Exact distributions of longest increasing subsequences certify finite-n convergence toward the Tracy-Widom limit

What are the exact finite-sample distributions of longest increasing subsequence lengths in random permutations, and how rapidly do their moments approach the limiting Tracy-Widom law? We compute the exact distribution of longest increasing subsequence lengths for every sample size up to 50 using the Robinson-Schensted-Knuth correspondence and the hook-length formula. At the largest sample size, the exact expected length is 11.309389 with variance 1.94134, corresponding to a normalized centered mean of -1.475863 and normalized variance of 0.526961. Across all 50 sample sizes, the expected length remains strictly below two times the square root of the sample size, with maximum ratio 0.799695. Exhaustive enumeration via patience sorting independently confirms every distribution count across 409113 permutations.