%%  This article has been submitted and accepted for publication in
%%  THE FOATA FESTSCHRIFT, 
%%  a special issue of THE ELECTRONIC JOURNAL OF COMBINATORICS
%%  dedicated to DOMINIQUE FOATA at the occasion of his 60th birthday.
%%
%%  Author(s):		GILBERT LABELLE AND PIERRE LEROUX
%%  Title:		AN EXTENSION OF THE EXPONENTIAL FORMULA IN 
%%			ENUMERATIVE COMBINATORICS
%%  Date of submission:	March 11, 1995
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%%  Number of pages:	14
%%  Author's email:	leroux@catalan.math.uqam.ca
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{\bf R\'esum\'e  \vspace{-.5em}} %\vspace{\z@}}%
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\title{An Extension of the Exponential Formula\\in Enumerative Combinatorics}
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\author{Gilbert Labelle and Pierre Leroux \\
LACIM, D\'ep. de math\'ematiques, Universit\'e du Qu\'ebec \`a Montr\'eal \\
C.P. 8888, Succ. Centre-Ville, Montr\'eal (Qu\'ebec), Canada  H3C 3P8}   
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\date{Submitted: March 11, 1995; Accepted: July 15, 1995}   % Deleting this command produces today's date.
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\begin{document}
\pagestyle{myheadings} 
\markright{\sc the electronic journal of combinatorics 3 (2) (1996), \#R12\hfill} 
\thispagestyle{empty}
         
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\maketitle                 % Produces the title.
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%\parskip
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En hommage \`a Dominique Foata, \`a l'occasion de son soixanti\`eme anniversaire.
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%%%%%%%%%%%%%%%%%%%%%% ABSTRACT %%%%%%%%%%%%%%%%%%
%
\begin{abstract} 
%
   Let $\alpha$  be a formal variable and 
$F_w$ be a weighted species of structures
(class of structures closed under weight-preserving isomorphisms) of the form
${F}_{w} = E({F}_{w}^{c})$, where $E$ and $F_w^c$ respectively denote the species of
{\em sets} and of {\em connected} $F_w${\em-structures}.
Multiplying by $\alpha$ the weight of each $F_w^c$-structure yields 
the species ${F}_{{w}^{( \alpha   )}} = E({F}_{ \alpha w}^{c})$.  
We introduce a ``universal'' virtual weighted species,
$\Lambda ^{(\alpha)}$, such that 
    $F_{w^{(\alpha)}} = \Lambda^{( \alpha)}\, \circ \, F_w^+$,  
where   $F_w^+$ denotes the species of non-empty $F_w$-structures. 
 Using general properties of $\Lambda^{( \alpha)}$ , 
 we compute the various enumerative power series 
$G(x)$, $\widetilde{G}(x)$, $\overline{G}(x)$, 
$G(x;q)$, $G\langle{x;q}\rangle$, 
${Z}_{G}(x_1,x_2,x_3,\ldots)$, 
${\Gamma }_{G}(x_1,x_2,x_3,\ldots)$, 
for $G = F_{w^{(\alpha)}}$, in terms of $F_w$. 
Special instances of our 
formulas include the exponential formula, 
${F}_{{w}^{(\alpha )}}(x)=\exp(\alpha F_{w}(x))=({F}_{w}(x){)}^{\alpha }$, 
cyclotomic identities, and their $q$-analogues.  
The virtual weighted species, $\Lambda ^{(\alpha)}$, is, in fact, 
a new combinatorial lifting of the function ${(1+x)}^{\alpha }$.
%
\end{abstract}
%
%\parskip

%%%%%%%%%%%%%%%%%%%%% R\'ESUM\'E %%%%%%%%%%%%%%%%%%%
\begin{resume}
 Soit $\alpha$ une variable formelle et $F_w$ une esp\`ece de structures 
pond\'er\'ee (classe de structures ferm\'ee sous les 
isomorphismes pr\'eservant les poids) de la forme ${F}_{w} = E({F}_{w}^{c})$,  o\`u $E$ 
et $F_w^c$ d\'esignent respectivement l'esp\`ece des {\em ensembles} et celle des 
$F_w$-{\em structures connexes}. 
En multi\-pliant par $\alpha$ le poids de chaque $F_w^c$-structure, on obtient 
l'esp\`ece ${F}_{{w}^{( \alpha   )}} = E({F}_{ \alpha w}^{c})$.  
Nous introduisons une esp\`ece virtuelle ``universelle'', $\Lambda ^{(\alpha)}$, telle que  
$F_{w^{(\alpha)}} = \Lambda^{( \alpha)}\, \circ \, F_w^+$,  o\`u $F_w^+$  
d\'esigne l'esp\`ece des  $F_w$-structures non-vides.  En faisant appel \`a des propri\'et\'es 
g\'en\'erales de $\Lambda ^{(\alpha)}$, nous calculons les diverses s\'eries 
formelles \'enum\'eratives
$G(x)$, $\widetilde{G}(x)$, $\overline{G}(x)$, 
$G(x;q)$, $G\langle{x;q}\rangle$, 
$Z_G(x_1,x_2,x_3,\ldots)$, 
$\Gamma_G(x_1,x_2,x_3,\ldots)$, 
de $G = F_{w^{(\alpha)}}$, en fonction de $F_w$.  Comme cas sp\'eciaux des formules 
que nous d\'eveloppons, on retrouve la formule exponentielle,
$F_{w^{(\alpha )}}(x)=\exp(\alpha F_w(x))=(F_w(x))^\alpha $,  
les identit\'es cyclotomiques, ainsi que leurs $q$-analogues.  L'esp\`ece 
virtuelle pond\'er\'ee, $\Lambda ^{(\alpha)}$,  est, en fait, un nouveau rel\`evement 
combinatoire de la fonction $(1+x)^\alpha$.
%
\end{resume}%
%
%
%%%%%%%%%%%%%%%%%%%%%%% Introduction %%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}

Let $f(x)$ and $g(x)$ be two functions or formal power series such that
%
\begin{equation}
f(x)\ =\ \exp(g(x)) .  \label{eq:fexpg}
\end{equation}
%
The expression ${f(x)}^{\alpha}$ can then be given a meaning, for any variable $\alpha$, 
by means of the ``exponential formula'' 
%
\begin{equation}
f(x{)}^{\alpha }\ =\ \exp(\alpha \,g(x)) .  \label{eq:expalpha}
\end{equation}
%
D. Foata played a pioneer role in the interpretation and use of these formulas in enumerative 
combinatorics; see for example \cite{CaF}, \cite{Fo1}, \cite{FSc}. 
Essentially, if $f(x)$ and $g(x)$ are the exponential generating series for the structure
classes $F$ and $G$, then equation (\ref{eq:fexpg}) expresses the fact that the $F$-structures
can be identified with assemblies of $G$-structures, the latter being considered as ``connected''
$F$-structures. In that case, the ``weight'' $\alpha^{c(s)}$ can be assigned to any 
$F$-structure $s$, where $c(s)$ denotes the number of ``connected components'' of $s$;
one says that the variable $\alpha$ acts as a {\em counter} for
(or {\em marks}) the connected components. 
Formula (\ref{eq:expalpha}) then gives the exponential generating series of this new 
weighted class.  For example, it is immediately seen that the function 
$(1-x)^{-\alpha}$
is the generating series for permutations, where $\alpha$ acts as a cycle counter.
This approach is fundamental in the development of combinatorial models for 
the classical orthogonal polynomials; see, for example, 
%, \cite{FLa}, 
\cite{Fo2}--\cite{FLe}, \cite{FSt}.

