_1,\ldots,m_r)$ let
$$
B_{\bfm}(t,q,Z):=\sum_{w\in B_\bfm}
t^{\fdes w}q^{\fmaj w}Z^{\negg w}\leqno(1.9)
$$
be the generating polynomial for the class~$B_{\bfm}$ of signed
words by the three-variable statistic $(\fdes,\fmaj,\negg)$. Then
$$
\sum_{\bfm}(1+t)B_{\bfm}(t,q,Z){u_1^{m_1}\cdots
u_r^{m_r}\over (t^2;q^2)_{1+|\bfm|}}
=\sum_{s\ge 0}t^s L_s(u_1,\ldots,u_r),\leqno(1.10)
$$
where $|\bfm|:=m_1+\cdots+m_r$ and $L_s(u_1,\ldots, u_r)$ is
defined in
$(1.4)$.

\goodbreak
It is worth noticing that Reiner [Re93a] has calculated the
generating polynomial for~$B_n$ by another 5-variable statistic
involving each signed permutation and its inverse. The bibasic
series he has used are normalized by products of the form
$(t_1;q_1)_{n+1} (t_2;q_2)_n$ instead of
$(t_1^2;q_1^2)_{n+1}(t_2^2;q_2^2)_{n+1}$.

Using the properties of the fundamental transformation on signed words
described in our first paper [FoHa05a] we obtain the following
specialization of Theorem~1.1. Let
$${}^{\finv\kern-2pt}B_n(t,q)
:=\sum_{w\in B_n}t^{\fdes w}q^{\finv w}
\leqno(1.11)
$$
be the generating polynomial for the group $B_n$ by the pair
$(\fdes,\finv)$. Then, the $q$-factorial generating function for the
polynomials ${}^{\finv\kern-2pt}B_n(t,q)$ $(n\ge 0)$ has the following
form:
$$\displaylines{(1.12)\quad
\sum_{n\ge 0}{v^n\over (q^2;q^2)_{n}}
{}^{\finv\kern-2pt}B_n(t,q)\hfill\cr
\noalign{\vskip -7pt}
\hfill{}={1-t\over -t^2+(v(1-t^2);q)_\infty}
\bigl(t+(v(1-t^2)q;q^2)_\infty\bigr).\quad\cr}
$$
From identity (1.12) we deduce the generating function for
the polynomials ${}^{\dess\kern-2pt}B_n(t,q):=
\sum_{w\in B_n}t^{\dess w}q^{\finv w}$, where ``dess" is the
traditional number of descents for signed permutations defined by
$$\dess w:=\des w+\chi(x_1<0)\quad\hbox{instead of}\quad
\fdes w:=2\des w+\chi(x_1<0)
$$
and recover Reiner's identity [Re93a]
$$
\sum_{n\ge 0} {u^n\over (q^2;q^2)_n}
{}^{\dess\kern-2pt}B_n(t,q)
={1-t\over 1-t\,e_q(u(1-t))}
{1\over (v(1-t);q^2)_\infty},\leqno(1.13)
$$
where $e_q(v(1-t^2))=1/(v(1-t^2);q)_\infty$ is the traditional
$q$-exponential. This is done in section~4.

As in our second paper [FoHa05b] we make use of the {\it
MacMahon Verfahren} technique to prove the two theorems, which
consists of transferring the topology of the signed permutations
or words measured by the various statistics, ``fdes", ``fmaj",  to a
set of pairs of matrices with integral entries in the case of
Theorem~1.1 and a set of plain words in the case of Theorem~1.2
for which the calculation of the associated statistic is easy.
Each time there is then a combinatorial bijection between signed
permutations (resp. words) and pairs of matrices (resp. plain
words) that has the adequate properties. This is the content of
Theorem~3.1 and Theorem~4.1.

In all our results we have tried to include the variable~$Z$
that takes the number ``neg" of {\it negative} letters of each signed
permutation or word into account. This allows us to re-obtain the
classical results on the symmetric group and sets of
words.

\goodbreak
In the next section we recall the {\it MacMahon Verfahren} technique,
which was developed in our previous paper [FoHa05b] for signed
permutations. Notice that Reiner [Re93a, Re93b, Re93c, Re95a, Re95b],
extending Stanley's [St72] $(P,\omega)$-partition approach,
has successfully developed a $(P,\omega)$-partition theory for 
the combinatorial study of the
hyper\-octahedral group, which could have been used in this paper. Section~3
contains the proof of Theorem~1.1, whose specializations are given in
Section~4. We end the paper with the proof of Theorem~1.2 and its
specializations. Noticeably, the generating polynomial for the
class~$B_{\bfm}$ of signed {\it words} by the two-variable statistic
$(\fdes,\fmaj)$ is completely explicit, in the sense that we derive the factorial
generating function for those polynomials and also a recurrence
relation, while only the generating function given by~(1.10) has
a simple form when the variable~$Z$ is kept.

\bigskip

\centerline{\bf 2. The MacMahon Verfahren}

\medskip


Let ${\bboard N}^n$ (resp. $\NIW(n)$) be the set of  words
(resp. nonincreasing words) of length~$n$, whose letters are
nonnegative integers. As done in [FoHa05b] the MacMahon
Verfahren consists of mapping each pair $(b,w)\in \NIW(n)\times
B_n$ onto a word $c\in {\bboard N}^n$ as follows. Write the
signed permutation~$w$ as the linear word $w=x_1x_2\ldots
x_n$, where
$x_k$ is the image of the integer~$k$ $(1\le k\le n)$. For
each $k=1,2,\ldots,n$ let $z_k$ be the number of descents in the right
factor $x_kx_{k+1}\ldots x_n$ and $\epsilon_k$ be equal to~0 or~1
depending on whether $x_k$ is positive or negative.  Next, form the
words $z(w):=z_1z_2\ldots z_n$ and
$\epsilon(w):=\epsilon_1\epsilon_2\ldots\epsilon_n$.

Now, take a nonincreasing word $b=b_1b_2\ldots b_n$ and define
$a_k:=b_k+z_k$, $c'_k:=2a_k+\epsilon_k$
$(1\le k\le n)$, then $a(b,w):=a_1a_2\ldots a_n$
and $c'(b,w):=c'_1 c'_2\ldots c'_n$.
Finally, form the two-row matrix
$\left(\matrice{c'_1& c'_2&\ldots&c'_n\cr
|x_1|&|x_2|&\ldots&|x_n|\cr}\right)$.
Its bottom row is a permutation of $1\,2\,\ldots\,n$;
rearrange the columns in such a way that the
bottom row is precisely $1\,2\,\ldots\,n$. Then the word
$c(b,w)=c_1c_2\ldots c_n$ which corresponds to the pair $(b,w)$ is
defined to be the top row in the resulting matrix.

\medskip
{\it Example}.\quad Start with the pair $(b,w)$ below and
calculate all the necessary ingredients:
$$
\matrice{b&=&4&2&2&1&1&0&0\cr
w&=
&3&\overline 5&\overline 1&6&7&\overline 4&2\cr
z(w)&=&2&1&1&1&1&0&0\cr
\epsilon(w)&=&0&1&1&0&0&1&0\cr
a(b,w)&=&6&3&3&2&2&0&0\cr
c'(b,w)&=&12&7&7&4&4&1&0\cr
c(b,w)&=&7&0&12&1&7&4&4\cr}
$$

\goodbreak
For each $c=c_1\ldots c_n\in {\bboard N}^n$ and let 
$\tot c:=c_1+\cdots +c_n$,
$\max c:=\max\{c_1, \ldots,c_n\}$
and let $\odd c$ denote
the number of {\it odd} letters in~$c$ .
The proof of the following theorem can be found in
[FoHa05b, Theorem 4.1].


