I have been reading about the topic motivated by a problem I read that asked for the first three digits of the sum of the LIS lengths in all permutations of length $n$. It is easy to see that we are really interested in the expected value of the LIS length in a random permutation of the first $n$ positive integers. A paper by Mike Phulsuksombati states $$l_n=2\sqrt{n}+cn^{1/6}+O(n^{1/6})$$ Where $c\approx-1.771088$ as a known asymptotic.

Questions: Is there a better asymptotic known? Is there a polynomial or logartihmic(in terms of $n$) time algorithm for finding $l_n$ or approximating it to a desired precision?

I have been reading about the topic motivated by a problem I read that asked for the first three digits of the sum of the LIS lengths in all permutations of length $n$. It is easy to see that we are really interested in the expected value of the LIS length in a random permutation of the first $n$ positive integers. A paper by Mike Phulsuksombati states $$l_n=2\sqrt{n}+cn^{1/6}+o(n^{1/6})$$ Where $c\approx-1.771088$ as a known asymptotic.

Questions: Is there a better asymptotic known? Is there a polynomial or logartihmic(in terms of $n$) time algorithm for finding $l_n$ or approximating it to a desired precision?