In the following, we consider weighted structure classes (called {\em species}, see 
\cite{BLL}, \cite{Jo1}, \cite{La4}) that are closed under
isomorphisms induced by relabellings, that is, by bijections of the underlying sets.
In that case, new generating series, symmetric functions (cycle index series), 
and $q$-analogues can be associated, in relation with the enumeration of {\em unlabelled} 
(i.e.,\ up to isomorphism) or {\em asymmetric}  (i.e.,\ whose automorphism group is reduced
to the identity) structures. When moreover the concept
of connected components can be introduced, there arises the question of
deter\-mining the relations analogous to (\ref{eq:expalpha}) for these various series.
%In this article, we show that there is a {\em universal}  solution to this problem,
%which yields an exponential formula adapted to each kind of series. The solution 
%takes the form of a weighted virtual species, denoted by $\Lambda^{(\alpha)}$,
%obtained by conjugation of the class of singletons of weight $\alpha$ by the non-empty 
%sets.
As we will see, more complex exponential formulas
appear, involving polynomials in $\alpha$ related to the classical ``cyclotomic identity''
and to a new ``cocyclotomic identity'' (see Corollary \ref{th:seriesfwalpha}).
%
%%%%%%%%%%%%%%%%% Figure %%%%%%%%%%%%%%%%%
%
\begin{figure}     \label{fig}
%  \vspace{4cm}
\begin{center}
\setlength{\unitlength}{0.008in}%{0.009375in}%
\begin{picture}(600,300)(0,10)
%
\thicklines
\put(20,10){\framebox(560,280){}}
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\thinlines
%
\put(80,220){\begin{picture}(260,140) % quadrilatere
\put(0,0){\circle*{10}} \put(-17,-17){$a$}  % a
\put(150,25){\circle*{10}}  \put(155,32){$c$} % c
\put(60,-60){\circle*{10}}  \put(43,-77){$i$} % i
\put(210,-35){\circle*{10}}  \put(211,-23){$d$} % d
\put(0,0){\line( 6, 1){150}} % a-c
\put(0,0){\line( 1, -1){60}} % a-i
\put(60,-60){\line( 6, 1){150}} % i-d
\put(150,25){\line( 1, -1){60}} % c-d
\end{picture}}
%
\put(380,245){\circle*{10}} \put(390,250){$e$}% e
%
\put(80,110) {\begin{picture}(60,60) % arete f-p
\put(0,0){\circle*{10}}  \put(-20,-18){$f$} % f
\put(60,-60){\circle*{10}}  \put(43,-77){$p$} % p
\put(0,0){\line(1,-1){60}} % f-p
\end{picture}}
%
\put(470,190) {\begin{picture}(60,60) % arete k-n
\put(0,0){\circle*{10}}  \put(-19,-17){$k$} % k
\put(50,50){\circle*{10}}  \put(55,60){$n$} % n
\put(0,0){\line( 1, 1){50}} % k-n
\end{picture}}
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\put(300,50) {\begin{picture}(220,70) % arbre
\put(-10,0){\circle*{10}}  \put(-27,-17){$h$} % h
\put(80,0){\circle*{10}}  \put(85,-17){$q$} % q
\put(150,35){\circle*{10}}  \put(145,50){$b$} % b
\put(80,70){\circle*{10}}  \put(63,53){$r$} % r
\put(220,0){\circle*{10}}  \put(203,-17){$m$} % m
\put(220,70){\circle*{10}}  \put(225,80){$j$} % j
\put(-10,0){\line( 1, 0){90}} % h-q
\put(80,0){\line( 2, 1){70}} % q-b
\put(150,35){\line( 2, 1){70}} % b-j
\put(150,35){\line( -2, 1){70}} % b-r
\put(150,35){\line( 2,-1){70}} % b-m
\end{picture}}
%
%
\end{picture}
%
\caption{A graph $g$ with $w(g) = y^{11}$ and $w^{(\alpha)}(g) = \alpha^5 y^{11}$}
\end{center}
\end{figure}%
%\begin{figure}
%      \label{fig}
%     \vspace{4cm}
%      \caption{}
%\end{figure}
%

Let us illustrate our approach with the species ${\rm Gr}_w$ of (simple) graphs, weighted 
by an edge counter, $y$. For instance, the graph $g$  of Figure 1 
has weight $w(g)=y^{11}$ since it has 11 edges. Note that this weighting is
invariant under relabellings and {\em multiplicative} with respect to the natural
concept of connected components, in the sense that the weight 
of a graph is equal to the product of the weights of its components.  These facts
are summarized by the equation ${\rm Gr}_w = E({\rm Gr}^c_w)$, 
where $E$ denotes the species of sets and ${\rm Gr}^c$ denotes the connected graphs.
Now the weight $w$ can be refined to a new weight $w^{(\alpha)}$ by adjoining 
a counter $\alpha$ of connected components. Hence the previous graph $g$ has 
a refined weight $w^{(\alpha)}(g)=\alpha^5 y^{11}$ (see Figure 1).  
This refinement $w \mapsto w^{(\alpha )}$ transforms the species ${\rm Gr}_w$
of graphs weighted by the number of edges into the species  ${\rm Gr}_{w^{(\alpha)}}$ 
of graphs weighted by the numbers of edges {\em and} of connected components. Since any graph
is an assembly of connected graphs, the transformation can be described as
%
\begin{equation}  \label{eq:Gralphaw}
{\rm Gr}_w = E({\rm Gr}^c_w) \  \mapsto \ 
             {\rm Gr}_{w^{(\alpha)}} = E({\rm Gr}^c_{\alpha w}), 
\end{equation}
%
where $\alpha w$ expresses that the weight $w$ of any connected component is multiplied
by $\alpha$.

It is clear that the transformation (\ref{eq:Gralphaw}) can be extended to any weighted
species $F_w$, of the form $F_w = E(F_w^c)$, where $F_w^c$ denotes the species 
of connected $F_w$-structures:
%
\begin{equation}  \label{eq:Falphaw}
F_w = E(F^c_w) \  \mapsto \ 
             F_{w^{(\alpha)}} = E(F^c_{\alpha w}), 
\end{equation}
%
Our goal is to analyse the general transformation $F_w \mapsto F_{w^{(\alpha)}}$
and to deduce a number of enumerative results. We show, in section 2, that there exists 
a ``universal'' virtual (i.e.,\ formal difference of species, see \cite{BLL}, \cite{Jo2}, \cite{Ye})
weighted species, $\Lambda^{(\alpha)}$, such that 
%
\begin{equation}
F_{w^{(\alpha )}}\ =\ \Lambda ^{(\alpha )}\circ F_w^{+} ,
\end{equation}
where $F_w^{+} = F_w - 1$ denotes the species of non-empty $F_w$-structures, and 
we compute the main enumerative series associated with  $\Lambda^{(\alpha)}$. From this
follows an explicit general form for the series $G(x)$, $\widetilde{G}(x)$, $\overline{G}(x)$, 
$G(x;q)$, $G\langle{x;q}\rangle$, ${Z}_G(x_1,x_2,x_3,\ldots)$, 
${\Gamma }_G(x_1,x_2,x_3,\ldots)$, associated with the species $G = F_{w^{(\alpha )}}$, 
in terms of the series associated with the species $F_w$. In section 3, we give a number
of identities (and $q$-identities) which follow %--- automatically --- 
for some particular choices 
of species $F_w$. We conclude, in section 4, with an analysis of the species  
$\Lambda ^{(\alpha )}$. In particular, we give the first terms of the molecular decomposition
of $\Lambda ^{(\alpha )}$ which constitutes a new combinatorial lifting of the binomial
formula for a formal exponent $\alpha$: 
%
\begin{equation}  \label{eq:binomial}
(1+x{)}^{\alpha }\ = \ 1+ \alpha \, x+\alpha (\alpha -1)\,\frac{x^2}{2!}+
    \alpha(\alpha -1)(\alpha -2)\,\frac {x^3}{3!}+\alpha(\alpha-1)(\alpha-2)(\alpha-3)\,
    \frac{x^4}{4!}+\cdots \ .
\end{equation}
%
%
%
%%%%%%%%%%%%%%%%%%%%%%%%% Section 2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\section{The transformation $F_w \mapsto F_{w^{(\alpha)}}$}
%
Many formal power series can be assigned to a weighted species $F=F_w$. 
The two main ones, $Z_F$ and $\Gamma_F$ are called
respectively the {\em cycle index series of $F$} (see \cite{Jo1}) and the {\em asymmetry 
index series of $F$} (see \cite{La3}--\cite{La5}).  They are power series in an infinity 
of variables $x_1$, $x_2$, $x_3$, $\ldots$ :
%
\begin{equation}  \label{eq:zf}
Z_F(x_1,x_2,x_3,\ldots)\ =\ \sum\limits_{n=0}^{\infty} \frac 1{n!} 
\sum\limits_{\sigma\in S_n}^{} f_\sigma \  x_1^{\sigma_1}x_2^{\sigma_2}
x_3^{\sigma_3}\cdots\ ,
\end{equation}
%
\begin{equation}  \label{eq:gammaf}
\Gamma_F(x_1,x_2,x_3,\ldots)\ =\ \sum\limits_{n=0}^{\infty} \frac 1{n!} 
\sum\limits_{\sigma\in S_n}^{} f_\sigma^\ast\  x_1^{\sigma_1}x_2^{\sigma_2}
x_3^{\sigma_3}\cdots\ .
\end{equation}
%
In these formulas, the coeficients $f_\sigma$ (resp. $f_\sigma^\ast$) are 
{\Cp N}-linear (resp. {\Cp Z}-linear) combinations of the weights given to the $F$-structures
on $[n] = \{1,2,\ldots,n\}$, $S_n$ is the symmetric group of degree $n$ and $\sigma_k$
denotes the number of cycles of length $k$ in the cyclic decomposition of any permutation 
$\sigma\in S_n$. In fact, $Z_F$ and $\Gamma_F$ can be seen as symmetric functions of 
auxiliary variables $\xi_1,\xi_2,\xi_3,\ldots$, by making the substitutions
%
\begin{equation}    \label{eq:powersums}
x_i\ :=\ p_i(\xi_1,\xi_2,\xi_3,\ldots )\ =\ \xi_1^i+ \xi_2^i+\xi_3^i+\cdots \ , 
\;\; i\geq 1 \;\;\; (\mbox{power sums}).
\end{equation}
%

