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\newtheorem{prop}[T]{Proposition}
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\def\tde{table de diff\'erence d'Euler }
\def\sii{si et seulement si }
\def\kc{{$k$-succession circulaire }}
\def\kcs{{$k$-successions circulaires }}
\def\kl{{$k$-succession lin\'eaire }}
\def\kls{$k$-successions lin\'eaires }
\def\kld{$k$-succession lin\'eaire de  deuxi\`eme \'esp\`ece  }
\def\klds{$k$-successions lin\'eaires de deuxi\`eme \'esp\`ece  }
\def\phib{\Phi }
\def\si{\pi }
\def\sid{\widehat{\sigma}^d  }
\def\sieps{\si=(\varepsilon, \sigma)}
\def\siep'{\si'=(\varepsilon',\sigma')}
\def\sig'{sign_{\si'}}
\def\sig{sign_{\si }}
\def\sign{\textrm{couleur}}
\def\phi1b{\varphi^b }
\def\phid{\delta}
\def\S{{\mathcal S}}
\newcommand\RISE{\mathop{\rm RISE}}
\newcommand\rise{\mathop{\rm rise}}
%\newcommand\des{\mathop{\rm des}}
\newcommand\exc{\mathop{\rm exc}}
\newcommand\fexc{\mathop{\rm fexc}}
\newcommand\couleur{\mathop{\rm couleur}}
%\newcommand\maf{\mathop{\rm maf}}
\newcommand\mafz{\mathop{\rm mafz}}
\newcommand\length{\mathop{\rm length}}
\newcommand\Der{{ \rm Der}}
%\newcommand\DES{{ \rm DES}}
\newcommand\Zder{{ \rm Zder}}
\newcommand\fix{\mathop{\rm fix}}
\newcommand\SIGN{{\textsc{COL}}}
\newcommand\FIX{\mathop{\rm FIX}}
\newcommand\bmaj{\mathop{ \rm bmaj}}
\newcommand\binv{\mathop{ \rm binv}}
\newcommand\bmajmil{\mathop{ \rm bmajMil}}
\newcommand\cbmaj{\mathop{ \rm cbMaj}}
\newcommand\cbinv{\mathop{ \rm cbinv}}
\newcommand\Rfinv{\mathop{ \rm Rfinv}}
%\newcommand\Maj{\mathop{ \rm Maj}}
\newcommand\bDes{\mathop{ \rm bDes}}
\newcommand\ZDer{{\mathop{\rm ZDer}}}
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\def\Rise{\mathop{\hbox{\small \rm RISE}}\nolimits}
\def\RIZE{\mathop{\hbox{\small \rm RIZE}}\nolimits}
%%%%%%%%%%%%%%%%%%%%%%%
\def\gln{G_{\ell,n}}
\def\glm{G_{\bf m}^\ell}
\def\sln{\Sigma_{\ell,n}}
\def\cln{C_{\ell} \wr S_n}
\def\c{\mathcal{C}}
\def\Exp{\textrm{Exp}}
\def\glnm{G_{\ell,n+m}}
\def\ns{ non-subexcedent }
\def\F{{\bf F}}
\def\f{{\bf\Phi^{-1}}}
\def\ph{{\bf\Phi}}
\def\s{ subexcedent }
\def\n{ negative }
\def\fh{Foata  and Han }
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%\DeclareMathOperator{\dep}{\it dp}
\DeclareMathOperator{\des}{des}
\DeclareMathOperator{\maj}{maj}
\DeclareMathOperator{\maf}{maf}
\DeclareMathOperator{\fdes}{fdes}
\DeclareMathOperator{\fmaj}{fmaj}
\DeclareMathOperator{\fmaf}{fmaf}
\DeclareMathOperator{\ndes}{ndes}
\DeclareMathOperator{\nmaj}{nmaj}
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\newcommand\mb{\boldmath}
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\newcommand\mc{\cal}
\def\maj{\mathop{\rm maj}\nolimits}
\def\fix{\mathop{\rm fix}\nolimits}
\def\maf{\mathop{\rm maf}\nolimits}
\def\mag{\mathop{\rm mag}\nolimits}
\def\inv{\mathop{\rm inv}\nolimits}
\def\des{\mathop{\rm des}\nolimits}
\def\FIX{\mathop{\rm FIX}\nolimits}
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\def\permn#1{#1_1\cdots #1_n}
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\def\resp{{\it resp}}
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\def\cl#1{{\mc #1}}
\def\mychoose#1#2{{\binom{#1}{#2}}}
\def\H{{\bf H}}
\def\p-{\Phi^{-1}}
%\newcommand\fmaj{\mathop{ \rm fmaj}}
\newcommand\finv{\mathop{ \rm finv}}
\newcommand\pcol{\mathop{ \rm pcol}\nolimits}
\newcommand\desbis{\mathop{ \rm des^*}}
\newcommand\finvbis{\mathop{ \rm finv^*}}
\def\qfd{\quad\rule[-1mm]{1.5mm}{3.5mm}}

\begin{document}
\title[Flag-major index and flag-inversion number on colored words]
{Flag-major index and flag-inversion number \\on colored words
and wreath product}

