Geodesic
Solution to a Combinatorial Puzzle Arising from Mayer's Theory of Cluster Integrals
Séminaire lotharingien de combinatoire, Tome 59 (2008-2010) 
Olivier Bernardi. Solution to a Combinatorial Puzzle Arising from Mayer's Theory of Cluster Integrals. Séminaire lotharingien de combinatoire, Tome 59 (2008-2010). http://geodesic.mathdoc.fr/item/SLC_2008-2010_59_a3/
@article{SLC_2008-2010_59_a3,
     author = {Olivier Bernardi},
     title = {Solution to a {Combinatorial} {Puzzle} {Arising} from {Mayer's} {Theory} of {Cluster} {Integrals}},
     journal = {S\'eminaire lotharingien de combinatoire},
     year = {2008-2010},
     volume = {59},
     url = {http://geodesic.mathdoc.fr/item/SLC_2008-2010_59_a3/}
}
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Voir la notice de l'acte provenant de la source Séminaire Lotharingien de Combinatoire website

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.