Other specializations of the variables $x_1,x_2,x_3,\ldots$, in the series 
(\ref{eq:zf})--(\ref{eq:gammaf}) give (see \cite{Jo1}, \cite{La3})
%
\begin{equation}
Z_F(x,0,0,\ldots)\ =\ \Gamma_F(x,0,0,\ldots)\ =\ F(x)\ =
  \ \sum\limits_{n\ge 0}^{} {f}_n\ \frac {x^n}{n!}\ ,
\end{equation}
%
\begin{equation}
Z_F(x,x^2,x^3,\ldots )\ =\ \widetilde{F}(x)\ =
  \ \sum\limits_{n\ge 0}^{} {\widetilde{f}}_nx^n,
\end{equation}
%
\begin{equation}
\Gamma_F(x,x^2,x^3,\ldots )\ =\ \overline{F}(x)\ =
  \ \sum\limits_{n\ge 0}^{} {\overline{f}}_nx^n,
\end{equation}
%
where
%
\begin{eqnarray}  
f_n& = &\mbox{total weight of the $F_w$-structures on } [n] = \{1,2,\ldots,n\},
   \label{eq:fn}   \\
\widetilde{f}_n& = &\mbox{total weight of the unlabelled 
   $F_w$-structures on $n$ points},
  \label{eq:fniso}  \\ 
\overline{f}_n& = &\mbox{total weight of the unlabelled asymmetric  $ F_w$-structures 
on $n$ points}.   
  \label{eq:fnasym}
\end{eqnarray}
%
Of course, when the weighting $w$ is trivial (i.e.,\ $w=1$ for any structure), then the total
weights (\ref{eq:fn}) -- (\ref{eq:fnasym}) give the numbers of corresponding structures.

Finally, two $q$-series $F(x;q)$ and $F\langle{x;q}\rangle$ can be canonically
associated to the species $F$ (see \cite{De1}--\cite{DeL}) by setting
%
\begin{equation}   \label{eq:fxq}
F(x;q)\ =\ \sum\limits_{n=0}^{\infty } f_n(q)\frac {x^n}{n!_q}\ =
\ Z_F\left(\frac{(1-q)}{(1-q)}x,\frac{(1-q)^2}{(1-q^2)}x^2,\frac {(1-q)^3}{(1-q^3)}x^3,
\ldots \right),
\end{equation}
%
\begin{equation}   \label{eq:fcrochetxq}
F\langle{x;q}\rangle\ 
=\ \sum\limits_{n=0}^{\infty } f_n\left\langle{q}\right\rangle\frac {x^n}{n!_q}\ 
=\ \Gamma_F\left(\frac{(1-q)}{(1-q)}x,\frac{(1-q)^2}{(1-q^2)}x^2,\frac {(1-q)^3}{(1-q^3)}x^3,
\ldots \right),
\end{equation}
%
where
%
\begin{equation}   \label{eq:nqfact}
n!_q\ =\ \frac{(1-q)}{(1-q)}\cdot\frac{(1-q^2)}{(1-q)}\cdot\frac {(1-q^3)}{(1-q)}\cdots
\frac {(1-q^n)}{(1-q)}
\end{equation}
%
denotes the $q$-analogue of $n!$ ($n! = \lim\limits_{q\rightarrow1} n!_q$). These $q$-series satisfy
the following properties:
%
\begin{equation}   \label{eq:limfxq}
\lim\limits_{q\rightarrow 1}^{}F(x;q)\ =\ F(x),\hspace{1cm}
\lim\limits_{q\rightarrow 0}^{}F(x;q)\ =\ \widetilde{F}(x).
\end{equation}
%
%
\begin{equation}   \label{eq:limcrochet}
\lim\limits_{q\rightarrow 1}^{}F\langle{x;q}\rangle\ =\ F(x),\hspace{1cm}
\lim\limits_{q\rightarrow 0}^{}F\langle{x;q}\rangle\ =\ \overline{F}(x).
\end{equation}
%
Moreover their coefficients, $f_n(q)$ and $f_n\left\langle{q}\right\rangle$, are 
polynomials in $q$ of degree $\leq n(n-1)/2$. For $F=E$, the species of sets, the $q$-series
(\ref{eq:fxq}) and (\ref{eq:fcrochetxq}) take the form of the two classical $q$-analogues
of the exponential function:
%
\begin{eqnarray}  
E(x;q) & = & \sum\limits_{n=0}^{\infty } \frac {x^n}{n!_q}\ 
  =\  \frac {1}{(1-(1-q)x)(1-(1-q)qx)(1-(1-q)q^2x)\cdots}\ ,
 \label{eq:exq}      \\
E\langle{x;q}\rangle &
 = & \sum\limits_{n=0}^{\infty } {q}^{n(n-1)/2}\,\frac {{x}^{n}}{n{!}_{q}}\ 
 =\  (1+(1-q)x)(1+(1-q)qx)(1+(1-q){q}^{2}x)\cdots.
\label{eq:ecrochetxq}    
\end{eqnarray}
%
The definition of the series ${Z}_{F}$, ${\Gamma }_{F}$, $F(x)$, $\widetilde{F}(x)$, 
$\overline{F}(x)$, $F(x;q)$, $F\langle{x;q}\rangle$, 
can be extended to any {\em virtual} weighted species $F=F_w = A_u - B_v$, where
$A_u$ and $B_v$ are two weighted species, by setting $Z_F=Z_{A_u} - Z_{B_v}$,
$\Gamma_F=\Gamma_{A_u} - \Gamma_{B_v}$, and so on.

The transformations $F\mapsto Z_F$ and $F\mapsto \Gamma_F$, from species to index
series, have the remarkable property of preserving the main combinatorial operations
(\cite{Jo1}, \cite{La3}): for any (virtual weighted) species $F$ and $G$, one has
%
\begin{equation}   \label{eq:plus}
Z_{F+G}\ =\ Z_F + Z_G\ , \hspace{1cm} \Gamma_{F+G}\ =\ \Gamma_F + \Gamma_G\ ,
\end{equation}
%
\begin{equation}   \label{eq:produit}
Z_{F\cdot G}\ =\ Z_F \cdot  Z_G\ , \hspace{1cm}
  \Gamma_{F\cdot G}\ =\ \Gamma_F \cdot  \Gamma_G\ ,
\end{equation}
%
 \begin{equation}   \label{eq:compose}
Z_{F\circ G}\ =\ Z_F \circ  Z_G\ , \hspace{1cm}
  \Gamma_{F\circ G}\ =\ \Gamma_F \circ  \Gamma_G\ ,
\end{equation}
%
 \begin{equation}   \label{eq:derivee}
Z_{F'}\ =\ \frac{\partial Z_F}{\partial x_1} \ , \hspace{1cm} 
\Gamma_{F'}\ =\ \frac{\partial \Gamma_F}{\partial x_1} \ ,
\end{equation}
%
In the right hand sides of (\ref{eq:compose}), the operation used is the {\em plethystic
substitution} of index series. It is defined, for two arbitrary index series 
$f_w(x_1,x_2,x_3,\ldots)$ and $g_v(x_1,x_2,x_3,\ldots)$, with $g_v(0,0,0,\ldots) = 0$,
arising from weighted virtual species $F_w$ and $G_v$, by
%
 \begin{equation}   \label{eq:plethysme}
  (f_w\circ g_v)(x_1,x_2,x_3,\ldots)\ =\ f_w((g_v)_1,(g_v)_2,(g_v)_3,\ldots)\ ,
\end{equation}
%
where $(g_v)_k := g_{v^k}(x_k,x_{2k},x_{3k},\ldots)$ denotes the index series obtained
by raising to the power $k$ any weight appearing in $g_v(x_1,x_2,x_3,\ldots)$ and
by multiplying the index of any variable by $k$. The following general principle follows
immediately from the equations (\ref{eq:plus})--(\ref{eq:derivee}):
%
\begin{principe}   \label{th:principe}
Any combinatorial equation {\rm (that is, a natural isomorphism, see \cite{BLL})} between 
species $F$, $G$, $H$, $\dots$ gives rise---automatically---to seven corresponding 
equations between the series
%
\begin{eqnarray}   
& F(x), G(x), H(x), \ldots , & \nonumber  \\
Z_F, Z_G, Z_H, \ldots\ ,  & \ \widetilde{F}(x),   \widetilde{G}(x), \widetilde{H}(x),  \ldots\ , & 
\ F(x;q), G(x;q), H(x;q), \ldots , \label{eq:series}  \\
\Gamma_F, \Gamma_G, \Gamma_H, \ldots\ ,  
& \ \overline{F}(x),   \overline{G}(x), \overline{H}(x), \ldots\ , & \ F\langle{x;q}\rangle, 
G\langle{x;q}\rangle, H\langle{x;q}\rangle, \ldots . \nonumber
\end{eqnarray}
\end{principe}
%

