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 提供了 GUE 大矩阵极限的缩放最大级别分布的渐进性。然而，作为数值近似，这种渐进对于较小的n来说是不准确的，并且收敛速度很慢，推测其阶数仅为\(n^{-1/3}\)。在这里，我们基于海曼对斯特林公式的推广提出了一种不同类型的近似。对于小至 20 的n，这样的公式已经给出了长度分布的几个正确数字，但允许数值评估，具有表观阶数\(n^{-2/3}\)的统一误差，对于n大至\(10^{12}\)，从而缩小了精确值表（编译最多\(n=1000\)）与随机矩阵限制之间的差距。斯特林型公式比蒙特卡罗模拟更加高效和准确，可以精确地数值理解随机矩阵极限的前几个有限尺寸校正项。由此我们得出期望值和长度方差的扩展，展示出比之前提出的更多的项。