Asymptotics of the Longest Increasing Subsequence in Random Permutations

In this paper, we examine the asymptotic behavior of the longest increasing subsequence (LIS) in a uniformly random permutation of $n$ elements. We rely on the Robinson--Schensted--Knuth correspondence, Young tableaux, and key classical results -- including the Erd\H{o}s--Szekeres theorem and the Hook Length Formula -- to demonstrate that the expected LIS length grows as $2\sqrt{n}$. We review the essential variational principles of Logan--Shepp and Vershik--Kerov, which determine the limiting shape of the associated random Young diagrams, and summarize the Baik--Deift--Johansson theorem that links fluctuations of the LIS length to the Tracy--Widom distribution. Our approach focuses on providing conceptual and intuitive explanations of these results, unifying classical proofs into a single narrative and supplying fresh visual examples, while referring the reader to the original literature for detailed proofs and rigorous arguments.