Simulations: The longest increasing subsequence (LIS) of random separable permutations

Let LIS(n) denote the length of the longest increasing subsequence of a uniform random separable permutations with a given size n. I carried out simulations of LIS(n) for n=10^k for k=1,..,8 with more than 10^7 samples for each k. The obtained data suggests that LIS(n) grows at the rate n^alpha for alpha=0.815..., and LIS(n) / n^alpha approaches a limiting random variable with mean approximate 0.663...

This agrees with a recent conjecture by Adhikari, Borga, Budzinski, Da Silva, and Sénizuerges in their paper on the longest increasing subsequence of permutations sampled from the Brownian permuton.

LIS of uniform random separable permutations of size 1M

LIS of uniform random separable permutations of size 10M

LIS of uniform random separable permutations of size 100M