\proclaim Theorem 2.1. For each nonnegative integer $s$ the
above mapping is a bijection $(b,w)\mapsto c(b,w)$ of the set of pairs
$$(b,w)=(b_1b_2\ldots b_n,\,x_1x_2\ldots x_n)
\in \NIW(n)\times B_n$$ such that $2b_1+\fdes w=s$ onto the set
of words~$c=c_1c_2\ldots c_n\in {\bboard N}^n$ such that
$\max c=s$. Moreover,
$$\displaylines{(2.1)\quad
2b_1+\fdes w=\max c(b,w);\quad
2\tot b+\fmaj w=\tot c(b,w);\hfill\cr
{\rm neg}\, w=\odd c(b,w).\cr}
$$

We end this section by quoting the following classical identity, where $b_1$ is
the first letter of~$b$.
$$
{1\over (u;q)_{n+1}}=\sum_{s\ge 0}u^s
\sum_{b\in \NIW(n),\,b_1\le s}q^{\tot b}.\leqno(2.2)
$$

\bigskip
\centerline{\bf 3. Proof of Theorem 1.1}

\medskip
We apply the MacMahon Verfahren just described to the two pairs
$(b,w)$ and $(\beta,w^{-1})$, where~$b$ and~$\beta$ are two
nonincreasing words. The pair $(b,w)$ (resp. $(\beta,w^{-1})$)
is mapped onto a word $c:=c(b,w)=c_1c_2\ldots c_n$ (resp.
$C:=c(\beta,w^{-1})=C_1C_2\ldots C_n$) of length~$n$, with the
properties
$$\displaylines{(3.1)\hfill
2b_1+\fdes w=\max c;\quad
2\tot b+\fmaj w=\tot c;\hfill\cr
(3.2)\hfill
2\beta_1+\ifdes w=\max C;\quad
2\tot \beta+\ifmaj w=\tot C.\hfill\cr}
$$
There remains to investigate the relation between the two
words~$c$ and $C$ before pursuing the calculation. Form the
two-row matrix
$$
\left(\matrix{c'\cr C\cr}\right)
=\left(\matrix{c'_1&c'_2&\ldots&c'_n\cr
C_1&C_2&\ldots &C_n\cr}\right),
$$
where $c'=c'_1c'_2\ldots c'_n$ designates the
nonincreasing rearrangement of the word~$c$. The following lemma 
is readily verified.

\proclaim Lemma.
The following properties hold:
\hfill\break
{\rm (i)} {for each $i=1,2,\ldots,n$ the letters $c'_i$ and $C_i$
are of the same parity;}
\hfill\break
{\rm (ii)} {if $c'_i=c'_{i+1}$ is even (resp. is
odd), then
$C_i\ge C_{i+1}$ (resp. $C_i\le C_{i+1}$).}
\hfill\break
{\rm (iii)} {if $c'\choose C$ is a
two-row matrix of length~$n$ having properties~{\rm (i)} and~{\rm
(ii)}, there exists one and only one signed permutation~$w$ and
two nonincreasing words~$b$, $\beta$ satisfying} $(3.1)$ and $(3.2)$.

\goodbreak
\medskip
{\it Example}.\quad We take the same example for $w$ as in the previous
section, but we also calculate $C=c(\beta,w^{-1})$.
$$\displaylines{
\matrice{b&=&4&2&2&1&1&0&0\cr
w&=
&3&\overline 5&\overline 1&6&7&\overline 4&2\cr
z(w)&=&2&1&1&1&1&0&0\cr
\epsilon(w)&=&0&1&1&0&0&1&0\cr
a(b,w)&=&6&3&3&2&2&0&0\cr
c':=c'(b,w)&=&12&7&7&4&4&1&0\cr
c:=c(b,w)&=&7&0&12&1&7&4&4\cr}
\qquad
\matrice{\beta&=&3&2&1&1&0&0&0\cr
w^{-1}&=&\overline 3&7&1&\overline 6&\overline
2&4&5\cr 
z(w^{-1})&=&2&2&1&0&0&0&0\cr
\epsilon(w^{-1})&=&1&0&0&1&1&0&0\cr
a(\beta,w^{-1})&=&5&4&2&1&0&0&0\cr
C':=c'(\beta,w^{-1})&=&11&8&4&3&1&0&0\cr
C:=c(\beta,w^{-1})&=&4&1&11&0&0&3&8\cr}\cr
{c'\choose C}=\left(\matrice{12&7&7&4&4&1&0\cr
4&1&11&0&0&3&8\cr}\right).\cr}
$$

Let $(i_1<i_1<\cdots<i_r)$ (resp. $(j_1<j_2<\cdots< j_s)$) be the
increasing sequence of the integers~$i$ (resp. the integers~$j$)
such that $c'_i$ is {\it even} (resp. $c'_j$ is {\it
odd\/}). Define (``e" for ``even" and ``o" for ``odd")
$$\leqalignno{
2d^e={2f^e\choose 2g^e}&:=
\left(\matrix{c'_{i_1}&c'_{i_2}&\ldots&c'_{i_r}\cr
C_{i_1}&C_{i_2}&\ldots &C_{i_r}\cr}\right);\cr
2d^o+1={2f^o+1\choose 2g^o+1}&:=
\left(\matrix{c'_{j_1}&c'_{j_2}&\ldots&c'_{j_s}\cr
C_{j_1}&C_{j_2}&\ldots &C_{j_s}\cr}\right);\cr}
$$
so that the two two-row matrices
$$\leqalignno{
d^e={f^e\choose g^e}&:=
\left(\matrix{c'_{i_1}/2&c'_{i_2}/2&\ldots&c'_{i_r}/2\cr
C_{i_1}/2&C_{i_2}/2&\ldots &C_{i_r}/2\cr}\right),\cr
d^o={f^o\choose g^o}&:=
\left(\matrix{(c'_{j_1}-1)/2&(c'_{j_2}-1)/2&\ldots&(c'_{j_s}-1)/2\cr
(C_{j_1}-1)/2&(C_{j_2}-1)/2&\ldots
&(C_{j_s}-1)/2\cr}\right),\cr }
$$
may be regarded as another expression for the two-row matrix
$c'\choose C$. 


Define the integers $i_{\max}$ and $j_{\max}$ by:
$$\displaylines{
i_{\max}:=\cases{(\max c-1)/2,&if $\max c$ is odd;\cr
\max c/2,&if $\max c$ is even;\cr}\cr
j_{\max}:=\cases{(\max C-1)/2,&if $\max C$ is odd;\cr
\max C/2,&if $\max C$ is even.\cr}\cr}
$$
Then, to  the pair $d^e$, $d^o$ there corresponds a pair of unique
{\it finite} matrices $D^e =(d^e_{ij})$, $D^o=(d^o_{ij})$ $(0\le
i\le i_{\max}, 0\le j\le j_{\max})$ (and no other pair of smaller
dimensions), where $d_{ij}^e$ (resp. $d_{ij}^o$) is equal to the
number of the two-letters in~$d^e$ (resp. in $d^o$) that are equal
to~$i\choose j$. 


On the other hand, with $|f|$ designating the length of the
word~$f$,
$$
\eqalignno{
\tot c&=\tot 2f^e+\tot (2f^o+1)=2\tot f^e+2\tot f^o+|f^o|\cr
&=2\,{\textstyle\sum\limits_{i,j}}\,i\,d_{ij}^e
+2\,{\textstyle\sum\limits_{i,j}}\,
i\,d_{ij}^o+{\textstyle\sum\limits_{i,j}}\,d_{ij}^o;\cr
\tot C&=\tot 2g^e+\tot (2g^o+1)=2\tot g^e+2\tot g^o+|g^o|\cr
&=2\,{\textstyle\sum\limits_{i,j}}\,j\,d_{ij}^e
+2\,{\textstyle\sum\limits_{i,j}}\,
j\,d_{ij}^o+{\textstyle\sum\limits_{i,j}}\,d_{ij}^o.\cr
}
$$
The following proposition follows from Theorem 2.1.

