  The permanent of unitary matrices and their blocks has attracted increasing
attention in quantum physics and quantum computation because of connections
with the Hong-Ou-Mandel effect and the Boson Sampling problem. In that context,
it would be useful to know the distribution of the permanent or other immanants
for random matrices, but that seems a difficult problem.We advance this program
by calculating the average of the squared modulus of a generic immanant for
blocks from random matrices in the unitary group, in the orthogonal group and
in the circular orthogonal ensemble. In the case of the permanent in the
unitary group, we also compute the variance. Our approach is based on
Weingarten functions and factorizations of permutations. In the course of our
calculations we are led to a conjecture relating dimensions of irreducible
representations of the orthogonal group to the value of zonal polynomials at
the identity.
