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\begin{document}
\bibliographystyle{plain}
\title{A combinatorial formula for the linearization \\
coefficients of general Sheffer polynomials}
\author{Dongsu Kim\\
{\small Department of Mathematics, KAIST, Taejon 305-701, Korea}\\ {\small
Email address: dskim@math.kaist.ac.kr}\\ and\\ Jiang Zeng\\ {\small Institut
Girard Desargues, Universit\'e Claude Bernard (Lyon I), France}\\ {\small Email
address: zeng@desargues.univ-lyon1.fr}}
\date{June 13, 2000}
\maketitle
\begin{abstract}
We prove a formula for the linearization coefficients of the general Sheffer
polynomials, which unifies all the special known results for Hermite, Charlier,
Laguerre, Meixner and Meixner-Pollaczek polynomials. Furthermore, we give a new
and explicit real version of the corresponding formula for Meixner-Pollaczek
polynomials. Our proof is based on some explicit bijections and sign-reversing
weight-preserving involutions.
\end{abstract}
{\bf Running head}: \emph{Combinatorics of Sheffer polynomials}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Given a sequence of formal orthogonal polynomials $p_n(x)$ ($n\geq 0$) with
respect to linear functional~$\L$, the values $\L(\prod_{i=1}^mp_{n_i}(x))$ are
usually called the linearization coefficients of the polynomials $p_n(x)$ (see
\cite{askey}). The problem of finding interesting formulas for the
linearization coefficients of the classical polynomials have attracted much
attention in the last three decades (see \cite{AI,DV, EG,FZ,Ge,Vi,Ze1} and the
references cited there). Once the moment sequence $\L(x^n)$ is determined, this
problem can be considered as a formal algebraic calculus. From this point of
view, Viennot~\cite[Ch. 1]{Vi} combinatorially proved the Favard theorem, using
the combinatorial interpretations of the moments and polynomials.

One important class of polynomials is the {\em Sheffer polynomials}, which
includes the classical Hermite, Charlier, Laguerre, Meixner and
Meixner-Pollaczek polynomials as special cases. Recall that the monic Sheffer
polynomials $P_n(x)$ ($n\geq0$) are defined by the three-term recurrence
relations \cite[p. 164]{Ch}:
\begin{equation}
P_{n+1}(x)=\bigl(x-(\alpha\beta+nu_3+nu_4)\bigr)P_n(x)- n(\beta
+n-1)u_1u_2P_{n-1}(x),\quad n\geq0,
\label{sheffer}
\end{equation}
with $P_{-1}(x)=0$ and $P_0(x)=1$.

There are two known combinatorial interpretations of the linear functional
corresponding to the above Sheffer polynomials. The first interpretation using
\emph{linear permutation statistics} is due to
Viennot~\cite[Ch. 2]{Vi}, who proved the orthogonality by a bijection of
Fran\c{c}on and Viennot. Another interpretation using
\emph{permutations statistics} related to
cycle structures was given by Zeng~\cite{Ze3}. As mentioned in \cite{Ze3} the
two interpretations are transformed from one to another through Foata's first
fundamental transformation~\cite{Fo}. For our purpose we need only the second
interpretation related to the cycle structures of permutations, that we 
recall below.

For any positive integer $n$, let $\S_n$ denote
the set of permutations of $[n]:=\{1,\ldots, n\}$. 
Given a permutation $\sigma\in \S_n$, a value
$i\in [n]$ is called a
\emph{peak or ascent-descent}, if $\sigma^{-1}(i)<i>\sigma(i)$;
\emph{valley or descent-ascent}, if $\sigma^{-1}(i)>i<\sigma(i)$;
\emph{double-ascent or ascent-ascent}, if $\sigma^{-1}(i)<i<\sigma(i)$;
\emph{double-descent or descent-descent}, if $\sigma^{-1}(i)>i>\sigma(i)$;
\emph{fixed point}, if $\sigma(i)=i$.
Let $\pic\sigma$ (resp. $\cc\sigma$, $\dm\sigma$, $\dd\sigma$ and $\fix\sigma$)
be the number of peaks (resp. valleys, double-ascents, double-descents and
fixed points) of $\sigma$. Note that $\pic\sigma=\cc\sigma$. Furthermore, a
value $i$ is called \emph{excedance} (resp. \emph{decedance}) if $\sigma(i)>i$
(resp. $\sigma(i)<i$). Let $\dec\sigma$ (resp. $\exc\sigma$) be the number of
decedances (resp. excedences) of $\sigma$. It is easy to see that
$\dec\sigma=\ad\sigma+\dd\sigma$ and $\exc\sigma=\aa\sigma+\da\sigma$. Finally
each permutation can be decomposed into disjoint {\em cycles}. Let $\cyc\sigma$
denote the number of cycles of $\sigma$. Consider the linear functional ${\L}:
K[x]\to K$, where $K$ is a commutative ring of characteristic 0, defined by its
values on the monomials $x^n$ :
\begin{equation}
{\L}(x^n)=\sum_{\sigma\in\S_n}u_1^{\pic\sigma}u_2^{\cc\sigma}
u_3^{\dm\sigma}u_4^{\dd\sigma}\alpha^{\fix \sigma}\beta^{\cyc\sigma}.
\label{momsheff}
\end{equation}
For instance, the first three values of ${\L}(x^n)$ are ${\L}(x)=\alpha \beta$,
${\L}(x^2)=\alpha^2\beta^2+u_1u_2\beta $ and
${\L}(x^3)=\alpha^3\beta^3+3u_1u_2\alpha\beta^2+u_1u_2(u_3+u_4)\beta$. It was
then proved in \cite{Ze3} that the general Sheffer polynomials defined
by~(\ref{sheffer}) are orthogonal with respect to the linear functional ${\L}$:
\begin{equation}\label{eq:orth}
{\L}(P_{n_1}(x)P_{n_2}(x))={n_1}!(\beta)_{n_1}(u_1u_2)^{n_1}\delta_{n_1\,n_2}.
\end{equation}

The following important definitions follow Foata and Zeilberger~\cite{FZ}.
Given $m$ nonnegative integers $\n=(n_1,\ldots,n_m)$, let $A_1,\ldots,A_{m-1}$
and $ A_m$ be $m$ \emph{disjoint totally ordered} sets such that $|A_i|=n_i$
for $i\in [m]$ and set$A_{\n}=A_1\cup\cdots\cup A_m$. For $a\in A_{\n}$, an
integer $i\in[m]$ is called \emph{color} of $a$ if $a\in A_i$, written as
$c(a)=i$. Clearly the set $A_{\n}$ is totally ordered with respect to the
lexicographical order, i.e., for two elements $a$ and $b$ in $A_{\n}$ we say
that $a<b$ iff $c(a)<c(b)$ or $c(a)=c(b)$ and $a<b$. So the above classical
weight function on $\S_n$ is still valid for any permutation of $A_{\n}$
considered as a totally ordered set. A permutation $\sigma$ of $A_{\n}$ is
called a \emph{generalized derangement} if $c(a)\ne c(\sigma(a))$ for all $a\in
A_{\n}$. Let $\D_{\n}$ be the set of generalized derangements of $A_{\n}$. Note
that a generalized derangement $\pi$ of $A_{\n}$ must also be an \emph{ordinary
derangement}, i.e. $\pi(a)\not=a$ for all $a\in A_{\n}$ and that $a>\pi(a)$ if
and only if $c(a)>c(\pi(a))$.

Foata and Zeilberger~\cite{FZ} then proved that the linearization coefficients
of Laguerre polynomials are the generating functions of $\D_{\n}$ with respect
to the number of cycles. In \cite{Ze2} the second author proved that for each
classical class of Sheffer polynomials the linearization coefficients are the
generating functions of derangements with respect to the corresponding weight
functions of their moment sequences. In this paper, we will refine the above
results and give a formula for the values $\L(\prod_{i=1}^mP_{n_i}(x))$ for
$m\geq 1$ in its full generality. To this end, we need to introduce a new
statistic on the set of generalized derangements.
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\caption{The lattice path representation of a cyclic permutation
$\pi=(1\,4\,5\,8\,10\,9\,2\,7\,11\,6\,3)$ of $A_{\mathbf n}$ with $\mathbf
n=(1,3,2,3,1,1)$, where the elements of same color are placed on the same
level.}
\label{Fig:match}
\end{figure}

Consider a generalized derangement $\pi$ of $A_{\n}$, factor it into disjoint
cycles and identify each cycle of $\pi$ with a path $P : s_1\to s_2\to
\dots\to s_l\to s_1$ if $\sigma(s_i)=s_{i+1}$ for $1\leq i\leq l-1$, where
$s_1$ is the smallest element in the cycle. Clearly the valleys (local minima)
and peaks (local maxima) appear alternatively in the path $P$ and their numbers
are equal. Suppose there is a double-ascent $s_p$ and a double-descent $s_q$
($p<q$) in the path $P$ such that $c(s_{p})=c(s_{q})$. Then the pair $(s_p,\
s_q)$ is called a {\em color match} of $\pi$ if $c(s_{j}) \geq c(s_{p})$
($=c(s_{q})$) for all $j\in \{p,\ldots, q\}$. Let $\mat\sigma$ denote the
number of {\em color matches} of $\pi$.

\begin{example} For $\mathbf n=(1,3,2,3,1,1)$, let $A_1=\{1\}$,
$A_2=\{2,\, 3,\,4\}$, $A_3=\{5,\, 6\}$, $A_4=\{7,\, 8,\,9\}$, $A_5=\{10\}$ and
$A_6=\{11\}$. Consider the cyclic permutation $\pi\, :\, 1\to 4\to 5\to 8\to
10\to 9\to 2\to 7\to 11\to 6\to 3\to 1$ of $A_{\mathbf n}$. In order to
illustrate the different statistics on $\pi$, it is instructive to draw a
\emph{lattice path} in the plan ${\bf N}\times {\bf N}$ by placing $s_i$ at
$(i,j)$ if $s_i\in A_j$. We give such a representation in
Figure~\ref{Fig:match}, called \emph{lattice path representation} of $\pi$. It
follows that $\aa\pi=4$, $\dd\pi=3$, $\da\pi=2$, $\ad\pi=2$ and $\mat\pi=2$,
for there are two color matches: $(4,\, 3)$ of color `2' and $(8,\, 9)$ of
color `4'. Note that (5,\, 6) is not a color match.
\end{example}

We are now in a position to state our main result.
\begin{theorem} The linearization coefficient of Sheffer polynomials is the
generating functions of generalized derangements, i.e.
\begin{equation}
{\L}\left(\prod_{i=1}^mP_{n_i}(x)\right)=\sum_{\sigma\in{\D}_{\n}}
u_1^{\pic\sigma+\mat\sigma}u_2^{\cc\sigma+\mat\sigma}
u_3^{\dm\sigma-\mat\sigma}u_4^{\dd\sigma-\mat\sigma}\beta^{\cyc\sigma}.
\label{eq:D}
\end{equation}
\label{thm:D}
\end{theorem}

\begin{remark}
The number of color matches is at most the number of double-ascents or
double-descents, i.e. $\aa\sigma,\dd\sigma\geq\mat\sigma$, so each term in the
above sum is a monomial with nonnegative exponents.
\end{remark}

The proof of theorem~\ref{thm:D} is of combinatorial nature, this is in the
same vein as \cite{DV,deM,Ge}, where the authors were looking for a unified and
more transparent approach to the linearization problems. More precisely, we
first set up a combinatorial model for the Sheffer polynomials, combining with
the combinatorial interpretation for the moment $\L$ in (\ref{momsheff}) we
then interpret the linearization coefficients as the generating functions of
some finite structures. Theorem~\ref{thm:D} is finally proved by constructing
some explicit weight-preserving sign-reversing involutions on these structures
(See lemma~\ref{thm:E} and theorem~\ref{thm:Phi}). Recall that a {\em
weight-preserving sign-reversing} (wpsr) involution~$\phi$ on a set $S$ is an
involution such that for all $x\in S$, $\phi(x)\ne x$ implies
$w(\phi(x))=-w(x)$. The set $\Fix\Phi=\{x\in S:\Phi(x)=x\}$ is called the {\em
fixed set of $\Phi$}.

