  We study the shape of the Young diagram \lambda associated via the
Robinson-Schensted-Knuth algorithm to a random permutation in S_n such that the
length of the longest decreasing subsequence is not bigger than a fixed number
d; in other words we study the restriction of the Plancherel measure to Young
diagrams with at most d rows. We prove that in the limit n\to\infty the rows of
\lambda behave like the eigenvalues of a certain random matrix (traceless
Gaussian Unitary Ensemble) with d rows and columns. In particular, the length
of the longest increasing subsequence of such a random permutation behaves
asymptotically like the largest eigenvalue of the corresponding random matrix.