\newbox\Aut
\setbox\Aut\vbox{
\centerline{\sc
Hilarion L. M. Faliharimalala$^1$}
\centerline{and}
\centerline{\sc Arthur Randrianarivony$^2$}
\vskip18pt
\centerline{$^1$Universit\'e de Lyon, Universit\'e Lyon 1,}
\centerline{Institut Camille Jordan, UMR 5208 du CNRS,}
\centerline{F-69622, Villeurbanne Cedex, France}
\centerline{\footnotesize E-mail: \footnotesize\tt
heritianamihanta@yahoo.fr}
\vskip18pt
\centerline{$^2$D\'epartement de Math\'ematiques,
     Universit\'e d'Antananarivo}
\centerline{BP 906 Antananarivo Madagascar}
\centerline{\footnotesize WWW: \footnotesize\tt
arthur@univ-antananarivo.mg}
}
\author[H. L. M. Faliharimalala and A. Randrianarivony]{\box\Aut}
\date{}
\begin{abstract}
In [{\it Proc. Amer. Math. Soc.} {\bf 19} (1968), 236--240],
Dominique Foata constructed a map $\Phi$, called {\it
second fundamental transformation},  exchanging the integer-valued
statistics {\it inversion number} ``inv'' and {\it major index}
``maj'' on  words whose letters are  integers. Later,  Foata and Han
introduced the {\it flag-inversion number} ``finv'' and extended
$\Phi$ on signed words and permutations, showing that the {\it flag
major index} ``fmaj'' and  ``finv'' were equidistributed. In this
paper we give an extension of   $\Phi$  to {\it $\ell$-colored
words}. Using this extension, we show that the bistatistics
$(\fmaj,\desbis)$ and $(\finv,\pcol)$ are equidistributed,  where
\lq\lq$\pcol$\rq\rq\  is the sum of  color powers and
\lq\lq$\desbis$\rq\rq\ is a new statistic derived from
\lq\lq$\des$\rq\rq .
\end{abstract}

\maketitle

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\markright{\its S\'eminaire Lotharingien de
Combinatoire \bfs 62 \rms (2010), Article~B62c\hfill}
\def\thepage{}

\section{Introduction}
The {\it second fundamental transformation}, denoted by $\Phi$ and
described in~\cite{Fo} by Foata, is defined on finite
words whose letters are integers. If $\textbf{m}=(m_1,\cdots,m_r)$ is
a sequence of nonnegative integers, let $R_{\textbf{m}}$ be the
set of all rearrangements $w=x_1\,x_2\cdots x_m$ of the sequence
$1^{m_1}2^{m_2}\dots n^{m_r}$ where $m=m_1+m_2+\cdots +m_r$. The
transformation $\Phi$ maps each word $w$ to another word $\Phi(w)$ and
has the following properties:
\begin{enumerate}
 \item
 $\maj\,w=\inv\,\Phi(w);$
\item  $\Phi(w)$ is a rearrangement of $w$ and the restriction of
$\Phi$  to $R_{\textbf{m}}$ is a bijection of $R_{\textbf{m}}$ onto
itself.
\end{enumerate}
Further properties were proved later on by Foata and Sch\"utzenberger
\cite{FoSc}, and by Bj\"orner and Wachs \cite{BjWa}, in particular, when the
transformation is restricted to act on the symmetric group $S_r$.

The purpose of this paper is to extend the transformation $\Phi$ to
{\it $\ell$-colored words}.

Let $C_{\ell}$ be the $\ell$-cyclic group  generated by $\zeta = e^{2i
\pi/ \ell}$. By an {\it $\ell$-colored word}, we understand a pair
$(\varepsilon,x)$, where $\varepsilon\in (C_{\ell})^m$ and $x$
is a word of length $m$ whose letters are nonnegative integers.
For reasons which will appear later, if $w:=(\varepsilon,x)$ is an
$\ell$-colored word where
$\varepsilon=(\varepsilon_{1},\varepsilon_{2},\ldots,\varepsilon_{m})$
and $x=x_1\,x_2\cdots x_m$, we write $w:=w_1w_2\cdots w_m$ where
$w_j=\varepsilon_{j}x_j$ $(1\le j\le m)$. For any $j$ with $1\le j\le m$,
$\varepsilon_{j}$ is called the {\it color} of $w_j$ and, if
$\varepsilon_{j}=\zeta^{k_j}$, $k_j$ is the power of this color. For
small values of $\ell$, we shall use $k_j$ bars over $x_j$ instead of
$\zeta^{k_j}x_j$.