For $\n=(n_1,\ldots, n_m)$, set $|\n|=n_1+\cdots+n_m$. For convenience, we will
often identify $A_\n$ with $[|\n|]$, that is
$A_i=\{n_1+\cdots+n_{i-1}+1,\dots,n_1+\cdots+n_i\}$ for $1\leq i\leq m$.

This paper is organized as follows. In section 2 we derive all the known
results about the linearization coefficients of the classical Hermite,
Charlier, Laguerre, Meixner and Meixner-Pollaczek polynomials from
theorem~\ref{thm:D}. We also prove a new formula for the linearization
coefficients of Meixner-Pollaczek polynomials. In section 3 we set up a
preliminary combinatorial model, called $L$-graphs, for the linearization
coefficients of Sheffer polynomials. In section 4 we construct a
weight-preserving and sign-reversing involution on the $L$-graphs in order to
prove theorem~\ref{thm:D}. Finally we conclude this paper with some remarks on
further extensions in section 5.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Applications}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Orthogonality of Sheffer polynomials}
First of all the special $m=2$ case of theorem~\ref{thm:D} corresponds to the
orthogonality relation of Sheffer polynomials~\cite{Vi,Ze3}. Indeed, if $n_1\ne
n_2$, then $\D_{\n}=\emptyset$; if $n_1=n_2=n$, each derangement $\pi$ in
$\D_{\n}$ can be identified with two permutations $\pi_1$ and $\pi_2$ in
$S_{n}$, determined by $\pi(1,i)=(2,\pi_1(i))$ and $\pi(2,i)=(1,\pi_2(i))$ for
each $i\in [n]$. It is easy to check that $\cyc\pi=\cyc\pi_2$,
$\ad\pi=\da\pi=n$ and $\mat\pi=\aa \pi=\dd \pi=0$. It follows from (\ref{eq:D})
that
$$
\L(P_n(x)P_n(x))=(u_1u_2)^n\left(\sum_{\pi_1\in\S_n}1\right)
\left(\sum_{\pi_2\in\S_n}\beta^{\cyc\pi}\right)=n!(\beta)_n(u_1u_2)^n,
$$
where $(\beta)_n=\beta(\beta+1)\cdots(\beta+n-1)$. By combining the above two
cases, we obtain (\ref{eq:orth}).


\subsection{Hermite polynomials}
Set $u_1=u_2=\sqrt{u}$, where $u$ is a positive real number, $u_3=u_4=0$,
$\alpha=0$ and $\beta=u^{-1}$, then the moment sequence (\ref{momsheff})
becomes
$$
\L(x^n)=\sum_{\sigma\in \D_n^*}u^{\ad\sigma-\cyc\sigma},
$$
where $\D_n^*$ is the set of derangements of $[n]$ without double-descents or
double-ascents. Now, letting $u\to 0$, the above sum reduces to 0, if $n$ is
odd, and is the number of involutions without fixed points, if $n$ is even,
i.e.
$$
\L(x^n)=\left\{\begin{array}{ll}
0&\textrm{if}\quad n \quad \textrm{odd},\\
1\cdot 3\cdot \cdots \cdot (n-1)&\textrm{if}\quad n \quad \textrm{even}.
\end{array}\right.
$$
Besides, the three-term recurrence (\ref{sheffer}) reduces to that of Hermite
polynomials:
$$
H_{n+1}(x)=xH_n(x)-nH_{n-1}(x).
$$
We recover then from theorem~\ref{thm:D} the following result of Azor et al.
\cite{AGV}:
$$
\L\left(\prod_{i=1}^mH_{n_i}(x)\right)=\sum_{\pi \in \I_{\n}}1,
$$
where $\I_{\n}$ is the set of all involutions in $\D_{\n}$.

\subsection{Charlier polynomials}
Set $\alpha=u_1=u_4=u$, $u_2=u_3=1$ and $\beta=a/u$, then the moment sequence
(\ref{momsheff}) becomes
$$
\L(x^n)=\sum_{\sigma\in \S_n}
u^{\fix \sigma+\dec \sigma -\cyc\sigma}a^{\cyc \sigma}.
$$
Setting $u=0$ the above sum reduces to the generating function of permutations
of which each cycle contains only one decedance or is a singleton. It is easy
to see that such permutations are in bijection with the set of partitions of
$[n]$ by identifying each cycle with its underlying set, called \emph{block}.
 Let $\bloc\pi$ denote the number
of blocks in a partition $\pi$. For a positive integer $n$, let $\Pi_n$ denote
the set of partitions of $[n]$. Hence
$$
\L(x^n)=\sum_{\pi\in\Pi_n}a^{\bloc \pi},
$$
and the three-term recurrence becomes that of Charlier polynomials:
$$
C_{n+1}(x)=(x-(a+n))C_n(x)-anC_{n-1}(x).
$$
We derive then from theorem~\ref{thm:D} the following result of Zeng
\cite{Ze1}:
$$
\L\left(\prod_{i=1}^mC_{n_i}(x)\right)=\sum_{\pi\in\Pi_{\n}}a^{\bloc\pi},
$$
where $\Pi_{\n}$ denotes the set of partitions $\pi$ of $A_{\n}$ such that the
elements of each block of $\pi$ have distinct colors.

\subsection{Laguerre polynomials}
Set $u_1=u_2=u_3=u_4=\alpha=1$, then the three-term recurrence (\ref{sheffer})
becomes
$$
L_{n+1}(x)=(x-(\beta+2n))L_n(x)-n(\beta+n-1)L_{n-1}(x),
$$
and the moment sequence (\ref{momsheff}) reduces to
$$
\L(x^n)=\sum_{\sigma\in \S_n}\beta^{\cyc \sigma}.
$$
We recover then from theorem~\ref{thm:D} the following result of
Foata-Zeilberger \cite{FZ}:
$$
\L\left(\prod_{i=1}^mL_{n_i}(x)\right)=
\sum_{\pi \in \D_{\n}}\beta^{\cyc \sigma}.
$$

\subsection{Meixner polynomials}
Set $\alpha=\frac{c}{1-c}$, $u_1=u_4=\frac{c}{1-c}$ and
$u_2=u_3=\frac{1}{1-c}$, then the three-term recurrence (\ref{sheffer}) becomes
that of Meixner polynomials:
$$
M_{n+1}(x)=(x-(c\beta+(1+c)n)/(1-c))M_n(x)-((cn(\beta+n-1))/(1-c)^2)M_{n-1}(x),
$$
and the moment sequence (\ref{momsheff}) reduces to
\begin{equation}\label{meixmom}
\L(x^n)=
\left(\frac{c}{1-c}\right)^{n}\sum_{\sigma\in\S_n}\beta^{\cyc\sigma}
c^{-\exc\sigma}.
\end{equation}
We recover then from theorem~\ref{thm:D} the following result of Zeng
\cite{Ze1}:
\begin{equation}
\L\left(\prod_{i=1}^mM_{n_i}(x)\right)=\left(\frac{c}{1-c}\right)^{n}
\sum_{\sigma\in D_{\n}}\beta^{\cyc\sigma}c^{-\exc\sigma}.
\label{eq:linmeix}
\end{equation}
Another proof of this result was given by de M\'edicis \cite{deM}.

\subsection{Meixner-Pollaczek polynomials}
Let $i$ be a complex number such that $i^2=-1$. Set $\alpha=\delta$,
$u_1=u_4=\delta+i$, $u_2=u_3=\delta-i$ and $\beta=\eta$, then the three-term
recurrence (\ref{sheffer}) becomes that of Meixner-Pollaczek polynomials:
$$
P_{n+1}(x)=(x-(\delta\eta+2\delta n))P_n(x)-n(\eta+n-1)(1+\delta^2)P_{n-1}(x),
$$
and the moment sequence (\ref{momsheff}) reduces to
\begin{equation}
\L(x^n)=\sum_{\sigma\in\S_n}\eta^{\cyc\sigma}\delta^{ \fix\sigma}
(\delta+i)^{\dec\sigma}(\delta-i)^{\exc\sigma}.
\label{momMP}
\end{equation}
We recover then from theorem~\ref{thm:D} the following result of Zeng
\cite{Ze2}:
\begin{equation}
\L\left(\prod_{k=1}^m P_{n_k}(x)\right)=\sum_{\sigma\in\D_{\n}}
(\delta+i)^{\dec\sigma}(\delta-i)^{\exc\sigma}\eta^{\cyc\sigma}.
\label{eq:linmp}
\end{equation}
It is interesting to note that we naturally obtain a {\em complex} expression
instead of {\em real} one. Although it was proved in \cite{Ze2} that the
right-hand side of (\ref{eq:linmp}) is a polynomial of $\delta$ and $\eta$ with
nonnegative integers, an explicit real expression was missing. We now proceed
to give such an explicit {\em real} formula.

Recall that we identify a cyclic permutation $\sigma$ with the path $P : s_1\to
s_2\to\dots\to s_k\to s_1$ if $\sigma(s_i)=s_{i+1}$ for $1\leq i\leq k-1$,
where $s_1$ is the smallest element. Suppose that $k\geq 2$. Clearly the local
minima and maxima appear alternatively in the above path and their numbers are
equal. The sequence of its extreme elements
$(s_{j_1},s_{i_1},\dots,s_{j_l},s_{i_l})$ is called \emph{extreme sequence},
i.e., $1=j_1<i_1<j_2<i_2<j_3<\cdots<j_l<i_l\leq k$ and $i_r$ (resp. $j_r$) for
$1\leq r\leq p$ are the positions of local maxima (resp. minima). Let
$\C_{\n}({\mathbf\alpha})$ be the set of $|\n|$-cycles in $\D_{\n}$ with fixed
extreme sequence $\mathbf\alpha=(\alpha_1,\ldots,\alpha_{2l})$.
\begin{theorem}
There holds
\begin{equation}\label{eq1}
\sum_{\sigma\in \C_{\n}({\mathbf \alpha})}(1+ui)^{\dec\sigma}
(1-ui)^{\exc\sigma}=\sum_{\sigma\in \C_{\n}({\mathbf\alpha})}\,
(1+u^2)^{\ad\sigma+\mat\sigma}.
\end{equation}
\end{theorem}
\begin{proof}
Consider a cyclic permutation $\sigma$ in $\C_{\n}({\mathbf\alpha})$ identified
with the path $p:s_1\to s_2\to\cdots\to s_{n}\to s_1$, where $s_1$ is the
smallest element. The vertex $s_j$ is called an \emph{excedance} if
$c(s_j)<c(s_{j+1})$, a \emph{decedance} if $c(s_j)>c(s_{j+1})$. Weight each
vertex $s_i$ by $w_i=1$ or $-ui$ (resp. $ui$) if it is an excedance (resp.
decedance). Let $w=(w_1,\ldots, w_n)$ be the weight sequence and define the
weight of the pair $(p,w)$ by $v(p,w)=w_1\cdots w_n$. Clearly the left-hand
side of (\ref{eq1}) is the generating function of all such pairs.

