Strings from Feynman Graph counting : without large N

A well-known connection between n strings winding around a circle and permutations of n objects plays a fundamental role in the string theory of large N two dimensional Yang Mills theory and elsewhere in topological and physical string theories. Basic questions in the enumeration of Feynman graphs can be expressed elegantly in terms of permutation groups. We show that these permutation techniques for Feynman graph enumeration, along with the Burnside counting lemma, lead to equalities between counting problems of Feynman graphs in scalar field theories and Quantum Electrodynamics with the counting of amplitudes in a string theory with torus or cylinder target space. This string theory arises in the large N expansion of two dimensional Yang Mills and is closely related to lattice gauge theory with S_n gauge group. We collect and extend results on generating functions for Feynman graph counting, which connect directly with the string picture. We propose that the connection between string combinatorics and permutations has implications for QFT-string dualities, beyond the framework of large N gauge theory.