For example, if $w=\zeta^23\,\zeta^21\,\zeta^04\,\zeta1\,\zeta^23$,
then we write $w=\bar{\bar{3}} \bar{\bar{1}} 4 \bar{1}
\bar{\bar{3}}$.

\smallskip
Any $\ell$-colored word can be considered as a finite word over the
alphabet $\Sigma_\ell:=\{\xi j;\ \xi\in C_{\ell},\ j\ge 1\}$.

 Let $w:=w_1w_2\cdots
w_m:=\varepsilon_{1}x_1\,\varepsilon_{2}x_2\cdots
\varepsilon_{m}x_m$ be an $\ell$-colored word. We write
\begin{equation}\label{def_|w|}
\begin{matrix}
\hfill |w_i|\kern-8pt&:=x_i,\quad \quad\quad \quad  1\le i\le m;\hfill\\
\hfill |w|\kern-8pt&:=|w_1||w_2|\cdots |w_m|;\hfill
\end{matrix}
\end{equation}
and we define the statistic \emph{power-color} \lq\lq$\pcol$\rq\rq\  by
\begin{equation}\label{def_pcol}
\begin{matrix}
\hfill {\rm pcol}_iw\kern-6pt&:=\displaystyle\sum_{0\le
j\le\ell-1}j\chi(\varepsilon_{i}=\zeta^j),\quad \quad \quad
1\le i\le m;\hfill\\
\hfill {\rm pcol}\,w\kern-6pt&:=\displaystyle\sum_{1\le i\le m}\pcol_iw.\hfill
\end{matrix}
\end{equation}
If $\textbf{m}=(m_1,\cdots,m_r)$ is a sequence of nonnegative integers
such that $m_1+\cdots+m_r~=~m$, let $G_{\ell,\textbf{m}}$ be the
set of all $\ell$-colored words $w=w_1\,w_2\cdots w_m$ such that
$|w|\in R_{\textbf{m}}$. The class $G_{\ell,\textbf{m}}$ contains
$\ell^m{\binom m{ m_1,m_2,\ldots,m_r}}$ $\ell$-colored words. When
$m_1=m_2=\dots=m_r=1$,  the class $G_{\ell,\textbf{m}}$  is the wreath
product $C_{\ell} \wr S_r$ denoted by $G_{\ell,r}$. We define an
order relation on $\Sigma_\ell$ as follows:
\begin{equation}\label{ordre}
\zeta^j i >  \zeta^{j'}i' \Longleftrightarrow [j<j']
\quad\text{or}\quad [(j=j') \quad\text{and} \quad(i>i')].
\end{equation}
The restriction of this order  to the class of ordinary words (with
nonnegative letters) is the usual order.

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\pagenumbering{arabic}
\addtocounter{page}{1}
\markboth{\SMALL H. L. M. FALIHARIMALALA AND
A. RANDRIANARIVONY}{\SMALL FLAG-MAJOR INDEX AND FLAG-INVERSION
NUMBER ON COLORED WORDS}
%
%

As in \cite{FoHa}, the statistics ``inv'' and ``maj'' must be adapted
to {\it $\ell$-colored words} and correspond to classical statistics
when applied to ordinary words. Let
$$
(\omega;q)_n:=\begin{cases}
 1&{\rm if}\ n=0;\\
(1-\omega)(1-\omega q)\cdots (1-\omega q^{n-1})&{\rm if}\ n\ge 1;
\end{cases}
$$
denote the usual {\it $q$-shifted factorial}, and let
$$\begin{bmatrix}
m_1+m_2+\cdots +m_r\\
m_1,m_2,\ldots,m_r
\end{bmatrix}_q:=\frac{(q;q)_{m_1+m_2+\cdots
+m_r}}{(q;q)_{m_1}(q;q)_{m_2}\cdots(q;q)_{m_r}}
$$
be the {\it $q$-multinomial coefficient}.

With the order relation defined in (\ref{ordre}), the natural
extensions of the {\it flag-major index} ``fmaj'' and the {\it
flag-inversion number} ``finv'' introduced by Foata and Han
\cite{FoHa} to {\it $\ell$-colored words} are defined as follows: for
all $\ell$-colored word $w:=w_1w_2\cdots w_m$,
\begin{align*}
 {\rm fmaj}\,w&:=\ell\displaystyle\sum_{i=1}^{m-1}i\chi(w_i>w_{i+1})+\pcol\,w;\\
{\rm finv}\,w&:=
\underset{\xi\in
C_{\ell}}{\sum_{1\le i<j\le m}}\chi(\xi w_i>w_j)+{\pcol}\,w.
\end{align*}
Foata and Han defined $(-q;q)_m\left[\begin{smallmatrix}
m\\
m_1,m_2,\ldots,m_r
\end{smallmatrix}\right]_q$ as a $q$-analog of $2^m
\left(\begin{smallmatrix} m\\
{m_1,m_2,\ldots,m_r}
\end{smallmatrix}\right) $.
By analogy,
$\frac{(q^\ell;q^\ell)_m}{(q;q)_m}\left[\begin{smallmatrix}
m\\
m_1,m_2,\ldots,m_r
      \end{smallmatrix}\right]_q$ is a natural $q$-analog of
$\ell^m\left(\begin{smallmatrix} m \\{ m_1,m_2,\ldots,m_r}
\end{smallmatrix}\right) $.

