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Quantum field theory is not the first place a combinatorialist is likely to look for interesting problems or to look to apply their techniques. On the other hand, from the physics side, too often combinatorics is viewed as a kind of uninteresting messy detail. However, there is actually a lot of beautiful and useful combinatorics in quantum field theory, and the discrete structures illuminate the physical structure. Neither side is necessarily well positioned to penetrate the literature of the other.