  Lattice gauge theories of permutation groups with a simple topological action
(henceforth permutation-TFTs) have recently found several applications in the
combinatorics of quantum field theories (QFTs). They have been used to solve
counting problems of Feynman graphs in QFTs and ribbon graphs of large $N$,
often revealing inter-relations between different counting problems. In another
recent development, tensor theories generalizing matrix theories have been
actively developed as models of random geometry in three or more dimensions.
Here, we apply permutation-TFT methods to count gauge invariants for tensor
models (colored as well as non-colored), exhibiting a relationship with
counting problems of branched covers of the 2-sphere, where the rank $d$ of the
tensor gets related to a number of branch points. We give explicit generating
functions for the relevant counting and describe algorithms for the enumeration
of the invariants. As well as the classic count of Hurwitz equivalence classes
of branched covers with fixed branch points, collecting these under an
equivalence of permuting the branch points is relevant to the color-symmetrized
tensor invariant counting. We also apply the permutation-TFT methods to obtain
some formulae for correlators of the tensor model invariants.
