  $U(N)^{\otimes r} \otimes O(N)^{\otimes q}$ invariants are constructed by
contractions of complex tensors of order $r+q$, also denoted $(r,q)$. These
tensors transform under $r$ fundamental representations of the unitary group
$U(N)$ and $q$ fundamental representations of the orthogonal group $O(N)$.
Therefore, $U(N)^{\otimes r} \otimes O(N)^{\otimes q}$ invariants are tensor
model observables endowed with a tensor field of order $(r,q)$. We enumerate
these observables using group theoretic formulae, for arbitrary tensor fields
of order $(r,q)$. Inspecting lower-order cases reveals that, at order $(1,1)$,
the number of invariants corresponds to a number of 2- or 4-ary necklaces that
exhibit pattern avoidance, offering insights into enumerative combinatorics.
For a general order $(r,q)$, the counting can be interpreted as the partition
function of a topological quantum field theory (TQFT) with the symmetric group
serving as gauge group. We identify the 2-complex pertaining to the enumeration
of the invariants, which in turn defines the TQFT, and establish a
correspondence with countings associated with covers of diverse topologies. For
$r>1$, the number of invariants matches the number of ($q$-dependent) weighted
equivalence classes of branched covers of the 2-sphere with $r$ branched
points. At $r=1$, the counting maps to the enumeration of branched covers of
the 2-sphere with $q+3$ branched points. The formalism unveils a wide array of
novel integer sequences that have not been previously documented. We also
provide various codes for running computational experiments.
