Asymptotics of the Longest Increasing Subsequence in Random Permutations

TL;DR

This paper surveys the asymptotics of the longest increasing subsequence (LIS) in a uniformly random permutation, drawing connections between combinatorics (RSK correspondence, Young diagrams, Hook Length Formula) and probabilistic limit laws. It synthesizes classical results (Logan–Shepp, Vershik–Kerov) on the limit shape to establish $\mathbb{E}[L(\sigma_n)]\sim 2\sqrt{n}$ and, via the Baik–Deift–Johansson theorem, shows fluctuations follow the Tracy–Widom distribution on the $n^{1/6}$ scale. The narrative unifies proofs around the Plancherel measure and variational calculus, illustrating how the dominant partition controls the LIS and how Poissonisation facilitates analytic handling. The work also contextualizes these findings with practical perspectives (airplane boarding, patience sorting) and highlights deep links to random matrix theory and integrable systems, underscoring the interdisciplinary significance of LIS asymptotics.

Abstract

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 Erdős--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.

Figures (6)

• Figure 1: A Standard Young Tableau for the partition $(4,3,2)$ of $n=9$. Entries increase across rows and down columns.
• Figure 2: A hook walk from cell $u$ to the corner cell $v$.
• Figure 3: A $2\times2$ rectangle with $h(a)+h(d)=h(b)+h(c)$.
• Figure 4: The cell labeling scheme for the probabilistic hook-walk proof labels each cell $w$ by the reciprocal of its available moves, $1/(h(w)-1)$. The left panel shows the general algebraic pattern determined by the co-hook of a corner cell, while the right provides a concrete numerical example with boundary values $x=(4,2,1)$ and $y=(4,3,1)$ derived from a specific Young diagram.
• Figure 5: Sum over the two lattice paths equals $\frac{1}{x_1y_1}$.

Theorems & Definitions (22)

• Example 1.1
• Example 1.2
• Definition 2.1
• Lemma 2.2: Strict Decrease in Each Basic Subsequence
• proof
• Lemma 2.3: Linking Adjacent Basic Subsequences
• proof
• Theorem 2.4: LIS and the Number of Columns schensted1961longest
• proof
• Theorem 2.5: LDS and the Number of Rows schensted1961longest