\proclaim Proposition 3.1. For each pair of nonnegative integers
$(r,s)$ the map $(b,\beta,w)\mapsto (D^e,D^o)$ is a bijection of the
triples $(b,\beta,w)$ such that $2b_1+\fdes w\le r$,
$2\beta_1+\ifdes w\le s$ onto the pairs of matrices
$D^e=(d_{i,j}^e)$, $D^o=(d_{i,j}^o)$
$(0\le i\le r, 0\le j\le s)$. Moreover,
$$
\displaylines{
\rlap{$(3.3)$}\hfill
2\tot b+\fmaj w=2\,{\textstyle\sum\limits_{i,j}}\,i\,d_{ij}^e
+2\,{\textstyle\sum\limits_{i,j}}\,
i\,d_{ij}^o+{\textstyle\sum\limits_{i,j}}\,d_{ij}^o;\hfill\cr
\rlap{$(3.4)$}\hfill
2\tot \beta+\ifmaj w=2\,{\textstyle\sum\limits_{i,j}}\,j\,d_{ij}^e
+2\,{\textstyle\sum\limits_{i,j}}\,
j\,d_{ij}^o+{\textstyle\sum\limits_{i,j}}\,d_{ij}^o;\hfill\cr
\rlap{$(3.5)$}\hfill{\rm neg}\,
w={\textstyle\sum\limits_{i,j}}\,d_{ij}^o.\hfill\cr
\rlap{$(3.6)$}\hfill
|w|=
{\textstyle\sum\limits_{i,j}}\,d_{ij}^e+
{\textstyle\sum\limits_{i,j}}\,d_{ij}^o.\hfill\cr
}
$$

Again work with the same example as above. To the two-row matrix
$$\displaylines{
{c'\choose C}=\left(\matrice{12&7&7&4&4&1&0\cr
4&1&11&0&0&3&8\cr}\right)\cr
\noalign{\hbox{there corresponds the pair}}
d^e=\left(\matrice{6&2&2&0\cr
2&0&0&4\cr}\right),\quad
d^o=\left(\matrice{3&3&0\cr
0&5&1\cr}\right).\cr}
$$

\goodbreak
As $\max c=12$ is even (resp. $\max C=11$ is odd),
we have
$i_{\max}=6$, $j_{\max}=5$ and
$$
D^e=\bordermatrix{&0&1&2&3&4&5\cr
0&.&.&.&.&1&.\cr
1&.&.&.&.&.&.\cr
2&2&.&.&.&.&.\cr
3&.&.&.&.&.&.\cr
4&.&.&.&.&.&.\cr
5&.&.&.&.&.&.\cr
6&.&.&1&.&.&.\cr};\quad
D^o=\bordermatrix{&0&1&2&3&4&5\cr
0&.&1&.&.&&.\cr
1&.&.&.&.&.&.\cr
2&.&.&.&.&.&.\cr
3&1&.&.&.&.&1\cr
4&.&.&.&.&.&.\cr
5&.&.&.&.&.&.\cr
6&.&.&.&.&.&.\cr}.
$$
Also verify that $2\tot b+\fmaj w=35=2\times (2+2+6)+2\times
(3+3)+3$;
$2\tot\beta+\ifmaj w=27=2\times (2+4)+2\times (1+5)+3$ and
$\negg w=3$.

\goodbreak
In the following summations $b$ and $\beta$ run over
the set of nonincreasing words of length~$n$. By using
identity (2.2) we have
$$
\displaylines{\quad
\sum_{n\ge 0}{u^n\over (t_1^2;q_1^2)_{n+1}(t_2^2;q_2^2)_{n+1}}
(1+t_1)(1+t_2)B_n(t_1,t_2,q_1,q_2,Z)\hfill\cr
\kern.3cm{}=\sum_{n\ge 0}u^n
(1+t_1)(1+t_2)B_n(t_1,t_2,q_1,q_2,Z)\kern-20pt
\sum_{\scriptstyle r'\ge 0,\,s'\ge0\atop
{\scriptstyle b, \beta,b_1\le r'\!,\,\beta_1\le s'}}\kern-20pt
t_1^{2r'}t_2^{2s'}q_1^{2\tot b}q_2^{2\tot \beta}
\hfill\cr
\kern.3cm{}=\sum_{n,r',s'}u^n(t_1^{2r'}+t_1^{2r'+1})
(t_2^{2s'}+t_2^{2s'+1})\hfill\cr
\noalign{\vskip-10pt}
\kern 2.5cm{}\times\kern-20pt
\sum_{\scriptstyle w\in B_n\atop
{\scriptstyle b, \beta,b_1\le r'\!,\,\beta_1\le s'}}\kern-20pt
t_1^{\fdes w}t_2^{\ifdes w}q_1^{\fmaj w+2\tot b}
q_2^{\ifmaj w+2\tot \beta}Z^{{\rm neg}\,w}\hfill\cr
\kern.3cm{}=\sum_{n,r',s'}\kern-5pt u^nt_1^{r'}
t_2^{s'}\kern-25pt
\sum_{\scriptstyle w\in B_n\atop
{\scriptstyle b,\beta, 2b_1\le r'\!,\, 2\beta_1\le s'}}\kern-20pt
t_1^{\fdes w}t_2^{\ifdes w}q_1^{\fmaj w+2\tot b}
q_2^{\ifmaj w+2\tot \beta}Z^{{\rm neg}\,w}\hfill\cr
\kern.3cm{}=\sum_{n,r,s} u^nt_1^{r}
t_2^{s}\kern-20pt
\sum_{\scriptstyle w\in B_n,b,\beta\atop
{\scriptstyle  2b_1+\fdes w\le r,\, 2\beta_1+\ifdes w\le
s}}\kern-20pt q_1^{\fmaj w+2\tot b}
q_2^{\ifmaj w+2\tot \beta}Z^{{\rm neg}\,w}.\hfill\cr
}
$$
We can continue the calculation by using
(3.3), (3.4) and (3.5), the inner summation being over matrices $D^e$,
$D^o$ of dimensions
$(r+1)\times (s+1)$
with
$\sum_{i,j}d^e_{ij} + \sum_{i,j} d^o_{ij} = n$, that is,
$$
\displaylines{\kern.1cm{}=\sum_{n,r,s} u^nt_1^{r}t_2^s
\sum_{D^e,D^o}q_1^{2\Sigma\,id_{ij}^e
+2\Sigma\,id_{ij}^o+\Sigma\,d_{ij}^o}
q_2^{2\Sigma\,jd_{ij}^e
+2\Sigma\,jd_{ij}^o+\Sigma\,d_{ij}^o}Z^{\Sigma\,d_{ij}^o}
\hfill\cr
\kern.1cm{}=\!\sum_{r,s} t_1^{r}t_2^s
\sum_{D^e}u^{\Sigma\,d_{ij}^e}
q_1^{\Sigma\,(2i)d_{ij}^e}q_2^{\Sigma\,(2j)d_{ij}^e}\kern-2pt
\sum_{D^o}u^{\Sigma\,d_{ij}^o}
q_1^{\Sigma\,(2i+1)d_{ij}^o}q_2^{\Sigma\,(2j+1)d_{ij}^o}
Z^{\Sigma\,d_{ij}^o}\hfill\cr
\kern.1cm{}=\sum_{r,s} t_1^{r}t_2^s\sum_{D^e}
{\textstyle\prod\limits_{i,j}}(uq_1^{2i}q_2^{2j})^{d^e_{ij}}
\sum_{D^o}
{\textstyle\prod\limits_{i,j}}(uZq_1^{2i+1}q_2^{2j+1})^{d^o_{ij}}
\quad (0\!\le\! i\!\le\! r,0\!\le\! j\!\le\! s)
\hfill\cr
\kern.1cm{}=\sum_{r,s} t_1^{r}t_2^s
{1\over
\prod\limits_{i,j}(1-uq_1^{2i}q_2^{2j})} {1\over
\prod\limits_{i,j}(1-uZq_1^{2i+1}q_2^{2j+1})}
\quad (0\!\le\! i\!\le\! r,0\!\le\! j\!\le\! s)\hfill\cr
\kern.1cm{}=\sum_{r\ge 0, s\ge 0}
t_1^rt_2^s 
{1\over \kern-12pt\prod\limits_{0\le i\le r,
0\le j\le s} 
\kern-15pt (1-uZ_{ij}q_1^{i}q_2^{j})}.\qed\hfill\cr }
$$

