The discrete distribution of the length of the longest increasing subsequence in a random permutation of order $n$ is closely related to random matrix theory. In a seminal work, Baik, Deift, and Johansson provided an asymptotic method for the maximum level distribution of the large matrix limit of GUE. However, as a numerical approximation, this asymptote is inaccurate for short lengths and has a slow convergence rate, estimated to be on the order of $n^{-1/3}$. Here we propose another type of approximation based on Hayman’s generalization of Stirling’s formula. Such a formula already gives the correct number of digits for the length distribution of $n$ as small as $20$, but allows numerical evaluation. The size of $10^{12}$. So it bridges the gap between a table of exact values (recently compiled with a maximum of $n=1000$) and a random matrix limit. Much more efficient and accurate than Monte Carlo simulations of larger $n$, the Stirling-type formula allows an accurate numerical understanding of the first few finite-size correction terms down to the random matrix limit. to Forrester and Mays were the first to visualize the form of such terms. We also show the second one of order $n^{-2/3}$, derive (heuristically) the expansion of the expectation and the variance of the length, and add some more terms than previously known. indicate.
