<?xml version="1.0" encoding="UTF-8"?>
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</assembly>ve equivariant statistics. We have 
\begin{equation}
\mathbf{EGF}(S;z)=c(\epsilon)-1+e^{\mathbf{EGF}({\mathtt{atoms}(S)};z)}\ ..
\end{equation}
In particular if $c(\epsilon)=1$ (for instance if $c$ is proper and $R$ is an integral domain), 
\begin{equation}
\mathbf{EGF}(S;z)=e^{\mathbf{EGF}({\mathtt{atoms}(S)};z)}\ .
\end{equation}
\end{thm}

\Proof
Let $n=0$. Then the unique element of $S_{\emptyset}$ is $\epsilon$. Therefore $c(S_{\emptyset})=c(\epsilon)$. Now suppose that $n> 0$ and let $\omega\in S_{[1..n]}$. According to proposition~\ref{decomposition_atomique_unique}, there is a unique finite set $\{\alpha_1,\dots,\alpha_k\}\subseteq {\mathtt{atoms}(S)}$ such that $\omega=\displaystyle\oplus_{i=1}^k \alpha_i$. By 
lemma~\ref{lemme_somme_generale_et_support}, $\{\sigma(\alpha_i):1\leq i\leq k\}$ is a partition of $[1..n]$ into $k$ blocks. Therefore $\omega\in {\mathtt{atoms}(S)}_{P_1}\oplus\dots\oplus {\mathtt{atoms}(S)}_{P_k}$ where $P_i=\sigma(\alpha_i)$ for $i=1,\dots,k$. We can remark that $\alpha_1\oplus\dots\oplus\alpha_k$ is well-defined for each $(\alpha_1,\dots,\alpha_k)\in {\mathtt{atoms}(S)}_{P_1}\times\dots\times {\mathtt{atoms}(S)}_{P_k}$ since the supports are disjoint. 
Now, one has, thanks to the partitions of $[1..n]$
\begin{eqnarray}
\displaystyle S_{[1..n]}=\bigsqcup_{\pi\in \Pi_n}\bigoplus_{p\in\pi}{\mathtt{atoms}(S)}_p \\
c(S_{[1..n]})=\sum_{\pi\in\Pi_n}\prod_{p\in \pi}c({\mathtt{atoms}(S)}_{p})
\end{eqnarray}
as, for disjoint (finite) sets $F$ and $G$ of $\mathbb{N}^+$, it is easy to check that $c(X_F\oplus X_G)=c(X_F)c(X_G)$ for every $X\subseteq S$ and because the disjoint union as only a finite number of factors. 
Therefore due to equivariance of $c$ on sets of the form ${\mathtt{atoms}(S)}_{F}$, one has 
\begin{equation}
\displaystyle c(S_{[1..n]})=\sum_{\pi\in\Pi_n}\prod_{p\in \pi}c({\mathtt{atoms}(S)}_{[1..\card(p)]})\ .
\end{equation}
But $c({\mathtt{atoms}(S)}_{[1..\card(p)]})$ is the $\card(p)$th coefficient of the series $\mathbf{EGF}({\mathtt{atoms}(S)};z)$. Therefore due to the usual exponential formula, $\mathbf{EGF}(S;z)=c(\epsilon)-1+e^{\mathbf{EGF}({\mathtt{atoms}(S)};z)}$. Now if  $c(\epsilon)=1$, then we obtain $\mathbf{EGF}(S;z)=e^{\mathbf{EGF}({\mathtt{atoms}(S)};z)}$. 
\cqfd

\section{Two examples}

The examples provided here pertain to the class of labelled graphs where the ``classic" exponential formula applies, namely \emph{Burnside's Classes\footnote{The name is related to the notion of \emph{free Burnside semigroups}, namely the quotient of the free semigroup $A^{+}$, where $A$ is a finite alphabet, by the the smallest congruence that contains the relators $\om^{n+m}=\om^n$, $\om\in A^{+}$. For more details see~\cite{PdLS01}.}} $\mathit{Burn}_{a,b}$, defined, for $0\leq a < b$ two integers, as the class of graphs of numeric endofunctions $f$ such that 
\begin{equation}
f^a = f^b
\end{equation}
where $f^n$ denotes the $n$th power with respect to functional composition. Despite of its simplicity, there are still (enumerative combinatorial) open problems for this class and only $B_{1,\ell+1}$ gives rise to an elegant formula \cite{GJ,St} (see also  \cite{HS67}, for the idempotent case: $\ell=1$ and compare to exact but non-easily tractable formulas in \cite{DM97} for the general case in the symmetric semigroup, and in \cite{Krat} for their generalization to the wreath product of the symmetric semigroup and a finite group).\\