We claim that
\begin{equation} \label{claim}
\frac{(q^\ell;q^\ell)_m}{(q;q)_m}\begin{bmatrix}
m\\
m_1,m_2,\ldots,m_r
\end{bmatrix}_q=\sum_{w\in G_{\ell,\textbf{m}}}q^{{\rm finv}\,w}.
\end{equation}
This can be established by induction on $m$.
Indeed, let us consider the bijective transformation
\begin{align*}
\varphi:\quad& G_{\ell,\textbf{m}}\longrightarrow
\{0,1,\ldots,\ell-1\}\times
\bigcup_{k=1}^{r}G_{\ell,\textbf{m-1}_k},\\
& w:=w_1w_2\cdots w_m\longmapsto (s,w'):=(\pcol_mw,w_1w_2\cdots w_{m-1})
\end{align*}
where $\textbf{m-1}_k=(m_1,m_2,\ldots,m_{k-1},m_k-1,m_{k+1},\ldots,m_r)$.
We have  $k=|w_m|$ and
\begin{align*}
\finv\,w&=\finv\,w'+s+
\underset{0\le j\le \ell-1}{\sum_{1\le i\le m-1}}
\chi(\zeta^{j}|w_i|>w_m)\\
&=\finv\,w'+ms+(m_{k+1}+\cdots+m_r)\chi(k<r).
\end{align*}
So,
\begin{align*}
 \displaystyle\sum_{w\in G_{\ell,\textbf{m}}}q^{{\rm
finv}\,w}&=\displaystyle\sum_{0\le s\le
\ell-1}q^{ms}\left(\displaystyle\sum_{w'\in
G_{\ell,\textbf{m-1}_r}}q^{{\rm finv}\,w'}+\displaystyle\sum_{1\le
k\le r-1}q^{m_{k+1}+\cdots+m_r}\displaystyle\sum_{w'\in
G_{\ell,\textbf{m-1}_k}}q^{{\rm finv}\,w'}
\right)\\
&=\displaystyle\frac{1-q^{\ell
m}}{1-q^m}\left(\displaystyle\sum_{w'\in
G_{\ell,\textbf{m-1}_r}}q^{{\rm finv}\,w'}+\displaystyle\sum_{1\le
k\le r-1}q^{m_{k+1}+\cdots+m_r}\displaystyle\sum_{w'\in
G_{\ell,\textbf{m-1}_k}}q^{{\rm finv}\,w'}
\right).
\end{align*}
By induction, for each $1\le k\le r$, we have
\begin{align*}
\displaystyle\sum_{w'\in G_{\ell,\textbf{m-1}_k}}q^{{\rm
finv}\,w'}&=\frac{(q^\ell;q^\ell)_{m-1}}{(q;q)_{m-1}}
\frac{(q;q)_{m-1}}{(q;q)_{m_1}\cdots(q;q)_{m_{k-1}}
(q;q)_{m_k-1}(q;q)_{m_{k+1}}\cdots(q;q)_{m_r}}\\
&=\frac{(q^\ell;q^\ell)_{m-1}(1-q^{m_k})}{(q;q)_{m}}
\frac{(q;q)_{m}}{(q;q)_{m_1}\cdots(q;q)_{m_k}\cdots(q;q)_{m_r}}.
\end{align*}
Thus,
\begin{align*}
\displaystyle\sum_{1\le k\le r-1}&q^{m_{k+1}+\cdots+m_r}\sum_{w'\in
G_{\ell,\textbf{m-1}_k}}q^{{\rm finv}\,w'}\\
&=\frac{(q^\ell;q^\ell)_{m-1}}{(q;q)_{m}}\frac{(q;q)_{m}}
{(q;q)_{m_1}\cdots(q;q)_{m_k}\cdots(q;q)_{m_r}}\sum_{1\le
k\le r-1}q^{m_{k+1}+\cdots+m_r}(1-q^{m_k})\\
&=\frac{(q^\ell;q^\ell)_{m-1}(q^{m_r}-q^m)}{(q;q)_{m}}
\frac{(q;q)_{m}}{(q;q)_{m_1}\cdots(q;q)_{m_k}\cdots(q;q)_{m_r}}
\end{align*}
and
\begin{align*}
\displaystyle\sum_{w\in G_{\ell,\textbf{m}}}q^{{\rm
finv}\,w}&=\displaystyle\frac{1-q^{\ell m}}{1-q^m}\left(
\frac{(q^\ell;q^\ell)_{m-1}(1-q^{m_r})}{(q;q)_{m}}
\frac{(q;q)_{m}}{(q;q)_{m_1}\cdots(q;q)_{m_k}\cdots(q;q)_{m_r}}
\right.\\
&\qquad\qquad+\left.\frac{(q^\ell;q^\ell)_{m-1}(q^{m_r}-q^m)}
{(q;q)_{m}}\frac{(q;q)_{m}}{(q;q)_{m_1}\cdots
(q;q)_{m_k}\cdots(q;q)_{m_r}}\right)\\
&=\frac{(q^\ell;q^\ell)_{m}}{(q;q)_{m}}
\frac{(q;q)_{m}}{(q;q)_{m_1}\cdots(q;q)_{m_k}\cdots(q;q)_{m_r}}\\
&=\frac{(q^\ell;q^\ell)_{m}}{(q;q)_{m}}\begin{bmatrix}
m\\
m_1,m_2,\ldots,m_r
\end{bmatrix}_q.
\end{align*}
This concludes the proof of the claim in \eqref{claim}.