A pair $(p,w)$ is said to be a \emph{bad guy} if there are two vertices $s_j$,
$s_{j'}$ with $j'>j$ satisfying one of the following three conditions:
\begin{itemize}
\item[(i)] $s_j$ is a double-ascent with weight $-ui$ and $j'$ is the smallest
integer such that $c(s_{j'-1})>c(s_j)>c(s_{j'})$,
\item[(ii)] $s_{j'}$ is a double-descent with weight $ui$ and $j$ is the
largest integer such that $c(s_{j-1})<c(s_{j'})<c(s_j)$,
\item[(iii)] $s_j$ is a valley and $j'$ is the smallest integer such that
$s_{j'}$ is a peak, or $s_j$ and $s_{j'}$ form a color match and
$(w_j,w_{j'})=(1, ui)$ or $(-ui,1)$.
\end{itemize}

%\thicklines
\thinlines
\setlength\unitlength{0.2em}
\begin{figure}
\begin{picture}(166,60)(0,-5)
%\linethickness{0.2em}
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 \put(20,40){\vector(1,-1){10}}
 \put(30,30){\vector(1,2){10}}
 \put(40,50){\vector(1,-3){10}}
 \put(50,20){\vector(1,-2){10}}

 \put(0,-1.5){\makebox(0,0)[ct]{\footnotesize$1$}}
 \put(11,9){\makebox(0,0)[lt]{\footnotesize$2$}}
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 \put(30,28.5){\makebox(0,0)[ct]{\footnotesize$4$}}
 \put(40,51.5){\makebox(0,0)[cb]{\footnotesize$6$}}
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 \put(60,-1.5){\makebox(0,0)[ct]{\footnotesize$1$}}

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 \put(50,20){\circle*{1.5}}
 \put(60,0){\circle*{1.5}}

 \put(3,4){\makebox(0,0)[rb]{\footnotesize$-ui$}}
 \put(14.5,25){\makebox(0,0)[rb]{\footnotesize$w_1$}}
 \put(25.5,34){\makebox(0,0)[rt]{\footnotesize$w_2$}}
 \put(34.5,41){\makebox(0,0)[rb]{\footnotesize$w_3$}}
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 \put(4.5,21){\makebox(0,0)[rb]{\footnotesize$w_1$}}
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 \put(24.5,41){\makebox(0,0)[rb]{\footnotesize$w_3$}}
 \put(34.5,35){\makebox(0,0)[rt]{\footnotesize$w_4$}}
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\end{picture}
\caption{Involution $\kappa:\sigma=(1\,2\,5\,4\,6\,3)\mapsto\sigma^*=
(1\,5\,4\,6\,3\,2)$}
\label{Fig:*}
\end{figure}

We now set up a killing involution $\kappa$ on the set of all bad guys. Let
$(p,w)$ be a bad guy and $(j,j')$ be the pair such that $s_j$ and $s_{j'}$
satisfy one of the above three conditions in $(p,w)$ and $\min\{s_i,s_j\}$ is
the smallest among all such pairs. We form a new pair $(p^*,w^*)$ as follows:
\begin{itemize}
\item[(i)] If $s_j$ is a double-ascent with weight $-ui$, then we define the
path
$$
p^*: s_1\to\cdots\to s_{j-1}\to s_{j+1}\to\cdots\to s_{j'-1}\to s_j\to
s_{j'}\to\cdots\to s_n\to s_1.
$$
The weights of vertices are directly inherited from $(p, w)$ except the weight
of $s_{j}$ in $p^*$, which is $ui$.
\item[(ii)] If $s_{j'}$ is a double-descent in $p$ with weight $ui$, then we
define $p^*$ by moving $s_{j'}$ to the left of $s_j$ and changing the weight to
$-ui$.
\item[(iii)] If $s_j$ is a valley and $s_{j'}$ a peak or $s_j$ and $s_{j'}$
form a color match, then we define $p^*=p$ and switch $(w_j,w_{j'})=(1, ui)$ to
$(w_j^*,w_{j'}^*)=(-ui,1)$ or $(w_j,w_{j'})=(-ui,1)$ to
$(w_j^*,w_{j'}^*)=(1,ui)$.
\end{itemize}
Clearly this is an involution such that $v(p,w)=-v(p^*, w^*)$ on the set of bad
guys. In the remaining pairs $(p,w)$, each double-ascent (resp. valley) $s_j$
can be associated to a unique double-descent (resp. peak) $s_{j'}$ ($j>j'$)
such that $(s_j,s_{j'})$ is a color match (resp. (valley, peak)) and
$w_jw_{j'}=1$ or $u^2$.
\end{proof}

\begin{remark} By definition we see immediately that
$$
\sum_{\sigma\in \C_{\bf n}({\mathbf \alpha})}
(1+ui)^{\dec\sigma}(1-ui)^{\exc\sigma}=(1+u^2)^{\ad\sigma}
\sum_{\sigma\in \C_{\bf 
n}({\mathbf\alpha})}\,(1+ui)^{\dd\sigma}(1-ui)^{\aa\sigma}.
$$
Hence identity (\ref{eq1}) is equivalent to
$$
\sum_{\sigma\in \C_{\bf 
n}({\mathbf\alpha})}\,(1+ui)^{\dd\sigma}(1-ui)^{\aa\sigma}=
\sum_{\sigma\in \C_{\bf n}({\mathbf\alpha})}\,(1+u^2)^{\mat\sigma}.
$$
If ${\n}=(1,1,\ldots,1)$, then $\mat\sigma=0$ for any
$\sigma\in \C_{\n}(\alpha)$ and we have the following result:
$$
\sum_{\sigma\in \C_n({\mathbf \alpha})}
(1+\mu i)^{\dd\sigma}(1-\mu i)^{\aa\sigma}=
|\C_n({\mathbf \alpha})|.
$$
\end{remark}

Recall that the cycle index $z_\sigma$ of a permutation $\sigma$ is defined by
$z_\sigma=\prod_{i\geq 1}z_i^{c_i(\sigma)}$, where $c_i(\sigma)$ is the number
of cycles of length $i\geq 1 $ of $\sigma$.

\begin{corollary}
For any sequence $\n=(n_1,\dots,n_m)$ of nonnegative integers, we have
\begin{equation}
\sum_{\sigma\in\D_{\n}}(\delta+i)^{\dec\sigma}(\delta-i)^{\exc\sigma}z_\sigma=
\sum_{\sigma\in\D_{\n}}\delta^{|\n|-2\ad\sigma-2\mat\sigma}
(\delta^2+1)^{\ad\sigma+\mat\sigma}z_\sigma,
\label{eq:RC}
\end{equation}
and for any positive integer $n$, we have
\begin{equation}
\sum_{\sigma\in\S_{n}}
(\delta+i)^{\dec\sigma}(\delta-i)^{\exc\sigma}\delta^{\fix\sigma}z_\sigma=
\sum_{\sigma\in\S_{n}}\delta^{n-2\ad\sigma}
(\delta^2+1)^{\ad\sigma}z_\sigma.
\label{eq2}
\end{equation}
\end{corollary}
\begin{proof}
Since each permutation in $\D_{\n}$ can be decomposed as a set of disjoint
cycles of length at least 2 and the weight functions $\dec$, $\exc$, $\ad$ and
$\mat$ are additive with respect to the cycle-decomposition, we easily derive
(\ref{eq:RC}) from identity (\ref{eq1}). Since there is no color match in any
permutation of $\S_n$, it follows from (\ref{eq:RC}) that for any subset $T$ of
$[n]$,
$$
\sum_{\sigma\in\D_{T}}(\delta+i)^{\dec\sigma}(\delta-i)^{\exc\sigma}z_\sigma=
\sum_{\sigma\in\\D_{T}}\delta^{|T|-2\ad\sigma}(\delta^2+1)^{\ad\sigma}z_\sigma.
$$
Multiplying by $(\delta z_1)^{n-|T|}$ and summing over all subsets $T$ of $[n]$
we obtain (\ref{eq2}).
\end{proof}

Setting $z_i=\eta$ for all $i\in[n]$ in (\ref{eq2}), we recover from
(\ref{momMP}) the classical interpretation \cite{Vi} for the moment sequence of
Meixner-Pollaczek polynomials:
$$
\L(x^n)=\sum_{\sigma\in\S_{n}}\delta^{n-2\ad\sigma}
(\delta^2+1)^{\ad\sigma}\eta^{\cyc\sigma}.
$$

Finally, we derive from (\ref{eq:linmp}) and (\ref{eq:RC}) a new interpretation
for the linearization coefficient of Meixner-Pollaczek polynomials.
\begin{theorem} We have
$$
\L\left(\prod_{k=1}^m P_{n_k}(x)\right)=\sum_{\sigma\in\D_{\n}}
\delta^{|\n|-2\ad\sigma-2\mat\sigma}
(\delta^2+1)^{\ad\sigma+\mat\sigma}\eta^{\cyc\sigma}.
$$
\end{theorem}

Notice that, comparing with (\ref{eq:linmp}), there is no occurrence of
$\sqrt{-1}$ in the above expression.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Combinatorial models}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Sheffer Polynomials and $P$-graphs}
In what follows a {\em tree} is a directed rooted tree, whose edges are all
directed toward the root. Let $S$ be a finite set of positive integers. A {\em
$P$-tree} on vertex set $S$ is an edge labeled tree such that each edge
$i\Arrow{w}j$ satisfies the following:
\begin{enumerate}
\setlength{\itemsep}{0mm}
\item If $i>j$, then $i$ is the greatest vertex in the path from $i$ to the
root and $w=-u_3$, $-u_4$ or $u_2$ and if $w=u_2$, then $k\Arrow{-u_1}i$
appears in the tree, for a unique $k$ less than $i$.
\item If $i<j$, then $w=-u_1$ and $j\Arrow{u_2}k$ appears in the tree,
for a unique $k$ less than $j$.
\end{enumerate}
\begin{remark}
There are only four types of edges in a $P$-tree: $i\Arrow{-u_1}j$,
$i\Arrow{u_2}j$, $i\Arrow{-u_3}j$, $i\Arrow{-u_4}j$. In addition, edges
$i\Arrow{-u_1}j$ and $j\Arrow{u_2}k$ can appear only as the path
$i\Arrow{-u_1}j\Arrow{u_2}k$.
\end{remark}

\begin{definition}
A \emph{$P$-graph} on a finite set $S$ is a pair $(T,\,V)$, where $T$ is a set
of $P$-trees whose underlying vertices form a partition of $S$ and $V$ is a
family of subsets of $T$ of size 1 or of size 2 satisfying the following
condition: If two $P$-trees $g_1$ and $g_2$ constitute a 2-subset in $V$ then
their roots are smaller than vertices adjacent to each root. A 1-subset in $V$
is called a \emph{$P$-singleton}, and a 2-subset in $V$ is called a
\emph{$P$-doubleton}.
\end{definition}

A $P$-graph can be visualized by drawing a \emph{loop} at the root of a
$P$-tree in a {\em $P$-singleton}, and a \emph{2-cycle} formed with the roots
of two $P$-trees in a $P$-doubleton. A $P$-singleton or loop has weight
$-\alpha\beta$ and a $P$-doubleton or 2-cycle has weight $-u_1u_2\beta$. The
{\em weight of a $P$-graph} $g$, denoted by $w(g)$, is the product of weights
of its edges, loops and 2-cycles. For instance, the weight of $P$-graph in
Figure~\ref{fig:Pgraph} is $u_1^4u_2^4u_3^5u_4^6\alpha\beta^2$. The number of
isolated $P$-trees of $g$ (i.e. the $P$-trees not belonging to any
$P$-singleton or $P$-doubleton) is denoted by $\tr g$. Let $\P_{n}$ be the set
of $P$-graphs on $[n]$.

