  Mayer's theory of cluster integrals allows one to write the partition
function of a gas model as a generating function of weighted graphs. Recently,
Labelle, Leroux and Ducharme have studied the graph weights arising from the
one-dimensional hard-core gas model and noticed that the sum of the weights
over all connected graphs with $n$ vertices is $(-n)^{n-1}$. This is, up to
sign, the number of rooted Cayley trees on $n$ vertices and the authors asked
for a combinatorial explanation. The main goal of this article is to provide
such an explanation.