\smallskip
We construct the extension $\widehat\Phi$ of the second fundamental
transformation $\Phi$ to $\ell$-colored words in the next
section. Define
\begin{equation}\label{colr*}
    \desbis\,w=\ell\,\des\,w-\des\,|w|+\pcol_1w,
\end{equation}
where $\des\,w:=\sum_{i=1}^{m-1}\chi(w_i>w_{i+1})$.

\smallskip
The main purpose of this paper is to prove the following theorem.
\begin{T}\label{equi}
The transformation $\widehat\Phi$ constructed in
Section~\ref{hat_Phi} has the following properties
 \begin{enumerate}
   \item For  every $\ell$-colored word $w$, $(\fmaj,\desbis)\,w
=(\finv,\pcol)\,\widehat\Phi(w)$;
   \item The restriction of $\widehat\Phi$ to each class
$G_{\ell,\textbf{m}}$ is a bijection of $G_{\ell,\textbf{m}}$ onto
itself.
 \end{enumerate}
 \end{T}

\begin{cor}
For each $m=(m_1,m_2,\dots,m_r)$, the bistatistics $(\fmaj,\desbis)$
and $(\finv,\break\pcol)$ are equidistributed on $G_{\ell,\textbf{m}}$.
\end{cor}
\begin{exe}{\rm
 Let us consider the hyperoctahedral group of order 2.
$$
\begin{array}{|c|c|c|c|c|c|c|c|c|}
  \hline
  &&&&&&&&\\
   w&1\,2 &\bar{1}\,2 & 1\,\bar{2} &\bar{1} \,\bar{2} &2 \,1 &\bar{2}
\,1 &2 \,\bar{1} &\bar{2} \,\bar{1} \\
    \hline
    \hline
\fmaj\,w &0&1&3&2&2&1&3&4\\
    \hline
    \desbis w&0&1&2&1&1&0&1&2\\
        \hline
        \hline
   \finv\,w&0&1&2&3&1&2&3&4\\
       \hline
    \pcol\,w&0&1&1&2&0&1&1&2\\
        \hline
\end{array}
$$}
\end{exe}
Now consider the  statistic $\Rfinv$  defined on the hyperoctahedral group
of order $n$ as follows:
 $$\Rfinv\,w={\rm inv}\,w+\sum_{i=1}^n|w_i|\chi(w_i<0).$$
If one uses the natural order relation on $[-n,n]$ given by
\begin{equation}\label{ordre naturel}
-n\,<\,-(n-1)\,<\,\cdots\,<\,-1\,<\,1\,<\,\cdots\,<\,(n-1)\,<\,n,
\end{equation}
 Brenti \cite{Br} shows that  $\finv$ coincides with the traditional
\emph{length function}, and
Adin  and Roichman \cite{AdRo}  proved that $\Rfinv$  and $\fmaj$ are
equidistributed on the hyperoctahedral group.

