$O(N)$ invariants are the observables of real tensor models. We use regular
colored graphs to represent these invariants, the valence of the vertices of
the graphs relates to the tensor rank. We enumerate $O(N)$ invariants as
$d$-regular graphs, using permutation group techniques. We also list their
generating functions and give (software) algorithms computing their number at
an arbitrary rank and an arbitrary number of vertices. As an interesting
property, we reveal that the algebraic structure which organizes these
invariants differs from that of the unitary invariants. The underlying
topological field theory formulation of the rank $d$ counting shows that it
corresponds to counting of coverings of the $d-1$ cylinders sharing the same
boundary circle and with $d$ defects. At fixed rank and fixed number of
vertices, an associative semi-simple algebra with dimension the number of
invariants naturally emerges from the formulation. Using the representation
theory of the symmetric group, we enlighten a few crucial facts: the
enumeration of $O(N)$ invariants gives a sum of constrained Kronecker
coefficients; there is a representation theoretic orthogonal base of the
algebra that reflects its dimension; normal ordered 2-pt correlators of the
Gaussian models evaluate using permutation group language, and further, via
representation theory, these functions provide other representation theoretic
orthogonal bases of the algebra.