Here is the main result of this article.
%
\begin{theorem} \label{th:Lambdaalpha}
There exists a ``universal'' virtual weighted species, $\Lambda^{(\alpha)}$, such that 
for any species of the form $F_w = E(F_w^c)$, we have
%
\begin{equation}  \label{eq:fwalpha}
F_{w^{(\alpha )}}\ =\ \Lambda ^{(\alpha )}\circ F_w^{+} ,
\end{equation}
%
where $F_w^{+} = F_w - 1$ denotes the species of non-empty $F_w$-structures and
$F_{w^{(\alpha )}}=E(F_{\alpha w}^c)$ denotes the species of 
$F_w$-structures with the adjunction to the weight of a counter $\alpha$ of connected
components. The main series associated to $\Lambda^{(\alpha)}$ are given by the
formulas
%
\begin{eqnarray}   
& \Lambda^{(\alpha)}(x)\ =\ (1+x)^\alpha,  & \nonumber  \\
Z_{{\Lambda }^{(\alpha)}}=\prod\limits_{n\ge 1}^{} (1+x_n)^{\lambda_n(\alpha)},
 & \widetilde{\Lambda ^{(\alpha)}}(x)=\prod\limits_{n\ge 1}^{} (1+x^n)^{\lambda_n(\alpha)},
 & \Lambda^{(\alpha)}(x;q)=\prod\limits_{n\ge 1}^{} {\left({1+\frac{(1-q)^n}
 {(1-q^n)}\,x^n}\right)}^{\lambda_n(\alpha)}
\label{eq:serieslambda}  \\
\Gamma_{{\Lambda }^{(\alpha)}} = \prod\limits_{n\ge 1}^{} (1+x_n)^{\gamma_n(\alpha)},
 & \overline{\Lambda ^{(\alpha)}}(x) = \prod\limits_{n\ge 1}^{} (1+x^n)^{\gamma_n(\alpha)},
 & \Lambda^{(\alpha)}\langle{x;q}\rangle = \prod\limits_{n\ge 1}^{} {\left({1+\frac{(1-q)^n}
 {(1-q^n)}\,x^n}\right)}^{\gamma_n(\alpha)}, \nonumber
\end{eqnarray}
%
where the exponents $\lambda_n(\alpha)$ and $\gamma_n(\alpha)$ are polynomials of 
degree $n$ in $\alpha$ defined by 
%
\begin{eqnarray}   
\lambda_n(\alpha) & = & \frac {1}{n}\sum\limits_{d|n}^{} \mu (n/d)\alpha^d,
\label{eq:lambdanalpha} \\
\gamma_n(\alpha) & = & -\lambda_{n}(-\alpha) - \lambda_{n/2}(-\alpha) 
 - \lambda_{n/4}(-\alpha) - \cdots - \lambda_{n/{2^k}}(-\alpha) - \cdots\ .
\label{eq:gammanalpha}
\end{eqnarray}
%
In {\rm(\ref{eq:lambdanalpha})},  $\mu$ denotes the classical M\"obius function and 
in {\rm(\ref{eq:gammanalpha})}, the sum ranges over all integers $k\ge 0$ for which $n/{2^k}$
is an integer.
\end{theorem}

Before proving Theorem \ref{th:Lambdaalpha}, observe that when $\alpha$ is a positive integer, 
$\lambda_n(\alpha)$ , given by (\ref{eq:lambdanalpha}), is equal to the number of
{\em primitive circular words (Lyndon words)} of length $n$ over an alphabet of $\alpha$
letters (c.f.\ \cite{Lo}). It also represents the number of irreducible unitary polynomials of degree $n$
over the finite field $F_q$ of cardinality $q$, when $\alpha = q$ (c.f.\ \cite{DS3}, 
\cite{Re}).
The polynomials (\ref{eq:lambdanalpha}) are known as {\em Lyndon polynomials} (see \cite{St1})
and also as (primitive) {\em necklace polynomials} (see \cite{MeR}). 
Finally, note that formula (\ref{eq:fwalpha}) permits the
extension of the transformation $F_w \mapsto F_{w^{(\alpha)}}$ to {\em all} the weighted 
species $F_w$ for which $F_w(0)=1$. Indeed, it suffices to substitute $F_w^{+} = F_w - 1$
into $\Lambda^{(\alpha)}$.  The species $F_w$ can even be virtual.  In fact, as long as $F_w(0)=1$, 
a concept of connected components can be defined (see (\ref{eq:fwc}) below) and the 
associated series can be computed (see \cite{BLL}, \cite{Jo2}, \cite{La2}, \cite{La5}).
%

