  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.