\bigskip
\centerline{\bf 4. Specializations and Flag-inversion number}
\medskip
Define
$$
(u; q_1, q_2)_{r+1,s+1} = \kern -7pt \prod_{\scriptstyle 0\leq i\leq r \atop
\scriptstyle 0\leq j\leq s} \kern -5pt (1-u q_1^i q_2^j);
\quad
(u; q_1, q_2)_{\infty,\infty} = \kern -5pt \prod_{i,j\geq 0}\kern -4pt (1-u q_1^i q_2^j).
$$
\goodbreak\noindent
When $Z:=0$ and $t_1$, $t_2$, $q_1$, $q_2$ are replaced by
their square roots, identity (1.8) becomes
$$
\sum_{n\ge0}{u^n\over (t_1;q_1)_{n+1}(t_2;q_2)_{n+1}}
A_n(t_1,t_2,q_1,q_2)\kern-2pt
=\kern-8pt\sum_{r\ge 0, s\ge 0}
{t_1^rt_2^s \over (u;q_1,q_2)_{r+1,s+1}}
,\leqno(4.1)
$$
where $A_n(t_1,t_2,q_1,q_2)$ is the generating polynomial for the
symmetric group~${\goth S}_n$ by the quadruple
$(\des,\ides,\maj,\imaj)$, an identity first derived by Garsia and
Gessel [GaGe78]. Other approaches can be found in [Ra79],
[DeFo85].

Remember that when a variable is missing in
$B_n(t_1,t_2,q_1,q_2,Z)$ it means that the variable has been
given the value~1. Multiply both sides of (1.8) by $(1-t_2)$
and let $t_2=1$. We get:
$$\displaylines{(4.2)\quad
\sum_{n\ge 0}{u^n\over (t_1^2;q_1^2)_{n+1}(q_2^2;q_2^2)_{n}}
(1+t_1)B_n(t_1,q_1,q_2,Z)\hfill\cr
\noalign{\vskip-10pt}
\hfill{}
=\sum_{r\ge 0}
t_1^r {1\over \kern-5pt\prod\limits_{0\le i\le r,j\ge
0} \kern-5pt (1-uZ_{ij}q_1^{i}q_2^{j})}.\quad\cr}
$$
Again multiply both sides by $(1-t_1)$ and let $t_1=1$:
$$
\sum_{n\ge 0}{u^n\over (q_1^2;q_1^2)_{n}(q_2^2;q_2^2)_{n}}
B_n(q_1,q_2,Z)
={1\over \kern-5pt\prod\limits_{i\ge 0,j\ge
0} \kern-5pt (1-uZ_{ij}q_1^{i}q_2^{j})}.\leqno(4.3)
$$
Also the (classical) generating function for
the polynomials
$A_n(q_1,q_2)$ can be derived from identity (4.3) and reads:
$$
\sum_{n\ge0}{u^n\over (q_1;q_1)_{n}(q_2;q_2)_{n}}
A_n(q_1,q_2)={1\over (u;q_1,q_2)_{\infty,\infty}}.$$

With $q_1=1$ the denominator of the fraction on the right side
of identity (4.2) becomes
$$
\cases{(u;q_2^2)_\infty^{(r/2)+1}(uq_2Z;q_2^2)_\infty^{r/2},&if
$r$ is even;\cr
\noalign{\medskip}
(u;q_2^2)_\infty^{(r+1)/2}(uq_2Z;q_2^2)_\infty^{(r+1)/2},&if
$r$ is odd.\cr}
$$
Hence,
$$\displaylines{\quad
\sum_{n\ge 0}{u^n\over (1-t_1^2)^{n+1}(q_2^2;q_2^2)_{n}}
(1+t_1)B_n(t_1,q_2,Z)\hfill\cr
\kern.2cm{}=
\sum_{s\ge 0}
{t_1^{2s+1}\over ((u;q_2^2)_\infty(uq_2Z;q_2^2)_\infty)^{s+1}}
+\sum_{s\ge 0}
{t_1^{2s}\over((u;q_2^2)_\infty(uq_2Z;q_2^2)_\infty)^{s}}
{1\over (u;q_2^2)_\infty}.\hfill\cr
\kern.2cm{}=
{1\over
-t_1^2+(u;q_2^2)_\infty(uq_2Z;q_2^2)_\infty}
\bigl(t_1+(uq_2Z;q_2^2)_\infty\bigr).\hfill\cr
 }
$$
Now replace $u$ by
$v(1-t_1^2)$. This implies the following result stated as a theorem.

\goodbreak
\proclaim Theorem 4.1. Let $B_n(t_1,q_2,Z)$ be the generating
polynomial for the group~$B_n$ by the triple $(\fdes,\ifmaj,{\rm
neg})$. Then,
$$\displaylines{(4.4)\quad
\sum_{n\ge 0}{v^n\over (q_2^2;q_2^2)_{n}}
B_n(t_1,q_2,Z)\hfill\cr
\noalign{\vskip -8pt}
\kern.2cm{}
={1-t_1\over -t_1^2
+(v(1-t_1^2);q_2^2)_\infty\,(v(1-t_1^2)q_2Z;q_2^2)_\infty}
\bigl(t_1+(v(1-t_1^2)q_2Z;q_2^2)_\infty\bigr).
\hfill\cr}
$$

\goodbreak
Several consequences are drawn from Theorem 4.1.
First, when $Z=0$ and when $t_1$, $q_2$ are replaced by their square
roots, we get 
$$
\sum_{n\ge 0}{v^n\over (q_2;q_2)_{n}}
A_n(t_1,q_2)
={1-t_1\over -t_1+(v(1-t_1);q_2)_\infty},\leqno(4.5)
$$
where $A_n(t_1,q_2)$ is the generating polynomial for the group
${\goth S}_n$ by the pair $(\des,\imaj)$, but also by the pair
$(\des,\inv)$ (see [St76], [FoHa97]).

\medskip
Let ${\bf i}\,w:=w^{-1}$ denote the inverse of the signed
permutation~$w$. At this stage we have to remember that the
bijection $\Psi$ of~$B_n$ onto itself that we have constructed in
our first paper [FoHa05a] preserves the {\it inverse ligne of
route} [FoHa05a, Theorem~1.2], so that the chain
$${
\def\rightarrowh#1{\smash{\mathop{\hbox to 9mm{\rightarrowfill}}
\limits^{\textstyle#1}}}
\matrix{
w & \rightarrowh{{\bf i}} & w_1 & \rightarrowh{\Psi} & w_2 &
  \rightarrowh{{\bf i}} & w_3 \cr
\noalign{\medskip}
\displaystyle{\fdes\choose\ifmaj} &&
\displaystyle{\ifdes\choose\fmaj} &&
\displaystyle{\ifdes\choose\finv} &&
\displaystyle{\fdes\choose\finv}\cr
}
}
$$
shows that the four pairs $(\fdes,\ifmaj)$, $(\ifdes,\fmaj)$,
$(\ifdes,\finv)$ and $(\fdes,\finv)$ are all {\it equidistributed}
over~$B_n$. Therefore,
$$\sum_{w} t^{\fdes w}q^{\ifmaj w}\!=\!
\sum_w t^{\ifdes w}q^{\fmaj w}\!=\!
\sum_w t^{\ifdes w}q^{\finv w}\!=\!\sum_w t^{\fdes w}q^{\finv
w},$$
where $w$ runs over $B_n$. The rightmost generating polynomial
was designated by
${}^{\finv\kern-2pt}B_n(t,q)$ in (1.11). Therefore we can
derive a formula for ${}^{\finv\kern-2pt}B_n(t,q)$
by using its $(\fdes,\ifmaj)$
interpretation. We have then
${}^{\finv\kern-2pt}B_n(t,q)=B_n(t_1,q_2,Z)$ with
$t_1=t$, $q_2=q$ and
$Z=1$. Let $Z:=1$ in (4.4); as
$
(v(1-t_1^2);q_2^2)_\infty\,(v(1-t_1^2)q_2;q_2^2)_\infty
=(v(1-t_1^2);q_2)_\infty$, identity (4.4) implies identity
(1.12).