\vspace{\baselineskip}
%\begin{proof}{\bf  of Theorem \ref{th:Lambdaalpha}}.
\noindent
{\bf  Proof of Theorem \ref{th:Lambdaalpha}}.
Consider the species $X_\alpha$ of singletons of weight $\alpha$ and $E^+=E-1$, of non-empty
sets.  A. Joyal \cite{Jo2} has shown that the species $E^+$ possesses an inverse $(E^+)^{<-1>}$ under
substitution, given by the formula
%
\begin{equation}  \label{eq:einverse}
(E^+)^{<-1>}\ =\ X- \Delta X+ \Delta^{2}X- \Delta^{3}X+\cdots,
\end{equation}
%
where $X$ is the species of singletons (of weight $1$) and $\Delta$ is the combinatorial
difference operator defined by $\Delta G = G\circ E^+ - G$, for any (virtual) species $G$.
Since $F_w = E(F_w^c) = 1+E^+(F_w^c)$, we get $F_w^+ = F_w-1 = E^+(F_w^c)$, whence
%
\begin{equation}  \label{eq:fwc}
F_w^c\ =\ (E^+)^{<-1>}\circ F_w^+ .
\end{equation}
%
Let us define $\Lambda^{(\alpha)}$ by the formula
%
\begin{equation}  \label{eq:Lambdaalphadefinition}
\Lambda^{(\alpha)}\ :=\ E \circ X_\alpha \circ (E^+)^{<-1>}.
\end{equation}
%
A short computation, based on (\ref{eq:einverse}), gives the first few terms of the molecular 
expansion of $\Lambda^{(\alpha)}$ (see also Section 4):
%
\begin{equation}  \label{eq:Lambdaalphaformule}
\Lambda^{(\alpha)}\ =\ 1+X_\alpha+(E_2)_{\alpha^2}-(E_2)_\alpha+\cdots ,
\end{equation}
%
where $(E_2)_\alpha$ and $(E_2)_{\alpha^2}$ denote the species of sets of
cardinality $2$ weighted by $\alpha$ and by $\alpha^2$, respectively. This shows that
$\Lambda^{(\alpha)}$ is a virtual weighted species. We then have successively
%
\begin{equation}  \label{eq:fwalphaformule}
F_{w^{(\alpha )}} = E(F_{\alpha \,w}^c) = E\circ {X}_{\alpha }\circ F_w^c
=E\circ {X}_{\alpha }\circ {(E}^{+}{)}^{<-1>}\circ F_{w}^{+}
={\Lambda }^{(\alpha )}\circ F_w^{+}
\end{equation}
%
which gives (\ref{eq:fwalpha}).  By taking the generating series of both sides of
(\ref{eq:Lambdaalphadefinition}), we get 
%
\begin{equation}  \label{eq:Lambdaalphax}
\Lambda^{(\alpha)}(x)\ =\ {e}^{\alpha\log(1+x)}\ =\ (1+x)^\alpha.
\end{equation}
%
The explicit formula for the cycle index series $Z_{\Lambda^{(\alpha)}}$ is obtained by
substituting the known expressions (see \cite{Jo1}, \cite{Jo2}, \cite{La2})
%
\begin{equation}  \label{eq:ze}
Z_E\ =\ \exp(\sum\limits_{n\ge1}^{}{x_n/ n})\ , \hspace{1cm}
Z_{(E^+)^{<-1>}}\ =\ \sum\limits_{n=1}^{\infty} \frac{\mu (n)}{n}\log(1+x_n)
\end{equation}
%
in the formula 
${Z}_{\Lambda^{(\alpha )}}={Z}_{E\circ X_\alpha\circ {(E}^{+}{)}^{<-1>}}
=({Z}_E)\circ (\alpha x_{1})\circ ({Z}_{(E^{+}{)}^{<-1>}})$.
The explicit computation of the asymmetry index series 
$\Gamma_{\Lambda^{(\alpha )}}
=\Gamma_{E\circ X_\alpha\circ {(E}^{+}{)}^{<-1>}}
=(\Gamma_E)\circ (\alpha x_{1})\circ (\Gamma_{(E^+)^{<-1>}})$
is more delicate.  It is known \cite{La3} that
%
\begin{equation}  \label{eq:Gammae}
\Gamma_E\ =\ \exp(\sum\limits_{n\ge1}^{}(-1)^{n-1}{x_n / n})
\end{equation}
%
and we must determine $\Theta := \Gamma_{(E^+)^{<-1>}}$. Since $E\circ(E^+)^{<-1>} 
= 1 + E^+\circ (E^+)^{<-1>} = 1 +X$, we have $\Gamma_E\circ \Theta = 1 + x_1$ .
Substituting $\Theta$ in both sides of (\ref{eq:Gammae}) and taking logarithms, we 
find
%
\begin{equation}  \label{eq:Thetalog}
\Theta_1-\frac{1}{2}\Theta_2+\frac {1}{3}\Theta_3-\cdots
+\frac{(-1)^{i-1}}{i}\Theta_i+\cdots\ =\ \log(1+x_1),
\end{equation}
%
where $\Theta_i = \Theta(x_i, x_{2i}, x_{3i}, \cdots)$, $i\geq 1$.  By summation, we obtain
the remarkable relation 
%
\begin{equation}  \label{eq:ThetaOmega}
\Theta_1-\Theta_2\ =\ \Omega\ , \hspace{.5cm} {\rm where} \hspace{.5cm} 
\Omega\ :=\ Z_{(E^+)^{<-1>}}\ =\ \sum\limits_{k=1}^{\infty} \frac{\mu (k)}{k}\log(1+x_k)
\end{equation}
%
since the M\"{o}bius function satisfies the property
%
\begin{equation}  \label{eq:mu}
{1\over n}\sum\limits_{d|n}^{} \mu({n/d})(-1)^{d-1}\ 
=\ \left\{ \begin{array}{rcl}
	     1,   &   \mbox{if}    &  n = 1,  \\
     -1,   &   \mbox{if}   &  n = 2,\\
       0,   &   \mbox{if}   &  n \ge 3.         \end{array}\right.
\end{equation}
%

Another summation, based on (\ref{eq:ThetaOmega}), gives
%
\begin{equation}  \label{eq:ThetaOmegabis}
\Theta\ =\ \Omega_1 + \Omega_2 + \Omega_4 + \cdots + \Omega_{2^i} + \cdots.
\end{equation}
%
We then get, by collecting terms properly,
%
\begin{equation}  \label{eq:GammaLambdaalpha}
\Gamma_{\Lambda^{(\alpha )}}
=(\Gamma_E)\circ (\alpha x_{1})\circ (\Gamma_{(E^+)^{<-1>}})
=\Gamma_E\circ(\alpha\Omega_1 + \alpha\Omega_2 + \alpha\Omega_4 + \cdots)
=\prod\limits_{n\ge 1}^{} (1+x_n)^{\gamma_n(\alpha)}
\end{equation}
%
where
%
\begin{equation}  \label{eq:Gammanalphabis}
{\gamma }_{n}(\alpha )=\sum\limits_{2^i\cdot j\cdot k=n}^{} 
\frac {(-1)^{j-1}\mu(k)}{kj}\alpha^j
\end{equation}
%
and equation (\ref{eq:gammanalpha}) is satisfied. The other series associated to 
$\Lambda_{(\alpha)}$ follow directly from the formulas
for $Z_{\Lambda_{(\alpha)}}$ and $\Gamma_{\Lambda_{(\alpha)}}$. \qed
%\end{proof}
%

The following corollary gives the general explicit form of the series associated to
the species $F_{w^{(\alpha)}}$ in terms of the series for $F_w$. The proof is straightforward 
and left to the reader.
%
\begin{corollaire}    \label{th:seriesfwalpha}
Let $F_w$ be a weighted virtual species such that $F_w(0)=1$ and set 
$F_{w^{(\alpha )}}\ =\ \Lambda ^{(\alpha )}\circ F_w^{+}$. Then we have
%
\begin{eqnarray}
{F}_{{w}^{(\alpha )}}(x) & = & F_{w}(x)^{\alpha }, 
\label{eq:fwalphax}    \\
Z_{F_{w^{(\alpha )}}}(x_1,x_2,x_3,\ldots ) & 
= & \prod\limits_{n\ge 1}^{} Z_{F_{w^n}}(x_n,x_{2n},x_{3n},{\ldots )}^{\lambda_n(\alpha )},
\label{eq:zfwalpha}    \\
\widetilde{F_{w^{(\alpha )}}}(x) & 
= & \prod\limits_{n\ge 1}^{} \widetilde{F_{w^n}}({x}^n)^{\lambda_n(\alpha)},
\label{eq:ftildewalpha}    \\
F_{w^{(\alpha )}}(x;q) &
= & \prod\limits_{n\ge 1}^{} F_{w^n}{\left(\frac{(1-q)^n}{(1-q^n)}x^n;q^n\right)}^{\lambda_n(\alpha)},
\label{eq:fwalphaq}    \\
\Gamma_{F_{w^{(\alpha )}}}(x_1,x_2,x_3,\ldots ) & 
= & \prod\limits_{n\ge 1}^{} \Gamma_{F_{w^n}}(x_n,x_{2n},x_{3n},{\ldots )}^{\gamma_n(\alpha )},
\label{eq:gammafwalpha}    \\
\overline{F_{w^{(\alpha )}}}(x) & 
= & \prod\limits_{n\ge 1}^{} \overline{F_{w^n}}({x}^n)^{\gamma_n(\alpha)},
\label{eq:fbarwalpha}    \\
F_{w^{(\alpha )}}\langle{x;q}\rangle &
= & \prod\limits_{n\ge 1}^{} F_{w^n}{\left\langle\frac{(1-q)^n}{(1-q^n)}x^n;q^n
\right\rangle}^{\gamma_n(\alpha)}.
\label{eq:fwalphaqcrochet}  
\end{eqnarray}
%
\end{corollaire}
\qed

