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

Solution to a combinatorial puzzle arising from Mayer's theory of cluster integrals