To recover Reiner's identity (1.13) we make use of (1.12) by
sorting the signed permutations according to the parity of their
flag descent numbers: $B_n(t,q)=:B_n'(t^2,q)+t\,B_n''(t^2,q)$, so
that
${}^{\dess\kern-2pt}B_n(t^2,q)=B_n'(t^2,q)+t^2\,B_n''(t^2,q)$.
Hence
$$\eqalignno{
\sum_{n\ge 0}{v^n\over (q^2;q^2)_{n}}\kern-5pt
{}^{\dess\kern-2pt}B_n(t^2,q)
\!&=\!{(v(1-t^2)q;q^2)_\infty-t^2\over
-t^2+(v(1-t^2);q)_\infty}
+t^2{1-(v(1-t^2)q;q^2)_\infty\over
-t^2+(v(1-t^2);q)_\infty}\cr
&=\!{1-t^2\over -t^2+(v(1-t^2);q)_\infty}
(v(1-t^2)q;q^2)_\infty.\cr
}
$$
As $1/(v(1-t^2);q)_\infty$ can also be expressed as the
$q$-exponential $e_q(v(1-t^2))$, we then get identity (1.13).

\bigskip
\centerline{\bf 5. The MacMahon Verfahren for signed words}

\medskip
Let $w=x_1x_2\ldots x_m$ be a signed {\it word} belonging to
the class $B_{\bfm}$, where $\bfm=(m_1,m_2,\ldots, m_r)$ is a
sequence of nonnegative integers such that
$m_1+m_2+\cdots+m_r=m$.  Remember that this
means that~$w$ is a rearrangement of
$1^{m_1}2^{m_2}\ldots r^{m_r}$, with the convention that some
letters~$i$ $(1\le i\le r)$ may be replaced by their opposite values
$\overline i$. Again, let
$\epsilon:=\epsilon(w):=\epsilon_1\epsilon_2\ldots \epsilon_m$ be the
binary word defined by $\epsilon_i=0$ or~1, depending on
whether
$x_i$ is positive or negative.
As before, let $b\in\NIW(m)$.

The MacMahon Verfahren bijection
for signed words is constructed as follows. First, compute the
word~$z=z_1z_2\ldots z_m$, where~$z_k$ is equal to the
number of descents in the right factor
$x_kx_{k+1}\ldots x_m$, as well as the word
$\epsilon=\epsilon_1\epsilon_2\ldots
\epsilon_m$ mentioned above, so that, as in the case of signed
permutations,
$$\fmaj w=2\tot z+\tot
\epsilon.\leqno(5.1)
$$ 
Next, define
$a_k:=b_k+z_k$, $c'_k:=2a_k+\epsilon_k$
$(1\le k\le m)$, then $a:=a_1a_2\ldots a_m$
and $c':=c'_1c'_2\ldots c'_m$.
Finally, form the two-row matrix
$\left(\matrice{c'\cr \abs w\cr}\right)
=\left(\matrice{c'_1&c'_2&\ldots&c'_m\cr
|x_1|&|x_2|&\ldots&|x_m|\cr}\right)$. Its bottom row is a
rearrangement of the word
$1^{m_1}2^{m_2}\,\ldots\,r^{m_r}$.

Make the convention that two
biletters $c'_k\choose |x_k|$ and
$c'_l\choose |x_l|$ commute if and only if
$|x_k|$ and $|x_l|$ are different and rearrange the biletters of that
biword in such a way that the bottom row is precisely
$1^{m_1}2^{m_2}\,\ldots\,r^{m_r}$. The top row in the resulting
two-row matrix is then the juxtaposition product
of~$r$ {\it nonincreasing} words
$b^{(1)}=b^{(1)}_1\ldots b^{(1)}_{m_1}$, $b^{(2)}=b^{(2)}_1\ldots
b^{(2)}_{m_2}$, \dots~, $b^{(r)}=b^{(r)}_1\ldots b^{(r)}_{m_r}$, of
lengths~$m_1$, $m_2$, \dots~, $m_r$, respectively. Moreover,
$$\leqalignno{
\tot b^{(1)}+\tot b^{(2)}+\cdots+
\tot b^{(r)}&=\tot c'=2\tot a+\tot\epsilon\cr
&=2\tot b+2\tot z+\tot\epsilon\cr
&=2\tot b+\fmaj w.&(5.2)\cr}
$$

\goodbreak\noindent
On the other hand,
$$
\leqalignno{
2b_1+\fdes w&=2b_1+2z_1+\epsilon_1=c'_1\cr
&=\max\nolimits_{i}b^{(i)}_1
\  (1\le i\le r).&(5.3)\cr}
$$

As in the case of signed permutations, we can easily see that for
each nonnegative integer~$s$ the map $(b,w)\mapsto (b^{(1)},
b^{(2)},\ldots,b^{(r)})$ is a bijection~of the set of pairs
$(b,w)\in\NIW(m)\times R_{\bfm}$ such that $2b_1\!+\fdes
w\!=\!s$ onto~the set of juxtaposition products
$b^{(1)}b^{(2)}\ldots b^{(r)}$ such that
$\max\nolimits_{i}b^{(i)}_1=s$. The reverse bijection is
constructed in the same way as in the case of signed
permutations.

\medskip
{\it Example}.\quad Start with the pair $(b,w)$, where~$w$ belongs to
$B_{\bfm}$ with $\bfm=(2,2,2,2,2,2)$, $r=6$, $m=12$, the negative
elements being overlined.
$$
\matrice{{\rm
Id}&=&\,1\,&\,2\,&\,3\,&\,4\,&\,5\,&\,6\,&\,7\,&\,8\,&\,9\,&
10&11&12\cr
b&=&4&4&4&4&4&4&4&2&2&1&1&1\cr
w&=&1&\overline 5&\overline 2&2&\overline 3&
\overline 3&\overline {\bf1}&4&\overline 6&\overline 5&\bf
4&\bf6\cr 
z&=&3&2&2&2&1&1&1&1&0&0&0&0\cr
\epsilon&=&0&1&1&0&1&1&1&0&1&1&0&0\cr
a&=&7&6&6&6&5&5&5&3&2&1&1&1\cr
c'&=&14&13&13&12&11&11&\bf11&6&5&3&\bf2&\bf2\cr
\abs w&=&1& 5& 2&2& 3&
 3& {\bf1}&4& 6& 5&\bf4&\bf6\cr
b^{(1)}\ldots b^{(r)}
&=&14&\bf11\,\vrule depth 2pt height
10pt&13&12\,\vrule&11&11\,\vrule&6
&\bf2\,\vrule&13&3\,\vrule&5&\bf2\cr
1^{m_1}\ldots r^{m_r}
&=&1&\; \bf1\kern4.5pt\vrule depth 2pt height
10pt&2&\,2\kern5pt\vrule&3&\,3\;\kern2pt\vrule&4&\bf4\,\vrule&
5&5\,\vrule&6&\bf6\cr
}
$$
We verify that $2\tot b+\fmaj w=2\tot b+2\tot
z+\tot\epsilon =2\times 35+2\times 13
+7=103=\tot b^{(1)}+\cdots+\tot b^{(6)}$
and $2b_1+\fdes w=2b_1+2z_1+\epsilon_1=
2\times 4+2\times 3+0=14=c'_1=\max\nolimits_{i}b^{(i)}_1$.
\medskip

The combinatorial theorem for signed words that corresponds to
Theorem~2.1 is now stated.