The special case $\alpha=-1$ is particularly interesting. In the simplest case of the series
(\ref{eq:fwalphax}), it corresponds to the computation of the excess weight of the 
$F_w$-structures having an even number of connected components over those having
an odd number:
%
\begin{equation}     \label{fwdifference}
F_{w^{(-1)}}(x)\ =\ \sum\limits_{n\ge 0}^{} (f_n^+-f_n^-)\frac {x^n}{n!},
\end{equation}
%
where
%
\begin{eqnarray}    
f_n^+\!\! &=&\!\!\mbox{total weight of $F_w$-structures on $[n]$ with an even number of 
connected components},
\label{fnplus}   \\
f_n^-\!\! &=&\!\!\mbox{total weight of $F_w$-structures on $[n]$ with an odd number of 
connected components}.
 \label{fnmoins} \end{eqnarray}
%
The behavior of the seven main series with respect to the transformation 
$F_w\mapsto F_{w^{(-1)}}$ is described in the following corollary.
%
\begin{corollaire}   \label{th:wmoinsun}
Let $F_w$ be a weighted virtual species such that $F_w(0)=1$ and set 
$F_{w^{(-1)}}\ =\ \Lambda ^{(-1)}\circ F_w^{+}$. Then we have
%
\begin{eqnarray}
{F}_{{w}^{(-1)}}(x) & = & 1/F_{w}(x), 
\label{eq:fwmoinsunx}    \\
Z_{F_{w^{(-1)}}}(x_1,x_2,x_3,\ldots ) & 
= &  \frac{Z_{F_{w^2}}(x_2,x_4,x_6,\ldots)} {Z_{F_w}(x_1,x_2,x_3,\ldots)},
\label{eq:zfwmoinsun}    \\
\widetilde{F_{w^{(-1)}}}(x) & 
= &  \frac{\widetilde{F_{w^2}}(x^2)} {\widetilde{F_w}(x)},
\label{eq:ftildewmoinsun}    \\
F_{w^{(-1)}}(x;q) &
= & \frac{F_{w^2}(\frac{(1-q)}{(1+q)}x^2;q^2)} {F_w(x;q)},
\label{eq:fwmoinsunq}    \\
\Gamma_{F_{w^{(-1)}}}(x_1,x_2,x_3,\ldots ) & 
= & 1/ \prod\limits_{k\ge 0}^{}\Gamma_{F_{w^{2^k}}}(x_{2^k},x_{2\cdot {2^k}},
  x_{3\cdot {2^k}},\ldots),
\label{eq:gammafwmoinsun}    \\
\overline{F_{w^{(-1)}}}(x) & 
= & 1/ \prod\limits_{k\ge 0}^{} \overline{F_{w^{2^k}}}({x}^{2^k}),
\label{eq:fbarwmoinsun}    \\
F_{w^{(-1)}}\langle{x;q}\rangle &
= & 1/ \prod\limits_{k\ge 0}^{} F_{w^{2^k}}
   \langle{\frac{(1-q)^{2^k}}{(1-q^{2^k})}x^{2^k};q^{2^k}\rangle}.
\label{eq:fwmoinsunqcrochet}  
\end{eqnarray}
%
\end{corollaire}
%
%\begin{proof}

\noindent  {\bf Proof.}
It suffices to use the relations
%
\begin{equation}  \label{eq:lambdanmoinsun}
\lambda_n(-1)\ 
=\ \left\{ \begin{array}{rcl}
	   -1,   &   \mbox{if}    &  n = 1  \\
       1,   &   \mbox{if}   &  n = 2  \\
       0,   &   \mbox{if}   &  n \ge 3         \end{array}\right. 
\ \ \  \mbox{and} \ \ \ \ 
\gamma_n(-1)\ 
=\ \left\{ \begin{array}{rl}
	   -1,   &   \mbox{if} \   n = 2^k, k\ge 0 \\
       0,   &   \mbox{otherwise}  \ .       \end{array}\right.
\end{equation}
%
\begin{flushright}
\qed
\end{flushright}
%
%\end{proof}

Observe that the transformations described by (\ref{eq:fwmoinsunx})--(\ref{eq:fwmoinsunqcrochet}) 
are involutive since $w^{(-1)(-1)}=w$.  It is easily verified
that the series associated to the universal virtual species $\Lambda^{(-1)}$ are given by
%
\begin{eqnarray}   
& \Lambda^{(-1)}(x)\ =\ (1+x)^{-1},  & \nonumber  \\
Z_{{\Lambda }^{(-1)}}= \frac{1+x_2} {1+x_1},
 & \widetilde{\Lambda ^{(-1)}}(x) = {\displaystyle \frac{1+x^2} %{\displaystyle 
{1+x}},
 & \Lambda^{(-1)}(x;q) = \frac{1+\frac{1-q}{1+q}\,x^2}{1+x}\ ,
\label{eq:seriesLambdamoinsun}  \\
\Gamma_{\Lambda^{(-1)}} = \prod\limits_{k\ge 0} (1+x_{2^k})^{-1},
 & \overline{\Lambda ^{(-1)}}(x) = 1-x,
 & \Lambda^{(-1)}\langle{x;q}\rangle = \prod\limits_{k\ge 0}^{} 
({1+ %{\scriptstyle    }
\frac{(1-q)^{2^k}} {1-q^{2^k}}\,x^{2^k}})^{-1}. \nonumber
\end{eqnarray}
%
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Section 3 %%%%%%%%%%%%%%%%%%%%%%%%%%
%
\section{Examples arising from particular choices of $F_w$}
%
The corollaries \ref{th:seriesfwalpha} and \ref{th:wmoinsun} give rise to a multitude of 
identities and of interesting expressions, by choosing particular species $F_w$. For example, 
in the case of the species $F_w=E$, of sets (trivially weighted by $w=1$), we have
$Z_E = \exp (\sum_{n\ge 1} {x_n/n})$ and 
$\Gamma_E = \exp (\sum_{n\ge 1} (-1)^{n-1} {x_n/n})$.
The formulas (\ref{eq:fwalphax})--(\ref{eq:fwalphaqcrochet}) of Corollary \ref{th:seriesfwalpha}
produce respectively the identities
%
\begin{eqnarray}
e^{\alpha x} & = & (e^x)^\alpha, 
\label{eq:ealphax}    \\
\exp (\sum_{k\ge 1} \alpha^k {x_k/k}) & 
= & \prod\limits_{n\ge 1}^{} \exp ({\lambda_n(\alpha)}\sum_{k\ge 1} {x_{kn}/k}),
\label{eq:zealpha}    \\
(1-\alpha x)^{-1} & 
= & \prod\limits_{n\ge 1}(1-x^n)^{-\lambda_n(\alpha)},
\label{eq:etildealpha}    \\
E(\alpha x;q) &
= &  E{(x;q)}^{\alpha}\ E( {\scriptstyle \frac{(1-q)^2}{(1-q^2)}}x^2;q^2)^{\lambda_2(\alpha)} 
\ E({\scriptstyle \frac{(1-q)^3}{(1-q^3)}}x^3;q^3)^{\lambda_3(\alpha)} \cdots ,
\label{eq:ealphaq}    \\
\exp (\sum_{k\ge 1} (-1)^{k-1} \alpha^k {x_k/k}) & 
= & \prod\limits_{n\ge 1}^{} \exp ({\lambda_n(\alpha)}\sum_{k\ge 1} (-1)^{k-1} {x_{kn}/k}),
\label{eq:gammaealpha}    \\
1+\alpha x & 
= & \prod\limits_{n\ge 1}^{} ( 1 +x^n)^{\gamma_n(\alpha)},
\label{eq:ebaralpha}    \\
E\langle{\alpha x;q}\rangle &
= & E\langle{x;q}\rangle^{\alpha}\ E\langle{{\scriptstyle \frac{(1-q)^2}{(1-q^2)}}x^2;q^2}\rangle^{\gamma_2(\alpha)} 
\ E\langle{{\scriptstyle \frac{(1-q)^3}{(1-q^3)}}x^3;q^3}\rangle^{\gamma_3(\alpha)} \cdots .
\label{eq:ealphaqcrochet}  
\end{eqnarray}
%
The reader will recognize, in (\ref{eq:etildealpha}), the classical cyclotomic identity of
Gauss (see \cite{DS1}, \cite{MeR}).  The identity (\ref{eq:ebaralpha}) constitutes a kind of dual of
(\ref{eq:etildealpha}) and could be called the {\em cocyclotomic} identity.
The identities (\ref{eq:zealpha}) (due to F. Bergeron, see \cite{Be}, p.\ 300) and  
(\ref{eq:gammaealpha}) are extensions of (\ref{eq:etildealpha})
and (\ref{eq:ebaralpha}) which also yield (\ref{eq:ealphaq}) and 
(\ref{eq:ealphaqcrochet}) by specialization. Expressed in terms of the $q$-exponentials, the latter can be seen
as $q$-analogues of the cyclotomic and cocyclotomic identities.

