Séminaire Lotharingien de Combinatoire, B59e (2008), 10 pp.

Olivier Bernardi

Solution to a Combinatorial Puzzle Arising from Mayer's Theory of Cluster Integrals

Abstract. 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.

Received: April 1, 2008. Accepted: October 15, 2008. Final Version: October 16, 2008.

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