On polynomial integrals over the orthogonal group

We consider integrals of type ∫ O n u 11 a 1 ⋯ u 1 n a n u 21 b 1 ⋯ u 2 n b n d u , with respect to the Haar measure on the orthogonal group. We establish several remarkable invariance properties satisfied by such integrals, by using combinatorial methods. We present as well a general formula for such integrals, as a sum of products of factorials.