The Mayer series: two explicit expressions for its coefficients and the combinatorial problem

We derive the (formal) Mayer series of the pressure of a system of classical particles interacting via a pair potential from the standard grand canonical partition function. We give two alternative representations of its coefficients and we discuss the combinatorial problem involved towards the question of its convergence. We conclude the talk by giving a brief review on results obtained and methods proposed , since 1963,to face this combinatorial problem.