Séminaire Lotharingien de Combinatoire, B59e (2008), 10 pp.
Solution to a Combinatorial Puzzle Arising from Mayer's Theory of Cluster Integrals
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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