In the case of the species $F_w=S$, of permutations, we have $Z_S = \prod_{n\ge 1}(1-x_n)^{-1}$ 
and $\Gamma_S = (1-x_2)/(1-x_1)$.  Let us then denote by $S_{(\alpha)}$, the transformed
species of permutations, weighted by the cycle counter $\alpha$. Applications of Corollary 
\ref{th:seriesfwalpha} give for example, after a proper term collection,
%
\begin{equation}   \label{eq:zsalpha}
Z_{S_{(\alpha)}}\ =\ \prod_{n\ge 1}(1-x_n)^{-\nu_n(\alpha)},
\end{equation}
%
where $\nu_n(\alpha)$ denotes the polynomial defined by
%
\begin{equation}   \label{eq:nunalpha}
\nu_n(\alpha)\ = \ \frac {1}{n}\sum\limits_{d|n}^{} \phi (n/d)\alpha^d,
\end{equation}
%
where  $\phi(n)$ is the Euler totient function. When $\alpha$ is a positive integer,
$\nu_n(\alpha)$ is the number of circular words of length $n$ over an alphabet with 
$\alpha$ letters or, equivalently, the number of (not necessarily primitive) necklaces
of length $n$ with $\alpha$ colors.
Formula (\ref{eq:zsalpha}), due to
H. D\'ecoste, is the basis for the computation of the cycle index series %(the characters)
of combinatorial models of the classical orthogonal polynomials 
of Laguerre, Charlier,  Meixner,  Meixner-Pollaczek, and Jacobi,
(see \cite{De1}, \cite{De2}, and also \cite{Co}, \cite{St1} and  \cite{St2})
and their canonical $q$-analogues.  We also obtain
%
\begin{equation}   \label{eq:gammasalpha}
\Gamma_{S_{(\alpha)}}\ =\ \prod_{n\ \mbox{\scriptsize{odd}}}(1-x_n)^{-\gamma_n(\alpha)}
 \prod_{n\ \mbox{\scriptsize {even}}}(1-x_n)^{\lambda_n(-\alpha)} .
\end{equation}
%
Taking the particular value $\alpha=-1$, Corollary \ref{th:wmoinsun} gives the following 
formulas:
%
\begin{eqnarray}
S_{(-1)}(x) & = & 1-x , 
\label{eq:swmoinsunx}    \\
Z_{S_{(-1)}}(x_1,x_2,x_3,\ldots ) & 
= &  (1-x_1)(1-x_3)(1-x_5) \cdots ,
\label{eq:zswmoinsun}    \\
\widetilde{S_{(-1)}}(x) & 
= &  (1-x)(1-x^3)(1-x^5) \cdots ,
\label{eq:stildewmoinsun}    \\
S_{(-1)}(x;q) &
= & (1-x)(1-{\scriptstyle \frac{(1-q)^3}{(1-q^3)}}x^3)
(1-{\scriptstyle \frac{(1-q)^5}{(1-q^5)}}x^5) \cdots,
\label{eq:swmoinsunq}    \\
\Gamma_{S_{(-1)}}(x_1,x_2,x_3,\ldots ) & 
= & 1-x_1 ,
\label{eq:gammaswmoinsun}    \\
\overline{S_{(-1)}}(x) & 
= & 1-x ,
\label{eq:sbarwmoinsun}    \\
S_{(-1)}\langle{x;q}\rangle &
= & 1-x .
\label{eq:swmoinsunqcrochet}  
\end{eqnarray}
%
The reader is invited to give direct combinatorial proofs of these formulas. 

In the case of the species $F_w = {\rm Gr}_w$ of %simple 
graphs, weighted by an edge counter $y$, already seen in the introduction, the series
${\rm Gr}_{w^{(\alpha)}}(x)$, $Z_{{\rm Gr}_{w^{(\alpha)}}}$, 
$\widetilde{{\rm Gr}_{w^{(\alpha)}}}(x)$, ${\rm Gr}_{w^{(\alpha)}}(x;q)$, of the associated
species ${\rm Gr}_{w^{(\alpha)}}$ can be computed with the help of 
(\ref{eq:fwalphax})--(\ref{eq:fwalphaq}), by using the known formula (see \cite{BLL}, \cite{HaP})
%
\begin{equation}   \label{eq:zgr}
Z_{\rm Gr}(x_1,x_2,x_3,\ldots )\ =\ \sum\limits_{n=0}^{\infty } \frac 1{n!}
{\sum\limits_{\sigma\in {S}_n} (1+y)^{c_1}(1+y^2)^{c_2}(1+y^3)^{c_3}\cdots
x_1^{{\sigma }_1}x_2^{{\sigma }_2}x_3^{{\sigma }_3}\cdots},
\end{equation}
%
where
%
\begin{equation}   \label{eq:ck}
c_k\ =\ \frac {1} {2} \sum\limits_{[i,j]=k} (i,j)\sigma_i\sigma_j + \sigma_{2k}- \sigma_k
+ {1\over 2}(k \bmod 2)\sigma_k ,
\end{equation}
%
and where $[i,j]$ denotes the smallest common multiple of the integers $i$ and $j$.
However, the explicit computation of the other series  $\Gamma_{{\rm Gr}_{w^{(\alpha)}}}$, 
$\overline{{\rm Gr}_{w^{(\alpha)}}}(x)$, ${\rm Gr}_{w^{(\alpha)}}\langle x;q\rangle$ is an open problem since
no closed formulas for the series $\Gamma_{{\rm Gr}_{w}}$, 
$\overline{{\rm Gr}_{w}}(x)$, ${\rm Gr}_{w}\langle x;q\rangle$ are known up to now.

Other applications are possible, in addition to the various combinatorial models of
orthogonal polynomials %
already mentioned.
An example is given by the species $F_w = E(A) = A/X$, of forests of rooted trees 
(where $A$ is the species of rooted trees). 
The transformation $F_w\mapsto F_{w^{(\alpha)}}$ is then equivalent to the introduction
of a counter for trees in the forests. Taking the coefficients of the seven associated series
yields families of polynomials in $\alpha$: $a_n(\alpha)$, $a_\sigma(\alpha)$, 
$\tilde{a}_n(\alpha)$, $a_n(\alpha;q)$, 
$a^*_\sigma(\alpha)$, $\overline{a}_n(\alpha)$ and $a_n\langle\alpha;q\rangle$.
The first is the family of classical Abel polynomials $a_n(\alpha) 
= \alpha(\alpha + n)^{n-1}$
while the others constitute generalized versions of it. %$a_n(\alpha)$. 
Note that
these generalized Abel polynomials are distinct from those introduced in \cite{La6},
since the species $F_{w^{(\alpha)}}$ is different from the simple exponentiation 
$(F_w)^\alpha$, even when $\alpha$ is an integer.
%
%
%
%%%%%%%%%%%%%%%%%%%%%%%%% Section 4 %%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\section{Some properties of the species $\Lambda^{(\alpha)}$}
Recall that the universal virtual weighted species $\Lambda^{(\alpha)}$ is defined by
$\Lambda^{(\alpha)} = E\circ X_\alpha \circ (E^+)^{<-1>}$ and that we have 
$F_{w^{(\alpha)}} = \Lambda^{(\alpha)} \circ F_w^+$, for any weighted species $F_w$ such that 
$F_w(0)=1$. Since the exponential generating series of $\Lambda^{(\alpha)}$ given by
$\Lambda^{(\alpha)}(x) = (1+x)^\alpha$, it is natural to ask which properties of the function
$(1+x)^\alpha$ carry over to the species $\Lambda^{(\alpha)}$. For example, the algebraic
identity ${((1+x)^\beta)}^\alpha = (1+x)^{\alpha\beta}$ can be lifted to the combinatorial
equality
%
\begin{equation}    \label{eq:Lambdaalphabeta}
\Lambda^{(\alpha)} \circ {(\Lambda^{(\beta)})}^+\ =\ \Lambda^{(\alpha\beta)},
\end{equation}
%
where ${(\Lambda^{(\beta)})}^+ = \Lambda^{(\beta)} - 1 = E^+ \circ X_\beta \circ {(E^+)}^{<-1>}$
is the conjugate (for composition) of the species of singletons of weight $\beta$ by that of 
non-empty sets. Indeed, since $X_\alpha \circ X_\beta = X_{\alpha\beta}$, we have
immediately
%
\begin{equation}    \label{eq:exalphaemoinsun}
E \circ X_\alpha \circ {(E^+)}^{<-1>} \circ E^+ \circ X_\beta \circ {(E^+)}^{<-1>}\ 
=\ E \circ X_{\alpha\beta} \circ {(E^+)}^{<-1>}.
\end{equation}
%
By taking the cycle and asymmetry index series of both sides of (\ref{eq:Lambdaalphabeta})
the following identities are easily obtained:
%
\begin{equation}    \label{eq:lambdanalphabeta}
\lambda_n(\alpha \beta )\ 
=\ \sum\limits_{i\cdot j=n} \lambda_i(\alpha^j)\lambda_j(\beta), \hspace{.5cm}
\gamma_n(\alpha \beta )\ 
=\ \sum\limits_{i\cdot j=n} \gamma_i(\alpha^j)\gamma_j(\beta), \hspace{.5cm}
n\ge 0.
\end{equation}
%
The identity on the left hand side of (\ref{eq:lambdanalphabeta}) is a variant of the
well known relation (see \cite{MeR})
%
\begin{equation}    \label{eq:lambdanalphabetabis}
\lambda_n(\alpha \beta )\ 
=\ \sum\limits_{[i,j]=n} (i,j)\lambda_i(\alpha)\lambda_j(\beta).
\end{equation}
%

