TITLE: How is the longest increasing subsequence a matrix integral?
QUESTION [8 upvotes]: 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?

REPLY [3 votes]: There is a roundabout way of putting this.
A discrete analogue of random matrix spectra is random partitions (=Young diagrams). There clearly is a 'gadget' relating random permutations with random Young diagrams - the celebrated Robinson-Schensted correspondence. 
On the other hand, the passage from random Young diagrams to random matrices is rather well understood (mainly, from the algebraic point of view). In particular, many random matrix ensembles arise from random Young diagrams via certain degenerations. And of course, in various limit regimes the asymptotic distributions of random partition and random matrix ensembles are precisely the same (the Airy ensembles, etc.).
On the other hand, maybe a look on some generalizations of your identity (arXiv:math/9905083) can help?