A Solution to the Combinatorial Puzzle of Mayer's Virial Expansion

Mayer's second theorem in the context of a classical gas model allows us to write the coefficients of the virial expansion of pressure in terms of weighted two-connected graphs. Labelle, Leroux and Ducharme studied the graph weights arising from the one-dimensional hardcore gas model and noticed that the sum of these weights over all two-connected graphs with $n$ vertices is $-n(n-2)!$. This paper addresses the question of achieving a purely combinatorial proof of this observation.