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On the distribution of the length of the longest increasing subsequence of random permutations. (English) Zbl 0932.05001
If $$\pi(1), \dots, \pi(n)$$ is a permutation of $$1, \dots, n$$, the subsequence $$\pi(i_1), \dots, \pi(i_k)$$ is increasing if $$i_1< \cdots <i_k$$ and $$\pi(i_1) <\cdots <\pi (i_k)$$. 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 $$n\times n$$ matrix $$M$$ with a probability density proportional to $$\exp[-\text{trace}(M^2)]$$.

##### MSC:
 05A05 Permutations, words, matrices 15B52 Random matrices (algebraic aspects) 33D45 Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.) 45E05 Integral equations with kernels of Cauchy type 60F99 Limit theorems in probability theory
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