On Chen's conjecture for the log-concavity of the longest increasing subsequence distribution

Let $P_{n,k}$ count permutations of $[n]=\{1,\dots ,n\}$ whose longest increasing subsequence has length $k$. Chen conjectured that $(P_{n,k}{)}_{k=1}^n$ is log-concave for every positive integer $n$. In this paper, we prove that Chen’s conjecture is true when $n\ge {10}^{{10}^{65}}$. The proof combines Gessel’s determinantal formula with quantitative coefficient extraction, Tracy–Widom density estimates, and exact inequalities for the extreme lengths.