Back to the order relation \eqref{ordre} on $[-n,n]$, i.e.,
$$
-1\,<\,\cdots\,<\,-(n-1)<\,-n\,<\,1\,<\,\cdots\,<\,(n-1)\,<\,n
$$
one has
$$\
\text {\it length function}  \not=\Rfinv \quad \text{and}\quad \finv\not=\Rfinv,
$$
but we observe that  $\Rfinv$ remains equidistributed with $\fmaj$, and
we prove that its extension  to the wreath product  is also Mahonian. We
have the following theorem.
\begin{T}\label{t Rfinv}
The statistic $\Rfinv$ defined on the wreath product $C_{\ell} \wr S_n$ by
 \begin{equation}\label{Rfinv}
    \Rfinv\,w= \inv\,w+\sum_{i=1}^n|w_i|\pcol_iw
\end{equation}
is Mahonian.
\end{T}
By a result of Haglund, Loehr and Remmel  \cite{HaLoRe},
we obtain the following corollary.
\begin{cor}We have
\begin{equation}\label{haglund}
\sum_{\sigma \in G_{\ell,n}}q^{\Rfinv\sigma}=\sum_{\sigma \in
G_{\ell,n}}q^{\finv\sigma}=\frac{(q^{\ell};q^{\ell})_n}{(1-q)^n}.
\end{equation}
\end{cor}
\section{ The construction of the transformation $\widehat\Phi$}\label{hat_Phi}
Let us  recall the second fundamental transformation $\Phi$ from
\cite{Fo}. First, for each  integer  $x$,  we recall the
transformation $\gamma_x$. Let $w=x_1\,x_2\cdots x_m$ be a  word with
positive letters.  The first (respectively last) letter $x_1$ (respectively $x_m$)
is denoted by $F(w)$ (respectively $L(w)$). If $L(w)\le x$ (respectively $L(w)>x$),
$w$ admits the unique factorization
$$
(u_1y_1,u_2y_2,\cdots,u_py_p)
$$
called its {\it x-right-to-left factorization} having the following properties:
\begin{enumerate}
 \item each $y_i$ $(1\le i\le p)$ is a letter verifying $y_i\le x$
(respectively $y_i>x$);
\item each $u_i$ $(1\le i\le p)$ is a factor which is either empty or
has all its letters greater than (respectively smaller than or equal to)
$x$.
\end{enumerate}
Then, the bijective transformation $\gamma_x$  maps
$w=u_1y_1u_2y_2\dots u_py_p$ to the  word
$$
\gamma_x(w)=y_1u_1y_2u_2\cdots y_pu_p.
$$
Foata defined $\Phi(w)$  by induction on the length of $w$. If $w$ has
length one, then
$\Phi(w)=w$.
If it  has more than one letter, write the word as $vx$ where $x$ is
the last letter and define $\Phi(vx)$ to be the juxtaposition product
\begin{equation}\label{def_Phi}
\Phi(vx):=\gamma_x(\Phi(v))x.
\end{equation}
We now define $\widehat\Phi$ as follows.
For each   word $u=x_1\,x_2\,\cdots\,x_m$ with nonnegative letters and
each element $\epsilon:=(\epsilon_1,\cdots,\,\epsilon_m)$ of
$(\mathcal{C}_{\ell})^m$, we denote by $\Psi_{u}(\epsilon)$ the element
$\epsilon'=(\epsilon'_1,\cdots,\,\epsilon'_m)$ of
$(\mathcal{C}_{\ell})^m$ defined as follows:
\begin{equation}\label{grand psi}
\begin{cases}
    \epsilon'_i=\frac{\epsilon_i}{\epsilon_{i+1}}
\zeta^{-\chi(x_i>x_{i+1})}&\text {if } i<m,\\
      \epsilon'_m=\epsilon_m&\text {if }i=m.
    \end{cases}
\end{equation}
Let $w:=(\epsilon,u)$ be an $\ell$-colored word $(u=|w|)$. Define
\begin{equation}\label{def_hat_Phi}
\widehat\Phi(w)=(\Psi_{u}(\epsilon),\Phi(u)).
\end{equation}
\begin{exe}{\rm
Let us take  $\ell=4$ and $w=\bar{\bar{3}} \bar{1} 4 \bar{1}
\bar{\bar{3}}$. We have
$$
w=((\zeta^2,\zeta, 1, \zeta, \zeta^2),3 1 4 1 3).
$$
By construction of $\Phi$ (relation (\ref{def_Phi})), we have
\begin{align*}
\Phi(3)&=3,\\
 \Phi(31)&=\gamma_1(\Phi(3))1=\gamma_1(3)1=31,\\
\Phi(314)&=\gamma_4(\Phi(31))4=\gamma_4(31)4=314,\\
\Phi(3141)&=\gamma_1(\Phi(314))1=\gamma_1(314)1=3411,\\
\Phi(31413)&=\gamma_3(\Phi(3141))3=\gamma_3(3411)3=31413.
\end{align*}
In the other hand,
 $$
\epsilon'_1=\frac{\zeta^{2}}{\zeta} \zeta^{-1}=1,\quad
\epsilon'_2=\frac{\zeta}{1} =\zeta,\quad
\epsilon'_3=\frac{1}{\zeta^{1}} \zeta^{-1}=\zeta^2,\quad
\epsilon'_4=\frac{\zeta^{1}}{\zeta^{2}}=\zeta^3,\quad
\epsilon'_5=\zeta^{2}.
$$
Therefore,
$$
w'=\widehat\Phi(\bar{\bar{3}} \bar{1} 4 \bar{1}
\bar{\bar{3}})=3\bar{1}\bar{\bar{4}}\bar{\bar{\bar{1}}}\bar{\bar{3}}.
$$}
We have $(\fmaj,\desbis)w=(\finv,\pcol)w'=(34,8)$.
\end{exe}
\section{ Proof of Theorem \ref{equi}}
\begin{lem}\label{lemme central}
Let $w:=w_1w_2\cdots w_m$ be an $\ell$-colored word. With notation
in relation $(\ref{def_pcol})$, we have
\begin{equation}\label{def equi finv}
\finv\,w= \inv\,|w|+ \sum_{i=1}^m i\pcol_iw.
\end{equation}
\end{lem}

