$O(N)$ 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.

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