\begin{figure}
\xymatrix@=5mm@!R@!C{
{1}\ar@{=>}^{-u_1}[r]&{4}\ar@{=>}^{u_2}[r]&*+[o][F]{2}&&
{7}\ar@{=>}^{-u_3}[dr]&&&{16}\ar@{=>}_{-u_3}[dr]&&&{13}\ar@{=>}_{u_2}[dl]&{9}\ar
@{=>}_{-u_1}[l]\\
&{15}\ar@{=>}^{-u_4}[u]&{6}\ar@{=>}^{-u_3}[u]&{3}
\ar@{=>}^{-u_1}[r]&{14}\ar@{=>}^{u_2}[r]&*+[o][F]{5}\ar@(ur,dr)@{=>}
[]^{-\alpha\beta}&&&*+[o][F]{12}\ar@/^/@{=>}
[r]^{-u_1u_2\beta}&*+[o][F]{11}\ar@/^/@{=>}[l]&&{19}\ar@{=>}^{-u_3}[u]\\
&&{8}\ar@{=>}^{-u_4}[u]&{17}\ar@{=>}_{-u_4}[u]&{10}\ar@{=>}_{-u_4}[ur]&&&{18}
\ar@{=>}^{-u_4}[ur]&&&{20}\ar@{=>}^{-u_4}[ul]&{21}\ar@{=>}_{-u_3}[l]
}
\caption{A $P$-graph with vertex set $[21]$, where the roots 2, 5, 11 and 12
are circled.}
\label{fig:Pgraph}
\end{figure}

\begin{proposition}\label{prop:P}
The Sheffer polynomial $P_n(x)$ is the generating function of $P$-graphs on
$[n]$, i.e.
$$
P_n(x)=\sum_{g\in\P_n}w(g)\,x^{\tr g}.
$$
\end{proposition}
\begin{proof} For $n=1$ and 2, we note first that $P_1(x)=x-\alpha\beta$ and
$P_2(x)=x^2-(2\alpha\beta+u_3+u_4)x+\alpha^2\beta^2+
u_3\alpha\beta+u_4\alpha\beta-\beta u_1u_2$. On the other hand, it is easy to
see that there are exactly two and nine $P$-graphs on $\{1\}$ and $\{1,\,2\}$
respectively, whose weights correspond to the monomials in the above
polynomials. For the general case we proceed by induction on $n\geq 3$. By
definition the greatest vertex in a $P$-tree must be of degree 0, 1 or 2. In
the last case, it should be adjacent to a vertex of degree 1. In order to
compute the generating function of the $P$-graphs on $[n{+}1]$, we classify
them in five cases and, by inductive hypothesis, write the corresponding
generating function next to each case:
\begin{itemize}
\setlength{\itemsep}{0mm}
\item $n{+}1$ forms an isolated $P$-tree: $xP_{n}(x)$,
\item $n{+}1$ forms a separate $P$-singleton: $-\alpha\beta P_{n}(x)$,
\item $n{+}1$ is attached to a $P$-graph on $[n]$ at
some $i\in[n]$, creating an edge $n{+}1\Arrow{-u_3}i$
or $n{+}1\Arrow{-u_4}i$:
$-nu_3P_{n}(x)-nu_4P_{n}(x)$,
\item $n{+}1$ forms a 2-cycle with some $i\in[n]$:
$-nu_1u_2\beta P_{n-1}(x)$,
\item for some $i\in[n]$,
$n{+}1$ is attached to a $P$-graph on
$[n]\setminus\{i\}$ at some $j\in[n]\setminus\{i\}$,
creating two edges
$i\Arrow{-u_1}n{+}1\Arrow{u_2}j$: $-n(n{-}1)u_1u_2 P_{n-1}(x)$.
\end{itemize}
Summing all the cases, we get the recurrence relation~(\ref{sheffer}).
\end{proof}

\begin{remark}
Viennot \cite{Vi} has given a general model for general orthogonal polynomials
based on the three term recurrence relations. The $P$-graphs are more
structured ad hoc models, which reflect literally the three-term recurrence
relation~(\ref{sheffer}) and can be seen somehow as an enriched version of
Viennot's model.
\end{remark}

\subsection{Linearization coefficients and $L$-graphs}
Given a linearly ordered set $E$ of cardinality $n$, we can identify each
permutation $\sigma\in\S_n$ with its functional digraph on $E$: for each $x$
and $y$ in $E$, there is an edge $x\to y$ iff $y=\sigma(x)$. We put weight on
each edge $x\to\sigma(x)$ as follows:
\begin{itemize}
\setlength{\itemsep}{0mm}
\item $\alpha$, if $x$ is a fixed point, i.e. $\,\sigma(x)=x$,
\item $u_1$, if $\sigma(x)$ is a peak, i.e. $\,x<\sigma(x)>\sigma^2(x)$,
\item $u_2$, if $\sigma(x)$ is a valley, i.e. $\,x>\sigma(x)<\sigma^2(x)$,
\item $u_3$, if $\sigma(x)$ is a double-ascent, i.e. 
$\,x<\sigma(x)<\sigma^2(x)$,
\item $u_4$, if $\sigma(x)$ is a double-descent, i.e.
$\,x>\sigma(x)>\sigma^2(x)$.
\end{itemize}
The weight of a cycle is $\beta$ times the product of the weight of its edges.
The weight of a permutation $\sigma$, denoted by $w(\sigma)$, is the product of
weights of all cycles of $\sigma$. Weighting the functional digraph of a
permutation this way, the resulting generating function of $\S_n$ is equal to
the moment $\L(x^n)$ in (\ref{momsheff}).

\begin{definition} An $L$-graph on $A_{\n}$ is an $(m{+}1)$-tuple
$(g_1,\dots,g_m,\sigma)$, where $g_i$ is a $P$-graph on $A_i$ for each $i\in
[m]$ and $\sigma$ a permutation of isolated trees in $g_1,\,\ldots,\,g_{m-1}$
and $g_m$, linearly ordered according to their roots.
\end{definition}

We naturally identify $\sigma$ with its functional digraph on the set of roots
of isolated trees in $g_1,\,\ldots,\,g_m$. The weight of an $L$-graph
$(g_1,\dots,g_m,\sigma)$ is $w(g_1)\cdots w(g_m)w(\sigma)$. We now describe an
$L$-graph in graph terminology. A component of a graph on vertex set $A_{\n}$
is  called \emph{monochromatic}, if all its vertices are of the same color. An
$L$-graph on $A_{\n}$ is then a digraph whose connected components are
monochromatic $P$-singletons, monochromatic $P$-doubletons or cycles of
monochromatic $P$-trees, i.e., (r{+}1)-tuples $(t_1,\ldots,t_r,\sigma)$ for
some integer $r\geq 1$, where $t_i$ ($1\leq i\leq r$) are monochromatic
$P$-trees and $\sigma$ is a cyclic permutation of $S_r$. A vertex adjacent to
the root in a $P$-tree is called a {\em pendant} of the root. Any pendant of a
root of a $P$-tree in an $L$-graph $g$ is called a {\em pendant} of $g$. A
cycle of $P$-trees is called an {\em $L$-singleton}, if it is a 1-cycle of a
monochromatic $P$-tree; and an {\em $L$-doubleton}, if it is a monochromatic
2-cycle whose pendants, if any, are greater than both of its roots. For
example, the functional digraph of a permutation in~$\S_{\n}$ is an $L$-graph
without pendants.

The {\em weight of an $L$-graph} is then equal to the product of the weight
inherited from $P$-graphs and the weight of the permutation of roots of its
isolated $P$-trees. Each edge in a cycle of $P$-trees is weighted as in the
above. Let $\L_{\n}$ denote the set of all $L$-graphs on vertex set $A_{\n}$.

\begin{example}
Figure~\ref{fig:L-graph1} illustrates an $L$-graph, which has a 6-cycle, two
2-cycles, an $L$-singleton, a $P$-singleton, an $L$-doubleton, a $P$-doubleton,
with weight $u_1^8u_2^8u_3^9u_4^6\alpha^2\beta^7$.
\end{example}

\begin{figure}
\xymatrix@=5mm@!R@!C{
{12}\ar@{=>}^{-u_3}[r]&*+[o][F]{10}\ar@{->}^{u_3}[r]&
*+[o][F]{15}\ar@{->}^{u_1}[r]&
*+[o][F]{21}&&{17}\ar@{=>}^{-u_3}[dr]&&{23}\ar@{=>}^{-u_3}[dr]&&&
{27}\ar@{=>}_{-u_3}[d]&&*+[o][F]{28}\ar@(ul,dl)@{=>}[]_{-\alpha}\\
{9}\ar@{=>}^{u_2}[r]&*+[o][F]{1}\ar@{->}^{u_3}[u]\ar@{<-}^{u_2}[r]&
*+[o][F]{3}\ar@{<-}^{u_4}[r]&*+[o][F]{11}\ar@{<-}^{u_4}[u]&
{14}\ar@{=>}^{-u_1}[r]&{18}\ar@{=>}^{u_2}[r]&
*+[o][F]{16}\ar@(ur,dr)@{->}[]^{\alpha}&&
*+[o][F]{22}\ar@/^/@{->}[r]^{u_2}&*+[o][F]{4}\ar@/^/@{->}[l]^{u_1}&
*+[o][F]{25}\ar@/^/@{->}[r]^{u_1}&*+[o][F]{26}\ar@/^/@{->}[l]^{u_2}&
*+[o][F]{29}\ar@/^/@{=>}[r]^{-u_1}&*+[o][F]{30}\ar@/^/@{=>}[l]^{u_2}\\
{8}\ar@{=>}^{-u_1}[u]&{5}\ar@{=>}^{-u_4}[u]&{6}\ar@{=>}^{-u_3}[u]&
{13}\ar@{=>}^{-u_4}[u]&{20}\ar@{=>}_{-u_3}[u]&{19}\ar@{=>}_{-u_4}[u]&&
{24}\ar@{=>}^{-u_4}[ur]&{2}\ar@{=>}^{-u_1}[r]&{7}\ar@{=>}_{u_2}[u]&&
*+[o][F]{33}\ar@/^/@{->}[r]^{u_2}&*+[o][F]{31}\ar@/^/@{->}[l]^{u_1}&{32}\ar@{=>}
_{-u_3}[l]}
\caption{An $L$-graph on $A_{\n}$ with ${\n}=(9,4,7,7,6)$.}
\label{fig:L-graph1}
\end{figure}

\begin{proposition}
The linearization coefficient of the Sheffer polynomials is the generating
function of $L$-graphs, i.e.
\begin{equation}\label{eq:L}
\L\left(\prod_{i=1}^mP_{n_i}(x)\right)=\sum_{g\in\L_{\n}}w(g).
\end{equation}
\end{proposition}
\begin{proof}
Applying (\ref{momsheff}) and using proposition \ref{prop:P}, we obtain
\begin{eqnarray*}
\L\left(\prod_{i=1}^mP_{n_i}(x)\right)&=&
\sum_{(g_1,\dots,g_m)}w(g_1)\cdots w(g_m)\,\L(x^{\tr g_1+\cdots+\,\tr g_m})\\
&=&\sum_{(g_1,\dots,g_m, \sigma)}w(g_1)\cdots w(g_m)w(\sigma),
\end{eqnarray*}
where the sum is over all $L$-graphs $(g_1,\dots,g_m, \sigma)$ on $A_{\n}$.
\end{proof}

\begin{definition}
A skew-derangement is a permutation $\pi$ of $A_{\n}$ without color matches
such that if there is any $a\in A_{\n}$ such that $c(a)=c(\pi(a))$, then
$c(\pi^{-1}(a))\ne c(a)$ and $c(\pi(a))\ne c(\pi^2(a))$, and the smaller in
$\{a,\pi(a)\}$ is a valley and the larger a peak.
\end{definition}
Note that any generalized derangement without color matches is a
skew-derangement and any skew-derangement is a derangement in the classical
sense. Let $\D'_{\n}$ denote the set of skew-derangements of $A_{\n}$. We now
give a useful variant version of main theorem~\ref{thm:D}.
\begin{theorem}
There is a one-to-one correspondence $\pi\mapsto\pi'$ from $\D_{\n}$ onto
$\D'_{\n}$ such that
$$
(u_1u_2/u_3u_4)^{\mat\pi}w(\pi)=w(\pi'),
$$
where $w(\pi)=u_1^{\ad\pi}u_2^{\da\pi}u_3^{\aa\pi}u_4^{\dd\pi}\beta^{\cyc\pi}$.
Hence
\begin{equation}\label{eq:quasi}
\sum_{\pi\in{\D}_{\n}}(u_1u_2/u_3u_4)^{\mat\pi}w(\pi)=
\sum_{\pi\in\D'_{\n}}w(\pi).
\end{equation}\label{thm:E}
\end{theorem}
\begin{proof}
It suffices to describe the correspondence for cyclic permutations. Recall that
we identify a cyclic permutation $\pi$ with the path
$$
\pi:\quad s_1\arrow{<}s_2\arrow{}\cdots\arrow{}s_l\arrow{>}s_1,
$$
where $\pi(s_i)=s_{i+1}$ for $1\leq i\leq l-1$, and $s_1$ is the smallest
element. If $\pi$ has no color match then set $\pi'=\pi$, otherwise suppose it
has only one color match $(s_p,s_q)$, $p<q$, where $s_p$ is a double-ascent and
$s_q$ a double-descent. If $s_p<s_q$, then define $\pi'$ by
$\pi'(s_{p-1})=s_q$, $\pi'(s_{q-1})=s_{q+1}$, $\pi'(s_q)=s_p$, and
$\pi'(s_i)=\pi(s_i)$, for $i\ne p{-}1$, $q{-}1$,~$q$, which corresponds to the
path
$$
\pi':\quad 
s_1\arrow{<}\cdots\arrow{}s_{p-1}\arrow{u_1}s_q\arrow{u_2}s_p\arrow{<}
\cdots\arrow{}s_{q-1}\arrow{>}s_{q+1}\arrow{}\cdots\arrow{}s_1;
$$
if $s_p>s_q$ by $\pi'(s_{p-1})=s_{p+1}$, $\pi'(s_p)=s_{q+1}$, $\pi'(s_q)=s_p$,
and $\pi'(s_i)=\pi(s_i)$, for $i\ne p{-}1$, $p$,~$q$, which corresponds to the
path
$$
\pi':\quad s_1\arrow{<}\cdots\arrow{}s_{p-1}\arrow{<}s_{p+1}\arrow{}
\cdots\arrow{u_2}s_{q}\arrow{u_1}s_{p}\arrow{>}s_{q+1}\arrow{}\cdots\arrow{}s_1.
$$
If $\pi$ has more than one color match, then we apply the above process to each
color match. It remains to show that the resulting skew-derangement $\pi'$ is
independent of the order of color matches chosen. Indeed, suppose
$(s_{p_1},s_{q_1})$ and $(s_{p_2},s_{q_2})$ are two color matches such that
$p_1<p_2$. Then either $p_1<q_1<p_2<q_2$ or $p_1<p_2<q_2<q_1$. In both cases it
is easy to see that we get the same $\pi'$.