The second example: the class of finite partitions which can be (and should here) identified as graphs of equivalence relations on finite subsets $F\subseteq \mathbb{N}^+$. Call this class ``Stirling class" as the number of such graphs with support $[1..n]$ and $k$ connected components is exactly the Stirling number of the second kind $S_2(n,k)$ and, using the statistics $x^{(\mathit{number\ of\ points})}y^{(\mathit{number\ of\ connected\ components})}$, one obtains 
\begin{equation}\label{stirling-class-classique}
\displaystyle\sum_{n,k\geq 0}S_2(n,k)\frac{x^n}{n!}y^k=e^{y(e^x-1)}\ .
\end{equation}

Examples of this kind bring us to the conclusion that bivariate ststistics like $Burn_{a,b}(n,k)$, 
$S_2(n,k)$ or $S_1(n,k)$ (Stirling numbers of the second and first kind) are better understood 
through the notion of one-parameter group, conversely such groups naturally arinsing in 
Combinatorial Physics lead to such statistics and new ones some of which can be interpreted 
combinatorially.

\section{Generalized Stirling numbers in Combinatorial Physics}

In Quantum Mechanics, many tools boil down to the consideration of creation and annihilation operators which will be here denoted respectively $a^{\dagger}$ and $a$. These two symbols do not commute and are subject to the unique relation 
\begin{equation}
[a,a^{\dagger}]=1\ .
\end{equation}
The complex algebra generated by these two symbols and this unique relation, the \emph{Heisenberg-Weyl algebra}, will be here denoted by $HW_{\mathbb{C}}$. The consideration of evolution (one-parameter) groups $e^{\lambda \Omega}$ where 
$\Omega=\displaystyle\sum_{\omega\in HW_{\mathbb{C}}}\alpha(\omega)\omega$ is an element of $HW_{\mathbb{C}}$, with all - but a finite number of them - the complex numbers $\alpha(\omega)$ equal to $0$, and $\omega$ a word on the alphabet $\{a,a^{\dagger}\}$ leads to the necessity of solving the \emph{Normal Ordering Problem}, \emph{i.e.}, the reduction of the powers of $\Omega$ to the form 
\begin{equation}
\Omega^n = \displaystyle\sum \beta_{i,j}(a^{\dagger})^i a^j\ .
\end{equation}
In the sequel, $\mathit{Normal}(\Omega^n)$ denotes such a sum. 
This problem can be performed with three indices in general and two in the case of homogeneous operators that is operators for which the ``excess" $e=i-j$ is constant along the monomials $(a^{\dagger})^ia^j$ of the support (for which $\beta_{i,j}\not=0$). Thus, for 
\begin{equation}
\Omega = \displaystyle\sum_{i-j=e}\beta_{i,j}(a^{\dagger})^ia^j
\end{equation}
one has, for all $n\in\mathbb{N}$, 
\begin{equation}
\mathit{Normal}(\Omega^n)=(a^{\dagger})^{ne}\displaystyle\sum_{k=0}^{\infty}S_{\Omega}(n,k)(a^{\dagger})^ka^k
\end{equation}
when $e\geq 0$, and 
\begin{equation}
\mathit{Normal}(\Omega^n)=\displaystyle\left ( \sum_{k=0}^{\infty}S_{\Omega}(n,k)(a^{\dagger})^ka^k\right )a^{n|e|}
\end{equation}
otherwise. It turns out that, when there is only one annihilation, one gets a formula of the type ($x,y$ are formal commutative variables) 
\begin{equation}
\displaystyle\sum_{n,k\geq 0}S_{\Omega}(n,k)\frac{x^n}{n!}y^k = g(x)e^{y\sum_{n\geq 1}S_{\Omega}(n,1)\frac{x^n}{n!}}
\end{equation} 
which is a generalization of formula (\ref{stirling-class-classique}). A complete study of such a procedure and the details to perform the solution of the normal ordering problem may be found in \cite{GHED}. 

\section{Conclusion}

In this paper, we have broadened the domain
of application of the exponential formula, a
tool originated from statistical physics.
This broadening reveals us together with the
essence of ``why this formula works" a possibility
of extension to denominators other than the
factorial and, on the other hand, provide a 
link with one-parameter groups 
whose infinitesimal generators are (formal) vector
fields on the line. The general combinatorial
theory of the correspondence
(vector fields $\leftrightarrow$ bivariate ststistics)
is still to be done despite the fact that we have
already a wealth of results in this direction.

\section*{Acknowledgements}
The research of this work was supported, in part, by the Agence Nationale de la Recherche (Paris, France) under Program No. ANR-08-BLAN-0243-2. We would like also to acknowledge support from “Projet interne au LIPN 2009” ``Polyz\^eta functions''. 

\footnotesize

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