\proclaim Theorem 5.1. For each nonnegative integer $s$ the
above mapping is a bijection of the set of pairs
$(b,w)=(b_1b_2\ldots b_m,\,x_1x_2\ldots x_m)\in \NIW(m)\times
B_{\bfm}$ such that
$2b_1+\fdes w=s$ onto the set of juxtaposition products
$b^{(1)}\ldots b^{(r)}\in \NIW(m_1)\times\cdots\times
\NIW(m_r)$ such that $\max\nolimits_{i}b^{(i)}_1=s$. Moreover,
$(5.2)$ and $(5.3)$ hold, together with
$$
{\rm neg}\,w=\odd (b^{(1)}\ldots b^{(r)})\leqno(5.4)
$$

Relation (5.4) is obvious, as the negative letters of~$w$ are in
bijection with the odd letters of the juxtaposition product.
Now consider the generating polynomial $B_\bfm(t,q,Z)$, as
defined in (1.9). Making use of (2.2) and of the usual
$q$-identities
$$
\leqalignno{
{1\over (u;q)_N}&=\sum_{n\ge 0} {N+n-1\brack
n}_q\,u^n;\cr
{N+n\brack n}_q&=\sum_{b\in \NIW(N),\,b_1\le
n}q^{\tot b};\cr
\noalign{\vskip-8pt}
}
$$
we have
$$\eqalignno{\noalign{\vskip-6pt}
{1+t\over (t^2;q^2)_{m+1}}\,B_\bfm(t,q,Z)
&=\sum_{s'\ge 0}(t^{2s'}+t^{2s'+1}){m+s'\brack s'}_{q^2}
\,B_\bfm(t,q,Z)\cr 
&=\sum_{s'\ge 0}t^{s'}{m+\lfloor s'/2\rfloor
\brack \lfloor s'/2\rfloor}_{q^2}
\,B_\bfm(t,q,Z)\cr
&=\sum_{s'\ge 0}t^{s'}\kern-8pt
\sum_{\scriptstyle b\in \NIW(m),\atop\scriptstyle 2b_1\le s'}
\kern-10pt q^{2\tot b}\,\sum_{w\in B_{\bfm}}t^{\fdes w}q^{\fmaj
w}Z^{{\rm neg}\,w}\cr
&=\sum_{s\ge 0}t^s\kern-5pt
\sum_{\scriptstyle b\in \NIW(m),\,w\in B_{\bfm}
\atop \scriptstyle 2b_1+\fdes w\le s}\kern-10pt
q^{2\tot b+\fmaj w}Z^{{\rm neg}\,w}.\cr}
$$
But using (5.2), (5.3), (5.4) we can write
$$\displaylines{
{1+t\over (t^2;q^2)_{m+1}}\,B_\bfm(t,q,Z)
=\sum_{s\ge 0}t^s\kern-8pt
\sum_{\scriptstyle b^{(1)},\ldots, b^{(r)},\atop\scriptstyle
\max\nolimits_{i}b^{(i)}_1\le s}\kern-10pt
q^{\tot b^{(1)}+\cdots+\tot b^{(r)}}\!\!
Z^{\odd b^{(1)}+\cdots+\odd b^{(r)}}\cr
\noalign{\vskip -8pt}
\noalign{\hbox{and}}
\sum_{\bfm}(1+t)B_{\bfm}(t,q,Z){u_1^{m_1}\cdots
u_r^{m_r}\over (t^2;q^2)_{1+|\bfm|}}
=\sum_{s\ge 0}t^s\prod_{1\le i\le r}\sum_{m_i\ge0}
\sum_{b^{(i)}}u_i^{m_i}q^{\tot b^{(i)}}\!\!Z^{\odd
b^{(i)}}\!\!,\cr 
\noalign{\vskip -5pt}}
$$
using the notation:
$|\bfm|:=m_1+\cdots+m_r$.

There remains to evaluate
$\sum_m\sum_{b}u^mq^{\tot b}Z^{\odd b}$, where the second sum
is over all nonincreasing words $b=b_1\ldots b_m$ of length~$m$
such that $b_1\le s$. Let $1\le i_1<\cdots<i_k\le m$ (resp.
$1\le j_1<\cdots<j_l\le m$) be the sequence of all integers~$i$
(resp.~$j$) such that $b_i$ is even (resp. $b_j$ is odd). Then,~$b$
is completely characterized by the pair $(b^e, b^o)$, where
$b^e:=(b_{i_1}/2)\ldots (b_{i_k}/2)$ and
$b^o:=((b_{j_1}-1)/2)\ldots((b_{j_l}-1)/2)$. Moreover,
$\tot b=2\tot b^e+2\tot b^o+|b^o|$. Hence,
$$\leqalignno{
\sum_{m\ge 0}\sum_{b, |b|=m, b_1\le s}\kern-12pt
u^mq^{\tot b}Z^{\odd b}
&=\kern-6pt\sum_{b,  b_1\le s}u^{|b|}q^{\tot b}Z^{\odd b}\cr
&=\kern-6pt\sum_{b^e,\, 2b^e_1\le s}u^{|b^e|}q^{2\tot b^e}
\kern-15pt
\sum_{b^o,\, 2b^o_1\le s-1}\kern-10pt
(uqZ)^{|b^o|}q^{2\tot b^o}\cr
&={1\over (u;q^2)_{\lfloor s/2\rfloor +1}}
{1\over (uqZ;q^2)_{\lfloor (s+1)/2\rfloor}}.\cr
}
$$
This completes the proof of Theorem~1.2.

\goodbreak
\centerline{\bf 6. Specializations}

\medskip
As has been seen in this paper a {\it $t$-graded}
form such as (1.10) has an ungraded (infinite) version, obtained
by multiplying the formula by $(1-t)$ and letting $t:=1$. This gives 
$$\leqalignno{
\sum_{\bfm}B_{\bfm}(q,Z){u_1^{m_1}\cdots
u_r^{m_r}\over (q^2;q^2)_{|\bfm|}}
&= \prod_{1\le i\le r}
{1\over (u_i;q^2)_{\infty}}
{1\over (u_iqZ;q^2)_{\infty}}&(6.1)\cr
&=\prod_{1\le i\le r}e_{q^2}(u_i)\,e_{q^2}(u_iqZ),\cr}
$$
where $e_{q^2}(u)$ denotes the usual $q$-exponential with
basis~$q^2$ ([GaRa90], p.~9).
Notice that
$B_{\bfm}(q,Z)$ is the generating polynomial for the class
$B_{\bfm}$ by the pair
$(\fmaj,\negg)$, namely
$$B_{\bfm}(q,Z)=
\sum_w q^{\fmaj w}Z^{\negg w}\quad(w\in B_{\bfm}).\leqno(6.2)
$$


On the other hand, Let
$$
{}^{\finv\kern-2pt}B_{\bfm}(q,Z):=
\sum_{w\in B_{\bfm}}q^{\finv w}Z^{\negg w}\leqno(6.3)
$$
be the generating polynomial for the class~$B_\bfm$ by the pair
$(\finv,\negg)$. Using a different approach (the derivation is not
reproduced in the paper; it is not quite straightforward), 
we can prove  the identity
$$\leqalignno{
{}^{\finv\kern-2pt}B_{\bfm}(q,Z)
&=(-Zq;q)_{m_1+\cdots+m_r}
{(q;q)_{m_1+\cdots+m_r}\over
(q;q)_{m_1}\cdots(q;q)_{m_r}}&(6.4)\cr
&=(-Zq;q)_{m_1+\cdots+m_r}
{m_1+\cdots+m_r\brack m_1,\ldots, m_r}_q,&\cr}
$$
using the traditional notation for the $q$-multinomial
coefficient. In general, $B_{\bfm}(q,Z) \not= 
{}^{\finv\kern-2pt}B_{\bfm}(q,Z)$. This can be shown by means of a
combinatorial argument. 

