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{\bf Piotr \'Sniady}
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{\bf Permutations Without Long Decreasing Subsequences and Random Matrices}
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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
(namely the traceless Gaussian Unitary Ensemble random matrix) 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.
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