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From the perturbative expansion of the exact Green function, an exactcounting formula is derived to determine the number of different types ofconnected Feynman diagrams. This formula coincides with the Arqu\`es-Walshsequence formula in the rooted map theory, supporting the topologicalconnection between Feynman diagrams and rooted maps. A classificatorysumming-terms approach is used, in connection to discrete mathematical theory.

journal of mathematical physics

Equivalence between the Arqu\`es-Walsh sequence formula and the number  of connected Feynman diagrams for every perturbation order in the fermionic  many-body problem