Skip to content
DraconiticResearch
Search Sources API
201,642 studies31 sourcesUpdated 16 min ago
PreprintarXiv licencearXiv

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

  • Ernest X. W. Xia
Open at sourcePDFDOICite

Abstract

Let Pn,k count permutations of [n]={1,…,n} whose longest increasing subsequence has length k. Chen conjectured that (Pn,k)k=1n is log-concave for every positive integer n. In this paper, we prove that Chen's conjecture is true when n≥ 101065. The proof combines Gessel's determinantal formula with quantitative coefficient extraction, Tracy--Widom density estimates, and exact inequalities for the extreme lengths.

Related

  1. Log-concavity of longest increasing subsequence
    Yupeng Li
    arXivPreprintarXiv licenceFull textCombinatorics & discrete mathematicsProbabilityarXiv
  2. Longest increasing subsequences of dyadic-type chaotic orbits
    Shinsuke Iwao, Fumihiko Nakamura, Yushi Nakano
    arXivPreprintCC BY-SA 4.0Full textCombinatorics & discrete mathematicsDynamical systemsarXiv
  3. The longest increasing subsequences of Luce permutations and permutons
    Jacopo Borga, Victor Dubach
    HALPreprintCC BY 4.0Full textCombinatorics & discrete mathematicsProbabilityHALarXiv
  4. An Improved Bound for Smith's Longest Cycles Conjecture via a Forbidden Subdivision
    Douglas M. Chen
    arXivPreprintCC BY 4.0Full textCombinatorics & discrete mathematicsarXiv
  5. A counterexample to a log-concavity conjecture of Brenti
    Christian Gaetz
    arXivPreprintarXiv licenceFull textCombinatorics & discrete mathematicsarXiv

Cite

BibTeXRIS
BibTeX
@misc{xia2026chens,
  title = {{On Chen's conjecture for the log-concavity of the longest increasing subsequence distribution}},
  author = {Ernest X. W. Xia},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.18235},
  eprint = {2610.18235},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.18235},
}
RIS
TY  - UNPB
TI  - On Chen's conjecture for the log-concavity of the longest increasing subsequence distribution
AU  - Ernest X. W. Xia
PY  - 2026
DA  - 2026/10/13
PB  - arXiv
DO  - 10.48550/arxiv.2610.18235
UR  - https://arxiv.org/abs/2610.18235
AB  - Let P_n,k count permutations of [n]={1,…,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≥ 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.
ER  -