  We classify a large set of melonic theories with arbitrary $q$-fold
interactions, demonstrating that the interaction vertices exhibit a range of
symmetries, always of the form $\mathbb{Z}_2^n$ for some $n$, which may be $0$.
The number of different theories proliferates quickly as $q$ increases above
$8$ and is related to the problem of counting one-factorizations of complete
graphs. The symmetries of the interaction vertex lead to an effective
interaction strength that enters into the Schwinger-Dyson equation for the
two-point function as well as the kernel used for constructing higher-point
functions.
