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.

翻译：在随机随机调整的美元整数中,最长增长次序列长度的离散分布与随机矩阵理论密切相关。在一项重要工作中,Baik、Deift和Johansson提供了GUE大矩阵限制最大基数最大比例分布的零点。然而,作为一个数字近似值,这种零点分布对小美元是不准确的,其趋同速度缓慢,因此推测准确值表(最多为1美元=1/3美元)与随机矩阵限制之间的差额。根据Hayman对Stirling公式的概括化,这种公式已经给出了长度分布的几位正确数字,其数额小于20美元,但允许进行数字评价,但表面的错误为10美元=3美元,因此缩小了准确值表(最多为1 000美元)与随机矩阵限制之间的差距。从Monte-Carloiming 公式的第一次精确和准确数字数字数位数数字值,使得我们之前的精确度模型的精确度模型具有一定的弹性。