A representation of the Mayer expansion coefficients and virial coefficients

A new representation of the Mayer expansion coefficients and the virial coefficients is given. The essence of the new representation of the Mayer expansion coefficients is the replacement of the sum of integrals over all connected graphs with $n$ vertices by a sum of integrals over all trees with $n$ vertices. The essence of the new representation of the virial coefficients consists of replacement of the sum of integrals over all blocks with $n$ vertices by a sum of integrals over all blocks that are characteristic for certain blocks with $n$ vertices. As an application, a simple derivation is given of the well-known estimates for the radius of convergence of the Mayer expansion in the case of a non-negative potential.