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Mayer’s second theorem in the context of a classical gasmodel allows us to write the coefficients of the virial expansion of pressure in terms of weighted two-connected graphs. Labelle, Leroux and Ducharme studied the graph weights arising from the one-dimensional hardcore gas model and noticed that the sum of these weights over all two-connected graphs with vertices is . This paper addresses the question of achieving a purely combinatorial proof of this observation and extends the proof of Bernardi for the connected graph case.
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Stephen James Tate, A solution to the combinatorial puzzle of Mayer’s virial expansion. Ann. Inst. Henri Poincaré Comb. Phys. Interact. 2 (2015), no. 3, pp. 229–262