Since the above application changes each color match of $\pi$ into a
generalized fixed point of $\pi'$, which induces a weight change by the factor
$u_1u_2/u_3u_4$ for each color match, the weight change amounts to
$(u_1u_2/u_3u_4)^{\mat\pi}$.
\end{proof}

Therefore to prove theorem~\ref{thm:D} it suffices to prove the right hand side
of (\ref{eq:L}) is equal to that of (\ref{eq:quasi}).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Involutions on $L$-graphs}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Involution $\Phi$ on $\L_{(1,k,1)}$}
For clarity, we first consider the special case $\n=(1,k,1)$, where $k$ is some
positive integer, and identify the underlying set $A_\n$ with
$\{a,1,\dots,k,b\}$ , where $c(a)=1$, $c(1)=\cdots=c(k)=2$ and $c(b)=3$. If
there is no risk of confusion, we indicate only the root of a $P$-tree,
omitting possible pendants attached to each root. For example, the following
cycle
$$
\xymatrix@=3mm@!R@!C{
*+[][]{1}\ar@{->}^{}[r]&*+[][]{10}\ar@{->}^{}[r]&*+[][]{15}
\ar@{->}^{}[r]&*+[][]{21}\ar@{->}^{}[r]&*
+[][]{11}\ar@{->}^{}[r]&*+[][]{3}\ar@{->}^{}[r]&*+[][]{1}}
$$
represents the left-most component of the $L$-graph in
Figure~\ref{fig:L-graph1}. The following facts will be useful.

\begin{fact}
The connected components of an $L$-graph in
$\L_{(1,k,1)}\setminus\D'_{(1,k,1)}$ are of the following types: monochromatic
$P$-singletons, $L$-singletons, monochromatic $P$-doubletons, $L$-doubletons,
monochromatic 2-cycles which are not $L$-doubletons, two color 2-cycles and
r-cycles ($r\geq 3$).
\end{fact}
We proceed to define a {\em wpsr} involution $\Phi$ on $\L_{(1,k,1)}$. We first
present in Figure~\ref{fig:table} rules, i.e. pairs of patterns, involved in
the involution $\Phi$ for connected components. For example, rule $c_1$
associates each cycle which has a double-ascent ``2'' of color 2 with a
preimage ``1'' of color 2 to a cycle by moving ``2'' as a pendant of ``1'' with
weight $-u_3$.

Each rule $c_i$, $1\leq i\leq9$, induces another rule, $c_i'$, called the {\em
dual} of $c_i$, by reversing the arrows in the cycles, with appropriate
weights. For example, the dual of rule $c_1$ associates each cycle which has a
double-descent ``2'' of color 2 with a image ``1'' of color 2 to a cycle by
moving ``2'' as pendant of ``1'' with weight $-u_4$.

We include only the dual of $c_1$ in Figure~\ref{fig:table} and omit the duals
of other rules. In $e_3$, for simplicity, we break the convention of writing a
cycle with the smallest element in the first position.

\begin{fact}
Let $\sigma$ be an $r$-cycle ($r\geq 3$) without pendants of an $L$-graph in
$\L_{(1,k,1)}\setminus\D'_{(1,k,1)}$. If $\sigma$ has no movable root under
rules $c_1$ or $c_1'$, then it is one of the following types:
\begin{enumerate}
\setlength{\itemsep}{0mm}
\item[(i)] $a\to i_1\to\cdots\to i_p\to b\to j_1\to\cdots\to j_q\to a$,
$p,q\geq1$, and if $p>1$ ($q>1$, resp.), then the
sequence $i_1,\dots,i_p$ ($j_1,\dots,j_q$, resp.) alternates, with
$i_{p-1}>i_p$ ($j_1<j_2$, resp.).
\item[(ii)] $a\to i_1\to\cdots\to i_p\to b\to a$, $p\geq3$, and the sequence
$i_1,\dots,i_p$ alternates, with $i_{p-1}>i_p$.
\item[(iii)] $a\to b\to j_1\to\cdots\to j_q\to a$, $q\geq3$, and the sequence
$j_1,\dots,j_q$ alternates, with $j_1<j_2$.
\item[(iv)] $a\to i_1\to\cdots\to i_p\to a$, $p\geq2$, and the sequence
$i_2,\dots,i_{p-1}$ alternates.
\item[(v)] $b\to i_1\to\cdots\to i_p\to b$, $p\geq3$, $p$ is odd, and the
sequence $i_1,\dots,i_p$ alternates, with $i_{p-1}>i_p$.
\item[(vi)] $i_1\to\cdots\to i_p\to i_1$, $p\geq4$, $p$ is even, and the
sequence $i_1,\dots,i_p$ alternates, and $i_1$ is the greatest.
\end{enumerate}
\end{fact}

\begin{figure}
\fbox{
$\xymatrix@R=2.5mm@C=5mm@!R@!C{
\ar@{.>}^{}[r]&*+[][]{1}\ar@{->}^{u_3}[r]&*+[][]{2}\ar@{->}^{<}[r]&*+[][]{v}\ar@{.>}^{}[r]&\\\\
\ar@{<.}^{}[r]& *+[][]{1}\ar@{<-}^{<}[r]&*+[][]{2}\ar@{<-}^{u_4}[r]&*+[][]{v}\ar@{<.}^{}[r]&\\\\
*+[][]{a}\ar@{->}^{u_3}[r]&*+[][]{2}\ar@{->}^{u_1}[r]&*
+[][]{3}\ar@{->}^{>}[r]&*+[][]{1}\ar@{.>}^{}[r]&*+[][]{g}\ar@{.>}^{}[r]&\\\\
*+[][]{a}\ar@{->}^{< }[r]&*+[][]{1}\ar@{.>}^{}[r]&*
+[][]{g}\ar@{.>}^{u_4}[r]&*+[][]{2}\ar@{->}^{>}[r]&*+[][]{a}\\\\
*+[][]{a}\ar@{->}^{u_1}[r]&*+[][]{3}\ar@{->}^{u_2}[r]&*
+[][]{1}\ar@{.>}^{}[r]&*+[][]{g}\ar@{.>}^{u_4}[r]&*
+[][]{2}\ar@{->}^{>}[r]&*+[][]{a}\\\\
*+[][]{a}\ar@{->}^{u_3}[r]&*+[][]{1}\ar@{->}^{u_1}[r]&*+[][]{3}\ar@{->}^{u_2}[r]&*
+[][]{2}\ar@{.>}^{<}[r]&*+[][]{g}\ar@{.>}^{}[r]&\\\\
*+[][]{a}\ar@{.>}^{u_1}[r]&*+[][]{r}\ar@{->}^{u_2}[r]&*+[][]{p}\ar@{->}^{u_1}[r]&*
+[][]{t}\ar@{->}^{u_2}[r]&*+[][]{q}\ar@{.>}^{}[r]&*+[][]{g}\ar@{.>}^{}[r]&\\\\
*+[][]{a}\ar@{->}^{u_3}[r]&*+[][]{1}\ar@{.>}^{<}[r]&*+[][]{g}\ar@{.>}^{u_2}[r]&*
+[][]{2}\ar@{->}^{u_1}[r]&*+[][]{3}\ar@{->}^{>}[r]&*+[][]{a}\\\\
*+[][]{a}\ar@{->}^{u_1}[r]&*+[][]{1}\ar@{->}^{u_2}[r]&*+[][]{p}\ar@{.>}^{<}[r]&*
+[][]{g}\ar@{.>}^{u_2}[r]&*+[][]{q}\ar@{->}^{u_1}[r]&*+[][]{2}\ar@{->}^{>}[r]&*
+[][]{a}\\\\
*+[][]{a}\ar@{->}^{u_3}[r]&*+[][]{1}\ar@{->}^{u_1}[r]&*
+[][]{2}\ar@{->}^{u_2}[r]&*+[][]{a}\\\\
*+[][]{a}\ar@{->}^{u_1}[r]&*+[][]{2}\ar@{->}^{u_2}[r]&*+[][]{1}\ar@{->}^{u_1}[r]&*+[][]{3}\ar@{->}^{>}[r]&*+[][]{a}\\\\
*+[][]{a}\ar@{->}^{u_1}[r]&*+[][]{3}\ar@{->}^{u_2}[r]&*+[][]{1}\ar@{->}^{u_1}[r]&*+[][]{2}\ar@{->}^{>}[r]&*+[][]{a}\\\\
*+[][]{g}\ar@{.>}^{u_2}[r]&*+[][]{p}\ar@{->}^{u_1}[r]&*+[][]{q}\ar@{->}^{u_2}[r]&*+[][]{r}\ar@{.>}^{<}[r]&*+[][]{g}
}
\qquad
\xymatrix@R=2.5mm@C=5mm@!R@!C{
\ar@{<->}^{c_1}[r]&&\ar@{.>}^{}[r]&*+[][]{1}\ar@{->}^{<}[r]&*+[][]{v}\ar@{.>}^{}[r]&\\&&{2}\ar@{=>}_{-u_3}[ur]\\
\ar@{<->}^{c_1'}[r]&& \ar@{<.}^{}[r] &*+[][]{1}\ar@{<-}^{<}[r]&*+[][]{v}\ar@{<.}^{}[r]&\\&&{2}\ar@{=>}_{-u_4}[ur]\\
\ar@{<->}^{c_2}[r]&&*+[][]{a}\ar@{->}^{u_1}[r]&*+[][]{2}\ar@{->}^{>}[r]&*+[][]{1}\ar@{.>}^{}[r]&*
+[][]{g}\ar@{.>}^{}[r]&\\&&{3}\ar@{=>}_{-u_3}[ur]\\
\ar@{<->}^{c_3}[r]&&*+[][]{a}\ar@{->}^{< }[r]&*+[][]{1}\ar@{.>}^{}[r]&*+[][]{g}\ar@{.>}^{>}[r]&*
+[][]{a}\\&&*+{2}\ar@{=>}_{-u_4}[ur]\\
\ar@{<->}^{c_4}[r]&&*+[][]{a}\ar@{->}^{u_1}[r]&*+[][]{2}\ar@{->}^{u_2}[r]&*
+[][]{1}\ar@{.>}^{}[r]&*+[][]{g}\ar@{.>}^{>}[r]&*+[][]{a}\\
&&&*+{3}\ar@{=>}_{-u_4}[ur]\\
\ar@{<->}^{c_5}[r]&&&*+[][]{a}\ar@{->}^{u_3}[r]
&*+[][]{2}\ar@{.>}^{<}[r]&*+[][]{g}\ar@{.>}^{}[r]&\\
&&*+{1}\ar@{=>}^{-u_1}[r]&*+{3}\ar@{=>}_{u_2}[ur]\\
\ar@{<->}^{c_6}[r]&&*+[][]{a}\ar@{.>}^{u_1}[r]&*+[][]{r}\ar@{->}^{u_2}[r]
&*+[][]{p}\ar@{.>}^{}[r]&*
+[][]{g}\ar@{.>}^{}[r]&\\&&*+{q}\ar@{=>}^{-u_1}[r]&*+{t}\ar@{=>}_{u_2}[ur]\\
\ar@{<->}^{c_7}[r]&&&*+[][]{a}\ar@{->}^{u_3}[r]&*+[][]{1}\ar@{.>}^{<}[r]&*+[][]{g}\ar@{.>}^{>}[r]&*+[][]{a}\\&&*+{2}\ar@{=>}^{-u_1}[r]&*+{3}\ar@{=>}_{u_2}[ur]\\
\ar@{<->}^{c_8}[r]&&&*+[][]{a}\ar@{->}^{u_1}[r]&*+[][]{1}\ar@{->}^{u_2}[r]&*+[][]{p}\ar@{.>}^{<}[r]&*+[][]{g}\ar@{.>}^{>}[r]&*+[][]{a}\\&&*+{q}\ar@{=>}^{-u_1}[r]&*+{2}\ar@{=>}_{u_2}[ur]\\
\ar@{<->}^{c_9}[r]&&*+[][]{a}\ar@{->}^{u_1}[r]&*+[][]{1}\ar@{->}^{u_2}[r]&*+[][]{a}\\&&*+{2}\ar@{=>}_{-u_3}[ur]\\
\ar@{<->}^{e_1}[r]&&&*+[][]{a}\ar@{->}^{u_1}[r]&*+[][]{2}\ar@{->}^{>}[r]&*+[][]{a}\\&&*+{1}\ar@{=>}^{-u_1}[r]&*+{3}\ar@{=>}_{u_2}[ur]\\
\ar@{<->}^{e_2}[r]&&&*+[][]{a}\ar@{->}^{u_1}[r]&*+[][]{1}\ar@{->}^{>}[r]&*+[][]{a}\\&&*+{2}\ar@{=>}^{-u_1}[r]&*+{3}\ar@{=>}_{u_2}[ur]\\
\ar@{<->}^{e_3}[r]&&&*+[][]{g}\ar@{.>}^{u_2}[r]&
*+[][]{p}\ar@{.>}^{<}[r]&*+[][]{g}\\&&*+{r}\ar@{=>}^{-u_1}[r]&*+{q}\ar@{=>}_{u_2}[ur]
}$
}
\caption{Integers $1,2,3$ and $4$ denote labels of the same color in that order,
$p,q,r,t$ denote integers in $[k]$, $v>2$, and $g$ with color 2 or 3 is the
greatest on the cycle. There can be more elements around dotted arrows. Each
integer is the root of a P-tree. }
\label{fig:table}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In each rule, the unique root (pendant, resp.) which appears as a pendant
(root, resp.) in the other is called the {\em movable root} (pendant, resp.).
By a {\em movable element} we mean a movable root, a movable pendant, the root
of a $P$- or $L$-singleton or the larger of the roots of a $P$- or
$L$-doubleton.