\begin{proof}
 For all integers $i,j,k$ such that $1\le i<j\le m$ and $0\le k\le
\ell-1$, one has:
$$
\chi(\zeta^{k}|w_i|>w_j)=\chi(k<\pcol_jw)+\chi(k=\pcol_jw)\chi(|w_i|>|w_j|).
$$
Thus,
$$
\finv\,w=\sum_{j=1}^m(j-1)\pcol_jw+\inv\,|w|+\sum_{j=1}^m\pcol_j\,
w=\inv\,|w|+\sum_{j=1}^mj\pcol_jw.
$$
\end{proof}

Now, slightly abusing notation,
let $w=(\epsilon,|w|)$ be an $\ell$-colored word of length $m$ and
$w'=(\epsilon',|w'|):=\widehat\Phi(w)$.
For each $i$ such that  $1\le i\le m-1$, we have
\begin{enumerate}
\item[--] if $|w_i|\le |w_{i+1}|$, then
$\epsilon'_i=\displaystyle\frac{\epsilon_i}{\epsilon_{i+1}}
=\zeta^{\pcol_iw-\pcol_{i+1}w}$;
\item[--] if $|w_i|>|w_{i+1}|$, then
$\epsilon'_i=\displaystyle\frac{\epsilon_i}{\epsilon_{i+1}}\zeta^{-1}
=\zeta^{\pcol_iw-\pcol_{i+1}w-1}$.
\end{enumerate}
\noindent
So,
\begin{align*}
\pcol_iw'&=[\pcol_iw-\pcol_{i+1}w
+\ell\chi(\pcol_iw<\pcol_{i+1}w
)]\chi(|w_i|\le |w_{i+1}|)\\
&\kern2cm+[\pcol_iw-\pcol_{i+1}w-1
+\ell\chi(\pcol_iw\le \pcol_{i+1}w
)]\chi(|w_i|>|w_{i+1}|)\\
&= \pcol_iw-\pcol_{i+1}w+\ell\chi(\pcol_iw<\pcol_{i+1}w)\\
&\kern2cm +\ell\chi(\pcol_iw=\pcol_{i+1}w
)\chi(|w_i|>|w_{i+1}|)-\chi(|w_i|>|w_{i+1}|).
\end{align*}
Thus, we have
\begin{align*}
\pcol\,w'&=\sum_{i=1}^m\pcol_iw'\\
&=\sum_{i=1}^{m}\pcol_iw-\sum_{i=2}^{m}\pcol_iw
+\ell\sum_{i=1}^{m-1}[\chi(\pcol_iw<\pcol_{i+1}w)\\
&\kern2cm +\chi(\pcol_iw=\pcol_{i+1}w)
\chi(|w_i|>|w_{i+1}|)]-\sum_{i=1}^{m-1}\chi(|w_i|>|w_{i+1}|)\\
&= \pcol_1w+\ell\des\,w-\des\,|w|\\
&=\desbis\,w,
\end{align*}
and, by $\Phi$,
\begin{align*}
\finv\,w'&=\inv\,|w'|+\sum_{i=1}^m i\pcol_iw'\\
&=\maj\,|w|+\sum_{i=1}^{m}i\pcol_iw-\sum_{i=1}^{m}(i-1)\pcol_iw\\
&\kern1cm +\ell\sum_{i=1}^{m-1}i[\chi(\pcol_iw<\pcol_{i+1}w)
+\chi(\pcol_iw=\pcol_{i+1}w)
\chi(|w_i|>|w_{i+1}|)]\\
&\kern1cm -\sum_{i=1}^{m-1}i\chi(|w_i|>|w_{i+1}|)\\
&=\maj\,|w|+\sum_{i=1}^{m}\pcol_iw+
\ell\sum_{i=1}^{m-1}i\chi(w_i>w_{i+1})-\maj\,|w|\\
&= \ell\sum_{i=1}^{m-1}i\chi(w_i>w_{i+1})+\pcol\,w\\
&=\fmaj\,w.
\end{align*}
Finally, we show that
$\widehat\Phi$ is a bijection of $G_{\ell,\textbf{m}}$ onto itself.
Indeed, let $w':=(\epsilon', u')$ be an element of
$G_{\ell,\textbf{m}}$. By the relation (\ref{grand psi}), if
$w:=(\epsilon,u)$ is an element of $G_{\ell,\textbf{m}}$ such that
$\widehat\Phi(w)=w'$, then
$u=\Phi^{-1}(u')$, $\epsilon_m=\epsilon'_m$ and, for $i<m$,
$$\epsilon_i=\displaystyle\epsilon'_m\displaystyle\prod_{i\le j\le
m-1}\epsilon'_j\zeta^{\chi(x_j> x_{j+1})},$$
where $u:=x_1x_2\cdots x_m$.

