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On the distribution of the length of the longest increasing subsequence of random permutations. (English) Zbl 0932.05001
If π(1),,π(n) is a permutation of 1,,n, the subsequence π(i 1 ),,π(i k ) is increasing if i 1 <<i k and π(i 1 )<<π(i k ). Let l n be the length of the longest increasing subsequence in a random permutation assigning equal probabilities 1/n! to the permutations. The limiting distribution of l n is determined, and all moments of l n are shown to converge to the corresponding moments of the limiting distribution. This limiting distribution is equal to the limiting distribution of the largest eigenvalue of a random Hermitian n×n matrix M with a probability density proportional to exp[-trace(M 2 )].

MSC:
05A05Permutations, words, matrices
15A52Random matrices (MSC2000)
33D45Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)
45E05Integral equations with kernels of Cauchy type
60F99Limit theorems (probability)