\begin{lemma}
Any $L$-graph in $\L_{(1,k,1)}\setminus\D'_{(1,k,1)}$ has a movable element.
\label{claim}
\end{lemma}
\begin{proof}
Any $L$-graph in $\L_{(1,k,1)}\setminus\D'_{(1,k,1)}$ contains a connected
component $\sigma$, whose restriction on its underlying set is not a
skew-derangement.

1) If $\sigma$ is a $P$- or $L$-singleton, the rule $s$ is:
$$
\xymatrix@R=1mm@C=5mm@!R@!C{
*+[][]{i}\ar@(ur,dr)@{=>}[]^{-\alpha}&&\ar@{<->}^{s}[r]&&
*+[][]{i}\ar@(ur,dr)@{->}[]^{\alpha}
}
$$

2) If $\sigma$ is a $P$- or $L$-doubleton, the rule $d$ is :
$$
\xymatrix@R=1mm@C=5mm@!R@!C{
*+[][]{1}\ar@/^/@{=>}^{-u_1}[r]&*+[][]{2}\ar@/^/@{=>}^{u_2}[l]
&\ar@{<->}^{d}[r]&&
*+[][]{1}\ar@/^/@{->}^{u_1}[r]&*+[][]{2}\ar@/^/@{->}^{u_2}[l]
}.
$$

3) If $\sigma$ is a monochromatic 2-cycle whose larger root is greater than a
pendant of the smaller root, or a monochromatic 3-cycle, or a monochromatic
4-cycle with one peak and one valley, then $\sigma$ has a movable element under
one of rules $c_1$, $c_1'$.

4) If $\sigma$ is a two color 2-cycle (with pendants), or two color 3-cycle
then $\sigma$ has a movable element under one of rules $c_1$, $c_1'$, $c_9$
and $c_9'$,

In the remaining cases $\sigma$ is an $r$-cycle ($r\geq 4$). We will show that
at least one of the rules or their duals in Figure~\ref{fig:table} is
applicable to $\sigma$. We distinguish two types: I) $\sigma$ has no pendants
and II) $\sigma$ has pendants.

I) $\sigma$ has no pendants. Suppose that $\sigma$ has no movable element under
rules $c_1$ and $c_1'$. In view of Fact 4.2 we indicate for each case
applicable rules, which induce the corresponding movable elements.

\noindent{Case (i):} Assume that $p\geq q$.
\begin{itemize}
\setlength{\itemsep}{0mm}
\item $(p,q)=(1,1)$: rule $c_3$ or $c_3'$
\item $(p,q)=(2,1)$: rule $c_3$, $c_4$ or $c_7'$
\item $(p,q)=(2,2)$: rule $c_8$ or $c_8'$
\item $(p,q)$ with $p=3$: rule $c_2$ or $c_5$
\item $(p,q)$ with $p\geq4$: rule $c_6$
\end{itemize}
\noindent{Case (ii):} If $p=3$, then either $c_2$ or $c_5$ is applicable;
otherwise, $c_6$ is applicable.

\noindent{Case (iii):} If $p=3$, then either $c_2'$ or $c_5'$ is applicable;
otherwise, $c_6'$ is applicable.

\noindent{Case (iv):} If $p=2$, then either $c_9$ or $c_9'$ is applicable;
if $p=3$, then $c_3$, $c_3'$, $e_1$ or $e_2$ is applicable; otherwise, we
pretend that the greatest element has a greater color and apply case (i).

\noindent{Case (v):} Rule $e_3$ is applicable.

\noindent{Case (vi):} Rule $e_3$ is applicable.

II) $\sigma$ has pendants. If $\sigma$ has any movable roots under above rules,
then we are done; otherwise, the roots of $\sigma$ form a skew-derangement. If
a pendant $p$ is attached to $v$ with an edge of weight $-u_3$ or $-u_4$, then
it is movable under one of rules $c_1$, $c_1'$,
\dots, $c_4$, $c_4'$, $c_9$ and $c_9'$; otherwise, $\sigma$ has a pattern,
$t\Arrow{-u_1}p\Arrow{u_2}v$, and $s$ is movable under one of rules $c_5$,
$c_5'$,\dots, $c_8$, $c_8'$, $e_1$, $e_2$ and $e_3$.
\end{proof}

For each $\sigma\in\L_{(1,k,1)}$ we define $\Phi(\sigma)$ as follows: if it is
a skew-derangement then $\Phi(\sigma)=\sigma$; otherwise $\Phi(\sigma)$ is
obtained by applying the appropriate rule to the smallest movable element in
$\sigma$.
\begin{theorem}
The map $\Phi$ defined as above is a weight-preserving sign-reversing
involution on $\L_{(1,k,1)}$ such that $\Fix\Phi=D'_{(1,k,1)}$.
\end{theorem}
\begin{proof}
Since $\Phi$ is clearly weight-preserving sign-reversing, it suffices to show
that an element $i$ is the smallest movable element in $\sigma$ if and only if
it is the smallest movable element in $\sigma'$. It can be shown by examining
each rule. We omit the details.
\end{proof}

\begin{example}
Figure~\ref{fig:L-graph2} illustrates a correspondence under rule $c_1'$ and
Figure~\ref{fig:L-graph3} a correspondence under rule $e_3$.
\end{example}

\begin{figure}
\xymatrix@=5mm@!R@!C{
{12}\ar@{=>}^{-u_3}[r]&*+[][]{10}\ar@{->}^{u_3}[r]&
*+[][]{15}\ar@{->}^{u_1}[r]&*+[][]{21}&&&
{12}\ar@{=>}^{-u_3}[r]&*+[][]{10}\ar@{->}^{u_3}[r]&
*+[][]{15}\ar@{->}^{u_1}[r]&*+[][]{21}\\
{9}\ar@{=>}^{u_2}[r]&*+[][]{1}\ar@{->}^{u_3}[u]\ar@{<-}^{u_2}[r]&
*+[][]{3}\ar@{<-}^{u_4}[r]&*+[][]{11}\ar@{<-}^{u_4}[u]&\ar@{<->}^{c_1'}[r]&&
{9}\ar@{=>}^{u_2}[r]&*+[][]{1}\ar@{->}^{u_3}[u]\ar@{<=}_{-u_4}[r]&
{3}&*+[][]{11}\ar@/_3mm/@{->}_{u_2}[ll]\ar@{<-}^{u_4}[u]\\
{8}\ar@{=>}^{-u_1}[u]&{5}\ar@{=>}^{-u_4}[u]&{6}\ar@{=>}^{-u_3}[u]&
{13}\ar@{=>}^{-u_4}[u]&&&
{8}\ar@{=>}^{-u_1}[u]&{5}\ar@{=>}^{-u_4}[u]&{6}\ar@{=>}^{-u_3}[u]&
{13}\ar@{=>}^{-u_4}[u]
}
\caption{The smallest movable element of the $L$-graph in
Figure~\ref{fig:L-graph1} is root 3.}
\label{fig:L-graph2}
\end{figure}
\begin{figure}
\xymatrix@R=1mm@C=5mm@!R@!C{
{5}\ar@{=>}^{-u_3}[dr]&&&& &&&
{5}\ar@{=>}^{-u_3}[dr]&&*+[][]{1}\ar@{->}[dr]^{u_1}&\\
&*+[][]{4}\ar@/^/@{->}[r]^{u_2}&*+[][]{2}\ar@/^/@{->}[l]^{u_1}&
{3}\ar@{=>}_(.4){u_2}[l]&{1}\ar@{=>}_(.4){-u_1}[l] &\ar@{<->}^{e_3}[r]&&
&*+[][]{4}\ar@{->}[ur]_{u_2}&&*+[][]{3}\ar@{->}[dl]^{u_2}\\
{6}\ar@{=>}_{-u_4}[ur]&&&& &&&
{6}\ar@{=>}_{-u_4}[ur]&&*+[][]{2}\ar@{->}[ul]_{u_1}&
}
\caption{The correspondence $\Phi$ for two $L$-graphs on vertex set $[6]$
with ${\n}=(3,3)$.}
\label{fig:L-graph3}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Involution $\Phi$ for the general case}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
We now describe how to apply the rules for a cycle in $\L_{(1,k,1)}$ to a cycle
$\sigma$ in $\L_{\n}, $where $\n=(n_1,\dots,n_m)$. Since $m=1$ case has been
dealt with in the proof of lemma~\ref{claim}, we may assume that $m>1$.