This concludes the proof of Theorem~\ref{equi}.

\section{Proof of Theorem \ref{t Rfinv}}
Consider the following transformations:

\medskip\noindent
$\rhd$ \emph{\textbf{Transformation $\rho$}}
$$\rho:\gln \longrightarrow \gln$$
$$
w=(\epsilon,|w|) \longmapsto\rho(w)=w'=(\epsilon',|w'|),
$$
where
$$
|w'|=|w|^{-1} \quad {\rm and} \quad \epsilon'_i=\epsilon_{|w|^{-1}(i)};
$$

\medskip\noindent
$\rhd$ \emph{\textbf{Transformation $\tau$}}\\
For each  $\alpha=(\alpha_1,\alpha_2,\dots,\alpha_n)$ in
$[0,\ell-1]^n$, put
$\Sigma_\alpha=\{\zeta^{\alpha_1},\,
\zeta^{\alpha_2}2,\,\dots,\,\zeta^{\alpha_n}n\}$,
and let $G_\alpha$ be the class of {\it $\ell$-colored permutations}
whose letters are in $\Sigma_\alpha$, i.e.,
$$
G_\alpha=\{w=w_1w_2\cdots w_n\in \gln: \pcol_iw=\alpha_i
\text { for all } i\in[n]\}.
$$
Note that $\sharp G_\alpha=n!$. We denote by $I_\alpha$ the increasing
bijection from $[n]$ to $\Sigma_\alpha$, and we define $\tau$ for
each class $G_\alpha$ by $\tau(w)=w'_1w'_2\dots w'_n$, where
$$
w'_i=I_\alpha(|w_i|).
$$
\begin{lem}\label{lemme central2}
For all $w\in \gln$, we have
$$ \finv\,w=\Rfinv \tau\circ\rho(w). $$
\end{lem}
\begin{proof}[Proof of Lemma~\ref{lemme central2}]
Let $w=w_1w_2\cdots w_n\in \gln$. Consider the auxiliary statistics
\begin{align*}
\wp(w)&:=\displaystyle\sum_{i=1}^ni\pcol_iw;\\
\Im(w)&:=\displaystyle\sum_{i=1}^n|w_i|\pcol_iw;\\
|\text {inv}|\,w&:=\inv\,|w|.
 \end{align*}
It is easy to see that $\rho$  is an involution preserving $|\text
{inv}|$
and transforming $\wp$ into $\Im$ and vice versa:
$$
(|\text {inv}|,\wp)w=(|\text {inv}|,\Im)\rho(w),
$$
and $\tau$ preserves $\Im$ and transforms $|\text {inv}|$ into $\inv$,
i.e.,
$$(|\text {inv}|,\Im)w=(\inv,\Im)\rho(w).$$
By Lemma~\ref{lemme central}, we have
\begin{align*}
\finv\,w&=|\text {inv}|\,w+\wp(w)\\
&=|\text {inv}|\rho(w)+\Im\,\rho(w)=\inv\, \tau\circ \rho(w)+\Im\,\tau\circ \rho(w)\\
&=\Rfinv \tau\circ \rho(w).
\end{align*}
\end{proof}

\begin{exe}{\rm
$w=5\bar{\bar{3}}\bar{1}2\bar{4}$, $\finv\,w=18$;
$\rho(w)=\bar{3}4\bar{\bar{2}}\bar{5}1$. Let
$\alpha=(0,2,1,0,1)$. $I_\alpha$ is defined as follows:
$$
\begin{array}{|c|c|c|c|c|c|}
  \hline
i&1&2&3&4&5\\
  \hline
  I_\alpha(i)&\bar{\bar{2}}&\bar{3}&\bar{5}&1&4\\
  \hline
\end{array}
$$
So
$$
\tau\circ\rho(w)=w'=\bar{5}1\bar{3}4\bar{\bar{2}}\qquad {\rm
and}\qquad \Rfinv\,w'=18.
$$
}
\end{exe}
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\end{document}
 