Let $Z:=1$ in (6.1) and make use of the $q$-binomial theorem
(see [An76], p.~17, or [GaRa90], chap.~1), on the one hand, and let $Z:=1$ in
(6.4), on the other hand. We get the evaluations
$$
B_{\bfm}(q)=(-q;q)_{m_1+\cdots+m_r}
{m_1+\cdots+m_r\brack m_1,\ldots, m_r}_q
={}^{\finv\kern-2pt}B_{\bfm}(q).\leqno(6.5)
$$
This shows that ``fmaj" and ``finv" are equidistributed over each
class $B_\bfm$, a property proved ``bijectively" in our first
paper [FoHa05a]. 

Next, let $q:=1$ in (6.4). We obtain
$$\displaylines{(6.6)\qquad
{}^{\finv\kern-2pt}B_{\bfm}(Z)
=(1+Z)^{m_1+\cdots+m_r}
{m_1+\cdots+m_r\choose m_1,\ldots, m_r}
\quad \bigl(=B_\bfm(Z)\bigr),\hfill\cr
\noalign{\hbox{an identity which is equivalent to}}
(6.7)\qquad
\sum_{\bfm}B_{\bfm}(Z){u_1^{m_1}\cdots
u_r^{m_r}\over |\bfm|!}
=\prod_{1\le i\le r}
\exp(u_i)\,\exp(u_iZ).\hfill\cr}
$$
Thus, the $q$-analog of (6.6) yields (6.4) with a combinatorial
interpretation in terms of the flag-inversion number ``finv,"
while (6.1) may be interpreted as $q$-analog of (6.7)
with an interpretation in terms of the flag-major index
number ``fmaj."

Finally, for $Z=0$, formula (1.10)  yields the identity
$$\sum_{\bfm}A_{\bfm}(t,q){u_1^{m_1}\cdots
u_r^{m_r}\over (t;q)_{1+|\bfm|}}
=\sum_{s\ge 0}t^s \prod_{1\le i\le r}
{1\over (u_i;q)_{s+1}},
$$
where $A_{\bfm}(t,q)$ is the generating polynomial for the class of
the rearrangements of the word $1^{m_1}2^{m_2}\ldots r^{m_r}$
by $(\des,\maj)$. As done by Rawlings [Ra79], [Ra80], the
polynomials
$A_{\bfm}(t,q)$ can also be defined by a recurrence relation
involving either the polynomials themselves, or their coefficients.

\bigskip
\centerline{\bf 7. The Signed-Word-Euler-Mahonian
polynomials}


\medskip
We end the paper by showing that the polynomials
$B_\bfm(t,q)=B_\bfm(t,q,Z)\mid_{Z=1}$ can be calculated
not only by their factorial generating function given by (1.10)
for $Z:=1$, but also by a {\it recurrence formula}.

\rem Definition|A sequence
$\Bigl(\kern-1.5pt B_{\bfm}(t,q)=\kern-3pt\sum \limits_{k\ge
0}\kern-2pt  t^k
B_{\bfm,k}(q)\kern-1.5pt\Bigr)\,(\bfm=(m_1,\ldots,m_r);\break
m_1\ge 0,\ldots, m_r\ge 0)$) of polynomials in two variables~$t$,
$q$, is said to be {\it signed-word-Euler-Mahonian}, if one of the
following four  {\it equivalent} conditions holds:

\goodbreak
\medskip
(1) The $(t^2,q^2)$-factorial generating function for the
polynomials 
$$
C_{\bfm}(t,q):=(1+t)B_{\bfm}(t,q)\leqno(7.1)
$$
is given by identity (1.10) when $Z=1$, that is,
$$\leqalignno{
\sum_{\bfm}C_{\bfm}(t,q){u_1^{m_1}\cdots
u_r^{m_r}\over (t^2;q^2)_{1+|\bfm|}}
&=\sum_{s\ge 0}t^s \prod_{1\le i\le r}
{1\over (u_i;q)_{ s+1}}.&(7.2)\cr}
$$

(2) For each multiplicity $\bfm$ we have:
$$
{C_{\bfm}(t,q)\over (t^2;q^2)_{1+|\bfm|}}
=\sum_{s\ge 0}t^s{m_1+s\brack s}_q\cdots {m_r+s\brack s}_q.
\leqno(7.3)
$$

\goodbreak\noindent

Let $\bfm+1_r:=(m_1,\ldots,
m_{r-1},m_r+1)$.

\medskip
(3) The following recurrence relation 
$$\displaylines{(7.4)\quad
(1-q^{m_r+1})B_{\bfm +1_r}(t,q)\hfill\cr
\hfill{}
=(1-t^2q^{2+2|\bfm|})B_{\bfm}(t,q)
-q^{m_r+1}(1-t)(1+tq)B_{\bfm}(tq,q),\cr}
$$
holds with $B_{(0,\ldots,0)}(t,q)=1$.

\medskip
(4) The following recurrence relation for the coefficients
$B_{\bfm,k}(q)$ 
$$\displaylines{(7.5)\quad
(1+q+\cdots+q^{m_r})B_{\bfm +1_r,k}(q)
=(1+q+\cdots+q^{m_r+k})B_{\bfm,k}(q)\hfill\cr
\hfill{}+q^{m_r+k}B_{\bfm,k-1}(q)
+(q^{m_r+k}+q^{m_r+k+1}+\cdots+
q^{2|\bfm|+1})B_{\bfm,k-2}(q),\cr}
$$
holds with $B_{(0,\ldots,0),0}(q)=1$ and
$B_{(0,\ldots,0),k}(q)=0$ for every~$k\not =0$.

\goodbreak
\proclaim Theorem 7.1. The conditions $(1)$, $(2)$, $(3)$ and
$(4)$ in the previous definition are equivalent.

