Permutations without long decreasing subsequences and random matrices
  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. 
Piotr Sniady