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

From the perturbative expansion of the exact Green function, an exact counting formula is derived to determine the number of different types of connected Feynman diagrams. This formula coincides with the Arqu\`es-Walsh sequence formula in the rooted map theory, supporting the topological connection between Feynman diagrams and rooted maps. A classificatory summing-terms approach is used, in connection to discrete mathematical theory.