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The exponential-log transformation of exponential generating functions (see OEIS A036040 and A127671) relate the classical cumulants to their associated moments.

Who were some of the first to introduce diagrammatic illustrations of the relationship between the classical cumulants and moments?

(A excellent discussion of the connections can be found in "Three lectures on free probability" by Novak and LaCroix.)

Agata Fronczak in the "The microscopic meaning of the grand potential ..." mentions Mayer, Riddel, and Uhlenbeck in relation to the cluster expansion theorem.

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