To determine movable elements of color $c(q)$, we proceed as follows:
\begin{itemize}
\item Replace each vertex $v$ with $c(v)<c(q)$ (resp., $c(q)<c(v)$) by the
letter $a$ (resp., $b$) and contract the cycle by replacing each segment
containing repeated $a$'s (resp., $b$'s) by a single $a$ (resp., $b$);
\item Assume that the resulting cycle has $k$ ($k\geq 1$) letters $a$'s, which
divide the cycle into $k$ intervals. Clearly, by identifying the two $a$'s at
its ends, each interval defines a cycle, called \emph{subcycle}.
\item Furthermore, if any subcycle has more than one letter $b$, then using
these $b$'s instead of $a$'s as before, divide it into cycles, again called
subcycles. Each of the resulting subcycles has then at most one $a$ and at most
one $b$. Determine movable elements of each subcycle by the rules in
Figure~\ref{fig:table}.
\item Repeat this procedure, color by color, starting from the smallest color,
until we find a movable element, which should exist by lemma~\ref{claim}, and
apply to $\sigma$ the appropriate rule involving the smallest movable element,
and define $\Phi(\sigma)$ to be the resulting cycle.
\end{itemize}

\begin{example}
Suppose $\sigma$ is the following cycle:
$$
\xymatrix@=3mm@!R@!C{
*+[][]{o_1}\ar@{->}^{}[r]&*+[][]{p_1}\ar@{->}^{}[r]&*+[][]{q_1}\ar@{->}^{}[r]&
*+[][]{q_2}\ar@{->}^{}[r]&*+[][]{r_1}\ar@{->}^{}[r]&*+[][]{q_3}\ar@{->}^{}[r]&
*+[][]{r_2}\ar@{->}^{}[r]&*+[][]{q_4}\ar@{->}^{}[r]&*+[][]{q_5}\ar@{->}^{}[r]&
*+[][]{s_1}\ar@{->}^{}[r]&*+[][]{s_2}\ar@{->}^{}[r]&*+[][]{q_6}\ar@{->}^{}[r]&
*+[][]{o_2}\ar@{->}^{}[r]&*+[][]{o_3}\ar@{->}^{}[r]&*+[][]{q_7}\ar@{->}^{}[r]&
*+[][]{q_8}\ar@{->}^{}[r]&*+[][]{p_2}\ar@{->}^{}[r]&*+[][]{o_1}
},
$$
where $o_1,\,o_2,\,o_3$ have color 1; $p_1,\,p_2$ have color $2$;
$q_1,\ldots,q_8$ have color $3$; $r_1$, $r_2$ have color 4; $s_1$, $s_2$ have
color 5. To determine movable elements of color $3$, replace first $o_1$,
$o_2$, $o_3$, $p_1$, $p_2$ with the letter $a$, and $r_1$, $r_2$, $s_1$, $s_2$
with the letter $b$. The result is
$$
\xymatrix@=3mm@!R@!C{
*+[][]{a}\ar@{->}^{}[r]&*+[][]{a}\ar@{->}^{}[r]&*+[][]{q_1}\ar@{->}^{}[r]&
*+[][]{q_2}\ar@{->}^{}[r]&*+[][]{b}\ar@{->}^{}[r]&*+[][]{q_3}\ar@{->}^{}[r]&
*+[][]{b}\ar@{->}^{}[r]&*+[][]{q_4}\ar@{->}^{}[r]&*+[][]{q_5}\ar@{->}^{}[r]&
*+[][]{b}\ar@{->}^{}[r]&*+[][]{b}\ar@{->}^{}[r]&*+[][]{q_6}\ar@{->}^{}[r]&
*+[][]{a}\ar@{->}^{}[r]&*+[][]{a}\ar@{->}^{}[r]&*+[][]{q_7}\ar@{->}^{}[r]&
*+[][]{q_8}\ar@{->}^{}[r]&*+[][]{a}\ar@{->}^{}[r]&*+[][]{a}
}.
$$
Contracting the cycle yields
$$
\xymatrix@=3mm@!R@!C{
*+[][]{a}\ar@{->}^{}[r]&*+[][]{q_1}\ar@{->}^{}[r]&*+[][]{q_2}\ar@{->}^{}[r]&
*+[][]{b}\ar@{->}^{}[r]&*+[][]{q_3}\ar@{->}^{}[r]&*+[][]{b}\ar@{->}^{}[r]&
*+[][]{q_4}\ar@{->}^{}[r]&*+[][]{q_5}\ar@{->}^{}[r]&*+[][]{b}\ar@{->}^{}[r]&
*+[][]{q_6}\ar@{->}^{}[r]&*+[][]{a}\ar@{->}^{}[r]&*+[][]{q_7}\ar@{->}^{}[r]&
*+[][]{q_8}\ar@{->}^{}[r]&*+[][]{a}
}.
$$
This cycle is further divided into two subcycles by the two $a$'s:
$$
\xymatrix@=3mm@!R@!C{
*+[][]{a}\ar@{->}^{}[r]&*+[][]{q_1}\ar@{->}^{}[r]&*+[][]{q_2}\ar@{->}^{}[r]&
*+[][]{b}\ar@{->}^{}[r]&*+[][]{q_3}\ar@{->}^{}[r]&*+[][]{b}\ar@{->}^{}[r]&
*+[][]{q_4}\ar@{->}^{}[r]&*+[][]{q_5}\ar@{->}^{}[r]&*+[][]{b}\ar@{->}^{}[r]&
*+[][]{q_6}\ar@{->}^{}[r]&*+[][]{a}
}\,,\quad
\xymatrix@=3mm@!R@!C{
*+[][]{a}\ar@{->}^{}[r]&*+[][]{q_7}\ar@{->}^{}[r]&*+[][]{q_8}\ar@{->}^{}[r]&
*+[][]{a}
}.
$$
Now, the first subcycle has three $b$'s, which produce three subcycles, and
finally $\sigma$ is split into four subcycles:
$$
\xymatrix@=3mm@!R@!C{
*+[][]{a}\ar@{->}^{}[r]&*+[][]{q_1}\ar@{->}^{}[r]&*+[][]{q_2}\ar@{->}^{}[r]&
*+[][]{b}\ar@{->}^{}[r]&*+[][]{q_6}\ar@{->}^{}[r]&*+[][]{a}
}\,,\quad
\xymatrix@=3mm@!R@!C{
*+[][]{b}\ar@{->}^{}[r]&*+[][]{q_3}\ar@{->}^{}[r]&*+[][]{b}
}\,,\quad
\xymatrix@=3mm@!R@!C{
*+[][]{b}\ar@{->}^{}[r]&*+[][]{q_4}\ar@{->}^{}[r]&*+[][]{q_5}\ar@{->}^{}[r]&
*+[][]{b}
}\,,\quad
\xymatrix@=3mm@!R@!C{
*+[][]{a}\ar@{->}^{}[r]&*+[][]{q_7}\ar@{->}^{}[r]&*+[][]{q_8}\ar@{->}^{}[r]&
*+[][]{a}
}.
$$
Determine movable elements of each subcycle by the rules in
Figure~\ref{fig:table}. An application of a rule in a subcycle induces an
application of the rule to the original $\sigma$, by tracing the procedure
backward.
\end{example}

In summary we obtain the following result.
\begin{theorem}
The map $\Phi$, defined as above, is a weight-preserving sign-reversing
involution on $\L_{\n}$ with fixed set $\D'_{\n}$.
\label{thm:Phi}
\end{theorem}

Now our main theorem~\ref{thm:D} follows from theorems \ref{thm:E}
and~\ref{thm:Phi}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Further extensions}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
We provide some partial results on two possible generalizations of the
linearization coefficients for the Sheffer polynomials and make some comments.
\subsection{Permutation enumerations}
For any permutation $\pi\in {\S}_{\bf n}$, set
$$
\Fix\pi=\{a\in A_{\bf n}: c(\pi(a))=c(a)\}, \quad
\textstyle{\Fix_i}\pi=\Fix\pi\cap A_i.
$$
For the Laguerre polynomials $L_n(x)$, Foata and Zeilberger \cite{FZ}
proved that
\begin{equation}\label{eq:fz}
{\L}\left(\prod_{i=1}^mL_{n_i}(\lambda_i x)\right)=
\sum_{\pi\in{\S}_{\n}}\beta^{\cyc \pi}
\prod_{i=1}^m(\lambda_i-1)^{|\Fix_i \pi|}\lambda_i^{|A_i\setminus \Fix_i\pi|}.
\end{equation}
So it is natural to ask whether there is any interesting result about
${\L}\left(\prod_{i=1}^mP_{n_i}(\lambda_i x)\right)$. In what follows we will
show that a reasonable generalization of (\ref{eq:fz}) does not seem exist in
general. We shall denote by $e_k$ the $k$-th $(0\leq k\leq m)$ elementary
symmetric polynomial of $x_1,\ldots, x_m$, that is,
$$
\sum_{j=0}^m e_j x^j=\prod_{k=1}^m(1+x_k x).
$$
By convention, we define $e_k=0$ if $k>m$. For $\n=(n_1,\dots,n_m)$, set
$\mathbf x^\n=x_1^{n_1}\cdots x_m^{n_m}$ and $\n!=n_1!\cdots n_m!$.

For the Hermite polynomials $H_n(x)$, the generating function and moments
\cite[p. 145]{Ch} are
\begin{eqnarray*}
\sum_{n=0}^\infty H_n(x){t^n\over n!}&=&\exp{(xt-t^2/2)},\\
\L(x^n)&=& {1\over \sqrt{\pi}}\int_{-\infty}^\infty
x^ne^{-x^2}dx=\left\lbrace\begin{array}{ll}
(2m-1)!!& \mbox{ if $n=2m$};\\
0& \mbox{ if $n=2m+1$}.
\end{array}
\right.
\end{eqnarray*}
\begin{proposition} There holds
$$
\L \left( \prod_{i=1}^m H_{n_i}(\lambda_i x)\right)=
\sum_{\pi\in {\cal I}_{\n}} \prod_{i=1}^m(\lambda_i^2-1)^{|\Fix_i \pi|}
\lambda_i^{|A_i\setminus \Fix_i \pi|},
$$
where ${\cal I}_{\n}$ is the set of involutions without singleton of $A_{\n}$.
\end{proposition}
\begin{proof}
Using the generating function and moment of Hermite polynomials we have
\begin{eqnarray*}
\sum_{\n}\L\left(\prod_{i=1}^mH_{n_i}(\lambda_i x)\right)
{{\mathbf x}^{\n}\over {\n}!}
&=& \L\left({\prod_{i=1}^m}\exp(\lambda_ix_ix-x_i^2/2)\right)\\
&=&\exp {1\over 2}
\left[(\lambda_1 x_1+\cdots +\lambda_m x_m)^2-(x_1^2+\cdots x_m^2)\right]\\
&=&\exp\left[\sum_{i=1}^m(\lambda_i^2-1){x_i^2\over 2}
+\sum_{i\not=j}\lambda_i\lambda_jx_ix_j\right].
\end{eqnarray*}
It is then easy to derive the desired equation from the above formula.
\end{proof}

For the Charlier polynomials $C_n(x)$, the generating function and moments
\cite[p. 170]{Ch} are
\begin{eqnarray*}
\sum_{n=0}^\infty C_n(x){t^n\over n!}&=&e^{-at}(1+t)^x,\\
\L(x^n)&=&e^{-a}\sum_{x=0}^\infty x^n{a^x\over x!}.
\end{eqnarray*}
\begin{proposition} There holds
$$
\L \left(\prod_{i=1}^m C_{n_i}(\lambda_i x)\right)=
\sum_{\pi\in {\P}_{\n}}a^{\bloc\pi}\prod_{i=1}^m(\lambda_i-1)^{\sing_i \pi}
\prod_{B}\prod_{i=1}^m \lambda_i(\lambda_i-1)\cdots (\lambda_i-|A_i\cap B|+1),
$$
where $\sing_i \pi$ is the number of singletons of color $i$ in $\pi$ and $B$
is a non-singleton bloc of $\pi$.
\end{proposition}
\begin{proof}
Using the generating function and moments of Charlier polynomials we have
\begin{eqnarray*}
\sum_{\n}\L\left(\prod_{i=1}^mC_{n_i}(\lambda_i
x)\right)
{{\mathbf x}^{\n}\over{\n}!}&=& \L\left(\prod_{i=1}^m
e^{-ax_i}(1+x_i)^{\lambda_ix}
\right)\\
&=&\exp\left[-a-a\sum_{i=1}^mx_i+a\prod_{i=1}^m(1+x_i)^{\lambda_i}\right]\\
&=&\exp\left[a\sum_{i=1}^m(\lambda_i-1)x_i
+a\sum_{|{\mathbf k}|\geq
2}\prod_{i=1}^m(-\lambda_i)_{k_i}{(-x_i)^{k_i}\over k_i!}\right].
\end{eqnarray*}
It is then easy to derive the desired equation from the last result.
\end{proof}

