In "Random Matrices Random Permutations", the longest increasing subsequence of a permutation is related to an expectation over Hermitian matrices.

$$ \frac{1}{2^{|k|} n^{|k|/2}} \left\langle \prod_{j=1}^s \mathrm{tr}H^{k_j} \right\rangle$$

Can anyone clarify this relation? I vaguely remember this coming from a paper of Gessel.

In general, I wonder is there a "gadget" turning permutation statistics (such as inversion number, or number of cycles) into integrals over unitary matrices?