  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.