For the Meixner polynomials $M_n(x;\beta, c)$, the generating function and
moments \cite[p. 176]{Ch} are
$$
\sum_{n\geq 0}M_n(x;\beta, c){t^n\over n!}=
(1-t/c)^x(1-t)^{-x-\beta},
$$
$$
\L(x^n)= (1-c)^\beta\sum_{x\geq 0}x^n{c^x(\beta)_x\over x!}.
$$
Hence
\begin{eqnarray*}
\sum_{\n}\L\left(\prod_{i=1}^m
(-c)^{n_i}M_{n_i}(\lambda_ix;\beta,c)\right) {{\mathbf x}^{\n}\over \n!}&=&
(1-c)^\beta \sum_{x=0}^\infty \prod_{i=1}^m (1+x_i)^{\lambda_i
x}(1+cx_i)^{-\lambda_ix-\beta}{c^x(\beta)_x\over x!},
\end{eqnarray*}
which is equal to
$$
\left[(1-c)^{-1}{\prod_{i=1}^m(1+cx_i)-c(1-c)^{-1}\prod_{i=1}^m(1+x_i)
\left(1-{(1-c)x_i\over 1+x_i}\right)^{1-\lambda_i}}\right]^{-\beta}.
$$
It turns out that the expression inside the above brackets can be further
written as
$$
1-\sum_{k=2}^m(c+\cdots +c^{k-1})e_k({\mathbf x})
-c\sum_{|{\mathbf k}|\geq 1}(1-c)^{|{\mathbf k}|-1}\prod_{i=1}^m
{(\lambda_i-1)_{k_i}x_i^{k_i}\over (1+x_i)^{k_i-1}},
$$
which reduces to the case of Laguerre polynomials \cite{FZ} when $c=1$ :
$$
1-\sum_{k=2}^m(k-1)e_k({\mathbf x})-
\sum_{i=1}^m(\lambda_i-1)x_i\prod_{j\not=i}(1+x_j)=
\prod_{i=1}^m(1+x_i)-\sum_{i=1}^m\lambda_ix_i\prod_{j\not=i}(1+x_j),
$$
for $\sum_{i=1}^mx_i\prod_{j\not=i}(1+x_j)=\sum_{k=1}^mke_k$. However, if
$c\not=1$, the expression is not a polynomial in $x_1,\ldots, x_m$, so we
cannot expect to have a reasonable interpretation as for Hermite, Charlier and
Laguerre polynomials.

\subsection{$q$-analogs}
Another interesting question is to find a $q$-analog of theorem~\ref{thm:D}.
Till now a satisfying $q$-analog has been obtained only for Hermite polynomials
by Ismail et al. \cite{ISV}. The problem for $q$-Charlier polynomials was
studied by de M\'edicis et al. \cite{MSW}. We define the $q$-Laguerre
polynomials \cite{MV,SS} by
\begin{equation}
L_n^{(\alpha)}(x;q)=\sum_{k=0}^n(-1)^{n-k}\qbi{n}{k}{q}q^\bi{n-k}{2}
\prod_{i=k}^{n-1}(\alpha+[i+1]_q)x^k,
\end{equation}
where $\qbi{n}{k}{q}$ is the usual $q$-binomial coefficient. It is not hard to
see that this is a rescaled version of Wall polynomials
\cite[p.198]{Ch}:
$$
L_n^{(\alpha)}(x;q)=\left(\frac{1+\alpha(1-q)}{1-q}q\right)^n
W_n\left(\frac{(1-q)qx}{1+\alpha(1-q)};\frac{q}{1+\alpha(1-q)},q\right).
$$
It follows that $L_{-1}^{(\alpha)}(x;q)=0$, $L_0^{(\alpha)}(x;q)=1$, and
$$
L_{n+1}^{(\alpha)}(x;q)=(x-b_n)L_n^{(\alpha)}(x;q)-
\lambda_nL_{n-1}^{(\alpha)}(x;q),
$$
where $b_n=q^n([n]_q+\alpha+1+q[n]_q)$ and
$\lambda_n=q^{2n-1}[n]_q(\alpha+1+q[n-1]_q)$. The corresponding $n$-th moment
of $q$-Laguerre polynomials \cite[p.198]{Ch} reads as follows:
$$
\L_q(x^n)=\prod_{i=0}^{n-1}(\alpha+1+q[i]_q).
$$
Although we have the following $q$-analog of the orthogonality:
\begin{equation}\label{qlagorth}
\L_q(L_m^{(\alpha)}(x;q)L_n^{(\alpha)}(x;q))=
q^{n(n+1)}[n]_q!\prod_{i=0}^{n-1}(\alpha+1+q[i]_q)\delta_{m,n},
\end{equation}
the usual $q$-analog of the linearization coefficients doesn't give a
combinatorial refinement of the ordinary case ($q=1$), as the polynomial
$$
\L_q((L_{2}^{(0)}(x;q))^3)={q}^{15}+5\,{q}^{14}+14\,{q}^{13}+26\,{q}^{12}+
31\,{q}^{11}+20\,{q}^{10}-{q}^{9}-12\,{q}^{8}-6\,{q}^{7}+{q}^{6}+{q}^{5}
$$
has negative coefficients. On the other hand, applying the linear functional
$\L_q$ to the following $q$-binomial formula:
$$
(x-1)(x-q)\cdots(x-q^{n-1})
=\sum_{k=0}^{n}(-1)^k\qbi{n}{k}{q}q^\bi{k}{2}x^{n-k},
$$
we obtain
\begin{equation}
\L_q\left(q^{\bi{n}{2}}L_1^{(0)}(x)L_1^{(0)}(x/q)\cdots
L_1^{(0)}(x/q^{n-1})\right)=[n]_q!\sum_{k=0}^n\frac{(-1)^kq^\bi{k}{2}}{[k]_q!},
\end{equation}
which is the well-known $q$-derangement numbers \cite{Wachs}. This suggests to
consider the following $q$-analog:
\begin{eqnarray*}
\L_q^*(0,\n)&=&\L_q\left(q^{\sum_{i=1}^m(i-1)n_i}
L_{n_1}^{(0)}(x)L_{n_2}^{(0)}(x/q)\cdots L_{n_m}^{(0)}(x/q^{m-1})\right)\\
&=& \prod_{i=1}^{m}[n_i]_q!
\sum_{\mathbf k}(-1)^{|\n|-|\mathbf k|}
\qbi{k_1+\cdots+k_m}{k_1,\,\dots,\,k_m}{q}
\prod_{i=1}^m\qbi{n_i}{k_i}{q}
q^{\bi{n_i-k_i}{2}+(i-1)(n_i-k_i)}.
\end{eqnarray*}
\begin{conj}
The polynomial $\L_q^*(0,\n)$ has nonnegative integral coefficients if $n_1\geq
n_2\geq\cdots\geq n_m$.
\end{conj}
\begin{remark} For $\n=(1,2,1)$, we get $\L_q^*(0,\n)=q^6+q^5-q^4+2q^2+q$.
Further Maple experiments seem to suggest that the quotient
$\L_q^*(0,\n)/\prod_{i=1}^{m}[n_i]_q!$ is a polynomial of $q$ with nonnegative
integral coefficients if $n_1=n_2=\cdots=n_m$. Note that the latter has
negative coefficients when $\n=(8,5,4)$.
\end{remark}

For $m=3$ and $n_1\geq n_2\geq n_3$, it is easy to see by applying
$q$-Chu-Vandermonde formula that
$$
\L_q^*(0,\n)=[n_1]_q![n_2]_q![n_3]_q!\sum_{l\geq0}(-1)^{n_2+n_3-l}
\qbi{l}{n_1}{q}\sum_{k\geq0}\qbi{n_2}{l-k}{q}\qbi{n_3}{k}{q}\qbi{l}{k}{q}
q^{M(k,l)},
$$
where $M(k,l)=n_1^2+\bi{n_2-l+k+1}{2}+\bi{n_3-k+1}{2}+n_3-k$. If $n_3=0$, it
reduces to
$$
\L_q^*(0,\n)=[n_1]_q![n_2]_q!\sum_{l\geq0}(-1)^{n_2-l}
\qbi{l}{n_1}{q}\qbi{n_2}{l}{q}q^{n_1^2+\bi{n_2-l}{2}+n_2-l+k}=
[n_1]_q![n_2]_q!q^{n_1^2}\delta_{n_1\,n_2},
$$
which is clearly a polynomial of $q$ with nonnegative integral coefficients.
\section*{Acknowledgment}
The first author was partially supported by KOSEF: 971-0106-038-2.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\small
\begin{thebibliography}{99}
\bibitem{askey} Askey (R.),
Orthogonal polynomials and special functions, CBMS Regional Conference Series
in Applied Mathematics 21, SIAM, Philadelphia, 1975.

\bibitem{AGV} Azor (R.), Gillis (J.) and Victor (J.D.),
Combinatorial applications of Hermite polynomials, SIAM. J. Math. Anal., 13
(1982), 879--890.

\bibitem{AI} Askey (R.), Ismail (M.),
Permutation problems and special functions, Canad. J. Math., 28
(1976), no.4, 853--874.

\bibitem{Ch} Chihara (T.S.), An introduction to orthogonal polynomials,
Mathematics and its Applications, Vol. 13. Gordon and Breach Science
Publishers, New York-London-Paris, 1978.

\bibitem{deM} de M\'edicis (A.), Combinatorics of Meixner polynomials,
Europ. J. Comb. (1998) 19, 355--367.

\bibitem{MSW} de M\'edicis (A.), Stanton (D.) and White (D.),
The combinatorics of $q$-Charlier polynomials,
J. Combin. Theory Ser. A 69 (1995), no. 1, 87--114.

\bibitem{MV} de M\'edicis (A.) and Viennot (X.G.),
Moments des $q$-polyn\^omes de Laguerre et la bijection de Foata-Zeilberger,
Adv. in Appl. Math. 15 (1994), no. 3, 262--304.

\bibitem{DV} de Sainte-Catherine (M.) and Viennot (X.G.),
Combinatorial interpretation of integrals of products of Hermite, Laguerre and
Tchebycheff polynomials, Lecture Notes in Math., Vol. 1171, Springer-Verlag,
1985, 120--128.

\bibitem{EG} Even (S.) and Gillis (J.), Derangements and Laguerre polynomials,
Proc. Cambridge Phil. Soc., 31 (1976), 1355--143.

\bibitem{Fo} Foata (D.), Rearrangements of Words, in M. Lothaire,
Combinatorics on Words, CUP, Cambridge, 1997.

\bibitem{FZ} Foata (D.) and Zeilberger (D.), Laguerre polynomials,
weighted derangements and positivity, SIAM J. Disc. Math. 1 (1988), 425--433.

\bibitem{Ge} Gessel (I.), Generalized rook polynomials and orthogonal
polynomials, 159--176, in D. Stanton (ed.), $q$-Series and Partition, IMA
Volumes in Mathematics and its Application, Vol. 18, Springer-Verlag, New York,
1989

\bibitem{ISV} Ismail (M.), Stanton (D.) and Viennot (G.),
The combinatorics of $q$-Hermite polynomials and the Askey-Wilson integral,
European J. Combin. 8 (1987), no. 4, 379--392.

\bibitem{SS} Simion (R.) and Stanton (D.),
Octabasic Laguerre polynomials and permutation statistics,
J. Comp. and Appl. Math. 68 (1996), 197--329.

\bibitem{Vi} Viennot (X.G.), Une th\'eorie combinatoire des polyn\^omes
orthogonaux g\'en\'eraux, Notes de conf\'erences donn\'ees \`a l'UQAM,
Montr\'eal, 1983.

\bibitem{Wachs} Wachs (M.L.),
On $q$-derangement numbers, Proc. Amer. Math. Soc. 106 (1989), no. 1, 273--278.

\bibitem{Ze1} Zeng (J.), Lin\'earisation de produits de polyn\^omes de Meixner,
Krawtchouk, et Charlier, SIAM J. Math. Anal., Vol. 21, No. 5 (1990),
1349--1368.

\bibitem{Ze2} Zeng (J.), Weighted derangements and the linearization
coefficients of orthogonal Sheffer polynomials, Proc. London Math. Soc. (3) 65
(1992), 1--22.

\bibitem{Ze3} Zeng (J.), Enum\'erations de permutations et $J$-fractions
continues, Europ. J. Combinatorics (1993) 14, 373--382.
\end{thebibliography}

\end{document}

