A combinatorial problem considered by H. D. Ursell in his seminal paper on cluster theory [Proc. Camb. Phil. Soc. 23, 685 (1927)] is studied. Ursell’s analysis, which is not rigorous, is described by Fowler and Guggenheim as being far from simple. In this paper we arrive at Ursell’s result using a method which is straightforward, yet completely rigorous.

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