Of course, the algebraic identity 
$\Lambda^{(\alpha)}(x) \cdot \Lambda^{(\beta)}(x)
= (1+x)^\alpha(1+x)^\beta = (1+x)^{\alpha + \beta} = \Lambda^{(\alpha + \beta)}(x)$,
between generating series, is satisfied. However, we have, at the species level,
%
\begin{equation}    \label{eq:Lambdaalphabetabis}
\Lambda^{(\alpha)} \cdot \Lambda^{(\beta)}\ \neq \ \Lambda^{(\alpha + \beta)}
\end{equation}
%
This behavior follows from the fact that $X_\alpha + X_\beta \neq X_{\alpha + \beta}$. 
It is also reflected by the fact that although $\lambda_1(\alpha + \beta) 
= \alpha +\beta = \lambda_1(\alpha) + \lambda_1(\beta)$, we have
%
\begin{equation}    \label{eq:lambdanalphaplusbeta}
\lambda_n(\alpha + \beta )\ 
\neq\ \lambda_n(\alpha) + \lambda_n(\beta), \hspace{.5cm}  \mbox{if }  n \ge 2\ .
\end{equation}
%

The difference operator $\Delta$, defined by $\Delta G = G\circ E^+ - G$, permits the
expression of $\Lambda^{(\alpha)}$ in the form
%
\begin{equation}    \label{eq:Lambdaalphabis}
\Lambda^{(\alpha)}\ =\ E(X_\alpha) - \Delta E(X_\alpha) 
+ \Delta^{2}E(X_\alpha) - \Delta^{3}\,E(X_\alpha) + \cdots
\end{equation}
%
By using the computer algebra software Maple \cite{CGG}, Y. Chiricota  (personal
communication, see also \cite{Ch}) has 
computed the molecular decomposition of $\Lambda^{(\alpha)}$, i.e.,\ its expression  as
a {\Cp Z}-linear combination of molecular (i.e.,\ indecomposable under addition) species,
up to degree 7 . Here are the first terms, up to degree 5:
%
\begin{eqnarray}  
\Lambda^{(\alpha)} & = & E \circ X_\alpha \circ {(E^+)}^{<-1>}
 \nonumber     \\
 & =  &  1 + X_\alpha - (E_2)_\alpha + (E_2)_{\alpha^2}
-(E_3)_\alpha + (XE_2)_\alpha
-(XE_2)_{\alpha^2} + (E_3)_{\alpha^3}  \nonumber \\
 &  & \ \ + \ (E_2\circ E_2)_\alpha - (E_4)_\alpha + (XE_3)_\alpha
-(X^2E_2)_\alpha    \nonumber \\
 &  & \ \ %% \hspace{1cm}
+\ (E_2^2)_{\alpha^2} - (XE_3)_{\alpha^2} 
+ (X^2E_2)_{\alpha^2} - (E_2\circ E_2)_{\alpha^2} 
- (E_2^2)_{\alpha^3} + (E_4)_{\alpha^4} \label{eq:Lambdaalphater}  \\
 &  & \ \ + \  (E_2 E_3)_\alpha + (XE_4)_\alpha + (X^3 E_2)_\alpha - (X^2E_3)_\alpha 
- (XE_2^2)_\alpha - (E_5)_\alpha  \nonumber  \\
 &  & \ \ + \ (E_2E_3)_{\alpha^2} + (X^2E_3)_{\alpha^2} - (XE_4)_{\alpha^2}
-  (X^3 E_2)_{\alpha^2} + (X\ E_2\circ E_2)_{\alpha^2} - (X E_2^2)_{\alpha^2}  \nonumber  \\
 &  & \ \ + \ 2\,(X E_2^2)_{\alpha^3} - (E_2E_3)_{\alpha^3} - (X\ E_2\circ E_2)_{\alpha^3} 
- (E_2E_3)_{\alpha^4} + (E_5)_{\alpha^5}
\nonumber   \\
 &  & \ \ + \  \cdots\ \nonumber ,
\end{eqnarray}
%
where $E_k$ denotes the species of sets of cardinality $k$. 
%Since $E_k(x) = x^k/k!$,
The molecular decomposition (\ref{eq:Lambdaalphater}) can be seen as a combinatorial
lifting of the binomial formula
%
\begin{eqnarray}  
\Lambda^{(\alpha)}(x) & = & (1+x)^\alpha
 \nonumber     \\
 & =  & \ 1+ \alpha \,x+\alpha (\alpha -1)\frac {x^2}{2!}
+\alpha (\alpha -1)(\alpha -2)\frac {x^3}{3!}
+\alpha (\alpha -1)(\alpha -2)(\alpha -3)\frac {x^4}{4!}+\cdots .
\label{eq:binomialbis} 
\end{eqnarray}
%

The formulas (\ref{eq:Lambdaalphabis}) and (\ref{eq:Lambdaalphater}) show that 
$\Lambda^{(\alpha)}$ is a species of {\em set-like} type: all its molecular components 
are obtained by products and compositions of species of the form $E_k$ . 
For all $n \ge 0$, set
%
\begin{eqnarray}  
m_n & = & \mbox{number of set-like molecular species of degree $n$,}
 \label{eq:mn}   \\
 a_n & =  & \mbox{number of set-like atomic species of degree $n$.}
\label{eq:an} 
\end{eqnarray}
%
A species is {\em atomic} if it is molecular and indecomposable under the product. It 
can be shown that the sequences $(m_n)_{n\ge 0}$ and $(a_n)_{n\ge 0}$ satisfy the 
double recursive scheme
%
\begin{equation}  \label{eq:schema}
\begin{array}{cc}
	   m_0=1, & m_1=1,   \\
     a_0=0,  & a_1=1,
\end{array}             \hspace{.6cm}
\left\{ \begin{array}{ccl}
	m_n & = & \frac {1}{n}\ \sum\limits_{k=1}^{n} (\sum\limits_{d|k} d\,a_d)m_{n-k} \\
  a_n & =  & \sum\limits_{{k\,|\,n}\atop {k<n}} m_k
\end{array}  \right.      \hspace{.6cm}    n\ge 2,
\end{equation} 
%
which is equivalent to the following relations:
%
\begin{equation}  \label{eq:relations}
\sum\limits_{n \ge 0}^{} {m}_{n}\,x^{n}\ =\ \prod\limits_{k \ge 1} \frac {1}{(1-x^k)^{a_k}}\ , 
 \hspace{1cm}
\sum\limits_{n\ge 1} \frac {{a}_{n}}{{n}^{s}}\ =\ 1+(\zeta (s)-1) \cdot 
\sum\limits_{n\ge 1}^{} \frac {{m}_{n}}{{n}^{s}}\ ,
\end{equation} 
%
where $\zeta(s)$ denotes the Riemann zeta function.  In the case of all atomic and
molecular species, the first of these relations has an analogue (see \cite{LaJ}, \cite{LaY}) 
but not the second one.  Here are the values of $m_n$ and $a_n$, for $0\le n\le 40$:
%
\begin{eqnarray}  
 m_n & =  & 
 	\mbox{\footnotesize{1, 1, 2, 3, 7, 9, 20, 26, 54, 74, 137, 184, 356, 473, 841, 1154, 2034, 
 2742, 4740, 6405, 10874,}}   \nonumber \\
 & &  \mbox{\footnotesize{ 14794,	24515, 33246, 54955, 74380, 120501, 163828, 263144, 356621, 567330, 
768854, 1212354, }}  \label{eq:mnliste}  \\
 & &  \mbox{\footnotesize{ 1644335, 2567636, 3478873, 5403223, 7314662, 11265825, 15258443, 23363143,}}
 \nonumber
\end{eqnarray}
%
\begin{eqnarray}  
 a_n & = &	\mbox{\footnotesize{0, 1, 1, 1, 3, 1, 6, 1, 10, 4, 12, 1, 33, 1, 29, 13, 64, 1, 
100, 1, 156, 30, 187, 1, 443, 10, 476, 78,}}  \nonumber  \\
	 & & \mbox{\footnotesize{ 877, 1, 1326, 1, 2098, 188, 2745, 36, 5203, 1, 6408, 477, 11084.}}
\label{eq:anliste} 
\end{eqnarray}
%

These sequences do not appear in the 1973 edition of Sloane's book \cite{Sl} but they do in
the new edition \cite{SlP}. The analytical properties of the functions (\ref{eq:relations})
should prove useful for the asymptotic analysis of the numbers $m_n$, for 
$n\rightarrow \infty$.
%
%
%
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\end{document}          






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