\dem
The proofs of the equivalences $[(1)\Leftrightarrow (2)]$
and $[(3)\Leftrightarrow (4)]$ are easy and therefore omitted. For
proving the equivalence
$[(1)\Leftrightarrow (3)]$ proceed as follows. Let
$C(t,q;u_1,\ldots,u_r)$ denote the {\it right} side of (7.2) and
form the $q$-{\it difference}
$C(t,q;u_1,\ldots,u_r)-C(t,q;u_1,\ldots,u_{r-1},u_rq)$
applied to the sole variable~$u_r$. We get
$$\displaylines{(7.6)\quad
C(t,q;u_1,\ldots,u_r)-C(t,q;u_1,\ldots,u_{r-1},u_rq)\hfill\cr
\kern 1.5cm{}= \sum _{s\ge 0}{t^s\over (u_1;q)_{s+1} \ldots
(u_r;q)_{s+1} } -\sum _{s\ge 0}{t^s\over (u_1;q)_{s+1} \ldots
(u_rq;q)_{s+1} }\hfill\cr
\kern 1.5cm{}=\sum _{s\ge 0}{t^s\over (u_1;q)_{s+1} \ldots (u_r;q)_{s+1}}
\Bigl[1-{1-u_r\over 1-u_rq^{s+1}}\Bigr]\hfill\cr
\kern 1.5cm{}=u_r
\sum _{s\ge 0}{t^s\over (u_1;q)_{s+1} \ldots (u_r;q)_{s+1}}
\Bigl[1-q^{s+1}{1-u_r\over 1-u_rq^{s+1}}\Bigr]\hfill\cr
\kern 1.5cm{}=u_r\bigl(C(t,q;u_1,\ldots, u_r)-qC(tq,q;u_1,\ldots,
u_{r-1},u_rq)\bigr).\hfill\cr
}
$$
Now, let
$C(t,q;u_1,\ldots,u_r):=\sum_{\bfm}C_{\bfm}(t,q)u_1^{m_1}\cdots
u_r^{m_r}/ (t^2;q^2)_{1+|\bfm|}$ and express each
term~$C(\ldots)$ occurring in identity (7.6) as a factorial series
in the~$u_i$'s.
We obtain
$$\displaylines{\quad
\sum _{\bf m} (1-q^{m_r+1})
C_{\bfm+1_r}(t,q)  
{u_1^{m_1}\cdots u_r^{m_r+1}\over (t^2;q^2)_{2+|{\bf m}|}}
\hfill\cr
\kern1.5cm{}=\sum _{{\bf m}} (1-t^2q^{2+2\left|\bfm\right|})
C_{\bf m}(t,q) 
{u_1^{m_1}\cdots u_r^{m_r+1}\over (t^2;q^2)_{2+|{\bf m}|}}
\hfill\cr
\kern2.5cm{}-\sum _{{\bf m}} q^{m_r+1}
(1-t^2)C_{\bf m}(tq,q)
{u_1^{m_1}\cdots u_r^{m_r+1}\over (t^2;q^2)_{2+|{\bf m}|}}.
\hfill\cr
}
$$
Taking the coefficients of $u_1^{m_1}\ldots u_{r-1}^{m_{r-1}}
u_r^{m_r+1}$ yields
$$
(1-q^{m_r+1})C_{\bfm +1_r}(t,q)
=(1-t^2q^{2+2|\bfm|})C_{\bfm}(t,q)
-q^{m_r+1}(1-t^2)C_{\bfm}(tq,q),
$$
which in turn is equivalent to (7.4) in view of (7.1). All
the steps of the argument are reversible.\cqfd

\goodbreak
\rem Remark $1$|The fact that $B_{\bfm}(t,q)$ is the generating
polynomial for the class $B_{\bfm}$ by the pair $(\fdes,\fmaj)$
can also be proved by the insertion technique using (7.5). The
argument has been already developed in [ClFo95a, \S\kern2pt 6] for
ordinary words. Again let $m:=|\bfm|=m_1+\cdots+m_r$. With
each word from $B_{\bfm+1^r}$ associate~$(m_r+1)$ new words
obtained by {\it marking} one and only one letter equal to~$r$
or~$\overline r$. Let
$B_{\bfm+1^r}^*$ denote the class of all those {\it marked}
signed words. If $w^*=x_1\ldots x_i^*\ldots x_{m+1}$ is such a
word, where the $i$-th letter is marked (accordingly, equal to
either~$r$ or~$\overline r$), let $\mark_i$ be the number of
letters equal to~$r$ or~$\overline r$ in the right factor
$x_{i+1}x_{i+2}\ldots x_{m+1}$ and define:
$$
\fmaj^*w^*:=\fmaj w+\mark_i w^*.
$$
On the other hand, let
$$
\displaylines{
B_{\bfm ,k}(q):=\sum_{w\in B_{\bfm},\,\fdes w=k}q^{\fmaj w}.\cr
\noalign{\vskip-8pt}
\noalign{\hbox{Clearly,}}
\sum_{\scriptstyle w^*\in B_{\bfm+1^r}^*,
\,\scriptstyle \fdes w=k}
q^{\fmaj^* w^*}=
(1+q+\cdots+q^{m_r})B_{\bfm +1_r,k}(q).\cr}
$$
Now each word $w$ from the class $B_{\bfm}$ gives rise
to $2(m+1)$ {\it distinct} marked signed
words of length $(m+1)$, when the marked letter~$r$ or~$\overline
r$ is inserted between letters of~$w$, as well as in the beginning
of and at the end of the word. As in the case of the signed
permutations, we can verify that for each
$j=0,1,\ldots, 2m+1$ there is one and only one marked signed
word~$w^*$ of length~$(m+1)$ derived by insertion
such that $\fmaj^*w^*=\fmaj w+j$.

On the other hand, ``fdes" is not modified if~$r$ is inserted to the
right of~$w$, or if~$r$ or~$\overline r$ is inserted into a descent
$x_i>x_{i+1}$.  Furthermore, ``fdes" increases by one, if $x_1>0$
(resp. $x_1<0$) and~$\overline r$ (resp.~$r$) is inserted to the
left of~$w$. For all the other insertions ``fdes" increases by~2.

\goodbreak
Hence, all the marked signed words~$w^*$ from
$B_{\bfm+1^r}^*$, such that $\fdes w^*=k$ are derived by
insertion from three sources:

(i) the set $\{w\in B_{\bfm}:\fdes w=k\}$ and the contribution is:
\hfil\break
$(1+q+\cdots+q^{m_r+k})B_{\bfm,k}(q)$;

(ii) the set $\{w\in B_{\bfm}:\fdes w=k-1\}$ and the contribution
is: \hfil\break $q^{m_r+k}B_{\bfm,k-1}(q)$;

(iii) the set $\{w\in B_{\bfm}:\fdes w=k-2\}$
and the contribution is: \hfil\break $(q^{m_r+k}+\cdots+
q^{2|\bfm|+1})B_{\bfm,k-2}(q)$.\cqfd

\goodbreak
\rem Remark $2$|Let $\bfm:=1^{n}$ and $B_n(t,q)=B_{\bfm}(t,q)$, so that
$B_n(t,q)$ is now the generating polynomial for the set of signed {\it
permutations} of order~$n$.
Then (7.3), (7.4) and (7.5) become
$$
\displaylines{\rlap{(7.7)}\hfill
{(1+t)B_n(t,q)\over (t^2;q^2)_{n+1}}
=\sum_{s\ge 0}t^s(1+q+\cdots+q^s)^n\hfill\cr
(7.8)\ 
(1-q)B_n(t,q)=(1-t^2q^{2n})B_{n-1}(t,q)-q(1-t)(1+tq)B_{n-1}(tq,q).
\hfill\cr
(7.9)\quad
B_{n,k}(q)=(1+q+\cdots +q^k)B_{n-1,k}(q)\hfill\cr
\hfill{}+q^kB_{n-1,k-1}(q)
+(q^k+q^{k+1}+\cdots+q^{2n-1})B_{n-1,k-2}(q).\quad\cr}
$$
The last three relations have been derived by Brenti et al.
[ABR01], Chow and Gessel [ChGe04], Haglund et al. [HLR04].

\rem Concluding remarks|The statistical study of the
hyperoctahedral group~$B_n$ was initiated by Reiner ([Re93a],
[Re93b], [Re93c], [Re95a], [Re95b]). It had been rejuvenated by
Adin and Roichman [AR01] with their introduction of the
flag-major index, which was shown [ABR01] to be equidistributed
with the length function. See also their recent papers on the subject
[ABR05], [ReRo05]. Another approach to Theorems~1.1
and~1.2 would be to make use of the Cauchy identity for the
Schur functions, as was done in [ClFo95b].

\rem Acknowledgements|We should like to thank Christian Krattenthaler
and the anonymous referee for their careful readings and knowledgeable 
remarks.

\vglue 1cm\vskip0pt plus 5pt minus 5pt
\centerline{\bf References}
\medskip
{\eightpoint


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}


\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







