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arXiv · 2610.18235

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

Ernest X. W. Xia
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Abstract

Let $P_{n,k}$ count permutations of $[n]=\{1,\ldots,n\}$ 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.

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Ernest X. W. Xia. 2026-10-13. On Chen's conjecture for the log-concavity of the longest increasing subsequence distribution. https://arxiv.org/abs/2610.18235

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