Invariant polynomial functions on tensors under the action of a product of orthogonal groups

Let K be the product O(n_1) x O(n_2) x ... x O(n_r) of orthogonal groups. Let V the r-fold tensor product of defining representations of each orthogonal factor. We compute a stable formula for the dimension of the K-invariant algebra of degree d homogeneous polynomial functions on V. To accomplish this, we compute a formula for the number of matchings which commute with a fixed permutation. Finally, we provide formulas for the invariants and describe a bijection between a basis for the space of invariants and the isomorphism classes of certain r-regular graphs on d vertices, as well as a method of associating each invariant to other combinatorial settings such as phylogenetic trees.