\documentclass[11pt]{article}
%
%
\usepackage{amsmath}
\usepackage{amsfonts}
%
\newtheorem{df}{Definition}
\newtheorem{pr}{Proposition}
\newtheorem{teo}{Theorem}
\newtheorem{lem}{Lemma}
\newtheorem{co}{Corollary}
\newtheorem{re}{Remark}
\newtheorem{no}{Note}
\newtheorem{ex}{Example}

%%%%%%%%%%%%%%%%%%%%%%
% Sauts
%%%%%%%%%%%%%%%%%%%%%%

\def\hs{\hbox to 3mm{}}
\def\hhs{\hbox to 5cm{}}
\def\ss{\smallskip}
\def\ms{\medskip}
\def\bs{\bigskip}
\def\bbss{\vskip 6mm}


%%%%%%%%%%%%%%%%%%%%%%
% Symboles 
%%%%%%%%%%%%%%%%%%%%%%

\def\F#1{\mathcal{F}_#1}
\def\Q{\mathbb{Q}}
\def\Z{\mathbb{Z}}
\def\A{{\mathfrak A}}
\def\SG{{\mathfrak S}}
\def\ncs#1#2{#1\langle\langle #2\rangle\rangle}

%
\def\ann{{\rm ann}}
\def\be{{\rm behaviour}}
\def\di{{\rm dimension}}
\def\Endo{{\mathfrak E}{\rm nd}}
%
%
\title{\large\bf Some (joyful) interactions between Combinatorics, Computer Science and Physics}


\begin{document}
\maketitle

\begin{abstract}
We consider two aspects of the product formula for formal power series 
applied to combinatorial field theories. Firstly, we remark that the 
case when the functions involved in the product formula have a constant 
term  is of special interest as often these functions give rise to 
substitutional groups. The groups arising from the normal ordering 
problem of boson strings (and more generally homogeneous operators in 
the Weyl algebra) are naturally associated with explicit vector 
fields, or their conjugates, in the case when there is only one 
annihilation operator. We also consider one-parameter groups of 
operators when several annihilators are present. 
Secondly, we discuss the Feynman-type graph representation resulting 
from the product formula. We show that there is a correspondence 
between the packed integer matrices of the theory of noncommutative 
symmetric functions and these Feynman-type graphs. 
In particular, we obtain a new Hopf algebra structure over the space 
of matrix quasi-symmetric functions and a natural commutative Hopf 
algebra structure on the space of diagrams themselves. 
It is shown that this Hopf algebra originates from the formal 
doubling of variables in the product formula. Hilbert series of these to Hopf 
algebras are discussed.
\end{abstract}

\end{document}
