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

TL;DR

This paper provides a combinatorial explanation for a specific graph weight sum in Mayer's cluster integral theory, linking it to rooted Cayley trees, thus clarifying a mathematical connection in statistical mechanics.

Contribution

It offers the first combinatorial interpretation of the sum of graph weights in Mayer's theory for the one-dimensional hard-core gas model.

Findings

Sum of graph weights equals (-n)^{n-1} for connected graphs with n vertices.

This sum corresponds to the number of rooted Cayley trees on n vertices.

The paper establishes a direct combinatorial link between Mayer's graph weights and Cayley trees.

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.