The discrete distribution of the length of longest increasing subsequences in random permutations of n integers is deeply related to random matrix theory. In a seminal work, Baik, Deift and Johansson provided an asymptotics in terms of the distribution of the scaled largest level of the large matrix limit of GUE. As a numerical approximation, however, this asymptotics is inaccurate for small n and has a slow convergence rate, conjectured to be just of order \(n^{-1/3}\). Here, we suggest a different type of approximation, based on Hayman’s generalization of Stirling’s formula. Such a formula gives already a couple of correct digits of the length distribution for n as small as 20 but allows numerical evaluations, with a uniform error of apparent order \(n^{-2/3}\), for n as large as \(10^{12}\), thus closing the gap between a table of exact values (compiled for up to \(n=1000\)) and the random matrix limit. Being much more efficient and accurate than Monte Carlo simulations, the Stirling-type formula allows for a precise numerical understanding of the first few finite size correction terms to the random matrix limit. From this we derive expansions of the expected value and variance of the length, exhibiting several more terms than previously put forward.

n 个整数的随机排列中最长递增子序列长度的离散分布与随机矩阵理论密切相关。在一项开创性的工作中，Baik、Deift 和 Johansson 根据 GUE 大矩阵极限的缩放最大级别的分布提供了渐近。然而，作为数值近似，这种渐近对于较小的n是不准确的，并且收敛速度较慢，推测只是\(n^{-1/3}\)阶。在这里，我们基于海曼对斯特林公式的概括提出了一种不同类型的近似。这样的公式已经给出了n的长度分布的几个正确数字小至 20 但允许数值评估，具有表观阶\(n^{-2/3}\)的统一误差，对于n大至\(10^{12}\)，从而缩小 a 之间的差距精确值表（编译最多\(n=1000\)）和随机矩阵限制。斯特林型公式比蒙特卡罗模拟更有效和准确，允许对随机矩阵极限的前几个有限尺寸校正项进行精确的数值理解。由此我们推导出期望值的扩展和长度的方差，展示了比之前提出的更多的项。